Nutron transport Equation

8/10/2019 Nutron transport Equation

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Chapter 1

THE NEUTRON TRANSPORT EQUATION

1.1 Introduction

Two methods exist for simulating and modeling neutron transport and interactions in the reactor

core, or neutronics. Deterministic methods solve the Boltzmann transport equation in a

numerically approximated manner everywhere throughout a modeled system. Monte Carlo

methods model the nuclear system (almost) exactly and then solve the exact model statistically

(approximately) anywhere in the modeled system. Although deterministic methods are fast for

one-dimensional models, both methods are slow for realistic three-dimensional problems.

1.1.1 Deterministic methods

Deterministic neutronics methods play a fundamental role in reactor core modeling and

simulation. A first-principles treatment requires solution of the linearized Boltzmann transport

equation. This task demands enormous computational resources because the problem has seven

dimensions: three in space, two in direction, and one each in energy and time.

Nowadays reactor core analyses and design are performed using nodal (coarse-mesh) two-group

diffusion methods. These methods are based on pre-computed assembly homogenized cross-

sections and assembly discontinuity factors obtained by single assembly calculation with

reflective boundary conditions (infinite lattice). These methods are very efficient and accurate

when applied to the current low-heterogeneous Light Water Reactor (LWR) cores. With the

progress in the Nuclear Engineering field new generations of Nuclear Power Plants (NPPs) are

considered that will have more complicated rector core designs, such as cores loaded partially

with mixed-oxide (MOX) fuel, high burn-up loadings, and cores with advanced fuel assembly

and control rode designs. Such heterogeneous cores will have much more pronounced leakage

between the unlike assemblies, which will introduce challenges to the current methods for core

calculation. First, the use of pre-computed fuel-assembly homogenized cross-section could lead

to significant errors in the coarse-mesh solution. On the other hand, the application of the two-

group diffusion theory for such heterogeneous cores could also be source for large errors. New

core calculation methodologies based on fine-mesh (pin-by-pin), higher-order neutron transport

treatment, multi-group methods are needed to be developed to address the difficulties the

diffusion approximation meets by improving accuracy, while preserving efficiency of the

current reactor core calculations.

The diffusion theory model of neutron transport has played a crucial role in reactor theory

since it is simple enough to allow scientific insight, and it is sufficiently realistic to study many

important design problems. The mathematical methods used to analyze such a model are the

same as those applied in more sophisticated methods such as multi-group neutron transporttheory. The neutron flux () and current (J) in the diffusion theory model are related in a simple

way under certain conditions. This relationship between and J is identical in form to a law

used in the study of diffusion phenomena in liquids and gases: Ficks law. The use of this law in

reactor theory leads to the diffusion approximation, which is a result of a number of simplifying

assumptions. On the other hand, higher-order neutron transport codes have always beendeployed in nuclear engineering for mostly time-independent problems out-of-core shielding


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calculations. With all the limitations of two-group diffusion theory, it is desirable to implement

the more accurate, but also more expensive multi-group transport approach.

Replacing two group diffusion nodal (assembly-based) neutronics solvers with multi-group

transport (pin-based) solutions can be performed in direct and embedded manner. While studies

have been performed utilizing the two deterministic types of methods (depending on techniques

to treat angular flux dependence) - the discrete ordinates (SN) and spherical harmonics (PN)

approximations for core analysis, the majority of the efforts have been focused on replacing

diffusion approximation with Simplified P3 (SP3) approach. The SP3 method has gained

popularity in the last 10 years as an advanced method for neutronics calculation. The SP3approximation was chosen because of its improved accuracy as compared to the diffusion

theory for heterogeneous core problems and efficiency in terms of computing time as compared

to higher order transport approximations. The SP3approximation is more accurate when applied

to transport problems than the diffusion approximation with considerable less computation

expense than the discrete ordinates (SN) or spherical harmonics (PN) approximations. Another

advantage of SP3 approximation is that it can be solved by straightforward extension of the

common nodal diffusion methods with little overhead. It has been shown also that the multi-

group SP3pin-by-pin calculations can be used for practical core depletion and transient analyses.

Recent results obtained elsewhere indicated the importance of consistent cross-section

generation and utilization in the SP3equations in terms of modeling the P1scattering. Since this

is an important issue for practical utilization of SP3approximation for reactor core analysis.

1.1.2 Stochastic methods

Numerical methods that are known as Monte Carlo methods can be loosely described as

statistical simulation methods, where statistical simulation is defined in quite general terms. It

can be any method that utilizes sequences of random numbers to perform the simulation.

Statistical simulation methods may be contrasted to conventional numerical discretization

methods, which typically are applied to ordinary or partial differential equations that describe

some underlying physical or mathematical system. In many applications of Monte Carlo, the

physical process is simulated directly, and there is no need to even write down the differential

equations that describe the behavior of the system. The only requirement is that the physical (or

mathematical) system be described by probability density functions (pdfs). Once the pdfs are

known, the Monte Carlo simulation can proceed by random sampling from the pdfs. Many

simulations are then performed (multiple trails or histories) and the desired result is taken

as an average over the number of observations (which may be a single observation or perhaps

millions of observations). In many practical applications, one can predict the statistical error

(the variance) in this average result, and hence an estimate of the number of Monte Carlotrials that are needed to achieve a given error.

When it is difficult (impossible, impractical) to describe a physical phenomenon via

deterministic equations (balance equations, distribution functions, differential equations, etc.)

Monte Carlo method becomes useful (or necessary). In Nuclear Engineering, in the past themethod was mainly used for complex shielding problems and for benchmarking of

deterministic calculations. Today, however, because of the advent of faster computers and

parallel computing, the technique is being used more extensively for normal calculations. But,

as will be discussed later, one should not (cannot) only rely on Monte Carlo calculations

because the technique is very expensive/impractical (time and money) for obtaining detailinformation about a physical system or for performing sensitivity/perturbation studies. In such


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situation, deterministic methods are necessary for aiding Monte Carlo and/or for performing the

actual simulation.

