Nuclear Shell Model amp Alpha Decay Stanley Yen TRIUMF

Nuclear Shell Model

Electrons in atoms occupy well-defined shells of discrete well-separatedenergy Do nucleons inside a nucleus do the same or not

Evidence for electron shells in atoms sudden jumps in atomicproperties as a shell gets filled up eg atomic radius ionization energy chemical properties

from Krane IntroductoryNuclear Physics

atomicradius

atomicionizationenergy

chemical reactivity

Not evident a priori that nuclei should show shell structure Why not

1 Success of liquid drop model in predicting nuclear binding energies Liquid drops have smooth behaviour with increasing size and do not exhibit jumps

2 No obvious center for nucleons to orbit around unlike electronsin an atom

one pionexchangetwo pion

andheavy mesonexchange

overlap of 3-quark bagscomplicatedshort-rangebehaviour

3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other

before

after

Many theoretical reasons were given why nuclei shouldnot show any shell structure

but experiment says otherwise

2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms

Note jumps atnucleon numbers8 20 28 50 82 126

from KraneIntroductoryNuclear Physics

from Gadioli ampGadioliIntroNuclearPhysics

abundancesof even-evennuclides

neutron capturecross section--analogous tochemicalreactivity

dips at N=20 50 82 126

local maximaat 50 82 126

from KraneIntroductoryNuclear Physics

alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)

neutroncapturecross section(repeat)

nuclearchargeradius

minima atN=20 28 50 82 126

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

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Nuclear Shell Model

Electrons in atoms occupy well-defined shells of discrete well-separatedenergy Do nucleons inside a nucleus do the same or not

Evidence for electron shells in atoms sudden jumps in atomicproperties as a shell gets filled up eg atomic radius ionization energy chemical properties

from Krane IntroductoryNuclear Physics

atomicradius

atomicionizationenergy

chemical reactivity

Not evident a priori that nuclei should show shell structure Why not

1 Success of liquid drop model in predicting nuclear binding energies Liquid drops have smooth behaviour with increasing size and do not exhibit jumps

2 No obvious center for nucleons to orbit around unlike electronsin an atom

one pionexchangetwo pion

andheavy mesonexchange

overlap of 3-quark bagscomplicatedshort-rangebehaviour

3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other

before

after

Many theoretical reasons were given why nuclei shouldnot show any shell structure

but experiment says otherwise

2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms

Note jumps atnucleon numbers8 20 28 50 82 126

from KraneIntroductoryNuclear Physics

from Gadioli ampGadioliIntroNuclearPhysics

abundancesof even-evennuclides

neutron capturecross section--analogous tochemicalreactivity

dips at N=20 50 82 126

local maximaat 50 82 126

from KraneIntroductoryNuclear Physics

alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)

neutroncapturecross section(repeat)

nuclearchargeradius

minima atN=20 28 50 82 126

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
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• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

from Krane IntroductoryNuclear Physics

atomicradius

atomicionizationenergy

chemical reactivity

Not evident a priori that nuclei should show shell structure Why not

1 Success of liquid drop model in predicting nuclear binding energies Liquid drops have smooth behaviour with increasing size and do not exhibit jumps

2 No obvious center for nucleons to orbit around unlike electronsin an atom

one pionexchangetwo pion

andheavy mesonexchange

overlap of 3-quark bagscomplicatedshort-rangebehaviour

3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other

before

after

Many theoretical reasons were given why nuclei shouldnot show any shell structure

but experiment says otherwise

2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms

Note jumps atnucleon numbers8 20 28 50 82 126

from KraneIntroductoryNuclear Physics

from Gadioli ampGadioliIntroNuclearPhysics

abundancesof even-evennuclides

neutron capturecross section--analogous tochemicalreactivity

dips at N=20 50 82 126

local maximaat 50 82 126

from KraneIntroductoryNuclear Physics

alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)

neutroncapturecross section(repeat)

nuclearchargeradius

minima atN=20 28 50 82 126

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 47

chemical reactivity

Not evident a priori that nuclei should show shell structure Why not

1 Success of liquid drop model in predicting nuclear binding energies Liquid drops have smooth behaviour with increasing size and do not exhibit jumps

