Source Neutron Amplification and "Criticality Safety Margin" of an Energy Amplifier

Transcription

1 united nations educational, scientific and cultural organization the international centre for theoretical physics international atomic energy agency SMR/ Workshop on Hybrid Nuclear Systems for Energy Production, Utilisation of Actnides & Transmutation of Long-Lived Radioactive 3-7 September 2001 Waste Miramare - Trieste, Italy Source Neutron Amplification and "Criticality Safety Margin" of an Energy Amplifier Yacine Kadi CERN, Switzerland strada costiera, I I trieste italy - tel I I fax scijnfo@ictp.trieste.it -

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3 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 k e ff, 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 k e ff is however a meaningful measure of the actual safety characteristics of the device, that is l-k e ff is a proper gauge of the distance from criticality. An important quantity for the analysis of an EA is then the ratio F*= (l-k e ff)/(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* = , 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 k e ff. Accurate Montecarlo simulations are planned to verify the present results. Geneva, 16th March 1998 # ENEA, Divisione Fusione, Centro Ricerche di Frascati, Italy.

4 TABLE OF CONTENTS 1. INTRODUCTION AND OVERVIEW 1 2. BASIC FORMALISM Source neutron amplification Diffusion equation Solution by mode expansion (homogeneous system with arbitrary source) Solution for a multi-region spherical system with central point source HOMOGENEOUS SPHERE Harmonic analysis of the amplification Scaling of F* with the source size and the sphere properties Flux distribution (point source) MULTI-SHELL SPHERICAL SYSTEM WITH POINT SOURCE CONCLUSIONS 24 6 REFERENCES 26 APPENDIX A 27 -APPENDIX B 28 -APPENDIX C 29

5 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 the interaction of the proton beam with a suitable target, initiate a cascade process. The source is then "amplified" by a factor M x and the beam power is "amplified" by a factor G = G 0 M. (typically, for protons with kinetic energy of 1 GeV, Go ~ 2.7, as predicted by simulations [2] and confirmed by the FEAT experiment [3]). If we assume that all generations in the cascade are equivalent, we can define an average criticality factor k (ratio between the neutron population in two subsequent generations), so that M = l +Jfc =! (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 and geometry) and of the source (energy spectrum, position); it is independent on the intensity of the source. It is worth anticipating that k is, in general, different from (and typically larger than) the effective multiplication coefficient k e ff used in reactor studies. The analysis of the relevance of the two coefficients, and the possibility of taking advantage of this distinction are the objects of this paper. Although the analysis of an EA is based on the same basic physicalmathematical model describing a critical reactor (i.e. some appropriate approximation 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 classical textbook 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 or eigenstates) of oscillation which satisfy appropriate conditions at the cavity boundary. 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 meaning in reactor theory (where the infinite multiplication factor koo, and the effective multiplication factor k e ff are introduced), here we refer to M as to the "neutron source amplification factor", and to k as to the "criticality factor".

6 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 B n +i > B n (e.g., B n = nrc/rextr f r a sphere of extrapolated radius 2 R e xtr^ 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 eigenstates 3. It has been shown that, for homogeneous systems, the amplitude ( ) n of the series expansion of the flux, 0(r) = 0 n V n ( 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 M n = (1-kn)" 1, where k n is the criticality factor of the n-th eigenstate: k - k p nonesca P e - fr» (rt\ Here k^ is the infinite multiplication factor of the medium, /> nonesca w P e i s the nonescape 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. 2 The definition of the extrapolated distance can be found in Sec Formally, the flux is expressed in the orthormal, complete basis made by the eigenvectors of the characteristic wave equation. 4 The discussion here, which follows the classical treatment of Ref. [4], first applied to the EA in Ref. [5], applies strictly to the case of homogeneous. A detailed treatment is given in Sec. v 2.3 below.

