Variational characterizations of the effective multiplication factor of a nuclear reactor core

We derive some new inf-sup and sup-inf formulae for the so-called effective multiplication factor arising in the study of reactor analysis. We treat in a same formalism the transport equation and the energy-dependent diffusion equation.


INTRODUCTION
The aim of this paper is to give some variational characterizations of the effective multiplication factor arising in nuclear reactor theory. This work follows a very recent paper by M. Mokhtar-Kharroubi [22] devoted to the leading eigenvalue of transport operators.
In practical situations, the power distribution in a stable nuclear reactor core is determined as the steady-state solution φ of a linear transport equation for the neutron flux. Because of interactions between neutrons and fissile isotopes, a fission chain reaction occurs in the reactor core. Precisely, when an atom undergoes nuclear fission, some neutrons are ejected from the reaction and subsequently shall interact with the surrounding medium. If more fissile fuel is present, some may be absorbed and cause more fissions (see for details Refs. [5,6,11,25]). The linear stationary transport equation is therefore of non-standard type in the sense that the source term is itself a function of the solution. When delayed neutrons are neglected, this equation reads with free-surface boundary condition (i.e. the incoming flux is null). Here, the unknown φ(x, v) is the neutron density at point x ∈ D and velocity v ∈ V, where D is an open subset of R N (representing the reactor core) and the velocity space V is a closed subset of R N , dµ(·) being a positive Radon measure supported by V . For the usual cases, dµ(·) is either the Lebesgue measure on R N (continuous model) or on spheres (multigroup model). The transfer cross-sections κ s (·, ·, ·) and κ f (·, ·, ·) describe respectively the pure scattering and the fission process. The nonnegative bounded function σ(·, ·) is the absorption cross-section [5,11,25]. The positive ratio k eff is called the criticality eigenvalue (or the effective multiplication factor). It represents the average number of neutrons that go on to cause another fission reaction. The remaining neutrons either fail to induce fission, or never get absorbed and exit the system. Consequently, k eff measures the balance between the number of neutrons in successive generations (where the birth event separating generations is the fission process). The interpretation of the effective multiplication factor k eff is related to the following three cases (see [5,11,25]): • If k eff = 1, there is a perfect balance between production and removal of neutrons. The reactor is then said to be critical.
• The reactor is sub-critical when k eff < 1. This means that the removal of neutrons (at the boundary or due to absorption by the surrounding media) excesses the fission process and the chain reaction dies out rapidly.
• When k eff > 1, the fission chain reaction grows without bound and the reactor is said to be super-critical.
Up to now, in practical applications, the effective multiplication factor k eff was usually given by r σ (T − K s ) −1 K f = k eff , where the precise definition of the operators T , K s and K f is given subsequently and r σ [B] denotes the spectral radius of any generic bounded operator B: r σ [B] = lim n→∞ B n 1/n . Because its requires the computation of the resolvent (T − K s ) −1 , such a characterization makes the analysis of the effective multiplication factor quite difficult to handle. In particular, practical estimates of k eff as well as computational approximations are rather involved and merely rely on (direct or inverse) power method [2]. Our main objective in this paper is to provide tractable variational characterizations of k eff in terms of the different data of the system and suitable test functions. We hope that such a characterization shall be of interest for practical computations or for the homogenization of the above criticality transport equation in periodic media [1,4,21]. We also believe that our characterization can be useful in the delicate optimization problem of the assembly distribution in a nuclear reactor (see the recent contribution [28] based upon the homogenization method and where the criticality eigenvalue k eff plays a crucial role).