Using the Monte Carlo method, a model of the medium under study is set up in the computer

and individual particles are thrown at the medium or are generated in it, as if coming from a

source. The particles are followed, one by one, and the various events in which they participate

(collision, absorption, fission, escape, etc.) are recorded. All the events associated with one

particle constitute the history of that particle. The present trend in advanced and next generation

nuclear reactor core designs is towards increased material heterogeneity and geometry

complexity. The continuous energy Monte Carlo method has the capability of modeling such

core environments with high accuracy. Because of its statistical nature, Monte Carlo core

calculations usually involve a considerable computer time to attain reliable converged results

for both integral parameters and local distributions. Number of coupled Monte Carlo depletion

code systems have been developed and successfully used in assembly/core depletion reference

calculations. Now the tendency is to couple Monte Carlo neutronic calculation with a thermal-

hydraulics code to obtain three-dimensional (3D) power and thermal-hydraulic solutions for a

reactor core. The investigations performed at PSU showed that performing Monte Carlo based

coupled core steady state calculations are feasible. This conclusion indicates that in the future it

will be possible to use Monte Carlo method to produce reference results at operating conditions.

1.1.3 Conclusions

For reactor core modeling and simulation, deterministic methods will be used principally in the

short term (35 years) with Monte Carlo as a benchmarking tool. In the intermediate term (510

years) Monte Carlo methods could be used as a hybrid tool with multi-physics coupling to

deterministic neutronics and thermal hydraulics codes. In the long term (>10 years) multi-

physics codes using non-orthogonal grids will provide complete, high-accuracy design tools,

fully integrated into reactor core design and operation.

1.1.4 Basic terms and definitions

Please review Sections 1-1, 1-2, and 1-3 of Chapter 1 of E. E. Lewis and W. F. Miller,

Computational Methods of Neutron Transport, American Nuclear Society (1993).

1.2 Neutron Boltzmann Equation

Several choices are possible for describing neutron behavior in a medium filled with nuclei. A

neutron is a subatomic particle called a baryon having the characteristic strong nuclear force ofthe standard model. Thus, a quantum mechanical description seems appropriate, leading to an

involved system of Schrdinger equations describing neutron motion between and within

nuclei. A neutron is also a relativistic particle with variation of its mass over time when

travelling near the speed of light. Finally, a neutron possesses wave and classical particle

properties simultaneously and therefore a collective description like that of Maxwells equationsalso seems appropriate. In reality, a neutron displays all of the above characteristics at one time

or another. When a neutron collides with a nucleus, its strong force interacts with all of the

individual nucleons. However, between nuclear collisions, neutrons move ballistically. Neutrons

with energies above 20 MeV with speeds of more than 20% the speed of light (c), exhibit

relativistic motion, but most in a reactor are rarely above 0.17c. The neutron wavelength is mostimportant for ultra-low-energy neutrons mainly existing in the laboratory. Fortunately, the


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classical neutral particle description with quantum mechanics describing collisions emerges as

the most appropriate for the investigation of neutron motion within a nuclear reactor.

We now derive the neutron Boltzmann equation, also called the neutron transport equation, to

characterize a relatively small number of neutrons colliding in a vast sea of nuclei.

Mathematically, a neutron is a neutral point particle, experiencing deflection from or capture by

a nucleus at the center of an atom. If conditions are just right, the captured neutron causes a

fissile nucleus to fission producing more neutrons. The statistically large number of neutrons

interacting in a reactor allows for a continuum-like description through averaging resulting in

the linear Boltzmann equation. Also, a statistical mechanical formulation, first attempted by

Boltzmann for interacting gases, provides an appropriate description. Boltzmanns equation,

based on physical arguments, such as finite particle size, gives a more physically precise picture

of particle-particle interaction than is presented here. In his description, all particles, including

nuclei, are in motion with like particle collisions allowed. Because of the low density of

neutrons in comparison to nuclei in a reactor however, these collisions are infrequent enabling a

simplified physical/mathematical theory.

Several forms of the neutron transport equation exist. The integro-differential formulation,

arguably the most popular form in neutron transport and reactor physics applications, is

presented in Section 1.2. However, other forms of the transport equation exist and are being

used such as integral and adjoint forms.

1.2.1 Integro-differential neutron Boltzmann equation

A primary goal of nuclear reactor design is the reliable prediction of neutron production and

loss rates. Predictions come from the solution of the neutron conservation equation - hence, the

importance of the neutron Boltzmann equation. Of the several possible physical descriptions,

we consider the neutron as a classical interacting particle and formulate neutron conservation in

a medium.

1.2.1.1 Independent variables

Figure 1.1(a) shows a neutron moving at the position denoted by the vector r. Figure 1.1(b)

shows a second vector in a coordinate system (dotted axes) superimposed on the neutron

itself to indicate neutron direction. The -vector specifies a point on the surface of a fictitious

sphere of unit radius surrounding the neutron and pointing in its direction of travel. The neutron

velocity is therefore, V v, where v is the neutron speed giving the classical neutron kinetic

energy:

for a neutron of mass m. Along with time, represented by t and measured from a reference time,

r,,E symbolize the six independent variables of the classical description of neutron motion.


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1.2.1.2 Dependent variables

For convenience, we combine the position and velocity vectors into a single generalized six-

dimensional vector P(r,V) defining the neutron phase space. Furthermore, decomposing the

velocity vector into speed and direction gives the equivalent representations, P = (r,,v) =

(r,,E). Note that a Jacobian transformation is required between the various phase space

transitions. The following derivation uses the latter representation exclusively. The derivationwill be a purely heuristic one, i.e. coming from physical intuition rather than from precise

mathematical rigor. In this regard, we largely follow the approach of Boltzmann.

