united nationseducational, scientific

and culturalorganization

the

international centre for theoretical physics

international atomicenergy agency SMR/1326-12

Workshop onHybrid Nuclear Systems for Energy Production,

Utilisation of Actnides & Transmutation of

Long-Lived Radioactive Waste

3 - 7 September 2001

Miramare - Trieste, Italy

Source Neutron Amplification and"Criticality Safety Margin"

of an Energy Amplifier

Yacine KadiCERN, Switzerland

strada costiera, I I - 34014 trieste italy - tel. +39 04022401 I I fax +39 040224163 - scijnfo@ictp.trieste.it - www.ictp.trieste.it

SOURCE NEUTRON AMPLIFICATION AND"CRITICALITY SAFETY MARGIN"

OF AN ENERGY AMPLIFIER

S. Atzeni# and C. Rubbia

Abstract

The main parameter characterizing the neutron economy of an accelerator driven subcritical

fission device, like the Energy Amplifier (EA), is the factor M by which the "source" spallation

neutrons are multiplied by the fission dominated cascade. A related quantity is the criticality factor k =

(M-l)/M, that is the average ratio of the neutron populations in two subsequent generations of the

source-initiated cascade. Such a factor k, depending on both the properties of the source and of the

medium, is in general conceptually and numerically different from the effective criticality factor keff,

commonly used in reactor theory, which is in fact only relevant to the fundamental mode of the

neutron flux distribution, and is independent on the source. The effective criticality factor keff is

however a meaningful measure of the actual safety characteristics of the device, that is l-keff is a

proper gauge of the distance from criticality. An important quantity for the analysis of an EA is then

the ratio F*= (l-keff)/(l-k).

In this paper the issue of the dependence of F* on the source size and on the characteristics of

the subcritical device is addressed analytically by using diffusion theory for mono-energetic neutrons

and considering simple, model geometries. While such a model necessarily neglects any details of the

energy dependence of cross sections and drastically simplify the real geometry, they allow for insight

in some aspects of the relevant physics and may provide guidance to realistic simulations. It is found

that in all cases of practical interest F* is larger than unity (approaching the value F* = 2 in some

idealized situations), and increases with the "containment" of the source, with the criticality factor, and

with the ratio of the diffusion length to the fissile-core size. The presence of an absorbing layer (even a

breeding material) enclosing the fissile core is also favourable. According to the present treatment, for

an EA one may expect F* = 1.4-1.5, meaning, e.g., that when M = 50 (and then k = 0.98), the actual

criticality margin is about 2.8-3%, instead of 2%, as would have been naively estimated. Equivalently,

one can say that the energy gain is 40%-50% larger than the value which would have been computed

from the effective criticality factor keff. Accurate Montecarlo simulations are planned to verify the

present results.

Geneva, 16th March 1998

# ENEA, Divisione Fusione, Centro Ricerche di Frascati, Italy.

TABLE OF CONTENTS

1.— INTRODUCTION AND OVERVIEW 1

2.—BASIC FORMALISM 92.1. Source neutron amplification 92.2. Diffusion equation 92.3. Solution by mode expansion (homogeneous system with arbitrarysource) 102.4 Solution for a multi-region spherical system with central point source 13

3.—HOMOGENEOUS SPHERE 143.1 Harmonic analysis of the amplification 163.2 Scaling of F* with the source size and the sphere properties 183.3 Flux distribution (point source) 19

4.—MULTI-SHELL SPHERICAL SYSTEM WITH POINT SOURCE 20

5.—CONCLUSIONS 24

6—REFERENCES 26

—APPENDIX A 27

-APPENDIX B 28

-APPENDIX C 29

1.— INTRODUCTION AND OVERVIEW

In an accelerator driven, sub-critical fission device, like the Energy Amplifier(EA) [1], the "primary" (or "source") neutrons produced via spallation by theinteraction of the proton beam with a suitable target, initiate a cascade process. Thesource is then "amplified" by a factor M x and the beam power is "amplified" by afactor G = G0M. (typically, for protons with kinetic energy of 1 GeV, Go ~ 2.7, aspredicted by simulations [2] and confirmed by the FEAT experiment [3]).

If we assume that all generations in the cascade are equivalent, we can define anaverage criticality factor k (ratio between the neutron population in two subsequentgenerations), so that

M = l + Jfc + £2+£3+... =—!— (1)l-k V ;

and then k can be computed from M, according to

* ( 2 )

The factor k depends both on the properties of the medium (composition andgeometry) and of the source (energy spectrum, position); it is independent on theintensity of the source. It is worth anticipating that k is, in general, different from(and typically larger than) the effective multiplication coefficient keff used in reactorstudies. The analysis of the relevance of the two coefficients, and the possibility oftaking advantage of this distinction are the objects of this paper.

Although the analysis of an EA is based on the same basic physical-mathematical model describing a critical reactor (i.e. some appropriateapproximation of the neutron transport equation), it requires the use of concepts that,even if already outlined in early works on chain reactors (see, e.g., the classicaltextbook by Weinberg and Wigner [4]), have in the following often been overlooked.

