thermal neutron scattering by debye solids: a synthetic scattering function

Pergamon Ann. Nucl. Energy, Vol. 24, No. 10, pp. 763 783, 1997

© 1997 Elsevier Science Ltd. All rights reserved P l h S0306-4549(96)00050-3 Printed in Great Britain

0306-4549/97 $17.00 + 0.00

THERMAL NEUTRON SCATTERING BY DEBYE SOLIDS: A SYNTHETIC SCATTERING FUNCTION

G. J. C U E L L O l* and J. R. G R A N A D A 2

~Comisirn Nacional de Energia At6mica, Centro At6mico Baritoche and Instituto Balseiro, 8400 Bariloche, Argentina

2Consejo Nacional de Investigaciones Cientificas y Tbcnicas, Argentina

(Received 21 May 1996)

Abstract--In most neutronic calculations, either to account for the effect of lattice vibration on nuclear resonances, the contributions due to multiple scattering processes or the presence of sample container in scattering experiments, the neutron-solid interaction is represented in a rather crude way, usually through a simple gas model. In this work we introduce a synthetic scattering kernel to describe incoherent neutron scattering processes in Debye solids. The model is based on a new prescription to represent the many-phonon contribution, whereas the first three terms of the phonon expansion are explicitly included. This prescription, which contains no adjustable parameters, is given in terms of analytical functions that produce a very good approximation to the sum of multiphonon terms. We present a few selected applications of this new method, using vanadium as a case to illustrate its merits and limitations. © 1997 Elsevier Science Ltd.

1. INTRODUCTION

Thermal neutrons have different properties that make them a unique tool for studies in many fields, including materials science, condensed matter physics, chemistry, biology, fundamental and applied physics. Those properties permit the experimental exploration of a wide range in the Dynamics - Structure space of the materials in their different physical forms, concerning the arrangement and movement of both the atomic nuclei as well as the unpaired spins in their electronic structures. It is that set of properties, unique in many circumstances, that allowed the enormous development and the vast field of applications that neutron scattering techniques offer at present (Funahashi et al., 1995).

Many pioneering studies involving thermal neutrons were motivated by the need to develop the basic body of knowledge required for thermal reactor physics calculations. A large part of the experimental work was then devoted to measurements of cross sections

*Also at CRUB, Universidad Nacional del Comahue, Argentina.

763

764 G.J. Cuello and J. R. Granada

and neutron diffusion parameters over the thermal energy region, for moderator, fuel, and structural materials (Parks et al., 1970), whereas the theoretical efforts were essentially aimed at solving neutron slowing-down and thermalization problems emerging from the Boltzmann equation for specific conditions (Williams, 1966). Those studies created the basis of our present understanding in the field of Neutron and Reactor Physics (Dickens, 1994).

The central quantity to describe the interaction of slow neutrons with condensed matter is the Van Hove scattering function S(Q,o~) (Van Hove, 1954), as it embodies all the structural and dynamical properties of the scattering system. However, although a large portion of the relevant features in the energy (ho~) and momentum transfer (h Q) plane can be revealed by double differential cross section measurements, there will always be a particular problem for which the experimental data do not cover exactly the required material and physical conditions. While the Zemach-Glauber formalism (Zemach and Glauber, 1956) provides an essentially exact frame for the representation of S(Q,w), with a proper account of the quantum nature of the scattering system and the temperature-dependent distribution of its energy states, the resultant expressions are not quite amenable for calculations. A great deal of effort was therefore devoted to the development of simple, approximate expressions, to describe the scattering function for condensed matter systems of special interest. In particular, a significant progress in this direction was already attained many years ago for the case of molecular systems, prompted by their importance as mod- erating materials in nuclear reactors, as well as the basic interest in the study of atomic motion in matter, for which they provided a suitable test bench. As an example of a more recent contribution along this line we may mention the Synthetic Model for molecular gases (Granada, 1985), devised to describe integral magnitudes that stem from the double- differential cross section, and which are the quantities of interest in many circumstances.

Similarly, different models and approximations were developed to describe the neutron scattering by solid systems, notably the Gaussian approximation (Vineyard, 1958), the phonon (Sj61ander, 1958) and the mass (Placzek, 1952) expansions. Those classic methods were devised to deal with the incoherent component of the scattering process, and due to their importance from both conceptual and practical points of view, a relevant part of the associated theoretical foundation in many textbooks is deserved to them (Williams, 1966; Lovesey, 1987).

The phonon and mass expansions have shown to be highly successful in different applications, for example in determining the frequency spectrum of elementary excitations 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 (Parks et al., 1970; Lovesey, 1987). Of course, in the flame of the Gaussian approximation to the (incoherent) intermediate scattering function, equations (1) and (2) below, it is always possible to generate S(Q,w) from the actual frequency spectrum when the latter is known, but the computation of the required double Fourier transform could be a too lengthy process in many cases. This is especially true in those situations where the detailed shape of the scattering function over a large portion in the (Q,w) plane is not needed, but rather a fast, yet accurate, evaluation of quantities involving integrals of the double differential cross section.

