Some bounds for the effective multiplication factor

annals ofUCLEAR ENERGY
N
www.elsevier.com/locate/anucene

Annals of Nuclear Energy 31 (2004) 1207–1218

Some bounds for the effectivemultiplication factor

Richard Sanchez

Commissariat �a l’Energie Atomique, Direction de l’Energie Nucl�eaire, Service d’Etudes de R�eacteurs etde Mod�elisation Avanc�ee, CEA de Saclay, France

Received 20 February 2004; accepted 12 March 2004

Available online 14 April 2004

Abstract

The effective multiplication factor (keff ) is the dominant eigenvalue for the stationary linear

particle transport problem. Using positivity, global balance and a general duality formula

bounds for the keff are derived in terms of the reactivity of the medium, perturbation of cross-

sections and change of the size of the domain and/or its surface albedo. A different bound is

also obtained with the help of a local keff value that is solution of an adjoint source problem.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

The stationary linear transport equation (Bell and Glasstone, 1970; Williams,1971) gives an accurate description of the steady-state neutron distribution in a

nuclear reactor. The angular flux solution of the homogeneous transport equation is

always zero except for a particular size of a neutron multiplying medium, called

critical size. To characterize how far from criticality is a multiplying medium of fixed

size it is customary to introduce a critical parameter that divides the production term

in the transport equation, artificially increasing or decreasing the number of neu-

trons produced from fission. The introduction of this critical parameter transforms

the equation in an eigenvalue problem. The effective multiplication constant (keff ) isthe dominant eigenvalue that is also characterized for having an everywhere positive

solution and describes, hence, the only physical solution. The main aim of this paper

E-mail address: [email protected] (R. Sanchez).

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1208 R. Sanchez / Annals of Nuclear Energy 31 (2004) 1207–1218

is to establish bounds for the effective multiplication factor for several physical

configurations. Although most physicists would accept these results on the basis of

physical intuition, I will show that all there is needed in the proofs is the positivity of

the angular flux and a general duality relation between a pair of direct and adjoint

transport equations. This last relation is also used in perturbation theory and

therefore is familiar to physicists. On the other hand, the proof of the existence of aleading positive eigenvalue with a positive eigenvector (the fundamental mode) re-

quires techniques of functional analysis and is much harder to establish (Vladimirov,

1963; Krasnoselkii, 1964; Kaper et al., 1982; Dautray and Lions, 1984; Greenberg

et al., 1987; Mokhtar-Kharroubi, 1997; Agoshkov, 1998).

This paper is organized as follows. The eigenvalue problem in linear stationary

transport is described in Section 2, where I introduce physical assumptions on the

cross-sections, basically boundness and positivity. In this section I also present a

general result on duality that has been obtained elsewhere (Sanchez, 1998). Positivityof the fundamental mode and duality are used in Section 3 to establish bounds for

the keff in different physical situations. First, I use global balance to show that the keffcannot exceed the maximum value of m, the mean number of neutrons produced by a

fission event, a result that is obviously intuitive. A small difficulty here is the fact that

neutron multiplication by ‘scattering’ (n; 2n) and (n; 3n) at high-energy is not ac-

counted for by the eigenvalue, and one has to make sure that radiative capture

compensates for the extra neutron multiplication. In Section 3 I also obtain a general

perturbation formula that relates the keff to cross-section perturbations, includingalbedo perturbations. Next, I consider the effect of a modification of the size of a

domain, either by adding material on the periphery of the domain of by increasing

the value of the albedo, and prove the intuitive expected result that the keff ought toincrease or remain constant. A similar result has been already established by Ar-

zhanov (2002) for the case with vacuum boundary conditions. In the last part of the

section, I introduce a local multiplication factor keffðxÞ and show that the keff can be

obtained from the local keff and from the fission source distribution and that,

therefore, the maximum and minimum values of the local keff are natural bound forthe keff . Since the former obeys a source adjoint equation it is then possible, in

principle, to compute these bounds. Section 4 is dedicated to a discussion of the

solvability of the source problem. This material is classical and results from an ap-

plication of Fredholm’s alternative and I have added it for completeness. However, I

have generalized the analysis to include a general albedo condition and chosen to

analyze solvability by using positivity and duality and the fact that the inverse of

the transport operator admits a Neumann expansion, that is, a multiple collision

expansion. Finally, a summary of the results and some conclusions are given inSection 5.