The key element of the derivation is to maintain the point particle nature of a neutron while

taking advantage of the statistics of large numbers. Thus, point collisions occur in a statisticallyaveraged phase space continuum realized by defining the phase volume element

E rP aboutP as shown in Figure 1.2. Note that the unit of solid angle, , is thesteradian (sr), and that 4steradians account for all possible directions on a unit sphere.

The volume element of phase space P plays a central part in the overall neutron balance. Onephysically accounts for neutrons entering or leaving through the boundaries of r or that arelost or gained within r . In addition, neutrons enter or leave the phase element throughdeflection into or out of the direction range , or by slowing down or speeding up throughthe energy boundaries of E .

Let ),,,( tEN r be the neutron density distribution:

Since tv is the distance travelled per neutron of speed v, called the neutron track length, thetotal track length of all neutrons in P is:


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As shown, the track length serves as the bridge between point-like and continuum particle

behavior when characterizing collisions. In the above definition, it is convenient to define the

following quantity as the neutron angular flux at time t:

representing the total track length per second of all neutrons in P per unit of phase space.While the neutron density and flux are associated with a phase space volume element, the

neutron angular current vector, ),,,( tEJ r , is associated with an area. The magnitude of theangular current vector is the rate of neutrons per steradian, energy and area, in the direction

that pass through an area perpendicular to at time t. Neutrons, passing through the area A

oriented with the unit normal n , as shown in Figure 1.3, during t, enter the volume element

tA = v nr . Thus, the total number of neutrons in the phase volume element[ ] EtA = nP v is:

We then conveniently define the angular current vector to be:

such that EtEJ rn ),,,( is the rate of neutrons per area passing through an area of

orientation n .


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N, and J are called dependent variables since they depend on the independent variables of

phase space and time.

1.2.1.3 Nuclear data

Neutrons primarily experience scattering, capture and fission characterized through probabilities

represented by microscopic and macroscopic cross sections. In the following, we assume

neutrons interact with inert stationary nuclei. The macroscopic cross section for interaction of

type i with the nucleus of nuclidej is:

where i =scatter, capture,fission, absorption(= c +f)ij(r,E,t) is a product of a microscopic cross section

ij(E) which depends on the nuclidej and reaction type i and the nuclear atomic density,Nj(r,t),

which can vary spatially. The microscopic cross section nominally represents the area presented

to the neutron for an interaction to occur and is a measure of the probability of occurrence. The

cross section is fundamental to neutron transport theory and provides the essential element in

the continuum characterization of point collisions. Since, as previously shown, the total path

length of all neutrons in the element P is ttE Pr ),,,( , the number of interactions of

type i per time, or the reaction rate for interaction i in P with nuclide j is simply from thedefinition of cross section:

If we assume individual interactions to be independent events, the total interaction rate of type i

is the sum over allJparticipating nuclear species giving the total macroscopic cross section:

Note that the macroscopic interaction rate is independent of neutron direction and depends only

on neutron motion amongst interaction centers (the nuclei). When we next consider neutron

deflection or scattering, directional information is essential, however.


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The law of deflection, or the differential scattering kernel, is:

We mostly require, however, the differential scattering cross section for thejthnuclide:

Scattering, assumed rotationally invariant, depends only on the cosine of the angle between the

incoming ' and scattered neutron directions, ' . We normalize the scattering kernelsuch that neutrons must appear in some direction and energy range:

Assuming that the speed of light is an upper limit, the integration is actually over a finite range

[0,E0].

The scattering lawfsjis generally independent of time and position. The scattering cross section

for an individual nuclide, however, includes the atomic density, which can provide a time

variation of the differential scattering cross section.

Neutrons from fission are either prompt or delayed through subsequent nuclear decay and

emission. Here, we consider only prompt neutrons. The neutrons, produced isotropically, appear

in an energy and direction range according to the distribution )(E :

with normalization:

In addition, we shall require the average number of neutrons produced per fission, )(E .

With the notation now in place, an accounting of the total number of neutrons in the phase space

volume element P becomes our focus.


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1.2.1.4 Contributions to the total neutron balance

We now account for the net number of neutrons within an arbitrary volume V (shown in Figure

1.4) and in the partial phase space element E during a time t . In words, the neutronbalance in the entire volume V and partial element E is:

All contributions to the neutron balance, which we now consider separately, are in terms of the

dependent variables.

1.1.4.1.1 Total number of neutrons

The total number of neutrons in EV at times t and tt + is:


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1.1.4.1.2 Scattering gain

Neutrons from all points within V are scattered into the element E . The total numberscattered during t from any element '' ddE (at time t) is therefore the total scattering rate inV multiplied by t :

reach E . Thus, the total number scattering into E during t within V is theintegration over all possible contributions from all differential phase space elements '' ddE :

1.1.4.1.3 Fission production

The number of fissions occurring within V during t in any differential element '' ddE is:

For each fission, EEE

)'(4

)(

neutrons appear in E giving a total gain from

fission in EV of:

1.1.4.1.4 Losses from absorption and scattering

The number of neutrons lost due to absorption in EV is:


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Here, absorption refers to loss independently of whether a neutron causes a fission or not.

In addition, a neutron is lost when it scatters out of the direction range or energyrange E . The number lost from scattering out of EV is therefore:

1.1.4.1.5 Losses from streaming out of V

Consider the elemental area dA on the surface of V with outward normal sn (see Figure 1.4).

By definition of the current vector, the number of neutrons leaking out of dA from the

element E during t is:

Thus, over the entire surface of V, the total leakage is:

where the divergence theorem [2] has transformed the surface integration into a volume

integration.

