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A new method of determining partial radial distribution functions for amorphous alloys
Journal of Non-Crystalline Solids 79 (1986) 1-17 North-Holland, Amsterdam
1
A NEW M E T H O D OF D E T E R M I N I N G PARTIAL RADIAL D I S T R I B U T I O N F U N C T I O N S FOR A M O R P H O U S ALLOYS
|. The quasibinary problem
Yu.A. BABANOV, N.V. ERSHOV, V.R. SHVETSOV and A.V. SERIKOV Institute of Metal Physics, Ural Science Research Centre of the USSR Academy of Sciences, 6202 l 9 Sverdlovsk GSP- 170, USSR
A.L. AGEEV and V.V. VASIN Institute of Mathematics and Mechanics, Ural Science Research Centre of the USSR Academv of Sciences, 620219 Sverdlovsk GSP- 384, USSR
Received 9 May 1984 Revised manuscript received 9 August 1984
To determine the atomic structure of multicomponent alloys we propose a regular algorithm to solve a system of Fredholm integral equations of the first kind which describes independent experiments. For Fe-B-type alloys a quasibinary problem has been formulated which is a system of two equations: one for X-ray scattering, the other for EXAFS. To test the new method numerical model experiments were carried out. The partial radial distribution functions have been determined for metallic glass Fes0B20.
1. Introduction It is known that to obtain partial radial atomic distribution functions (RDF) for amorphous multicomponent systems a number of independent experiments should be performed. In the case of a binary system it is common practice to use data from three experiments. As a rule, the latter are diffraction (X-ray and neutron scattering) experiments [1-3]. Analysis of the experimental evidence for the atomic structure of binary alloys shows that the qualitative agreement of total radial distribution functions is often the case. However, partial R D F differ not only quantitatively (different coordination numbers and peak positions) but also in the forms of curves. This difference makes it difficult to unambiguously select a theoretical atomic structure model for amorphous alloys. One of the causes of the differences observed may be the two-step R D F derivation procedure (1 - obtaining partial structure factors, 2 - Fourier transformation thereof) [4]. 0022-3093/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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Yu.A. Babanoo et aL / A new method for determining RDFs. 1
Along with diffraction studies, the method of X-ray spectral structural analysis (EXAFS technique) has recently found extensive applications in the determination of local atomic order [5-7]. However, the peculiar procedure of extracting structural information (a combination of the Fourier filtration method and the least-squares method) does not allow the EXAFS to be included as an independent experiment into the technique employed by Waseda [1]. Apart from this, strong correlation between the parameters being varied, the number of which amounts to 8 for the binary problem, leads to instable solutions and unreliable results. To determine partial radial distribution functions the present paper proposes a regular method of solving a system of integral equations. Earlier, this method was advantageously applied to one-component systems both in the case of the EXAFS [8,9] and in the case of X-ray diffraction [10]. The method consists of using a special regularization procedure for a system of integral equations, which permits reduction of the initial problem to a system of algebraic equations with a well-conditioned matrix. A regular method of solution possesses stability with respect to perturbations of input data. This makes it possible to use true experimental data without a preliminary smoothing procedure. In contrast to the Fourier transformation technique, this method is not associated with a particular kind of kernel of integral equation. Therefore it is applicable to equations with kernels which are more complicated than that of an integral equation describing the X-ray (neutron, electron) scattering, for example to an EXAFS equation. This permits the inclusion of an EXAFS equation into a system of integral equations along with diffraction equations. Such an extension of the number of independent experiments indispensible for determining partial RDF will in all likelihood allow an appropriate combination of experiments. The present paper (Part I) deals with the solution of the quasibinary problem, which is a system of two equations: the first equation describes the X-ray scattering, the second equation describes the EXAFS. Section 2 formulates the problem of finding partial radial distribution functions for an n-component system and presents a scheme to solve this problem. To test the new method a model quasibinary problem is solved in section 3. In the same section the solution algorithm is described in detail. The final section furnishes results of studies on the atomic structure of metallic glass Fes0B20. Experimental values of the coordination numbers and interatomic distances for Fe-Fe and Fe-B pairs have been found. They are compared with results of experimental investigations using diffraction methods [2,3] and the EXAFS method in its conventional treatment [5] and with results of atomic structure simulation for transition metal-metalloid-type metallic glass [2,11,12]. Part II of this work will treat the solution of a binary problem, as a result of which three partial RDF are found from the experimental data including two EXAFS experiments and one diffraction experiment.
Yu.A. Babanov et al. / A new method for determining RDFs. I
3
2. Theory of the method Nearly all of the direct-method data relating to the arrangement of atoms in a condensed medium are described by a pair correlation function g(r~2 ). For a spatially homogeneous isotropic system, of which glass or liquid is an example, this function depends on the modulus of the vector [ r~z [ connecting points 1 and 2, i.e. g( try2 I) - g(r). If the system contains n elements then to describe the atomic structure requires the knowledge of N = n(n + 1)/2 partial functions go(r). The function go(r) relates the density of the probability of finding a particular pair of atoms at a particular atomic separation and possesses the property
g,j(r) = g,,(r).
(1)
It is known [13] that a total interference function I(s) describing a diffraction expriment may be expressed in terms of partial functions gij(r), i.e.
[I(s) - 1] s = 4~roo
~_, (ci cj F~ Fj)/Fo a
g,j(r) - 1] r sin(sr) dr.
i=1 j=l
(2) Here s = 4~r sin 0 / ~ is the vector of scattering of a quantum with a wavelength )~, O0 is the mean atomic density, c, the atomic concentration of element i, F, the amplitude of coherent scattering of element i, n
F0 = E c, U . i=l
To a single-scattering approximation for K absorption spectra, the integral equation in the EXAFS technique is [8] j=l
k
(3) Here x , ( k ) is the normalized oscillating part of the X-ray absorption spectrum of the ith-element, ] fj(k, ~r)] the modulus of the backscattering amplitude for a photoelectron with a momentum k on the neighboring atom j, ~p,/(k) the total phase shift, and ~ ( k ) the mean free path of a photoelectron. It should be noted that the normalized oscillating part is determined by the sum of the contributions from n partial functions gij, whereas the total interference function I(s) is determined by the sum of n(n + 1)/2 functions. This circumstance is essential to the formulation of the quasibinary approximation. Thus, to experimentally determine N partial functions it is necessary to have N sets of experimental data. Let us introduce a consecutive numbering. Let i = 1, 2 . . . . . n, and j = i, i + 1 , . . . , n then
hm = gm -- 1, (4) where m = i + ( j - 1 ) j / 2 , (m = 1, 2 .... N). We introduce a unified notation
4
Yu.A. Babanov et al. / A new method for determining RDFs. 1
for the set of experimental data [UI], where l = 1, 2 . . . . . N. Equations such as (2) and (3) may be combined into a system of Fredholm integral equations of the first kind N
Ut = ~2 ~',mh,~ ( l = 1, 2 , . . . , N ) ,
(5)
rn=l
where sJ~,~ is an integral operator. In matrix representation eq. (5) is of the form U1
U2
=
UN
J~ll
,5~12
"" "
J~IN
~21
~z2
"'"
d2N
J~NI
,5~N2
,-.