A neutron multiplying system of finite size, is analogous to a resonant cavity,which can "support" the numerable infinity of independent modes (eigenmodes oreigenstates) of oscillation which satisfy appropriate conditions at the cavityboundary. Each eigenmode has shape only dependent on the geometry of the system,

1 The quantity M measures the multiplication of the source neutrons by the cascade process.Since, on the other hand the term multiplication is usually employed with a different meaningin reactor theory (where the infinite multiplication factor koo, and the effective multiplicationfactor keff are introduced), here we refer to M as to the "neutron source amplification factor",and to k as to the "criticality factor".

and is the solution of the relevant wave equation corresponding to a certain proper

frequency or eigenvalue, again only dependent on the geometry. Analogously, a

neutron multiplying system can "support" a numerable infinity of independent

eigenmodes of the flux distribution, each corresponding to a different eigenvalue [4].

The eigenvalues and the eigenfunctions only depend on the geometry of the system

and on the boundary conditions.

For simplicity, in the following we refer to a one dimensional geometry (e.g. a

spherically symmetric system), so that we have a single infinity of eigenmodes, \|/n,

and eigenvalues, B n , with n = 1,2,..., ordered in such a way that Bn+i > Bn (e.g., Bn =

n rc/Rextr f°r a sphere of extrapolated radius2 Rextr^ We also make the further, drastic

approximation, of mono-energetic neutron population.

If we now consider a steady, spherical subcritical system, driven by an outer,

spherically symmetric source C=C(r), then both the known source and the unknown

flux distribution c(>(r) can be written as a linear superposition of the above

eigenstates3. It has been shown that, for homogeneous systems, the amplitude (|)n of

the series expansion of the flux, 0(r) = £0n Vn(r)' depend on the geometry and on

the properties of the fissionable system, as well as on the amplitude of the

corresponding mode of the expansion of the source.4 Each mode n of the external

source is amplified by a different factor M n , while the global amplification M of the

source is obtained as a linear combination of the amplification factors of the

individual modes, each weighted by the corresponding (space integrated) source

contribution. The mode amplification is given by Mn= (1-kn)"1, where kn is the

criticality factor of the n-th eigenstate:

k - k pnonescaPe - fr» (rt\

Here k^ is the infinite multiplication factor of the medium, />wnonescaPe is the non-

escape probability for neutrons in the n-th mode, and L is the usual diffusion length

(defined by Eq. (17) below). Equation (3) makes it clear that the escape probability

grows with the mode number, and then the criticality factor and hence the

amplification decrease with the mode number.

2The definition of the extrapolated distance can be found in Sec. 2.2.

3Formally, the flux is expressed in the orthormal, complete basis made by the eigenvectors ofthe characteristic wave equation.

4The discussion here, which follows the classical treatment of Ref. [4], first applied to the EAin Ref. [5], applies strictly to the case of homogeneous. A detailed treatment is given inSec.v2.3 below.

The case of the critical reactor is substantially different, and can be addressed asfollows. Let us consider again a subcritical system driven by a steady outer source,and assume that in some way (movement of control bars, change of position of someelements, change of temperature, etc.) the "reactivity" of the system is slowlyincreased (going through a sequence of quasi-static conditions); then, when thecondition lq = 1 is achieved, the amplitude of flux eigenmode n = 1 will diverge,while all other modes (having kn < ki) will stay finite. Equivalently, in absence of anexternal source, the fundamental mode takes a finite amplitude (because the internalfission source just suffices to sustain the chain reaction), while the higher harmonics(having criticality factor smaller then unity) are infinitesimally small. Therefore, in acritical reactor the flux shape is that the fundamental eigenstate of the relevantcharacteristic equation and (if the system is homogeneous) is uniquely defined by thegeometry and the boundary conditions. The criticality factor is unity and is related tothe smallest eigenvalue, Blr the so-called "geometrical buckling", of the characteristicequation.

It is worth observing that it is a common practice in reactor studies to computean effective multiplication factor keff for a subcritical system, as the factor by whichthe fission rate has to be divided to make the system critical (see § 4.56 and 4.57 ofRef. [6] and Sec. 2.VIII of Ref. [7]). In following this process one replaces the actual,subcritical system with an "associated critical reactor". It is apparent that kgff differsfrom the criticality factor k = (M-D/M characterizing the actual multiplication of thedriven subcritical system; in fact, kgff is just the criticality factor kj[ of the fundamentalmode of the subcritical system. In order for the two factors to coincide, the outersource should be homotetic to the fundamental eigenmode of the system. Using keffonly, and the standard theory of critical reactors, to characterize an EA wouldtherefore lead to significant qualitative and quantitative errors.

It is also important to analyze what happens in a subcritical system when thesource driving the steady operation is suddenly quenched. In this case the amplitudeof the n-th mode of the flux distribution decreases exponentially5, as exp(-t/tn),where tn is the relevant "prompt neutrons lifetime", which is related to the eigenvalueBn [4,5]. For instance, for a homogeneous system with macroscopic absorption crosssection Ea, and neutrons with a single velocity v, one has tn = [v Xa(l-koo+Bn

2L2)]~ =Mn[v £a(l+Bn

2L2)]~ . Since tn decreases rapidly with n, after a short transient thecharacteristic time scale will be set by ti (and then by the eigenvalue Bi).

5Delayed neutrons have been neglected here for simplicity, in view of the qualitative nature ofthe discussion.

As far as safety is concerned, one is also interested in the behaviour of thesubcritical system in absence of the outer source. This concerns for instance the casein which the accelerator is switched-off and the system reactivity undergoes changesdue to radioactive decays6. The preceding discussion indicates that, since now theonly neutron source is the internal fission source, and since the response of thesystem to sudden changes is dominated by the first eigenmode, then the criticalityfactor relevant to this case is that of the fundamental mode, ki = keff. In thisconnection, of particular interest is the evaluation of the "safety margin" of thesystem, which we can measure by means of the parameters

Akcrit=l-kc{{ (4)

and

M*rit(M) = k™1 - ^(M) = k^M) I — - 1 1 (5)

This last quantity expresses the increase of the fuel k^ necessary to achieve criticality.