Motivated by those ideas, we have developed a synthetic model to describe the interaction of thermal neutrons with solids, which involves analytical functions to represent the inelastic contribution to the scattering cross sections. Bearing in mind that the aim of this work has been the development of a formalism able to allow a good description of

Thermal neutron scattering by Debye solids 765

integral properties rather than the scattering function itself, we use the Debye model to characterize the frequency spectrum of the solid, and its characteristic parameter together with the mass of the constituent nuclei are in fact the only input parameters required by our model. It is then possible to evaluate microscopic cross sections at any temperature of the scattering system, from the neutron moderation energy region down to cold neutron energies.

We hope that the present model will be useful to produce fast and reliable differential and total cross section predictions which can be used in reactor physics as well as in neutron diffraction work. This prescription, together with the above mentioned model for molecular gases and liquids (Granada, 1985; Granada and Gillette, 1995), allow the evaluation of energy-transfer kernels for fuel, structural and moderator materials required by reactor calculations, and the evaluation of inelasticity effects, multiple scattering processes, sample container contribution, and normalizing spectra in neutron scattering experiments, among other applications.

2.1. General expressions

2. BASIC FORMULAS

We consider the scattering system to be a crystal composed of atoms harmonically bounded to their equilibrium positions, which are centres of inversion symmetry. For a lattice of defined symmetry, the elastic (coherent and incoherent) components of the cross sections have simple, easy to compute expressions determined by the structure and the temperature (Debye-Waller) factor. Concerning the inelastic components, we will treat them in the incoherent approximation, which implies neglecting the usually very small interference effects on the inelastic scattering of slow neutrons (Parks et al., 1970), although even those small effects should be properly accounted for before extracting reliable thermal parameters from the observed reflection intensities in a diffraction pattern (Wilson, 1995).

The second basic hypothesis in our model is the validity of the Gaussian approximation to the intermediate scattering function [equation (2)], a form which is verified exactly by some simple systems (Williams, 1966). As far as a crystalline system is concerned, non- Gaussian effects on its scattering function may arise as a consequence of anisotropy in the atomic vibrations.

Under this hypothesis, we can write the microscopic incoherent double-differential c~oss section for the scattering of unpolarized neutrons as (Lovesey, 1987)

~r(E, E', f2) - O ' i n c kt N / ~ 4zr k 2 r r h x(Q't)e-i°~tdt' (1)

O(3

where k and k' denote the (modulus of) incident and scattered neutron wave vectors, respectively, while hw = E - E' and hQ = h k - hk' are the energy and momentum transferred from the neutron to the system in the scattering process. In terms of the phonon density of states Z(w), the intermediate scattering function is expressed as

x(Q, t) e x p { - 2 W } e x - f h Q 2 y ( t ) ] (2)

766

where

G. J. Cuello and J. R. Granada

~'(t) = f ~ Z(w) n(w) e-i°'tdw, J-oo 09

n(w) is the thermally a v e r a g e ~ u p a t i o n number

n(w) = exp - 1 ,

(3)

(4)

2W hQZY(0) (5) 2M

is the Debye-Waller factor, kB is the Boltzmann's constant, T is the system temperature and M is the mass of the scattering nucleus.

In this context, the phonon expansion is obtained by expanding the intermediate scattering function as follows

x(Q, t) = e-2W~ --~ (214/)' [Y(t)] p p! [y(0)J ' (6)

where each term represents the interaction of the neutron with p phonons. In the manner of Sj61ander (1958), we can define the functions

foc 1 Gp(w) = ~ ~ [y(0)J dt, (7)

which allow the scattering cross section to be written in the form:

tr(E, E', g2)= N °'inc k'e-2W~--~(214qpGp(w) 4rr k z...~, , hp! "

p=O

(8)

The integration limit in equation (7) is formally extended to infinity, although the expression under the integral sign differs from zero only within a finite interval, determined by the condition

Io 1 poem (9)

for a given p, where Wm is the maximum frequency of the normal modes. It is evident from the same equation that the Gp(w) are normalized to unity.

The angular dependence of equation (8) is completely contained in the Debye-Waller factor and thus, by integration over the solid angle the phonon expansion of the energy- transfer kernel is obtained:

o o trine

tr(E, E') = N-~s E Gp(w)Kp(E, E'), (10) p=0


where

k'\2" " ¢'2 + k ) e-~('+~) Kp(E,E')= Pn~=O~. 1- -~ - ) e-~(i-r) - 1 (11) 9

E s = ~-~- y(0) (12)

and A is the ratio between the nucleus and the neutron (m) masses.

2.2. Debye model

We assume that the solid has a phonon density of states Z(@ described by the Debye spectrum

Z(u)={~ u2 ,uliU'~<l> 1 ' (13)

where we have defined the dimensionless variable u = w/OgD, (.O D being the cut-off frequency of the spectrum, related to the Debye temperature (gD by the expression hWD = kB(gD.

According to this model, equation (3) transforms to

and

3 i I xe -ix°)ot Y(t) = ~D D i e~-7-6- Z 7 dx, (14)

×(o) = 3 ((9), (15) COD

where we are using the functions

f t X n (_l)n+l

~"((9) = e:'/~-- 1 dx - i n + l

LI r e + I , /,/.] dx exl ~-- 1 (16)

with (9 representing the ratio between the system's physical and Debye temperatures, (9 = T / ( g D .