5~NN
hi ×
h2
(6)
The problem of finding the functions h m belongs to a family of inverse problems which are ill-posed [14,15]. A feature peculiar to such problems is that negligible input-data distortions lead to arbitrarily large distortions of the solution. Furthermore, to the same experimental data there corresponds an infinite set of solutions equivalent in exactness. Consequently, a principle of selecting solutions is required. In the selection it is quite natural to take account of the a priori physical information concerning the solution sought. The functions should conform to the following requirements: (~). The solution [hm] should be such that when acted upon by a matrix of integral operators J~lm the residual norm u
d, mhm-
u~
m=l
should not exceed the experimental-data error 6, i.e.
Lz[cl,dl]
=
'l Y~ ~¢~mh~(r) - U~(k)[ 2 d k
<~8 (l = 1, 2 . . . . . N ) .
m=l
Here C t and d l are the boundaries of the data set Ul. This major criterion for the correct choice of an approximate least-squares method. However, this method is instable with data perturbations, which presents certain difficulties in the results. (~). The solution should be sufficiently smooth. (~). The solution hm >~ - 1 for any r. @ . hm(r)--'O when r ~ oc. @ . The functions hm(r ) are normalized as follows
1 bm - a m )
rb'n
2
J. 47rr hm(r) d r - (W,,, hm) = 0 am
requirement is a solution in the respect to inputinterpretation of
Yu.A. Babanov et al. / A new methodfor determining RDFs. I
5
with the boundaries a,~ and bm chosen from the condition
g m ( r ) = O at r < am, gin(r) = 1 at r >_ bin.
(7)
The conventional method of constructing a solution to a system of Fredholm integral equations of the first kind is to approximate the functions h", and Ut by vectors, and the integral operator ~lm by the matrix, and then to solve a system of linear algebraic equations. This approach turns out to be unacceptable because of the instability of the problem. The matrix that arises is ill-conditioned or even degenerate, and the solution obtained does not approximate, generally speaking, the solution of the initial problem when the magnitude of the discretization step tends to zero (i.e. when the exactness of the approximation is improved). A regular algorithm, i.e. an algorithm stable with respect to input-data perturbations, cannot be constructed following this path. The problem (5) solution algorithm expounded below possesses the property of stability with respect to small variations of input data and permits taking account of the a priori information formulated in requirements @ to
@. A large contribution to the theory of solving ill-posed problems has been made by Tikhonov who introduced the fundamental concept of a regularization operator and a regularized solution [16]. By convention, three regular-solution construction stages can be singled out. (i) Regularization. As a regularized approximate solution we take the vector [ h~ (r)] minimizing in space L 2 the Tikhonov functional
min
~
/=1 m=l d
+
,S~lmhm-U l
+ ~
(amllh",-h~]l 2
m=l
II Zrr (hm - h'nr) [[2 --[- V., I(mm, hm) [2)]
L 2 [ a ....
b,.].
(8)
In eq. (8) the last term, introduced additionally, is a penalty function. Here h t~, is some trial function or zero, 7/the vector of the regularization parameters a",, tim, and of the penalty parameters ~'m. The presence of terms with small positive parameters a,, and tim in functional (8) renders the problem stable and assures the fulfilment of requirement ( ~ . It has been proved [15] that when the parameters a,~ and /3., are decreased an approximate solution h~,, tends to an exact one. The parameters ~'m are responsible for the satisfaction of normalization condition @ . (i 0 Discretization. To be solved numerically problem (8) should be preliminarily discretized. Approximating the space L 2 by a discrete space l 2 and prescribing in space l 2 the functions h,,, and Ut as vectors and prescribing the integral operators Jaet,. as matrices we obtain a discrete analog of problem (8). It follows from the general results of ref. [17] that the solution of the discrete problem converges to that of problem (8) when the magnitude of the discreti-
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Yu,A. Babanov et al. / A new methodfor determining RDFs. 1
zation step tends to zero. Using the requisite extremum condition - the equality to zero of the partial derivatives with respect to the components of the vectors hm of functional (8) in discrete form - we come to a system of linear algebraic equations. The matrix of this system is positive definite. This allows well-known linear algebra methods to be used for matrix inversion. (iii) Iterational refinement. The conditions (~) and (~) imposed on the solution [hm] are attained by the selection of the boundaries am, bm (see formula (7)) by virtue of the specific features of the radial distribution function for amorphous systems. However, if the approximate solution [hm] does not meet requirement (~) or if the exactness of this solution is unsatisfactory the iteration procedure [9,18] should be used h {"+') = P ( a g * a g + B ) - I ( a g * U + Rh {")) (n = 0, 1,... ).
(9)
Eq. (9) is represented in symbolic form, with ~ ' being the matrix of the resulting system of algebraic equations, U the total vector approximating the vector function [U~U2 . . . . , UN], h the total solution vector approximating the vector function [h~, h 2. . . . . hN], P the operator of metric projection to the set Q = { h: h >j - 1 } ; n = 0, 1, 2 . . . . the iteration number, B three-diagonal and diagonal matrices containing the parameters al, % , . . . ~N and fi~, f12. . . . . /~N.
3. Model quasibinary problem If one chooses (1) X-ray diffraction, (2) an EXAFS spectrum of element 1, and (3) an EXAFS spectrum of element 2 as independent experimental data when determining the partial functions of a binary system then eq. (6) assumes the form
U3
Lo'11 131ihl] 'J~32 3~33
(lO)
h3
where hl(r ) = gll(r) - 1, h2(r ) = g12(r) - 1, h3(r ) = g22(r) - 1, b SC~mh m = B a , , ( k ) f °'hm(r) s i n ( k r ) r dr (m = 1, 2, 3), a m
bm allmhm = B,,,( k ) ~ m h m ( r ) e -2r/x(k) sin[2kr + ~Pm(k)]dr
(I=2, •=3,
m=1,2) rn=2,3 '
B n ( k ) = 4rrpoe~F12(k)/Fff(k), B,2(k ) = S ~ r p o q c 2 F l ( k ) F 2 ( k ) / F o 2 ( k ) ,
(11)
Yu.A. Babanov et al. / A new method for determining RDFs. I
7
B,3( k ) = 4~rpoc~F?( k ) flFoa( k ), Bz~ ( k ) = 4 ~rOoc~ [A ( k, 7r) Il k , B22(k ) = 4~rPoC2 rf2( k, ~r) Ilk,
B32(k ) = B2,(k), B33(k ) = B22(k ).
(12)
The functions Ut(k) include the experimental values and asymptotes of elements dtm, which arises from the fact that the interval of h m functions is finite.
Ul(k ) = ( I(k ) - 1 ) k + Tu(k ) + T,2(k ) + T,3(k ), Uz(k ) = x,(k) + T21(k ) + T22(k ), U3(k) = x2(k) + T32(k ) + T33(k ).
(13)
Here ~a
Tlm(k ) = B l m ( k ) j o ' r sin(kr) dr (m = 1, 2, 3), OC
Tt..(k)= -Bzm(k) f
e-Zr/x(k) sin[2kr + ~ m ( k ) l d r a m
(l=2, •=3,
rn=l,2) m=2,3
(14) "
The quasibinary problem may be formulated for amorphous transition metal (TM)-metalloid (M) systems, of which the metallic glass Felo0_~B ~ (x = 15 to 25) may be an example. Fig. 1 presents ratios of the coefficients Blm for F e - B and F e - F e pairs as f Fe-B ~
f ~ )
o I
CB FB
= 2 CFe FF e
5 . . . .
