Neutron Moderation Spectrum Considering Inelastic Scattering

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Neutron moderation spectrum considering inelastic scattering

V.D. Rusova,∗ V.A. Tarasova, S.A. Chernezhenkoa, A.A. Kakaeva, V.V. Urbanevicha, V.M. Vashchenkob

aOdessa National Polytechnic University, Shevchenko av., 1, 65044, Odessa, Ukraine. bState Ecological Academy of Post-Graduate Education and Management, V. Lypkivskogo str., 35, bldg.2, 03035, Kyiv, Ukraine

Abstract For the first time an analytic expression was obtained for the inelastic neutron scattering law with an isotropic neutron source within the gas model, considering moderating medium temperature as a parameter. The inelastic scattering law is obtained, based on the solution of the kinematic problem of neutron inelastic scattering on a nucleus in laboratory coordinate system (L-system) in general case. I.e. in case not only a neutron but also a nucleus have arbitrary velocity vector in L-system. Analytic expressions are found for the neutron flux density and moderation spectrum in reactor fissile medium, both in case of the elastic scattering law, obtained earlier by the authors, and in case of the inelastic scattering law obtained in this paper. Both elastic and inelastic scattering laws are considered to be dependent on the medium temperature. The obtained expressions for neutron moderation spectra enable rein- terpretation of physical nature of the processes that determine the shape of neutron spectrum in a wide energy range.

1 Introduction

Neutron moderation theory is an important issue in nuclear reactor physics [1–9]. Conven- tional nuclear reactor physics theory of neutron moderation does not consider temperature of the moderating medium, because it is based on neutron scattering law not considering medium nuclei thermal motion. Within this theory the neutron moderation spectrum is

arXiv:1801.06142v2 [nucl-th] 27 Oct 2020 described by an analytic expression known as Fermi moderation spectrum [1–9]. Fermi moderation spectrum also does not depend on moderating medium temperature and therefore it cannot describe experimental neutrons moderation spectrum correctly in low-energy range, e.g. neutron spectra of thermal nuclear power reactors. Due to the fact that experimental neutron spectra for thermal nuclear reactors are similar to Maxwell distribution [1,3,7, 10, 11], an assumption has been made that neutron moderation spectrum can be described by this distribution in low-energy range. This assumption enabled development of widely used con- temporary semi-empiric calculation algorithm for neutron moderation spectra calculation. According to this algorithm the neutron moderation spectrum is a combination of Fermi moderation spectrum at high energies and the Maxwell’s distribution at low energies. Hence it contains a formula to determine neutron gas temperature based on moderating medium

∗Corresponding author: [email protected]

1 temperature which determines low-energy spectrum range [1–9]. And the formula itself was obtained in the past by numerical approximation of experimental spectra of several different thermal nuclear reactors available at that moment [1], and it is still widely used in nuclear reactor physics (see e.g. [5–9, 12–16]). Therefore in the strict sense there is no neutron moderation theory available, and creating one is an essential task. Let us also note, that the neutron moderation spectra in different media can be mod- eled by Monte-Carlo approach. This method is the basis for neutron moderation spectra calculation implemented in different reactor computer algorithms. This includes software with the corresponding interfaces designed to model the neutron transport, photons, etc in different media by Monte-Carlo method, like MCNP4, GEANT4 [17, 18] and others. They enable determination of neutron moderation spectrum by numerical solution of kinematics problem of neutron scattering scattering, taking into account a possible neutron absorption by moderating medium. This computation is repeated multiple times for random neutron and nucleus initial velocities to obtain resulting neutron moderation spectrum by averaging of the accumulated results across all samples. But this is rather a modeling approach than a theory. Lack of neutron moderation theory first of all hinders investigation of emergency modes of nuclear reactors and development of new generation of reactors physics, e.g. soliton fission reactors [19, 20], impulse reactors, neutron breeders (boosters), subcritical assemblies [21–23] and natural-occurring nuclear reactors, e.g. georeactor [24]. This paper proposes a further development of our recently published work [25] which provided a base theory for neutron moderation taking into account the temperature of the fissile medium. The base theory model of neutron moderation does not fully consider inelastic scattering reactions and therefore it is applicable to moderating media consisting of nuclei with negligibly small inelastic scattering cross-section. The current paper develops a neutron moderation theory for neutrons emitted by an isotropic neutron source considering inelastic scattering. Account for inelastic scattering reactions is especially important for moderating media consisting of heavy nuclei. According to [11], inelastic neutron scattering by heavy nuclei manifests itself at neutron energy higher than several hundreds keV, in contrast to the light nuclei, where inelastic scattering manifests itself at energy higher than few MeV. Our main interest is the physics of fissile neutron-multiplying media (reactor fuel), and therefore the neutron moderation theory was developed first of all to be applicable to such media. The assumption of neutron source isotropy corresponds to a common way of introducing the neutron source in fissile materials. Any fission medium (nuclear fuel during the reactor operation, but in general case any medium wherein a chain reaction takes place) is thermodynamically unstable due to fission processes followed by release of high amounts of energy and emission of neutrons and other particles. Moreover, the change in nuclide composition, thermal conductivity and radiation defects dynamics may sometimes lead to the medium deformation or even disruption. This way, the fission medium of the reactor where fission processes take place is an open physical system in non-equilibrium thermodynamical state. Such system may be described within the framework of non-linear, non-equilibrium thermodynamics of open physical systems. The non-equilibrium stationary states, satisfying the Prigogine criterion of minimum entropy pro- duction, may exist in such systems [26–28]. From non-linear non-stationary thermodynamics it is known that the emergence and the type of such stationary mode depend not only on system internal parameters (internal entropy) but also on boundary conditions (entropy flow at boundaries). For example, implementation of a stationary mode in a non-equilibrium system (so called non-equilibrium stationary state) demands constancy of boundary conditions, e.g. [27]. In this paper the following simplifications are used for building the model of neutron