7 The case of the critical reactor is substantially different, and can be addressed as follows. 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 some elements, change of temperature, etc.) the "reactivity" of the system is slowly increased (going through a sequence of quasi-static conditions); then, when the condition lq = 1 is achieved, the amplitude of flux eigenmode n = 1 will diverge, while all other modes (having k n < ki) will stay finite. Equivalently, in absence of an external source, the fundamental mode takes a finite amplitude (because the internal fission source just suffices to sustain the chain reaction), while the higher harmonics (having criticality factor smaller then unity) are infinitesimally small. Therefore, in a critical reactor the flux shape is that the fundamental eigenstate of the relevant characteristic equation and (if the system is homogeneous) is uniquely defined by the geometry and the boundary conditions. The criticality factor is unity and is related to the smallest eigenvalue, B lr the so-called "geometrical buckling", of the characteristic equation. It is worth observing that it is a common practice in reactor studies to compute an effective multiplication factor k e ff for a subcritical system, as the factor by which the fission rate has to be divided to make the system critical (see 4.56 and 4.57 of Ref. [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 differs from the criticality factor k = (M-D/M characterizing the actual multiplication of the driven subcritical system; in fact, kgff is just the criticality factor kj[ of the fundamental mode of the subcritical system. In order for the two factors to coincide, the outer source should be homotetic to the fundamental eigenmode of the system. Using k e ff only, and the standard theory of critical reactors, to characterize an EA would therefore lead to significant qualitative and quantitative errors. It is also important to analyze what happens in a subcritical system when the source driving the steady operation is suddenly quenched. In this case the amplitude of the n-th mode of the flux distribution decreases exponentially 5, as exp(-t/t n ), where t n is the relevant "prompt neutrons lifetime", which is related to the eigenvalue B n [4,5]. For instance, for a homogeneous system with macroscopic absorption cross section E a, and neutrons with a single velocity v, one has t n = [v X a (l-koo+b n 2 L 2 )]~ = M n [v a (l+b n 2 L 2 )]~. Since t n decreases rapidly with n, after a short transient the characteristic time scale will be set by ti (and then by the eigenvalue Bi). 5 Delayed neutrons have been neglected here for simplicity, in view of the qualitative nature of the discussion.

8 As far as safety is concerned, one is also interested in the behaviour of the subcritical system in absence of the outer source. This concerns for instance the case in which the accelerator is switched-off and the system reactivity undergoes changes due to radioactive decays 6. The preceding discussion indicates that, since now the only neutron source is the internal fission source, and since the response of the system to sudden changes is dominated by the first eigenmode, then the criticality factor relevant to this case is that of the fundamental mode, ki = k e ff. In this connection, of particular interest is the evaluation of the "safety margin" of the system, which we can measure by means of the parameters and Ak crit =l-k c{{ (4) M* rit (M) = k 1 - ^(M) = k^m) I -11 (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, is characterized by a criticality factor k, depending on the system as well as on the source, which differs from (and, as shown later in details, is typically larger than) the effective multiplication coefficient k e ff = ki, which only characterizes the fundamental eigenmode. The coefficient ki= k e ff, instead, is relevant to some safety properties of the system. Given the fact that k and k e ff are, to some extent, independent, one can try to design an EA taking advantage of the distinction between 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 to M eff =l/(l- eff ), i.e. the amplification computed by using Eq. (1), but replacing k with k e ff, or as the ratio of the actual criticality margin to (1-k). For comparison with recent simulations of subcritical devices [8,9], we also introduce the function (p*, defined as the ratio <p* = \xl li^a, of the number (I of outer neutrons per fission that are required to provide steady operation, to the number (Li e ff 6 e.g. caused by the decay of Pa-233 to U-233.