To be precise, we provide here variational characterizations of the effective multiplication factor of the type where W + p is a suitable class of positive test-functions in L p (D × V, dxdµ(v)) (1 p < ∞). This result (Theorem 3.3) holds true under compactness assumption on the full collision operator and under positivity assumptions on the fission cross-section κ f (·, ·, ·). The main strategy to derive (1.2) is adapted from [22] where M. Mokhtar-Kharroubi proved similar variational characterizations for the leading eigenvalue of perturbed transport operators. Note that the above characterization still holds true for transport equations with general boundary conditions modeled by some nonnegative albedo operator (see Remark 3.6).
At this point, one recalls that, besides the critical eigenvalue k eff , it is also possible to investigate the reactivity of the nuclear reactor core through another physical parameter, namely, the leading eigenvalue s(A) of the operator A = T +K s +K f , also associated to positive eigenfunctions. The two parameters k eff and s(A) are related by the following: if s(A) < 0 then the reactor is subcritical (i.e. k eff < 1), while it is super-critical whenever s(A) > 0. The reactor is critical when s(A) = 0. The paper [22] provides a variational characterization of the leading eigenvalue of the transport operator A. However, for practical calculations in nuclear engineering, the critical eigenvalue k eff is a more effective parameter. Actually, the existence of the leading eigenvalue s(T + K s + K f ) is not always ensured but is related to the size of the domain D and the possibility of small velocities (for more details on this disappearance phenomenon, see e.g. [19,Chapter 5]). Since the existence of the effective multiplication factor is not restricted by such physical constraints, it appears more efficient to measure the reactivity of nuclear reactor cores by k eff . This is what motivated us to generalize the result of [22] and provide variational characterization of k eff .
In this paper, we also give a characterization of the criticality eigenvalue associated to the energydependent diffusion model used in nuclear reactor theory [5,24,25]. For this description, the critical problem reads complemented by the Dirichlet boundary conditions ̺(·, ξ) |∂D = 0 a.e. ξ ∈ E. Here E is the set of admissible energies ξ = 1 2 mv 2 (m being the neutron mass and v the velocity), i.e. E is a subset of [0, +∞[. The diffusion coefficient D(·, ·) is a matrix-valued function over D × E and the unknown ̺(·, ·) is nonnegative.
The derivation of diffusion-like models for some macroscopic distribution function ̺(x, ξ) (corresponding to some angular momentum of the solution φ to (1.1)) is motivated in nuclear engineering by the necessity to provide simplified models tractable numerically. Such a energy-dependent diffusion model can be derived directly from a phenomenological analysis of the scattering models or it can be derived from the above kinetic equation (1.1) through a suitable asymptotic procedure (see [10] and the references therein for more details on that matter).
For this energy-dependent diffusion model, we give a variational characterization of k eff in terms of sup-inf and inf-sup criteria in the spirit of (1.2).
Actually, to treat the two above problems (1.1) and (1.3) it is possible to adopt a unified mathematical formalism. Precisely, let us denote by K s the integral operator with kernel κ s (·, ·, ·) and denote by K f the integral operator with kernel κ f (·, ·, ·). Then, problems (1.1) and (1.3) may be written in a unified abstract way: where the unbounded operator T refers to, according to the model we adopt: • the transport operator: associated to the absorbing boundary conditions φ |Γ − = 0.
The abstract treatment of the above problem is performed in Section 2 and relies mainly on positivity and compactness arguments. The main abstract result of this paper (Theorem 2.15) characterizes the criticality eigenvalue of a large class of (abstract) unbounded operators in L p -spaces. Besides this main analytical result, we also prove abstract results with their own interest. In particular, we provide in Theorem 2.12 an approximation resolution for the criticality eigenfunction φ eff which shall be hopefully useful for practical numerical approximations.
The outline of the paper is as follows. In Section 2, we describe the unified and abstract framework which allows us to treat in a same formalism Problems (1.1) and (1.3) with the aim of establishing general inf-sup and sup-inf formulae for the criticality eigenvalue of a class of unbounded operator. In Section 3, we are concerned with the characterization of the effective multiplication factor k eff associated to the transport problem (1.1). In Section 4, we investigate the effective multiplication factor associated to the energy-dependent diffusion model (1.3).