1.2.1.4 Total balance: the neutron Boltzmann equation

Putting all contributions together, dividing by Et and taking the limit as t , , E approach 0 gives the overall balance in any partial element dEd over the entire volume attime t:


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where the total macroscopic and the differential scattering cross sections are:

Finally, because the volume is arbitrary and assuming a continuous integrand, the above integral

is zero only if the integrand is zero yielding the following neutron Boltzmann equation:

Eq.(1-1)

We include an external volume source rate of emission from non-flux related events,

),,,( tEQ r , for completeness.

Equation (1-1) is a linear integro-differential equation and as such requires boundary and initial

conditions. We will fashion these conditions after the particular benchmark considered.

For steady state problems we assume that all nuclear data to be time independent and:

Then through integration over all time, the steady state form of Eq. (1-1) becomes:


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Eq.(1-2)

where

The steady state source distribution ),,( EQ r now incorporates the initial condition.

The neutron transport equation given by Eq. (1-2) is the source of all the other transportequations in this compendium. Obviously, Eq. (1-2), being an integro-differential equation in

six independent variables, does not lend itself to simple analytical or numerical solutions.

Therefore, we require further simplification to enable analytical solution representations for

eventual numerical evaluation.

Monte Carlo and other sophisticated deterministic methods solve Eq. (1-2) numerically; but

these methods contain numerical and sampling errors. It is also possible to solve Eq. (1-2) in the

most analytically pure way possible. Most of the solutions come from other forms of Eq. (1-2)

that provide the necessary simplicity for further analytical and numerical investigation. We now

consider these forms.

1.2.2 Additional forms of the neutron transport equation

It is of interest to reformulate Eq. (1-2) in the various versions used for further analytical and

numerical investigation. There are at least eight (and probably more) equivalent forms of Eq. (1-

2) not counting Monte Carlo including:

integral;

even/odd parity;

slowing down kernel;

multiple collision;

invariant embedding;

singular integral;

Greens function;

pseudo flux.

Each form has a particular mathematical property facilitating a class of solutions. For example,

the multiple collision form is appropriate for highly absorbing media. Invariant embedding is

useful for half-space problems, whereas the pseudo flux form is appropriate for isotropic


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scattering in multi-dimensions. The Greens function representation is best suited for highly

heterogeneous 1-D plane media in the multi-group approximation.

In the following sections, we consider the integral transport equation, and derivative forms of

neutron transport equation such as monoenergetic and multi-group approximations in 3-D and

specifically in 1-D geometries, as they enable analytical and simple numerical solutions and are

ubiquitous in the literature.

1.2.2.1 The integral equation

The integral transport equation is essentially integration along the characteristic defining the

neutron path. If a neutron travels in direction , shown in Figure 1.5, between the points

specified by vectors 'r and r , the following relationship holds:

where s is the magnitude of the vector 'rr . Since the derivative along the neutron path is:

where the gradient is with respect to 'r , evaluation of Eq. (1-2) at 'r gives:

Eq.(1-3)

where the collision source is

q.(1-4)

Further, if we define the optical depth, or mean free path, as:

and assume we know ),,( Eq r , the solution to Eq. (1-3), viewed as a first order ordinary

differential equations, is:


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Eq.(1-5)

and becomes:

Eq.(1-6)

for srr=' on the boundary of the medium with outward normal surface sn shown in Figure 1.6.

Note that the distance to the point on the surface from the origin, )(rs ,depends on the

neutron direction. )( n s is the Heaviside step function representing only incoming

neutrons where ),,( Es r is known for all incoming directions. We do not consider re-entrant

boundaries. Therefore, Eqs. (1-6) and (1-4) constitute a Fredholm integral equation of the

second kind for the angular flux. We obtain the scalar, or angularly integrated, flux:

Eq.(1-7)

through integration of Eq. (1-5):


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q.(1-8)

By substitution of the differential volume element, ''' 2 dsdsd r= , Eq. (1-8) becomes:

Eq.(1-9)

Note that Eq. (1-9) is not an integral equation for the scalar flux since the scalar flux does not

generally appear under the integral. However, we can find an integral equation for the scalar

flux under certain circumstances. If scattering is isotropic and the source is isotropically

emitting, then:

and Eq. (1-9) reduces to:


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Eq.(1-10)

with

and

Equation (4) then gives the angular flux for the collision source:

Eq.(1-11)

If there is no flux passing through the boundary from the vacuum into the medium,

0),,( Es r , for 0


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Because of its simplicity, the monoenergetic approximation, also called one-group

approximation, is the most widely considered neutron transport model both analytically and

numerically. The model serves as a relatively simple test case to observe the efficiency of new

mathematical and numerical methods. The one-group approximation retains the fundamental

features of the neutron transport process yet lends itself to theoretical consideration.

As with all approximations, we begin with Eq. (1-2) rewritten here:

Eq.(1-2)

A physically based derivation gives the one-group form directly, while the multigroup form of

the next section straightforwardly produces the same result as a degenerate multigroup

approximation.

To obtain the one-group form, we assume neutrons scatter elastically from a nucleus of infinite

mass, and therefore do not lose energy. The details of elastic scattering are found in nearly

every reactor physics text; however, the derivation provided here is more mathematically and

physically intuitive than most.

An elastic collision between a neutron and nucleus of masses m andM respectively is one that

leaves the internal (binding) energy of the nucleus unchanged. The collision occurs

instantaneously between two mathematical point particles as shown in Figure 1.7(a) in the

physical laboratory (L) reference frame. In what follows, a primed quantity refers to before

collision and unprimed after collision. The vectors r(t) andR(t) define the neutron and nucleus

positions at time t and the vectorRcm(t):

Eq.(1-13)

defines the position of the center of mass in L whereA is the mass ratioM/m. To the colliding

particles of constant velocity, the center of mass remains stationary before and after collision.

