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Nuclear Reaction Energetics. Conservation Laws Charge, Baryon number , total energy, linear momentum, angular momentum, parity, ( isospin ??) ……. b. p b. . gs. a. p a. X. . +ve Q -value exoergic reaction. -ve Q -value endoergic reaction. p Y. Y. Stationary X ??. - PowerPoint PPT Presentation
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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 ……
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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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
Nuclear Reaction Energetics
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
3
Nuclear Reaction Energetics
• 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)
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
4
Nuclear Reaction Energetics
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 )(
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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Nuclear Reaction Energetics
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
6
Nuclear Reaction Energetics
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
7
Nuclear Reaction Energetics
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
8
Nuclear Reaction Energetics
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
9
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.
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
11
n-TOFn-TOFCERNCERN
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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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
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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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
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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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
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
17
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
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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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
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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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
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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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.
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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Neutron Resonance Reactions
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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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 ?
Radiation Detection and Measurement, JU, 1st Semester, 2008-2009 Saed Dababneh
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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