XII. NEUTRON TRANSPORT THEORY

Our goal is to determine the distribution of neutrons as a function of time, their position in space,

and their velocities, i.e., we want to know . If we know the distribution of neutrons in a

nuclear reactor, we can determine the space/time distribution of various nuclear reactions. For

example, the power production and its space/time distribution in the reactor core is determined by

the space/time distribution of fission reactions.

Based on the concepts of the neutron number density, the neutron cross sections, and the neutron

reaction rates that were introduced in Chapter II, we can derive the so called “neutron transport

equation” in order to describe the behavior of neutrons (“neutron transport”) mathematically. We

will find out that the equation that governs the behavior of neutrons has the form of a linearized

Boltzmann equation. First, we will introduce the basic assumptions, and then give the basic defi-

nitions of various quantities and their physical interpretation. We will then derive the most gen-

eral time-dependent form of neutron transport equation and discuss the applicable boundary

conditions. Several simplified examples will be discussed. In the case of isotropic scattering and

neutron sources we will derive the integral form of the neutron transport equation for angular and

scalar flux, as well as for infinte and finite geometries. Finaly, we will present the relationship

between the adjoint and forward space in a unified, yet simple formulation that might be helpful

for understanding the physical meaning of adjoint functions, deducing the equation satisfied by

these functions and finding the adjoint space formulation for a variety of applications.

In the chapters that follow we will discuss various approximations of the general neutron trans-

port equations, their validity, and their usefulness in numerical solution methods of neutron trans-

port problems.

CHAPTERTWELVE

n r v t, ,( )

XII.1

XII.1. Basic Assumptions, Definitions, and Physical Interpretations

XII.1.1 Basic Assumptions

In order to simplify our derivation of the neutron transport equation, we introduce the following

assumptions and simplifications:

• The Boltzmann equation that is used in the kinetic theory of gases is non-linear. However, in the case of neutron transport, the neutron density is so much smaller than that of the scattering atoms, that the neutron-neutron interactions can be neglected. In addition, we can assume that the equilibrium distribution of the scatter-ing centers (atoms) is unaffected by the presence of the neutrons. These assumptions enable the Boltzmann equation to be linearized.

• Neutrons may be considered as “mathematical” points, that travel in straight lines between the collision points. In addition, the collisions may be considered instanta-neous.

• The gravitational force (and any other force) acting on “free” neutrons is neglected in comparison with the nuclear forces.

• The decay of a “free” neutron is neglected (T1/2 ~ 11 minutes).

• Thermal motion of the target nuclei is neglected.

• The material properties are assumed to be isotropic (thus, the differential scattering cross section depends only on the angle between incoming direction and outgoing neutron direction).

• The cross sections for all materials are assumed to be know and independent of time.

• We are solving for the statistically “expected” or “mean” values of neutron density distribution.

XII.1.2 Neutron Number Density

The total “expected” number of neutrons in the six-dimensional “volume element” (dr dv) at time

t is given by

(XII. 1)

which defines the expected number of neutrons in dr around r with velocities in dv around v at time

t. The graphical representation of this six-dimensional phase-space volume element is given in

Figure XII.1.

However, the total number of neutrons in the six-dimensional “volume element” must be the

same, regardless of our definition of the phase-space volume, namely

n r v t, ,( )drdv

XII.2

. (XII. 2)

FIGURE XII.1. Graphical representation of the six-dimensional phase-space volume element (dr dv)

In our derivation of the neutron transport equation, we will be using the energy-dependent angu-

lar neutron density as defined in Chapter II:

(XII. 3)

as the number of neutrons per unit volume around r, per unit energy around E, per unit solid angle

around Ω, at time t. The graphical representation of this six-dimensional phase-space volume ele-

ment (dr dE dΩ)is presented in Figure XII.2.

n r v t, ,( )drdv n r v Ω t, , ,( )drdvdΩ n r E Ω t, , ,( )drdEdΩ= =

z

yx

r

vz

v

vx

y

v = v Ω

dv

dr

dΩ

n r E Ω t, , ,( ) neutrons

cm3

MeV( )steradian-----------------------------------------------------=

z

y

x

r

Ωz

Ωy

Ωx

E, Ω

dEdΩ

dr

XII.3

FIGURE XII.2. Graphical representation of the six-dimensional phase-space volume element (dr dE dΩ)

XII.1.3 Angular neutron flux

We have already introduced the angular neutron flux as

ψ(r,E,Ω,t) = v(E) n(r,E,Ω,t) [neutrons / (cm2 s MeV ster)]

There are two common definitions for the angular neutron flux

ψ(r,E,Ω,t) dΩ dE is the total number of neutrons passing at r through an area of 1 cm2, perpendicular to Ω, per second with energy in dE around E with directions in dΩ round Ω at time t, or is the path length per unit volume about r, passed by neutrons with energies in dE about E and direction dΩ about Ω, per second at time t.

FIGURE XII.3. Illustration of the physical interpretation of angular flux

XII.1.4 Scalar neutron flux

In most cases for the calculation of the reaction rates we are not interested in angular dependence

of the flux, but in the combined effect of all neutrons coming from all directions. In this case, we

define the scalar neutron flux, or simply the neutron flux:

φ(r,E,t) = scalar flux [neutrons/(cm2 s MeV)]

(XII. 4)

Physically,

φ(r,E,t) dE = path length per unit volume about r, traveled by neutrons per second with energies in dE about E, at time t.

Ω

dΩr

1 cm2

φ r E t, ,( ) ψ r E Ω t, , ,( ) Ωd4π∫=

XII.4

We can also interpret the neutron flux as the number of neutrons that penetrates a sphere of a 1-

cm2 cross sectional area, located at r, per second with energies in dE about E, at time t.

FIGURE XII.4. Illustration of the physical interpretation of the scalar flux

The total neutron flux is obtained when the energy-dependent scalar flux is integrated over

energy:

in [neutrons/(cm2 s)] (XII. 5)

We can also define the thermal or the fast neutron flux by integrating over thermal or fast energy

range:

(XII. 6)

(XII. 7)

The scalar flux can be multiplied by macroscopic cross sections to obtain the reaction-rate densi-

ties. (Recall from Chapter II that the macroscopic cross sections are reaction probabilities per unit

path length.) Thus, we can define the energy dependent fission rate density at time t as:

[reactions/(cm3 s MeV)] (XII. 8)

The total fission rate in the volume V at time t is defined as:

[reactions/s] (XII. 9)

1 cm2

φ r t,( ) φ r E t, ,( ) Ed

0

∞

∫=

φth r t,( ) φ r E t, ,( ) Ed

0

Eth

∫=

φf r t,( ) φ r E t, ,( ) Ed

Eth

∞

∫=

F f r E t, ,( ) Σ f r E,( )φ r E t, ,( )=

F f t( ) r EΣ f r E,( )φ r E t, ,( )d

0

∞

∫dV∫=

XII.5

XII.1.5 Angular current density

Now, let us consider an arbitrary volume V surrounded by surface S (Figure XII.5).

FIGURE XII.5. An arbitrary volume V with surface S

At each point let es be the outer surface normal (a unit vector). Thus, if > 0, the neu-

trons are streaming out of the volume V through the surface element dS, and for < 0, the

neutrons are streaming into the volume V through the surface element dS. We can define the

angular neutron current (a vector) as

j(r,E,Ω,t) = Ω ψ(r,E,Ω,t) [neutrons/(cm2 s MeV sterad)]

Physically, the following dot product (a scalar)

es•j(r,E,Ω,t) dS dΩ dE

can be interpreted as the number of neutrons crossing the surface element dS about r, with ener-

gies in dE about E and direction dΩ about Ω, per unit second at time t. Positive value means the

neutron flow is across dS in direction of es; negative value means the flow is in direction of –es.

We can also define the angular current density (a scalar) as:

j (r,E,Ω,t) = es•j(r,E,Ω,t) = es•Ω ψ(r,E,Ω,t) [neutrons/(cm2 s MeV sterad)]

The angular current density can be used to determine the neutron leakage across surfaces.

