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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 (=nvNt)

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).

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).

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).

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

Surface effect

Coulomb effect

~200 MeV

Fission

Fusion

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

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 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 Fission

Liquid Drop

Shell

Activation Energy (MeV)

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

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 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 Fission

235U + n

93Rb + 141Cs + 2n

Not unique.

Low-energy fission processes.

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

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 Fission

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

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

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 Fission

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

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 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 Fission

Why not use it?

f,Th584 2.7x10-6 700 0.019 b

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

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 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 Fission

Enge

Distribution of fission energy

Krane sums them up as decays.

Lost … !

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

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