For direct numerical computations it is often more convenient to use a symmetrized form of Gp(@:

Gp(u)=e-"12°Gp(u). (17)

These functions are no longer normalized to unity, and can be evaluated through the recursion relations

~p(U) nG°~6J~l) [' xe~lZO = 4~1 d-le~7°---1 ~p-l(u+x)dx' (18)


which involve the unit-step function H ( x ) and the initial condition

G0(u) = 8(u). (19)

The first orders of these functions, corresponding to vanadium (19o = 390 K) at room temperature, are shown with solid lines in Fig. 1. To approximate them, Sjflander (1958) proposed polynomial forms

(p-lul)/2 nGo - lul)p (p - 2 v - lul ) p -1

(Tp(u) = (419sinh(1/219)) p E (-1)v (20) v=0 v!(p-v)! '

where the summation extends over all integers from zero to the largest one below (p- lu0/2, and we have added a factor that ensures a correct normalization. As it is shown with dotted lines in Fig. 1, those functions are very good approximations to the high-order Fourier transforms of y(t), but their evaluation becomes increasingly involved for high values of p. A further approximation, based on the central limit theorem has been developed (Sj61ander, 1958; Williams, 1966) to produce compact, analytical forms for the Gs valid for (large p) multiphonon terms:

(Tp(u) = (4® 2 v, (®)'~' e-"2/2~ (°) v A (o) ' (21)

0.6

0.0 -6 6

~ , 0.4

0.2

I I I l

O = 0.75 p = 1

I

-4 -2 0 2 4

U

Fig. 1. Symmetrized forms of the functions intervening in the phonon expansion [equation (8)], calculated at ®=0.75 for several orders (p = 1, 2, 3, 5 and 7). ( ) Exact calculation defined by the recurrence relation indicated in equations (18) and (19). ( . . . . ) SjSlander approximation with the correct normalization factor [equation (20)]. ( - - - ) Gaussian forms indicated in equations

(21) and (22).


where

20 / v3(®) ap(o) = V p v (O)

(22)

and

f l/2® X n

v.(®) = (23) ./0 sinh xdX"

The Gaussian form of the r.h.s in equation (21) is in fact the leading term of an asymptotic expansion, written in the symmetric formulation proposed by Schofield and Hassitt (1958) which automatically guarantees satisfaction of the detailed balance princi- ple, and it is accurate for most cases of interest as long as the medium temperature T is not very low compared with the Debye temperature ®D. The approximate functions defined by equation (21) and corresponding to ® = 0.75 are also included (dashed lines) in Fig. 1, showing a good representation of the 'exact' ones for p greater than 3.

Throughout the rest of this work, we will use as a reference a phonon expansion of the scattering function that includes the exact expression for GI (u), Sj61ander's polynomial approximation for G2(u) and a3(u), and the modified Gaussian form, equation (21), for Gp(u) with p >/4.

3. THE MODEL

In the previous section we have outlined the general features of the standard phonon expansion for the scattering law of a Bravais lattice, and reproduced the specific forms resulting from the assumption of a Debye frequency spectrum. Although such expansion can be easily evaluated, even for relatively high incident neutron energies when many phonon terms are required, there always exist situations wherein exact detail of the first- order effects is not called for, but some simple model is needed. The development of such a model has been the objective of the present work, under the premises of a good description of its integral properties rather than the scattering function itself, and the attainment of compact, very fast to calculate, expressions for those magnitudes. Bearing in mind that the elastic component of the cross sections are already given in terms of analytical expressions, we will treat in this paper the contribution due to the inelastic components alone, unless explicitly stated.

3.1. Energy-transfer kernel

In two classic papers Nelkin and Parks (1960) and Egelstaff and Schofield (1962) introduced asymptotic expansions for the scattering law, valid in regions of the (Q,w) space where the phonon expansion converges very slowly. A short-collision time approximation combined with the use of the central limit theorem in one case (Nelkin and Parks, 1960), or the use of the steepest-descent method to define the integration path through a


saddle point in the complex t-plane (Egelstaff and Schofield, 1962), demonstrate that the asymptotic form of the complete S(Q,w) tends to the result corresponding to an ideal gas of mass M and temperature T*. The range of validity of this limiting form has been thoroughly discussed by Williams (1966) and Gunn and Warner (1984).

For a Debye solid, the effective temperature is given by

kaT* = 3ho~4~3(®), (24)

and the energy-transfer kernel can be written in this approximation as

°rincA {erf(r/x' - px) + erf(rlx' + px) o'g(E, E') = N - - ~ x

+ exp(x 2 -x'2)[erf(r/x - Ox')q: erf(r/x + pxl)] ) .