XII.1.6 Net current density

The "net motion" of neutrons through some surface element is described by the net neutron cur-

rent density (a vector), which is defined as the integral over all directions of the angular current

density (a vector):

Ω

es

VS

dS

r S∈ es Ω•

es Ω•

XII.6

(XII. 10)

Physically, the following dot product [ es•J(r,E,t) dS dE ] can be interpreted as the net number of

neutrons crossing the surface element dS about r, with energies in dE about E and direction dΩabout Ω, per unit second at time t. Positive value means flow is across dS in direction of es; neg-

ative value means flow is in direction of –es.

The net current density (a scalar) can be defined as

(XII. 11)

The net current density can be used to determine a net leakage across surfaces. The detailed

energy dependence can be reduced by integrating over all energies, in order to obtain the energy-

independent net current density (a vector):

[neutrons/(cm2 s)] (XII. 12)

or the energy-independent net current density (a scalar):

(XII. 13)

XII.1.7 Partial current density (a scalar)

We can also define the neutron flow through the surface element dS in positive or negative direc-

tion with respect to the surface normal es

J±(r,E,t) = partial current density in direction ± es

Thus,

J±(r,E,t)dS dE

can be interpreted as the number of neutrons crossing the surface element dS in the direction ±es,

where es is normal to dS about r, with energies in dE about E, per second at time t. Partial cur-

rents are always positive scalars.

J r E t, ,( ) j r E Ω t, , ,( ) Ωd4π∫ Ωψ r E Ω t, , ,( ) Ωd

4π∫= =

J r E t, ,( ) es J r E t, ,( )• es Ω• ψ r E Ω t, , ,( ) Ωd4π∫= =

J r t,( ) J r E t, ,( ) Ed

0

∞

∫ E ΩΩψ r E Ω t, , ,( )d4π∫d

0

∞

∫= =

J r t,( ) es J r E t, ,( )• Ed

0

∞

∫ E Ω es Ω•( )ψ r E Ω t, , ,( )d4π∫d

0

∞

∫= =

XII.7

(XII. 14)

(XII. 15)

Then, the net current density (a scalar) can be calculated from the partial current densities as:

(XII. 16)

or

(XII. 17)

where we integrated over all energy range. The partial current density can be used to determine a

leakage across surfaces in a particular direction (in or out).

XII.1.8 Net neutron current (a scalar)

In the reactor physics we often use the "net current" for the quantity that is actually the net current

density as defined above. The "net current" is defined as

(XII. 18)

which is the total number of neutrons that are crossing the surface S per unit time, or the net rate

with which the neutrons are crossing the surface S. The dimension of the "net current" is [neu-

trons/s].

J+

r E t, ,( ) es Ω• ψ r E Ω t, , ,( ) Ωdes Ω 0>•

∫=

J–

r E t, ,( ) es Ω• ψ r E Ω t, , ,( ) Ωdes Ω 0<•

∫=

J r E t, ,( ) es J r E t, ,( )• J+

r E t, ,( ) J–

r E t, ,( )–= =

J r t,( ) es J r t,( )• J+

r t,( ) J–

r t,( )–= =

J t( ) E J r E t, ,( ) dS•S∫d

0

∞

∫ E J r E t, ,( ) es

• SdS∫d

0

∞

∫= =

XII.8

XII.2. Integro-Differential Form of Neutron Transport Equation

XII.2.1 Derivation

We would like to write a balance equation that describes the time rate of change of the neutron

population in the volume V (but only those with energies in dE around E and with directions in

dΩ round Ω), as the difference between the production rate and the loss rate:

(1) = = Production rate - Loss rate = (3) - (2) (XII. 19)

There are several contributions to the loss (2) and production (3) rates:

(2) Loss Rate = (4) Rate at which neutrons collide and leave V by absorption

or by scattering out of dE dΩ+ (5) Net leakage rate of neutrons with energies in dE around E and

with directions in dΩ round Ω through S.

(3) Production Rate = (6) Rate at which neutrons in V scatter into dE dΩ+ (7) Rate at which prompt fission or (8) delayed neutrons are born in V

with energies in dE around E and with directions in dΩ round Ω+ (9) Rate at which neutrons from “external” source [source that does

not depend on the n(r,E,Ω,t] are born in V with energies in dE around E

and with directions in dΩ round Ω.

Let’s determine each of these contributions:

(4) Loss rate due to neutron collisions in V for neutrons with energies in dE around E and with

directions in dΩ round Ω at time t:

(XII. 20)

(5) Net leakage rate of neutrons through S:

We need an expression for the net rate at which neutrons stream across a surface. Consider neu-

trons impinging on a surface element dS at time t, and count how many have crossed at time t+∆t

(Figure XII.6). The number of neutrons that pass through the surface element dS in the direction

of Ω (with speed v) during ∆t is equal to the number of neutrons contained in the volume element

t∂∂

n r E Ω t, , ,( ) rdV∫ dEdΩ

Σt r E,( )vn r E Ω t, , ,( ) rdV∫ dEdΩ

XII.9

(∆d dS). The “effective” distance ∆d that neutrons traveled in the direction of the normal to the

surface is

(XII. 21)

Thus, the number of neutrons contained in the volume element (∆d dS) at any time t with energy

in dE around E and direction in dΩ around Ω is:

(XII. 22)

Then, the number of neutrons at r with energy in dE around E and direction in dΩ around Ω that

pass through a surface element dS with outer normal es per unit time at time t is

(XII. 23)

FIGURE XII.6. Neutron streaming through the surface element dS

The net leakage rate of neutrons with energies in dE around E and with directions in dΩ around Ωthrough S is then:

(XII. 24)

Let’s recall the Divergence Theorem:

If V is a simple solid region whose boundary surface S has positive (outward) ori-

entation, and F(r) is a vector function whose (x,y,z) components have continuous

partial derivatives on an open region that contains V, then

∆d v∆t αcos( ) v∆t Ω es•( )= =

n r E Ω t, , ,( ) v∆t Ω es•( )( )dS

vn r E Ω t, , ,( ) Ω es•( )dS[ ] dEdΩ

α

time ttime t + ∆t

es

Ω

distancetravelled = v∆t

∆d = v∆t cosα= v∆t Ω•es

dS

∆d

“effective”distance

vn r E Ω t, , ,( ) Ω es•( )dSS∫ dEdΩ

XII.10

(XII. 25)

where the vector differential operator "del"is defined as

(XII. 26)

Also, the divergence of F(r) is a scalar function given by

.

Now we can write Eq. XII.24 in terms of a volumetric integral:

or, having in mind that "del" operator is the spatial derivative that does not operate on Ω or v, we

can rewrite it as

(XII. 27)

where is the gradient vector of the neutron nnumber density.

(6) Production rate in V due to neutrons scattering into dE around E, dΩ around Ω from other

energies E’ and directions Ω’ at time t:

(XII. 28)

(7) Production rate at which prompt fission neutrons are born in V with energies in dE around E

and with directions in dΩ round Ω at time t:

(XII. 29)

where: is the total number of neutrons produced in a fission event that is caused by

a neutron with energy E’

es F r( )• SdS∫ ∇ F r( )• rd

V∫=

∇

∇ ix∂

∂j

y∂∂

kz∂

∂++=

divF r( ) ∇ F r( )•x∂

∂Fx

y∂∂Fy

z∂∂Fz+ += =

vn r E Ω t, , ,( ) Ω es•( )dSS∫ dEdΩ ∇ Ω vn r E Ω t, , ,( )( )• rd

V∫ dEdΩ=

vn r E Ω t, , ,( ) Ω es•( )dSS∫ dEdΩ vΩ ∇• n r E Ω t, , ,( ) rd

V∫ dEdΩ=

∇ n r E Ω t, , ,( )

dE' Σs r E, ' E Ω' Ω•,→( )v'n r E' Ω' t, , ,( ) Ω'd4π∫

0

∞

∫ rdV∫

dEdΩ

dE' 1 β E'( )–( )ν E'( )Σ f r E, ' E→( )v'n r E' Ω' t, , ,( ) Ω'd4π∫

0

∞

∫ rdV∫

χ p E( )

4π---------------dEdΩ

ν E'( )

XII.11

is the total fraction of delayed neutrons in a fission event caused by a neu-

tron with energy E’,

Is the total fraction of all fission neutrons that are prompt

The spectrum (energy distribution) of the prompt fission neutrons. It is nor-

malized to 1:

(XII. 30)

We also assumed that the fission neutrons are emitted isotropically (the 1/4π term).