I
10 . . . .
I
15 . . . .
I
2O
0 ....
i'o ....
2'o .... s (~")
io
Fig. 1. Weighting factors for X-ray scattering method (1) and EXAFS method (2) for alloy Fe85-Bl 5.
8
Yu.A. Babanov et al. / A new method for determining RDFs. 1
for X-ray scattering * and as f/ F e - B )
c.
IfB(k,~')[
for the EXAFS ** The relative contribution of the B-B pair to the structure factor of X-ray scattering is small since (cBFB)Z/(cwFFe)2~ 0.001 (0.1%). Comparing this value with that of f ( ( F e - B ) / ( F e - Fe)) (curve 1 in fig. 1) we assume that the contribution made by the B-B pair to eq. (10) may be neglected, i.e. we set z¢~3 = 0. Then eq. (10) becomes
A similar approximation was made by Chen and Waseda [2]. However, they neglected the contribution of the B - B pair not only in the case of X-ray scattering but also for neutrons, although in the latter case this contribution is not so small ( - 1.7%). For the quasibinary problem (15) the Tikhonov functional (8) may be written as
M"Ih,, +
E E2
u,
2 l=1
22[ct,d A L
+ E2 {< ,.llho,-h ll m=l
hm-ht~)
+Tml(Wm, hm)l 2 h m ~ L 2 [ a m , bin].
(16)
To numerically solve problem (16) one has to pass to the discretely prescribed functions h m, U~ and integral operators ~t,,- To this end, we will divide the segment [al, bl] into Pl parts with a step Ar 1 = (b 1 - a l ) / p 1, the segment [a 2, b2] into P2 parts with a step Ar a = (b 2 - a 2 ) / p 2, and also the segment [c 1, dll into ql parts with a step Ak I = (d 1 - c l ) / q l , and the segment [c2, d2] into q2 parts with a step A k 2 = (d a - c2)/q 2. In our numerical algorithm Ar 1 = Ar 2 and Ak 1 = Ak 2. The discrete approximation of the functions and integral operators was carried out using the collocation method [21]. Replacing the space L 2 by a discrete space l f ( p - dimension), and the derivative d / d r by a difference equation we obtain a discrete analog of problem (16) Mn[hi,
h 2 ] -~ II z ~ ' l l h l + ~ ¢ 1 2 h 2 -
U1
I1~ +
+alllhl-hl(llJl+a2llh2 2
+/7'l-~r(h'-ht()],,,+
B
~ll ~¢21ha +,-~¢22h2 - U2 11~2 _ /,,tr II 2 " 2 11/~2
d
2 -~r ( h 2 - h ~ ) 1 2
t~'~
+Tal(W~,hl)12 +'/21(W2,h2)l 2" h~lP2 ', h 2 E l p~. * The values of the scattering amplitudes F have been borrowed from the tables of ref. [19] ** The values of the scattering amplitude modulus I f ( k , Ir)l and of the phase shifts have been borrowed from the tables of ref. [20].
(17)
Yu.A. Babanov et at / A new method for determining RDEs. 1
9
Here all, ~ is a ql x Pm matrix with the elements a i d = B,m(k,)fri'~'r sin(kr) dr ~=., is
a q= ×
(18)
Pm matrix with the elements
ai.i = Bzm(k,) fr['+l e ( - 2rn h2 kir-~) sin[2kir +~/m(ki)] dr.
(19)
In eqs. (18) and (19) the coefficients B,, are given by formulas (12). The phase shifts +re(k) in eq. (19) are determined as ~b,(k) =- 2 3 , ( k ) + qq (k, ~r),
2(k) =
+ 02(k,
(20)
where 31(k ) is the phase shift on the central atom and %,(k, or) is the phase shift of the scattering of a photoelectron on the surrounding atoms of species 1 and 2. The parameter ~ determines the relative weight of the EXAFS equation. The reason why this parameter 'is introduced will be explained in what follows. The vector h,,(r) has elements hma, the vector Ut(k ) elements Ut, - Ut(k,), the vector W., elements IV,,i -- (53+1 - r,3)/(b 3 - a3m), and the vector (d/dr)hm(r) elements (d/dr)(hmj) = (hmj+l - h,w)/Ar. The norm h m in space l p is given by the formula ][ hm J[~ = E~=lArh2j. Similarly, q
itu, tl, = i=1 E The v e c t o r s h 1 and ha minimize the functional M'[hl, h2] (17) if and only if the first variations of the functional in hi and h 2 are equal to zero. This leads to a system of linear algebraic equations which when represented in matrix form is
d,*2g, +
]
----- - d l ~ d l l + ~d2~';~21 + B1 ,.~1~11 q-- ~.5J2~.5~/21
"5~1~J~12 q- ~JJ2~ J~22 .5J1"2,-~12 "Jr-~..~/2"2J~/22 + B 2
×
. (21)
h2
Here .-~u and ad22 are matrices with elements a,j which are determined by formulas (18) and (19) for all i,j except for aq+lj = Wmj~--£- The symbol (*) denotes transposition. The vector G = [G,, U/:, .... ~,, 0]. The Pm × P,,, matrix B., has the form
I
0 -~,,,
Bm =
g~m + 2fim
-.- 0
--fl,,,
0
--~m
am
0
0
0
+ 2fi~
"'" .-.
0 0
•••
ffm + 2tim
(22)
10
Yu.A. Babanov et al. / A new method for determining RDFs: I
where
~,, = ( amAr ) / a k , ~',, = flm/( ArAk ), ~,, = Vm/ak.
(23)
The matrix involved in eq. (21) is of the dimension (P1 q ' P 2 ) X ( P l + P 2 ) " It is nondegenerate due to the presence of matrices B,, which contain small positive parameters ~,, and /9,,. As has already been stated above, standard methods may be used to solve the system of linear algebraic equations (21). In our calculations, for example, the square root method was applied. For solution refinement the iteration procedure according to formula (9) was employed. What was chosen as an initial approximation in the iteration process was the solution to system (21) specially transformed to meet requirements (~) and (~), i.e. h,,(r) >__ - 1 for any r, hm(r )--) 0 when r --* oo. To ascertain the efficiency of the algorithm described above numerical experiments were carried out. As model partial functions of Fe-Fe, Fe-B, and B-B we selected functions qualitatively similar to those calculated according to the relaxed dense random packing (DRP) model for metallic glass Fe85B15 [2] (fig. 4, curves la, 2a, and 3a). For these functions we calculated the normalized oscillating part x ( k ) according to eq. (3) and the structure factor I(s) according to eq. (2). The plots for the normalized oscillating part x ( k ) and structure factor I(s) are presented in fig. 2 on a single scale (s = 2k). Structure factor I(s) oscillations at low s, as is known, arose from model RDF cutoffs. Two peculiarities of diffraction and EXAFS data must be noted. (1) The X-ray scattering signal is approximately two orders of magnitude greater than the EXAFS signal. That is why we introduce the parameter ~ into functional (17).
I
~il, 3'
0 t_..._.z
¸1102
2-
10
0
)0
,,('A")
s(#')
r5
~3
r(~)
Fig. 2. Model interference function l ( s ) and normalized oscillating part of X-ray absorption x(k). Fig. 3. Partial radial distributions functions g,j(r) for amorphous state model (a) and result of solving a system of two integral equations for model l ( s ) without taking account of B-B contribution and model x(k) (b).