9 that would be required if the outer neutrons were distributed just as the fundamental mode of the system (that is if their number were computed by replacing k with k e ff). Such a function is often referred to as the importance of the (outer) neutron source. [8] or the effectiveness of the outer source [9] 7, and can also be written as *2L l ^ l ) = -*-F». (7) The operational safety margin on k^ for an EA with multiplication M, can then be written as ^ ^ (8) M Notice that if one had evaluated this last margin by using the factor k computed from the amplification M, one would have got Just to give the flavour of some of the aspects involved, we anticipate here the results of some of the simple, analytic calculations described in Sec. 4. We refer to the case of a spherical device, consisting of an inner diffusing zone, followed by the fissile region, in turn enclosed by a breeder shell or by an outer diffusive shell (more details on the parameters are given in Sec. 4). The system is driven by a central point source. Such a schematization is taken as a rough, "spherical equivalent" of the geometry of typical Energy Amplifiers. In Fig. 1 we show the behaviour of the quantity F* as function of k e ff. It is seen that F* is a growing function of keff and is always well above unity. In the range of k e ff of interest for an EA, namely keff = , from curve I of the figure one has F* « This means that once a given value of k e ff is fixed, the system amplification is considerably larger than naively estimated from (l-keff^1. The same results are presented in two alternative forms in Figs. 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 are discussing can be either viewed as an "increased amplification" at a given k e ff, or as an "increased safety margin" in correspondence to a given value of M. (p * 7 More properly (p* is the importance of the outer neutron source normalized to that of the internal source.

10 II LL Figure 1. Function F* (ratio of the actual safety margin to (1-k) vs k e ff, as computed by a simple diffusive model for two cases with geometry shematizing that of a typical energy 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]. In both cases F* is well above unity and grows with k e jf. keff 1 -k Figure 2. For the same system as in case I of Fig. 1, we show the "real" safety margin (1- k e ff) 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.

11 200 g 100- CL CO I 0 - ^ ^ <. - " " 1/(1-keff) 10-h Figure 3. For the same system as in case I of Fig, 1, we show the neutron amplification versus the criticality factor k e ff. The dashed line is the amplification that one would compute using k e ffinstead ofk in Eq. (1). The vertical and horizontal arrows indicate, respectively, the difference in the amplification at given k e jf and the difference in k (and hence in the safety margin) at given amplification, due to the distinction between the two criticality factors, k and k e ff. keff It is worth noticing that, despite the oversimplified nature of the model used to generate the results of Figs 1-3, they are in qualitative agreement with those of the few available complex Montecarlo simulations of similar systems [8]. Just to appreciate typical values of the relevant parameters, we recall that an EA optimized for power production (e.g. the 1500 MWt unit described in Ref. [1]) operates with M = 50 (and then k = 0.98). If F* = 1.5, then, according to Eq. (5) the effective multiplication factor is k e ff ~ Hence, Ak cni = 0.03, 50% larger than what would be assumed by naively confusing k e ff with k. As far as the margin on k^ is concerned, since EA fuels have k^ ~ 1.20, we compute a "safety margin" Akoo crit (M=50) = Outline of the paper. The rest of this paper is devoted to a more detailed analysis of the effects outlined above. Very simple analytical models, based on the diffusion equation for monoenergetic neutrons, are used to estimate the multiplication, k e ff, F*, etc. of simple systems, and to gain insight in the parameters controlling the function F*. To make the presentation smoother, the formalism is

12 described in Sec. 2 (with details given in the Appendices), while the physical results are presented and discussed in Sees. 3 and 4. First, the simplest case of a homogeneous, bare sphere, with a finite-size source is studied in detail, by means of the harmonic expansion of the solution of the diffusion equation. This treatment, in particular, allows to identify the contribution of the different harmonics to the amplification. The dependence of F* on the size of the source, on the criticality factor, and on the properties of the subcritical sphere is then discussed. Although useful for gaining insight, the homogeneous sphere model cannot be taken as a reasonable simplification of the structure of an EA, which is characterized by several "regions" of materials with different properties. Spherical systems, with concentric "zones" of different materials, and with a central, point-like neutron source, are then considered, as schematic representations of the geometrical structure of an EA. Despite their simplicity such models yield a spatial distribution of the neutron flux, and its dependence on k in qualitative agreement with accurate simulations. They also reproduce trends for source importance observed in some Montecarlo calculations of realistic systems [8]. Such results justify using the above simple models for getting further insight. In particular we consider how the shape of the flux can be controlled and the function F* maximized. In general, it is found that F* 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 is increased In conclusion, it is worth stressing that the treatment presented here, being based on over-simplified models, only aims at getting insight in some aspects of the problem. Any quantitative study of an accelerator driven subcritical fission device necessarily requires using treatments which can handle both the full energy dependence of the neutron flux (and make use of accurate cross sections) and the actual geometry of the system. It is hoped that the present results may provide guidance to such complex simulations.