2. The eigenvalue problem

The eigenvalue problem for the stationary transport equation can be stated as

follows:

R. Sanchez / Annals of Nuclear Energy 31 (2004) 1207–1218 1209

Bw ¼ 1k Pw; in X ;

w ¼ bw; on C�;

�ð1Þ

where x ¼ ðr; vÞ indicates a point in phase space X ¼ fx; r 2 D; v 2 V g, D and V are

the geometrical and velocity domains, and wðxÞ is the angular flux at x. The transportoperator B � L� H comprises the streaming and the scattering operators, L and H .

These two operators and the production operator P are defined as follows:

L � X � r þ RðxÞ;ðHwÞðxÞ ¼

RX dyRsðy ! xÞwðyÞ;

ðPwÞðxÞ ¼RX dyRfðy ! xÞwðyÞ;

8<: ð2Þ

where X ¼ v=v is the angular direction of the particle and dx ¼ drdv is the ordinary

Lebesgue measure in phase space.

The boundary condition of Eq. (1) is defined by a general albedo operator

(Sanchez et al., 2002),

ðbwÞðxÞ ¼ZCþ

dbybðy ! xÞwðyÞ; x 2 C�; ð3Þ

where, with n denoting the outward normal to the surface oD of D, C� ¼fx; r 2 oD; v 2 V ; v � n < 0g is the entering boundary of X and the integration is done

over the exiting boundary of X , Cþ ¼ fx; r 2 oD; v 2 V ; v � n > 0g, with the measure

dbx ¼ jX � njdS dv. The fraction of particles reentering the domain for one particle

leaving at x 2 Cþ is characterized by the local albedo value

bðxÞ ¼ZC�

dbybðx ! yÞ; x 2 Cþ: ð4Þ

For easy of notation I have considered that collisions, productions and reentering,

as expressed by the kernels of operators H , P and b, are not localized. The usual

assumption in particle transport is that collisions and productions are localized. For

instance, for localized collisions the kernel of the collision operator contains a delta

function in space, Rsðy ! xÞ ! dðry � rÞRsðvy ! vÞ; and the scattering operator

becomes an integral operator over only the velocity variable. Thus, localized colli-sions are characterized by the familiar collision operator ðHwÞðxÞ ¼

RV dvyRsðr; vy !

vÞwðr; vyÞ. All the results will apply, of course, to the restricted case of localized

collisions and productions. I also write the total cross-section RðxÞ as a function of

space and velocity and make no restrictive assumption on the angular dependence of

the scattering kernel so the results apply to non-isotropic media as well.

In this work I will only use simple physical assumptions on the cross-sections such

as positivity and boundness. With the usual notation R ¼ Rs þ Ra with Ra ¼ Rc þ Rf ,

where Rs represents the cross-section for all scattering events including productionby (n; 2n), (n; 3n) and any other multiplicative interaction with the exclusion of fis-

sion, Rc comprises all type of reactions with no reemission of neutrons and Rf is the

fission cross-section. I will also introduce a physically meaningful factorization of the

scattering and production kernels as follows:


Rsðx ! yÞ ¼ cðxÞRðxÞPsðx ! yÞ;Rfðx ! yÞ ¼ tðxÞRfðxÞPfðx ! yÞ;