2 moderation process in a fissile medium. Two thermodynamic sub-systems are distinguished: moderating neutrons sub-system and moderating medium nuclei sub-system. These are the open physical systems interacting with one another. According to the above argumentation, in reality both systems are in non-equilibrium state. However, in our model we use a simplification that the moderating medium nuclei sub-system is in state close to equilibrium due to its inertness with respect to perturbations. Therefore by neglecting the influence of neutrons on the nuclei sub-system, we assume that the latter is in thermodynamic equilibrium. This simplification enables us to introduce the moderating medium temperature into the model and to describe the nuclei kinetic energy by Maxwell’s distribution. We consider the moderating neutrons sub-system in non-equilibrium state, so we do not introduce the temperature for it in our model. Let us emphasize that, as noted above, in traditional approach to deriving neutron moderation spectrum a neutron gas sub-system temperature is introduced. This implies a simplified assumption that neutrons sub-system is also in thermodynamic equilibrium state which we avoided. Moreover, as noted above, the formula for the neutron gas temperature is connected with the moderating medium temperature through an empirical relation. Let us note that the anisotropy of elastic and inelastic scattering reactions was not considered, and this task is postponed for further theory development together with the problem of accounting for moderating medium nuclei interaction. Let us also note that the previous history of this topic is given in [25] in detail.

2 Kinematics of inelastic neutron scattering at moderating medium nucleus

If the neutron inelastic scattering is relevant to the medium (e.g. for heavy nuclei such as uranium-238), the kinematics presented in our paper [25] should be somewhat reﬁned. According to [5,6] neutron inelastic scattering reaction at a nucleus progresses through an excited compound nuclei state, i.e. at the ﬁrst phase of the reaction the neutron moderating medium nucleus captures a neutron:

1 ∗ (A, Z) + 0n → (A + 1,Z) , (1) 1 where (A, Z) is neutron moderating medium nucleus with mass number A and charge Z; 0n – neutron (0 – zero charge, 1 – neutron mass number); (A + 1,Z)∗ – a compound nucleus with mass number A + 1 and charge Z; ∗ – denotes nucleus excited state. During the next phase of the reaction the compound nucleus decomposes producing a nucleus of the neutron moderating medium in excited state and emitting a neutron. The nucleus is de-excited by emitting a γ-quantum:

∗ ∗ 1 1 (A + 1,Z) → (A, Z) + 0n → (A, Z) + 0n + γ. (2) Here we should note that the compound nucleus decomposition is a two-particle nuclear reaction, so the solution of the corresponding kinematics problem is unambiguous (see e.g. [29]). Let us consider neutron inelastic scattering at a moderating medium nucleus. Neutron moderating medium (following [25]) is described within the framework of gas model. I.e. it is assumed that the medium nuclei do not interact with one another, but possess kinetic energy due to thermal motion. Following [25] Let us introduce two laboratory coordinates systems (ﬁg.1): • a stationary laboratory coordinate system, the L-system

3 z0

z ~r (L0) (L) ~r (L) ~r O0 O0 y0

x0 O y x

Figure 1: L and L0 laboratory coordinate systems.

• a non-stationary laboratory coordinate system or L0-system that moves relative to L- system with a constant velocity equal to the velocity of thermal motion of the moderating medium nucleus at which the neutron is scattered. Let us note that here we consider a special case when the orientation of L and L0 coordinate systems is the same, and the radius vector of L0-system origin in L-system coincides with the moderator nucleus radius vector in L-system. So the moderator nucleus initially rests in L0-system. Here we use the same notation as in [25]: m1 = mneutron – neutron mass; m2 = mnucleus – nucleus mass; (L) ~r1 – neutron radius vector in L-system; (L) ~r2 – nucleus radius vector in L-system; (L) ~rC – inertia center radius vector in L-system; 0 (L ) 0 ~r1 – neutron radius vector in L -system; 0 (L ) 0 ~r2 – nucleus radius vector in L -system; 0 (L ) 0 ~rC – radius vector of inertia center in L -system ~ (L) V10 – neutron velocity before interaction with the nucleus in L-system; ~ (L) V1 – neutron velocity after interaction with the nucleus in L-system; ~ (L) V20 – nucleus velocity before interaction with the neutron in L-system; ~ (L) V2 – nucleus velocity after interaction with the neutron in L-system; 0 ~ (L ) 0 V10 – neutron velocity before interaction with the nucleus in L -system; 0 ~ (L ) 0 V1 – neutron velocity after interaction with the nucleus in L -system; 0 ~ (L ) 0 V20 – nucleus velocity before interaction with the neutron in L -system; 0 ~ (L ) 0 V2 – nucleus velocity after interaction with the neutron in L -system; 0 ~ (L ) 0 VC – inertia center velocity in L -system. The radius vectors of a point in L and L0 systems are connected as follows:

(L) (L) (L0) ~r = ~rO0 + ~r , (3)

