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Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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1/v
235U thermal cross sectionsfission 584 b.scattering 9 b.radiative capture 97 b.
Fast neutrons should be moderated.
Fission Barriers
Neutron Cross Section (Different Features)
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Neutron Induced Reactions
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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bbplL
lb 222
1max, )12( lbb lll
)()(
7.656)(2
keVEub
CM HW 3HW 3
)1()12)(12(
122max aX
XaaX JJ
J
Statistical Factor (Introduction)
Generalization
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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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
HW 4HW 4
More Generalization
Reaction Cross Section
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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
Resonance Reactions
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (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 5HW 5 When does R take its maximum value?
ba
Usually a >> b.
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Resonance Reactions
Ja + JX + l = J(-1)l (Ja) (JX) = (J)
(-1)l = (J) Natural parity.
ExitChannelb + Y
Compound Nucleus C*
ExcitedState
ExJ
Entrance Channela + X
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Resonance Reactions
Cro
ss s
ecti
on
EC a Energy
What is the “Resonance Strength” …?What is its significance?In what units is it measured?
ba
aXXa JJ
J)1(
)12)(12(
12
Charged particleradiative capture (a,)(What about neutrons?)(What about neutrons?)
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Neutron Resonance Reactions
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Neutron Activation Analysis
(Z,A) + n (Z, A+1)-
(Z+1, A+1)
(-delayed -ray)
Project 1Project 1
NAA and UNAA and U
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Recall Ft = n v t N = I t Simultaneous beams, different intensities, same energysame energy.
Ft = t (IA + IB + IC + …) = t (nA + nB + nC + …)vIn a reactorreactor, if neutrons are moving in all directionsall directions
n = nA + nB + nC + …
Ft = t nv
neutron flux = nv
Reaction Rate Rt Ft = t = /t (=nvNt)
Neutron Flux and Reaction Rate
Not talking about a beam
anymore.
same energysame energy
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Different energiesDifferent energiesDensity of neutrons with energy between E and E+dEn(E)dEReaction rate for those “monoenergetic” neutronsdRt = t(E) n(E)dE v(E)
0
)( dEEnn
00
)()()( dEEEndEE
00
)()()()()( dEEEdEEEnER ttt
0
)()( dEEER ii
Neutron Flux and Reaction Rate
Units!
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Neutron Flux and Reaction Rate
In general, neutron flux depends on:• Neutron energy, E.• Neutron spatial position, r. • Neutron angular direction, • Time, t.
Various kinds of neutron fluxes (depending on the degree of detail needed).
Time-dependent and time-independent angular neutron flux.
),,( Er),,,( tEr