Jul 28, 1994 - BY F. L. CUMBRERA AND F. SANCHEZ-BAJO. Department of Physics, Faculty of Science, University of Extremadura, 06071 Badajoz, Spain.
408
J. Appl. Cryst. (1995). 28, 408-415
An Improved Method for the Normalization of the Scattered Intensity and Accurate Determination of the Macroscopic Density of Noncrystalline Materials BY F. L. CUMBRERA AND F. SANCHEZ-BAJO
Department of Physics, Faculty of Science, University of Extremadura, 06071 Badajoz, Spain AND A. MU~OZ
Department of Condensed Matter Physics, Faculty of Physics, University of Sevilla, Sevilla, Spain (Received 28 July 1994; accepted 23 January 1995) Abstract
During the past few years, several studies have been carried out of the structural characterization of amorphous materials using the Monte Carlo method. It has been shown that a variety of errors gives rise to pronounced artifacts in the R-space correlation functions, which may hinder the accurate interpretation of the Monte Carlo results. The elimination of these ambiguities, particularly for heteroatomic systems, demands very careful experiments in association with careful error analysis. Recently, Kaszkur [j Appl. Cryst. (1990), 23, 180-185] presented a theory describing some convolutional properties of the reduced interference function. This procedure enables an estimation of the normalization constant (in all probability, nowadays, one of the more serious limitations in obtaining significant and reproducible radial distribution functions) with a high degree of accuracy. Since the Kaszkur formulation is limited to monoatomic substances, the present work recalls his basic arguments with the aim of extending them to the most general case of heteroatomic materials. In addition, the proposed procedure allows an overall insight into the quality of the measured data and determination of the macroscopic density on the basis of the scattered intensities. The success of the procedure is proved by its application to two experimental data sets and one simulated example. 1. Introduction
X-ray and neutron scattering experiments have become the most relevant tools for structural studies of liquids and amorphous solids. The Fourier transform of the scattered intensity allows us to derive a set of closely related correlation fimctions in direct R space, the most representative of them being the radial distribution function, RDF(R). By performing RDF analysis of these kinds of substances some preliminary details of the shortrange structure may be derived: (i) the characteristic distances associated with the first coordination shells; © 1995 International Union of Crystallography Printed in Great Britain - all rights reserved
(ii) the mean bond angle, as deduced from the firstand second-neighbour distances; (iii) the coordination number, or the number of atoms included in the first coordination shell. Otten, in order to obtain a more detailed picture of the structure of the substance, a model is proposed that accounts for these previously derived features and is consistent with the known physical or chemical properties of the material. Subsequently, the model is randomly modified according to the Monte Carlo method (e.g. see Henderson, 1971; Rechtin, Renninger & Averbach, 1974; Woodcock, 1975; Barker & Henderson, 1976; Sanz & D'Anjou, 1978; Esquivias & Sanz, 1985; and others). Atter each iteration, the corresponding RDF of the model is evaluated and compared with the experimental one. Once some figure of merit drops below a prescribed value, the iterative process is stopped. Nevertheless, a significant fit of the calculated to the experimental RDF demands very precise knowledge of the correlation functions in R space, which in turn must be obtained from high-quality scattered intensities. Indeed, the reduction process between the raw data and the final significant RDF includes a large number of corrections such as background, Lorentz-polarization, multiple scattering, absorption, incoherent scattering, inelastic effects etc., besides the small-angle extrapolation and the critical normalization process that converts the intensities from arbitrary units to absolute ones. Therefore, we are faced with a difficult task if we wish to obtain significant RDF functions for comparison with model results. It is now well recognized that the two main points limiting the reliability of the final results are the normalization of intensities and the termination effects related to the upper limit of the experimentally available intensities. Termination errors introduce spurious details into the calculated RDF and, from early works, a number of more or less efficient methods of correction have been suggested (e.g. see D'Antonio, George & Karle, 1971; Leadbetter & Wright, 1972; Narayan & Ramaseshan, 1979; Lovell, Mitchel & Windle, 1979; Janot, 1983;
Journal of Applied Crystallography ISSN 0021-8898 © 1995
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E L. CUMBRERA, E SANCHEZ-BAJO AND A. MUlqOZ
