A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions 1 NEWS Lecture1: Chapter 0 is already on my Website

A. Dokhane, PHYS487, KSU, 2008

Chapter1- Neutron Reactions

1

NEWS• Lecture1: Chapter 0 is already on my

Website



2

Chapter 1

Neutron Reactions

March 2008



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1. Review

2.2. Neutron ReactionsNeutron Reactions3. Nuclear Fission

4. Thermal Neutrons

5. Nuclear Chain Reaction

6. Neutron Diffusion

7. Critical Equation



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2.8 Neutron Flux and Reaction Rate

For large number of neutronslarge number of neutrons, it is physically convenient to consider them as a gas gas description of movements and random motion based on molecular theory of gases.

Assymptions: neutrons as a gas move with a speed (all neutrons have the same speed) mean free path

Time interval between two successive collisions for a given neutron is:

t

Hence: number of collisions per second by neutron

For n neutrons per cm3 total number of collisions/cm3 sec = number of collisions per neutron/sec X number of neutrons/cm3

nrV

this similar to that found for neutron beam traveling in a given direction I



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Definition: neutron flux =

Hence, this formula is valid for an assembly of neutrons traveling in different and arbitrary directions which is the current case.

nrV

nThen , the reaction rate/cm3 is: Vr

Reaction rate for a medium of volume V : VVrR v

Reality: neutrons in a reactor do not all have the same speed, hence it is necessary to generalize the definition of the neutron flux? How this can be done?

Define a flux element d for a small range of neutron velocities between and d

hence dn )( is the number of neutrons per cm3 having velocities within this range, then

dnd )(

0

)( dn SEE FIGURE 4.5



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Alternative definition: in term of energies: number of neutrons with energies between

E and E +dE is dEEn )(

Then the flux is:

0

)( dEEn

Another useful definition of the flux:

0

)( dEE

Where ddEE )( is the flux in the small energy range between E and E +dE




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2.9 Energy dependence of Neutron Cross Sections

So far, Neutron cross sectionsNeutron cross sections depends on: nature of target nucleus + energy of interacting neutrons.

Let us focus now on the interacting neutrons…

It is convenient to classify neutrons that are involved in nuclear reactions according to the general behaviour of the various cross sections and devide the neutron energies into several regions to take into account these general trends.



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There are 4 principal regions:

1. High-energy region neutron: energies between 10 Mev to 0.1 Mev fast neutrons

2. Intermediate-energy region: energies between 0.1 Mev and 1000ev intermediate neutrons.

3. Epithermal region: neutrons energies between 1000ev to 1 ev epithermal neutrons.

4. Thermal region: neutrons energies between 1 ev and less thermal neutrons or slow neutrons.

What are the most probable reaction cross sections for each region??



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Fast Neutrons: most probable interaction is scattering ((n,n) interaction). Absorption

cross section is very small, sa . Hence the total cross section is entirely due to

scattering: 22 Rst .

for medium and heavy nuclei, 32

125.0 At

Intermediate Neutrons: scattering is still dominant for intermediate and heavy nuclei and of the order of 1 barn. While, radiative capture is also probable (n, ) with a cross section of 1 millibarn.



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Epithermal Neutrons: known also as resonance region. In this region, the neutron cross sections of most elements show many distinct and high maxima in the total cross section. peaks point out the existence of resonance levels in the compound nucleus.

they appear to be superimposed on a background that varies as 21

E ( 1 )

number of peaks and their mutual separations vary with different nuclei SEE FIGURE 4.7



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The maxima are resonances in the capture cross section c superimposed to background which is scattering cross section.

scattering cross section for energies between individual resonances is of order of barn, while capture cross section is of order of milibarn

See Figure 4.7, page 98

Total background cross section between resonances is: 24 Rt

double comparing to that of fast neutron!

Continuation of Epithermal region…..



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Energy dependence of in resonance region is given by Breit-Wigner formula:c

220

2

24 EE

nc

called single-level formula, since it describes resonances well separated and not overlapping

de Broglie wavelength for neutrons, mh

0E resonance energy and E is neutron energy.

width of resonance line at half the maximum value of the cross section

n and are partial level widths for (n,n) scattering and (n, ) radiative capture

measure the probability for the corresponding reaction to take placeprobability for the corresponding reaction to take place.



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2

h is decay constant

Since the two processes are the only modes of decay of the compound nucleus in this energy region that need to be considered, then:

n

This means: probability for (n,n) reaction is :

n



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In most cases, when E is small then is much larger than n

ev1.0 and 43 1010 ton ev

Another form for Breit-Wigner formula

2

20

21

00

2

1

1

EEE

Ec

Where 0 is the maximum value of the resonance capture cross section for 0EE

Example: neutron absorption of Cadmium with maximum value of 7200 barns for a neutron energy of 0.18 ev. See Figure 4.9, page 100.



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For a broad resonance, i.e. 0EE , then

tcon

E

Ec

tan21

00

This is well known dependence of capture cross section for slow neutrons. See Figure 4.10, page 100



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For light nuclei resonances are very broad and widely separated from each other so that is satisfied 0EE

Thermal Neutrons:

s rises steadily as we go from lighter to heavier elements from 1 barn to 10 barns.

c follows 1 law and Breit-Wigner formula may be applied to (n, ) cross section.

no resonances present in thermal energy region

for 0E we use the value of the nearest resonance in the low-energy region.



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Schematic representation of the variation of neutron cross section with energy for typical nucleus is given in Figure 4.7, page 98.

Thermal neutron absorption cross section sections for some representative nuclides are listed in Table 4.1 .



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2.10 Fission Cross Section

Some of the very heavy nuclei undergo fission as a result of neutron absorption.

Natural fissionable nuclides are: U235 (fissionable by thermal and fast neutrons), U238 (fast neutrons more than 1 Mev) and Th232 (fast neutrons more than 1 Mev).



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U233 is produced artificially by neutron capture of Th232

Pu239 is produced artificially by neutron capture of U238

U233 and Pu239 are fissionable by thermal and fast neutrons as well.

See Figures 4.11 to 4.13, page 104



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BECAREFUL: Nonfissionable materialsNonfissionable materials: absorption = capture fissionable materialsfissionable materials: absorption differentdifferent from capture, hence:

fca

Capture refers to radiative capture only.

See Table 4.2: Thermal neutron cross section for some reactor fuel materials

Example 4.8



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Homework

• Problems: 1, 3, 4, 6, 8, 11 of Chapter 4 in Text Book, Page 107

• To be submitted next week.

Documents

A. Dokhane, PHYS487, KSU, 2008 Chapter1- Neutron Reactions 1 NEWS Lecture1: Chapter 0 is already on my Website