In conclusion, we can say that the energy gain of an Energy Amplifier, ischaracterized by a criticality factor k, depending on the system as well as on thesource, which differs from (and, as shown later in details, is typically larger than)the effective multiplication coefficient keff = ki, which only characterizes thefundamental eigenmode. The coefficient ki= keff, instead, is relevant to some safetyproperties of the system. Given the fact that k and keff are, to some extent,independent, one can try to design an EA taking advantage of the distinctionbetween these two quantities.

For a quantitative discussion, it is useful to introduce the quantity F*, defined as

F*=JL = M o ^ ) _!=** _ ^eff 1 - ^ 1"" ^

that can be viewed either as the ratio of the actual source amplification toMeff =l/(l-£ e f f), i.e. the amplification computed by using Eq. (1), but replacing kwith keff, or as the ratio of the actual criticality margin to (1-k).

For comparison with recent simulations of subcritical devices [8,9], we alsointroduce the function (p*, defined as the ratio

that would be required if the outer neutrons were distributed just as the fundamentalmode of the system (that is if their number were computed by replacing k with keff).Such a function is often referred to as the importance of the (outer) neutron source. [8] orthe effectiveness of the outer source [9] 7, and can also be written as

* 2 L l ^ l ) = - * - F » . (7)

The operational safety margin on k^ for an EA with multiplication M, can then bewritten as

^ ^ (8)M —

Notice that if one had evaluated this last margin by using the factor k computedfrom the amplification M, one would have got

(p *

Just to give the flavour of some of the aspects involved, we anticipate here theresults of some of the simple, analytic calculations described in Sec. 4. We refer to thecase of a spherical device, consisting of an inner diffusing zone, followed by thefissile region, in turn enclosed by a breeder shell or by an outer diffusive shell (moredetails on the parameters are given in Sec. 4). The system is driven by a central pointsource. Such a schematization is taken as a rough, "spherical equivalent" of thegeometry of typical Energy Amplifiers. In Fig. 1 we show the behaviour of thequantity F* as function of keff. It is seen that F* is a growing function of keff and isalways well above unity. In the range of keff of interest for an EA, namely keff =0.95-0.98, from curve I of the figure one has F* « 1.4-1.5. This means that once a givenvalue of keff is fixed, the system amplification is considerably larger than naivelyestimated from (l-keff^1. The same results are presented in two alternative forms inFigs. 2 and 3, respectively. Fig. 2 makes clear that the safety margin defined by Eq. (4)is larger than 1-k; Fig. 3 shows the amplification M, evidencing that the effect we arediscussing can be either viewed as an "increased amplification" at a given keff, or asan "increased safety margin" in correspondence to a given value of M.

7More properly (p* is the importance of the outer neutron source normalized to that of theinternal source.

II

LL

10.875

keff

Figure 1. Function F* (ratio of the actual safety margin to (1-k) vs keff, as computed by asimple diffusive model for two cases with geometry shematizing that of a typicalenergy amplifier (see Sec, 4 for details): I) the fissile core is surrounded by a breeder[case (d) of table I]; II) the core is surrounded by a diffuser [case (b) of table I]. Inboth cases F* is well above unity and grows with kejf.

1 - k

Figure 2. For the same system as in case I of Fig. 1, we show the "real" safety margin (1-keff) vs (1-k), where k is the criticality factor characterizing neutron amplification.The "gain" in the safety margin, at a given value ofk, is evidenced by the arrow.

g

CL

CO

I0

200

100--

- ^ ^

described in Sec. 2 (with details given in the Appendices), while the physical resultsare presented and discussed in Sees. 3 and 4.

First, the simplest case of a homogeneous, bare sphere, with a finite-size sourceis studied in detail, by means of the harmonic expansion of the solution of thediffusion equation. This treatment, in particular, allows to identify the contribution ofthe different harmonics to the amplification. The dependence of F* on the size of thesource, on the criticality factor, and on the properties of the subcritical sphere is thendiscussed.

Although useful for gaining insight, the homogeneous sphere model cannot betaken as a reasonable simplification of the structure of an EA, which is characterizedby several "regions" of materials with different properties. Spherical systems, withconcentric "zones" of different materials, and with a central, point-like neutronsource, are then considered, as schematic representations of the geometrical structureof an EA. Despite their simplicity such models yield a spatial distribution of theneutron flux, and its dependence on k in qualitative agreement with accuratesimulations. They also reproduce trends for source importance observed in someMontecarlo calculations of realistic systems [8]. Such results justify using the abovesimple models for getting further insight. In particular we consider how the shape ofthe flux can be controlled and the function F* maximized. In general, it is found thatF* grows with k; at given k, F* increases with

• the "containment" of the neutron source;

• the ratio of the neutron diffusion length to the size of fissile core;

• the presence of an absorbing medium, "enclosing" the fissionable core, which,in a sense, limits the "widening" of the neutron flux distribution as k isincreased

In conclusion, it is worth stressing that the treatment presented here, beingbased on over-simplified models, only aims at getting insight in some aspects of theproblem. Any quantitative study of an accelerator driven subcritical fission devicenecessarily requires using treatments which can handle both the full energydependence of the neutron flux (and make use of accurate cross sections) and theactual geometry of the system. It is hoped that the present results may provideguidance to such complex simulations.