(L) 0 where ~rO0 is the radius vector of L -system origin in L-system (see ﬁg.1). Therefore, according to the problem deﬁnition:

(L) (L) (L) d~r (L) d~r V~ = 1 6= 0 and V~ = 2 6= 0. (4) 10 dt 20 dt

4 In L0-system before the impact the nucleus rests, i.e.:

~ (L0) V20 = 0. (5)

0 The relation between coordinates of m1 and m2 in L and L -systems is given by (3). And the relation between velocities (inertial systems velocity addition law, based on Galilean relativity principle):

~ (L0) ~ (L) ~ (L) ~ (L0) ~ (L) ~ (L) V1 = V1 − V20 and V2 = V2 − V20 . (6)

Due to the fact that the moderating medium nucleus rests in L0-system, the solution of kinematic problem of inelastic scattering of the neutron at the nucleus in L0-system may be expressed similar to the solution of two-particle nuclear reaction kinematics with non-zero reaction thermal eﬀect, given in e.g. [29]. Indeed, according to the inelastic neutron scattering reaction schema given by (1) and (2), the thermal eﬀect of such reaction equals to emitted γ-quantum energy, further denoted by Eγ. Next, let us consider our problem in analogy with [29]. It is convenient to solve a two-particle interaction problem in a center-of-mass frame, i.e. in C-system. For inelastic neutron scattering reaction in C-system the total isolated system momentum conservation law is valid. However, the total kinetic energy conservation law in this isolated system does not work. From momentum conservation law for two impacting particles in C-system we obtain:

~ (C) ~ (C) ~ (C) ~ (C) P10 + P20 = P1 + P2 = 0, (7) where: ~ (C) P10 – neutron momentum before interaction with the nucleus in C-system; ~ (C) P1 – neutron momentum after interaction with the nucleus in C-system; ~ (C) P20 – nucleus momentum before interaction with the neutron in C-system; ~ (C) P2 – nucleus momentum after interaction with the neutron in C-system; As follows from relations (7):

~ (C) ~ (C) ~ (C) ~ (C) m1 · V10 = −m2 · V20 and m1 · V1 = −m2 · V2 . (8)

Velocity modulus can be obtained from (8):

~ (C) (C) m2 ~ (C) m2 (C) V10 = V10 = V20 = V20 (9) m1 m1 and ~ (C) (C) m2 ~ (C) m2 (C) V1 = V1 = V2 = V2 . (10) m1 m1

By introducing neutron Aneutron = 1 and nucleus Anucleus = A mass numbers and assum- ing m1 = mneutron ≈ Aneutron = 1 and m2 = mnucleus ≈ Anucleus = A we obtain the following expressions for (8) and (9)-(10):

~ (C) ~ (C) ~ (C) ~ (C) V10 = −A · V20 and V1 = −A · V2 , (11) and (C) (C) (C) (C) V10 = +A · V20 and V1 = +A · V2 . (12)

5 The moderating medium nucleus and neutron inertia center coordinates may be expressed in the following way: (L0) (L0) (L0) 1 ~r = (1 · ~r + A · ~r ) · . (13) C 1 2 A + 1 Considering the fact, that in L0-system the initial nucleus velocity prior to impact is ~ (L0) V20 = 0, the inertia center velocity for an isolated system consisting of two particles (i.e. the neutron and the nucleus) in L0-system is the following:

(L0) 1 (L0) V~ = · V~ . (14) C A + 1 10 Due to total momentum conservation law the inertia center velocity in L0-system doesn’t change after the impact. Therefore we omit corresponding indexes denoting values prior and after the impact for the inertia center velocity. Due to the fact that inertia center coordinate system (C-system) moves relative to laboratory system with velocity of inertia center in L0-system, the neutron velocity prior to impact in C-system is: ~ (C) ~ (L0) ~ (L0) V10 = V10 − VC . (15) Substituting it into (14) we obtain:

(C) (L0) 1 (L0) A (L0) V~ = V~ − · V~ = · V~ . (16) 10 10 A + 1 10 A + 1 10 Using (8) and considering (14), we ﬁnd the nucleus velocity in C-system prior to impact:

(C) 1 (L0) V~ = − · V~ . (17) 20 A + 1 10 ∗ 1 The total kinetic energy of the system (two-particle system consisting of (A, Z) and 0n) after the reaction T (C) equals to:

(C) where T0 – total kinetic energy of the system before the reaction. Using (16) and (17), we can express the total kinetic energy of the system prior to reaction:

(C) 2 (L0)2 P~ µ V (C) 10 10 T = = , (19) 0 2µ 2 where µ = A/(A + 1) – reduced mass of the particles prior to reaction. Then kinetic energy of the system after the reaction according to (1) and (2) equals:

(C) 2 (C) 2 (L0)2 P~ P~ µ V (C) 1 10 10 T = = − Eγ = − Eγ. (20) 2µ 2µ 2 The neutron momentum modulus after the reaction can be expressed based on (20):

q r ~ (C) 0 (C) 0 (C) P1 = 2µ T = 2µ T0 − Eγ = v u  0 2  u (L ) u µ V10 = u2µ0  − E , (21) t  2 γ

6 ~ (C) V1

~ (L0) V1

~e1 θ ~ (L0) VC

Figure 2: Velocity parallelogram in L0-system after the impact.