Yu.A. Babanov et al. / A new method for determining RDFs. I
11
(2) The structure factor I(s) has the main contribution at 2 A-1 < s < 15 ~, i whereas x(k) has the main contribution in the interval 3.5 , - 1 < k < 15 A -~ (7 ,~ i < s < 30 A - l ) . It is therefore natural to regard these two experiments as mutually complementary. For the calculated functions x(k) and I(s) the quasibinary problem (21) was solved with the following parameters: the number of points for l(s) ql = 400, the number of points for x(k) q2 = 241, which corresponds to the intervals [Q, d~] = [0.05 ~k-1; 20 A -l] and [c 2, d 2 ] = [3.5 ,~-1; 15.5 A - I [ with the step Ak = 0.05 A - l ; the number of points in r space p~ =P2 = 120 with the step Ar = 0.08 ,A corresponds to the intervals in which the solutions [a 1, b l ] = [2.15 ,~; 11.75 A], [a2, b2]=[1.75 A; 11.35 A] are sought; the mean atomic density for Fe85B15 P0 = 0.09594 a t . / A 3. In the first version of the problem solution the contribution of two pairs, F e - F e and Fe-B, was taken into account in I(s). But since the EXAFS equation (10) involves only these same pairs the inverse quasibinary problem is exact in this respect. The results are presented as plots in fig. 3 and in table 1. Comparing curves l a and l b we see that the solution gve-Fe practically coincides with the model function over the entire interval prescribed. As far as the function gvc-~ is concerned, the contribution of the F e - B pair to both the diffuse scattering intensity I(s) and x(k) is appreciabloy smaller. Therefore the solution gw-B (fig- 3, curve 2b) is distorted at r > 5 A because of the errors introduced by the calculation procedure. Here we must note that the EXAFS data interval was restricted to k > 3.5 A-1. The information contained in the first peak of the R D F is evident at high k, but the information contained in the tail is present only for small k, and hence is lost. For a quantitative comparison of the model and of the solution, the following major characteristics of the first peak of the function gij(r) have been selected: the maximal value of g, peak coordinate r, coordination number Nq, first-peak asymmetry index x. The quantities g and r are determined as the value and position of the vertex of the parabola obtained from three points in the vicinity of the peak maximum. The number of nearest/-species neighbors is estimated according to the formula e
Nij = 4~rpocj f " r2ggj( r ) dr,
(24)
where a is determined from condition (7), e corresponds to the first minimum behind the first function gij(r) peak. The asymmetry parameter x is found as the ratio of the right-hand and left-hand first-peak halfwidths measured on the half-height. Comparison of the major characteristics of the model and of the solution (see table 1) testifies that the regularization solution of the inverse problem is practically exact. The choice of regularization parameters and the stability of regularized solutions were discussed in [10]. In the solution of a quasibinary problem the contribution of the B-B pair to the intensity I(s) may be regarded as a perturbation of input data. In the second version of the model calculations the contribution of this pair was
5 × 10 -8
5 × 10- 7
10 -7
103
104
1
2.56
2.56
2.56
3.63
3.66
3.66
3.64
relaxed D R P model X-ray diffraction Neutron diffraction X-ray diffraction Neutron diffraction with n~t Fesol~ B20 Neutron diffraction with 57Fe8o 11B2° EXAFS X-ray diffraction EXAFS 2.57 2.55 2.57 + 0.02
12.4 8.2 12.0 h) +- 0.3
10.7
2.56
10 9
2)<10 - 9
5 × 10 - 7
2.14 2.06 2.07 +_0.02
2.57
2.08
r (A)
12.2
r (,~) 2.56
Fe-B
10 -8
2 × 10- 7
5 × 10 -9
F e - Fe N (at.)
1.80
1.79
1.77
1.78
a) This value has been obtained from the relation CBNB_Fe = cveNve_ e where NB_ w = 6.9 at. [2]. b) Integration interval [2.15 ,~, 3.20 A]. ~) Integration interval [1.65 A, 2.55 A.].
Fes0 B2o Fes0 B20
Fes0132o
FessB15 Fes3 B17
13.82
14.02
13.95
14.00
(at.)
b) Integration interval [1.75 ik, 2.55 A].
2 × 1 0 -8
10- v
10 -8
2.56
Table 2 Structure parameters for oearest neighbors in amorphous F e - B alloys
a) Integration interval [2.15 ,~, 3.45 A].
Model Result (without B-B) Result (with B-B) Result (with B-B)
(A)
N ~
Fe-B g 0' 2
r
al
Pl
Fe-Fe
Table 1 Interatomic correlations of nearest neighbors - results of model numerical experiments
2.2 2.2 1.6 c) +_0.2
1 . 4 ~)
1.5
N (at.)
2.08
2.08
2.08
2.08
(A)
r
4.63
4.94
4.96
4.91
this paper
[3] [51
[21
[2]
Ref.
1.38
1.44
1.54
1,56
(at.)
N b}
1.16
1.22
1.24
1.31
Yu.A. Babanov et al. / A new method for determining RDFs. 1
13
~4
0 5
5
g+ ~3
c~
6 r(~)
8
1'0
Fig. 4. Partial radial distribution functions g u ( r ) for amorphous state model (a) and results of solving a system of two integral equations for model l ( s ) taking account of B - B contribution and model x(k) at ~ = 10 4 (b) and ~ = 1 (c).
taken into account and the effect of the above perturbation on the solution estimated. The solutions gFe-Fe and gve_ u in this case are given as plots in fig. 4. The label b pertains to the curves corresponding to the solution at ~ = 1 0 4, and the label c to the solution at ~ = 1. The parameter ~ is responsible for the weight of the EXAFS data in functional (17). Comparison of these curves as well as of the major characteristics of the first peak of partial functions with the model (see table 1) suggests that enhancing the role of an exact equation when a problem is solved approximately is a fairly efficient technique. The weight of the EXAFS data in eq. (17) should be increased for two more reasons: (1) the intensity of oscillations in x ( k ) is approximately two orders of magnitude smaller than that in the structure factor I(s), (2) the relative contribution of the F e - B pair to x ( k ) is considerably larger than the contribution in the case of X-ray scattering.
