Chapter 5

Diffusion Theory

J. Frybort, L. Heraltova

Department of Nuclear Reactors

November 28, 2014

Chapter 5

J. Frybort, L. Heraltova (CTU in Prague) Diffusion Theory November 28, 2014 1 / 58

Content

1 Fick’s Law

2 Continuity Equation

3 Diffusion EquationValidity and Boundary Conditions

4 Neutron Sources in Infinite Diffusive MediumPlanar Source in Infinite Diffusive MediumPoint Source in Infinite Diffusive MediumLine Source in Infinite Diffusive MediumDiffusion Length

5 Neutron Sources in Finite Diffusive Medium

6 Group and Thermal Neutron Diffusion


Fick’s Law

Introduction

It is necessary to predict neutron distribution inside a nuclearreactorThe exact description of all the processes of neutrons (collisions,transport, nuclear reactions) is very difficultThe first approximation describes the movement of neutrons as akind of diffusionThis approximation is called diffusion approximation and was usedin development of the first types of nuclear reactorsMore advanced methods are developed now, but still, diffusiontheory is widely used for the analysis of large nuclear reactorsThe complete theory describing all neutron properties with littleapproximation is Transport theory solving Boltzmann transportequation


Fick’s Law

Fick’s Law

The diffusion theory is based on Fick’s law originally used forchemical diffusionIt was observed in chemistry that if concentration of a solute inone region of solution is greater than in another, the solutediffuses from the region of higher concentration to the region oflower concentrationThe rate of solute flow is proportional to the negative of thegradient of the solute concentrationNeutrons behave to a good approximation in the same wayIf the density (neutron flux) of neutrons is higher in one part of areactor, there is a net flow of neutrons into a region with lowerneutron flux


Fick’s Law

Neutron Current Density

If neutron density varies along x-direction, the net flow of neutronsthat pass per unit of time through a unit area perpendicular to thex-direction can be expressed as:

Jx “ ´DdΦ

dx(5-1)

Jx has the same unit as flux (neutrons/cm2-sec)Parameter D is called diffusion coefficient and has unit of cmThe flux is generally function of three spatial variables, therefore:

J “ ´D gradΦ “ ´D ∇Φ (5-2)

Here J is called neutron current density


Fick’s Law

Diffusion Coefficient

We assume that D is not a function of spatial variablesThe diffusion coefficient can be approximately calculated as:

D “λtr

3, where λtr is transport mean free path (5-3)

λtr “1

Σtr“

1Σsp1´ µq

(5-4)

Transport mean free path (λtr) is an average distance a neutronwill move in its original direction after infinite number of collisionssµ “ Ěcosϑ is average value of the cosine of the angle at whichneutrons are scattered in the medium. It can be calculated formost of the neutron energies as:

µ “2

3A(5-5)


Fick’s Law

Validity of Fick’s Law

Fick’s law is approximation which is not valid under the following condi-tions

1 In a medium that strongly absorbs neutrons2 Within about three mean free paths from either neutron source or

the outer surface of the diffusive medium3 When the scattering of neutrons is strongly anisotropic

These conditions are present in every practical reactor problemFick’s law and diffusion theory is therefore only the first estimateMore advanced methods must be used near sources, boundariesand in strongly absorbing media


Continuity Equation

General Equation of Continuity

In an arbitrary volume V of a diffusive medium containing neutronsthe number of neutrons may changeThe change of the number of neutrons is a result of a net flow ofneutrons in or out of V, some neutrons are absorbed inside V andthere might be also neutron sources inside volume VThe equation of continuity is mathematical representation of thefact that neutrons cannot disappear unaccountably

«

rate of changein number ofneutrons in V

ff

“

»

—

–

rate ofproductionof neutronsin V

fi

ffi

fl

´

»

—

–

rate ofabsorptionof neutronsin V

fi

ffi

fl

´

»

—

–

rate ofleakageof neutronsfrom V

fi

ffi

fl


Continuity Equation

Rate of Change of Neutrons in V

If n is density of neutrons at any point in time in V, the totalnumber of neutrons in V is then:

ż

Vn dV

The rate of change in number of neutrons is:

ddt

ż

Vn dV , which can be also written as:

ż

V

BnBt

dV


Continuity Equation

Production and Absorption Rate in V

Let s be the rate at which neutrons are emitted from sources percm3/sec in VThe rate at which neutrons are produced through V is given as:

Production rate “

ż

Vs dV

The rate at which neutrons are lost by absorption per cm3/sec isequal to ΣaΦ

The total loss of neutrons through absorption in volume V is:

Absorption rate “

ż

VΣaΦ dV


Continuity Equation

Leakage Rate out of V

J is current density vector on the surface of V and n is a unitnormal pointing outward from the surfaceThen Jn is the net number of neutrons passing outwards throughthe surface per cm2/secThe overall leakage through the surface A of the volume V is:

Leakage rate “

ż

AJn dA

This surface integral can be converted into a volume integral bythe divergence theorem:

ż

AJn dA “

ż

VdivJ dV


Continuity Equation

Resulting Equation of Continuity

ż

V

BnBt

dV “

ż

Vs dV ´

ż

VΣaΦ dV ´

ż

VdivJ dV

Integral were carried out over the same volume, thus theirintegrands must also be equal:

BnBt“ s ´ ΣaΦ´ divJ (5-6)

The above equation is the equation of continuity, if the neutrondensity is not a function of time, this equation reduces to:

´divJ´ ΣaΦ` s “ 0


Diffusion Equation

Diffusion Equation

The continuity equation has two unknowns – the neutron density(n) and neutron current density (J)There is a relation between neutron flux and neutron currentdensityOne of these unknowns can be eliminated by Fick’s lawSubstitution of (5-2) into (5-6) leads to:

´divp´D gradΦq ´ ΣaΦ` s “BnBt

Diffusion coefficient (D) is spatially independent

D divpgradΦq ´ ΣaΦ` s “BnBt


Diffusion Equation

Diffusion Equation (cont’d)

The continuity equation can be further simplified by introducingsymbol ∇2 ” div grad called Laplacian operatorThe resulting equation is called diffusion equation

D∇2Φ´ ΣaΦ` s “1vBΦ

Bt(5-7)

If only time independent problems are considered, steady-statediffusion equation is formulated

D∇2Φ´ ΣaΦ` s “ 0 (5-8)


Diffusion Equation

Laplacian Operator

Formula for Laplacian depends on used coordinate systemin rectangular coordinates:

∇2 “B2

Bx2 `B2

By2 `B2

Bz2

in cylindrical coordinates:

∇2 “1rB

Br

ˆ

rB

Br

˙

`1r2B2

Bϑ2 `B2

Bz2

in spherical coordinates:

∇2 “1r2B

Br

ˆ

r2 B

Br

˙

`1

r2 sinϑB

Bϑ

ˆ

sinϑB

Bϑ

˙

`1

r2 sin2 ϑ

B2

Bϕ2


Diffusion Equation

1D Laplacian Operators

In the simplest examples in one-dimensional space, the Laplacianoperator reduces to the following formulas:

rectangular coordinates:

∇2 “B2

Bx2

cylindrical coordinates:

∇2 “1rB

Br

ˆ

rB

Br

˙

“B2

Br2 `1rB

Br

spherical coordinates:

∇2 “1r2B

Br

ˆ

r2 B

Br

˙

“B2

Br2 `2rB

Br


Diffusion Equation

Diffusion Length

The equation (5-8) is often divided by D, resulting in:

∇2Φ´1L2 Φ`

sD“ 0 (5-9)

Parameter L2 is defined as:

L2 “DΣa

The quantity L is called diffusion length with unit cmQuantity L2 is diffusion area, unit cm2

Physical meaning of the diffusion length will be given later


Diffusion Equation Validity and Boundary Conditions

Boundary Conditions

The diffusion equation was derived using Fick’s law, thereforeconditions for validity of Fick’s law are also valid for the diffusionequationSince the diffusion equation is a partial differential equation,boundary conditions are requiredThere are typical boundary conditions:

1 Neutron flux must be non-negative and finite:0 ď Φ ă 8

2 Both neutron flux and current must be continuous across boundaryof two diffusive media (A and B):

ΦA “ ΦB

pJAqn “ pJBqn

3 Boundary condition for an external boundary of a diffusive medium4 Source conditions



Extrapolation Distance

Fick’s law is not valid for area close to an external surfacebetween the diffusive medium and atmosphereIt was found that if the flux vanishes in a distance d from thesurface, then the flux calculated by diffusion theory is close to thereal fluxThe parameter d is known as extrapolation distance and in mostcases it is given by simple formula d “ 0.71 λtr, where λtr istransport mean free path of the mediumFrom relation for diffusion coefficient – D “ λtr{3 – results thatd “ 2.13 DThe extrapolation distance is usually in units of several cm andtherefore it can be in many cases neglected and assumed thatneutron flux diminishes at the physical boundary of the medium