2.— BASIC FORMALISM

2.1. Source neutron amplification

We want to compute the amplification M of the source neutrons in amultiplying medium of volume V and surface S. We start from observing that, bydefinition, at steady state the rate of neutron generation and that of neutrondisappearance (absorption, A, plus escape, E) are equal. Since the rate of generation isgiven by the source rate Q (spallation neutrons per unit time), times theamplification M, then

^ ^ (10)Q

The rates of absorption and escape are given, respectively, by

A =Jj5:a0 dSdV (11)vs

and

E =jjj-n dgdS (12)ss

where

10

/ = -£>V0, (14)

where D = lt r/3 is the diffusion coefficient, ltr is the neutron transport mean free path,given by \r = (Et - // 2s)"

1/ where St, Es, and Ea, are respectively the macroscopic totalcross section, the scattering cross section and the absorption cross section, and fi isthe average value of the cosine of the scattering angle in the laboratory system . Sincein an EA the fuel is cooled (and the neutrons diffused and moderated) by a high-Zmaterial, then one can take /tr = (Za + Es) .

The neutron flux is the solution of the equation

V20+ 2 ^ 0 + ^ = 0 , (15)

where C is the contribution of the external source (neutrons per unit volume and unittime), BM is the so-called material buckling

^ = ^ (16)

koo and L are, respectively, the infinite multiplication coefficient and the diffusionlength:

^ [£ (17)

v is the average fission multiplicity, and Za is the macroscopic cross section.

In the usual case of no incoming neutron current, Eq. (15) is solved by imposingthat the flux vanishes at the extrapolated surface, i.e. a virtual surface lying at adistance lextr ~ 0.72 ltr outside the material boundary [4,6].

2.3. Solution by mode expansion (homogeneous system with arbitrary source). 9

As it is well known , if we consider a finite, system, with vanishing flux at the(extrapolated) boundaries, and a source also vanishing at and outside the boundaries,we can write the solution to Eq. (15) in terms of the eigenvectors y/n of thecharacteristic "wave equation"

V V = 0 , (18)

9The material presented here up to Eq. (29) follows closely that of Ref. [6], where thisformalism, discussed e.g. in the textbook [4], was applied for the first time to the EA.

11

which form a complete orthonormal basis, each eigenvector y/n corresponding to an

eigenvalue Bn.10 We normalize the eigenfunctions in such a way that jyf^dV = l;n

and introduce the volume integrals of the eigenfunctions :

%=ly/n(x)dV. (19)

We then write the (known) outer source as

C(x)= icny/n(x) , (20)n=\

with the expansion coefficients given by

cn=jvC(x)y/ndV , (21)

so that the space integrated source neutron rate can be written as

Q = \vC{x) dV= fn=\

The (unknown) neutron flux can be expanded in the same basis, too,

and a straightforward solution is found for a homogeneous medium. Indeed, in thiscase, by substituting the expansions for the source and the flux [Eqs. (20) and (23),respectively] into Eq. (15) we obtain an independent equation for each n, giving thecoefficient of the flux as a function of that of the source :

Za l - f lL- J^L 2 )where

B'L2(25)

n

As anticipated in the introduction, we see from Eq. (25) that if all knfs are

smaller then unity, then the flux is given by a linear superposition of eigenmodes; as

10To the purpose of keeping the notation simpler we refer again to a one dimensionalgeometry, e.g. a spherically symmetric system, characterized by a single infinity ofeigenvalues.11 Notice that the integrals are over the actual volume of the system, not over the extrapolatedone.

12

soon as ki = 1 the system becomes critical; the source is no more needed to sustainthe system, and the only surviving mode is the fundamental one.

We turn our attention back to the subcritical system. The contribution of the n-th eigenmode to neutron absorption is

(26)

while, the contribution to neutron leakage is

En=-DjVy/n- n dS = -DjV(Vy/n) dV = -DjV2y/n dV (27)

S V V

where Gauss theorem has been applied to the surface of the medium. By using the"wave equation" (18), and the definition of *Fn we can eventually write

En = - DJVVn dV = DB2n jysndV = DB^fn . (28)

V V

From Eqs. (26) and (28) we immediately find the non-escape probability forneutrons in the n-th mode:

pnon-escape _ Az _ 1

" A - l

confirming that the factor kn defined by Eq. (25) is just the criticality factor of the n-thmode.

By introducing the above expressions for the absorption and the leakage intothe definition of the amplification factor [Eq. (13)] we get

l E A ^ d ^L 2 ) , (30)Q Qn=\

and then, using Eq. (24), we can write

M=lMn^, Mn=T1r, (31)

which make transparent that the volume-integrated n-th mode of the source isamplified by a factor Mn. and that the source neutron amplification is obtained as alinear combination of the individual mode amplifications, each weighted by thecorresponding normalized mode of the source It is also clear that the Mn 's form adecreasing succession (because the Bn's form an increasing succession and then thekn's a decreasing one, kn+i < kn).

13

The limit kx —> 1 is of some interest. In this case only the fundamental mode

survives, and k —»1 too; the function F*, instead, takes the limiting value

lim F*= Hm [M(l-*eff)] = M ^ ( l - ^ ) = ^ ± (32)

which is, in general, different from unity. This is due to the fact that F* measures the

ratio of the amplification of the outer source (with assigned space dependence) to

that of a source distributed as the fundamental eigenmode. It will be seen in Sec. 3.2

that in the limit of point source and sufficiently large sphere c i^ i /Q = 2.