13 2. BASIC FORMALISM 2.1. Source neutron amplification We want to compute the amplification M of the source neutrons in a multiplying medium of volume V and surface S. We start from observing that, by definition, at steady state the rate of neutron generation and that of neutron disappearance (absorption, A, plus escape, E) are equal. Since the rate of generation is given by the source rate Q (spallation neutrons per unit time), times the amplification M, then and ^ ^ (10) Q The rates of absorption and escape are given, respectively, by A =Jj5: a 0 dsdv (11) vs E =jjj-n dgds (12) ss where < ) is the neutron flux, Z a is the macroscopic absorption cross section, J is the neutron current, n is the outward directed unit vector normal to the system surface, and E is the neutron energy. For simplicity, from now on we refer to neutron source amplification can then be written as mono-energetic neutrons 8, The M Q J-iidS and the criticality factor k can eventually be computed by means of Eq. (1) Diffusion equation To compute the neutron flux and the neutron current we use diffusion theory, according to which the current is given by 8 Again, we stress the roughness of the approximation, also in view of the presence of the spallation source, with a spectrum radically different from that of a fission source.

14 10 /= - >V0, (14) where D = lt r /3 is the diffusion coefficient, lt r is the neutron transport mean free path, given by \ r = (E t - // 2s)" 1 / where St, E s, and E a, are respectively the macroscopic total cross section, the scattering cross section and the absorption cross section, and fi is the average value of the cosine of the scattering angle in the laboratory system. Since in an EA the fuel is cooled (and the neutrons diffused and moderated) by a high-z material, then one can take / tr = (Z a + E s ). The neutron flux is the solution of the equation V ^0+^ = 0, (15) where C is the contribution of the external source (neutrons per unit volume and unit time), BM is the so-called material buckling ^ = ^ (16) koo and L are, respectively, the infinite multiplication coefficient and the diffusion length: ^ [ (17) v is the average fission multiplicity, and Z a is the macroscopic cross section. In the usual case of no incoming neutron current, Eq. (15) is solved by imposing that the flux vanishes at the extrapolated surface, i.e. a virtual surface lying at a distance l ex tr ~ 0.72 lt r outside the material boundary [4,6] 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 the characteristic "wave equation" V V = 0, (18) 9 The material presented here up to Eq. (29) follows closely that of Ref. [6], where this formalism, discussed e.g. in the textbook [4], was applied for the first time to the EA.

15 11 which form a complete orthonormal basis, each eigenvector y/ n corresponding to an eigenvalue B n. 10 We normalize the eigenfunctions in such a way that jyf^dv = l; n and introduce the volume integrals of the eigenfunctions : We then write the (known) outer source as with the expansion coefficients given by %=ly/ n (x)dv. (19) C(x)= ic n y/ n (x), (20) n=\ c n =j v C(x)y/ n dv, (21) so that the space integrated source neutron rate can be written as Q = \ v C{x) dv= f n=\ The (unknown) neutron flux can be expanded in the same basis, too, and a straightforward solution is found for a homogeneous medium. Indeed, in this case, 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 the coefficient of the flux as a function of that of the source : where Zal-flL-J^L 2 ) B'L 2 (25) n As anticipated in the introduction, we see from Eq. (25) that if all k n f s are smaller then unity, then the flux is given by a linear superposition of eigenmodes; as 10 To the purpose of keeping the notation simpler we refer again to a one dimensional geometry, e.g. a spherically symmetric system, characterized by a single infinity of eigenvalues. 11 Notice that the integrals are over the actual volume of the system, not over the extrapolated one.