For a nucleus initially at rest (V0) and a neutron moving with velocity vtoward the nucleus,the velocity of the center of mass in the laboratory reference frame is the time derivative of

Rcm(t):

Eq.(1-14)

The most convenient view of elastic scattering is from the center of mass reference frame (C)

created by imposing a velocity of Vcm on the laboratory reference frame shown in Figure1.7(b). The particle velocities in Cbefore collision are therefore:


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Eq.(1-15)

respectively, and the total linear momentum in C is zero, since:

Figure 1.7(c) shows the particle velocity vectors v, and V in L after collision. Again, if weimpose -Vcm, also shown, the corresponding velocities in C after collision result. Since the totalmomentum after collision in C is also zero from conservation:

the velocity of the nucleus in C after collision therefore is:

Eq.(1-16)

which is in the exact opposite direction of the neutron. Conservation of kinetic energy in C

requires:

and since 0AVvM 'c'

c

'

c =+= , /A-vV '

c

'

c = , we have:

Eq.(1-17)

Thus, the particles speed before and after collision remains unchanged in C.

The scattering angles made by the velocity vectors with respect to the incident neutron direction

in L and C, 0and c, and their corresponding cosines, 0, care also shown in Figure 1.7(c).For future reference, in terms of the neutron direction before and after collision , wedefine:

Eq.(1-18)

Because of the rotational invariance experienced by scattering of perfectly spherical point

particles, the collision dynamics will depend only on0.

To obtain a fundamental relationship between the energy before and after collision and the

scattering angle in C, we form the following expression:

to give using Eq. (1-14)


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Eq.(1-19)

where the collision parameter is:


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Eq.(1-20)

Since from Figure 1.7(c), we deduce that:

and using Eqs. (1-14) and (1-17), we can relate the scattering angle in the laboratory and center

of mass frames:

Eq.(1-21)

or rearranging:

Eq.(1-22)

Now recall the differential scattering kernel [defined just prior to Eq. (1-1)]:

Eq.(1-23)

with normalization:

Eq.(1-24)

Based on energy and momentum conservation, we express the law for scattering through an

angle 0in the laboratory frame, and correspondingly through angle cin the center of mass

frame as:

Eq.(1-25)

where the delta function enforces the correct kinematics for a given scattering angle. The firstfactor maintains the normalization with:


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Eq.(1-26)

From conservation of probability between the two reference frames:

Eq.(1-27)

the scattering kernel then becomes:

Eq.(1-28)

where, from Eq. (22):

Note that we introduce angular dependence in the laboratory representation through the

transformation between C and L coordinates.

Finally, when A , 1 Eq. (28) gives:

Eq.(1-29)

whith

Eq.(1-30)

Equation (1-29) confirms the intuitive notion that a neutron scatters elastically without energy

loss on collision with an infinite mass.

If, in addition, neutrons from fission and the external source appear only at energyE0,then:

Eq.(1-31)


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When we introduce Eqs. (1-29) and (1-31) into Eq. (1-2) after integrating over the delta function

in the collision term, there results:

Eq.(1-32)

Since all scattered and fission source neutrons appear at a single energyE0, physical consistency

requires neutrons to be at that energy or expressed formally:

Eq.(1-33)

where mathematical consistency requires:

When we introduced Eq. (1-33) into Eq. (1-32), the one-group approximation emerges:

Eq.(1-34)

Energy is now just a passive parameter, which we can ignore.

In three dimensions, Eq. (1-34) is still mathematically intractable so we need to make additional

simplifications to realize analytical solutions.

1.2.3.2 Multigroup approximation

We initiate the multigroup approximation to Eq. (1-2) by partitioning the total energy interval of

interest into G energy groups:

whereg = 1 is the highest group. By expressing the energy integrals of the scattering and fission

sources as sums over all groups, without loss of generality, the integrals in Eq. (1-2) become:


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and:

By integrating Eq. (2) over Eg, we therefore have:

Eq.(1-35)

At this point, we introduce the multigroup approximation:

Eq.(1-36)

stating the flux and source within each group, gEE are separable functions of energy and r,

. The following normalizations:

Eq.(1-37)

are also required with f(E)and g(E)assumed to be piecewise smooth functions of E. Then, withthe multigroup approximation, Eq. (1-35) becomes:

Eq.(1-38)


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with the following definitions of the group parameters:

Eq.(1-39)

In more convenient vector notation, Eq. (1-38) is:

Eq.(1-40)

with the group flux and source vectors:

and the group constants:

Eq.(1-41)

The multigroup approximation is the basis of nearly all reactor physics codes, making it one of

most widely used approximations of neutron transport and diffusion theory. For inner/outer

iterative numerical algorithms, one can reformulate the multigroup approximation as a series of

one-group equations.


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1.2.3.3One-dimensional plane symmetry

We obtain the 1-D transport equation for plane symmetry when the medium is transversely

infinite (in theyzplane) with cross section and source variation only in thex direction. For this

case, Eq. (1-2) becomes:

Eq.(1-42)

When we integrate Eq. (1-42) over the transverseyzplane, there results:

Eq.(1-43)

Now:

and thex-direction (cosine) is:

Since the differential scattering cross section is rotationally invariant with respect to the angle of

scattering:

Eq.(1-44)

where we explicitly show the dependence on the polar and azimuthal angles of the particle

directions (see Figure 1.1). Upon integration over the azimuth of the scattered direction:


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Eq.(1-45)

Eq. (1-43) becomes:

Eq.(1-46)

with:

Commonly, we express the scattering kernel as in a Legendre expansion representation:

Eq.(1-47)

where the scattering coefficients are (from orthogonality):

Eq.(1-48)

Equation (1-45) then becomes:

Eq.(1-49)

We have applied the addition theorem for Legendre polynomials, in the form:


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Eq.(1-50)

to obtain Eq. (1-49). With Eq. (1-49), Eq. (1-46) becomes:

Eq.(1-51)

where the lth

Legendre moment of the angular flux is:

Similarly, the multigroup approximation [Eq. (1-51)] for plane geometry is:

Eq.(1-52)

with:

and


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1.3 The Neutron Kinetics Equations

The neutron transport equation without delayed neutrons,

',E',t),r('d,E')rdE'(E)

,E,t),r()'E,,E'r(ddE',E,t),r(q

,E,t),r(,E)r(t

sex

+

+=

++

(

'

1

rr

rrr

rr

Eq.(1-53)

assumes that all neurons are emitted instantaneously at the time of fission. In fact, small fraction

of neutrons is emitted later due to certain fission products. To ascertain the time-dependent

behavior of a nuclear reactor, one must account for the emission of these so-called delayed

neutrons, for the small fraction of neutrons that are delayed make the chain reaction far more

sluggish under most conditions than would first appear.

To derive the neutron kinetics equations, we return to the time-dependent equation

),,(),,,(),,,(

),,,(),(),,,(),,,(1

tErqtErqtErq

tErErtErtErt

fexs

rrr

rrr

r

r

++=

++

Eq.(1-54)

When delayed neutrons are considered, we express the fission contribution to the emission

density as

{ {

neutronsdelayedfor

densityemission

neutronspromptfor

densityemission

dpf qqq += Eq.(1-55)

To represent qp, we assume that for theithfissionable isotope, a fraction iof the fission

neutrons is delayed. This delayed fraction is small, ranging from 0.0065 for 235U to about

0.0022 for 239Pu. The fraction of prompt neutrons must then be (1-i).

The rate of prompt fission neutrons produced per unit volume from isotope iis just

)',()',(')1( ErErdEi

f

i rr Eq.(1-56)

To obtain the rate at which prompt neutrons are produced we sum over the isotope index iyielding

=i

i

f

i

pp ErErdEEtErq )',()',(')1()(),,( rrr

Eq.(1-57)


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The delayed neutrons arise from the decay of variety of fission products. For neutron kinetics

considerations these are usually divided into six groups, each with a characteristic decay

constantl, for each fissionable isotope i, with a characteristic yield il, where

=l

i

l

i Eq.(1-58)

If the concentration of the fission product precursors of type lis ),( ErClr

per unit volume, then

the number of delayed neutrons produced by precursor type lper unit time per unit volume will

just be ),( trCllr

. Now suppose that the probability that the delayed neutrons from precursor

type lwill have an energy betweenEandE+dEisl(E)dE. We find the contribution of the

delayed neutrons to the emission density to be

)',()(),,( ErCEtErq lll

ld

rr

= Eq.(1-59)

Hence,

)',()(

)',()',(')1()(

'',

1

ErCE

ErErdEE

,t),E,r()'E,,E'r('ddE't),E,r(q

,E,t),r(,E)r(t

ll

l

l

i

i

f

i

p

sex

r

rr

rrr

rr

+

+

+=

++

Eq.(1-60)

Before this equation may be solved, an additional equation must be available for each precursor

concentration. Since each precursor emits one delayed neutron, the rate of production of

precursor type lis

i

i

f

i

l tErErdE ),',()',(' rr

Eq.(1-61)

while the decay rate is ),( trCllr

. Hence the net rate of change in the precursor concentration is

),(),',()',('),( trCtErErdEtrCt

ll

i

i

f

i

ll

rrrr

= Eq.(1-62)

The Equations 60 and 61 are referred to as the neutron kinetics equations.

The foregoing equations may be generalized along two lines that are worthy to mention. First,

we have made the assumption that the prompt and delayed spectrai

p andi

l are independent


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of the isotope producing the fission. This ansatz is invariably employed in problems where

delayed neutrons are ignored, since the neglected deviations are insignificant. In kinetics

problems neglect of the isotope dependence of the delayed neutron spectra may sometimes lead

to significant errors. It is then necessary to derive somewhat more general forms of the kinetics

equations in which isotope dependent prompt and delayed spectra are used. Since this

generalization causes no generic changes in computational methods, the derivation of the

equations is left as an exercise.

Kinetics equations also frequently involve time-dependent cross-sections; a re-derivation of the

foregoing equations would then simply introduce the time parameter tin each of the cross-

sections appearing in the neutron kinetics equations. In kinetics calculations the time-

dependence would be predetermined as a function of time. This would be the case, for example,

if a control rod were to be inserted into a reactor core at a specified rate.

More generally, changes in cross-sections resulting from thermal-hydraulic or other feedback

mechanisms are induced by the fact that the heat of fission causes the temperature in the reactor

to change; this in turn results in changes in the number densities and therefore the macroscopic

cross-sections appearing in the kinetics equations. The inclusion of these feedback effects for

so-called reactor dynamics calculations is outside the scope of this text. However, we note that

in most dynamics calculations the kinetics equations are treated as a separate module into which

the time-dependent cross-sections are input.

1.4 The Time-Independent Transport Equation

The vast majority of numerical transport equations are carried out using a form of the equations

in which time dependence is not treated explicitly. For non-multiplying systems in which the

external sources are time-independent and nonnegative there will always be nonnegative

solutions to the time-independent form of the transport equation

)'()(')()()]([ ,E,r'E,,E'rddE',E,rq,E,r,Er sex +=+ rrrrr

Eq.(1-63)

if the boundary conditions are also time-independent. For multiplying systems, the situation isless straightforward, for the critical state of system must be considered.