Extrapolation Distance – Visualisation

If d is not negligible, physical dimensions of the reactor are increasedby d and extrapolated boundary is formulated with dimension a “ a` d



Source Condition

The diffusion equation is not valid for the neutron source location,but it is necessary to match the magnitude of the neutron flux tothe source intensityThe source is surrounded by area for which it holds that allneutrons flowing through this area must come from the neutronsource characterized by source emissivity SFormulation depends on the source geometry: planar (5-10a),point (5-10b), or line (5-10c)

limxÑ0

Jpxq “S2

(5-10a)

limrÑ0

4πr2Jprq “ S (5-10b)

limrÑ0

2πrJprq “ S (5-10c)

This condition will be illustrated by examples of neutron sources indiffusive media


Neutron Sources in Infinite Diffusive Medium

Infinite Diffusive Medium

Spatial distribution of neutron flux in an infinite media will becalculated using diffusion equation and boundary conditionsBasic source geometries will be calculated – plane, point and lineOnly monoenergetic sources of neutrons are analysed


Neutron Sources in Infinite Diffusive Medium Planar Source in Infinite Diffusive Medium

Planar Source in Infinite Diffusive Medium

Planar source emitting S neutrons per cm2/secThe flux is only function of distance from the plane, e.g. in xdirection. The source location is not part of the analysed areaThe plane is located in x “ 0 and there are two solution forpositive (x ą 0) and negative (x ă 0) directionsThe diffusion equation (5-9) has the following form:

d2Φ

dx2 ´1L2 Φ “ 0 , x ‰ 0

Solution for x ą 0 is expected in form:

Φpxq “ e´λx , second derivative:d2Φ

dx2 “ λ2e´λx

Substituted in the above equation leads to:

λ2e´λx “1L2 e´λx



Planar Source in Infinite Diffusive Medium (cont’d)

There are two possible solution for λ “ ˘1L

The diffusion equation for planar source has thus general solutionwith two constants to be determined by boundary conditions

Φpxq “ Ae´x{L ` Cex{L

The second term can be eliminated from the condition of finiteneutron flux, then the equation reduces to:

Φpxq “ Ae´x{L

The constant A is determined from the source conditionFrom Fick’s law:

J “ ´DdΦ

dx“

DAL

e´x{L



Planar Source in Infinite Diffusive Medium (cont’d2)

Source condition for the planar source:

limxÑ0

Jpxq “ limxÑ0

DAL

e´x{L “DAL“

S2

This gives constant A

A “SL2D

The final formula for spatial dependence of neutron flux from theplanar source in infinite medium is:

Φpxq “SL2D

e´x{L (5-11)

The solution is valid for x ą 0, but because of symmetry of theproblem similar formulation could be obtained for negativex-direction


Neutron Sources in Infinite Diffusive Medium Point Source in Infinite Diffusive Medium

Point Source in Infinite Diffusive Medium

Point source emitting S neutrons/sec isotropically in an infinitemediumThe source is located in the centre of a spherical coordinatesystem and neutron flux depends only on distance r from thesourceThe diffusion equation (5-9) in spherical coordinate systembecomes for r ‰ 0:

d2Φ

dr2 `2r

dΦ

dr´

1L2 Φ “ 0

The equation is solved by introducing substitution uprq “ rΦprqSubstitution into the above equation results in:

d2udr2 ´

1L2 u “ 0



Point Source in Infinite Diffusive Medium (cont’d)

The solution for function u is found in an identical way as in thecase of the planar source:

uprq “ Ae´r{L ` Cer{L

Transform to the original function gives:

Φprq “ Ae´r{L

r` C

er{L

rConstants A and C must be determined from boundary conditionsIt is clear that if the neutron density flux must remain finite, C mustequal zeroThe constant A is found from the source conditionFrom Fick’s law:

J “ ´DdΦ

dr“ DA

ˆ

1rL`

1r2

˙

e´r{L



Point Source in Infinite Diffusive Medium (cont’d2)

The source condition for point source is:

limrÑ0

4πr2Jprq “ limrÑ0

4πDA´ r

L` 1

¯

e´r{L “ S

This gives constant A:

A “S

4πDResulting equation for neutron flux distribution is following:

Φprq “Se´r{L

4πDr(5-12)


Neutron Sources in Infinite Diffusive Medium Line Source in Infinite Diffusive Medium

Line Source in Infinite Diffusive Medium

Line source emitting S neutrons/sec per unit length in an infinitemediumThe source is located in the centre of a cylindical coordinatesystem and neutron flux depends only on distance r from thesourceThe diffusion equation (5-9) in the cylindical coordinate systembecomes for r ‰ 0:

d2Φprqdr2 `

1r

dΦprqdr

´1L2 Φprq “ 0

The equation can be transformed by substitution u=r/L intomodified Bessel’s equation of the order zero:

u2 d2Φpuqdu2 ` u

dΦpuqdu

´ u2Φpuq “ 0



Ordinary Bessel’s Functions

Bessel’s equation is:

x2 d2Φ

dx2 ` xdΦ

dx` pα2x2 ´ n2qΦ “ 0 , where α and n are constants

n is order of the equation, in practical problems it is usually zeroGeneral solution to Bessel’s equation is:

Φpxq “ AJnpαxq ` CYnpαxq

Functions Jn and Yn are called ordinary Bessel’s functions of thefirst and second kind, respectively



Modified Bessel’s Functions

If α2 is negative, Bessel’s equation becomes:

x2 d2Φ

dx2 ` xdΦ

dx´ pα2x2 ` n2qΦ “ 0

General solution is in form of modified Bessel’s functions of thefirst and second kind, respectively – In and Kn

Φpxq “ AInpαxq ` CKnpαxq

The following figures will show Bessel’s functions used in reactorphysics



Ordinary Bessel’s Functions Plot



Modified Bessel’s Functions Plot



Line Source in Infinite Diffusive Medium (cont’d)

General solution of this kind of Bessel’s equation is in form ofmodified Bessel’s function of the first (I) and second (K) kind of theorder zero

Φpuq “ AI0puq ` CK0puq

Φprq “ AI0pr{Lq ` CK0pr{Lq

Given the fact that function I0 Ñ8 for rÑ8, constant A mustequal 0 and above equation reduces to:

Φprq “ CK0pr{Lq

The constant C is found from the source conditionFrom Fick’s law, using dK0{dr “ ´K1:

J “ ´DdΦ

dr“ ´DC

dK0pr{Lqdr

“DCK1pr{Lq

L



K1 Bessel’s Function Plot



Line Source in Infinite Diffusive Medium (cont’d2)

K1(x) behaves similar to 1x function

It can be used in the source condition:

limrÑ0

2πpr{LqDCK1pr{Lq “ S

It can be written that limrÑ0rpr{LqK1pr{Lqs „ 1Resulting value for constant C is:

C “S

2πD

Final form of a neutron flux distribution from the line source is:

Φprq “S

2πDK0pr{Lq (5-13)


Neutron Sources in Infinite Diffusive Medium Diffusion Length

The Diffusion Length

It is necessary to understand physical meaning of the diffusionlength L, which appears in solutions of neutron flux from neutronsourcesIn diffusive medium, neutrons are moving along complicatedpaths, however, every neutron is absorbed in the medium since itis infiniteNumber of neutrons absorbed from a point neutron source perunit path length can be expressed as:

dn “ ΣaΦprq dV “SΣa

Dre´r{L dr “

SL2 re´r{L dr

Probability that neutron is absorbed in dr is:

pprq dr “1L2 re´r{L dr


Neutron Sources in Infinite Diffusive Medium Diffusion Length

The Diffusion Length (cont’d)

It is now possible to calculate a square of an average distancefrom the source at which a neutron is absorbed

sr2 “

ż 8

0r2pprq dr “

1L2

ż 8

0r3e´r{L dr “ 6L2

It means that L2 is one-sixth of the average of the square of thestraight distance a neutron travels from the point at which it isemitted to the point where it is finally absorbed

With higher diffusionlength, neutrons aremoving further. It meansthe diffusive medium isless absorbing


Neutron Sources in Finite Diffusive Medium

Finite Diffusive Medium

Finite diffusive medium is a more realistic situationFor neutron flux spatial distribution computation it is necessary touse boundary conditionsFor this purpose, physical dimensions of the diffusive media willbe increased by extrapolation distance dNeutron flux will be calculated for monoenergetic neutron sourcesin basic geometries – slab, cylinder and sphereSince the method of solving these geometries is similar, it will beillustrated only for a bare slab with a plane source of neutrons