ABSTRACT VARIATIONAL CHARACTERIZATION
This section is devoted to several abstract variational characterizations of the criticality eigenvalue. It is this abstract material that shall allow us to treat in the same formalism Problems (1.1) and (1.3).
2.1. Setting of the problem and existence result. Let us introduce the functional framework we shall use in the sequel. Given a measure space (Ω, ν) and a fixed 1 p < ∞, define and denote by X q its dual space, i.e. X q = L q (Ω, dν) (1/p + 1/q = 1). We first recall several definitions and facts about positive operators. Though the various concepts we shall deal with could be defined in general complex Banach lattices, we restrict ourselves to operators in X p (1 p < ∞): where ·, · is the duality pairing between X p and the dual space X q .
Let us denote the set of quasi-interiors elements of X p by X + p , i.e.
p , then f, ψ > 0 for any nonnegative ψ ∈ X q \ {0}. Definition 2.2. A bounded operator B in X p will be said to be positivity improving if its maps Notice that, given a bounded operator B in X p , if some power of B is positivity improving, then B is irreducible. This provides a practical criterion of irreducibility.
Recall also several fact about power-compact operators.

Definition 2.4. A bounded operator B in a
Banach space X is said to be power-compact if there exists n ∈ N such that B n is a compact operator in X.
The following fundamental result is due to B. De Pagter [8]. Let T be a given densely defined unbounded operator (2.1) Let K s and K f be two nonnegative bounded operators in X p . We are interested in the abstract critical problem: Let us introduce the family of operators indexed by the positive parameter γ: Therefore, solving (2.2) is equivalent to prove the existence (and uniqueness) of k eff > 0 such that 1 is an eigenvalue of (0 − T ) −1 K(k eff ) associated to a nonnegative eigenfunction. Such an existence and uniqueness result can be found in [19,Theorem 5.30] (see also [18]). We set Then, the spectral problem (2.2) admits a unique solution k eff > 0 associated with a nonnegative eigenfunction φ eff if and only if Remark 2.7. Notice that our assumptions differs slightly from that of [19]. Actually, in [19], it is assumed that (0 − T ) −1 K(γ) is power-compact and irreducible for any γ > 0. Our assumption implies those of [19]. Indeed, in this case, there is an integer becomes compact is independent of γ > 0. In the same way,

Remark 2.8. Under the assumptions of the previous Theorem, we point out that the mapping
. By analyticity arguments (Gohberg-Shmulyan theorem), it is also strictly decreasing. Thus, k eff is characterized by Let us now give some variational characterizations of the criticality eigenvalue k eff appearing in Theorem 2.6.

Abstract variational characterization of
(2.4) We start with the following characterization of k eff in terms of super-solution to the spectral problem (2.2). Proposition 2.9. Assume that (0 − T ) −1 K(γ) is power-compact and irreducible for any γ > 0. For any ϕ ∈ W + p , let τ + (ϕ) := sup{γ > 0 such that (T + K(γ)) ϕ is nonnegative } with the convention sup ∅ = 0. Then is power-compact and irreducible, a wellknown consequence of Krein-Rutman Theorem is that is nonnegative so that τ + (ϕ) k eff and consequently Assume now that k eff < sup Thus, for any n ∈ N, The following illustrates the fact that the extremal value to the above variational result is reached only by the nonnegative solution to the spectral problem (2.2).
and let ψ ⋆ ∈ X q be a nonnegative eigenfunction of the dual operator Then, by (2.5) and (2.6) which leads to a contradiction.
Conversely, if ϕ is a nonnegative eigenfunction of T + K(k eff ) associated to the null eigenvalue, then τ + (ϕ) k eff and the identity τ + (ϕ) = k eff follows from Proposition 2.9.

