nuclear reaction energetics

Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh

1

Nuclear Reaction Energetics

Conservation Laws• Charge, Baryon number, total energy, linear momentum, angular momentum, parity, (isospin??) …….

apa X

pY

pb

Y

bQTTcmcm iffi 22

+ve Q-value exoergic reaction. -ve Q-value endoergic reaction.

aYb TQTT +ve Q-value reaction possible if Ta 0. -ve Q-value reaction not possible if Ta 0. (Is Ta > |Q| sufficient?).

Conservation of momentum ……


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• Conservation of momentum.• We usually do not detect Y.Show that:

• The threshold energy (for Ta): (the condition occurs for = 0º).

• +ve Q-value reaction possible if Ta 0.• Coulomb barriers…….!!!• Neutrons vs. charged particles.• -ve Q-value reaction possible if Ta > TTh.

bY

aaYYbYabaabab mm

TmmQmmmTmmTmmT

])()[(coscos 2

HW b1HW b1

abY

bYTh mmm

mmQT



3


• The double valued situation occurs between TTh and the upper limit Ta

\.

• Double-valued in a forward cone.

aY

Ya mm

mQT

\

aba

aaYYbY

Tmm

TmmQmmm ])()[(cos max

2

HW b1HW b1 (continued)(continued)


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HW b1HW b1 (continued)(continued)

• Discuss the elasticelastic and inelastic inelastic

scatteringscattering of neutronsneutrons using these relations.

• If the reaction reaches excited states of Y

exbexYaXex EQcmEcmcmcmQ 02222 )(


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6



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Neutron Interactions

• Chadwick’s discovery.• Neutrons interact with nuclei, not with atoms. (Exceptions).

• Recall:o Inelastic scattering (n,n\). Q = -E* Inelastic gammas.

Threshold?o Elastic scattering (n,n). Q = ?? (Potential and CN).

Neutron moderation?o Radiative capture (n,). Q = ?? Capture gammas.o (n,), (n,p). Q = ?? Absorption Reactions.o (n,2n), (n,3n) Q = ?? Energetic neutrons on heavy water can easily eject the loosely bound neutron.o Fission. (n,f).

HW b2HW b2 Examples of such exo- and endo-thermic reactions with Q calculations.


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11

n-TOFn-TOFCERNCERN


12


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• Probability.• Projectile a will more probably hit target X if area is larger.• Classically: = (Ra + RX)2. Classical = ??? (in b) n + 1H, n + 238U, 238U + 238U • Quantum mechanically: = 2.

• Coulomb and centrifugal barriers energy dependence of . What about neutrons?What about neutrons?

CMaXaXaaX

Xa

EEmm

mm

22

Reaction Cross Section


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Neutron Induced Reactions

22 nXHCCHbY IIIn X(n,b)Y

n(En)b(Q+En)

For thermal neutronsQ >> En

b(Q) constant

2

11

vE

)( nln EPvn

Probability to penetrate the potential barrier

Po(Ethermal) = 1P>o(Ethermal) = 0

vEnn

1)( Non-resonant


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Neutron Induced Reactions

Nuclear Fission

1/v

235U thermal cross sectionsfission 584 b.scattering 9 b.radiative capture 97 b.

Fast neutrons should be moderated.

Fission Barriers 16Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh


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Resonance Reactions

Entrance Channela + X

ExitChannelb + YCompound

Nucleus C*

ExcitedState

ExJ

a + X Y + b Q > 0b + Y X + a Q < 0

Inverse Reaction

22 )1()12)(12(

12XaHCCHbY

JJ

JIIIaX

XaaXaX

QM StatisticalFactor ()

Identicalparticles

• Nature of force(s).• Time-reversal invariance.

22 )1()12)(12(

12YbHCCHXa

JJ

JIIIbY

YbbYbY

??bY

aX


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Projectile

TargetQ-value

Projectile

Q-valueTarget

Direct Capture(all energies)

Resonant Capture(selected energies with large X-section)

E = E + Q - Eex

2XaHY

Q + ER = Er

22XaHEEHE CNrrf

ba

Resonance Reactions


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Resonance Reactions

22

2 )()(

o

fresponse

Damped OscillatorDamped Oscillator

eigenfrequency

Dampingfactor

Oscillator strength

22

2 )()()(

R

ba

EEE

0

1

t

ot


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Resonance Reactions

22

2

2

)()()1(

)12)(12(

12)(

R

baaX

XaaX EEJJ

JE

Breit-Wigner formulaBreit-Wigner formula

• All quantities in CM system• Only for isolated resonances.

a

b

e

R

aae

baR

Reaction

Elastic scattering

HW b3HW b3 When does R take its maximum value?

ba

Usually a >> b.


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Neutron Resonance Reactions


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Neutron Activation Analysis

(Z,A) + n (Z, A+1)-

(Z+1, A+1)

(-delayed -ray)

Show that, after elasticelastic scattering the ratio between the final neutron energy E\ and its initial energy E is given by:

For a head-on collision:

After n ss-wave-wave collisions:where the average change in lethargy lethargy is

HW b4HW b4

2

222

2

2\

)1(

sincos

)1(

cos21

A

A

A

AA

E

E CM

2

min

\

1

1

A

A

E

E

nEEn lnln \

1

1ln

2

)1(1ln

2

\

A

A

A

A

E

Eu

av

24Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh

Neutron Moderation

)ln( EEu M

Reference

Average decrease in ln(E) after one collision.

11H ?H ?


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Neutron Moderation

• Reproduce the plot.• Discuss the effect of the thermal motion of the moderator atoms.

On 12C.

Most Most probable probable

and average and average energies?energies?

Neutron Moderation

• How many collisions are needed to thermalize a 2 MeV neutron if the moderator was:

1H 2H 4He graphite 238U ?• What is special about 1H?• Why we considered elastic scattering?• When does inelastic scattering become important?

26Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh

nuclear reaction energetics

Documents

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inelastic scattering

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ta tth