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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
2
Chapter 1
Neutron Reactions
March 2008
A. Dokhane, PHYS487, KSU, 2008
Chapter1- Neutron Reactions
3
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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2.8 Neutron Flux and Reaction Rate
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
A. Dokhane, PHYS487, KSU, 2008
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2.8 Neutron Flux and Reaction Rate
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
2.8 Neutron Flux and Reaction Rate
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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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2.9 Energy dependence of Neutron Cross Sections
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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2.9 Energy dependence of Neutron Cross Sections
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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2.9 Energy dependence of Neutron Cross Sections
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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2.9 Energy dependence of Neutron Cross Sections
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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2.9 Energy dependence of Neutron Cross Sections
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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2.9 Energy dependence of Neutron Cross Sections
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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2.9 Energy dependence of Neutron Cross Sections
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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2.9 Energy dependence of Neutron Cross Sections
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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2.9 Energy dependence of Neutron Cross Sections
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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2.9 Energy dependence of Neutron Cross Sections
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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2.10 Fission Cross Section
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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2.10 Fission Cross Section
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.