nuclear reactors, bau, 1st semester, 2008-2009 (saed dababneh). 1 neutron attenuation (revisited)...

Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

1

Neutron Attenuation (revisited)

X

X

t

t

eXP

eXP

1)(

)(

ninteractio

ninteractiono

Recall t = N t

Probability per unit path length.

X

I0 I

Probability

mfp for scattering s = 1/s

mfp for absorption a = 1/a………….

total mfp t = 1/t

XteIXI 0)(


2

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


3

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



4


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 aaThermal

aa


5


Thermal

aa dEEvEnER )()()(

)()()( 0

0 Evv

EE

a

a

Reference

2200 m/s flux2200 m/s flux

Independent of Independent of n(E)n(E)..

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 6HW 6

2

222

2

2\

)1(sincos

)1(cos21

AA

AAA

EE CM

2

min

\

11

AA

EE

nEEn lnln \

11ln

2)1(1ln

2

\

AA

AA

EEu

av

6Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

Neutron Moderation

)ln( EEu M

Reference

Average decrease in ln(E) after one collision.

11H ?H ?


7

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?

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?


EE )1(21\

EEEE )1(21\

)1(21

EE

Nuclear Fission

~200 MeV

Fission

Fusio

n

Coulomb effectSurface effect


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(.44.12


Nuclear Fission

• 238U (t½ = 4.5x109 y) for -decay.• 238U (t½ 1016 y) for 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.


Nuclear Fission

Liquid Drop

Shell

Act

ivat

ion

Ene

rgy

(MeV

)


Nuclear Fission

Surface Term Bs = - as A⅔

Coulomb Term BC = - aC Z(Z-1) / A⅓

3

34 R

2

34 ab

=

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.


Nuclear Fission

48300

)120( 2

Extrapolation to 47 10-20 s.

Consistent with activation energy curve for A = 300.


Nuclear Fission

235U + n93Rb + 141Cs + 2nNot unique.

Low-energy fission processes.


Nuclear Fission

Z1 + Z2 = 92Z1 37, Z2 55A1 95, A2 140Large neutron excessMost 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.


Nuclear Fission

The average number of neutrons is different, but the distribution is Gaussian.



18

Why only left side of the

mass parabola?

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.


• Waste.• Poison.

In general, decay favors high

energy.

Nuclear Fission


Nuclear Fission

1/v

235U thermal cross sectionsfission 584 b.scattering 9 b.radiative capture 97 b.

Fast neutrons should be moderated.

Fission Barriers 21Nuclear Reactors, BAU, 1st Semester, 2008-2009 (Saed Dababneh).

22

Nuclear Fission

• Q for 235U + n 236U is 6.54478 MeV.• Table 13.1 in Krane: Activation energy EA for 236U 6.2 MeV (Liquid drop + shell) 235U can be fissioned with zero-energy neutrons.

• Q for 238U + n 239U is 4.??? MeV.• EA for 239U 6.6 MeV MeV neutrons are needed.• Pairing term: = ??? (Fig. 13.11 in Krane).• What about 232Pa and 231Pa? (odd Z).• Odd-N nuclei have in general much larger thermal neutron cross sections than even-N nuclei (Table 13.1 in Krane).



23

Nuclear Fission

f,Th 584 2.7x10-6 700 0.019 b

Why not use it?Why not use it?

nuclear reactors, bau, 1st semester, 2008-2009 (saed dababneh). 1 neutron attenuation (revisited)...

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