Finite Bare Slab

The slab has thickness 2a, it is infinite in vertical direction and thesource of neutrons is located in its centreIt can be solved for positive and negative direction withsymmetrical resultsBoundary condition must be utilisedThe neutron flux is required to vanish at the extrapolated surfaceof the slab, at x “ a` d (or at x “ ´d ´ a)The boundary condition is:

Φpa` dq “ Φpá´ dq “ 0

The general solution of the diffusion equation for the planar sourceof neutrons is the same as for infinite diffusive medium:

Φpxq “ Ae´x{L ` Cex{L



Finite Bare Slab (cont’d)

Boundary condition at a` d is applied:

Φpa` dq “ Ae´pa`dq{L ` Cepa`dq{L “ 0

One constant can be expressed as the function of the other:

C “ Áe´2pa`dq{L

Substituting this into the general solution gives:

Φpxq “ A”

e´x{L ´ ex{L´2pa`dq{Lı



Finite Bare Slab (cont’d2)

The constant A is found from the source condition in the usual wayand is:

A “SL2D

´

1` e´2pa`dq{L¯´1

For positive x-direction, function for Φ is given by:

Φpxq “SL2D

e´x{L ´ ex{L´2pa`dq{L

1` e´2pa`dq{L (5-14)


Group and Thermal Neutron Diffusion

Group Neutron Diffusion

So far only monoenergetic neutron sources were considered andenergy dependence was neglectedCommonly used form of neutron energy spectrum description is agroup method

By this approach, neutrons aredivided into energy groupsdepending on their energyIf a neutron gains energy, itmoves to a higher energygroupBy loosing energy neutronmoves into a lower energygroup



Group Neutron Diffusion (cont’d)

Flux in a specific energy range g is calculated as:

Φg “

ż

gΦpEqdE

Neutrons can disappear from a group by absorption and byscattering from one group to another (g Ñ h):

absorption rate = ΣagΦg , group transfer rate = ΣgÑhΦg

Total transfer rate from one group to another is obtained bysumming over all groups that can scatter to the other groupThe group diffusion equation for the gth group can be written inform:

Dg∇2Φg ´ ΣagΦg ´

Nÿ

h“g`1

ΣgÑhΦg `

g´1ÿ

h“1

ΣhÑgΦh “ ´sg



Thermal Neutron Diffusion

Important aspect of neutron diffusion is diffusion of thermalneutronsEnergy distribution of thermal neutrons is given by Maxwelliandistribution:

npEq “2πn

pπkT q3{2E1{2eÉ{kT (5-15)

Using the usual formula for speed vpEq “ p2E{mq1{2 anddefinition of neutron flux ΦpEq “ npEqvpEq an average thermalflux can be calculated as:

ΦT “

ż

TΦpEq dE

T represents all thermal energies up to 0.1 eV



Thermal Neutron Diffusion (cont’d)

Average thermal neutron flux is calculated as:

ΦT “2πn

pπkT q3{2

ˆ

2m

˙1{2 ż 8

0EeÉ{kT dE “

2n?π

ˆ

2kTm

˙1{2

(5-16)

Energy and velocity corresponding to energy kT is denoted as ETand vT

If temperature is given in Kelvins, it can be calculated as:

ET “ 8.617T ˆ 10´5 eV

vT “ 1.284T 1{2 ˆ 104 cm/sec

Equation (5-16) can be now written as:

ΦT “2?π

nvT (5-17)



Thermal Neutron Diffusion (cont’d2)

There is a concept of thermal energy of neutrons E0 = 0.0253 eV,which corresponds to energy of thermal motion for 20˝C(T0 = 293 K)Data are usually tabulated for this energy. Relation between ΦTand Φ0 can be written as:

ΦT

Φ0“

2?π

vT

v0

Given the mentioned relation for velocity, the above equationbecomes:

ΦT

Φ0“

2?π

ˆ

TT0

˙1{2

(5-18)



Thermal Weighted Cross-Section

Thermal group absorption cross-section is calculated as:

sΣa “1

ΦT

ż

TΣapEqΦpEq dE

This integral is total absorption rate equal to gapT qΣapE0qΦ0,where gapT q is non-1/v factor and ΣapE0q is absorptioncross-section for thermal energy 0.0253 eV and Φ0 thermal flux

sΣa becomes:

sΣa “ gapT qΣapE0qΦ0{ΦT “

?π

2gapT qΣapE0q

ˆ

T0

T

˙1{2

(5-19)