2 No obvious center for nucleons to orbit around unlike electronsin an atom

one pionexchangetwo pion

andheavy mesonexchange

overlap of 3-quark bagscomplicatedshort-rangebehaviour

3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other

before

after

Many theoretical reasons were given why nuclei shouldnot show any shell structure

but experiment says otherwise

2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms

Note jumps atnucleon numbers8 20 28 50 82 126

from KraneIntroductoryNuclear Physics

from Gadioli ampGadioliIntroNuclearPhysics

abundancesof even-evennuclides

neutron capturecross section--analogous tochemicalreactivity

dips at N=20 50 82 126

local maximaat 50 82 126

from KraneIntroductoryNuclear Physics

alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)

neutroncapturecross section(repeat)

nuclearchargeradius

minima atN=20 28 50 82 126

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
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• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Not evident a priori that nuclei should show shell structure Why not

1 Success of liquid drop model in predicting nuclear binding energies Liquid drops have smooth behaviour with increasing size and do not exhibit jumps

2 No obvious center for nucleons to orbit around unlike electronsin an atom

one pionexchangetwo pion

andheavy mesonexchange

overlap of 3-quark bagscomplicatedshort-rangebehaviour

3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other

before

after

Many theoretical reasons were given why nuclei shouldnot show any shell structure

but experiment says otherwise

2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms

Note jumps atnucleon numbers8 20 28 50 82 126

from KraneIntroductoryNuclear Physics

from Gadioli ampGadioliIntroNuclearPhysics

abundancesof even-evennuclides

neutron capturecross section--analogous tochemicalreactivity

dips at N=20 50 82 126

local maximaat 50 82 126

from KraneIntroductoryNuclear Physics

alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)

neutroncapturecross section(repeat)

nuclearchargeradius

minima atN=20 28 50 82 126

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

one pionexchangetwo pion

andheavy mesonexchange

overlap of 3-quark bagscomplicatedshort-rangebehaviour

3rd objection to a shell model Strong repulsivecore in the nucleon-nucleonpotential should scatterthe nucleons farout of their orbits --nucleons should nothave a well-definedenergy but shouldbehave more likemolecules in a gascolliding and exchanging energywith each other

before

after

Many theoretical reasons were given why nuclei shouldnot show any shell structure

but experiment says otherwise

2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms

Note jumps atnucleon numbers8 20 28 50 82 126

from KraneIntroductoryNuclear Physics

from Gadioli ampGadioliIntroNuclearPhysics

abundancesof even-evennuclides

neutron capturecross section--analogous tochemicalreactivity

dips at N=20 50 82 126

local maximaat 50 82 126

from KraneIntroductoryNuclear Physics

alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)

neutroncapturecross section(repeat)

nuclearchargeradius

minima atN=20 28 50 82 126

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Many theoretical reasons were given why nuclei shouldnot show any shell structure

but experiment says otherwise

2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms

Note jumps atnucleon numbers8 20 28 50 82 126

from KraneIntroductoryNuclear Physics

from Gadioli ampGadioliIntroNuclearPhysics

abundancesof even-evennuclides

neutron capturecross section--analogous tochemicalreactivity

dips at N=20 50 82 126

local maximaat 50 82 126

from KraneIntroductoryNuclear Physics

alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)

neutroncapturecross section(repeat)

nuclearchargeradius

minima atN=20 28 50 82 126

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

2-neutron separationenergy in nuclei-- analogousto ionization energyin atoms

Note jumps atnucleon numbers8 20 28 50 82 126

from KraneIntroductoryNuclear Physics

from Gadioli ampGadioliIntroNuclearPhysics

abundancesof even-evennuclides

neutron capturecross section--analogous tochemicalreactivity

dips at N=20 50 82 126

local maximaat 50 82 126

from KraneIntroductoryNuclear Physics

alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)

neutroncapturecross section(repeat)

nuclearchargeradius

minima atN=20 28 50 82 126

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
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• Slide 46
• Slide 47