The system criticality factor k can be computed by inserting M, computed by

Eq. (31), into Eq. (2). An alternative expression is

= -5=1 r =

It is worth observing that the fact that the kn's form a decreasing succession, and

therefore the crt's form an increasing succession, does not mean that k < ki = keff,

because the ^n can have arbitrary sign. In particular, in the case of a sphere12, we

shall see that x¥n keff, and F* > 1.

2.4 Solution for a multi-region spherical system with central point source A realistic

schematization of a subcritical device requires taking into account the presence of

several "zones" with different properties (e.g., central diffuser, fissile core, outer

diffuser, etc.); unfortunately the treatment illustrated in Sec. 2.2 cannot be applied to

non-homogeneous systems. In this case, however, one can obtain closed-form,

analytical solutions of the diffusion equation, at the expense of restricting attention to

the case of a central point source. We have then considered systems consisting of

three or four concentric homogeneous zones, and without any distributed neutron

source (other than the fission source). The external source is instead introduced as a

boundary condition, by imposing that at the origin the incoming neutron rate equals

that due to the outer source, releasing Q neutrons per unit time, that is

^-) = Q. (33)dr

12and in the limit in which the extrapolated boundary coincides with the physical boundary.

14

In addition, the usual boundary conditions that the neutron flux and current becontinuous at the interfaces between the different zones, and that the neutron fluxvanish at the extrapolated boundary, are imposed. From the knowledge of the fluxdistribution 0(r), the amplification M and the criticality coefficient k are computedby Eqs. (13) and (2), respectively. The effective multiplication coefficient keff, insteadis computed by the standard technique outlined in Sec. 1 (that is, by identifying keffwith the value by which the fission source must be divided to make the system justcritical). At last, the functions F* and

15

100000

unit-source radial distribution

= (Ac/r) exp(-r /Rs )

10 20 30 40 50 60 70 80 90 100

radius, r (cm)

Figure 4 Radial profile of the source term C(r) described by Eq. (34), for different values ofthe characteristic source size Rs.

We solve the diffusion equation according to the method illustrated in Sec. 2.3.The n-th eigenvalue and the n-th normalized eigenfunction are given, respectively,by

—^extr

Yn —sin Bnr

(35)

(36)

where Ay is a normalization constant (see Appendix A), while the volume integratedeigenfunctions, defined by Eq. (19), are

K

n(37)

where the function f% (see Appendix A) accounts for the difference between R andRextiv it can be shown since in all practical cases the extrapolation distance lextr is asmall fraction of the radius13, then f% is very close to unity for all the modescontributing appreciably to the global amplification. Equation (37) shows that(neglecting the small correction due to f%) the integrals ^ n form a succession ofalternate size and decreasing absolute value.

13E.g., in a typical EA, ltr ~ 3 cm, and lextr ~ 2 cm, while the size of the multiplying medium isabout 1 m.

16

The coefficients of the source expansion, computed according to Eq. (21) are

(38)

where /nc is a function analogous to f% (see Appendix A).

3.1 Harmonic analysis of the amplificationThe amplification of the source neutronscan then be readily computed by inserting Eqs. (37) and (38) into Eq. (31), whichgives

/

'• "T= hMn (f) ftf)with

Mn= - V = T = T • (40)1 ^2 1

2 2

Equations (38)-(40) allow for an analysis of the contributions of the differentharmonics to the neutron amplification. For concretness we refer to the caseillustrated by Fig. 5, with source characterized by a = 10, and a sphere withRextr/L=7, and keff = 0.963. Figure 5a shows that the first harmonics of the sourcehave opposite sign and decreasing amplitude. Such a behaviour only depends on thegeometry of the system and on the shape of the source. Equation (38) shows that thesharper the source, the slower the decay of the source modes. The amplification of theindividual modes is a more rapidly decreasing function of the mode number, asshown in Fig. 5b, so that only a relatively small number of modes contribute to thesource amplification (see Fig. 5c). It is to be observed that the overall amplification Mis larger than that of the first mode, Mi, or, equivalently k > ki = keff = 0.963 (hence

17

100

10

mode number, n

100

10

mode number, n

100

s up

to

r

O)

gCDO13

a

nca

tion

ampn

54^

52-

50-

48-

46-

44-

42-

4CV

^ MiCi^VQ

\ yv

10

mode number, n

100

Figure 5 Solution of the diffusion equation by mode expansion for a homogeneous, baresphere, driven by an external source (with the parameters given in the main text);a) harmonic expansion of the source; b) amplification factors of the individual modes;c) contribution to the total amplification of the modes up to n, versus n. It is apparentthat the amplification converges rather rapidly; also notice that the globalamplification is higher than the amplification Mj (that is, k is larger than kj = kejf).

18

3.2 Scaling of F* with the source size and the sphere properties

Next we study the variation of the quantity F*=(l-keff )/(l-k) with the "size" ofthe neutron source, and the criticality factor. Figure 6, referring to a sphere withRextr/L=7, shows that for "well contained" sources F* is well above unity and growswith keff. Typical values of F* for such sources are in the range 1.4-2. Only in theacademic case of a nearly uniform source, F* is smaller than unity and nearlyindependent on keff. The figure also shows that as criticality is approached F* tendsto a finite limit, which is in general different from 1 and is independent on the valueof Rextr/L. In the case of a point source, F* —» 2 (see also Fig. 7), as is also recoveredfrom Eq. (40). Indeed, when the difference between the sphere radius and theextrapolated radius is neglected, M= 2£Mn(-l)

n+1 . As criticality is approached,only the fundamental mode contributes to the sum, and then M —> 2 Mx, and F* —» 2.

