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Neutron Attenuation (revisited). X. I 0. I. Recall  t = N  t. mfp for scattering l s = 1/ S s mfp for absorption l a = 1/ S a total mfp l t = 1/ S t. Probability per unit path length. Probability. Neutron Flux and Reaction Rate. Recall F t =  t I N = I  t

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Neutron Attenuation (revisited)

X

I0

I

Recall t= N t

  • mfp for scattering ls = 1/Ss

  • mfp for absorption la= 1/Sa

  • total mfp lt = 1/St

Probability per unit path length.

Probability

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


Neutron Flux and Reaction Rate

Recall Ft= t I N = I t

Simultaneous beams, different intensities, same energy.

Ft= t (IA + IB + IC + …) =t (nA + nB + nC + …)v

In a reactor, if neutrons are moving in all directions

n =nA + nB + nC + …

Ft= t nv

neutron flux  =nv

Reaction Rate Rt Ft= t  =  /t (=nvNt)

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


Neutron Flux and Reaction Rate

Different energies

Density of neutrons with energy between E and E+dE

n(E)dE

Reaction rate for those “monoenergetic” neutrons

dRt= t(E) n(E)dE v(E)

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


Neutron Flux and Reaction Rate

  • In general, neutron flux depends on:

    • Neutron energy, E.

    • Neutron angular direction, W.

    • 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.

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


Neutron Flux and Reaction Rate

  • In Thermal Reactors, the absorption 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.

Reference

What if not?

Factor

2200 m/s flux

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


Neutron Moderation

Show that, after elastic scattering the ratio between the final neutron energy E\ and its initial energy E is given by:

For a head-on collision:

After ns-wave collisions:

where the average change in lethargy is

HW 6

Collision

Parameter

Reference

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


Neutron Moderation

HW 6 (continued)

  • Reproduce the plot.

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

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


Neutron Moderation

HW 6 (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?

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


Nuclear fission
Nuclear Fission

Surface effect

Coulomb effect

~200 MeV

 Fission

Fusion 

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


Nuclear fission1
Nuclear Fission

  • B.E. per nucleon for 238U (BEU) and 119Pd (BEPd) ?

  • 2x119xBEPd – 238xBEU = ?? K.E. of the fragments  1011J/g

  • Burning coal  105J/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…!

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


Nuclear fission2
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 Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).


Nuclear fission3
Nuclear Fission

Liquid Drop

Shell

Activation Energy (MeV)

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


Nuclear fission4
Nuclear Fission

=

Volume Term (the same)

Surface Term Bs = - as A⅔

Coulomb Term BC = - aC Z(Z-1) / A⅓

 fission

Crude: QM and original shape could be different from spherical.

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


Nuclear fission5
Nuclear Fission

Consistent with activation energy curve for A = 300.

Extrapolation to 47  10-20 s.

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


Nuclear fission6
Nuclear Fission

235U + n

93Rb + 141Cs + 2n

Not unique.

Low-energy fission processes.

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


Nuclear fission7
Nuclear Fission

Z1 + Z2 = 92

Z1  37, Z2  55

A1 95, A2  140

Large neutron excess

Most stable:

Z=45 Z=58

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


Nuclear fission8
Nuclear Fission

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

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


Higher than Sn?

Delayed neutrons

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


Nuclear fission9
Nuclear Fission

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


Nuclear fission10
Nuclear Fission

1/v

Fast neutrons should be moderated.

235U thermal cross sections

fission  584 b.

scattering  9 b.

radiative capture  97 b.

Fission Barriers

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


Nuclear fission11
Nuclear Fission

Fissile

  • Q for 235U + n 236U is 6.54478 MeV.

  • Table 13.1 in Krane: Activation energy EAfor 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).

Fissionable

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


Nuclear fission12
Nuclear Fission

Why not use it?

f,Th584 2.7x10-6 700 0.019 b

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


Nuclear fission13
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

Light

fragments

Heavy

fragments

miscalibrated

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


Nuclear fission14
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.

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


Nuclear fission15
Nuclear Fission

Enge

Distribution of fission energy

Krane sums them up as  decays.

Lost … !

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


Nuclear fission16
Nuclear Fission

Segrè

Lost … !

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


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