in thermal reactors , the absorption rate in a “medium” of thermal ( maxwellian ) neutrons

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, First Semester, 2010-2011 (Saed Dababneh).

1

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

Show that, after one 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

2Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

Neutron Moderation

)ln( EEu M

Reference

Average decrease in ln(E) after one collision.

11H ?H ?

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

3

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?

First collision.

Second collision.

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?

4Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

EE )1(21\

EEEE )1(21\

)1(21

E

E

Nuclear Fission

~200 MeV

Fission

Fusi

on

Coulomb effectSurface effect

5Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

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

6Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

Nuclear Fission

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

7Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

Nuclear Fission

Liquid Drop

Shell

Act

iva

tion

Ene

rgy

(MeV

)

8Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

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.

9Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

Nuclear Fission

48300

)120( 2

Extrapolation to 47 10-20 s.

Consistent with activation energy curve for A = 300.

10Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

Nuclear Fission

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

Low-energy fission processes.

11Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

Nuclear FissionZ1 + Z2 = 92Z1 37, Z2 55A1 95, A2 140Large neutron excess

Most 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.

12Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

Nuclear Fission

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

13Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

14

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.

15Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

• Waste.• Poison.

In general, decay favors high

energy.

Nuclear Fission

16Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).