In Thermal Reactors , the absorption rate in a “medium” of thermal ( Maxwellian ) neutrons
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In Thermal ReactorsThermal Reactors, the absorptionabsorption rate in a “medium” of thermal (MaxwellianMaxwellian) neutrons
Usually 1/v cross section, thus
then
The reference energy is chosen at 0.0253 eV. • Look for Thermal Cross Sections.• Actually, look for evaluated nuclear data.
000000 )()()()( EnvEdEEnvER aa
Thermal
aa
Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Neutron Flux and Reaction Rate
Thermal
aa dEEvEnER )()()(
)()(
)( 0
0 Ev
v
E
E
a
a
Reference
2200 m/s flux2200 m/s flux
Independent of Independent of n(E)n(E)..
ENDF
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Show that, after one 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 6HW 6
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
2Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
Neutron Moderation
)ln( EEu M
Reference
Average decrease in ln(E) after one collision.
11H ?H ?
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Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
3
Neutron Moderation HW 6 HW 6 (continued)(continued)
• 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?
First collision.
Second collision.
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Neutron Moderation HW 6 HW 6 (continued)(continued)
Neutron scattering by light nuclei then the average energy loss and the average fractional energy loss
• 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?
4Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
EE )1(21\
EEEE )1(21\
)1(21
E
E
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Nuclear Fission
~200 MeV
Fission
Fusi
on
Coulomb effectSurface effect
5Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Nuclear Fission• B.E. per nucleon for 238U (BEU) and 119Pd (BEPd) ?• 2x119xBEPd – 238xBEU = ?? K.E. of the fragments 1011 J/g• Burning coal 105 J/g• Why not spontaneous?• Two 119Pd fragments just touching The Coulomb “barrier” is:
• Crude …! What if 79Zn and 159Sm? Large neutron excess, released neutrons, sharp potential edge, spherical U…!
MeVMeVfm
fmMeVV 2142502.12
)46(.44.1
2
6Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Nuclear Fission
• 238U (t½ = 4.5x109 y) for -decay.• 238U (t½ 1016 y) for spontaneous fission.• Heavier nuclei??• Energy absorption from a neutron (for example) could form an intermediate state probably above barrier induced fission.• Height of barrier is called activation energy.
7Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Nuclear Fission
Liquid Drop
Shell
Act
iva
tion
Ene
rgy
(MeV
)
8Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Nuclear Fission
Surface Term Bs = - as A⅔
Coulomb Term BC = - aC Z(Z-1) / A⅓
3
3
4R
2
3
4ab=
1
)1(
Rb
Ra23 abR
...)1( 252
...)1( 251
Volume Term (the same)
32
31
52
51 )1( AaAZZa SC fission
47~2
A
Z
Crude: QM and original shape could be different from spherical.
9Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Nuclear Fission
48300
)120( 2
Extrapolation to 47 10-20 s.
Consistent with activation energy curve for A = 300.
10Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Nuclear Fission
235U + n93Rb + 141Cs + 2nNot unique.
Low-energy fission processes.
11Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Nuclear FissionZ1 + Z2 = 92Z1 37, Z2 55A1 95, A2 140Large neutron excess
Most stable:Z=45 Z=58 Prompt neutronsPrompt neutrons within 10-16 s.Number depends on nature of fragments and on incident neutron energy.The average number is characteristic of the process.
12Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
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Nuclear Fission
The average number of neutrons is different, but the distribution is Gaussian.
13Nuclear 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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Why only left side of the
mass parabola?
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Delayed neutronsDelayed neutrons
Higher than Sn?
~ 1 delayed neutron per 100 fissions, but essential for control of the reactor.
Follow -decay and find the most
long-lived isotope (waste) in this
case.
15Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).
• Waste.• Poison.
In general, decay favors high
energy.
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Nuclear Fission
16Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).