�ð5Þ

where all the quantities are positive, c is the number of secondaries per collision

excluding fissions and t the mean number of neutrons emitted by fission. Notice that

because of multiplication by (n; 2n) and other reactions cðxÞRðxÞ ¼ nðxÞRsðxÞ, wherenðxÞ, the mean number of particles produced by a ‘scattering’ event, may be greaterthan 1. Transfer kernels Ps and Pf give the density of probability for the distribution

of particles produced by scattering and fission. Also, according to (4) the albedo

kernel can be factorized as

bðx ! yÞ ¼ bðxÞPbðx ! yÞ: ð6Þ
All these kernels are positive and normalized to unity Z
XdyPsðx ! yÞ ¼

ZXdyPfðx ! yÞ ¼

ZC�

dbyPbðx ! yÞ ¼ 1: ð7Þ

2.1. Duality

Duality theory offers a general approach to obtain bounds for the keff eigenvalue.I recall here a general result obtained in Sanchez (1998). The starting point is a pairof direct and adjoint transport equations of the form

Bw ¼ S; in X ;w ¼ bwþ win; on C�;

�;

Bywy ¼ Sy; in X ;wy ¼ bywy þ wy

out; on Cþ;

�ð8Þ

where By � �X � r þ R� H y and by are, respectively, the adjoints of the transport

operator B � X � r þ R� H and of the albedo operator b for the scalar products

ðf ; gÞ ¼ZXdxðfgÞðxÞ; hf ; gi� ¼

ZC�

dbxðfgÞðxÞ:

Notice that hf ; bgi� ¼ hbyf ; giþ, where hf ; giþ ¼RCþ

dbxðfgÞðxÞ is the scalar producton the space of functions defined on the entering boundary. The adjoint albedo

operator,

ðbywyÞðxÞ ¼ZC�

dbybyðy ! xÞwyðyÞ; for x 2 Cþ;

sends incoming particles onto outgoing ones and its kernel is the dual of the directalbedo kernel, byðy ! xÞ ¼ bðx ! yÞ.

A general duality result can be established for the pair of transport Eq. (8). The

solutions w and wy of the direct and adjoint problems satisfy the general Green’s

formula (Sanchez, 1998):

ðwy; SÞ þ hwy;wini� ¼ ðSy;wÞ þ hwyout;wiþ: ð9Þ

This formula can be viewed as a generalized conservation relation between the direct

and the adjoint problem: the measure with weight wy of the sources of the direct

problem equals the measure with weight w of the sources of the adjoint problem.


I end this part with a remark on the ‘physical’ interpretation of the adjoint flux.

The adjoint transport equation describes a ‘world’ where particles move backward

and collisions and fission are inverted, Rysðx ! yÞ ¼ Rsðy ! xÞ and Ry

f ðx ! yÞ ¼Rfðy ! xÞ, that is, these processes send the exiting states into the incoming states of

the real world. In fact, the adjoint flux is a mathematical artifact and can only be

interpreted as an ‘importance’ function via the role that it plays in duality formula(9). However, trajectories are mechanically reversible and this reversibility is build

into the transport equation. If, furthermore, the medium is isotropic (i.e., its prop-

erties are invariant by rotations), then the angular part of the collision and pro-

duction operators is physically reversible. In one group transport theory this leads to

the well-known reciprocity theorem (Sanchez and Santandrea, 2001) which is an

essential property of transport.

3. Bounds on the keff

In this section I establish bounds for the keff value under several conditions. First,I exploit the fact that the keff is the critical parameter that helps equilibrate global

production with global absorption and leakage to derive a basic bound for the ef-

fective multiplication factor. Then I use classical perturbation theory to analyze the

effects of cross-sections perturbations and of a perturbation of the size of the domain

and/or its surface albedo. Finally, a new approach is obtained by writing the keff asan average of a local keff value, where the latter is shown to be the total neutron

production due to a localized source. The first result is solely based on the positivity

of the fundamental mode, all other results use also the positivity of the associated

adjoint flux and the duality relation between direct and adjoint fluxes.

3.1. A basic result from global particle conservation

Consider transport Eq. (1) for the dominant eigenvalue, keff ¼ max k. I shall as-sume here without further proof that this eigenvalue is real, simple and positive and

its associated eigenvector, i.e., the fundamental mode of the medium, is also positive.