0 2 where µ = (A + Eγ/c )/(A + 1) – reduced mass of the particles after the reaction, where c is the speed of light in vacuum. Denoting a unit vector along the neutron velocity in C-system after the reaction as ~e1, according to (21) we obtain: v u  (L0)2  u µ V (C) u 10 V~ = u2µ0  − E  · ~e . (22) 1 t  2 γ 1

Let us note that the angular distribution of inelastic neutron scattering may be almost always considered spherically-symmetric [5,7] just as for the elastic scattering. The neutron velocity vector after the reaction in L0-system may be obtained by adding neutron velocity vector after impact in C-system (22) and velocity vector in C-system (14):

(L0) (C) 1 (L0) V~ = V~ + V~ . (23) 1 1 A + 1 10 Based on velocity parallelogram shown in fig.2, square of neutron velocity modulus in L0-system after the impact is: 2 (L0)2 (C)2 (L0) 1 V~ = V~ + V · + 1 1 10 A + 1 (C) (L0) 2 · V1 · V10 + · cos θ, (24) A + 1 where θ is neutron exit angle in C-system (fig.2). The energy of the emitted γ-quantum, Eγ, is confined within the range:

0 2 ~ (L ) A V10 0 ≤ Eγ ≤ · < E , (25) A + 1 2 f where Ef – is the margin neutron energy, that causes ﬁssion of the compound nucleus (A + 1,Z)∗. As follows from neutron and nucleus (at rest in L0-system) isolated system total energy conservation law, the compound nucleus excitation energy equals to (see e.g. [5,6]):

∗ A (L0) E = εn + T , (26) A + 1 n (L0) where εn – neutron bond energy in the compound nucleus, Tn – neutron kinetic energy prior to scattering in L0-system.

7 During decomposition of the compound nucleus into a moderating medium nucleus in a non-excited state, neutron and γ-quantum (see reaction schema (2)) a fraction of the excited nucleus energy εn denotes the work against strong interaction forces during neutron emission, and the remaining excitation energy fraction equals to sum of system kinetic energy (nucleus and neutron) in C-system after scattering and γ-quantum energy:

(C)2 P~ A (L0) 1 T = + Eγ. (27) A + 1 n 2µ0 A stochastic multitude of reactions of neutrons inelastic scattering at moderating medium nuclei contribute to total neutron moderation spectrum, herewith the γ-quantum kinetic energy (and also square of the neutron momentum after scattering in C-system univocally related to it through (27)) varies randomly within the energy range determined by (25). There are no reasons to suggest that some value of γ-quantum kinetic energy is randomly preferred, i.e. that a gamma-quantum with some speciﬁc energy is somehow emitted more frequently than others. Therefore it may be assumed that a random γ-quants energy value within the inelastic scattering of a multitude of neutrons at nuclei of the moderating medium will be given by equally probable distribution and the probability density is determined by the following expression: ~ (C) 1 ρ(Eγ) = ρ P1 = . (28) (L0)2 A V10 A + 1 2 The average value of square of neutron momentum and average value of square of neutron velocity after scattering in C-system (which are equal because the neutron has Aneutron = 1) in case of equiprobable random distribution can be easily expressed as a sum of its maximal A (L0) (at Eγ = 0) and minimal (at Eγ = A+1 Tn ) values divided by 2:

 (L0)2  2 ~ (C)2 1 A V10 V = 2 + 0 = 1 2  A + 1 2 

2 2 2 1 A (L0)2 A B (L0)2 = V~ = V~ , (29) 2 A + 1 10 (A + 1)2 10 1 B2 = . (30) 2 Knowing average value of neutron velocity modulus after scattering in C-system, we can analogous to expression (22) multiply it by unit vector of neutron exit direction ~e1 in C-system and obtain an expression for average neutron velocity vector after scattering:

~ (C) AB (L0) V = V · ~e1. (31) 1 A + 1 10 Therefore, the average value of square of neutron velocity after scattering in L0-system can be obtained analogous to (31), i.e.:

0 2 2 2 (L ) (C) (L0) 1 V~ = V~ + V~ · + 1 1 10 A + 1 (C) (L0) 2 · V 1 · V10 + · cos θ. (32) A + 1

8 (C) Substituting V 1 from (31) into (32) we obtain the following expression:

0 2 (L ) · 2 2 + 2 cos + 12 (L0)2 V10 A B AB θ V = . (33) 1 (A + 1)2

(L0)2 The ratio of square of average neutron velocity after inelastic scattering V 1 to 0 2 (L ) 0 square of neutron velocity prior to scattering V10 in L -system can be derived from (33) considering inelastic scattering of multitude of neutrons that have equal velocity module prior to scattering. This ratio is also equal to ratio of average neutron kinetic energy after (L0) (L0) scattering E2 to neutron kinetic energy prior to scattering E1 :

0 2 (L ) (L0) V 1 E A2B2 + 2AB cos θ + 1 = 2 = . (34) 0 2 (L0) 2 (L ) E (A + 1) V10 1 By introducing inelastic scattering coeﬃcients in the following way:

AB − 12 AB + 12 α2 = and α2 = , (35) e1 AB + 1 e2 A + 1 expression (34) transforms into:

(L0) E α2 2 = e2 [(1 + α ) + (1 − α ) cos θ] . (36) (L0) 2 e1 e1 E1 Now, expressing the vectors of average neutron velocity after scattering in C-system through analogous vector of average neutron velocity after scattering in L-system, we can rewrite (36) in the following form:

2 ~ (L) (L) V − V~ (L0) 1 20 E = 2 = 2 (L0) ~ (L) ~ (L) E V10 − V20 1 α2 = e2 [(1 + α ) + (1 − α ) cos θ] . (37) 2 e1 e1 From expression (37) for inelastic scattering we obtain an expression analogous to elastic

9 scattering expression obtained in [25]:

2 ~ (L) V (L) 1 E α2 = 1 = e2 [(1 + α ) + (1 − α ) cos θ] × 2 (L) e1 e1 ~ (L) E 2 V10 10  2  (L) (L) (L) V V cos β V20 × 1 − 2 10 20 +  +  (L)2 (L)2  V10 V10 (L) (L) 2 V~ , V~ (L) 1 20 V20 + 2 − = (L)2 (L)2 V10 V10 α2 = e2 [(1 + α ) + (1 − α ) cos θ] − 2 e1 e1 2 (L) (L) αe2 V10 V20 cos β − [(1 + α1) + (1 − α1) cos θ] + 2 e e (L)2 V10 (L) (L) V V cos γ + 2 1 20 − (L)2 V10 (L) α2 1 · E − 1 − e2 [(1 + α ) + (1 − α ) cos θ] 20 , (38) 2 e1 e1 (L) A · E10

(L)2 (L) 1 ~ ~ (L) ~ (L) where E1 = 2 V 1 ; cos β – cosine of the angle between V10 and V20 , expressed ~ (L) ~ (L) through scalar product V10 , V20 in the following way: ~ (L) ~ (L) ~ (L) ~ (L) V10 , V20 V10 , V20 cos β = = , (39) ~ (L) ~ (L) (L) (L) V10 V20 V10 V20

(L) ~ ~ (L) cos γ – cosine of the angle between V 1 and V20 :

(L) (L) ~ ~ (L) ~ ~ (L) V 1 , V20 V 1 , V20 cos = = (40) γ (L) . ~ (L) (L) (L) V V~ V 1 V20 1 20

As is be shown below, for our purposes it is suﬃcient to limit oneself to expression (38) without performing the rest of the transformations of the Eq. (38)-(40) and obtaining the exact solution of the considered kinematic problem. It would only yield a set of lengthy expressions, since the intermediate solution of the problem (38) includes the angles between the neutron and nucleus velocity vectors (39) and (40). These cosines would also require the adaptation to the L-system. Therefore reducing the solution to a single analytic expression is pointless within the present paper.

10 3 Neutron inelastic scattering law considering moderator nuclei thermal motion

According to the kinematics of the neutron inelastic scattering on a moderator nucleus given in Section2 by (38), the probability density function for scattering to a certain energy interval is determined by several independent random variables (θ, β, γ, EN ). Thus:

(L) (L) P θ, β, γ, EN = Pθ(θ) · Pβ(β) · Pγ(γ) · PEN EN (41) As known in nuclear physics, as long as the compound nucleus decomposition does not depend on history of its formation, the inelastic scattering of neutrons in inertia center coordinate system is spherically symmetric (isotropic). Therefore for P (θ)dθ we obtain:

2π 2π Z Z r sin θdφ · rdθ P (θ)dθ = [P (θ, φ)] dφ = = 4πr2 0 0 2π sin θdθ Z 1 = dφ = sin θdθ, (42) 4π 2 0 where φ denotes azimuth angle of spherical coordinates r, θ, φ, introduced in inertia center coordinates system. Due to the fact that thermal motion of the moderating medium nuclei is chaotic and the neutron source is isotropic (neutron source emits a neutron group with given energy and isotropic spatial distribution of their velocity vectors directions), the distribution of velocity vectors directions in space for neutrons after inelastic scattering is given by equiprobable random distribution law along β and γ angles as a part of (38). I.e. it is also spherically symmetric (isotropic). Therefore analogous to the previous case we obtain: 1 P (β)dβ = sin βdβ, (43) 2 1 P (γ)dγ = sin γdγ. (44) 2 Let us average kinetic energy of the neutron after inelastic scattering at a nucleus (given by (38)) across spherically-symmetric distribution of thermal motion velocities of the moderating medium nuclei the and across isotropic neutron source (across isotropic spatial distribution of neutron velocity vectors with given energy, emitted by the neutron source). Sequentially we obtain the following expression:

π π (L) Z Z (L) E1 = E1 P (β)dβP (γ)dγ = 0 0 2 (L) α = E e2 [(1 + α ) + (1 − α ) cos θ] − 10 2 e1 e1 " (L) #) α2 E − 1 − e2 [(1 + α ) + (1 − α ) cos θ] N . (45) 2 e1 e1 (L) A · E10

(L) Here E10 – neutron energy for isotropic neutron source averaged across neutron mo- (L) mentum directions, which is equal to energy of neutrons emitted by the source E10 , i.e.