16 12 soon as ki = 1 the system becomes critical; the source is no more needed to sustain the 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 E n =-DjVy/ n - n ds = -DjV(Vy/ n ) dv = -DjV 2 y/ 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 *F n we can eventually write E n = - DJVVn dv = DB 2 n jys n dv = DB^f n. (28) V V From Eqs. (26) and (28) we immediately find the non-escape probability for neutrons in the n-th mode: pnon-escape _ Az _ 1 " A - l confirming that the factor k n defined by Eq. (25) is just the criticality factor of the n-th mode. By introducing the above expressions for the absorption and the leakage into the definition of the amplification factor [Eq. (13)] we get Q Qn=\ lea^d ^L 2 ), (30) and then, using Eq. (24), we can write M=lM n^, M n=t 1 r, (31) which make transparent that the volume-integrated n-th mode of the source is amplified by a factor M n. and that the source neutron amplification is obtained as a linear combination of the individual mode amplifications, each weighted by the corresponding normalized mode of the source It is also clear that the M n 's form a decreasing succession (because the B n 's form an increasing succession and then the k n 's a decreasing one, k n+ i < k n ).

17 13 The limit k x > 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 ci^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 k n 's form a decreasing succession, and therefore the c rt '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 sphere 12, we shall see that x n <*: {-\) n In; then the positivity of the c f n s, which occurs for any "reasonably well contained source", is a sufficient condition for k > keff, and F* > 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 12 and in the limit in which the extrapolated boundary coincides with the physical boundary.

18 14 In addition, the usual boundary conditions that the neutron flux and current be continuous at the interfaces between the different zones, and that the neutron flux vanish at the extrapolated boundary, are imposed. From the knowledge of the flux distribution 0(r), the amplification M and the criticality coefficient k are computed by Eqs. (13) and (2), respectively. The effective multiplication coefficient k e ff, instead is computed by the standard technique outlined in Sec. 1 (that is, by identifying k e ff with the value by which the fission source must be divided to make the system just critical). At last, the functions F* and <p* are computed by means of Eq. (6) and (7), respectively. 3. HOMOGENEOUS SPHERE In this section we consider the simple case of a subcritical, homogeneous, bare sphere of radius R, driven by a spherically symmetric external neutron source. This is a highly simplified schematization of a real system, but allows for insight concerning the effect of the spatial distribution of the source, as well as on the parameters controlling the function F*. In particular, it is found that, once the source shape is given, the system is fully characterized by two parameters only, namely the criticality factor k (or k e ff) and the ratio of its extrapolated radius Rextto the diffusion length L. We refer to a steady source with radial profile e- rlr s -ocrlr C(r) = A c = A c (34) c rlr c rlr V ' where R s is a characteristic dimension, a = R/R s and A c is a normalization constant (see appendix A). Such a profile has been chosen because allows for simple solutions, and at the same time can model several situations of interest. E.g. if R s «15-20 cm it reproduces with reasonable approximation (for not too small radii) the distribution of a typical spallation source (see, e.g. Ref. [8]); in the limit /? 5 -» 0 it simulates a nearly point source; for small negative values of a (a ~ -2) it models a source distributed rather uniformly over the whole volume (see Fig. 4).

19 unit-source radial distribution = (Ac/r) exp(-r/r s ) radius, r (cm) Figure 4 Radial profile of the source term C(r) described by Eq. (34), for different values of the characteristic source size R s. We solve the diffusion equation according to the method illustrated in Sec The n-th eigenvalue and the n-th normalized eigenfunction are given, respectively, by Yn ^extr sin B n r (35) (36) where Ay is a normalization constant (see Appendix A), while the volume integrated eigenfunctions, defined by Eq. (19), are K n (37) where the function f% (see Appendix A) accounts for the difference between R and Rextiv it can be shown since in all practical cases the extrapolation distance l ex tr is a small fraction of the radius 13, then f% is very close to unity for all the modes contributing appreciably to the global amplification. Equation (37) shows that (neglecting the small correction due to f%) the integrals ^n form a succession of alternate size and decreasing absolute value. 13 E.g., in a typical EA, l tr ~ 3 cm, and l ex tr ~ 2 cm, while the size of the multiplying medium is about 1 m.