(8) Production rate at which delayed fission neutrons are born in V with energies in dE around E

and with directions in dΩ round Ω at time t:

(XII. 31)

where is the probable number of fission products in precursor group j, in dr

around r, at time t.

is the decay constant of the delayed neutron precursor in group j,

the fraction of delayed neutrons in a fission event, caused by a neutron with

energy E’, that are emitted from the j-th precursor group, with j = 1, 2, ..., 6,

is the total fraction of delayed neutrons in a fission event,

caused by a neutron with energy E’,

The spectrum (energy distribution) of the delayed neutrons in a fission

event emitted from precursor group j. It is normalized to 1:

(XII. 32)

β E'( )

1 β E'( )–

χ p E( )

χ p E( ) Ed

0

∞

∫ 1=

χd j, E( )4π

------------------ λ jC j r t,( ) rdV∫ dEdΩ

C j r t,( ) rd

λ j

β j E'( )

β E'( ) β j E'( )j 1=

6

∑=

χd j, E( )

χp,j E( ) Ed

0

∞

∫ 1=

XII.12

(9) “External” source rate at which neutrons are born in V with energies in dE around E and with

directions in dΩ round Ω at time t:

(XII. 33)

Often we can assume that the external source is isotropic:

such as the presence of Cf source. Another posibility is a monodirectional beam of neutrons com-

ing from some other source (for example, an accelerator):

.

Now, collecting the terms (1) through (9), we have:

(1) + (4) + (5) - (6) - (7) - (8) - (9) = 0 (XII. 34)

or

(XII. 35)

Equation XII.3 states that the integral of a certain function, over an arbitrary volume V, is zero. If

the integral of a function is zero, the function itself must be zero regardless of the range of inte-

Qext r E Ω t, , ,( ) rdV∫ dEdΩ

14π------ Qext r E t, ,( ) rd

V∫ dEdΩ

Qext r E Ω t, , ,( ) rdV∫ dEdΩ Qext r E t, ,( )δ Ω' Ω–( ) rd

V∫ dEdΩ=

rt∂

∂n r E Ω t, , ,( ) Σt r E,( )vn r E Ω t, , ,( ) vΩ ∇• n r E Ω t, , ,( )+ + –d∫

dE' Σs r E, ' E Ω' Ω•,→( )v'n r E' Ω' t, , ,( ) Ω'd4π∫

0

∞

∫–

χ p E( )4π

--------------- dE' 1 β E'( )–( )ν E'( )Σ f r E, ' E→( )v'n r E' Ω' t, , ,( ) Ω'd4π∫

0

∞

∫–

χd j, E( )4π

------------------λ jC j r t,( ) 14π------Q

extr E t, ,( )–

j 1=

6

∑– dEdΩ 0=

XII.13

gration. This equation can be rewritten in terms of the angular neutron flux instead of the angular

neutron density, having in mind that:

ψ(r,E,Ω,t) = v(E) n(r,E,Ω,t) = angular neutron flux [neutrons / (cm2 s MeV sterad)]

Thus, we have:

(XII. 36)

In order to account for the delayed neutrons, we introduced the delayed neutron precursor con-

centrations, Cj(r,t), that depend on the neutron flux and need to be determined. Thus, for each pre-

cursor group j we need to write a rate of change equation:

(XII. 37)

with j = 1, 2, ...6. In Eq. XII.37 we assumed that the delayed neutron precursors are produced

only by fission, and that they decay. The production term on the right-hand-side of Eq. XII.37

represents the fraction of all neutrons that are emitted from the j-th delayed neutron precursor

group per unit volume and per unit time, which is equal to the rate at which the j-th delayed neu-

tron precursor group is generated per unit volume.

Equations XII.36 and XII.37 form a set of 7 coupled integro-differential equations that we need to

solve. The good thing is that they are linear. The bad thing is that there are seven independent

variables. We cannot solve Eqs. XII.36 and XII.37 analytically in the general case. There are only

a few very simple cases in which we can find an analytic solution to these equations. For all other

1v---

t∂∂ ψ r E Ω t, , ,( ) Σt r E,( )ψ r E Ω t, , ,( ) Ω ∇• ψ r E Ω t, , ,( )+ + =

dE' Ω'Σs r E, ' E Ω' Ω•,→( )ψ r E' Ω' t, , ,( ) +d4π∫

0

∞

∫

χ pE

4π----------+ dE' 1 β E'( )–( )ν E'( )Σ f r E, '( )ψ r E' Ω' t, , ,( ) Ω'd

4π∫

0

∞

∫

χd j, E( )4π

------------------λ jC j r t,( ) 14π------Q

extr E t, ,( )+

j 1=

6

∑+

t∂∂

C j r t,( ) λ jC j r t,( )– dE

0

∞

∫ ' Ω'β j E'( )ν E'( )Σ f r E, '( )ψ r E' Ω' t, , ,( )d4π∫+=

XII.14

cases which are more realistic, we have to solve these equations using various NUMERICAL

methods and modern computers. However, analytic solutions of very special problems are of

great interest to us, because they give us insight into the behavior of solutions in more realistic

problems, and can provide excellent benchmarks for numerical solutions.

In the process of simplifying Eqs. XII.36 and XII.37 we may neglect some terms, drop out some

of the independent variables, or try to solve them in uniform homogeneous one dimensional geo-

metric domains. For example, we usually assume that all neutrons produced in a fission event are

prompt. In this case, we have only one equation known as the time-dependent neutron transport

equation (in integro-differential form):

(XII. 38)

This equation is also known as the integro-differential form of neutron transport equation. We

would like to be able to solve this equation for angular neutron flux. However, the angular neutron

flux is a function of seven independent variables! In the chapters that follow we will discuss vari-

ous approximations of Eq. XII.38, their validity and their usefulness in numerical solution

method of neutron transport problems.

XII.2.2 Initial and Boundary Conditions for the Transport Equation

(1) Initial Condition

The initial condition requires that the angular neutron flux is known everywhere in phase space at

the beginning of the problem:

(XII. 39)

for all , 0 < E < oo, and .

(2) Boundary Conditions

1v---

t∂∂ ψ r E Ω t, , ,( ) Σt r E,( )ψ r E Ω t, , ,( ) Ω ∇• ψ r E Ω t, , ,( )+ + =

dE' Ω'Σs r E, ' E Ω' Ω•,→( )ψ r E' Ω' t, , ,( ) +d4π∫

0

∞

∫

χ pE

4π----------+ dE' Ω'ν E'( )Σ f r E, '( )ψ r E' Ω' t, , ,( ) Qext r E Ω t, , ,( )+d

4π∫

0

∞

∫

ψ r E Ω t0, , ,( ) ψ0 r E Ω, ,( )=

r V∈ Ω 0 4π,( )∈

XII.15

The boundary condition requires that the angular neutron flux is given on the surface of the spa-

tial domain V, for all angles that are incoming:

(XII. 40)

for and , where f is a known function. There are several special case of the

boundary conditions:

(2.a) Mirror (or specular) reflection

In this cases, the boundary condition requires:

(XII. 41)

for and , where Ω' is the angle that would reflect (mirror-like) onto the angle Ω.

Mirror-like reflection is called specular reflection. This boundary condition is often used in reac-

tor analysis, because reactors often have approximate symmetry across a centerline.

FIGURE XII.7. Mirror reflection

(2.b) Isotropic reflective boundary condition

In this case, the reflected ray may have any angle with equal probability:

(XII. 42)

where the scalar flux is defined as

(XII. 43)

ψ r E Ω t, , ,( ) f r E Ω t, , ,( )=

r S∈ es Ω 0<•

ψ r E Ω t, , ,( ) ψ r E Ω' t, , ,( )=

r S∈ es Ω 0<•

Ω Ω’

es

ψ r E Ω' t, , ,( ) φ r E t, ,( )π

----------------------=

φ r E t, ,( ) ψ r E Ω t, , ,( ) Ωd4π∫=

XII.16

FIGURE XII.8. Isotropic reflective boundary condition

(2.c) Free surface (vacuum) boundary conditions

This type of surface is also called a non-reentrant surface, because a neutron streaming out

through the surface will never reenter.

