A new approach to computing the incoherent neutron scattering function

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Abstract

A new method for calculating the incoherent neutron scattering function for solids is introduced, based on the traditional phonon expansion, where the terms are calculated using the saddle point approximation for each term. The method is simple enough to be used in calculation systems of practical application. It provides very accurate results even for the tails of the scattering function, and is superior to the Gaussian approximation to the phonon terms. Numerical and experimental comparisons are presented in different types of magnitudes derived from the double differential cross section, using vanadium as a test case.

Introduction

Accurate representations of neutron scattering cross-sections for materials is an ongoing requirement for many subfields of pure and applied physics. From the experimental point of view, the constant progress in the neutron facilities allows increasingly refined measurements of different properties of the interaction of neutrons with matter. However, the experimental determination of cross sections (with different degrees of detail) has lagged behind in the general interest of neutron installations. Even when this activity has been revitalized as an urgent need arising from the calculation and design of new neutron facilities, the experimental knowledge of cross sections will always be possible only in limited ranges of neutron energy and impulse transfer. For this reason, it is essential that the experiments be accompanied by precise calculations based on physical models. Thus, over the years, many calculation methods based on first principles have been developed, among which those that can be applied in a practical way to computational calculation systems stand out.

The application that probably arouses the greatest interest in the knowledge of cross sections, is the accurate calculation of the neutron spectra, an essential issue in the design of neutron facilities in their different aspects such as moderation, shielding, background estimation and radiation protection, as well as in Nuclear Engineering calculations for reactor design.

In the field of cross-section modeling, the approach used for the treatment of the elastic component is clearly distinguished from that of the inelastic in terms of the strategies used to tackle the problem. While the elastic components (coherent and incoherent) are, in general, adequately treated, the inelastic ones depend on models that have been developed over the years for different types of systems.

The study of inelastic cross sections for thermal neutrons is determined by a detailed understanding of the dynamics of the system. In fact this is due to the distinctive feature of neutrons, which allows a direct experimental access to the details of the individual and collective dynamics of atomic movements in matter. From a theoretical point of view, these properties are contained in the dynamic structure factor or scattering law [1]. For a single crystal material, the inelastic scattering function has an extremely rich structure of peaks originated by the coherent scattering from single phonons. This structure is smoothed down in the case of multi-phonon scattering, and the multi-phonon part of the scattering function can be approximately obtained in the so called incoherent approximation, in which coherence effects are neglected [2]. Despite the simplifications that this approach implies, it has a wide field of applications in the calculation of cross sections, for instance, in a polycrystalline material, where the coherence effects can also be neglected in the cross section due to single phonons, because of the blurring caused by the average on the microstructure. Therefore, the scattering function can be calculated in the incoherent approximation [3], with the consequent simplification in the calculations that this entails. If the anisotropies are not severe, the incoherent scattering function is completely determined by the density of phonon states (DoS) in the case of harmonic solids, or more generally in the velocity frequency spectrum applicable to the description of liquids and anharmonic solids. In general reducing the incoherent dispersion function to a simple expression containing a density of phonon states is a reasonable and very practical approximation. This understood within the framework of the Gaussian Approximation, exact in the short and long time limits of the intermediate scattering function, and of wide application in the general theory of neutron scattering [4].

Some programs that require the calculation of scattering cross sections are traditionally based on pre-calculated data tables. The alternative to this procedure is to obtain the scattering function ”on the fly” in the calculation code. This alternative becomes a necessity in some cases, for instance when there are temperature variations and precision in the calculation is required. The complete evaluation of the scattering function requires performing a Fourier transform. Numerically, it is necessary to introduce a sufficiently dense grid with N points, in a large enough range of the integration variable. In addition, for high energy transfer, the integrand is highly oscillating and the convergence of the numerical procedure is slow. This means that N has to be large. Even using Fast Fourier Transform techniques, which allows to determine the integral for N values of energy with only a number of operations of the order of Nlog(N), instead of N2, the computational cost would be prohibitively high. Therefore, alternative ways to evaluate the scattering function have to be devised.

In practice, to calculate cross sections in the incoherent approximation, different mathematical techniques are often used. One of the most widely used, the phonon expansion, consists in developing the scattering law in terms that represent the number of phonons exchanged in the scattering process. Despite its name, the expansion is possible for harmonic crystals as well as for anharmonic crystals and liquids (in which the phonon concept does not apply), as a result of the formal equivalence between the vibrational density of states of a harmonic lattice and the generalized frequency spectra of the other systems. The basic way to calculate the expansion is through the recurrent convolutions of the successive terms. However, this procedure is not computationally efficient, since the calculation time for computing p terms increases roughly as Cp2, with a large prefactor C that is proportional to the number of operations needed to evaluate an integral with sufficient accuracy. For this reason, a standard simplification often used to compute the multi-phonon expansion is the Gaussian approximation [5], [6], [7] (which we will call MPGA so as not to be confused with the one mentioned above in the context of the development of the scattering function theories), in which each term of the phonon expansion beyond the one-phonon term is obtained as a Gaussian function with mean and standard deviation that increase with the order of the multi-phonon term, and which are easily calculable from the DoS. The MPGA works generally very well, although inaccuracies have been noticed in some ranges of momentum and energy transfer [8]. On the other hand, for sufficiently high momentum transfers the scattering function can be accurately obtained (without resorting to serial expansions) from the leading term of a saddle point expansion in powers of the inverse momentum transfer [9]. This approach is however very inaccurate at low momentum transfer.