20 16 The coefficients of the source expansion, computed according to Eq. (21) are (38) where / n c is a function analogous to f% (see Appendix A). 3.1 Harmonic analysis of the amplificationthe amplification of the source neutrons can then be readily computed by inserting Eqs. (37) and (38) into Eq. (31), which gives with ' "T = h Mn (f) ftf) / M n = -V = T = T (40) 1 ^ Equations (38)-(40) allow for an analysis of the contributions of the different harmonics to the neutron amplification. For concretness we refer to the case illustrated by Fig. 5, with source characterized by a = 10, and a sphere with R ex tr/l=7, and k e ff = Figure 5a shows that the first harmonics of the source have opposite sign and decreasing amplitude. Such a behaviour only depends on the geometry of the system and on the shape of the source. Equation (38) shows that the sharper the source, the slower the decay of the source modes. The amplification of the individual modes is a more rapidly decreasing function of the mode number, as shown in Fig. 5b, so that only a relatively small number of modes contribute to the source amplification (see Fig. 5c). It is to be observed that the overall amplification M is larger than that of the first mode, Mi, or, equivalently k > ki = keff = (hence

21 17 10 mode number, n mode number, n ^ s up to r O) g CD O 13 a ampnncation ^ MiCi^VQ CV \ yv mode number, n Figure 5 Solution of the diffusion equation by mode expansion for a homogeneous, bare sphere, 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 apparent that the amplification converges rather rapidly; also notice that the global amplification is higher than the amplification Mj (that is, k is larger than kj = k e jf).

22 Scaling of F* with the source size and the sphere properties Next we study the variation of the quantity F*=(l-k e ff )/(l-k) with the "size" of the neutron source, and the criticality factor. Figure 6, referring to a sphere with R ex tr/l=7, shows that for "well contained" sources F* is well above unity and grows with k e ff. Typical values of F* for such sources are in the range Only in the academic case of a nearly uniform source, F* is smaller than unity and nearly independent on k e ff. The figure also shows that as criticality is approached F* tends to a finite limit, which is in general different from 1 and is independent on the value of Rextr/L. In the case of a point source, F*» 2 (see also Fig. 7), as is also recovered from Eq. (40). Indeed, when the difference between the sphere radius and the extrapolated radius is neglected, M= 2 M n (-l) n+1. As criticality is approached, only the fundamental mode contributes to the sum, and then M > 2 M x, and F*» 2. The effect of the size of the sphere (relative to the diffusion length L) is evidenced, for the case of a point source 14, by Fig. 7: for a given value of the criticality factor k e ff < 1, the smaller R/L the larger F*. uniform, bare sphere of radius R (with R/L = 7) source C(r) = (1/r) exp (-ar/r) "distributed" source, a = Figure 6 F* vs k e ff, for different values of the parameter a characterizing the dimension of the source relative to the sphere radius R, and for R extr /L = 7. It is apparent that for reasonably well contained sources, F* is greater than 1 and is a growing function of k keff 14 Such a case can either be obtained by using the solution of Sec. 3.1 and taking the limit of large a, or by the closed form solution illustrated in Appendix A.

23 19 uniform, bare sphere of radius R; point neutron source Figure 7 F* vs k e ff, for a homogeneous, bare sphere with a central point source, and different values of the ratio R/L of the sphere radius to the diffusion length. It is seen that the longer the diffusion length, the higher is the importance of the neutron source, that is the amplification for a given value ofk e ff. keff 3.3 Flux distribution (point source) In Fig. 8 we show the radial flux shapes for cases with a central point-source and different values of the criticality factor. It is seen that, far from the centre, the flux profile becomes less peaked as k approaches unity. On the other hand, the extremely high value of the flux close to the source demands the adoption of a more suitable geometry, limiting the fluence on the most exposed elements of the fuel (and of the supporting and confining structural materials). In particular, in the EA the goals of a more uniform utilization of the fuel and of the limitation of the peak flux to the fuel are achieved by shaping the multiplying fissile core (possibly surrounded by a breeder 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 the spallation neutrons are produced and partially moderated, followed by the fissile core, by the breeder, and by the outer lead. Schematizations of such a geometry are discussed in the next Section.