(XII. 44)

for and .

There are two types of non-reentrant boundary: a surface facing a vacuum, and a surface facing a

perfect absorber, or so-called "black body". Equation XII.44 also implies that there will be no

neutron current crossing the free surface from outside, i.e., incoming partial current is zero:

(XII. 45)

(2.d) Periodic Boundary Condition

Another kind of spatial symmetry requires a periodic boundary condition over part of the surface

of the spatial domain:

(XII. 46)

for and , where (r–d) is the position on another part of the boundary. Periodic

boundary conditions are sometimes used in the study of fuel/moderator lattices:

Ω Ω’

ψ r E Ω t, , ,( ) 0=

r S∈ es Ω 0<•

J–

r t,( ) es Ω• ψ r E Ω t, , ,( ) Ωdes Ω 0<•

∫ Ed

E∫=

ψ r E Ω t, , ,( ) ψ r d– E Ω t, , ,( )=

r S∈ es Ω 0<•

XII.17

FIGURE XII.9. Periodic boundary condition

(3) Finiteness Condition

Our solution for the angular neutron flux must not be negative, and must be finite for all r, E, Ωand t, except at the localized sources:

(XII. 47)

(4) Interface Condition

At an interface between two regions of different materials (and different cross sections), continu-

ity of the angular neutron flux must be preserved, i.e.

(XII. 48)

for all , all energies, and all Ω. This condition states that the number of neutrons crossing

the interface surface must be conserved.

(5) Source Condition

Localized sources are introduced as mathematical singularities at the location of the source. For

example, a point source at is defined as

Fuel Moderator Fuel Moderator

r r - d r+d

0 ψ< r E Ω t, , ,( ) ∞<

ψ1 r E Ω t, , ,( ) ψ2 r E Ω t, , ,( )=

r Si∈

Interface

Medium 1 Medium 2

Ω

Ω

r0

XII.18

. (XII. 49)

Note that the unit for is [cm-3].

Thus, we have to assure that

(XII. 50)

i.e., the partial outgoing current integrated over the arbitrary surface S that surround the point

source will be equal to the source strength is we shirnk the surface.

XII.2.3 Simplified Forms of the Neutron Transport Equation

For each of the simplified forms of neutron transport equation the corresponding boundary condi-

tions should be added.

(a) Time-independent (steady-state) neutron transport equation

If the angular neutron flux is independent of time, Eq. XII.38 becomes:

(XII. 51)

(b) Steady-state neutron transport in purely-absorbing medium

In this case, there is no scattering, and there is no fission source of neutrons:

Q r0 E Ω t, , ,( ) Q0 E Ω t, ,( )δ r r0–( )=

r0

S

δ r r0–( )

es Ω•S∫r r0→

lim ψ r E Ω t, , ,( )dS Q0 E Ω t, ,( )=

Ω ∇• ψ r E Ω, ,( ) Σt r E,( )ψ r E Ω, ,( )+ =

dE' Ω'Σs r E, ' E Ω' Ω•,→( )ψ r E' Ω', ,( ) +d4π∫

0

∞

∫

χ pE

4π----------+ dE' Ω'ν E'( )Σ f r E, '( )ψ r E' Ω', ,( ) Qext r E Ω, ,( )+d

4π∫

0

∞

∫

XII.19

(XII. 52)

Equation XII.52 can be solved analytically, because there is no coupling between energies or

angles, i.e. the angular flux at energy E and angle Ω does not depend on angular flux at any other

energy E’ and angle Ω’.

(c)Steady-state neutron transport in vacuum

In this case, there are no interactions and neutron sources. Neutrons are streaming through vac-

uum:

(XII. 53)

Equation XII.53 can also be solved analytically.

(d) One-speed (monoenergetic) neutron transport equation

Now we will assume that all neutrons are moving with the same speed vo (or all neutrons have the same energy Eo). i.e.

(XII. 54)

where is the Dirac delta function, which has the following property:

We also assume that all neutron sources are monoenergetic and isotropic:

(XII. 55)

and that the differential scattering cross section can be written:

(XII. 56)

Substituting Eqs. XII.54 through XII.56 into Eq. XII.51 and integrating over E, we obtain:

(XII. 57)

Ω ∇• ψ r E Ω, ,( ) Σt r E,( )ψ r E Ω, ,( )+ Qext r E Ω, ,( )=

Ω ∇• ψ r E Ω, ,( ) 0=

ψ r E Ω, ,( ) ψ r Ω,( )δ E E0–( )=

δ E E0–( )

Ef E( )δ E E0–( )d

∞–

∞

∫ f E0( )=

Qext r E Ω, ,( ) 14π------Q

extr( )δ E E0–( )=

Σs r E, ' E Ω' Ω•,→( ) Σs r Ω' Ω•,( )δ E E0–( )=

Ω ∇• ψ r Ω,( ) Σt r( )ψ r Ω,( )+ =

XII.20

Having in mind the definition of the scalar flux, Eq. XII.4, we have

(XII. 58)

and

(XII. 59)

If, in addition, the scattering collisions are isotropic in the laboratory system, we have

(XII. 60)

and Eq. XII.59 becomes:

(XII. 61)

(e) Steady-state neutron transport equation in infinite homogeneous medium

In the case of infinite homogeneous medium nothing depends on the spatial position (r), and Eq.

XII.51 becomes

(XII. 62)

(f) One-dimensional (slab geometry), monoenergetic, steady -state neutron transport equation

If we assume that our medium is infinite and homogeneous in x and y directions, Then the angu-

lar flux will be the function of z only:

Ω'Σs r Ω' Ω•,( )ψ r Ω',( )νΣ f r'( )

4π------------------ Ω'ψ r Ω',( ) 1

4π------Q

extr( )+d

4π∫+d

4π∫

φ r( ) Ω'ψ r Ω',( )d4π∫=

Ω ∇• ψ r Ω,( ) Σt r( )ψ r Ω,( )+ =

Ω'Σs r Ω' Ω•,( )ψ r Ω',( ) 14π------+ νΣ f r'( )φ r( ) Qext r( )+( )d

4π∫

Σs r Ω' Ω•,( )Σs r( )

4π-------------=

Ω ∇• ψ r Ω,( ) Σt r( )ψ r Ω,( )+1

4π------ Σs r( )φ r( ) νΣ f r'( )φ r( ) Qext r( )+ +[ ]=

Σt E( )ψ E Ω,( ) dE' Ω'Σs E' E Ω' Ω•,→( )ψ E' Ω',( ) +d4π∫

0

∞

∫=

χ p E( )4π

---------------+ dE' Ω'ν E'( )Σ f E'( )ψ E' Ω',( ) Qext E Ω,( )+d4π∫

0

∞

∫

XII.21

Under these conditions, Eq. XII.59 becomes

(XII. 63)

If, in addition, we assume azimuthal symmetry

such that

, (XII. 64)

where

,

and

,

Eq. XII.63 can be integrated over to obtain

ψ r Ω,( ) ψ z Ω,( )=

Ωz z∂∂ ψ z Ω,( ) Σt z( )ψ z Ω,( )+ Ω'Σs z Ω' Ω•,( )ψ z Ω',( ) 1

4π------+ νΣ f z( )φ z( ) Qext z( )+( )d

4π∫=

y

z

x

θ

ϕ Ω

ψ z µ,( ) ψ z Ω,( ) ϕd

0

2π

∫ 2πψ z Ω,( )= =

dΩ θdθdϕsin dµdϕ–= =

Ωz θcos µ= =

ϕ

µz∂

∂ ψ z µ,( ) Σt z( )ψ z µ,( )+ ϕ Σs z µ0,( )ψ z µ',( )2π

------------------- Ω'd4π∫ 1

2---+ νΣ f z( )φ z( ) Qext z( )+( )d

2π∫=

XII.22

or

(XII. 65)

where .