The effect of the size of the sphere (relative to the diffusion length L) isevidenced, for the case of a point source14, by Fig. 7: for a given value of the criticalityfactor keff < 1, the smaller R/L the larger F*.

0.8-

0.60.85

uniform, bare sphere of radius R (with R/L = 7)source C(r) = (1/r) exp (-ar/R)

"distributed" source, a = -2

0.9 0.95

keff

Figure 6 F* vs keff, for different values of the parameter a characterizing the dimension ofthe source relative to the sphere radius R, and for Rextr/L = 7. It is apparent that forreasonably well contained sources, F* is greater than 1 and is a growing function ofk

14Such a case can either be obtained by using the solution of Sec. 3.1 and taking the limit oflarge a, or by the closed form solution illustrated in Appendix A.

19

uniform, bare sphere of radius R; point neutron source

0.85 0.95

keff

Figure 7 F* vs keff, for a homogeneous, bare sphere with a central point source, anddifferent values of the ratio R/L of the sphere radius to the diffusion length. It is seenthat the longer the diffusion length, the higher is the importance of the neutron source,that is the amplification for a given value ofkeff.

3.3 Flux distribution (point source)

In Fig. 8 we show the radial flux shapes for cases with a central point-sourceand different values of the criticality factor. It is seen that, far from the centre, the fluxprofile becomes less peaked as k approaches unity. On the other hand, the extremelyhigh value of the flux close to the source demands the adoption of a more suitablegeometry, limiting the fluence on the most exposed elements of the fuel (and of thesupporting and confining structural materials). In particular, in the EA the goals of amore uniform utilization of the fuel and of the limitation of the peak flux to the fuelare achieved by shaping the multiplying fissile core (possibly surrounded by abreeder layer) in nearly cylindrical, annular form, and immersing it in liquid lead [1].Moving from the symmetry axis outwards, one has an inner buffer region, where thespallation neutrons are produced and partially moderated, followed by the fissilecore, by the breeder, and by the outer lead. Schematizations of such a geometry arediscussed in the next Section.

20

0.01-

x

ce0

0.0001

k = 0.900

k = 0.949

k = 0.980

k = 0.994

0.001-

0.2 0.4 0.6 0.8 1

normalized radius, r/R

1.2

Figure 8 Radial flux profile for a homogeneous, bare sphere, with central point source, fordifferent values of the criticality factor k. (Here Rextr/L =7).

4.—MULTI-SHELL SPHERICAL SYSTEM WITH POINT SOURCE

As the simplest geometrical schematization which retain the multi-regionfeatures of an EA, we consider a spherical system with three or four concentric,homogeneous regions (see Fig. 9), and with a central point neutron source. Thesolution of the relevant diffusion equation is detailed in Appendix C.

The geometry of some of the cases considered in the following is described intable I, while the values of the absorption cross-sections assumed for the differentmaterials are listed in Table II (For the transport cross section we have taken the samevalue Xs= 0.33 cm"

1 in all materials). Here it is worth stressing that a drasticsimplification of our model is the assumption of monoenergetic neutrons, whichrequires the use of spectrum-averaged cross sections. Our choices are ratherarbitrary, but do not affect the observed qualitative trends. Two aspects deservehowever some comment. First, from table II one sees that we have taken differentabsorption cross section for the central Lead buffer and the outer Lead diffuser. Thishas been motivated by the difference in the "average" neutron spectra in the tworegions and the possible presence of additional structural materials. A second

21

comment concerns the procedure followed to compute the variation of themultiplication (and then of F*) with k. In practice, one can change the criticality factoreither varying the fuel enrichment or the fuel/moderator volume ratio (that is thepitch between fuel rods). Here, for simplicity, we have taken both the macroscopicabsorption cross section and the transport mean free path constant, and havechanged the koo of the (homogeneized) fuel core.

outer "diffuser"

or fertile ("absorber") shell

fissile core

central buffer

Figure 9 Sketch of the four-layer, spherical systems considered in this section. The neutronsource is located at the centre of the buffer.

Table I - geometry of the studied cases

a) bare system

outer radius

buffer 30 cm

fissile 100 cm

b) system withouter diffuser

outer radius

buffer 30 cm

fissile 100 cm

diffuser 200 cm

c) system withouter fertile layer

outer radius

buffer 30 cm

fissile 100 cm

fertile 150 cm

d) system withtwo fissile layers,outer fertile layer

outer radius

buffer 30 cm

fissile I 70 cm

fissile II 100 cm

fertile 150 cm

22

Table II - Macroscopic absorption cross sections

layer (material)

inner diffuser (Lead)

fissile core (U-Th + Lead)

outer shell of the fissile core (U-Th + Lead)

fertile shell (Th + Lead)

outer diffuser (Lead)

Ea (an"1)

2xlO"4

5 x 10"3

6 x 10"3

3 x 10"3

5 x 10"4

We start by considering a three shell system (buffer, fuel, diffuser): as shown byFig. 10 the flux distribution in the fuel region is now more uniform than in the case ofthe homogeneous, bare sphere. This is a well known feature [1], which however isworth being shown here, because it proves the qualitative agreement between thepresent, very simple treatment and the complex simulations required for the designof a device. Such a result supports using simple models for getting insight in thephysics of subcritical systems.

k = 0.990k = 0.975

k = 0.950

k = 0.900

1.2 1.4 1.6 1.8

radius (m)

buffer fissile core diffuser

Figure 10 Radial flux distribution for the spherical systems described as case a) of table I, fordifferent values of k. These profiles are analogous to those produced by completeMontecarlo simulations of a power producing Energy Amplifier (see Figs. 2.3 a-c inRef. [11).

