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TRANSCRIPT
Introduction to Nuclei Physics
1
Wednesday, Mach, 23, 2011
Arif Hidayat
1. Nature of the Nuclear Force • Shape of the Nuclear Potential
• Yukawa Potential
• Range of Yukawa Potential
2. Nuclear Models • Liquid Drop Model
• Fermi Gas Model
• Shell Model
• A square well nuclear potential provides the basis of quantum theory with discrete energy levels and corresponding bound state just like in atoms – Presence of nuclear quantum states have been confirmed
through • Scattering experiments • Studies of the energies emitted in nuclear radiation
• Studies of mirror nuclei and the scatterings of protons and neutrons demonstrate – Without the Coulomb effects, the forces between two
neutrons, two protons or a proton and a neutron are the same • Nuclear force has nothing to do with electrical charge • Protons and neutrons behave the same under the nuclear force
– Inferred as charge independence of nuclear force.
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Nuclear Potential
• Strong nuclear force is independent of the electric charge carried by nucleons – Concept of strong isotopic-spin symmetry.
• proton and neutron are the two different iso-spin state of the same particle called nucleon
– In other words, • If Coulomb effect were turned off, protons and neutrons
would be indistinguishable in their nuclear interactions
• Can you give another case just like this???
– This is analogues to the indistinguishability of spin up and down states in the absence of a magnetic field!!
• This is called Iso-spin symmetry!!!
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Nuclear Potential – Iso-spin symmetry
• EM force can be understood as a result of a photon exchange – Photon propagation is described by the Maxwell’s
equation – Photons propagate at the speed of light. – What does this tell you about the mass of the
photon? • Massless
• Coulomb potential is
• What does this tell you about the range of the Coulomb force? – Long range. Why?
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Range of the Nuclear Force
V r 1
r
Massless
particle
exchange
• For massive particle exchanges, the potential takes the form
– What is the mass, m, in this expression?
• Mass of the particle exchanged in the interaction – The force mediator mass
• This form of potential is called Yukawa Potential – Formulated by Hideki Yukawa in 1934
• What does Yukawa potential turn to in the limit m 0? – Coulomb potential
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Yukawa Potential
V r
mcr
e
r
• From the form of the Yukawa potential
• The range of the interaction is given by some characteristic value of r. What is this? – Compton wavelength of the mediator with
mass, m:
• What does this mean? – Once the mass of the mediator is known, range
can be predicted – Once the range is known, the mass can be
predicted 6
Ranges in Yukawa Potential
mcr
eV r
r
re
r
mc
• Let’s put Yukawa potential to work • What is the range of the nuclear force?
– About the same as the typical size of a nucleus • 1.2x10-13cm
– thus the mediator mass is
• This is close to the mass of a well known p meson (pion)
• Thus, it was thought that p are the mediators of the nuclear force
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Ranges in Yukawa Potential
2mc
mp
c
197164
1.2
MeV fmMeV
fm
2139.6 / ;MeV c
2
0 135 /m MeV cp
mp
• Experiments showed very different characteristics of nuclear forces than other forces
• Quantification of nuclear forces and the structure of nucleus were not straightforward
– Fundamentals of nuclear force were not well understood
• Several phenomenological models (not theories) that describe only limited cases of experimental findings
• Most the models assume central potential, just like Coulomb potential
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Nuclear Models
• An earliest phenomenological success in describing binding energy of a nucleus
• Nucleus is essentially spherical with radius proportional to A1/3. – Densities are independent of the number of nucleons
• Led to a model that envisions the nucleus as an incompressible liquid droplet – In this model, nucleons are equivalent to molecules
• Quantum properties of individual nucleons are ignored
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Nuclear Models: Liquid Droplet Model
• Nucleus is imagined to consist of
– A stable central core of nucleons where nuclear force is completely saturated
– A surface layer of nucleons that are not bound tightly
• This weaker binding at the surface decreases the effective BE per nucleon (B/A)
• Provides an attraction of the surface nucleons towards the core just as the surface tension to the liquid
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Nuclear Models: Liquid Droplet Model
• If a constant BE per nucleon is due to the saturation of the nuclear force, the nuclear BE can be written as:
• What do you think each term does?
