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)

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

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).

3

Neutron Induced Reactions

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

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

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).

6

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)

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).

7

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

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.

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

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).

11

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?)

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).

12

Neutron Resonance Reactions (revisited)

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).

13

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, Second Semester, 2008-2009 (Saed Dababneh).

14

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, Second Semester, 2008-2009 (Saed Dababneh).

15

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

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

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