The proof of this is discussed in the literature (Vladimirov, 1963; Kaper et al., 1982;

Dautray and Lions, 1984; Greenberg et al., 1987; Mokhtar-Kharroubi, 1997; Ag-

oshkov, 1998). By integration of Eq. (1) over phase space one obtains
ZCþ
dbx½1� bðxÞ�wðxÞ þZXdxRaðxÞwðxÞ ¼

1

keff

ZXdxðmRfÞðxÞwðxÞ; ð10Þ

where I have used (5)–(7), and Ra ¼ Ra � ðn� 1ÞRs is the effective absorption cross-

section. This equation expresses the global balance of particles in the domain. By

writing the equation as Lþ A ¼ ð1=keffÞF , where L, A and F stand, respectively, for

leakage, absorption and production from fission, I can write

keff ¼F

Lþ A6

FA6

FAF

¼RXF

dxðmRfÞðxÞwðxÞRXF

dxRaðxÞwðxÞ6 max

Xk1ðxÞ; ð11Þ


where k1 ¼ mRf=Ra 6 m is the local infinite-medium multiplication constant. The first

inequality is true only if the boundary does not create new particles, bmax ¼maxCþ bðxÞ6 1. The term AF indicates the integral of the effective absorption term

over the part of phase space, XF � X , where there is fission (Rf > 0). This chain of

inequalities makes sense only for RaðxÞP 0, that is, when the number of secondaries

per collision cðxÞ ¼ nRs=R is smaller than 1. Notice also that the upper boundkeff ¼ k1;max is reached only for a perfectly reflected domain containing a homoge-

neous material.

For a perfectly reflected domain the leakage term vanishes and (10) can be written

as
ZXdx
1

keffðmRfÞðxÞ

�� RaðxÞ

�wðxÞ ¼ 0:

Then, by using the positivity of the fundamental mode, w > 0, this equation entails

the following inequalities:

maxXF ½ 1keff

mRfðxÞ � RaðxÞ� > 0;

minX ½ 1keff

mRfðxÞ � RaðxÞ� < 0;

(ð12Þ

where clearly the maximum is reached in the fission domain XF and, if there is a part

of X without fission, i.e., if XF � X , then the minimum must happen there.

3.2. Cross-section perturbations

I analyze now the effects of cross-section perturbations on the value of the keff .The basic tools to establish keff bounds are duality formula (9) and the adjoint ei-

genvalue equation

Bywy ¼ 1

kyP ywy; in X ;

wy ¼ bywy; on Cþ;

(ð13Þ

for the leading eigenvalue.

As a first application of (9) I will use the positivity of the solutions to show that

the leading eigenvalue of the adjoint transport equations is equal to the leading ei-genvalue of the direct equation. In this case win ¼ wy

out ¼ 0, S ¼ ð1=keffÞPw,Sy ¼ ð1=kyeffÞP ywy and Eq. (9) yields

1

keff

� 1

kyeff

!ðwy; PwÞ ¼ 0:

Hence, from the positivity of the fundamental modes w and wy one concludes that

kyeff ¼ keff .Next, I analyze the effect of a general perturbation of cross-sections and albedo on

the main eigenvalue. By writing the perturbed equation in the form


Bw0 ¼ 1k0eff

P 0w0 � dBw0; in X ;

w0 ¼ bw0 þ dbw0; on C�;