– First term: volume energy for uniform saturated binding
– Second term corrects for weaker surface tension
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Liquid Droplet Model: Binding Energy
BE
• This can explain the low BE/nucleon
behavior of low A nuclei
– For low A nuclei, the proportion of the
second term is larger.
– Reflects relatively large number of
surface nucleons than the core.
1a A 2 3
2a A
• Small decrease of BE for heavy nuclei can be understood as due to Coulomb repulsion – The electrostatic energies of protons have destabilizing
effect
• Reflecting this effect, the empirical formula for BE takes the correction term
• All terms of this formula have classical origin. • This formula does not explain
– Lighter nuclei with the equal number of protons and neutrons are stable or have a stronger binding (larger –BE)
– Natural abundance of stable even-even nuclei or paucity of odd-odd nuclei
• These could mainly arise from quantum effect of spins.
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Liquid Droplet Model: Binding Energy
2 31 2BE a A a A 2 1 3
3a Z A
• Additional corrections to compensate the deficiency, give corrections to the empirical formula (again…)
– All parameters are assumed to be positive
– The forth term reflects N=Z stability
– The last term • Positive sign is chosen for odd-odd nuclei, reflecting
instability
• Negative sign is chosen for even-even nuclei
• For odd-A nuclei, a5 is chosen to be 0.
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Liquid Droplet Model: Binding Energy
2 3 2 1 31 2 3BE a A a A a Z A
2
4
N Za
A
3 45a A
• The parameters are determined by fitting experimentally observed BE for a wide range of nuclei:
• Now we can write an empirical formula for masses of nuclei
• This is Bethe-Weizsacker semi-empirical mass formula
– Used to predict stability and masses of unknown nuclei of arbitrary A and Z
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Liquid Droplet Model: Binding Energy
1 15.6a MeV 2 16.8a MeV 3 0.72 a MeV
4 23.3 a MeV 5 34 ; a MeV
2
, n p
BEM A Z A Z m Zm
c n pA Z m Zm
12
aA
c 2 32
2
aA
c 2 1 33
2
aZ A
c
2
42
N Za
Ac
3 452
aA
c
• An early attempt to incorporate quantum effects
• Assumes nucleus as a gas of free protons and neutrons confined to the nuclear volume
– The nucleons occupy quantized (discrete) energy levels
– Nucleons are moving inside a spherically symmetric well with the range determined by the radius of the nucleus
– Depth of the well is adjusted to obtain correct binding energy
• Protons carry electric charge Senses slightly different potential than neutrons
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Nuclear Models: Fermi Gas Model
• Nucleons are Fermions (spin ½ particles) so – Obey Pauli exclusion principle – Any given energy level can be occupied by at most
two identical nucleons – opposite spin projections
• For a greater stability, the energy levels fill up from the bottom to the Fermi level – Fermi level: Highest, fully occupied energy level (EF)
• Binding energies are given as follows: – BE of the last nucleon= EF since no Fermions above
EF – In other words, the level occupied by Fermion
reflects the BE of the last nucleon
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Nuclear Models: Fermi Gas Model
• Experimental observations show BE is charge independent
• If the well depth is the same for p and n, BE for the last nucleon would be charge dependent for heavy nuclei (Why?)
– Since there are more neutrons than protons, neutrons sit higher EF
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Nuclear Models: Fermi Gas Model
Same Depth Potential Wells
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Neutron Well Proton Well
Nuclear b-decay
nFE
pFE…
…
en e p
• Experimental observations show BE is charge independent • If the well depth is the same for p and n, BE for the last
nucleon would be charge dependent for heavy nuclei (Why?) – Since there are more neutrons than protons, neutrons sit higher
EF
– But experiments observed otherwise • EF must be the same for protons and neutrons. How do
we make this happen? – Make protons move to a shallower potential well
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Nuclear Models: Fermi Gas Model
• What happens if this weren’t the
case?