from Gadioli ampGadioliIntroNuclearPhysics

abundancesof even-evennuclides

neutron capturecross section--analogous tochemicalreactivity

dips at N=20 50 82 126

local maximaat 50 82 126

from KraneIntroductoryNuclear Physics

alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)

neutroncapturecross section(repeat)

nuclearchargeradius

minima atN=20 28 50 82 126

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

from KraneIntroductoryNuclear Physics

alpha particleemissionenergy vs neutron numberof parent nucleusmax for N=128 parent (N=126 daughter)

neutroncapturecross section(repeat)

nuclearchargeradius

minima atN=20 28 50 82 126

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
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• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Experimental data indicates local maxima in binding energy local minima in charge radiusminima in neutron absorption when proton or neutron numbers are one of thefollowing ldquomagic numbersrdquo

2 8 20 28 50 82 126

These numbers mark the shell closures for nucleons analogous to electron shellclosures for atoms

Were physicists we dont believe in magic So where do these ldquomagic numbersrdquocome from

It must be related to the shape of the potential well that the nucleons sit in

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Reminder 1D infinite square well V(x) = 0 for 0 lt x lt L V(x) = infin otherwise

Greg Goebel wwwvectorsitenettpqm_02html

wavefunction is like a guitar string constrainedat both ends (wavefunction excluded fromregion xlt0 and xgtL by infinitely high potentialbarrier)

wavelength λ = 2Ln n=1 2 3

De Broglie momentum p = hλ = nh2L

Potential energy V(x)=0 inside well

E = p22M = n2h2 (8ML2)

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

A different shape for the potential well would result in a different set of level spacings eg

Krane IntroductoryNuclear Physics

infinite squarewell potential

harmonic oscillatorpotential

Notations means L=0p means L=1d L=2f L=3

p state can have2L+1 =3 substates

each substate canhave spin up orspin down

Thus 6 protons and6 neutrons in 1plevel

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
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• Slide 28
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• Slide 33
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• Slide 35
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• Slide 37
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• Slide 39
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• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

From lecture 1 we learned that the charge distribution of a nucleus lookslike the figure below and since the nuclear forces are short rangethe nuclear potential must also be shaped like this

from KraneIntroductoryNuclear Physics

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

But this shape of potential doesnt give the right energy level spacings either

No reasonable shape of nuclear potential seems to work

Empirical magic numbers2 8 20 28 50 82 126

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Fermis suggestion ldquoAny evidence for a spin-orbit forcerdquo

rarr rarr rarrV(r) = V

0(r) + V

LS LS L = orbital angular momentum

rarr S = spin angular momentum

where the potentials V0 and V

LS are negative (ie attractive)

L and S paralleldeeper attractive potential

L and S anti-parallelshallower attractive potential

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 47

So for example if L=1 S=frac12 then by the rule of angular momentum additionin quantum mechanics the total angular momentum J = 1 + frac12 = 32 (parallel)or J = 1 - frac12 = frac12 (anti-parallel) In the presence of a spin-orbit force these twolevels are split the larger the L value the larger the splitting

J=32 and J=12

J=12

J=32

In the H atom this is a small effect which results in the ldquofine structurerdquo of thehydrogen atom spectral lines In nuclear physics the spin-orbit effect islarge enough to shift the energy levels up into the next major shell therebychanging the location of the large gaps

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
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• Slide 19
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• Slide 31
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• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Hydrogen atom

Small spin-orbitsplitting(greatly exaggeratedactual splittingof 2P

32 and 2P

12 is

only 0000045 eVcompared tothe ionization energyof 34 eV ie~ 1 part in 105)

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
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• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

With a nuclear spin-orbit force the levels now match what is observed experimentally

1g levelis split somuch thatthe twocomponentsnow liein differentshells

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 47

Maria Goeppert-MayerHans JensenNobel Prize in Physics 1963

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 47

Now we can build up the shell structure of nuclei just like we fill up electron shells in chemistry class only now there are separate proton and neutron shells

J=12 so there are2 possible orientationsnamely Jz = +12 -12Therefore maximumof 2 nucleonsin this sub- shell