(25)

The upper sign corresponds to E ~ ~< E and the lower to E ~ 1> E; also, we are using the notation

A + I A - 1 x 2 _ E (26) 77- 2x/-A ' P = 2v/-A ' kaT*

and a similar definition for x'. According to the previous remarks, and consistent with arguments based on simple

physical grounds, the free-particle gas kernel represents a good approximation to the actual one when the energy transferred in the scattering process is much larger than the characteristic excitation energies in the solid (hw > > hWD). This behaviour is illustrated in Fig. 2 for the case of vanadium at room temperature, where the free-gas model (dashed line) is compared with the full phonon expansion (circles) at four well separated energies. In fact, the comparison is not fair for the gas at the lowest energy shown, because the integral of this kernel is actually consistent with the total (inelastic plus elastic) cross section. On the other hand, even at the highest incident energy (E = 2 eV) we can observe that the gas model only gives a good description of the actual kernel at the wings of the distribution, where the 'many-phonon' (p >t 4) contribution dominates.

In order to establish the scenario for our approximations, we start from the identity resulting from a rearrangement of equation (10)

o(E, E') = cr3(E, E') + crg(E, E') - crc(E, E'), (27)

where we defined

Ntrinc o'3(E , E t) - - e u/2° y~ Gp(u)Kv(E, E') (28)

4ehWD p=l

and the 'correction' function

~rc(E, E') = ,Tg(E, E') Ntrinc eU/2 o ~ Gp(u)Kp(E, E') 4ehtOD p=4

N~rinc eU/2°Dc(E, E'). 4ehogD

(29)


The structure of equation (27) emphasizes one of the basic requirements in our development, i.e. to preserve the correct limiting forms at the low and high ends of the energy range. We believe that by keeping the first three phonon terms, a good description of the features due to any structure borne into a general frequency spectrum can be economic- ally achieved; besides, the large number of final states involved in multiphonon processes smears out the effects of such structure in a real spectrum, and their contributions can be effectively evaluated using a simple Debye model.

The function De(E, E') goes to zero at large incident neutron energies, when the multiphonon contribution is close to the gas kernel. Strictly speaking, it is the full phonon

, - , 6

t~

40

¢-,, 3O

b

I0 ........ I ........ I '. ...... I * I

8 E = ( ) . 0 1 e V _

.

2 .

', ~d.., ~ 0 ' '

10 -4 10 -3 10 -2 10 -1

E'(eV) 5 0 ' 1 . . . . I . . . . 2 . . . . I '

10

0 ~' 0 0.8 0.9 1.0 1.1

40 .... I

E = 0 .

30 -

m

10 -

4

0 ' ' ~ " 0.00 0.05

20

b

' " ' I ' ' ' I ' ' ' '

eV

0.10 0.15 0.20

E'(eV)

30

~ 20

t~ 10

I ' I ' I '

E = 2 e V

1 . 7 1 .8 1 .9 2.0

I ' I

2.1

E ' ( e V ) E ' ( e V )

Fig. 2. Energy-transfer kernels of vanadium at room temperature for a few selected energies. (O) Phonon expansion results, which are compared with the corresponding curves of the free gas model

( - - -) . ( ) The kernel obtained from equation (34).


expansion - - including the elastic term - - which tends to the ideal gas form, but of course the relative contribution of the low order terms becomes negligible at neutron energies much larger than hO~D. Conversely, in the low energy limit the multiphonon contribution is practically zero, Dc tends to the gas kernel and thus, in the total kernel expression only the first three terms of the phonon expansion remain. We show in Fig. 3(a) the behaviour of the function De(E, E') at several incident energies as a function of the ratio k' /k .

Bearing in mind the exponential factor that modifies Dc in equation (29), which leaves the latter in its most symmetrical form, we have chosen to approximate the down-scattering branch with a quadratic function, and reflect it over the up-scattering side. Thus, we propose to represent the correction function by the following expression

Dc(E,E') ~ e x p { A 0 - A ] 1 - - ~ - A 2 ( 1 k"~2~ - ~ - ) j , (30)

where the energy-dependent coefficients Ai are determined by the Taylor expansion of the actual function around the elastic energy. Thus, we have

Ao --- In{Dc(E, E)}, (31)

103 ' ' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' '

~ E = 0.01--...

102 .05 ~ . . :

j -0,\

100 1

2 J ) i , I J ) ! i i 103

~ 101

~ 10 o

10-1 , , ," , 0.8 0.9 1.0 1.1 1.2

k ' / k

Fig. 3. (a) Curves of De(E, E') for some neutron energies (in eV) as a function of the ratio k'/k, for vanadium at room temperature. (b) Same curves ( ), but affected by the exponential factor indicated in equation (29), compared with the approximation introduced in equation (30) ( . . . . ).

The incident neutron energies are the same as in (a).