24 20 k = k = k = k = x c e normalized radius, r/r 1.2 Figure 8 Radial flux profile for a homogeneous, bare sphere, with central point source, for different values of the criticality factor k. (Here R extr /L =7). 4. MULTI-SHELL SPHERICAL SYSTEM WITH POINT SOURCE As the simplest geometrical schematization which retain the multi-region features 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. The solution of the relevant diffusion equation is detailed in Appendix C. The geometry of some of the cases considered in the following is described in table I, while the values of the absorption cross-sections assumed for the different materials are listed in Table II (For the transport cross section we have taken the same value X s = 0.33 cm" 1 in all materials). Here it is worth stressing that a drastic simplification of our model is the assumption of monoenergetic neutrons, which requires the use of spectrum-averaged cross sections. Our choices are rather arbitrary, but do not affect the observed qualitative trends. Two aspects deserve however some comment. First, from table II one sees that we have taken different absorption cross section for the central Lead buffer and the outer Lead diffuser. This has been motivated by the difference in the "average" neutron spectra in the two regions and the possible presence of additional structural materials. A second

25 21 comment concerns the procedure followed to compute the variation of the multiplication (and then of F*) with k. In practice, one can change the criticality factor either varying the fuel enrichment or the fuel/moderator volume ratio (that is the pitch between fuel rods). Here, for simplicity, we have taken both the macroscopic absorption cross section and the transport mean free path constant, and have changed 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 neutron source is located at the centre of the buffer. Table I - geometry of the studied cases a) bare system b) system with outer diffuser c) system with outer fertile layer d) system with two fissile layers, outer fertile layer outer radius outer radius outer radius outer radius buffer 30 cm buffer 30 cm buffer 30 cm buffer 30 cm fissile 100 cm fissile 100 cm fissile 100 cm fissile I 70 cm diffuser 200 cm fertile 150 cm fissile II 100 cm fertile 150 cm

26 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) E a (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 by Fig. 10 the flux distribution in the fuel region is now more uniform than in the case of the homogeneous, bare sphere. This is a well known feature [1], which however is worth being shown here, because it proves the qualitative agreement between the present, very simple treatment and the complex simulations required for the design of a device. Such a result supports using simple models for getting insight in the physics of subcritical systems. k = k = k = k = radius (m) buffer fissile core diffuser Figure 10 Radial flux distribution for the spherical systems described as case a) of table I, for different values of k. These profiles are analogous to those produced by complete Montecarlo simulations of a power producing Energy Amplifier (see Figs. 2.3 a-c in Ref. [11).

27 23 The choice of the distribution of the fuel and of the material surrounding the fissile core results from consideration of different nature (fuel utilization, material damage, interest in greater breeding etc.), which are outside the scope of the present work. It is however worth seeing how such design features affect the flux distribution and 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 the flux distributions in the case of i) bare fuel, of purely academic relevance; ii) single shell of fuel surrounded by a diffuser (the same as in Fig. 10); iii) single shell of fuel surrounded 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 radius (cm) Figure 11 Effect of the material surrounding the fissile core on the spatial distribution of the flux. The three curves refer, respectively, to cases a),b), and c) of Table I. The behaviour of F* vs k e ff 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 the homogeneous 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 the most active portion of the sphere. It is also found that F* is higher for a bare core than for 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

28 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 time making use of the escaping neutrons). We have also considered an additional case in which 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 the fuel, has negligible influence on the values of the quantity F* (compare curves c) and d) in Fig. 12) a) c). # 1 LL (a) bare A (b) outer diffuser m (c) outer breeder (d) 2 fissile layers + breeder Figure 12 Function F* vs k e fffor the multi-shell systems described in table L It is apparent that surrounding the fuel by layer of moderately absorbing material (e.g. a breeding material) results in an increase of F* y and therefore in an improvement of the safety features of the system. keff 5. CONCLUSIONS Accelerator driven subcritical fission devices, like the Energy Amplifier, can be characterized 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 on the properties of the source and of the medium. The second quantity is the effective