(

where the prime indicates the perturbed quantities, P 0 ¼ P þ dP , dB � dR� dH is the

perturbation of B and db is the perturbation of albedo operator b, one can use (9)

with w ¼ w0, win ¼ dbw0, S ¼ ½ð1=k0effÞP 0 � dB�w0 and Sy ¼ 1keff

P ywy and wyout ¼ 0 to

obtain

1

keff� 1

k0eff¼

ðwy; 1keff

dP þ dH � dRh i

w0Þ þ hwy; dbw0i�ðwy; P 0w0Þ

: ð14Þ

The detailed knowledge of the perturbation and the positivity of the direct and

adjoint fluxes can now be used to ascertain whether k0eff is greater or smaller than keff ,providing thus a bound for k0eff . To facilitate the analysis it can be assumed that the

perturbations of the cross-sections dR, dRs and dRf correspond to adding or deleting

isotopes with cross-sections eRðxÞ, eRsðx ! yÞ and eRfðx ! yÞ and, similarly, that the

perturbation of the albedo dbðx ! yÞ can be written as ebðx ! yÞ. Hence, the nu-merator on the right hand side of (14) can be written as
Z
Xdxw0ðxÞ ðemeRfÞðxÞ

keff

ZXdywyðyÞePfðx

(! yÞ

þ eRðxÞ ZXdy½ecðxÞwyðyÞ � wyðxÞ�ePsðx ! yÞ

)þZCþ

dbxw0ðxÞebðxÞ

�ZC�

dbywyðyÞePbðx ! yÞ;

but the analysis cannot be carried further without the exact values or, at least, the

signs of the perturbations. Assume that only the fission cross-section and/or the

albedo are perturbed either positively or negatively, then formula (14) implies thatthe keff will increase or decrease according to whether emeRf and/or eb are positive or

negative. This observation is trivial but serves at least to show that (11) is a natural

maximum bound: the keff is always limited by the maximum value of m and by adding

fissile material (with m6 mmax) to a domain will get the keff closer and closer to its

maximum allowed value.

3.3. The effect of the size of the domain

When material is added on the periphery of a domain or the albedo on the

boundary of the domain increases, one intuitively expects an increase on the keffvalue. To demonstrate this I analyze the case of a domain D contained in another

domain D0, D � D0, with part or all of the boundary of D contained in the interior of

D0 (see Fig. 1). Consider the leading eigenvalue problem on domain D0 with phase

space X 0,

D

D'

Γ-

Γ-

ψin

β0

Fig. 1. A domain D contained in a domain D0. The two entering boundaries of D are indicated. Boundary

Cb� is closed under the action of albedo operators b and b0.


Bw0 ¼ 1k0eff

Pw0; in X 0;

w0 ¼ b0w0; on C0�;

(ð15Þ

and an associated eigenvalue problem on domain D:

Bw ¼ 1keff

Pw; in X ;

w ¼ bw; on Cb�;

0; on C0�:

�8<: ð16Þ

Here the incoming boundary C� has been split into two parts C� ¼ Cb� [ C0

� with

oDb ¼ oD \ oD0 � oD0 closed with respect to the action of the albedo b0. An albedo

boundary condition is applied on Cb� while a non-reentrant condition is applied on

C0�. Note that it is essential that the common boundary oDb be closed with respect to

the action of the albedos b0 and b. In Eqs. (15) and (16) I have used the same symbolsfor operators B and P to indicate that domain D is physically contained in domain D0

and therefore the cross-sections are the same. Further, I will assume that domain Dreflects lesser or equal than domain D0,

bðx ! yÞ6b0ðx ! yÞ; 8x 2 Cbþ; 8y 2 Cb

�:

To be able to use general duality result (9) one needs to consider both problems in

the same phase space. Hence, I write (15) on phase space X as

Bw0 ¼ 1k0eff

Pw0; in X ;

w0 ¼ b0w0; on Cb�;

w0in; on C0

�;

(8>><>>: ð17Þ

where w0in is the angular flux solution of (15) that enters domain D through the in-

coming boundary C0�. I am now able to compare problems (16) and (17). I use the

adjoint flux wy solution of the adjoint equation of (16) and the direct flux w0 of (17) to

obtain

1

keff� 1

k0eff¼

< wy; ðb0 � bÞw0 >Cb�þ < wy;w0

in >C0�

ðwy; Pw0ÞXP 0:


Notice that domain D0 can be viewed as formed from domain D by adding extra

material on the periphery of D and/or increasing the reflecting properties of the

boundary of D. The last result shows that the eigenvalue of domain D0 is greater or

equal to that of the original domain D. The main eigenvalue does not change,

k0eff ¼ keff , only if the reflectance has not changed, b0 � b, and the incoming flux is

zero, w0in ¼ 0 on C0

�. The later is true only in the case when non-reflecting material isadded on the periphery of D.