At = - 2 E D~c(E' E) Dc(E,E)

(32)

and

D'(E, E) (~ ~D'c(E, E) A2=ED---~,E) ztz D----~, E)

1) - 2E 2 D'~(E, E) (33) Dc(E, E) '

where the primes indicate derivation with respect to E'. Explicit expressions for the derivatives of Dc are given in the Appendix. In Fig. 3(b) we compare the actual correction function (solid lines) with the proposed approximation (dashed lines), both already mul- tiplied by the corresponding exponential factor [see equation (29)]

The integral of the total kernel over E' must give the total inelastic cross section, but due to the approximation performed [equation (30)] the correct value is not attained at low energies. To conform to the above requirement we introduce an energy dependent normalization factor, changing the kernel expression to

fi(E, E ' ) = o 3 ( E , E t) "~ F(E)[O-g(E,E')- fie(E, E')] . (34)

The factor F(E) is obtained from the equation

o-r(E) = fiel(E) + o-3(E) + F( E) [o-g( E) - f ie(E)] , 1. ,J (35)

which results after integration of the incoherent kernel over E'. In this equation, o-x(E) is the total cross section according to Placzek's mass expansion (Granada, 1984), fiej(E) is the elastic contribution

1 -- e -4e o-el(E) = No-inc 4--7--' (36)

O-3 (E) includes one-, two- and three-phonons terms, o-g (E) is the free-gas total cross section

-- Nfiinc 2-~x2) erf ( 'v/~x) "}- ~ x x o-g(E) (l + l/A) 2 1 + , (37)

and o-c(E) is the total cross section corresponding to the correction function. All those contributions are shown in Fig. 4 for vanadium at room temperature.

Normalization factors obtained from equation (35) for a given element at different temperatures will, in general, give rise to different curves as a function of energy. How- ever, with the use of the variable e [equation (l 2)] instead of E, all those curves practically collapse into a single one, as illustrated in Fig. 5(a) for vanadium, where we also include a fitted smooth function given by the expression

F(E) = (1 + ae-b) -1, (38)

with a and b being adjustable coefficients.

774 G.J. Cuelio and J. R. Granada

b

7

6

5

4

3

2

1

0 10 .4

\ ' : , ' . . . . . . . . ' . . . . . . . . ' . . . . . . . . ' . . . . . . . . ' . . . . . . . 1 . . . . . . . .

~ " ,5

" 1 10 -3 10 -2 10 -1 10 0 101 10 2

E (eV)

Fig 4. Different contributions to the incoherent total cross section of vanadium at room temperature (see text for details).

Following a similar process for several materials, we obtained using equation (38), a corresponding set of different curves as a function of e. Finally, the replacement of this variable by

e* -- ee '~(A), (39)

again produces an almost complete collapse of all curves into one, as shown in Fig. 5(b), where we included the new function. In this figure, the inset indicates the values of the parameter a for the elements considered, with an exponential function fitted to them. These fitted values of a change the corresponding values of the parameters in equation (38), thus we had to repeat the process until no changes occurred in the parameters or in the A-dependence of a. After three iterations, we obtained the final expression for the normalization function

F ( E ) = ( l+e*-5) -1, (40)

and for the shift in the logarithmic scale, equation (39),

ox ( (41)

It must be emphasized that the last two expressions, together with equations (12) and (15), allow the evaluation of the normalization factor and with it, the energy-transfer kernel for any material at any temperature, as a function of the incident neutron energy. As an example, we show in Fig. 2 some kernels obtained from equation (34) (solid lines),


1.0

0.8

0.6

0.4

0.2

0.0

10 -3

' ' ' " ' 1 ' ' ' ' ' " ' 1 ' ' ' ' ' " ' 1 . . . . . . . '1

........ 0=0.5 f " - - O = 0.75 1 ~' O =1

O = 1.5 t ~

i i i J H i r ~ , 10 -2 10 -1 lO 0 lO 1

g

I I I I I I

Ca) 10 z

' '''"'I ' '''"'I ' ' '''"'I ' ' '''"'I ' '''"'

0 i,f 10.,// 0.6 0.4

r.~ o.4

0.2

o . o . . . . . . . . , . . . . . . . . , . . . . . . . . , . . . . . . . . , ( b ) , , , ,H

10 -3 10 -2 10 -I 10 0 101 10 2

tea(A) Fig 5. (a) Normalization factor F(E) as a function of the dimensionless parameter e, for vanadium at several temperatures. With a thick solid line we show the function that we adopted to describe it [equation (38)]. (b) The same factor as a function of the modified parameter e e ~(A). In the inset the value required for each element considered is shown, as well as the fitted curve employed to repre-

sent it [equation (41)].

compared to the full phonon expansion and the free-gas kernel. The agreement between our prescription and the phonon expansion is good, especially for incident energies greater than 0.1 eV, where the multiphonon contribution begins to be important and the gas model is still too crude.

Total cross sections calculated by integration of our kernels are in very good agreement with the 'exact' results obtained from the Placzek's mass expansion. As a typical example, we plot in Fig. 6 the relative difference between both calculations for silver at different temperatures. The discrepancies observed, up to 5% for the lowest temperature considered, are representative of the limitations imposed by the use of a very simple form for the normalization factor. Those can be greatly removed in a specific case if the actual values of F(E) obtained from equation (35) are used in equation (34), but our analytical expressions to represent the energy-transfer kernel should still be adequate for many applications involving correction procedures or the evaluation of multiple scattering processes, providing instead for a considerable reduction in computing time as compared with the use of a full calculation procedure.

776 G.J. CueUo and J. R. Granada

Fig. 6.