4. Atomic structure of metallic glass FesoB2o The X-ray scattering experiments were performed in reflection geometry using an automated diffractometer [22]. Monochromatized radiation (Mo Kc~, 2~= 0.7107 ,~) was produced using pyrolytic graphite on the reflected beam. The time needed to gain 10 4 counts was measured. The recording was performed stepwise with a step As = 0+05 ,~-1 uniform in s space over the interval 0.8 ,~-1 < s < 16.6 ,~-t. This interval corresponds to the angular
14
Yu.A. Babanov et al. / A new method for determining RDFs. 1
interval 5 ° < 20 < 140 °. The specimen for diffraction experiments was composed of amorphous FesoB20 alloy ribbons 1.5 mrn wide produced by melt spinning. The thickness of the specimen was such that the beam attenuation coefficient at normal incidence was equal t o - 100. The preliminary experimental-data processing was effected following a program in which the ideas advanced in [23] were used. The structure factor I(s) for metallic glass FesoB20 is presented in fig. 5. In the interval [0 to 2 sk-t] the structure factor was extrapolated to zero. The same figure gives the normalized oscillating part of the X-ray absorption coefficient, x ( k ) . The Fe absorption K spectrum for a specimen in the form of a 12/~m thick ribbon was registered using the synchrotron radiation of a VEPP-3 storage ring (Novosibirsk). The experimental conditions and the preliminary processing are described in [24]. The preliminary processing was done using optimal parameters for pure iron. The result of solving the system of algebraic equations (21) was obtained with the following regularization parameters: al = 10-3,/31 = 10-3, a2 = 1 0 - 1 , /32 = 2 × 10 2, "/a = "72= 10-l, n = 3. The value of the mean atomic density for FesoB20 was 00 = 0.09515 at./,~ 3. The experimental values were prescribed in the limits: [ca, da] = [0.05 k -1, 16.6 j - i ] for l(s) and [c 2, d2] = [3.5 k -1, 15.5 j - 1 ] for x(k) with the step Ak = As = 0.05 A-~. The solutions were tried in
p. :.
gtO~ - ".
/':
t,.
/:7-
::
•
.
".-all'/
~'
..
• "
o¥
.-
V
"
-:
'.."
I
,
L
,
!
,~
:.
I
,
lo
I~
I
(l")
I
I
~5
#,,. •
:~, ; \.f"---"-%1/°
J |
0
i
i
i
i
|
i
i
,0
I
i
I.
2o
i
s (#i-')
Fig. 5. Experimental interference function l(s) and normalized oscillating part of X-ray absorp-
lion x(k) for amorphous alloy Feso-B2o.
15
Yu.A. Babanov et al. / A new method for determining RDFs. 1
~4
~2 1
0 4
1
o
t (4)
Fig. 6. Partial radial distributionfunctionsof amorphous alloy F%o-B2o.
the intervals: [a,, ba] = [2.15 ,~, 16.55 ,~] for g v e - w and [a2, b2]= [1.65 A, 16.05 A] for g v e - B with the step Ar = 0.12 A. The partial radial distribution functions g v e - V e and gve B are presented as plots in fig. 6. The curve g w - v e is similar to the total radial distribution function. However, one feature merits special consideration. The first coordination sphere is asymmetric in form (the asymmetry factor x = 1.6) and is separated from the other sphere by a minimum typical for coordination spheres of crystalline materials. As might be expected proceeding from model calculations, the function gFe-B is most distorted in the region of higher coordination spheres. It is next to impossible to single out the signal from these spheres from among random noise. As was shown earlier [5,9], the intensity of the EXAFS spectrum for amorphous systems is determined chiefly by the contribution of the nearest neighbors. By increasing the relative weight of the exact EXAFS equation in the quasibinary problem (21) to 104 we increase the relative weight of the EXAFS contribution by the Fe-B pair. Therefore it is hoped that trustworthier information may be obtained concerning the first coordination sphere gF~-B. The high narrow first peak is symmetric in form (K = 1). The total width of this peak on the halfheight is equal to 0.4 A, which is close to the value obtained in [3]. However, the other peak characteristics determined in the present paper differ from the data obtained in [3]. In table 2 the values of interatomic distances and coordination numbers for F e - F e and Fe-B pairs are given, obtained in Fujiwara's theoretical model [2] and from the data of different experiments [2,3,5]. First and foremost, it should be noted that the stability of the experimental results for the F e - F e pair is comparatively high. An exception is the coordination number NF~_Fe 8.2 at. in ref. [5] where only EXAFS data were analyzed, Our results are the closest to the data obtained by Nold et al. [3] and give a fairly good fit to the results of the theoretical relaxed DRP structure model [2]. ~---
16
Yu.A. Babanoo et al. / A new method for determining RDFs. 1
The largest spread in data occurs for gFe-~. The interatomic distance rFe-B = 2.07 A determined by us coincides within the experimental error with the data obtained by Haensel et al. [5] and with the result of theoretical calculations [2]. The difference in coordination numbers (12.0 at. and 8.2 at. for F e - F e ) (1.6 at. and 2.2 at. for F e - B ) apparently stems from the assumption, adopted by the authors of [5], as to the asymmetry of the first peak of the gFe-B function. As is well known, in the processing of EXAFS spectra a strong correlation arises between the parameters varied, specifically between the coordination number and the asymmetry parameter. A similar conclusion may be arrived at if one analyzes the results of EXAFS studies of metallic glass Fe~oB20 presented in [25]. In refs. [2,3] systems of two and three linear algebraic equations were solved. The matrix of those systems is ill-conditioned. In this case it may be expected that errors in initial data will lead to large distortions of the solution, especially for the function gFe-B whose contribution to the total intensity is comparatively small. The function g~e-B is particularly essential to our knowledge about the atomic structure of T M - M metallic glasses. Modern theoretical T M - M structure models [2,11,12,26] rest on the assumption: the interaction between metallic and metalloid atoms is several times stronger than that between metallic atoms. Therefore the metallic atoms surrounding the metalloid atom form a perfect configuration. This peculiar molecular unit is a major characteristic of the atomic structure of an alloy. It persists, as shown in [11,27], when the concentration of the components is varied. In the Boudreaux model each boron atom is surrounded by 6.6 Fe atoms. The coordination number obtained in the present paper, NBFe = 6.4 at., is close to the result of Boudreaux and to the experimental value NBFe = 6.9 at. of ref. [2]. In the Fujiwara relaxed D R P model [2] the value N B v e = 7.7 at. differs rather significantly from the data obtained by Nold et al. ( N B v e = 8.7 at.) [3] and Haensel et al. [5]. The symmetric form of the g w - B curve (see fig. 6) is also testimony to the presence of a perfect configuration of Fe atoms. On the strength of the experimental evidence for one alloy it is difficult to draw final conclusions concerning the atomic structure of amorphous T M - M alloys. The results of further investigations into the atomic structure of the F e - B system will be published elsewhere. We wish to thank Dr A.V. Serebryakov who kindly supplied us with a specimen for analysis.
References [1] Y. Waseda and S. Tamaki, Z. Phys. B23 (1976) 315. [2] T. Fujiwara, H.S. Chen and Y. Waseda, J. Phys. Fll (1981) 1327. [3] E. Nold, P. Lamparter, H. Olbrich, G. Rainer-Harbach and S. Steeb, Z. Naturf. 36a (1981) 1032.