3.4. Local multiplication factor

The fundamental mode is a global equilibrium state in which production from

fission compensates losses by absorption and leakage. Hence, the corresponding

eigenvalue, keff , is a global property of the system that depends on both its com-

position and shape. Nevertheless, the keff can be viewed as an average of the mul-tiplication properties of the medium due to localized sources. To show this I consider

Eq. (1) for the fundamental mode, k ¼ keff , and the associated equation for the

Green’s function with fission sources suppressed:

BGy ¼ dy ; in X ;Gy ¼ bGy ; on C�;

�ð18Þ

where GyðxÞ ¼ Gðy ! xÞ is the flux at x produced by a singular source at y,dyðxÞ ¼ dðy � xÞ ¼ dðry � rÞdðvy � vÞ. Then, duality between the fundamental mode

w, solution of (1), and the adjoint Green’s function GyyðxÞ, solution of adjoint Eq.

(18), yields:

wðxÞ ¼ZXF

dy1

keffðPwÞðyÞGðy ! xÞ; ð19Þ

where use has been made of the basic duality relation between direct and adjoint

Green’s functions (Sanchez, 1998), Gyðx ! yÞ ¼ Gðy ! xÞ, and where XF � X is the

part of phase space where there is fission.

The last equation shows that the fundamental flux at a given location x is the

compound result of the direct contributions from the fission sources 1keff

ðPwÞ dis-

tributed across the domain, where by ‘direct contribution’ I understand all transport

processes excluding fission. This equation can be used to obtain a local estimation of

the keff , but global estimations are preferred to local ones because they are numer-ically more stable. A global estimation can be obtained by multiplying Eq. (19) by

some weight and by integrating over a part of over the entire phase space. Here I

apply the production operator to Eq. (19) and integrate the result over phase space.

This yields

keff ¼RXF

dxðPwÞðxÞkeffðxÞRXF

dxðPwÞðxÞ ; ð20Þ


where

keffðxÞ ¼ZXF

dyðPGÞðx ! yÞ; ð21Þ

with the operator P in PG acting on the variable y.Note that the local eigenvalue keffðxÞ gives the average total number of fission

neutrons produced in one generation by one neutron emitted at x. Indeed, to calculate

keffðxÞ fission is suppressed, the flux Gðx ! yÞ is computed at each point, and finally

P is applied to this flux to estimate the number of neutrons that would be produced if

there were fission. Since there is one neutron emitted at x, keffðxÞ can be viewed as the

local keff value. Notice also that Eq. (20) illustrates the technique used to estimate the

eigenvalue by the so-called external iterations: from a given fission source distribu-

tion, ðPwÞðxÞ, the fission source distribution for the next generation, ðPwÞðxÞkeffðxÞ, iscomputed, and the eigenvalue is estimated from the ratio of the new to the old fissionsources.

Eq. (20) shows that the keff depends on the distribution Pw of fission sources for

the fundamental mode and on the local keff values throughout the medium. The Pwdistribution is a global property that requires the calculation of the fundamental

mode. However, the local albedo can be calculated without need of solving an ei-

genvalue problem and (20) can be used to obtain a priori bounds for the keff

minXF

keffðxÞ6 keff 6 maxXF

keffðxÞ:

I end this section by showing that the local keff obeys an adjoint transport equation.