1.4.1 Fixed Source and Eigenvalue Problems

Physically, a system containing fissionable material is said to be critical if there is a self-

sustaining time-independent chain reaction in the absence of external source of neutrons. That is

to say, if neutrons are inserted into a critical system, then after sufficient time has elapsed for

the decay of transient effects, a time-independent asymptotic distribution of neutrons will exists

in which the rate of fission neutron production is just equal to the losses due absorption and

leakage from the system. If such an equilibrium cannot be established, the asymptotic

distribution of neutrons called fundamental mode will not be steady state and will either

increase or decrease with time exponentially as indicated in Figure 1.8. The system then is said

to be supercritical or subcritical, respectively. If neutrons from a time-independent externalsource are supplied, a subcritical system will eventually come to an equilibrium state

characterized by a time-independent neutron flux distribution in which the production rate of

external and fission neutrons is in equilibrium with the absorption and leakage from the system.

However, if the system is critical or supercritical, no such equilibrium can exist in the presence

of an external source, and the neutron flux distribution will be an increasing function of time.

This flux behavior is qualitatively depicted in Figure 1.9.


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Figure 1.8.Time dependence of the flux for a source-free multiplying medium

Figure 1.9Time dependence of the flux a multiplying medium with a known source

Mathematically, a system is critical if a time-independent nonnegative solution to the source-

free transport equation can be found. Hence, from the transport equation without delayed

neutrons, Eq. 53, one must have a solution to


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Vr',E',rd,E'rdE'E

,E,r'E,,E'rddE'

,E,r,Er

f

s

+

=

+

rrr

rr

rr

),(')()(

)'()('

)()]([

Eq.(1-64)

with appropriate boundary conditions, for example,


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)(')()(

)'()('

)()(

',E',rd,E'rdE'E

,E,r'E,,E'rddE'

,E,r,Er

f

s

+

=

++

rr

rr

rr

Eq.(69)

In general there will be a spectrum of eigenvalues for which the equation has a solution.

However, at long times, only nonnegative solutions corresponding to the largest real component

of will predominate. From our physical definition of criticality we have, for 0, the eigenvalue

with the largest real part,

.subritical0

critical,0

cal,supercriti0

Re 0

Moreover, the magnitude of Re[0] provides a measure of how far from criticality the system is.

1.4.1.2 The k Eigenvalue

The k eigenvalue form of the criticality problem is formulated by assuming that , the average

number of neutrons per fission, can be adjusted to obtain a time-independent solution to Eq. 1-

64. Hence we replace by /kand write

)(')()(

)'()('

)()(

',E',rd,E'rdE'kE

,E,r'E,,E'rddE'

,E,r,Er

f

s

+

=

++

rr

rr

rr

Eq.(1-70)

Once again we have an eigenvalue problem. From our physical definition of criticality it is

apparent that any chain reaction can be made critical if the number of neutrons per fission can

be adjusted between zero and infinity. Thus there will be a largest value of k, for which a

nonnegative fundamental mode solution can be found. Clearly the system is critical if thislargest value of is k = 1. A value of k < 1implies that the hypothetical number of neutrons per

fission, /k, required to make the system just critical is larger than , the number available in

reality. Hence the system is subcritical. Conversely, for k > 1, fewer neutrons per fission are

required to make the system critical than are produced in reality, and thus the system is

supercritical. To summarize,

.subritical1

critical,1

cal,supercriti1

k

For most reactor criticality calculations the k rather than eigenvalue is used for several

reasons. First, as indicated by Eq. 69, the eigenvalue equation has the same form as the time-

independent transport equation but with /added to the absorption cross-section. For this


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reason this formulation is normally referred to as the time-absorption eigenvalue. If the system

is subcritical, however, will be negative, and situation is likely to arise, such as in voided

regions where the pseudo total cross-section that include/term will be negative. Such

negative cross-sections are problematic for many of the numerical algorithms used to solve the

transport equation.

A second drawback arises from the fact that the eigenvalue causes the removal of neutronsfrom the system to be inversely proportional to their speed. Hence for supercritical systems /

absorption will cause more neutrons to be removed from low energies and result in a neutron

energy spectrum that is shifted upward in energy compared to that of a just critical system with

= 0. Conversely, in a subcritical system the negative value of causes neutrons to be

preferentially added at lower energies, causing a downward shift in the energy spectrum. In

many calculations where it is desired to know not only the critical state of the system but also

the spectrum that the system would have if it were critical, these spectral shifts are undesirable.

Moreover, such shifts are insignificant when the keigenvalue formulation is used.

A third advantage of keigenvalue formulation lies in the physical interpretations of the

eigenvalue and the iterative technique derived from it. The technique is a powerful method for

carrying out criticality calculations. Also, kmay be interpreted as the asymptotic ratio of the

number of neutrons in one generation and the number in the next.

1.5The Adjoint Transport Equation

For each of the foregoing forms of the transport equation a related adjoint equation may be

formulated. Solutions for such adjoint equations may serve a variety of purposes. For fixed

source problems they may be used to expedite certain classes of calculations, particularly when

Monte Carlo methods are used. For eigenvalue problems probably the most extensive use of

adjoint solution is in perturbation theory estimates for changes in the neutron multiplication

caused by small changes in material properties.

In this section we first discuss the properties of adjoint operators and develop an adjoint

equation for non-multiplying systems. After demonstrating some uses of the resulting adjoint

equation, we turn to the criticality problem in order to formulate an adjoint equation for a

multiplying system and demonstrate its use for calculating perturbations in the multiplication.

1.5.1 Non-multiplying Systems

To begin, suppose that we have an operatorHand any pair of functions and +that meet

appropriate boundary and continuity conditions. For real-valued functions the adjoint operator

H+is defined by the identity

+++ = HH Eq.(1-71)

where signifies integration over all of the independent variables. IfH = H+, the operator is

said to be self-adjoint; otherwise it is non-self-adjoint.