If we assume that the scattering is isotropic in the laboratory system, i.e.,

Eq. XII.65 can be written as

or

, (XII. 66)

where we used the definition of scalar flux. In the case of purely absorbing, non-multiplying

medium, Eq. XII.66 could be further simplified to

. (XII. 67)

However, in the most general case, the differential scattering cross section is expanded into Leg-

endre polynomials, :

, (XII. 68)

where the angular components are defined as

. (XII. 69)

µz∂

∂ ψ z µ,( ) Σt z( )ψ z µ,( )+ 2π Σs z µ0,( )ψ z µ',( ) µ'd

1–

1

∫ 12---+ νΣ f z( )φ z( ) Qext z( )+( )=

µ0 Ω' Ω•=

Σs z µ0,( )Σs z( )

4π-------------=

µz∂

∂ ψ z µ,( ) Σt z( )ψ z µ,( )+Σs z( )

2------------- ψ z µ',( ) µ'd

1–

1

∫ 12--- νΣ f z( )φ z( ) Qext z( )+( )+=

µz∂

∂ ψ z µ,( ) Σt z( )ψ z µ,( )+Σs z( )

2-------------φ z( ) 1

2--- νΣ f z( )φ z( ) Qext z( )+( )+=

µz∂

∂ ψ z µ,( ) Σa z( )ψ z µ,( )+12---Qext z( )=

Pl µ0( )

Σs z µ0,( ) 2l 1+4π

--------------Σsl z( )Pl µ0( )l 0=

∞

∑=

Σsl z( ) 2π dµ0Σs z µ0,( )Pl µ0( )1–

1

∫=

XII.23

The linearly anisotropic scattering approximation means that only the first two terms in Eq.

XII.68 are considered, i.e.

. (XII. 70)

Using the definition of expansion coefficients from Eq. XII.69, we find:

and

.

We can now define the "average" angle of scattering as

. (XII. 71)

Equation XII.70 can be written in the following form

(XII. 72)

,

If we use this term in Eq. XII.65 we obtain

(XII. 73)

Σs z µ0,( ) 14π------ Σs0 z( ) 3µ0Σs1 z( )+[ ]≈

Σs0 z( ) 2π dµ0Σs z µ0,( )1

1–

1

∫ Σs z( )= =

Σs1 z( ) 2π dµ0µ0Σs z µ0,( )1–

1

∫=

µ0

2π dµ0µ0Σs z µ, 0( )1–

1

∫2π dµ0Σs z µ, 0( )

1–

1

∫-----------------------------------------------------

Σs1 z( )Σs0 z( )---------------= =

Σs z µ, 0( ) 14π------ Σs0 z( ) 3µ0µ0 z( )Σs0 z( )+[ ]≈

Σs z( )4π

------------- 1 3µ0µ0 z( )+[ ]=

µz∂

∂ ψ z µ,( ) Σt z( )ψ z µ,( )+Σs z( )

2------------- 1 3µ0µ0 z( )+[ ]ψ z µ',( ) µ'd

1–

1

∫ 12---+ νΣ f z( )φ z( ) Qext z( )+( )=

XII.24

(f) One-dimensional (spherical geometry), monoenergetic, steady -state neutron transport equa-

tion

XII.2.4 Examples of Analytical Solutions for Simple Problems

XII.25

XII.3. Integral Form of Neutron Transport Equation

The major idea behind the development of the integral form of neutron transport equation is to

integrate out the angular dependence and to solve the transport equation for the scalar flux

directly (and the partial currents on the region boundaries in some cases). Providing that the scat-

tering anisotropy is low (isotropic or linearly anisotropic), numerical methods based on the inte-

gral form can treat angular dependence EXACTLY. These methods (see Chapter XVI) are very

accurate and relatively simple to apply, if isotropic scattering can be assumed. Linear anisotropy

is usually handled by using the transport-corrected cross sections in the integral transport algo-

rithm developed for the isotropic scattering. Exact treatment of anisotropy of higher order is usu-

ally avoided in the integral transport methods for geometries that are not one-dimensional.

The integral transport equation is based on a global neutron balance in a given direction, leading

to a strong coupling of all regions, which is opposite of the case with the integro-differential

transport equation (XII.38) which is based on a local neutron balance, leading to a coupling

between the neighboring regions in space only (see Chapters XVII and XVIII).

In this Section we will derive the general integral form of neutron transport equation, and give

some examples of its applications.

XII.3.1 Derivation of Integral Form of Neutron Transport Equation for Angular Flux

The neutron transport equation, being a linear first-order partial integro-differential equation, can

be converted into an integral form by the method of characteristics. The integral transport equa-

tion can be derived either from the continuous energy or the monoenergetic form of the transport

equation, since the manipulation does not involve the neutron energy. In our work, we will be

mostly working with the multigroup form of the equation, so we will take as our starting point the

time-dependent monoenergetic transport equation derived from Eq. XII.38:

(XII. 74)

where the source includes all sources: scattering fission and external. Now we write a

similar equation:

(XII. 75)

where

1v---

t∂∂ ψ r Ω t, ,( ) Q r Ω t, ,( ) Σt r( )ψ r Ω t, ,( ) Ω ∇• ψ r Ω t, ,( )––=

Q r Ω t, ,( )

1v---

t∂∂ ψ r Ω t, ,( ) Q r Ω t, ,( ) Σt r( )ψ r Ω t, ,( ) Ω ∇• ψ r Ω t, ,( )––=

XII.26

, with , (XII. 76)

, with . (XII. 77)

Note that:

, and (XII. 78)

The variable R is the distance in the direction of neutron travel (i.e., in the direction of W) as

shown in Fig. XII.10

FIGURE XII.10. Definition of a new variable R

Now we can define two new functions of R:

, with . (XII. 79)

and

, with . (XII. 80)

The total derivative of is then given by:

(XII. 81)

Note that:

r R( ) r RΩ+= r 0( ) r=

t R( ) t R1v---+= t 0( ) t=

rddR------- Ω= dt

dR------- 1

v---=

Ω

r, t

r t( , )

R

ψ R( ) ψ r Ω t, ,( ) ψ r RΩ+ Ω t R1v---+, ,

= = ψ 0( ) ψ r Ω t, ,( )=

Q R( ) Q r Ω t, ,( ) Q r RΩ+ Ω t R1v---+, ,

= = Q 0( ) Q r Ω t, ,( )=

ψ R( )

dψ R( )dR

----------------ψ r Ω t, ,( )∂

r∂---------------------------

rdRd

------ψ r Ω t, ,( )∂

t∂--------------------------- td

Rd------+=

XII.27

(XII. 82)

which is the gradient of ψ. Substituting Eq. XII.78 and XII.82 into Eq.XII.81, we obtain

(XII. 83)

Equations XII.83 and XII.75 can be combined to get

. (XII. 84)

Being a first order linear differential equation, Eq. XII.84 has the solution:

(XII. 85)

where is known. Let us assume that we want to determine angular neutron flux in volume

V surrounded by a surface S as shown in Figure XII.11. If we change the variable R to (-R), and

let R go to zero, Eq. XII.85 becomes:

(XII. 86)

Using the definition of and (Eqs. XII.79 and XII.80) and assuming the time inde-

pendent case, we can rewrite Eq. XII.86 in the following form:

(XII. 87)

ψ r Ω t, ,( )∂r∂

--------------------------- ψ r Ω t, ,( )∇=

dψ R( )dR

---------------- Ω ψ r Ω t, ,( )∇ 1v---+• ψ r Ω t, ,( )∂

t∂---------------------------=

dψ R( )dR

---------------- Σt r( )ψ R( )+ Q R( )=

ψ R( ) ψ R0( ) Σt r R''Ω+( ) R''dRs

R

∫– + exp=

Q R'( ) Σt r R''Ω+( ) R''dR'

R

∫– R'dexp

Rs

R

∫+

ψ Rs( )

ψ 0( ) ψ R– s( ) Σt r R''Ω–( ) R''d0

Rs

∫– + exp=

Q R– '( ) Σt r R''Ω–( ) R''d0

R'