23

The choice of the distribution of the fuel and of the material surrounding thefissile core results from consideration of different nature (fuel utilization, materialdamage, interest in greater breeding etc.), which are outside the scope of the presentwork. It is however worth seeing how such design features affect the flux distributionand the value of F* (and then the previously defined "criticality margin").

In Fig. 11, which refers to cases with the same value of k = 0.98, we compare theflux distributions in the case of i) bare fuel, of purely academic relevance; ii) singleshell of fuel surrounded by a diffuser (the same as in Fig. 10); iii) single shell of fuelsurrounded by a breeder layer; and iv) two shells of fuels with different properties,surrounded by a breeder shell.

(a) bare

(b) outer diffuser

(c) outer breeder

120 140

radius (cm)

Figure 11 Effect of the material surrounding the fissile core on the spatial distribution of theflux. The three curves refer, respectively, to cases a),b), and c) of Table I.

The behaviour of F* vs keff is shown in Fig. 12 for the same cases as in Fig. 11:the trend is the same in all cases, and similar to what already observed for thehomogeneous sphere. In all cases F* is somewhat smaller than in the ideal-sphere,which can be explained by the fact that now we are missing amplification just in themost active portion of the sphere. It is also found that F* is higher for a bare core thanfor one surrounded by a diffuser. We have also tested that very similar values of F*are recovered for a bare core and a core surrounded by a strong absorber. Such

24

results indicate that a worst neutron "confinement" close to the boundary has positive"safety" features, in the sense that opposes the natural tendency towards "widening"the flux distribution as k increases. An interesting result is that, as shown in Fig. 12,an outer breeder also performs rather effectively such a task (at the same timemaking use of the escaping neutrons). We have also considered an additional case inwhich the core is made of two shells with different fractional volume of fuel.According to our model, such a choice, aiming at a more uniform utilization of thefuel, has negligible influence on the values of the quantity F* (compare curves c) andd) in Fig. 12).

1.7-

1.6-

1LL

1.5-

1.4-

1.3-

1.2-

1.1-

a)

• • • •

# 1• • • •

. • •

c)

• (a) bare

A (b) outer diffuser

m (c) outer breeder

• (d) 2 fissile layers + breeder

0.9 0.92 0.94 0.96 0.98

keff

Figure 12 Function F* vs kefffor the multi-shell systems described in table L It is apparentthat surrounding the fuel by layer of moderately absorbing material (e.g. a breedingmaterial) results in an increase of F*y and therefore in an improvement of the safetyfeatures of the system.

5.—CONCLUSIONS

Accelerator driven subcritical fission devices, like the Energy Amplifier, can becharacterized in terms of two distinct criticality factors. The first one, defined as k =(M-l)/M, is related to the amplification of the neutron cascade, and depends both onthe properties of the source and of the medium. The second quantity is the effective

25

criticality factor (keff) commonly used in the theory of critical reactors, which is

instead independent on the source. The difference between the two factors stems

from the fact that while the neutron flux distribution in a critical reactor corresponds

to the fundamental eigenstate of the system (and is therefore fully described by a

single eigenvalue, related to keff), that of a subcritical system results from a

superposition of modes, each characterized by its own eigenvalue, and then by its

own amplification and criticality factor. The total amplification is a properly

weighted sum of the amplifications of the individual modes.

We have argued that the effective criticality factor is a relevant measure of thecriticality margin of the system, since it characterizes its response to sudden sourcechanges, as well its behaviour once the source is switched off.

An important quantity for the design of an EA is therefore the ratio F* =(l-keff)/(l-k), whose analysis has been the main object of this paper. With the sole

purpose of understanding trends and isolating basic, essential physical effects, wehave used diffusion theory for mono-energetic neutrons.

We have shown that for any realistic source distribution F* is larger than unity(that is in any case an Energy Amplifier has an "additional safety margin", over thatcorresponding to its energy gain), and have shown how it is affected by features ofthe source and of the subcritical system. In general, at a given keff, F* grows as the

"containment of the source" is improved (a point source being more favourable thana distributed one). As far as the subcritical device is concerned, F* increases with keff,

and with the ratio of the diffusion length L to the core size, R. Regarding other designchoices, those making the flux more uniform, such as the presence of an inner buffer(where spallation neutrons are produced) and of an outer diffuser, result in somereduction of F*, while an external absorbing layer has favourable effect.

Our simple calculations seem to indicate that an Energy Amplifier could operatewith F* = 1.4-1.5, that is with a considerable "additional safety margin" over the naiveestimate based on the criticality factor k corresponding to the neutron amplificationM. Such estimates of F* are roughly consistent with a few Montecarlo simulations of asubcritical device, somehow modelling an Energy Amplifier [8].

The complexity of the physics of a real system, however demand for accuratesimulations taking into account both the energy dependence of the neutronpopulation (and the relevant reaction cross sections) and the geometry of the device.A systematic work in this direction is planned for the near future.

26

6—REFERENCES.