For the time-independent non-multiplying transport problem, given by Eq. 1-63, the transport

operator may be given by

)'()(')()]([ ,E,r'E,,E'rddE',E,r,ErH s += rrrr

Eq.(1-72)


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and the integration over the independent variable is

= dEddV Eq.(1-73)

The spatial domain of the problem Vis bounded by a surface across which we shall assume

that no particles enter (i.e., vacuum boundary conditions). Hence the functions upon whichH

operates are constrained to satisfy


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In the collision term of Eq. 1-75 we simply note that++ = , while in the scattering

term we interchange dummy variablesEandEas well as and ' to obtain

)''()''('')(

)''()('')(

,E,r,E,ErdEd,E,rdEd

,E,r'E,,E'rdEd,E,rdEd

s

s

=

+

+

rrr

rrr

Eq.(1-80)

Substituting the three preceding expressions into Eq. 1-75 we have

)''()''('

)]()()([)(

,E,r,E,ErddE'

,E,r,Er,E,r,E,rdEddVH

s

+=

+

+++

rr

rrrr

Eq.(1-81)

Hence Eq. 1-71 will be satisfied provided we define the adjoint transport operator as

)''()''(')()]([ ,E,r,E,ErddE',E,r,ErH s += ++++

rrrr

Eq.(1-82)

To illustrate the use of this adjoint identity suppose we solve a transport problem with a known

external source distribution qex,

exqH = Eq.(1-83)

where meets the boundary conditions of Eq. 1-74. Suppose further that we would like to know

the response of a small detector with a total cross-section dand a volume Vd, represented by the

reaction rate localized at point drr

,

= ),()( ErEdEVR ddd

r

Eq.(1-84)

We define the adjoint 1-problem as

)( ddd rrVH rr

=++ Eq.(1-85)

where the adjoint flux +is required to satisfy the boundary condition of Eq. 1-78. Now

multiply Eq. 1-83 by + and integrate over the independent variables to obtain

exqH +++ = Eq.(1-86)

Similarly, multiply Eq. 1-85 by and integrate to yield

RrrVH ddd ==++ )(

rr

Eq.(1-87)

Taking the difference of these two expressions yields

RqHH ex = ++++ Eq.(1-88)


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Replace and +with and +in the adjoint identity, Eq. 1-81. The left-hand side of Eq. 1-88

then vanishes to yield the desired result:

exqR += Eq.(1-89)

From this illustration, we see that the detector response is just given by the volume integral ofthe adjoint weighted source distribution. In particular, if we assume that the particles are emitted

at 0rr

in direction 0 at energyE0at a rate of one per second,

)()()( 000 = EErrqexrr

Eq.(1-90)

and obtain

),,( 000 ErR = + r Eq.(1-91)

This provides us with a physical interpretation of the adjoint flux defined by Eq. 1-85: it is justthe importance of particles produced at 000 ,

, Er r

to the detector response. The concept of

particle importance is elaborated elsewhere for a broad range of neutronics problems.

The foregoing methodology has widespread applications, particularly in Monte Carlo

calculations.

The foregoing formalism may be applied by slight modification to external surface source of

particles, or to calculate distributions of particles leaving the problem domain. For example,

suppose that qex= 0within Vbut that particles enter the volume across . Then we would have

0=H Eq.(1-92)

with the boundary condition


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= + 0for Vrr

, obeying the vacuum

boundary conditions. Likewise the adjoint equation is

01

= ++++ Gk

H Eq.(1-101)

where +is the fundamental mode solution +> 0for Vrr

, obeying the boundary conditionsof Eq. 1-78. If we multiply Eq. 1-100 by +, Eq. 1-101 by , and integrate the difference over

the independent variables, then

011

=+ +++

++++ Gk

Gk

HH Eq.(1-102)

After simplifying this expression with identities, Eq. 1-71 and Eq. 1-98, we have


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011

=

+

+ G

kk Eq.(1-103)

It is easily shown in Eq. 103 that if +and + are positive, the integral on the left will notvanish. Thus for this equation to hold the eigenvalues of the transport equation and its adjoint

must be equal:

+= kk Eq.(1-104)

We are now prepared to use the adjoint operators to calculate changes in reactivity, where

reactivity is defined by

k

k 1= Eq.(1-105)

For small changes, k, in k, we have

2

0k

k= Eq.(1-106)

where k0is unperturbed value.

For the unperturbed rector we define the fundamental mode equations as

01

00

0

00 = Gk

H Eq.(1-107)

01

00

0

00 = ++

+

++ Gk

H Eq.(1-108)

Let Eq. 1-100 be used to represent the perturbed state. Then for small changes we represent the

perturbations by

+=+=

+=+=

00

00

,

,

kkk

GGGHHH Eq.(1-109)

Substituting these expressions into Eq. 1-100, we have

( )( )( )

( )( ) 01

00

0

00 =+++

++

GGkk

HH Eq.(1-110)

For small perturbations we may disregard terms of order 2. Thus, using


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( ) ( )

+=

+ 000001

1

1

11

k

k

kkkkkk

Eq.(1-111)

and regrouping, we obtain from Eq. 1-110

+

+

0

0

00

0

00

0

0002

0

111G

kHG

kHG

kHG

k

k Eq.(1-112)

where 2terms are deleted. Comparing this expression with Eq. 1-107, we see that first term on

the right vanishes. Moreover, if we multiply by+0 and integrate over the independent

variables, we have

)()( 01

0000

1

000002

0

GkHGkHGk

k +++ + Eq.(1-113)

We may now use the properties of the adjoint operators to eliminate the last term. If in Eq. 1-71

and Eq. 1-98 we take =and +=0+, then we obtain

++++ = 001

000

1

000 )()( GkHGkH Eq.(1-114)

which , according to Eq. 1-108, must vanish. Hence

000

0

1

00

20

)(

G

HGk

k

k

+

+

= Eq.(1-115)

This is powerful result, for it allows us to estimate many different small reactivity changes using

only two unperturbed solutions, 0 and 0+, and the cross-section changes that appear as H

and G.

Nutron transport Equation

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