In this work we propose a different approach that is still simple enough to be used in practice but is far more accurate than the MPGA for the lowest order terms of the phonon expansion, and thus provide more accurate differential cross sections for some energy ranges. It is based on a saddle point approximation for each term of the phonon expansion. It is easily implementable in a calculation program, and has the advantage that each phononic term can be calculated independently of the other terms, unlike the exact expansion in which to calculate a term of a given order, all lower orders must be calculated through a recurrence operation. From the computational calculation point of view, this constitutes a great advantage, since it allows the simultaneous calculation of terms through a parallelized scheme.

As the order of phonon expansion increases our model tends to the MPGA. Both, the approach developed in this work and the one presented in Ref. [9] are based on the mathematical formalism of the saddle point expansion, although they are very different. To distinguish them we call the former the saddle point approximation for the phonon expansion (SPPE) and the latter the saddle point approximation for the full scattering function (SPFS).

The paper is organized as follows. In Section 2 the main properties of the scattering function for harmonic lattices are reviewed in order to set clearly the problem and to introduce the notation. In Section 3 the SPFS is briefly described as an introduction of the SPPE, which is developed in Section 4. The MPGA is derived from the SPPE in Section 5. Section 6 is devoted to describe the numerical computations performed to compare the different approximations considered in this work. The paper ends with a summary of the conclusions.

Section snippets

The incoherent scattering function

The incoherent differential scattering cross section for an incident neutron of wave vector k and energy E scattered into a neutron of wave vector k and energy E can be written in terms of the target scattering function [1] S(Q,ω)=dt2πħeiωteiQr(0)eiQr(t)as d2σdΩdE=Nkkσinc4πS(Q,ω),where N is the number of scattering centers in the target, Q=kk is the scattering vector, and ω=(EE)ħ. For simplicity, we assume that all atoms in the target are equivalent, so that r(t) represents the

The saddle point approximation for the full scattering function in the short-collision time approximation

There are several methods to evaluate the scattering function adapted to different energy regimes. The short collision time approximation is attained at high momentum transfers, i.e. when the recoil energy is much higher than the maximum characteristic energy of the system’s frequency spectrum. In this section we will describe a method to determine the scattering law under this regime (studied by Egelstaff and Schofield in Ref [9]), as it is illustrative for the purposes of this work.

In this

The saddle point expansion for the multiphonon expansion

The saddle point evaluation of the full scattering function, accurate at the short-collision-time limit, fails at low momentum transfer, when q21. For any value of q2, the well known multiphonon expansion [7] S(q,u)=exp(q2γ0)p=01p!(q2γ0)pFp(u)is convergent provided the DoS is well behaved [see Eq. (19) below]. In the above equation γ0=γ(0), with γ(0)=11duZ(u)un(uωm).

The pth term of the expansion represents an interaction where p phonons are exchanged, and Fp(u)=ds2πγ(s)γ0peius.The

The multiphonon Gaussian approximation

The Gaussian approximation to the pth multiphonon term [5], [6], [7], already defined as MPGA and not to be confused with the Gaussian approximation to the correlation function given by Eq. (3), assumes that in multi-phonon effects the integral in Eq. (24) gets the main contribution form the neighborhood of s=0, which by virtue of Eq. (27) is true in the case when τ=0, in which case ξ(0)=1γ0 as commented above. Thus, the MPGA can only be accurate for u in the neighborhood of pγ0. As can be

Results and discussion

In this section we will illustrate the developed theory with numerical calculations as well a comparisons with experimental data. Firstly we will show the behavior of the saddle point for the phonon expansion, that as we anticipated, is a calculation that must be done only once, before calculating the desired terms in the expansion. Then we will show several derived magnitudes, comparing them with different approaches, and if possible with experimental data.

Conclusions

In this work we have presented a new method that allows us to calculate the phononic terms for the calculation of the incoherent cross sections of condensed matter in general. Indeed, this method can be applied to any system whose dynamics can be described by a generalized density of states, (which includes both harmonic and anharmonic systems) in which a phonon expansion to describe the incoherent scattering function can be performed. This method is based on a mathematical method called the

CRediT authorship contribution statement

V. Laliena: Conceptualization, Methodology, Software, Formal analysis, Writing - review & editing, Supervision. J. Dawidowski: Conceptualization, Formal analysis, Writing - review & editing, Methodology, Visualization, Validation, Project administration. G.J. Cuello: Investigation. J. Campo: Investigation, Project administration.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This project has received funding from the EU H2020 research and innovation programme under grant agreement No 654000 SINE2020. Grants No PGC-2018-099024-B-I00-ChiMag from the Ministry of Science and Innovation of Spain, i-COOPB20524 from CSIC, Spain, DGA-M4 from the Diputación General de Aragón (Spain), and 06/C563 from Universidad Nacional de Cuyo (Argentina) are also acknowledged.

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