∫– R'dexp

0

Rs

∫+

ψ 0( ) Q 0( )

ψ r Ω,( ) ψ r RsΩ– Ω,( ) Σt r R'Ω–( ) R'd0

Rs

∫– + exp=

Q r R'Ω Ω,–( ) Σt r R''Ω–( ) R''d0

R'

∫– R'dexp

0

Rs

∫+

XII.28

FIGURE XII.11. Coordinates characterizing neutron transport in 3-D

Introducing:

, , (XII. 88)

and defining the optical length as

, (XII. 89)

(XII. 90)

we can write the integral form of the transport equation (or simply, integral transport equation) in

its final form for the angular flux as:

(XII. 91)

With the help of Fig. XII.11, we can interpret Eq. XII.91 as follows: the neutron flux at , in the

direction of , is the result of adding up all the uncollided source neutrons in the volume V, pro-

duced on the trajectory in the direction of (multiplied by an exponential attenua-

S

V

es'

ΩR’

r

r'

rs

es

rs'

Rs

dS

dS'

r' r R'Ω–= rs r RsΩ–=

τ r r',( ) τ r r R'Ω–,( ) Σt r R''Ω–( ) R''d0

R'

∫–= =

τ r rs,( ) τ r r RsΩ–,( ) Σt r R''Ω–( ) R''d0

Rs

∫–= =

ψ r Ω,( ) ψ rs Ω,( ) τ r rs,( )–[ ]exp Q r' Ω,( ) τ r r',( )–[ ] R'dexp

0

Rs

∫+=

r

Ω

r r RsΩ–,( ) Ω

XII.29

tion factor), plus all uncollided neutrons entering the volume V through the surface S at

in the direction of (also multiplied by an attenuation factor).

(a) Vacuum boundary conditions

In the case of vacuum boundary conditions, there are no neutrons entering V through S, i.e.

, (XII. 92)

and Eq. XII.91 becomes:

. (XII. 93)

(b) Infinite medium

In the case of an infinite medium, we let Rs in Eq. XII.93 go to infinity, i.e.

(XII. 94)

XII.3.2 Derivation of Integral Transport Equation for Scalar Flux:Vacuum Boundary Conditions

In order to determine reaction rates and power distribution in the nuclear reactor, we need to

know the scalar flux. Per definition, the scalar flux can be determined by integrating the angular

flux over all angles. Let us try to do it first for the case with vacuum boundary conditions, i.e. we

will integrate Eq. XII.93 over solid angle :

(XII. 95)

Note that both Rs and depend on , i.e. . The integration presented in Eq. XII.95

is in polar coordinates ( ) with the origin at . Figure XII.12 shows that by changing from

0 to , and by changing for each from o to , we are integrating over entire volume

V. Thus, we can change the variables from polar to Cartesian, in order to have a volumetric inte-

rs r RsΩ–= Ω

ψ rs Ω,( ) 0= es Ω• 0≤

ψ r Ω,( ) Q r' Ω,( ) τ r r',( )–[ ] R'dexp

0

Rs

∫=

ψ r Ω,( ) Q r' Ω,( ) τ r r',( )–[ ] R'dexp

0

∞

∫=

Ω

φ r( ) Ω R'Qd r' Ω,( ) τ r r',( )–[ ]exp

0

Rs Ω( )

∫d4π∫=

r' Ω r' r R'Ω–=

R' Ω, r Ω

4π R' Ω Rs Ω( )

XII.30

gral in Cartesian coordinates. The volume element at point in polar coordinates as shown in

Fig. XII.12 is given as

(XII. 96)

FIGURE XII.12. Changing the variables of integration

Using the definition of the solid angle, we can express the small solid angle as

(XII. 97)

The volume element in Cartesian coordinates can be expressed as:

(XII. 98)

Using Eqs. XII.96, XII.87, and XII.98 we can change the variables of integration as

(XII. 99)

and the double integral in Eq. XII.95 can be replaced by a volumetric integral:

(XII. 100)

r'

dV ' dR'dA=

S

V

ΩR’

r

r'dA

dR’

dΩ

dΩ

dΩ dA

R'( )2------------=

dV ' dx'dy'dz' dr'= =

dΩ R'ddr'

R'( )2------------

dr'

r r'–2

-----------------= =

φ r( ) r' Q r' Ω,( ) τ r r',( )–[ ]exp

r r'–2

---------------------------------- d

V∫=

XII.31

Note that this equation was derived in the case of vacuum boundary conditions. However, we can

see the problem: our source is still a function of ! Let us suppose that the total source in Eq.

XII.100 is isotropic (i.e., assuming isotropic scattering, and isotropic fission source and external

source):

(XII. 101)

Equation XII.100 becomes

(XII. 102)

Thus, if we know the spatial distribution of the source and the spatial distribution of the optical

lengths, we can determine the spatial distribution of the scalar flux using Eq. XII.102. This equa-

tion can be easily rewritten for the energy-dependent flux and sources:

(XII. 103)

The total (isotropic) source may consist of the following terms:

(XII. 104)

(a) Point isotropic source, vacuum boundary conditions

A point source at can be represented as

(XII. 105)

In the case of a point source, Eq.IV.103 can be integrated over volume V

(XII. 106)

(b) Point isotropic source in infinite homogeneous medium

If the medium is homogeneous, the total cross section will not depend on spatial variable, and the

optical length can be expressed as:

Ω

Q r' Ω,( ) Q r'( )4π

-------------=

φ r( ) r' Q r'( ) τ r r',( )–[ ]exp

4π r r'–2

---------------------------------- d

V∫=

φ r E,( ) r' Q r' E,( ) τ r r' E, ,( )–[ ]exp

4π r r'–2

----------------------------------------- d

V∫=

Q r' E,( ) E'Σs r' E' E→,( )φ r' E',( )d

0

∞

∫ χ E( ) E'νΣ f r' E',( )φ r' E',( ) Qext r' E,( )+( )d

0

∞

∫+=

r0

Qp r E,( ) Qp 0, E( )δ r r0–( ) Qp 0, E( )δ x x0–( )δ y y0–( )δ z z0–( )= =

φ r E,( ) Qp 0, E( )τ r r0 E, ,( )–[ ]exp

4π r r0–2

------------------------------------------=

XII.32

(XII. 107)

The scalar flux in this case is given as

(XII. 108)

If = 0 (i.e., the point source is located at the origin), Eq. XII.108 could be further simplified:

(XII. 109)

where r is the distance from the origin.

(c) Isotropic point source in vacuum

If the infinite homogeneous medium is vacuum, and the scalar flux is inversely pro-

portional to the square distance from the point source:

(XII. 110)

XII.3.3 Derivation of Integral Transport Equation for Scalar Flux: General Boundary Conditions

Now we go back to Eq. XII.91 and integrate it over solid angle :

(XII. 111)

from our analysis in Section XII.3.2, we know that the second integral can be replaced by the vol-

umetric integral in Cartesian coordinate system. Using our notation on Figs. XII.11, we can also

change the variables in the first integral from polar to Cartesian. Note that the angular flux in the

first integral represents the incoming flux from outside the volume V through the surface S, i.e.

. Thus, by integrating over in polar coordinate system, we integrate over

entire surface S in Cartesian coordinate system, i.e.