Thijsse, 1984; and others). Recently, the maximumentropy method (MEM) has been proposed (Root, Egelstaff& Nickel, 1985; Wei, 1986; Mufioz, Cumbrera & Marquez, 1988; Mufioz & Cumbrera, 1988) by applying the tools of information theory. In fact, the MEM, which does not involve arbitrary assumptions about the extrapolation of truncated intensities, has been a useful finding concerning tnmcation problems. With respect to the normalization of intensities, the problem remains open: the most widely accepted methods are slight modifications of some classical old methods (Krogh-Moe, 1956; Norman, 1957; Wagner, 1969)and involve the following equation for the high-angle intensities:
2. Mathematical background The experimentally available intensity, lexp(S), can be described as /exp(S) ---/(S)nsS~,
(2)
where l(s) is the ideal unrestricted spectrum, sl and s2 are the scattering-vector values of the lower and upper measurement limit, and HS22is a step-like function that is equal to 1 for Sl < s < s2 and to zero for other positive numbers. The above equation can also be expressed as lexp(S) = I(s)(1 - H~')H~ 2
(3)
and the same holds for the reduced interference function, I(s) = c [ ( f Z ( s ) ) + line(S)],
(1)
si(s): [si(s)]ex p = si(s)(1 -- H~ ~)H~2,
where c is the normalization constant, s is the length of the scattering vector, f (s) is the scattering factor, Ii,¢(s) is the incoherent scattering (mainly Compton scattering, for X-rays) and the brackets ( ) denote the statistical average. In practice, however, the accuracy of the determination of c is seriously limited because of the statistical errors associated with the experimental highangle scattered intensities. It is worth pointing out the interesting but exceedingly tedious procedures applied by Lorch (1969) and Thijsse (1984). In this approach, the parameter c is repeatedly varied until (i) the resulting interference function satisfies the socalled sum rule (Wagner, 1978; Thijsse, 1984); (ii) the magnitude of the first ripple in the RDF is cancelled out (Lorch, 1969). The efficiency of these methods may be substantially weakened by the influence of other error sources: e.g. termination errors etc. Recently, Kaszkur (1990) has suggested an excellent mathematical procedure allowing precise determination of both the normalization constant and the local average density. This method involves the whole measured spectrum of scattered intensities, instead of only the high-angle region of traditional methods. Nevertheless, the theory presented by Kaszkur is restricted to materials containing only one atomic species and, moreover, the usefulness of its 'repeated averaging procedure' has not yet been tested on experimental data. In this paper, the procedure of Kaszkur is extended to the general case of disordered substances containing several atom species. The importance and efficiency of the method is illustrated by application to the scattered intensities of liquid iron and to those of the A123Te77 and Si3N 4 glassy systems. In addition, one may remark, in passing, on the interest of a procedure allowing an accurate determination of the material density from the scattered intensity data. This procedure should be particularly useful when resorting to the classical Archimedes method fails: e.g. for porous or inhomogeneous materials etc.
(4)
where the interference function, i(s), is defined by i(s) = [ I ( s ) / c -- ( f 2(s)) - I i n e ( S ) ] / ( f (s)) 2.
(5)
If we perform the Fourier transform of (4), after some algebraic handling we obtain Q ( R ) -- 4~zR(p(R) - Po - ( 1 / R )
x {RLo(R) - P0] * #-(H~')}),
(6)
where Q ( R ) is the reduced radial distribution, p ( R ) is the radial density, P0 is the macroscopic average density, the symbol ~ denotes the full complex Fourier transform and the symbol • denotes the convolution defined as oo f ( x ) * g ( x ) = (21t)-1/2
J" f ( z ) g ( x - z) dr.
(7)
--00
The term Po.s, = ( 1 / R ) { R L o ( R ) - P0] * o~(H~')}
(8)
is a local density function taking into account ,the density fluctuations considered by Cargill (1971). In fact, the neglect of small-angle scattering gives rise to a reduced radial distribution function, which appears, as deduced from its slope at small R, to correspond to a material of greater average atomic density, p~ t
Po = Po + Po, s,-
(9)
In practice, the overestimation of Po may be as large as 17% (Cargill, 1971). It is assumed that p ( R ) = 0 for 0 < R < Re, where Rc is a given distance called the hard-core diameter. Then, Q(R)HoRC - - 4 r c R P o, H 6Rc ,
(10)
although the same could be written by substituting Hff~ for every function, ~p, fulfilling the condition ~pHffc = ~p. Taking account of this property of the ~p functions, and
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STRUCTURAL CHARACTERIZATION OF NONCRYSTALLINE MATERIALS
after performing the Fourier inversion of (10), one obtains
[si(s)].~(tp)=(2x)3/2po d~(tp)/ds.
(11)
From this set of preliminary equations, Kaszkur (1990) derived the basic and fundamental convolutional equation (valid only for monoatomic substances):
[sJ(s)] • ~(tp) = s + (2x)3/2p0 d ~ ( t p ) / d s ,
(12)
from which one obtains (1/s){s[I(s)/f2(s)]. ~-(tP)}
= c + (2x)3/Zc/sP'o d~(tp)/ds,
(13)
where J(s) - i(s) + 1. In the light of these results, we intend to extend the applicability to heteroatomic substances. In this case, one can write
J(s)=[l(s)/cIf(s))2]--[IL +linc(S)/(f(s))2],
(14)
IL being the Laue intensity;
IL(s) = (f2(s)) - (f(s)) 2.