"O

r¢

8 i

4

2

0

-2

-4 ' 10 -3

W I I g I ~ l w

O = 0 . 5 0

0=0.75

O = 1 . 0 0

~ - - O = 2 . 0 0

;°

. l

.o.'° s p ~ -

°',.o •

10 .2 i0 -I 10 °

E (eV)

"-\

,0 \

"o

°-o.

J

101 10 2

\

Relative difference between the total cross sections of Ag at different temperatures, as calculated from Placzek's mass expansion and the present model.

3.2 Double differential cross section

In the previous section we have presented-and discussed formulas to evaluate the probability for an incident neutron with energy E to emerge with energy E', after being (incoherently) scattered by a Debye solid. In doing so, we have seen once again the strong variation of that magnitude with both, the energies involved and the elastic or inelastic character of the scattering process (cf. Figs 2 and 4). The angular distribution of the scattered particles in each of those processes is the other quantity required for a complete description of them, as expressed by the double-differential cross section [equation (1)]. We will now discuss our prescription to represent this magnitude for a polycrystalline Debye solid.

In the same spirit of the approximations that led us to propose a compact form for the scattering kernel [equation (34)] and motivated by a natural extension of the intervening terms, we introduce a synthetic expression for the inelastic incoherent double-differential

~ross section

~r(E, E', a ) = cr3(E, E', a) + F(E)[crg(E, E', a ) - G(/z)¢re(E, E')], (42)

where /z is the cosine of the scattering angle, tra(E, E', f~) denotes the combined contributions due to one, two, and three phonons as given by equation (8), trg(E, E', f2) is the cross section of the ideal gas (Lovesey, 1987), and trc(E, E') and F(E) are the quantities defined by equations (27) and (35), respectively. Equation (42) represents the central expression of this paper.

The function G(/z) introduced in the last equation is intended to provide the bracketed term of equation (42) with an angular dependence as similar as possible to the actual one


contained in the multiphonon (p/> 4) cross section. Besides, G0z) must be properly normalized

f 2zr G(E, #)d# = 1, (43) 1 in order to ensure consistency between equation (34) and the integral over solid angles of equation (42). It is evident from the form of this latter equation, and from the energy- dependence of ere(E, E') and F(E) previously discussed, that this synthetic differential cross section will tend to the correct limiting forms, few-phonons or free-gas expressions, at low or high incident neutron energy plus high momentum transfer, respectively.

The shape of G(/z) is obtained by imposing on it the requirement to satisfy

or(E, a ) = cr3(E, a ) + F(E)[%(E, a) - G(#)crc(E)], (44)

which results after integration of equation (42) over all final energies E', and clearly that shape will depend on the incident energy E. In order to derive G(/z) in an economical manner, we adopted the following criteria to solve equation (44):

ag(E, fa) ~ m,(E, f2) (45) (a) E > hwD =~ a(E, f2) -~ crp(E, ~2) - Crel(E, a) ,

where

crp (E, ~) O'inc 1 = W Y - U f 2 u + - - 3~b3 ((D) 3~5(~))] 4x~ 64x 6 J

+ -- [ 3¢3((9) 1 + 3 ( 2 # 2 1) 1 + - - - 4,42 2x 2

3/'34~3(®)'~2+ 3 ]} (46)

is the total (elastic plus inelastic) differential cross section derived by Placzek (1952) for a Debye solid, written in our notation for the first two terms of the mass expansion, and

E (47) x~ -- hOJD"

It can be shown that this simple formula produces a remarkable agreement with the exact result for all neutron energies larger than hwD, and that both expressions in equation (45) can indeed be considered as identities over that range. Under those conditions, we have

O(u) = ooI(E, a ) + o3(E, a ) - [1 - F(E)J p(E, a )

F(E)ac(E) (E > htoo). (48)

(b) E < hco D =:~ tr(E, ~2) ~ t73(E, ~). (49)


This condition is based on the obvious fact that at energies below htOD, the differential cross section is completely dominated by the first phonon terms. Then, from equation (44) it results

G(#) = o'g(E, a) c(E) ' (E < h,oD). (50)

The formulas (48) and (50) allow the determination of G(#) over the full energy range, and the functions thus obtained satisfy the normalization condition [equation (43)], except over two small regions. At the lowest energies, the rise in the free-gas differential cross section cannot be 'followed' by trc(E) (see Fig. 4), and consequently equation (50) gen- erates a function with an anomalously large area. However, at those energies the function F(E) is already zero, and the synthetic double differential cross section reduces to that corresponding to few phonons as it should be, no matter the value of G(/z). On the other hand, at neutron energies slightly higher than ho~, the interplay between the strong energy variation of some of the terms in equation (48) and the approximations involved in others, may produce even negative values for G(tt) over a small energy interval. In those cases, we have adopted the criterion of taking them as zero, after testing the small effect of this choice on different magnitudes. The behaviour of G(#) over the full energy range is illustrated in Fig. 7(a), again for our standard vanadium case.