(XII. 112)

τ r r0 E, ,( ) Σt E( ) r r0–( )=

φ r E,( ) Qp 0, E( )Σt E( ) r r0–( )–[ ]exp

4π r r0–2

-----------------------------------------------------=

r0

φ r E,( ) Qp 0, E( )Σt E( )r–[ ]exp

4πr2

----------------------------------=

Σt E( ) 0=

φ r E,( )Qp 0, E( )

4πr2

--------------------=

Ω

φ r( ) Ωψ rs Ω,( ) τ r rs,( )–[ ]expd4π∫ Ω R'Qd r' Ω,( ) τ r r',( )–[ ]exp

0

R Ω( )s

∫d4π∫+=

rs S∈ Ω 0 4π,[ ]∈

dΩdS es Ω•

r rs– 2------------------------=

XII.33

to get

(XII. 113)

where

and

, with (XII. 114)

is the incoming angular current at on surface S, with being the outward normal to surface S

(See Fig. XII.11). In the case of either isotropic scattering or when linearly anisotropic scattering

is approximated by transport corrected cross sections, the scattering source depends on the scalar

flux only, and Eq.XII.113 becomes a self-contained balance equation, the so called Peierls’ equa-

tion. In the case of the general boundary conditions with incoming angular currents, an additional

equation is needed for the partial currents of neutrons leaving the volume V,

(XII. 115)

which was obtained by multiplying Eq. XII.91 by and integrating over for

XII.3.4 Transport Corrected Cross Sections

In order to evaluate Eqs. XII.100, XII.112, and XII.115, the angular dependency of the volumetric

sources must be known. The fission source does not usually depend on , but the differential

scattering cross sections in the scattering source depend on the cosine of the scattering angle

between incoming and outgoing neutron directions, . If isotropic scattering is

assumed, the volumetric sources do not depend on any more, and the evaluation of scalar flux

is simplified (Eq. XII.102, for example). If linearly anisotropic scattering is assumed, the volu-

metric sources and fluxes could be expanded into spherical harmonics, which may lead to more

φ r( ) r' Q r' Ω,( ) τ r r',( )–[ ]exp

r r'–2

---------------------------------- d

V∫ +=

+dS

r rs– 2------------------J

inrs Ω,( )exp τ r rs,( )–[ ] ,

S∫

r' r RΩ, r s – r R s Ω , Ω – r r

'–

r r

'–

--------------,= = =

Jin

rs Ω,( ) es Ω• ψ rs Ω,( )= es Ω• 0<

rs es

Jout

rs'( ) dr'r rs'– 2------------------- es' Ω•( )Q r' Ω,( )exp τ r rs,( )–[ ]

V∫=

+dS

rs rs'– 2--------------------- es' Ω•( )J

inrs Ω,( )exp τ rs' rs,( )–[ ] ,

S∫

es' Ω• Ω es' Ω• 0>

Ω

µ0 Ω Ω'•=

Ω

XII.34

complex computational algorithms and prolong the computational time. To remedy this, so-called

transport correction is used, i.e., it is assumed that problems with linearly anisotropic scattering

can be replaced by problems with isotropic scattering, provided that the total and scattering cross

sections are reduced.

The differential scattering cross section is usually expanded in Legendre polynomials, ,

where , is:

, (XII. 116)

where we suppressed the spatial variable, and where the angular components are defined as

. (XII. 117)

The linearly anisotropic scattering approximation means that only the first two terms in Eq.

XII.116 are considered, i.e.

, (XII. 118)

which after integration over gives

(XII. 119)

,

where

. (XII. 120)

Usually, linearly anisotropic scattering is not treated explicitly in the integral transport methods.

In fact, scattering is assumed to be isotropic, but the total and scattering cross sections are

reduced (transport corrected) as follows:

Pl µ0( )

µ0 Ω Ω'•=

Σs µ0 E E'→,( ) 2l 1+4π

--------------Σsl E E'→( )Pl µ0( )l 0=

∞

∑=

Σsl E E'→( ) 2π dµ0Σs µ0 E E'→,( )Pl µ0( )1–

1

∫=

Σs µ0 E E'→,( ) 14π------ Σs0 E E'→( ) 3µ0Σs1 E E'→( )+[ ]≈

E'

Σs µ0 E,( ) 14π------ Σs0 E( ) 3µ0µ0 E( )Σs0 E( )+[ ]≈

14π------ Σs0 E( ) 3µ0Σs1 E( )+[ ]=

µ0 E( )2π dµ0dE'µ0Σs µ0 E E'→,( )∫∫

2π dµ0dE'Σs µ0 E E'→,( )∫∫---------------------------------------------------------------------------

Σs1 E( )Σs0 E( )-----------------= =

XII.35

(XII. 121)

i.e., the diagonal of the scattering matrix is corrected, and the transport-corrected total cross

section is given by

. (XII. 122)

XII.3.5 Examples of Application of Integral Equation

Σs0tr

E E'→( )Σs0 E E'→( ), otherwise

Σs0 E E'→( ) µ0 E( )Σs0 E( ), for E = E'–

≈

P0

Σttr

E( ) Σt E( ) µ0 E( )Σs0 E( )– Σt E( ) Σs1 E( )–= =

XII.36

XII.4. Adjoint Transport Equation

The primary purpose of this Section is to present the relationship between the adjoint and forward

space in a unified, yet simple formulation that might be helpful for understanding the physical

meaning of adjoint functions, deducing the equation satisfied by these functions and finding the

adjoint space formulation for a variety of applications.

Starting with the most general time dependent transport equation with delayed neutron precursors

we derive, using formal mathematical techniques, a generalized total importance balance condi-

tion (Section XII.4.1). We then illustrate how this condition can be used for (1) interpreting the

physical meaning of different adjoint functions and finding the detailed form of the equation,

boundary conditions and time conditions that will assign the adjoint function desired physical

meaning. (Section XII.4.2); and (2) providing the mathematical foundation for many different

applications of adjoint formulations (Section XII.4.3).

XII.4.1 Total Importance Balance Condition

Consider the general time-dependent source-driven Boltzmann equation with delayed neutrons

that we derived earlier (Eq. XII.36):

(XII. 123)

It this case, we assumed that the fission source is isotropic, while the external source is not, and

that does not depend on energy. We also consider the equations for the density (or concentra-

tion) of the delayed neutron precursors (Eq. XII.37):

1v---

t∂∂ ψ r E Ω t, , ,( ) Σt r E,( )ψ r E Ω t, , ,( ) Ω ∇• ψ r E Ω t, , ,( )+ + =

dE' Ω'Σs r E, ' E Ω' Ω•,→( )ψ r E' Ω' t, , ,( ) +d4π∫

0

∞

∫

1 β–( )χ p r E,( )

4π--------------------+ dE'νΣ f r E, '( )φ r E' t, ,( )

0

∞

∫

χd i, r E,( )4π

-----------------------λ iCi r t,( ) Qext r E Ω t, , ,( )+i 1=∑+

β

XII.37

(XII. 124)

where

(XII. 125)

and represents an external source of delayed neutron precursors. Even though such a pre-

cursor source is not required for most of the problems encountered in practice, it is included in the

present formulation to make the forward equations symmetrical with the adjoint ones.

The corresponding adjoint equation is defined to be:

(XII. 126)

and

(XII. 127)

where

(XII. 128)

t∂∂

Ci r t,( ) λ iCi r t,( )– βi dE

0

∞

∫ 'νΣ f r E, '( )φ r E' t, ,( ) Qi r t,( )+ +=

φ r E t, ,( ) Ωψd r E Ω t, , ,( )4π∫=

Qi r t,( )

1v---–

t∂∂ ψ+

r E Ω t, , ,( ) Σt– r E,( )ψ+r E Ω t, , ,( ) Ω ∇• ψ +

r E Ω t, , ,( )+ =

dE' Ω'Σs r E, ' E Ω' Ω•,→( )ψ+r E' Ω' t, , ,( ) +d

4π∫

0

∞

∫

1 β–( )νΣ f r E,( )+ ddE'χ p r E',( )φ+r E' t, ,( )

0

∞

∫

βiνΣ f r E,( )Ci+

r t,( ) Qext+

r E Ω t, , ,( )+i 1=∑+

t∂∂– Ci

+r t,( )( ) λ iCi

+r t,( )– λ i dE

0

∞

∫ 'νΣ f r E, '( )φ+r E' t, ,( ) Qi

+r t,( )+ +=

φ+r E t, ,( ) 1

4π------ Ωψ+

d r E Ω t, , ,( )4π∫=

XII.38

Multiplying Eq. XII.123 by , equations XII.124 by , equation XII.126 by

, equations XII.127 by ; integrating over the independent variables and

subtracting the resulting adjoint equations from the flux and precursor equations is found that:

(XII. 129)

In the above equation , where is a radius vector to the outer sur-

face of the medium under the consideration; is a unit vector (normal) pointing outwards at the

surface, and dS is a unit surface area. The notation <,> stands for integration over phase space:

; (XII. 130)

is the neutron density; and and are, respectively, the initial and final

time considered. The surface integral in the first term of Eq. XII.129 is obtained from a volume

integral by the application of the divergence theorem.

