Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 1 Neutron Attenuation (revisited)...

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

1

Neutron Attenuation (revisited)

Xeraction

Xeractionno

t

t

eXP

eXP

1)(

)(

int

int

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 = t I 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


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

0

)()( dEEEn

00

)()()()()( dEEEdEEEnER ttt

0

)()( dEEER ii



4


In general, neutron flux depends on:• Neutron energy, E.• Neutron angular direction, • Neutron spatial position, r.• Time, t.

Various kinds of neutron fluxes (depending on the degree of detail needed).

Time-dependent and time-independent angular neutron flux. ),,,( tE Ωr

),,( ΩEr

In Thermal ReactorsThermal Reactors, the absorptionabsorption rate in a “medium” of thermal (Maxwellian) 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


5


Thermal

aa dEEvEnER )()()(

)()(

)( 0

0 Ev

v

E

E

a

a

Reference

2200 m/s flux2200 m/s flux

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

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

6Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

Neutron Moderation

)ln( EEu M

Reference


7

Neutron Moderation HW 6 HW 6 (continued)(continued)

• Reproduce the plot.• Discuss the effect of the thermal motion of the moderator atoms.

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

E

E

Nuclear Fission

~200 MeV

Fission

Fusi

on

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(.44.1

2


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 above g.s. is called activation energy.


Nuclear Fission

Liquid Drop

Shell

Act

iva

tion

Ene

rgy

(MeV

)


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.


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 excess

Most stable:Z=45 Z=58Prompt neutronsPrompt neutrons within 10-16 s.Number depends on nature of fragments and on incident particle energy.The average number is characteristic of the process.


Nuclear Fission

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


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.


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 20Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

21

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).



22

Nuclear Fission

f,Th 584 2.7x10-6 700 0.019 b

Why not use it?Why not use it?

23

Nuclear Fission

• 235U + n 93Rb + 141Cs + 2n• Q = ????• What if other fragments?• Different number of neutrons.• Take 200 MeV as a representative value.

66 MeV 98 MeV

miscalibrated

Heavyfragments

Lightfragments


24

Nuclear Fission• Mean neutron energy 2 MeV.• 2.4 neutrons per fission (average) 5 MeV average kinetic energy carried by prompt neutrons per fission.

• Show that the average momentum carried by a neutron is only 1.5 % that carried by a fragment. • Thus neglecting neutron momenta, show that the ratio between kinetic energies of the two fragments is the inverse of the ratio of their masses.

1

2

2

1

m

m

E

E

140

95

98

66


25

Nuclear Fission

Distribution of fission energy

Krane sums

them up as

decays.Lost … !

Enge


26

Nuclear Fission

Segrè

Lost … !


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Nuclear Reactors, BAU, 1 st Semester, 2007-2008 (Saed Dababneh). 1 Neutron Attenuation (revisited)...

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