J=32 so there are4 possible orientationsnamelyJz=32 12 -12 -32Therefore maximumof 4 nucleonsin this sub-shell

In general shell with angular momentum Jcan hold up to 2J+1 nucleons of each type

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
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• Slide 45
• Slide 46
• Slide 47

filled 1S shellcorrespondingto magic no 2

4He

so He-4 is a closed-shell nucleus with extra high bindingenergy extra-small radius extra-low reaction probability

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

16O

filled 1S shell

filled 1P shell

Oxygen-16 is another double closed-shell nucleus

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
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• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

The protons and neutron pair off to form angular momentum J=0 pairsTherefore every even-even nucleus (even number of protonseven number of neutrons) has a ground state with Jπ = 0+

eg 12C N=6 Z=6 has a 0+ ground state

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

If there is a single unpaired nucleon (either proton or neutron)then the angular momentum of that single nucleon is theangular momentum of the ground state of that nucleus

Carbon-13 has one unpaired neutron in the 1P12 shell so the C-13 ground state has J=12

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Carbon-11 has one unpaired neutronin the 1P32 subshell so C-11 groundstate has J=32

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
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• Slide 31
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• Slide 38
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• Slide 41
• Slide 42
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• Slide 44
• Slide 45
• Slide 46
• Slide 47

The ldquomagic numberrdquo that wehave seen so far were obtainedfor nuclei close to the valleyof stability

As we move further and furtherfrom the valley of stability byadding more neutrons theshape of the nuclear potentialchanges and so the locationof the ldquomagic numbersrdquomarking the shell closuresalso changes

This is one of the frontiersthat are being explored atISAC -- how do the magicnumbers evolve as we moveaway from the valley of stability

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
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• Slide 26
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• Slide 33
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• Slide 35
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• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Alpha Decay

α-particles are 4He nuclei

Spontaneous emission of α-particles occurs for heavy nucleias a way for the nucleus to reduce the ratio Z2A

eg 238U rarr 234Th + α

Z 92 90 A 238 234Z2A 3556 3461

Recall that it is the Coulombrepulsion ~ Z2 that reducesthe binding energy ofheavy nuclei while theattractive strong interactionterm ~ A soZ2A is the ratio of repulsiveCoulomb potential toattractive strong potential

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Well if shedding charge were the aim why doesnt the nucleus justspit out protons or deuterons then

The α particle is especially tightly bound (especially small mass)so that in the decay

238U rarr 234Th + α

this gives an especially large amount of available energy to the decayproduct

It is expected from ldquophase spacerdquo considerationsie counting the number of availablestates a la statistical mechanics that the decay rate

Rate ~ p where p=momentum of the particle

and for non-relativistic particles E = p22M

so Rate ~ E12

ie doubling the energy results in radic2 increase in the decay rate

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Q = energy released

compare 232Th and218Th

roughly double theenergy but 24 ordersof magnitude changein decay rate

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

This defied explanation until George Gamow realized it was the result of quantummechanical tunneling

Quantum mechanical tunneling is just the quantumanalog of evanescant waves in optics

Remember Snells law of refractionn1 sinθ1 = n2 sinθ2Total internal reflection can occur if lightmoves from a region of high refractive index to a regionof low refractive index when sinθ2 gt 1

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
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• Slide 45
• Slide 46
• Slide 47

If you can actually see a real ray of lightpropagating freely like this thenits electric field varies like a sinewave

But what about right hereDoes the electric fieldabruptly drop to zero atthe boundary if there istotal internal reflection

The answer is NOThe electromagnetic waveactually becomes adecaying exponentialand takes some distanceto completely die awayIt looks something like this

E

The bigger the difference in refractive index the faster theexponential drops off

E

rarr

rarr

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
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• Slide 19
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• Slide 35
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• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

If you hold a second block of glass very close to the first oneyou will see a faint ray of light propagating into the secondblock even though there is ldquototalrdquo internal reflection accordingto Snells law The smaller the air gap the stronger thelight in the second block and the weaker the internal reflectionIn the limit of zero air gap all the light passes through and there is no internal reflection