– Nucleus is unstable.
– All neutrons at higher energy levels
would undergo a b-decay and
transition to lower proton levels
• Fermi momentum: • Volume for momentum space up to Fermi level • Total volume for the states (kinematic phase space)
– Proportional to the total number of quantum states in the system
• Using Heisenberg’s uncertainty principle: • The minimum volume associated with a physical
system becomes • The nF that can fill up to EF is
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3
22
TOTF
Vn
p
Fermi Gas Model: EF vs nF
2F Fp mE
FpV
FTOT pV V V
x p
3
2stateV p
30
4
3r A
p
23
0
4
3FA r p
p
23
03
2 4
32FA r p
p
p
3
04
9
Fr pA
p
2 2F FE p m
34
3Fp
p
34
3Fp
p
2
Why?
• Let’s consider a nucleus with N=Z=A/2 and assume that all states up to Fermi level are filled
• What do you see about pF above? – Fermi momentum is constant, independent of the number
of nucleons
• Using the average BE of -8MeV, the depth of potential well (V0) is ~40MeV – Consistent with other findings
• This model is a natural way of accounting for a4 term in Bethe-Weizsacker mass formula
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Fermi Gas Model: EF vs nF
2
AN Z
1 3
0
9
8Fp
r
p
or
FE
3
04
9
Fr pA
p
2
2
Fp
m
2 2 3
0
1 9
2 8m r
p
2
20
2.32
2
c
rmc
2.32 19733
2 940 1.2
MeV fmMeV
fm
• Exploit the success of atomic model
– Uses orbital structure of nucleons
– Electron energy levels are quantized
– Limited number of electrons in each level based on available spin and angular momentum configurations • For nth energy level, l angular momentum (l<n), one
expects a total of 2(2l+1) possible degenerate states for electrons
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Nuclear Models: Shell Model
• Orbits and energy levels an electron can occupy are labeled by – Principle quantum number: n
• n can only be integer
– For given n, energy degenerate orbital angular momentum: l • The values are given from 0 to n – 1 for each n
– For any given orbital angular momentum, there are (2l+1) sub-states: ml
• ml=-l, -l+1, …, 0, 1, …, l – l, l
• Due to rotational symmetry of the Coulomb potential, all these sub-states are degenerate in energy
– Since electrons are fermions w/ intrinsic spin angular momentum , • Each of the sub-states can be occupied by two electrons
– So the total number of state is 2(2l+1) 23
Atomic Shell Model Reminder
2
• Exploit the success of atomic model
– Uses orbital structure of nucleons
– Electron energy levels are quantized
– Limited number of electrons in each level based on available spin and angular momentum configurations • For nth energy level, l angular momentum (l<n), one expects a
total of 2(2l+1) possible degenerate states for electrons
• Quantum numbers of individual nucleons are taken into account to affect the fine structure of spectra
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Nuclear Models: Shell Model
• Nuclei have magic numbers just like inert atoms
– Atoms: Z=2, 10, 18, 36, 54
– Nuclei: N=2, 8, 20, 28, 50, 82, and 126 and Z=2, 8, 20, 28, 50, and 82
– Magic Nuclei: Nuclei with either N or Z a magic number Stable
– Doubly magic nuclei: Nuclei with both N and Z magic numbers Particularly stable
• Explains well the stability of nucleus
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Nuclear Models: Shell Model
• To solve equation of motion in quantum mechanics, Schrödinger equation, one must know the shape of the potential
–
– Details of nuclear potential not well known
• A few shapes of potential energies tried out
– Infinite square well: Each shell can contain up to 2(2l+1) nucleons
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Shell Model: Various Potential Shapes
2
2
20
mE V r r
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Nuclear Models: Shell Model – Square
well potential case
NM n l=n-1 Ns=2(2l+1) NT
2 1 0 2 2
8 2 0,1 2+6 8
20 3 0,1,2 2+6+10 18
28 4 0,1,2,3 2+6+10+14 32
50 5 0,1,2,3,4 2+6+10+14+18 50
82 6 0,1,2,3,4,5 2+6+10+14+18+22 72
• To solve equation of motion in quantum mechanics, Schrödinger equation, one must know the shape of the potential
–
– Details of nuclear potential not well known