Remark 2.11. A careful reading of the proof here above shows that
Let us denote by φ eff the unique critical eigenfunction with unit norm, i.e. φ eff ∈ W + p satisfies Then one can prove the following approximation resolution for the criticality eigenfunction φ eff whose proof is inspired by [20,Theorem 7]. Such a result shall be hopefully useful for practical numerical approximation of the critical mode φ eff of the reactor.
where N is the integer given by Remark 2.7. Let us assume that 1 is a simple eigenvalue of (0 − T ) −1 K(k eff ). Moreover, in the case p = 1, let us assume that the dual operator (0 − T ) −1 K(k eff ) ⋆ admits an eigenfunction associated to its spectral radius which is bounded away from zero. Then, p is the unique positive eigenfunction of (0 − T ) −1 K(k eff ) associated to 1 and with unit norm .
Proof. According to the definition of γ k := τ + (ϕ k ), (T + K(γ k ))ϕ k is nonnegative. Therefore, This shows, according to (2.7), that ϕ k 1. Now, if 1 < p < ∞ there exists a subsequence (ψ k ) k which converges weakly to some ψ ∈ X p . If p = 1, the fact that γ k → k eff combined with the compactness of (0 − T ) −1 K(γ k ) N lead to the relative compactness of In particular, this sequence is equi-integrable and by domination (2.9), (ϕ k ) k is also equiintegrable. We can extract a subsequence (ψ k ) k converging weakly to some ψ ∈ X 1 . In both cases, the compactness of (0 − T ) −1 K(γ k ) N together with γ k → k eff yield to the strong convergence In particular ψ = 0, and, taking the weak limit in (2.8), ψ (0 − T ) −1 K(k eff )ψ. Now, according to Remark 2.11, this last inequality is actually an equality, i. e.
Iterating again, one gets Now, since 1 is a simple eigenvalue of (0−T ) −1 K(k eff ), the set of eigenfunctions of (0−T ) −1 K(k eff ) with unit norm reduces to a singleton. This shows that ψ is the (unique) weak limit of any subsequence of (ϕ k ) k so that the whole sequence (ϕ k ) k converges weakly to ψ ∈ X p . The remainder of the proof consists in showing that the convergence actually holds in the strong sense. Let us consider first the case 1 < p < ∞. To show now that ϕ k − ψ → 0, it suffices to prove that ϕ k → ψ . A consequence of the weak convergence leads to Since ϕ k 1 for any k ∈ N, this proves the Theorem for 1 < p < ∞. Let us now assume p = 1. Then ϕ k → ψ strongly in X 1 if and only if the convergence holds in measure, i. e., for any Ξ ⊂ Ω with finite dν-measure and every ǫ > 0 Arguing by contradiction, assume there exist Ξ ⊂ Ω with finite dν-measure, a subsequence still denoted (ψ k ) k and some δ > 0 and some ǫ 0 > 0 such that and, one deduces immediately from (2.10) that dν{ω ∈ Ξ ; |ψ k (ω) − ψ(ω)| > ǫ 0 /2} δ/2 for large k. (2.11) Now, let ψ ⋆ ∈ L ∞ (Ω) be a positive eigenfunction of (0 − T ) −1 K(k eff ) ⋆ associated to the eigenvalue 1, with ψ ⋆ bounded away from zero. One has Equivalently Now, the contradiction follows from the fact that ψ k , ψ ⋆ → ψ, ψ ⋆ .
The following characterizes k eff in terms of sub-solution to the spectral problem (2.2).
We are now able to characterize the effective multiplication factor k eff by means of Inf-Sup and Sup-Inf criteria, where we recall that K = K s + K f .

Theorem 2.15.
Under the assumptions of Theorem 2.6, if K f (X + p ) ⊂ X + p then the criticality eigenvalue k eff is characterized by the following: Proof. Let ϕ ∈ W + p be given, Since K f (X + p ) ⊂ X + p , one gets . By Proposition 2.9, and the infimum is attained for the criticality eigenfunction. Similarly, let Using Proposition 2.13, one proves that which ends the proof.

The class of regular collision operators.
We end this section by recalling the class of regular collision operators introduced in kinetic theory by M. Mokhtar-Kharroubi [19]. This class of operators will also be useful to study diffusion problems of type (1.3). We assume here that the measure space (Ω, dν) writes as follows: where dµ is a suitable Radon measure over V . Let K ∈ B(L p (Ω, dν)) be given by where the kernel k(·, ·, ·) is measurable. For almost every x ∈ D, define and assume that the mapping K : x ∈ D → K(x) ∈ B(L p (V, dµ)) is strongly measurable and bounded, i.e. ess sup x∈D K(x) B(L p (V,dµ)) < ∞.
The class of regular operators in L p spaces with 1 < p < ∞ is given by the following (see [ (1) For almost every x ∈ D, K(x) ∈ B(L p (V, dµ)) is a compact operator, In L 1 -spaces, the definition differs a bit. We have the following [17] Definition 2.17. Let K be defined by (2.15). Then, K is said to be a regular operator whenever The main interest of that classes of operators relies to the following (see Ref. [19] for 1 < p < ∞ and Ref. [17] for a similar result whenever p = 1): (2.15) be a regular operator in L p (D × V, dx ⊗ dµ(v)). Then, K can be approximated in the norm operator by operators of the form:

