PDF 3.4 Getting Ready to Calculate a Reaction Rate

7/28/2019 PDF 3.4 Getting Ready to Calculate a Reaction Rate

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Nuclear Reactionsand Radiation

3.4 Getting ready to calculate a

reaction rate but you gotta get

the flux

L. R. Foulke


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SUMMARY - MICROSCOPIC

CROSS SECTIONS MICROSCOPIC ()

Measure of Equivalent Nucleus "TargetSize

Multiplied with Atom Density (N) to YieldMacroscopic Cross Section ()

=N

Note: for this lecture, we will try to reserve capital N for numberdensity of a nuclide, and lower case n for neutron density.


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Microscopic Cross Section So far we have only considered the probability that

any type of reaction will occur:

This is referred to as the Total Microscopic CrossSection, t

We can also consider the probability that a specifictype of reaction will occur:

Microscopic Scattering Cross Section,s

Microscopic Absorption Cross Section, a Microscopic Capture (n,) Cross Section, c Microscopic Fission Cross Section, f


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Cross Section Hierarchy

t=

a+

s

a=c+f

s=

e+

i

Micros on eachlevel are the sum

of all constituent

micros on lower

levels.

Image Source: See Note 1


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Macroscopic Cross Sections Microscopic cross sections give us information about

the probability of interaction on a per nucleus basis

What if we want to know the probability of an interaction in amaterial with a known density.

In this case we need to multiply the probability of interactingwith a single nucleus by the number (density) of nuclei in the

material.

The quantity is called the macroscopic crosssection, it has units of 1/cm.

t= N

tUnits: [nuclei/cm3] x [cm2/nucleus]


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Macroscopic Cross Sections The macroscopic cross section gives the probability

that a neutron will undergo a reaction per distance

travelled (1/cm).

Since microscopic cross sections are energy dependentit follows that macroscopic cross sections are as well.

Macroscopic cross sections for individual reaction typescan be calculated from the corresponding microscopiccross sections.

a= N

a

s= N

s f = N f


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Calculating Number Density Calculating macroscopic cross sections requirescalculation of the number density of nuclei in a

material.

This information can be calculated using Avogadro'snumber.Avogadros NumberNA= 6.02210

23 atoms/mole

Atomic mass of atom in amu = Mass [g] per mole of atoms

Units: [atoms/cm3] = [g/cm3] [atoms/mole] [mole/g]

N = NA/A N= Atomic number density = Material densityNA = Avogadros Number

A = Atomic mass


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SUMMARY - MACROSCOPIC

CROSS SECTIONS MACROSCOPIC ()

Product of Microscopic Cross Section andAtom Density

Multiplied with Neutron Flux () to Yield aReaction Rate (R)

R =

R = (cm1)(neutrons / cm2 sec) = reactions / cm3 sec


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Reaction Rates Neutron radiation levels are usually measured in termsofneutron flux, denoted

Neutron flux is the rate at which neutrons pass through a spatialposition, per unit time.

Units: [neutrons / cm2 / sec] The rate of neutron interactions (per unit volume) in a

material is given by:

The rate of individual reactions can be calculated by substitutingthe reaction cross section for the total cross section.

R = t

Units: [neutrons/cm2/sec][reactions/cm]

= [reactions/cm3/sec]


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Neutrons at t = 0Neutrons at t = t

Total distance traveled by all neutrons during t= Total path length generated by all neutrons during t

What is Neutron Flux?



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Neutrons at t = 0Neutrons at t = t

Total distance traveled by all neutrons during t= Total path length generated by all neutrons during t

What is Neutron Flux?



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Neutron Fluxes



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Reaction Rate:

Reaction Rate Density (Energy Integrated)Rate at which neutrons at position , all

energies, undergo a reaction of typex.

This reaction rate density is what must becalculated to design a reactor; it governs where thefission energy is deposited.

Rx

r,t( ) = x

r,t( )

r,t( )

r


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Neutron Attenuation



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NEUTRON BEAM ATTENUATION

NEUTRON BEAM INTERACTION (x) = (x) x

d(x) = (x) dxd(x)

dx= (x)

(x) = (0) e x

(x)

0

= e x


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1) Adapted with permission from the AmericanNuclear Society. Nuclear Engineering

Theory and Technology of CommercialNuclear Powerby Ronald Allen Knief, 2ndEdition. Copyright 2008 by the AmericanNuclear Society, La Grange Park, Illinois.

Figure 2-10 (slide 4), 4-1 (slide 12), and 2-14(slide 14).

2) Reprinted with permission from DavidGriesheimer, University of Pittsburgh.

Image Source Notes

PDF 3.4 Getting Ready to Calculate a Reaction Rate

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