By expliciting the operator P in (21) and with the help of (7) one can write

keffðxÞ ¼ZXF

dyðmRfÞðyÞGðx ! yÞ:

Then, since by duality Gðx ! yÞ ¼ Gyðy ! xÞ, it can be seen that keffðxÞ satisfies thesource adjoint equation

Bykeff ¼ mRf ; in X ;keff ¼ bykeff ; on Cþ:

�
This reformulation provides a direct way to derive (20) from duality between keffðxÞ,as the adjoint flux, and the fundamental mode solution of (1). The fact that the local
keff satisfies an adjoint equation with source the production cross-section leads to an

interpretation of keffðxÞ as the importance function for neutron production.

4. Analysis of the source problem

Duality and positivity can also be applied to a discussion of the solution of thetransport source problem:

Bw0 ¼ 1k0 Pw

0 þ S; in X ;w0 ¼ bw0 þ win; on C�;

(ð22Þ


where k0 is fixed. By using in (9) the solution w0 of this problem together with one of

the eigen functions solution of adjoint Eq. (13) one can write

1

k

�� 1

k0ÞðP ywy;w0

�¼ ðwy; SÞ þ hwy;wini�; ð23Þ

where k is the complex conjugate of ky.Consider first the case when k0 coincides with the complex conjugate of some

eigenvalue ky. Then the previous equation yields a compatibility condition for thesolution

ðwy; SÞ þ hwy;wini� ¼ 0:

Hence Eq. (22) admits a solution only if S and win satisfy this condition for every

eigenvector wy associated to eigenvalue ky; this solution is the sum of a particular

solution plus an arbitrary eigenvector associated to k. In the case when k0 ¼ keff , thepositivity of the fundamental adjoint flux tell us that S and/or win must change of

sign. This type of problems with negative sources are characteristic of high-orderperturbation analysis (Baudron et al., 1998).

When k0 is not in the spectrum Eq. (22) admits a unique solution. I am mainly

interested on the case when k0 is real and positive. In the following I restrict the

analysis to the physical case with positive sources, S;win P 0, and with k0 real. I willalso exclusively use k ¼ keff in (23), so that P ywy;wy > 0 which makes positive the

right hand side of (23). Assume first that k0 < keff , then Eq. (23) shows that the so-

lution w0 to the source problem must change of sign. This corresponds to the

physically meaningless case when one looks for a steady-state solution in a super-critical medium in the presence of external sources. The only viable, positive,

physical solution to this problem is within the framework of the time-dependent

transport equation.

Finally, in the subcritical case with k0 > keff Eq. (22) admits a unique, positive

solution. I use here some results that are proved in the literature. Write Eq. (22) on

the form

ðk0 � B�1P Þw0 ¼ B�1S; in X ;w0 ¼ bw0 þ win; on C�:

�
Since the keff equals the spectral radius of operator B�1P , for k0 > keff , k
0 is in the

resolvent set of B�1P and the operator k0 � B�1P has an inverse. Moreover, the re-

solvent operator can be written as a Neumann series

ðk0 � B�1PÞ�1 � ðk0Þ�1XnP 0

½ðk0Þ�1B�1P �n;

and positivity follows from the positivity of B�1P . Furthermore, the non-homoge-

neous component of the boundary condition win can be treated as a surface source

and the albedo can be incorporated in the definition of B.


5. Conclusions

Linearity, positivity and particle balance are surely some of the basic properties of

the neutron field that a physicist would accept without doubts when dealing with

linear particle transport in a multiplying domain where leakage is important enough

to compensate for particle production. When this is not the case the physicist in-troduces an eigenvalue that directly weakens particle production from fission and

allows for an unique and positive solution. In this work I have used basic physical

insight to establish several bounds for the dominant eigenvalue, keff , in several sit-

uations comprising cross-section perturbations and changes of the shape of a domain

or of the reflecting properties of its external surface. Further, by introducing a local

multiplication factor, keffðxÞ, it was possible to obtain a separate technique to pro-

duce bounds on the main eigenvalue. The local keff satisfies an adjoint source

problem and can be interpreted as an importance function for particle generation.

References

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Some bounds for the effective multiplication factor

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