pdf 3.3 one-on-one probabilities-neutron vs nuclide

7/28/2019 PDF 3.3 One-On-One Probabilities-Neutron vs Nuclide

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Nuclear Reactionsand Radiation

3.3 One-on-One Probabilities

Neutron versus Nuclide;

Some neutrons are better thanothers

L. R. Foulke


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Nuclear Reaction Rates So far we have covered:

Atomic / nuclear structure Nuclear stability Types of radioactive decay / radiation Radiation interactions Neutron interactions

Remaining question: How can we predict the expected frequency of neutron

interactions in a medium? A substance such as nuclear fuel?


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Probability of an Interaction We start by considering theprobability that a neutron

will strike a single nucleus

Assume that the neutron andnucleus are both solid

spheres.

The neutron sees thenucleus as a round target with

area:A = r2 where ris theradius of the nucleus.

Probability of interaction isproportional to cross sectional

area of nucleus.

Atom

Free Neutron

Top View Neutron View

(Side View)

Image Source: See Note 1


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Probability of an Interaction A reaction will occur whenany part of the neutron

contacts the nucleus

May be straight-on or glancingblow.

This increases the effectivetarget area of the nucleus by

the radius of the neutron, rn.

The effective target area ofthe nucleus is referred to as

the microscopic cross

section for the nuclide.Atom

Free Neutron

Top View Neutrons View

(Side View)

EffectiveTarget Area

EffectiveTarget Area

miss



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Microscopic Cross Section Microscopic cross section

Cross sectional area of nucleus as seen by neutron, denotedby symbol (units of cm2/nucleus)

Has units of area, given in units ofbarns 1 barn = 1024 cm2

Proportional to the probability that a neutron will strike thenucleus and undergo a reaction

Nuclide dependent Seems like it should be proportional to the nuclear

radius


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Microscopic Cross SectionMicroscopic Cross Sections Calculated vs. MeasuredNeutron radius: rn=0.8510

15 [meters]

Nuclear radius: ra=1.21015 (A)1/3[meters] (A is atomic mass)

Theoretical Microscopic Cross Section: =(rn+r

a)2[meters2]

Nuclide Calculated Measured

1H 0.132 barns 20.43 barns4He 0.238 barns 0.759 barns235

U 2.140 barns ~500 barns

Theoretical (solid spheres) model is correct to within a couple oforders or magnitude, but it is not a good approximation


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Microscopic Cross Sections

Nuclide Calculated Measured* @ incident energy of 0.1 eV

1H 0.132 barns 20.43 barns4He 0.238 barns 0.759 barns235U 2.140 barns ~500 barns

U-235

He-4

H-1

Total microscopic cross sections as a function of

incident neutron energy



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Microscopic Cross Section In reality quantum effects play a large role in determining the effective

target area of the nucleus as seen by an approaching neutron

The microscopic cross section depends heavily on: The structure / stability of the target nucleus

Partially filled neutron shells are more receptive to a neutroninteraction than completely filled shells

The energy of the neutron

Typically low-energy (slow) neutrons are more likely to interactwith a nucleus than high-energy (fast) neutrons.

The neutron energy dependence is extremely complicated


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Microscopic Cross Section



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Microscopic Cross Section The energy dependence of microscopic cross sections can bedivided into three ranges:

High Energy (neutron energy > 1 keV) (aka FAST range) Quantum effects are less important and probability of

interaction shows little variation. Resonance Range (1 eV < neutron energy < 1 keV)

Quantum effects dominate and probability of interactiondepends on how closely the neutron energy matches an

unfilled nuclear shell in the target nuclide.

Thermal / 1-over-v Range (neutron energy < 1 eV) Probability of interaction increases as neutron energy

(velocity) decreases


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Microscopic Cross Section

1/v Range(Thermal range)

ResonanceRange

High-Energy

Range(Fast range)



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Analogy for

1-over-vNeutron

Absorption


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1 MeV

235U

238

U

Uranium Fission Cross Sections



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Uranium-235 Fission Cross Section

235U

1 barn0.025 eV

584 barns

10 MeV0.1 Mev

Thermal Advantage~ x500+



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1. Reprinted with permission from DavidGriesheimer, University of Pittsburgh.

2. Adapted with permission from the AmericanNuclear Society. Nuclear Engineering Theory and Technology of CommercialNuclear Powerby Ronald Allen Knief, 2nd

Edition. Copyright 2008 by the AmericanNuclear Society, La Grange Park, Illinois.Figure 2-12 (slides 9, 11, 13, and 14).

Image Source Notes

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