29 25 criticality factor (k e ff) 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 k e ff), 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 the criticality margin of the system, since it characterizes its response to sudden source changes, 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-k e ff)/(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, we have 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 that corresponding to its energy gain), and have shown how it is affected by features of the source and of the subcritical system. In general, at a given k e ff, F* grows as the "containment of the source" is improved (a point source being more favourable than a 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 design choices, 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 some reduction of F*, while an external absorbing layer has favourable effect. Our simple calculations seem to indicate that an Energy Amplifier could operate with F* = , that is with a considerable "additional safety margin" over the naive estimate based on the criticality factor k corresponding to the neutron amplification M. Such estimates of F* are roughly consistent with a few Montecarlo simulations of a subcritical device, somehow modelling an Energy Amplifier [8]. The complexity of the physics of a real system, however demand for accurate simulations taking into account both the energy dependence of the neutron population (and the relevant reaction cross sections) and the geometry of the device. A systematic work in this direction is planned for the near future.

30 26 6 REFERENCES. [1] C. Rubbia et al, "Conceptual Design of a Fast Neutron Operated High Power Energy Amplifier", CERN/AT/95-44 (ET), 29th September See also C. Rubbia, "A High Gain Energy Amplifier Operated with Fast Neutrons", AIP Conference Proceedings 346, International Conference on Accelerator-Driven Transmutation Technologies and Applications, Las Vegas, July [2] C. Rubbia et al. note in preparation, see also Sec. 5 of Ref. [1]; the high energy cascade is dealt with by the code FLUKA, described in A. 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, Nuclear Instruments and Methods A, 332, 459 (1993), also, CERN Divisional Report CERN/TIS-RP/93-2/PP (1993). [3] S. Andriamonje et al. Physics Letters. B 348 (1995) and J. Calero et al. Nuclear Instruments and Methods. A 376 (1996) [4] A. M. Weinberg and E. P. Wigner, "The Physical Theory of Neutron Chain Reactors", University of Chicago Press, Chicago (1958). [5] C. Rubbia, "An analytic approach to the Energy Amplifier", Internal Note CERN/AT/ET Internal Note (1994); see also Sec. 2.1 of Ref. [1]. [6] S. Glasstone and A. Sesonke, "Nuclear Reactor Engineering", 4th Ed., Chapman and Hall, New York (1991). [7] J. F. Breismeister (Editor), "MCNP - A General Montecarlo N-Particle Transport Code, Version 4A", Los Alamos National Laboratory Report, LA M (Nov. 1993). [8] C. Rubbia, F. Carminati and Y. Kadi, "EA Montecarlo Benchmark", to be published...??? [9] I. Slessarev et al., "IAEA-ADS Benchmark (Stage I), Results and Analysis", to appear in the Proceedings of the IAEA-TCM on ADS, Madrid, September 1997.

31 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 in Eqs. (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 radius and the extrapolated radius are CO i f /, sin nn XK V R extr v extr, and R extr { R extr ) na R ) y R extr

32 28 APPENDIX B Homogeneous, bare sphere, with central point source. Although this is just a special case of the multi-shell systems considered in Appendices C and D, we detail here the closed form solution for the case of a homogeneous, bare sphere, with radius R and extrapolated radius R e xtr = R + lextr/ and with a central point source, releasing Q neutrons per unit time. We define the positive quantities (Bl) The neutron flux can then be written as AnDr sinh[7 M (/? extr -r)] sinh(7 M /? extr ) sin[b M (R em - r)] sin(5 M /? extr ) for k^ < 1 / (B2) and the amplification is given by M = sinh(7 M i? extr ) - k co [y M Rcosh(y M l exti ) + sinh(7 M / ( (1-^) sinh(7 M /? extr ) extr/j for k <i,[%i?cos(5 M / extr )+sin(5 M / extr )]-sin(5 M /? extr ) 2 2 i : ^ for 1</: OO <1 + %L (B3) Notice that the criticality condition is immediately recovered from Eq. (B3); extr indeed, when M ~~ then M diverges; this is just the well known condition of equality between the material buckling and the geometrical buckling (the first eigenvalue of Eq. (18)); by using the definition of the material buckling (Eq. (Bl)) the criticality condition can be written in the usual forms [4,6] or