The decaying exponentialwave in the gap regionis called an evanescentwave

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Initial glass blockStrong light beam(large amplitude sinusoidal E-field)

Air gapAmplitude ofE-field dropsexponentiallywith distance

Second glass blockWeak light beam(small amplitudesinusoidal E-field)

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
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• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Other examples of classical evanescent waves

submarines hundreds of metres beneaththe sea receive communications with ELF (Extremely Low Frequency ndash 30-300 Hz λ=103-104 km) EM waves which penetrateinto the conductive sea-water as an exponentiallyattenuated evanescent wave

The better the conductingmedium the faster theexponential drops off

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Now consider a stream of particlesof energy E hitting a wall (a potential energy barrier) of height V

E

E

V

Classically if E lt V then NO particles can penetrate through the wall -- they dont have enough energy to climb over the wall

But if these are subatomic particles the wave-like nature of the particles is importantThe particles wavefunction does not abruptly end at the face of the wall ndash the wavefunction inside the barrier (the classically forbidden region) is a decaying exponential just like we had in opticsThe higher or thicker the barrier the faster the exponential falls off

And just like the light rays in the two blocks of glass a small fraction of theparticles will emerge on the other side of the barrier This is quantummechanical tunneling

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
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• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Heres what the wavefunction looks like

Initial sinusoidal wavewith large amplitude A

Barrier regiondecayingexponential

Final sinusoidal wavewith small amplitude B

Since the probability of finding a particle is given by the square of thewavefunctions amplitude |ψ(x)|2 the initial probability is A2 thefinal probability is B2 and the transmission probability T = B2 A2

It can be shown that T ~ exp( -2αd) where α=radic2m(V-E)and d = thickness of the barrier

ie the probability of penetrating the barrier drops exponentially with the barrier thickness and exponentially with the square root of the height of the barrier

So a small change in energy E or thickness d makes a huge change in penetration

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
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• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45
• Slide 46
• Slide 47

Because of the hyperbolic shape of the Coulombbarrier a higher α-particle energy E means botha lower barrier and a thinner barrier

Since penetration goes exponentially as thethickness and exponentially as the square rootof the barrier height a small increase in energyE means a MUCH MUCH larger probability thatthe α-particle will leak out

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

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• Slide 47

For a hyperbolic-shaped barrier it can be shown that

Transmission prob T = exp(-2G) = exp(-2Z E-12)

The half-life must vary inversely with the transmission prob(if the α particles leak out half as fast the nuclei will live twice as long)

Half-life t12 ~ 1T = exp(+2Z E-12)

ln(t12 ) ~ 2 Z E-12

from Segregrave Nuclei and Particles

Confirms linear relationship between

ln(t12 ) and E-12

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

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• Slide 47

Now we understand whythe half-life of α-decaydecreases 24 orders of magnitudewhen the energy justchanges by a factor of 2Its because the α particleshave to quantum mechanicallytunnel through the hyperbolic-shaped Coulomb barrier

The first example of quantumtunneling that was discoveredand an inevitable consequenceof wave-particle duality

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
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• Slide 47

Putting in some typical numbers

For α-particles of ~4 MeV velocity v ~ 046c ~ 14x107 msecrattling around inside a thorium nucleus (diameter=147 fm = 15x10-14 m)

those α-particles would traverse the diameter of the nucleus andstrike the walls ~1021 times per second

and yet it takes ~14x1010 years to leak out

It is evident that the tunneling probability is very low

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
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• Slide 47

Next week Nuclear reactions

The same process for alpha particles tunneling OUT of a nucleusalso works in reverse for charged particles trying to get INTO the nucleus

accelerator

projectiletarget

The projectiles have to quantum mechanically tunnel through theCoulomb barrier rarr reaction probability increases exponentiallywith the radic energy of the projectile rarr important astrophysical implications

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
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• Slide 47

Alvin the α-particle is the epitome of the sayingldquoIf at first you dont succeed try againrdquoHe rams his head against the wall 1021 times per secondfor up to 1010 years before escaping from his prison inside the nucleus

I hope you find learning nuclear physics easier than that

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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