Equation XII.129 is a total importance balance condition. The importance, in the present context,

pertains to the contributions of a neutron to a given (real or fictitious) detector reading. Applica-

tions of this balance condition are illustrated in Sections XII.4.2 and XII.4.3. To simplify nota-

tions it will be assumed that there is only one group of delayed neutrons, represented by the

spectrum .

ψ+r E Ω t, , ,( ) Ci

+r t,( )

ψ r E Ω t, , ,( ) Ci r t,( )

t E Ω Ses Ω ψ+xs t;( )ψ xs t;( )( )•d

A∫d

4π∫d

0

∞

∫d

ti

t f

∫ +

<ψ+x t f;( ) n x t f;( )>, <ψ+

x ti;( ) n x ti;( )>,– <Ci+

r t f;( ) Ci r t f;( )> <Ci+

r ti;( ) Ci r ti;( )>=,–,i

∑+

t <ψ+x t;( ) Qext x t;( )>, <ψ x ti;( ) Qext

+x ti;( )>,–[ ]d

ti

t f

∫=

t <Ci+

r t f;( ) Qi r t f;( )> <Ci r ti;( ) Q+

i r ti;( )>,–,[ ]i

∑d

ti

t f

∫+

x r E Ω, ,( )≡ xs rs E Ω, ,( )≡ rs

es

< f x( ) g x( )>, r E Ω f r E Ω, ,( )g r E Ω, ,( )d4π∫d

0

∞

∫dV∫≡

n x t;( ) ψ x t;( ) v⁄= ti t f

χd

XII.39

XII.4.2 The Physical Interpretation of Adjoint Functions

Assume the conventional boundary conditions:

, for , (XII. 131)

and

, for . (XII. 132)

With these conditions, the final integral of equation XII.129 vanishes. The boundary conditions of

equations XII.130, XII.131 and XII.132 describe physical situation when both Qext and Qext+

have no component outside the boundary of the region encompassed by . Suppose also that the

initial and final conditions are:

(XII. 133)

(XII. 134)

(XII. 135)

(XII. 136)

With these time conditions [and the boundary conditions of Eqs. XII.130, XII.131 and XII.132],

equation XII.129 becomes

(XII. 137)

(1) Accumulated detection probability

Suppose that represents the response function (or efficiency) of a given detector (real

or fictitious) and let

ψ xs t;( ) 0= Ω es• 0<

ψ+xs t;( ) 0= Ω es• 0>

rs

ψ x t;( ) 0=

C r t,( ) 0=

ψ+x t;( ) 0=

C+

r t,( ) 0=

t <ψ x t;( ) Q+

ext x t;( )>, <C r t;( ) Qd+

r t;( )>,+[ ]d

ti

t f

∫ =

t <ψ+x t;( ) Qext x t;( )>, <C

+r t;( ) Qd r t;( )>,+[ ]d

ti

t f

∫

Q+

ext x t;( )

XII.40

(XII. 138)

Then for source term

, for ti <t’ <tf (XII. 139)

and the initial conditions

or ti<t’ (XII. 140)

the total importance balance condition of Eq. XII.137 gives

(XII. 141)

Equation XII.141 shows that the adjoint function defined by Eqs. XII.136 and XII.137 with the

conditions defined by Eqs. XII.131, XII.132, XII.135, XII.136, and XII.138 is the number of neu-

trons counted by the detector in the period from t’ to tf as a result of one neutron introduced into

phase space point x’ at t’. This adjoint function can be referred to as the accumulated detection

probability.

(2) Instantaneous detection probability

If, in addition to the conditions specified in Section (1), we consider the adjoint source term of the

special time-dependence

, for (XII. 142)

then Eq. XII.141 becomes

(XII. 143)

Thus, the solutions of Eqs. XII.126 and XII.127 with the special source term of Eq. XII.142 (as

well as Eq. XII.138) and the final conditions of Eqs. XII.131, XII.132,and XII.135 is the average

count rate of the detector at t’’ as a result of the insertion of one neutron into x’ at t’ (< t’’). It can

be referred to as the instantaneous detection probability.

(3) Precursor detector

Setting , and

Qd+

r t;( ) Qd r t;( ) 0= =

Qext x t;( ) δ x x'–( )δ t t'–( )=

ψ x t;( ) C r t;( ) 0= =

ψ+x' t';( ) t <ψ x t;( ) Q

+ext x t;( )>,( )d

ti

t f

∫=

Q+

ext x t;( ) Q+

ext x( )δ t t''–( )= t' t'' t f< <

ψ+x' t';( ) <ψ x t'';( ) Q

+ext x( )>,=

Qd r t;( ) 0= Q+

ext x t;( ) 0=

XII.41

, for (XII. 144)

equation XII.137 becomes

(XII. 145)

Thus, under the condition stated, is the probability of finding precursors at as a

result of one neutron introduced at . This example illustrates the physical meaning of the

precursor adjoint source term Qd+ it represent the response function of a (fictitious) detector that

reacts to delayed neutron precursors (and not to neutrons). Such a detector can not be represented

by the conventional adjoint source term, Qext+.

(4) Selected examples for adjoint source terms

Table XII.1 presents examples for the response function of several fictitious neutron detectors

(represented by Qext+) and the corresponding adjoint functions that represent physical quantities

of interest in reactor theory.

TABLE XII.1.

Adjoint represents

0 Absorption rate

0 Fission rate

0 Collision rate

0 Slowing down density passed Ec

0 Birth rate density

Q+

d r t;( ) δ r r0–( )δ t t0–( )= ti t0 t f< <

t <ψ+x t;( ) Qext x t;( )>,( )d

ti

t f

∫ C r0 t0;( )=

ψ+x' t';( ) r0 t0;( )

x' t',( )

Qext+

r E Ω t, , ,( )ψ+

r E Ω t, , ,( )Ω es• 0>

Σa r E,( )

Σ f r E,( )

Σt r E,( )

E' Σs r E E'→,( )( )d0

Ec

∫

E' Ω'Σs r EΩ E'Ω'→,( ) νΣ f r E,( )+d4π∫d

0

∞∫

XII.42

XII.4.3 Applications of Adjoint Space Formulations

By a proper selection of source terms, boundary conditions and time conditions, the total impor-

tance balance condition of Eq. XII.129 can be reduced to relationships that provide useful formu-

lations for a variety of applications of adjoint functions.

(a) Detector response to different sources

If, in addition to the condition leading to Eq. XII.137, it is assumed that Qd = Qd+ = 0 and that

there is no time dependence, Eq. XII.129 (or Eq. XII.137) reduces to

(XII. 146)

This is, perhaps, the most well known special case of the total importance balance condition. The

right-hand side of Eq. XII.133 (i.e. the adjoint space) is the most convenient formulation for cal-

culating the effect of different sources (or components of a given source) on a given detector.

(b) Deep penetration problems

When the source (Qext and Qd) is outside of the material medium under consideration, equation

XII.133 is replaced by the relation:

(XII. 147)

obtained from Eq. XII.129 for time independent problems with the conditions of Eqs. XII.133 -

XII.136 and Qd+ = 0. The volume bounded by the surface S should contain all the detector and

none of the source. The right-hand side of Eq. XII.134 also gives the detector response for prob-

lems in which the detector is outside (and only outside) of the region under consideration and the

source is inside (and only inside) that region. It has been shown that right-hand side of Eq.

XII.134 is useful for problems in which the source is far away from the medium containing the

detector that is assumed to be in vacuum.

ψ x( ) Q+

ext x( )> = <ψ+x( ) Qext x( )>,,<

ψ x( ) Q+

ext x( )> = E Ω Ses Ω ψ+xs( )ψ xs( )( )•d

A∫d

4π∫d

0

∞

∫,<

XII.43

XII.5. Concluding Remarks

XII.44