It must be emphasized that the actual factor affecting the correction term in equation (42) is the product F(E)G(lz), that usually has a much smoother behaviour as displayed in Fig. 7(b) for the same case. The null values of that product for all angles at low energies, reflect the fact that the double differential cross section is completely described by the three phonon terms, because the gas term has been also turned off by F(E). The null values of F(E) G(/x) at high energies and large angles indicate the disappearance of the correction term and the approach of the cross section to the free-gas expression. At low scattering angles the values are large, the correction term partially compensates the gas contribution, and the few phonon terms will distinctly show up because their Debye- Waller factors are small, even for large incident neutron energies.

In order to test the angular dependence borne into our model, we evaluated integrals of equation (42) over U under different conditions. Angular distributions calculated for 'reactor measurements' on vanadium at different wavelengths are shown in Fig. 8(a), whereas differential cross sections corresponding to a few fixed scattering angles over a wide range of incident energies are plotted in Fig. 8(b). Those are 'ideal' spectra, in the sense that black detectors and a fiat incident spectrum in the case of the 'time-of-flight' measurement, Fig. 8(b), are assumed. Nonetheless, these results should serve the purpose of raising again a cautionary flag about the use of that material as a normalizer in neutron scattering experiments (Granada et al., 1981, 1982; Mayers, 1984, 1989).

4. SUMMARY AND CONCLUSIONS

The formulas derived in the previous section provide a scheme for a fast evaluation of an approximated double-differential cross section for a Debye solid. A few typical results are presented in Fig. 9, where calculations based on the present model are compared with those obtained from a full phonon expansion and the gas model. Even though the main objective of this work has been the development of compact expressions for integral


,02 iiiiii

,00,, 1 70 ,02

10 100 , , 120

- ~ e l / ) 10 -~ 10.3180 ~,.,"

(b)

101

10 °

0

~ ,o' ~ 2 o ; ~ °~ Io ~ . ~ ~

Fig. 7. (a) The function G(#) as obtained from equations (48) and (50). (b) The product F(E) G(#) over the complete angular and energy ranges. Both calculated for vanadium at room temperature.

See text for details.

magnitudes, rather than the double-differential cross section itself, we can observe that its main characteristics have been preserved. However, discrepancies are clearly visible at low scattering angles and intermediate energies [Fig. 9 (b) and (c)] where those compact expressions cannot describe the full complexity of the actual scattering function.

Although this model is heuristic in essence, following the initial requirement of replacing the sequence of multiphonon terms by a single function, the correct limiting forms are explicitly borne into its formulation. Over the intermediate region of the (Q,w) space, where several - - but not too many - - phonons intervene in the scattering process, the model provides a simple way to account for their effects. The penalties that we had to pay in exchange for simplicity are the discrepancies referred to in the previous paragraph with regard to the double-differential cross section. Furthermore, detailed balance is not rig- orously satisfied by our kernel at the intermediate energy region, where the up- to down-

780 G.J . Cuello and J. R. Granada

1.05

l.OO

"~ 0,95 ! 00 meV

30 60 90 120 ! 50 80 Angle (Degrees)

1.10 ' ' ' ' ' " ' 1 ' ' ' ' ' " ' 1 ' ' ' ' ' " ' 1 ' ' . . . . " 1 ' ' ' ' ' " ~ ]

1 lO° .t

" " 90 ° 500 1 0.95 ,,' , . , . A ~ . ~ 6 A A ~ . i J . . i /

~' - ] 7 0 ° 3 0.90 . . . . ~,,,a . . . . . . . . i . . . . . . . . ~ . . . . . . . . ~ . . . . . . . a

10-3 10-2 10 -I 100 101 102

E (eV)

Fig. 8. (a) Angular distribution of neutrons scattered by vanadium at room temperature for several incident energies in a reactor-type experiment, as calculated from the phonon expansion (O) and the present model ( ). (b) The same plot as a function of incident energy for several scattering angles. Our results are compared with the Placzek's mass expansion, [equation (46)] (. • • A • • .), and the phonon expansion, the latter shown over the range where the convergence is achieved with 100 terms.

scattering ratio in the approximated correction function departs from the exact one, as it is displayed in Fig. 3(b).

Some integral magnitudes obtained from it have been displayed and discussed throughout this work, specifically energy-transfer kernels, differential and total cross sections. In all those cases, the agreement of the model 's results with those based on a full phonon calculation have been highly satisfactory, and much less expensive in terms of computing time. In fact, a reduction by a factor ~ 2000 is typical for a calculation involving 1 eV neutrons. The attainment of this kind of time reductions while still keeping good accuracy in its predictions, should make this model appropriate for Monte Carlo simulations, for example to describe the effect of structural materials in reactor calculations or sample containers in neutron diffraction work.

Most of the figures presented here show results for vanadium at room temperature, yet we must emphasize that the model can be applied to any material at any temperature, the mass of the constituent a toms and its Debye temperature being the only input parameters required. Our particular choice of this material was motivated by a long standing discus- sion on the use of vanadium as spectrum normalizer in scattering experiments, trying to


' ° ' F . . . . . . . . ' . . . . . . . . ' . . . . . . . ' (a ) o I 0 ° .

lOq

~ 10 -2

10 -3

10-4 10-4 10 -3 10 -2 10 -1 0.05 0.10 0.15 0.20

E ' (eV) E'(eV)

102 ~--) " ' 1 ' ' ' ' E ( d ) ' ~ ' ' ' '

lOl - eV ~ I E = I O e V

10 °.

