ELSEVIER Physica B 234-236 (1997) 1114-1116

Thermal neutron scattering by solids: Development and applications of a synthetic

scattering law

G.J . C u e l l o 1, J. D a w i d o w s k i , J .R. G r a n a d a *

Comisidn Nacional de Energia Atrmica and CONICET, Centro Atrmico Bariloche, 8400 Bariloche, Argentina

Abstract

We have used a recently developed synthetic scattering function for solids [1] to evaluate some quantities of interest with regard to different problems in the field of Neutron Physics. Our model, based on a new prescription to represent the multiphonon contributions in the scattering process, has shown to produce excellent agreement with full multiphonon calculations for integral magnitudes of the scattering law. It contains no adjustable parameters and is given in terms of analytical functions, which makes it very convenient in term of computing time. In this work we present calculated results describing the differential cross section of Vanadium, as observed in typical pulsed neutron source measurements. These calculations emphasize the effects of inelasticity and anisotropy in the neutron scattering by that material, and consequently, the limitations of its use as a normalizing standard.

Keywords: Incoherent scattering; Lattice dynamics; Phonons; Scattering functions

The central quantity to describe the interaction of slow neutrons with condensed matter is the Van Hove scattering function S(Q, co) [2], as it em- bodies all the structural and dynamical properties of the scattering system. While the Zemach- Glauber formalism 1-3] provides an essentially exact frame for the representation of S(Q, co), the resultant expressions are not quite amenable for calculations. As a consequence, different models and approximations were developed to describe the neutron scattering by solid systems, notably the Gaussian approximation [4"], the phonon E5] and the mass 1-6"] expansions.

* Corresponding author. 1 Also at CRUB, Comahue University, Argentina.

The phonon and mass expansions have shown to be highly successful in different applications, for example, in determining the frequency spectrum in solids from slow neutron experiments, or in the evaluation of angular distributions and total cross sections, respectively, for which those methods are especially suited I-7, 8]. Of course, in the frame of the Gaussian approximation to the (incoherent) intermediate scattering function, it is always pos- sible to generate S(Q, co) from the actual frequency spectrum, but the computation of the required double Fourier transform could be too lengthy a process. This is especially true in those situations where the detailed shape of the scattering function is not needed, but rather a fast, yet accurate, evalu- ation of integrals of the double differential cross section.

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We have considered the scattering system to be a crystal composed of atoms harmonically bounded to their equilibrium positions, and we have treated the inelastic components of the cross sections in the incoherent approximation [7]. Fur- thermore, we have used the Gaussian approxima- tion to the intermediate scattering function, which is, in fact, an exact form for some simple systems [9]. Assuming a Debye frequency spectrum and using the standard phonon expansion, we can ob- tain an essentially exact scattering law of a Bravais lattice. It is possible to demonstrate that the asymp- totic form of the complete S(Q, co) tends to the result corresponding to an ideal gas with the mass of the scatterer and at an effective temperature [9, 10]. Therefore, we have proposed a new model [1] to replace the slow-convergent phonon expansion, that combines one-, two- and three-phonons terms with the free gas model to provide a scheme for a fast evaluation of an approximated double- differential cross section. Even though the main objective of that work [1] has been the develop- ment of compact expressions for integral magni- tudes, rather than the double-differential cross sec- tion itself, it has been shown that its main charac- teristics are preserved.

Bearing in mind the widespread use of Vanadium as a normalizer in scattering experiments, we have chosen this material to perform calculations repres- enting the spectra that should be observed in typi- cal pulsed neutron source measurements. For this purpose we assumed a well thermalized incident spectrum, characteristic of that produced by a stan- dard H20 moderator, together with a detector effi- ciency corresponding to 3He tubes with 40 atm. filling pressure. In addition, we have taken the value R = 0.057 for the ratio between the scattering and incident flight-paths.

In Fig. 1, we compare the ratios of the calculated spectra for Vanadium and an ideal elastic scatterer (a), and between Vanadium and a gas with corres- ponding mass and effective temperature (b), for scattering angles of 55 °, 90 °, and 150 °. It is evident from these curves that the non-elastic character of scattering by Vanadium causes distortions in the range of several percent at low Q values, whenever this material is considered either as an ideal elastic scatterer or a heavy gas from the point of view of

1.12 . . . . (a) ............ Vanadium / Incident spectrum

1 . 0 8

1 . 0 6

1.04 ..................................................................................................

1 . 0 0

1 . 0 6 t I ~ I ~ I ~ I ,

1 04 f (b) ........... ~ Vanadium/Gas ......

0.98 f ! i! i ~ r .................................................................. 0.96

0.94 , , 0 2 4 6 8 10

Q (/~-1)

Fig. 1. Ratios of the calculated spectra for Vanadium and (a) an ideally elastic scatterer, and (b) a gas with corresponding mass and effective temperature, for scattering angles of 55 °, 90 °, and 150 ° .

the interaction with thermal neutrons. Similar re- sults have been discussed by Mayers [11], on the basis of a full phonon calculation. We hope that our compact expressions to evaluate the scatter ing of neutrons by solid systems will be useful to attain a more accurate absolute normalization when Vanadium is used as a standard, or more reliable Monte Carlo calculations to correct for multiple scattering effects in solid samples or thick containers.

References

[1] G.J. Cuello and J.R. Granada, Ann. Nucl. Energy (1996), in press.

[2] L. Van Hove, Phys. Rev. 95 (1954) 249.

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[3] A.C. Zemach and R.J. Glauber, Phys. Rev. 101 (1956) 118, [4] G.H. Vineyard, Phys. Rev. 110 (1958) 999. [5] A. Sjflander, Ark. Fys. 14 (1958) 315. [6] G. Placzek, Phys. Rev. 86 (1952) 377. [7] D.E. Parks et al., Slow Neutron Scattering and Thermaliz-

ation (Benjamin, New York, 1970).

[8] S.W. Lovesey, Theory of Neutron Scattering from Con- densed Matter (Oxford Science, New York, 1987).

[9] M.M.R. Williams, The Slowing Down and Thermalization of Neutrons (North-Holland, Amsterdam, 1966).

[10] J.M.F. Gunn and M. Warner, Z. Phys. B 56 (1984) 13. 1-11] J. Mayers, Nucl. Instr. and Meth. A 281 (1989) 654.