THE CRITICAL TRANSPORT PROBLEM
3.1. Variational characterization. This section is devoted to the determination of the effective multiplication factor associated to the transport operator. We adopt the notations of Section 2.3, namely Ω = D × V and dν(x, v) = dx ⊗ dµ(v). Throughout this section, we assume D to be a convex and bounded open subset of R N while µ is the Lebesgue measure over R N or on spheres. In particular, our results cover continuous or multi-group neutron transport problems but do not apply to transport problems with discrete velocities. Let where n(x) denotes the outward unit normal at x ∈ ∂ D. Let T be the unbounded absorption operator Here, the nonnegative function σ(·, ·) ∈ L ∞ (D × V ) is the collision frequency. It is assumed to admit a positive lower bound σ(x, v) c > 0 a.e. (x, v) ∈ D × V.
(3.1) Define the (full) collision operator K as the bounded linear (partial) integral operator The collision kernel Σ(·, ·, ·) is assumed to be nonnegative. In nuclear reactor theory, in a fissile material, this collision kernel splits as where Σ s (x, v, v ′ ) describes the pure scattering phenomena and Σ f (x, v, v ′ ) describes the fission processes. Define the corresponding linear operators As we told it in Introduction, we are interested here in the critical problem: where the eigenfunction ϕ is nonnegative and satisfies the boundary condition ϕ |Γ − = 0. We recall that the spectral bound of T is given by [29] with τ (x, v) := inf{s > 0 ; x − sv / ∈ D}. Therefore, by (3.1), we have s(T ) < 0. Moreover, so that (0 − T ) −1 fulfills (2.1). Let us now recall the irreducibility properties of (0 − T ) −1 K f for the continuous and multigroup models. The following result may be found in Ref. [19], Theorem 5.15, Theorem 5.16, (see also [29]).

Remark 3.2.
In the above case (1), corresponding to continuous models, it is possible to provide different criteria ensuring the irreducibility of (0 − T ) −1 K f (see for instance Ref. [13]). In the second case (2), which corresponds to multigroup transport equation, several different criteria also exist [23].
Using the notations of Section 2, we have the following characterization of the effective multiplication factor of the transport operator.

Theorem 3.3. Let us assume that K is a regular collision operator and that one of the hypothesis of Theorem 3.1 holds. The critical problem (3.2) admits a unique solution k eff if and only if
Moreover, .

Remark 3.4.
Denote by φ eff the nonnegative solution of (3.2), one can check that φ eff ∈ W + p . Therefore, in (3.5), the supremum and the infimum are reached for φ = φ eff .

Remark 3.5.
Note that it is possible to provide practical criteria that are satisfied in nuclear reactor theory and that ensure the existence of k eff [4,28]. Such criteria usually rely on dissipative properties of the pure scattering operator. Remark 3.6. It is important to point out that the above characterization is not restricted to the case of absorbing conditions but also holds for general boundary conditions modeled by some suitable nonnegative albedo operator. Actually, if one considers a transport operator T H associated to general nonnegative albedo boundary operator H which relates the incoming and outgoing fluxes in D [16], then the above theorem holds true provided (0 − T H ) −1 K is a power-compact operator in X p (1 p < ∞) when K s and K f are regular operators. This is always the case whenever 1 < p < ∞ by virtue of the velocity averaging lemma [16]. The problem is more delicate in a L 1 -setting and is related to the geometry of the domain D [27].

