Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross...
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Transcript of Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross...
![Page 1: Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture.](https://reader038.fdocuments.us/reader038/viewer/2022102923/5513eb5f55034646298b5d9c/html5/thumbnails/1.jpg)
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
1
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)
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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
2
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
![Page 3: Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture.](https://reader038.fdocuments.us/reader038/viewer/2022102923/5513eb5f55034646298b5d9c/html5/thumbnails/3.jpg)
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
3
Neutron Induced Reactions
![Page 4: Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture.](https://reader038.fdocuments.us/reader038/viewer/2022102923/5513eb5f55034646298b5d9c/html5/thumbnails/4.jpg)
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
4
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 (revisited)
Generalization
![Page 5: Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture.](https://reader038.fdocuments.us/reader038/viewer/2022102923/5513eb5f55034646298b5d9c/html5/thumbnails/5.jpg)
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
5
Resonance Reactions (revisited)
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
![Page 6: Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture.](https://reader038.fdocuments.us/reader038/viewer/2022102923/5513eb5f55034646298b5d9c/html5/thumbnails/6.jpg)
Nuclear Reactor Theory, JU, Second 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
Resonance Reactions (revisited)
![Page 7: Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture.](https://reader038.fdocuments.us/reader038/viewer/2022102923/5513eb5f55034646298b5d9c/html5/thumbnails/7.jpg)
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
7
![Page 8: Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture.](https://reader038.fdocuments.us/reader038/viewer/2022102923/5513eb5f55034646298b5d9c/html5/thumbnails/8.jpg)
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
8
Resonance Reactions (revisited)
22
2 )()(
o
fresponse
Damped OscillatorDamped Oscillator
eigenfrequency
Dampingfactor
Oscillator strength
22
2 )()()(
R
ba
EEE
0
1
t
ot
![Page 9: Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture.](https://reader038.fdocuments.us/reader038/viewer/2022102923/5513eb5f55034646298b5d9c/html5/thumbnails/9.jpg)
Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
9
Resonance Reactions (revisited)
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.
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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
10
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
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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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Resonance Reactions (revisited)
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?)
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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
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Neutron Resonance Reactions (revisited)
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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (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
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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (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
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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (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
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Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).
16
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
![Page 17: Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh). 1 1/ v 235 U thermal cross sections fission 584 b. scattering 9 b. radiative capture.](https://reader038.fdocuments.us/reader038/viewer/2022102923/5513eb5f55034646298b5d9c/html5/thumbnails/17.jpg)
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, Second Semester, 2008-2009 (Saed Dababneh).
17
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