• A few models of potential tried out
– Infinite square well: Each shell can contain up to 2(2l+1) nucleons
• Can predict 2, 8 and 50 but no other magic numbers
– Three dimensional harmonic oscillator:
• Predicts 2, 8, 20, 40 and 70 Some magic numbers predicted
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Shell Model: Various Potential Shapes
V r 2 21
2m r
2
2
20
mE V r r
• Central potential could not reproduce all magic numbers
• In 1940, Mayer and Jesen proposed a central potential + strong spin-orbit interaction w/
– f(r) is an arbitrary empirical
function of radial coordinates and chosen to fit the data
• The spin-orbit interaction with the properly chosen f(r), a finite square well can split
• Reproduces all the desired magic numbers
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Shell Model: Spin-Orbit Potential
TOTV
Spectroscopic notation: n L j
Orbit number Orbital angular
momentum Projection of
total momentum
V r f r L S
• Spin-Parity of large number of odd-A nuclei predicted well – Nucleons are Fermions so the obey Pauli exclusion
principle
– Fill up ground state energy levels in pairs
– Ground state of all even-even nuclei have zero total angular momentum
• The shell model cannot predict stable odd-odd nuclei spins – No prescription for how to combine the unpaired
proton and neutron spins
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Predictions of the Shell Model
• Magnetic Moment of neutron and proton are
• Intrinsic magnetic moment of unpaired nucleons contribute to total magnetic moment of nuclei
– What does a deuteron consist of?
• Measured value is
– For Boron (10B5) , the 5 neutrons and 5 protons have the same level structure: (1S1/2)2(1P3/2)3, leaving one of each unpaired proton and neutron in angular momentum l=1 state
• Measured value is
• Does not work well with heavy nuclei 31
Predictions of the Shell Model
D
2.79p N 1.91n N
D
B
1.80B N
2.79 1.91N N N
p 2.79 N 1.91 N 0.88 Nn
0.86 N
2 N
el
m c N1
2 N
e
m c
p n orbit 1.88 N
• For heavy nuclei, shell model predictions do not agree with experimental measurements – Especially in magnetic dipole moments
• Measured values of quadrupole moments for closed shells differ significantly with experiments – Some nuclei’s large quadrupole moments suggests
significant nonspherical shapes – The assumption of rotational symmetry in shell model
does not seem quite right
• These deficiencies are somewhat covered through the reconciliation of liquid drop model with Shell model – Bohr, Mottelson and Rainwater’s collective model,
1953 32
Collective Model
• Assumption – Nucleus consists of hard core of nucleons in filled shells – Outer valence nucleons behave like the surface molecules in a
liquid drop – Non-sphericity of the central core caused by the surface motion of
the valence nucleon
• Thus, in collective model, the potential is a shell model with a spherically asymmetric potential – Aspherical nuclei can produce additional energy levels upon
rotation while spherical ones cannot
• Important predictions of collective model: – Existence of rotational and vibrational energy levels in nuclei – Accommodate decrease of spacing between first excite state and
the ground level for even-even nuclei as A increases, since moment of inertia increases with A
– Spacing is largest for closed shell nuclei, since they tend to be spherical
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Collective Model
• Nuclei tend to have relatively small intrinsic spins
• Particularly stable nuclei predicted for A between 150 and 190 with spheroidal character – Semi-major axis about a factor of 2 larger than semi-minor
• Heavy ion collisions in late 1980s produced super-deformed nuclei with angular momentum of
• The energy level spacings of these observed through photon radiation seem to be fixed
• Different nuclei seem to have identical emissions as they spin down
• Problem with collective model and understanding of strong pairing of nucleon binding energy
• Understanding nuclear structure still in progress 34
Super-deformed Nuclei
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