Necessary conditions of super-criticality and sub-criticality.
We shall use the result of the previous section to derive necessary conditions ensuring the reactor to be super-critical or sub-critical. Note that, for practical implications, a nuclear reactor can be operative and create energy only when slightly super-critical (i.e. 1 < k eff < 1 + δ with δ > 0 small enough), in this case, the whole chain fission being controlled by rods of absorbing matter. Throughout this section, we shall assume k eff to exist.
We shall provide lower and upper bounds on the effective multiplicative factor k eff only when the velocity space V is bounded away from zero. Recall that, since V is assumed to be closed, this means that 0 / ∈ V (see also Remark 3.14). For almost every x ∈ D, define K τ (x) as the following operator on L p (V, dµ): Then, one defines as in [22], the following Then, one has the following estimate: Proof. Assume ϑ < 1. Given ϑ ∈ (ϑ, 1), let ψ 0 ∈ L p + (V, dµ) be such that Let us consider then the following test-function is bounded and such an application ϕ 0 belongs to W + p since Then, for any γ > 0, one sees that Since Σ s 0 and 1 − ϑ 0, one sees that In particular, from the positivity of Σ f , one sees that, provided γ ϑ, −(T + K(γ))ϕ 0 (x, v) 0 for almost every (x, v) ∈ D × V . Then, from Proposition 2.13, this means that τ − (ϕ 0 ) ϑ and k eff ϑ.
Since ϑ > ϑ is arbitrary, one gets the result.
Remark 3.8. From the above result, one sees that the reactor is sub-critical whenever ϑ < 1. Note that the fact that ϑ < 1 implies k eff 1 is already contained in [22,Theorem 7].
The above result provides an upper bound of k eff leading to the sub-criticality of the reactor core. To get a lower bound of k eff , one defines a similar quantity Proposition 3.9. Under the assumptions of Theorem 3.3, if ϑ > 1, then k eff ϑ. In particular, for a reactor core to be sub-critical, it is necessary that ϑ 1.
where we recall that K τ (x) is an operator in L p (V, dµ).
In the same spirit, for almost every x ∈ D, define K τ f (x) as the following operator on L p (V, dµ): and let us define, as in [22], the set I f of all β 0 for which there exists ψ ∈ L p , for almost every (x, v) ∈ D × V. According to [22,Lemma 4] the set I is closed so that, if one defines When the velocity space is bounded away from 0 then, β f provides a lower bound for k eff : Proposition 3.11. Under the assumptions of Theorem 3.3, one has k eff β f .
Arguing as in the proof of Proposition 3.7, one sees that, since 0 / ∈ V , ϕ f ∈ W + p . Therefore, Theorem 3.3 ensures that .

Variational characterization.
In this section, we are concerned with the following where the unknown ̺(·, ·) is assumed to be nonnegative and to satisfy the Dirichlet boundary conditions where D is C 2 open bounded and connected subset of R N and E is an interval of ]0, ∞[. We will assume throughout this section that there exist some constants σ i > 0 (i = 1, 2) such that Moreover, we assume the measurable matrix-valued application D(·, ·) satisfies the following (uniform) ellipticity property and regularity assumption d ij (·, ξ) ∈ W 1,2 loc (D) for almost every ξ ∈ E. We will study Problem (4.1) in a Hilbert space setting for simplicity. Namely, set Let us assume the kernels Σ s (·, ·, ·) and Σ f (·, ·, ·) to be nonnegative and define the scattering operator and the fission operator We will assume K s and K f to be bounded operators in X 2 . Define then the full collision operator Let us introduce the diffusion operator with domain D(T ) = {ψ ∈ X 2 ; ψ(·, ξ) ∈ H 1 0 (D) ∩ H 2 (D) a.e. ξ ∈ E and T ψ ∈ X 2 } where H 1 0 (D) and H 2 (D) are the usual Sobolev spaces. With these notations, the spectral problem (4.1) reads According to the strong maximum principle, it is clear that s(T ) < 0 and (0 − T ) −1 (X + 2 ) ⊂ X + 2 . In order to apply Theorem 2.15, one has to make sure that (0 − T ) −1 K is power-compact and that (0 − T ) −1 K f is irreducible. Let us begin with the following compactness result which is similar to the usual velocity averaging lemma (see [14] and [19,Chapter 2]) for transport equations and is based on some consequence of the Sobolev embedding Theorem [7].
Proof. By Proposition 2.18, it suffices to prove the result for a collision operator K such that Moreover, by a density argument, one can also assume f and h to be continuous functions with compact support in E. Let us split K(0 − T ) −1 as: and M is the averaging operator It is enough to prove that M(0 − T ) −1 : X 2 → L 2 (D) is compact. Let B be a bounded subset of X 2 . One has to show that {Mg ; g ∈ (0 − T ) −1 (B)} is a relatively compact subset of L 2 (D). For any ϕ ∈ B, set g(x, ξ) = (0 − T ) −1 ϕ(x, ξ).