10 -2

10 -3

1 0 -4 i , ,' .

0.7 08 0.9 1.0 1.1 9.0 9.2 9.4 9.6 9.8 10.0 10.2

E' (eV) E' (eV)

Fig. 9. Double-differential cross sections for selected incident energies and scattering angles of 10, 50, 90, 130 and 170 °. For the two lowest energies we compare the present model ( ) with the phonon expansion (symbols). For the two highest energies we compare our prescription with the free gas model ( - - - ) , except at 10 °. All curves are calculated for vanadium at room temperature.

bring attention against the widespread practice of assigning to it properties of "an elastic and isotropic scatterer". Perhaps the availability of a compact formula to evaluate its scattered intensity will help to avoid this oversimplification, which in many cases may spoil an otherwise excellent set of experimental data.

In a forthcoming publication (Cuello et al., 1996) we will present other applications of this model, like the evaluation of Doppler broadening in nuclear resonances, predictions of spectrum distortions in measurements on vanadium using different instruments at stationary and pulsed neutron sources, multiple scattering corrections, etc.

Acknowledgements--We are most grateful to J. Dawidowski for his help in performing some calculations, and to V. H. Gillette for his continuous interest on this work.

R E F E R E N C E S

Cuello, G. J., Dawidowski, J. and Granada, J. R. (1996) In preparation. Dickens, J. K. (Ed.) (1994) Proc. Int. Conf. on Nuclear Data for Science and Technology.

ANS, Gatlinburg.


Egelstaff, P. A. and Schofield, P. (1962) Nucl. Sci. Eng. 12, 260. Funahashi, S., Katano, S. and Robinson, R. A. (Eds.) (1995) Neutron Scattering - - ICNS

'94, Proc. XLI Yamada Conf. Elsevier Sci. Publ. Yamada Sci. Found., Japan. Granada, J. R., Kropff, F. and Mayer, R. E. (1981) Nucl. Instr. Meth. 189, 555. Granada, J. R., Kropff, F. and Mayer, R. E. (1982) Nucl. Instr. Meth. 200, 547. Granada, J. R. (1984) Z. Naturf. 39a, 1160. Granada, J. R. (1985) Phys. Rev. B31, 4167. Granada, J. R. and Gillette, V. H. (1995) Physica B 213&214, 821. Gunn, J. M. F. and Warner, M. (1984) Z. Phys. B56, 13. Lovesey, S. W. (1987) Theory of Neutron Scattering from Condensed Matter. Oxford

Science, New York. Mayers, J. (1984) Nucl. Instr. Meth. 221, 609. Mayers, J. (1989) Nucl. Instr. Meth. A281, 654. Nelkin, M. S. and Parks, D. E. (1960) Phys. Rev. 119, 1060. Parks, D. E., Nelkin, M. S., Beyster, J. R. and Wikner, N. F. (1970) Slow Neutron Scat-

tering and Thermalization. Benjamin, New York. Placzek, G. (1952) Phys. Rev. 86 377. Schofield, P. and Hassitt, A. (1958) Proc. 2 "d UN Int. Conf. on Peaceful Uses of Atomic

Energy, Geneva Vol. 16, 217. Sj/51ander, A. (1958) Ark. Fys. 14, 315. Van Hove, L. (1954) Phys. Rev. 95, 249. Vineyard, G. H. (1958) Phys. Rev. 110, 999. Williams, M. M. R. (1966) The Slowing Down and Thermalization of Neutrons. North-

Holland, Amsterdam. Wilson, C. C. (1995) Nucl. Instr. Meth. A354, 38. Zemach, A. C. and Glauber, R. J. (1956) Phys. Rev. 101, 118.

APPENDIX

In order to obtain expressions for the coefficients defined by equations (31) to (33), we need to evaluate the derivatives of the function

o o

De(E, E') = 4ehtoD e_U/2• trg(E ' E') - E Gp(u)Kp(E, E'). Ntrinc p=4

(A1)

For E' = E we obtain, after some lengthy algebra, the following results:

De(E, E) = 301(®)eft - (0) 1 - e -4~ , (A2)

(A3)


and

1 x 2 [erf(x/Ax)- D~(E,E) -34~1(®) ~ e r f ( ~ ) + ~ erf(~A) ] (2hoJt~) z

1 + A 2 - 1 x21A] + ~/_~__----~ [ [2 (1- O ) - x-2] (e-A~ + e - ~ l a ) ~ e - j }

1 3~bl (®) -4~ Z(8e _ 2p + 1)(Tp(0) (4e)p (hoJo) z 2-~x2 e p=4 P!

I - Z_, A 2 1 - e -4e ~=4 r ~ - - ~ - ] '

where we have [equation (21)].

(A4)

used the approximated expressions valid for the multiphonon terms

thermal neutron scattering by debye solids: a synthetic scattering function

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