Yu.A. Babanov et al. / A new method for determining RDFs. 1
17
[4] 3. Hafner, in: Liquid and Amorphous Metals, eds., E. Luscher and H. Coufal, Proc. Nato Adv. Study Inst. (Sijthoff & Noordhoff Int. B.V., Alphen aan den Rijn, The Netherlands, 1980) p. 183 ]5] R. Haensel, R. Rabe, G. Tolkiehn and A. Werner, ibid, ref. 4, p. 459. [6] P.A. Lee, P.H. Citrin, P. Eisenberger and B.M. Kincaid, Rev. Mod. Phys. 53 (1981) 769. [71 P.H. Gaskell, Proc. of the study Weekend 13-14 Nov 1982 (DL/SCI/RI9, 1983) p. 28. [8] Yu.A. Babanov, V.V. Vasin. A.L. Ageev and N.V. Ershov, phys. stat. sol. (b)105 (1981) 747. [9] A.L. Ageev, Yu.A. Babanov, V.V. Vasin, N.V. Ershov and A.V. Serikov, phys. stat. sol. (b)l 17 (1983) 345. [10] N.V. Ershov, A.L. Ageev, A.V. Serikov, Yu.A. Babanov and V.V. Vasin, phys. star. sol. (b)121 (1984) 451. [11] D.S. Boudreaux, Phys. Rev. B18 (1978) 4039. [12] D.S, Boudreaux and H.J. Frost, Phys. Rev. B23 (1981) 1506. [13] C.N.J. Wagner, J. Non-Cryst. Solids 42 (1980) 3. [14] A.N. Tikhonov and V.Ya. Arsenin, Solution of ill-posed problems, translated from the Russian. Preface by translation, ed., Filtz John Scripta Series in Mathematics, (Winston. Washington DC; Wiley, New York, Toronto, London, 1977). [15] V.K. Ivanov, V.V, Vasin and V.P. Tanana, Teoriya lineynykh nekorrekmych zadach i eyo prilozheniya (Izd. Nauka, Moscow, 1978). [16] A.N. Tikhonov, Dokl. Akad. Nauk SSSR 151 (1963) 501. [17] V.V. Vasin, Dokl. Akad. Nauk SSSR, 258 (1981) 271. [18] V.V. Vasin, Proksimalnyi algoritm s proektirovaniem v zadachakh vypuklogo programmirovaniya (Preprint: Institute of Mathematics, Acad. Sci. USSR, Ural Sci. Res. Center, SverdIovsk, 1981). [19] P.A. Doyle and P.S. Turner, Acta Cryst. A24 11968) 390. [20] B.K. Teo and P.A. Lee, J. Amer. Chem. Soc. 101 (1979) 2815. [21] A.L. Ageev, Yu.A. Babanov, V.V. Vasin and N.V. Ershov, in: Chislennye i analiticheskie metody resheniya zadach mekhaniki sploshnoi sredy (Ural Sci. Res. Center, Acad. Sci. USSR. Sverdlovsk, 1981) p. 3. [22] E.Yu. Medvedev, V.A. Boshegov, Yu.A. Deryabin, Yu.A. Babanov, N.V. Ershov and A.V. Serikov, in: Apparatura i metody rentgenovskogo analiza 29 (Izd. Mashinostroenie, Leningrad. 1983) p.79. [23] S. Ergun, J. Bayer and W. van Buren, J. Appl. Phys. 38 (1967) 3540. [24] N.V. Ershov, Yu.A. Babanov and V.R. Galakhov, phys. stat. sol. (b)l17 (1983) 749. [25] M. de Crescenzi, A. Balzarotti, F. Comin, L. Incoccia, S. Mobilio and N. Motta, Solid. St. Commun. 37 (1981) 921. [26] P.H. Gaskell, J. Non-Cryst. Solids 32 (1979) 207. [27] T. Fujiwara and Y. Ishii, J. Phys. F10 (1980) 1901.
1
A NEW M E T H O D OF D E T E R M I N I N G PARTIAL RADIAL D I S T R I B U T I O N F U N C T I O N S FOR A M O R P H O U S ALLOYS
|. The quasibinary problem
Yu.A. BABANOV, N.V. ERSHOV, V.R. SHVETSOV and A.V. SERIKOV Institute of Metal Physics, Ural Science Research Centre of the USSR Academy of Sciences, 6202 l 9 Sverdlovsk GSP- 170, USSR
A.L. AGEEV and V.V. VASIN Institute of Mathematics and Mechanics, Ural Science Research Centre of the USSR Academv of Sciences, 620219 Sverdlovsk GSP- 384, USSR
Received 9 May 1984 Revised manuscript received 9 August 1984
To determine the atomic structure of multicomponent alloys we propose a regular algorithm to solve a system of Fredholm integral equations of the first kind which describes independent experiments. For Fe-B-type alloys a quasibinary problem has been formulated which is a system of two equations: one for X-ray scattering, the other for EXAFS. To test the new method numerical model experiments were carried out. The partial radial distribution functions have been determined for metallic glass Fes0B20.
1. Introduction It is known that to obtain partial radial atomic distribution functions (RDF) for amorphous multicomponent systems a number of independent experiments should be performed. In the case of a binary system it is common practice to use data from three experiments. As a rule, the latter are diffraction (X-ray and neutron scattering) experiments [1-3]. Analysis of the experimental evidence for the atomic structure of binary alloys shows that the qualitative agreement of total radial distribution functions is often the case. However, partial R D F differ not only quantitatively (different coordination numbers and peak positions) but also in the forms of curves. This difference makes it difficult to unambiguously select a theoretical atomic structure model for amorphous alloys. One of the causes of the differences observed may be the two-step R D F derivation procedure (1 - obtaining partial structure factors, 2 - Fourier transformation thereof) [4]. 0022-3093/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
2
Yu.A. Babanoo et aL / A new method for determining RDFs. 1
Along with diffraction studies, the method of X-ray spectral structural analysis (EXAFS technique) has recently found extensive applications in the determination of local atomic order [5-7]. However, the peculiar procedure of extracting structural information (a combination of the Fourier filtration method and the least-squares method) does not allow the EXAFS to be included as an independent experiment into the technique employed by Waseda [1]. Apart from this, strong correlation between the parameters being varied, the number of which amounts to 8 for the binary problem, leads to instable solutions and unreliable results. To determine partial radial distribution functions the present paper proposes a regular method of solving a system of integral equations. Earlier, this method was advantageously applied to one-component systems both in the case of the EXAFS [8,9] and in the case of X-ray diffraction [10]. The method consists of using a special regularization procedure for a system of integral equations, which permits reduction of the initial problem to a system of algebraic equations with a well-conditioned matrix. A regular method of solution possesses stability with respect to perturbations of input data. This makes it possible to use true experimental data without a preliminary smoothing procedure. In contrast to the Fourier transformation technique, this method is not associated with a particular kind of kernel of integral equation. Therefore it is applicable to equations with kernels which are more complicated than that of an integral equation describing the X-ray (neutron, electron) scattering, for example to an EXAFS equation. This permits the inclusion of an EXAFS equation into a system of integral equations along with diffraction equations. Such an extension of the number of independent experiments indispensible for determining partial RDF will in all likelihood allow an appropriate combination of experiments. The present paper (Part I) deals with the solution of the quasibinary problem, which is a system of two equations: the first equation describes the X-ray scattering, the second equation describes the EXAFS. Section 2 formulates the problem of finding partial radial distribution functions for an n-component system and presents a scheme to solve this problem. To test the new method a model quasibinary problem is solved in section 3. In the same section the solution algorithm is described in detail. The final section furnishes results of studies on the atomic structure of metallic glass Fes0B20. Experimental values of the coordination numbers and interatomic distances for Fe-Fe and Fe-B pairs have been found. They are compared with results of experimental investigations using diffraction methods [2,3] and the EXAFS method in its conventional treatment [5] and with results of atomic structure simulation for transition metal-metalloid-type metallic glass [2,11,12]. Part II of this work will treat the solution of a binary problem, as a result of which three partial RDF are found from the experimental data including two EXAFS experiments and one diffraction experiment.