11 (L) (L) (L) E10 = E10 . And EN is nucleus kinetic enerty, determined by Maxwell’s distribution for neutron moderating medium nuclei [30], which depends on temperature of the moderating medium as a parameter and has the following form:

" (L) #q (L) (L) 2 EN (L) (L) P EN dEN = p exp − EN dEN . (46) π(kT )3 kT Let us average the expression (45) across Maxwell’s distribution of the neutron moderating (L) (L) medium nuclei thermal motion (46), considering E10 = E10 and using known expression [30]

∞ Z (L) (L) (L) (L) 3 E = E P E dE = kT, (47) N N N N 2 0 we obtain the following:

(L) 2 (L) α E = E e2 [(1 + α ) + (1 − α ) cos θ] − (48) 1 10 2 e1 e1 " #) α2 3 kT − 1 − e2 [(1 + α ) + (1 − α ) cos θ] 2 . 2 e1 e1 (L) A · E10

(L) Since the functional relationship between E1 and θ is unique, as follows from (49), (L) (L) (L) the probability P E1 dE1 or the neutron with a kinetic energy E10 in the L-system (L) (L) (L) before scattering to possess the kinetic energy in the range from E1 to E1 + dE1 after the scattering on the chaotically moving moderator nuclei is determined by the P (θ)dθ distribution (42). Therefore we obtain the following relation (here we omit the symbols of averaging and the laboratory coordinate system L for simplicity), i.e.

dθ 1 P (E1) = P (θ) = sin θ× (49) dE1 2

" 2 2 3 # (L) α α kT E e2 (1 − α ) sin θ + e2 (1 − α ) sin θ 2 10 2 e1 2 e1 (L) A · E10 1 = . h (L) 1 3 i 2 E10 + A · 2 kT αe2(1 − αe1)

Therefore the obtained neutron inelastic scattering law considering thermal motion of moderating medium nuclei is the following:  dE1 P (E1)dE1 = h i ,  (L) 1 3 2  E10 + A · 2 kT αe2(1 − αe1)  2 (L) 1 3 (L) 1 3 in case αe2αe1 E10 + A · 2 kT ≤E1≤ E10 + A · 2 kT ; (50)  P (E1) = 0,   2 (L) 1 3 (L) 1 3  in case E1<αe2αe1 E10 + A · 2 kT ; E1> E10 + A · 2 kT .

12 In the conclusion of the section Let us stress that the law of inelastic scattering (50) is given for averaged neutron energy after scattering E1. The neutron energy averaging is made across thermal (chaotic) motion of the neutron moderating medium nuclei and across isotropic neutron source. As follows from the scattering law (50) all the neutrons emitted by the (L) isotropic neutron source and having energy E10 prior to scattering at the moderating medium nuclei, after inelastic scattering at moderating medium nuclei will have energy E1 averaged across thermal (chaotic) motion of the moderating medium nuclei and across isotropic neutron source with probability given by (50). As will be shown below, the obtained scattering law in form (50) enables obtaining expressions for neutron ﬂux density and neutron energy spectra in diﬀerent moderators as a function of temperature.

4 Neutron moderation in absorbing media containing several sorts of nuclei

According to [25], an analytic solution for stationary balance equation for moderated neutrons may be found as neutron flux density. In that case the the elastically scattered neutron flux is given in Eq. (51), where Q(E) is the quantity of neutrons generated with energy E per unit volume per unit time, Σel – total macroscopic cross section of elastic scattering of the P i i moderating medium (Σel = Σel, where Σel – macroscopic cross section of the i-th nuclide in i the moderator medium compound), Σt = Σs +Σa – total macroscopic cross section, Σs – total fission environment macroscopic cross section, Σa – total neutron absorption macroscopic cross section, ξel – modulus of average-logarithmic energy decrement for elastic scattering ξel, which is determined in analogy to standard neutron moderation theory (see e.g. [4–6, 25]), but through a new neutron elastic scattering law in [25]. Let us recall that Σs = Σel + Σin, where Σel = Σp + Σrs – total macroscopic cross section of elastic scattering, Σp – total macroscopic cross section of potential scattering, Σrs – total macroscopic cross section of resonant scattering, Σin – inelastic scattering macroscopic cross section, see e.g. [6].

∞ 0 " E0 # Σ (E ) 00 00 R el 0 R Σa(E )dE 0 Σel(E) Q(E ) · exp − 00 00 1 3 dE 0 |ξ |Σt(E )[E + · kT ] Q(E) E Σt(E ) E el A 2 Σ (E) Φ (E) = + t , (51) el 1 3 Σ ( ) E + A · 2 kT Σt(E) ξel t E The expression (51) contains probability function of resonant neutron non-absorption [4– 6, 25, 31], now containing moderating medium temperature:

 ∞  Z 0 0 Σa(E )dE ϕ(E) = exp − . (52)  0 0 1 3  ξel Σt(E ) E + A · 2 kT E Given the inelastic scattering law (50) and performing calculations analogous to the one made for elastic scattering in [25], we obtain an expression for inelastically scattered moderating neutrons ﬂux density in the following form:

∞ 0 " E0 # Σ ( ) 00 00 R in E 0 R Σa(E )dE 0 Σin(E) Q(E ) · exp − 00 00 1 3 dE 0 |ξ |Σt(E )[E + · kT ] Q(E) E Σt(E ) E in A 2 Σ (E) Φ (E) = + t , (53) in 1 3 Σ ( ) E + A · 2 kT Σt(E) ξin t E

13 where average-logarithmic energy decrement of inelastic scattering ξin is introduced by analogy to standard neutron moderation theory (see e.g. [4–6, 25]), but through inelastic neutron scattering law (50). After performing calculations similar to calculations performed for ξel in (24), we obtain the following expression:

2 ! α1 1 − α α1 E ξ = e ln α2α + e2 e + ln . (54) in 1 − e2 e1 2(1 − ) 1 3 αe1 αe2 αe1 E + A 2 kT Therefore, considering elastic and inelastic neutron scattering it is possible to obtain an expression for total moderating neutrons ﬂux density:

Φ(E) = Φel(E) + Φin(E). (55) And the neutron spectrum will be deﬁned by a standard expression:

√1 Φ(E) n(E) 2E ρ(E) = ∞ = ∞ . (56) R n(E0)dE0 R √ 1 Φ(E0)dE0 2E0 0 0 Let us note, that considering discreteness of energy levels of moderating medium nuclei at which neutrons are scattered inelastically, it is obvious that the inelastic scattering is possible not at any value of neutron kinetic energy, but with its kinetic energy above a threshold given by the following expression (see (26)): A + 1 Ethold = E∗ , (57) in A min ∗ where Emin is the minimal energy of excitation of the scattering nucleus. This is confirmed by experimental results, e.g. according to [11], neutron inelastic scattering at heavy nuclei is observed only at neutron energy higher than several hundreds kilo- electronvolts. And for light nuclei – at energies higher than one or several megaelectronvolts. For fission reactor environments Q(E) is determined by fission spectrum of the fission nuclide or fission nuclides combination, which according to [1,7, 25, 32, 33] may be given by the following expression: √ Q(E) = Qe · q · exp (−aE) · sh( bE), (58) where a, b and q – constants, given in table1 below, E – neutron energy nondimensionalized ∞ R by 1 MeV, Qe = Q(E)dE – total neutrons quantity generated in a unit volume in a unit 0 time.

Table 1: Constants determining the ﬁssion spectrum for main reactor ﬁssion nuclides [32]. U235 Pu239 U233 Pu241

a 1.036 1 1.05 ± 0.03 1.0 ± 0.05

b 2.29 2 2.3 ± 0.10 2.2 ± 0.05

q 2 q 0.4527 πe = 0.48394 0.46534 0.43892

Fig.3 shows the energy spectrum of neutrons moderating in hydrogen medium calculated using the Eq. (51). The neutron source was determined by the Eq. (58) for uranium-235

14 (Table1), the medium temperature was set 1200 K. The cross-sections of neutron reactions in hydrogen were taken from the ENDF/B-VIII.0 database [34]. This spectrum is compared to the one obtained by Monte-Carlo simulation using GEANT4 package [18], and demonstrates a rather good agreement.

108 107 eory GEANT4 106 105 104

Counts 103 102 101 100 10-6 10-4 10-2 100 102 104 106 108 Energy, eV

Figure 3: Comparison of the theoretical and Monte-Carlo spectra of moderating neutrons in hydrogen at 1200 K.

The comparison of the two neutron spectra in hydrogen moderator, obtained using different methods and presented in figure3 demonstrates a good agreement between them. As it was noted in our recent paper [25], the analytic expression obtained within the new theory of neutron moderation, describes the entire spectrum of moderating neutrons – i.e. in the entire band of possible energies depending on the moderator composition and temperature. It may contain two maxima (high-energy and low-energy) or a single maximum (high-energy for fast neutrons or low-energy for thermal neutrons, or an intermediate one). Such transformation of the theoretical spectrum is demonstrated below by the series of images depicting the spectra of neutrons moderating in uranium-carbon media of different composition. Fig.4 shows the energy spectra of neutrons moderating in homogeneous uranium-carbon medium. The theoretical curves were calculated using Eq. (56), and the Monte-Carlo simulated ones were obtained using GEANT4 software. The neutron source was determined by Eq. (58) for uranium-235 in all cases. The neutron reactions cross-sections for uranium-238 (Fig.6) and carbon (Fig.5) were taken from ENDF/B-VIII.0 [34]. From comparison of the corresponding curves in Fig.4 it may be seen that the theoretically calculated spectra are in good qualitative agreement with the Monte-Carlo simulated ones. Both curves match in a wide range for a 100% carbon medium. In the low energy range the curves somewhat differ, but the slope in log-log scale remains the same. With increase of the 238U portion in the moderator, our theoretical model gives noticeably lower values below few keVs, which is a subject of further investigation. The high energy part of the spectrum as well as the slopes in log-log scale are still very close though. The series of images demonstrates the gradual transformation of the neutron spectrum from fast to thermal with increase of the carbon percentage in the moderating mixture. Let us mention the sharp peak in the thermal part of the theoretical spectrum (seen in Fig.4). It is associated with the logarithmic energy decrement ξ crossing zero, since the neutron flux density formally tends to infinity in this case (see (51) or (53)). As it was noted in our recent paper [25], for the new scattering law, the energy decrement is not constant, in contrast to the standard moderation theory. Instead, it depends on the neutron energy and

15 109 109 8 8 10 eory 10 7 7 238 12 10 GEANT4 10 10% U + 90% C 106 106 105 105 4 4 ρ (E)dE 10 100%12C 10 103 103 102 102 101 101 10-6 10-4 10-2 100 102 104 106 108 10-6 10-4 10-2 100 102 104 106 108 109 109 8 8 10 10 238 12 107 107 20% U + 80% C 106 106 105 105 4 4 ρ (E)dE 10 10 238 12 103 1% U + 99% C 103 102 102 101 101 10-6 10-4 10-2 100 102 104 106 108 10-6 10-4 10-2 100 102 104 106 108 109 109 8 8 10 238 12 10 238 12 107 5% U + 95% C 107 50% U + 50% C 106 106 105 105 4 4 ρ (E)dE 10 10 103 103 102 102 101 101 10-6 10-4 10-2 100 102 104 106 108 10-6 10-4 10-2 100 102 104 106 108