(15)
Here, it is essential to realize that the information about c and p~) is contained within the above equation but, in order to derive such parameters in a straightforward way, we must have a well conceived choice for the tp function. In fact, we need some kind of mathematical function satisfying the following conditions: (i) t# may only differ from zero in the interval (0, Re); (ii) the Fourier transform of tp, ~(tp), must be simple enough that both its derivative, d~(tp)/ds, and its convolution with any function defined in reciprocal space, g(s), are likewise simple. In the choice for tp lies, in our opinion, most of the merit of the Kaszkur work: he assumed a function that does not strictly satisfy condition (i): tp(R) = sin (kaR)/kaR,
where ka > 7t/Rc. Anyway, the function tp,,(R) = tp"(R) (n integer) converges more quickly to zero for R > Rc the larger the integer n is; condition (i) then still holds approximately but reasonably. Moreover, as a counterweight, ~ ( t p ) has the convenient form of a 'window' function:
~[tP(R)]= {(~ 7t)l/2/2ka
By combining (12) and (14), one obtains
(20)
[s' < > ka
(21)
and the convolution of any function of reciprocal space, g(s), with ~-(~o) is so easy as to calculate an average: , do~(~0)
= c + (21r)3/2c/s PO
c +-
s /
(f(s)) 2
ds
x+ko
} • ~(tp) •
[g(s)] • ~-(tp) = 1/2ka (16)
by assuming the [h(s)+li,¢(s)]/(f(s)) 2 function to be smooth enough not to change noticeably after convolution with ~-(~0), we can rewrite (16) in the form 1 [
S
S
l(s) (f(s)) 2
*,~(~0)
]
, d#-(tp)
= C + (2~)3/2C/S PO
ds
(f(s)) 2
(17)
Thus, after regrouping the terms including the true normalization constant, c, one can write the following basic equation for further applications:
sM(s)
s ~
= c + (2x) 3/2
* ,~(¢p) c
, d~(cp)
(18)
M(s) = [(f2(s)) + Iinc(s)]/ ( f (s)) 2.
(19)
sM(s) Po ~ ,
g(t) dt.
(22)
Owing to this property, the convolution of any smooth function g(s) with ~-(tp) allows us to retrieve the same g(s) function. This property was used to establish (17). In addition, the convolution of g(s) with o~-(tp,,) can be treated with equal ease: it consists of a 'repeated average procedure' since ~(~o,) = ~(~o")
[IL(s) + Ii.¢] + c
f x-ka
where
As far as the incoherent scattering correction would already have been included in l(s), then the term line(S) would be omitted in (14) and (19). For a monoatomic substance, (fZ(s)) = (f(s)) 2, and so (18) reverts to (13).
= {~(~o) • ~ ( ~ ) • :~(~0) • ~ ( q , ) . . . • ~(~o)}, n times.
(23)
The evaluation of the left term of (18) demands a proper choice of the parameters n and k,,, both defining the tp,, function that will be henceforth used as the appropriate tp function. In fact, the 'repeated average procedure' is more efficient for larger k,, and integer n [better fulfilment of condition (i)], even though it is evident that each convolution (average) narrows the region of integration by ka for each side of the experimental s range [note, however, that the low-angle reduction does not raise any serious problem because we take advantage of the symmetry relationship g ( - s ) = g(s)]. Therefore, it is desirable to use a smaller initial value for ka (e.g. ka "~ it~Re) and with respect to n, a value n = 3 appears a reasonable compromise. Figs. l(a)-(c) show the Fourier transform, ~[tp,,(R)], from n = 1 to 3 when we assume
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F. L. CUMBRERA, E SANCHEZ-BAJO AND A. MUglOZ
the Kaszkur choice for the q9 functions. These functions are known as the spline functions (Greville, 1969). Once tp3 has been adopted as the most suitable tp function, we calculate its corresponding derivative, d~(q~)/ds; (18) then takes the form
Equation (25) was deduced with the relationship .~-(Hg') = (2/rt)l/2[sin (siR)/R] taken into account, besides the fact that
Po, s, (R) _~ P0,s~ (0) = lim Po, s, (R). R~0
[1/sM(s)l{s[l(s)/(f(s)) 21*,~-((p)}
_
[~c - e [ n z / M ( s ) k 3 a ] p ' o ' O ' s < k a + c[rr2/ZM(s)k3][(s - 3k,,)/s]p'o ka < s < 2ka 3k,~ < s