Yu.A. Babanov et al. / A new method for determining RDFs. I
3
2. Theory of the method Nearly all of the direct-method data relating to the arrangement of atoms in a condensed medium are described by a pair correlation function g(r~2 ). For a spatially homogeneous isotropic system, of which glass or liquid is an example, this function depends on the modulus of the vector [ r~z [ connecting points 1 and 2, i.e. g( try2 I) - g(r). If the system contains n elements then to describe the atomic structure requires the knowledge of N = n(n + 1)/2 partial functions go(r). The function go(r) relates the density of the probability of finding a particular pair of atoms at a particular atomic separation and possesses the property
g,j(r) = g,,(r).
(1)
It is known [13] that a total interference function I(s) describing a diffraction expriment may be expressed in terms of partial functions gij(r), i.e.
[I(s) - 1] s = 4~roo
~_, (ci cj F~ Fj)/Fo a
g,j(r) - 1] r sin(sr) dr.
i=1 j=l
(2) Here s = 4~r sin 0 / ~ is the vector of scattering of a quantum with a wavelength )~, O0 is the mean atomic density, c, the atomic concentration of element i, F, the amplitude of coherent scattering of element i, n
F0 = E c, U . i=l
To a single-scattering approximation for K absorption spectra, the integral equation in the EXAFS technique is [8] j=l
k
(3) Here x , ( k ) is the normalized oscillating part of the X-ray absorption spectrum of the ith-element, ] fj(k, ~r)] the modulus of the backscattering amplitude for a photoelectron with a momentum k on the neighboring atom j, ~p,/(k) the total phase shift, and ~ ( k ) the mean free path of a photoelectron. It should be noted that the normalized oscillating part is determined by the sum of the contributions from n partial functions gij, whereas the total interference function I(s) is determined by the sum of n(n + 1)/2 functions. This circumstance is essential to the formulation of the quasibinary approximation. Thus, to experimentally determine N partial functions it is necessary to have N sets of experimental data. Let us introduce a consecutive numbering. Let i = 1, 2 . . . . . n, and j = i, i + 1 , . . . , n then
hm = gm -- 1, (4) where m = i + ( j - 1 ) j / 2 , (m = 1, 2 .... N). We introduce a unified notation
4
Yu.A. Babanov et al. / A new method for determining RDFs. 1
for the set of experimental data [UI], where l = 1, 2 . . . . . N. Equations such as (2) and (3) may be combined into a system of Fredholm integral equations of the first kind N
Ut = ~2 ~',mh,~ ( l = 1, 2 , . . . , N ) ,
(5)
rn=l
where sJ~,~ is an integral operator. In matrix representation eq. (5) is of the form U1
U2
=
UN
J~ll
,5~12
"" "
J~IN
~21
~z2
"'"
d2N
J~NI
,5~N2
,-.
5~NN
hi ×
h2
(6)
The problem of finding the functions h m belongs to a family of inverse problems which are ill-posed [14,15]. A feature peculiar to such problems is that negligible input-data distortions lead to arbitrarily large distortions of the solution. Furthermore, to the same experimental data there corresponds an infinite set of solutions equivalent in exactness. Consequently, a principle of selecting solutions is required. In the selection it is quite natural to take account of the a priori physical information concerning the solution sought. The functions should conform to the following requirements: (~). The solution [hm] should be such that when acted upon by a matrix of integral operators J~lm the residual norm u
d, mhm-
u~
m=l
should not exceed the experimental-data error 6, i.e.
Lz[cl,dl]
=
'l Y~ ~¢~mh~(r) - U~(k)[ 2 d k
<~8 (l = 1, 2 . . . . . N ) .
m=l
Here C t and d l are the boundaries of the data set Ul. This major criterion for the correct choice of an approximate least-squares method. However, this method is instable with data perturbations, which presents certain difficulties in the results. (~). The solution should be sufficiently smooth. (~). The solution hm >~ - 1 for any r. @ . hm(r)--'O when r ~ oc. @ . The functions hm(r ) are normalized as follows
1 bm - a m )
rb'n
2
J. 47rr hm(r) d r - (W,,, hm) = 0 am
requirement is a solution in the respect to inputinterpretation of
Yu.A. Babanov et al. / A new methodfor determining RDFs. I
5
with the boundaries a,~ and bm chosen from the condition
g m ( r ) = O at r < am, gin(r) = 1 at r >_ bin.
(7)
The conventional method of constructing a solution to a system of Fredholm integral equations of the first kind is to approximate the functions h", and Ut by vectors, and the integral operator ~lm by the matrix, and then to solve a system of linear algebraic equations. This approach turns out to be unacceptable because of the instability of the problem. The matrix that arises is ill-conditioned or even degenerate, and the solution obtained does not approximate, generally speaking, the solution of the initial problem when the magnitude of the discretization step tends to zero (i.e. when the exactness of the approximation is improved). A regular algorithm, i.e. an algorithm stable with respect to input-data perturbations, cannot be constructed following this path. The problem (5) solution algorithm expounded below possesses the property of stability with respect to small variations of input data and permits taking account of the a priori information formulated in requirements @ to
@. A large contribution to the theory of solving ill-posed problems has been made by Tikhonov who introduced the fundamental concept of a regularization operator and a regularized solution [16]. By convention, three regular-solution construction stages can be singled out. (i) Regularization. As a regularized approximate solution we take the vector [ h~ (r)] minimizing in space L 2 the Tikhonov functional
min
~
/=1 m=l d
+
,S~lmhm-U l
+ ~
(amllh",-h~]l 2
m=l
II Zrr (hm - h'nr) [[2 --[- V., I(mm, hm) [2)]
L 2 [ a ....
b,.].
(8)
In eq. (8) the last term, introduced additionally, is a penalty function. Here h t~, is some trial function or zero, 7/the vector of the regularization parameters a",, tim, and of the penalty parameters ~'m. The presence of terms with small positive parameters a,, and tim in functional (8) renders the problem stable and assures the fulfilment of requirement ( ~ . It has been proved [15] that when the parameters a,~ and /3., are decreased an approximate solution h~,, tends to an exact one. The parameters ~'m are responsible for the satisfaction of normalization condition @ . (i 0 Discretization. To be solved numerically problem (8) should be preliminarily discretized. Approximating the space L 2 by a discrete space l 2 and prescribing in space l 2 the functions h,,, and Ut as vectors and prescribing the integral operators Jaet,. as matrices we obtain a discrete analog of problem (8). It follows from the general results of ref. [17] that the solution of the discrete problem converges to that of problem (8) when the magnitude of the discreti-
6
Yu,A. Babanov et al. / A new methodfor determining RDFs. 1
zation step tends to zero. Using the requisite extremum condition - the equality to zero of the partial derivatives with respect to the components of the vectors hm of functional (8) in discrete form - we come to a system of linear algebraic equations. The matrix of this system is positive definite. This allows well-known linear algebra methods to be used for matrix inversion. (iii) Iterational refinement. The conditions (~) and (~) imposed on the solution [hm] are attained by the selection of the boundaries am, bm (see formula (7)) by virtue of the specific features of the radial distribution function for amorphous systems. However, if the approximate solution [hm] does not meet requirement (~) or if the exactness of this solution is unsatisfactory the iteration procedure [9,18] should be used h {"+') = P ( a g * a g + B ) - I ( a g * U + Rh {")) (n = 0, 1,... ).