109 8 10 eory 7 10 GEANT4 106 105 4 ρ (E)dE 10 100%238U 103 102 101 10-6 10-4 10-2 100 102 104 106 108

Energy, eV

Figure 4: Energy spectra of the moderating neutrons in uranium-carbon media of diﬀerent composition calculated using Eq. (56) (Theory, solid line) and using GEANT4 software (black dots). The initial spectrum was that of ﬁssion (58). The medium temperature is 600 K.

moderator temperature (see e.g. (54)). In the energy range below the thermal energies the logarithmic energy decrement ξ becomes negative. This seems reasonable, since such neutrons would obviously gain some extra energy when interacting with the thermally moving nuclei. So the neutrons are ”pushed” away from the zero energy, and our new theory of neutron moderation ”captures” this feature. Let us consider the point which corresponds to the zero value of the logarithmic energy decrement. The theoretical expression for the neutron ﬂux is divergent at this point. Ac- cording to the deﬁnition of the logarithmic energy decrement (e.g. [4–6, 25]), the energy of such neutrons after the interaction remains the same as before. So the neutrons with such energy tend not to change it in the course of interaction with moderator nuclei. Therefore

16 4 10 capture 102 inelastic 100

σ , barn elastic 10-2 10-4

10-6 10-4 10-2 100 102 104 106 108 Neutron energy, eV

Figure 5: The microscopic cross-sections for the neutron reactions in carbon at 600 K.

4 10 capture inelastic 102

100 ﬁssion σ , barn 10-2 elastic

10-4

10-6 10-4 10-2 100 102 104 106 108 Neutron energy, eV

Figure 6: The microscopic cross-sections for the neutron reactions in uranium-238 at 600 K.

such neutrons would accumulate. However if we look at the kinematics of two-particle elastic scattering, the case when the light incident particle after collision has the same energy as before, corresponds to the only point of the momentum diagram – when both particles after collision move along the same (L) (L) (L) (L) line in the L-system. I.e. P1 k P2 , where P1 and P2 are the momenta of the incident and scattering particles in L-system after collision (e.g. [29]). This point corresponds to the (L) (L) (L) (L) solution P1 = P10 and P2 = 0, P10 is the incident particle momentum in L-system after collision. This solution is traditionally considered as valid, but it is obviously not physical, since the situation when the classical particle is scattered on another one, while preserving its momentum vector, is not possible. So the point at which the logarithmic energy decrement is zero, should be excluded from consideration as non-physical. It is interesting though that when considering the inelastic scattering through the stage of compound nucleus, this point cannot be excluded. The neutron can be ejected from the compound nucleus with the same momentum vector it had before. This problem is smoothed over by the fact that the inelastic scattering is a threshold reaction (with the threshold about 105 keV), and its cross-section is zero for the neutron energy corresponding to the point of ξ = 0.

5 Conclusions

For the ﬁrst time a general analytic expression for inelastic neutron scattering law considering temperature of the moderating medium as a parameter is obtained for isotropic neutron source. Also analytic expressions for neutron ﬂux density and neutron moderation spectrum are obtained for isotropic neutron source in neutron moderating and absorbing media (e.g.

17 different reactor media), containing nuclei of different kinds and also depending on medium temperature. The obtained expressions for moderating neutrons spectra open a new way to interpret the physical nature of processes that determine the shape of neutron spectrum in low-energy range. We also found how the scattering cross-sections and the logarithmic energy decrement behavior impact the formation of the low-energy maximum in the moderating neutron spectrum. The nature of this maximum is coupled with non-stationary neutron system moderation by scattering at thermalized moderation medium nuclei system. Therefore its peculiarities cannot be explained within classic approach considering neutrons system in thermodynamic equilibrium and obeying Maxwell’s distribution. It is noteworthy the presented theoretical model allows to obtain the neutron moderation spectra for the case of non-equilibrium moderator, i.e. without introducing the moderator temperature into the model. This is also true for the basic theoretical model [25]. Indeed, our model remains rather generic until the Eq. (45), where an assumption about the thermal equilibrium of the moderator nuclei is made, and the Maxwell distribution is applied (Eq. (46)). Then, by averaging of the expression (45) across the Maxwell’s distribution, the moderator temperature is introduced. However, such assumption is not so indispensable. Suppose, the (L) (L) kinetic energy distribution of the moderator nuclei PM EN dEN is known. Then the 3 averaging of the expression (45) over this distribution would only change the 2 kT in (49) ∞ (L) (L) R (L) (L) (L) to EN , where EN = EN PM EN dEN is the mean kinetic energy of the moderator 0 3 (L) nuclei. This way, substituting 2 kT by EN in all subsequent equations, one would obtain a more general form of neutron moderation theory. The energy spectra of neutrons moderating in homogeneous hydrogen and uranium-carbon media are presented. The graphs were obtained using the analytical expression (56) and the GEANT4 Monte-Carlo code [18]. The comparative analysis of the neutron spectra obtained using both methods reveals a good agreement. Finally, a significantly different behavior of elastic and inelastic scattering cross-sections in different moderators (see e.g. ENDF/B-VIII.0 or [7]) provides opportunities for experimental investigation of their influence on the shape of maximum in neutron moderation spectrum in low-energy range, and also for the experimental testing of the obtained analytic expressions.

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