[1] C. Rubbia et al, "Conceptual Design of a Fast Neutron Operated High PowerEnergy Amplifier", CERN/AT/95-44 (ET), 29th September 1995.See also C. Rubbia, "A High Gain Energy Amplifier Operated with FastNeutrons", AIP Conference Proceedings 346, International Conference onAccelerator-Driven Transmutation Technologies and Applications, Las Vegas,July 1994.

[2] C. Rubbia et al. note in preparation, see also Sec. 5 of Ref. [1]; the high energycascade is dealt with by the code FLUKA, described inA. Fasso et al, in "Intermediate Energy Nuclear Data: Models and Codes",Proceedings of a Specialists' Meeting, Issy les Moulineaux (France)30 May-1 June 1994, p. 271, published by OECD, 1994, and references therein;see also:A. Fasso, A. Ferrari, J. Ranft, P.R. Sala, G.R. Stevenson and J.M. Zazula, NuclearInstruments and Methods A, 332, 459 (1993), also, CERN Divisional ReportCERN/TIS-RP/93-2/PP (1993).

[3] S. Andriamonje et al. Physics Letters. B 348 (1995) 697-709 and J. Calero et al.Nuclear Instruments and Methods. A 376 (1996) 89-103.

[4] A. M. Weinberg and E. P. Wigner, "The Physical Theory of Neutron ChainReactors", University of Chicago Press, Chicago (1958).

[5] C. Rubbia, "An analytic approach to the Energy Amplifier", Internal NoteCERN/AT/ET Internal Note 94-036 (1994); see also Sec. 2.1 of Ref. [1].

[6] S. Glasstone and A. Sesonke, "Nuclear Reactor Engineering", 4th Ed., Chapmanand Hall, New York (1991).

[7] J. F. Breismeister (Editor), "MCNP™ - A General Montecarlo N-ParticleTransport Code, Version 4A", Los Alamos National Laboratory Report, LA-12625-M (Nov. 1993).

[8] C. Rubbia, F. Carminati and Y. Kadi, "EA Montecarlo Benchmark", to bepublished ...???

[9] I. Slessarev et al., "IAEA-ADS Benchmark (Stage I), Results and Analysis", toappear in the Proceedings of the IAEA-TCM on ADS, Madrid, 17-19 September1997.

27

—APPENDIX A

With reference to the case discussed in Sec. 3.1 (bare sphere, with source C(r)expressed by Eq. (37)) we give here the expressions of the quantities appearing inEqs. (35)-(39), that have been omitted in the main text.

The eigenfunction normalization constant is

and the source normalization constant is

The functions accounting for the difference between the actual sphere radiusand the extrapolated radius are

—CO| |i f / ,—sin nnXK V Rextrvextr,

and

Rextr { Rextr) nA R ) y Rextr

28

—APPENDIX B

Homogeneous, bare sphere, with central point source. Although this is just aspecial case of the multi-shell systems considered in Appendices C and D, we detailhere the closed form solution for the case of a homogeneous, bare sphere, with radiusR and extrapolated radius Rextr = R + lextr/ and with a central point source, releasingQ neutrons per unit time.

We define the positive quantities

(Bl)

The neutron flux can then be written as

sinh[7M(/?extr-r)]

AnDr

for k^ < 1sinh(7M/?extr)

/ (B2)sin[BM(Rem-r)]

sin(5M/?extr)

and the amplification is given by

sinh(7Mi?extr) - kco[yMRcosh(yMlextI) + sinh(7M/(

M =

ex t r / J for k < i( 1 - ^ ) sinh(7M/?extr)

(B3),[%i?cos(5M/extr)+sin(5M/extr)]-sin(5M/?extr) 2 2•i : ^ for 1

29

—APPENDIX C.

We consider here a spherical system with four concentric regions

• region "D" (inner diffuser)

0 < r < RDk= 0

rl = £D

• region "M" (multiplying, fissionable material)

RD < r < RM

and set

72

region "O"(any material; either multiplier, or diffuser, or absorber)

< r <

rl= r2*6k^n > 14

• region "A" (diffuser or absorber)

RQ < r < R

extrapolated radius:

v2 - T~X

Subscripts D, M, O and A are used to refer the values of the parameters y, B, L,and D, in the four regions "D", "M", "O", and "A" respectively.

30

The neutron flux is given by

where

y(r) =

region "D"

cos(BMr) + c2 sin(5Mr) region "M"

region "O";

region "O";

region "A"

c3e Y

c3 cos(Bor) + c4 si

-V r Y r

c5e 7A +c 6 e "

< 1

> 1

The expression of the coefficients a , c+, c\, ci, C3, c\, C5,and C6, are given below

for the cases kooR < 1, and kooR > 1, respectively.

A) kooR

31

Q 7Dsinh(yDi?D)cos(flMflD)

• x

IC2tanh[yD#D]

tan(5M/?D)M3

tanh[rD/?D]- + 5MZ)Mtan(BM/?D)-

1-1

XD

2sinh[yDRD]

c = -Q

= -c. 2 sinhfrA(/?A -

B) kooR > 1

B0D00 0

C3

tanh[7A(/?A-/?o)]

tanh[rA(i?A-/?0)]

32

C3

C3+ g tan(BMRM)

C3 M

7D

sinh(7D/?D)cos(fiM/?D)-x

I £2tanh[7D/?D]

- tan(£M /?D)-f lMZ)M-M>

1

i - i

2sinh[yDRD]

. . G

= c i

2 sinh[rA(/?A-/?0)]

27A