This is principle how to calculate thermal group cross-section fromdata tabulated at 0.0253 eV



Diffusion Equation for Thermal Energy Group

With thermal flux defined and possibility to calculate thermalgroup weighted cross-section, it is possible to formulate one-grouptime-independent diffusion equation for thermal neutrons:

sD∇2ΦT ´ sΣaΦT “ ´sT , where sT is thermal neutron source

Dividing by sD – thermal diffusion coefficient – gives:

∇2ΦT ´1

L2T

ΦT “ ´sTsD

(5-20)

LT is thermal diffusion length defined as L2T “

sDsΣa



Two-Group Diffusion Calculation

For most of the calculations, it is necessary to consider at leasttwo energetic groupsOne group describing thermal neutrons and the other groupdealing with neutron moderation (all neutrons above 5kT )The neutron source emits s neutrons of fast neutrons per secondThe source is placed inside infinite uniform diffusion medium



Diffusion Equations Formulation

The diffusion equation for the fast group has following shape (44):

Dg∇2Φg ´ ΣagΦg ´

Nÿ

h“g`1

ΣgÑhΦg `

g´1ÿ

h“1

ΣhÑgΦh “ ´sg

Absorption cross-section for fast neutrons is very small Ñ it ispossible to neglect the absorption altogetherWe are considering only 2 groups, therefore each neutronscattered in the fast group will reach thermal energy Ñ Σ1Ñ2 ” Σ1Possibility of scattering from the thermal group into the fast groupis neglectedThe equation is solved outside the point neutron source and thereare no other neutron sourcesThe diffusion equation is reduced to the following formulation:

D1∇2Φ1 ´ Σ1Φ1 “ 0 (5-21)



Diffusion Equations Formulation (cont’d)

Thermal neutron flux is described by equation (5-20)

∇2ΦT ´1

L2T

ΦT “ ´sT

D

Source of thermal neutrons are neutrons scattered from the fastgroup into the thermal group Ñ Σ1Φ1

Diffusion equation for thermal neutrons has the following form:

∇2ΦT ´1

L2T

ΦT “ ´Σ1Φ1

D(5-22)

For determination of the thermal flux, we must first determine thefast neutron flux



Neutron Age

We define parameter τT – neutron age:

τT “D1

Σ1(5-23)

Neutron age gives relation to the time required for slowing-downneutron to the thermal energyUnit of τT is cm2

By introducing neutron age into equation (5-21) we get:

∇2Φ1 ´1τT

Φ1 “ 0 (5-24)



Solution the Diffusion Equation for the Fast Group

After introducing the shape of Laplace operator for sphericalcoordinates, the diffusion equation for thermal neutrons has shapeidentical to equation (5-12)Neutron flux of fast neutrons is given by the following formula

Φ1 “Se´r{

?τT

4πD1r, r ‰ 0 (5-25)

The physical meaning of neutron age is similar to the case ofdiffusion lengthThe solution is identical except for the fact that neutrons do notdisappear by absorption, but only by scattering into the thermalgroup



Physical Meaning of Neutron Age

τT “16

r2 (5-26)

Neutron age gives 1/6 of square of the average distance neutron travels,from the place where it was released to the place where it was slowed-down to the thermal energy



Solution of Diffusion Equation for the Thermal Group

Introducing the solution for fast neutron flux (5-25) into thediffusion equation for thermal neutrons (5-22) a new equation withnonzero right side is obtainedThe solution is:

ΦT “S L2

T

4πDpL2T ´ τT q

´

e´r{LT ´ e´r{?τT

¯

(5-27)

This equation gives spatial distribution of thermal neutrons as aresult of presence of the source of fast neutrons



Important constants for moderators for the fast neutrongroup

Moderator D1 (cm) Σ1 (cm´1) τT (cm2)

H2O 0.92 0.0489 19D2O 1.26 0.0116 109Be 0.55 0.0090 61Graphite 1.04 0.0037 278



Diffusion parameters for moderators in the thermalneutrons group for temperature 20˝C

Moderator density (g/cm3) D (cm) LTM (cm)

H2O 1.00 0.13 2.62D2O 1.10 0.76 147Be 1.85 0.45 22Graphite 1.60 0.87 61


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