(9)
Eq. (9) is represented in symbolic form, with ~ ' being the matrix of the resulting system of algebraic equations, U the total vector approximating the vector function [U~U2 . . . . , UN], h the total solution vector approximating the vector function [h~, h 2. . . . . hN], P the operator of metric projection to the set Q = { h: h >j - 1 } ; n = 0, 1, 2 . . . . the iteration number, B three-diagonal and diagonal matrices containing the parameters al, % , . . . ~N and fi~, f12. . . . . /~N.
3. Model quasibinary problem If one chooses (1) X-ray diffraction, (2) an EXAFS spectrum of element 1, and (3) an EXAFS spectrum of element 2 as independent experimental data when determining the partial functions of a binary system then eq. (6) assumes the form
U3
Lo'11 131ihl] 'J~32 3~33
(lO)
h3
where hl(r ) = gll(r) - 1, h2(r ) = g12(r) - 1, h3(r ) = g22(r) - 1, b SC~mh m = B a , , ( k ) f °'hm(r) s i n ( k r ) r dr (m = 1, 2, 3), a m
bm allmhm = B,,,( k ) ~ m h m ( r ) e -2r/x(k) sin[2kr + ~Pm(k)]dr
(I=2, •=3,
m=1,2) rn=2,3 '
B n ( k ) = 4rrpoe~F12(k)/Fff(k), B,2(k ) = S ~ r p o q c 2 F l ( k ) F 2 ( k ) / F o 2 ( k ) ,
(11)
Yu.A. Babanov et al. / A new method for determining RDFs. I
7
B,3( k ) = 4~rpoc~F?( k ) flFoa( k ), Bz~ ( k ) = 4 ~rOoc~ [A ( k, 7r) Il k , B22(k ) = 4~rPoC2 rf2( k, ~r) Ilk,
B32(k ) = B2,(k), B33(k ) = B22(k ).
(12)
The functions Ut(k) include the experimental values and asymptotes of elements dtm, which arises from the fact that the interval of h m functions is finite.
Ul(k ) = ( I(k ) - 1 ) k + Tu(k ) + T,2(k ) + T,3(k ), Uz(k ) = x,(k) + T21(k ) + T22(k ), U3(k) = x2(k) + T32(k ) + T33(k ).
(13)
Here ~a
Tlm(k ) = B l m ( k ) j o ' r sin(kr) dr (m = 1, 2, 3), OC
Tt..(k)= -Bzm(k) f
e-Zr/x(k) sin[2kr + ~ m ( k ) l d r a m
(l=2, •=3,
rn=l,2) m=2,3
(14) "
The quasibinary problem may be formulated for amorphous transition metal (TM)-metalloid (M) systems, of which the metallic glass Felo0_~B ~ (x = 15 to 25) may be an example. Fig. 1 presents ratios of the coefficients Blm for F e - B and F e - F e pairs as f Fe-B ~
f ~ )
o I
CB FB
= 2 CFe FF e
5 . . . .
I
10 . . . .
I
15 . . . .
I
2O
0 ....
i'o ....
2'o .... s (~")
io
Fig. 1. Weighting factors for X-ray scattering method (1) and EXAFS method (2) for alloy Fe85-Bl 5.
8
Yu.A. Babanov et al. / A new method for determining RDFs. 1
for X-ray scattering * and as f/ F e - B )
c.
IfB(k,~')[
for the EXAFS ** The relative contribution of the B-B pair to the structure factor of X-ray scattering is small since (cBFB)Z/(cwFFe)2~ 0.001 (0.1%). Comparing this value with that of f ( ( F e - B ) / ( F e - Fe)) (curve 1 in fig. 1) we assume that the contribution made by the B-B pair to eq. (10) may be neglected, i.e. we set z¢~3 = 0. Then eq. (10) becomes
A similar approximation was made by Chen and Waseda [2]. However, they neglected the contribution of the B - B pair not only in the case of X-ray scattering but also for neutrons, although in the latter case this contribution is not so small ( - 1.7%). For the quasibinary problem (15) the Tikhonov functional (8) may be written as
M"Ih,, +
E E2
u,
2 l=1
22[ct,d A L
+ E2 {< ,.llho,-h ll m=l
hm-ht~)
+Tml(Wm, hm)l 2 h m ~ L 2 [ a m , bin].
(16)
To numerically solve problem (16) one has to pass to the discretely prescribed functions h m, U~ and integral operators ~t,,- To this end, we will divide the segment [al, bl] into Pl parts with a step Ar 1 = (b 1 - a l ) / p 1, the segment [a 2, b2] into P2 parts with a step Ar a = (b 2 - a 2 ) / p 2, and also the segment [c 1, dll into ql parts with a step Ak I = (d 1 - c l ) / q l , and the segment [c2, d2] into q2 parts with a step A k 2 = (d a - c2)/q 2. In our numerical algorithm Ar 1 = Ar 2 and Ak 1 = Ak 2. The discrete approximation of the functions and integral operators was carried out using the collocation method [21]. Replacing the space L 2 by a discrete space l f ( p - dimension), and the derivative d / d r by a difference equation we obtain a discrete analog of problem (16) Mn[hi,
h 2 ] -~ II z ~ ' l l h l + ~ ¢ 1 2 h 2 -
U1
I1~ +
+alllhl-hl(llJl+a2llh2 2
+/7'l-~r(h'-ht()],,,+
B
~ll ~¢21ha +,-~¢22h2 - U2 11~2 _ /,,tr II 2 " 2 11/~2
d
2 -~r ( h 2 - h ~ ) 1 2
t~'~
+Tal(W~,hl)12 +'/21(W2,h2)l 2" h~lP2 ', h 2 E l p~. * The values of the scattering amplitudes F have been borrowed from the tables of ref. [19] ** The values of the scattering amplitude modulus I f ( k , Ir)l and of the phase shifts have been borrowed from the tables of ref. [20].
(17)
Yu.A. Babanov et at / A new method for determining RDEs. 1
9
Here all, ~ is a ql x Pm matrix with the elements a i d = B,m(k,)fri'~'r sin(kr) dr ~=., is
a q= ×
(18)
Pm matrix with the elements
ai.i = Bzm(k,) fr['+l e ( - 2rn h2 kir-~) sin[2kir +~/m(ki)] dr.
(19)
In eqs. (18) and (19) the coefficients B,, are given by formulas (12). The phase shifts +re(k) in eq. (19) are determined as ~b,(k) =- 2 3 , ( k ) + qq (k, ~r),
2(k) =
+ 02(k,
(20)
where 31(k ) is the phase shift on the central atom and %,(k, or) is the phase shift of the scattering of a photoelectron on the surrounding atoms of species 1 and 2. The parameter ~ determines the relative weight of the EXAFS equation. The reason why this parameter 'is introduced will be explained in what follows. The vector h,,(r) has elements hma, the vector Ut(k ) elements Ut, - Ut(k,), the vector W., elements IV,,i -- (53+1 - r,3)/(b 3 - a3m), and the vector (d/dr)hm(r) elements (d/dr)(hmj) = (hmj+l - h,w)/Ar. The norm h m in space l p is given by the formula ][ hm J[~ = E~=lArh2j. Similarly, q
itu, tl, = i=1 E The v e c t o r s h 1 and ha minimize the functional M'[hl, h2] (17) if and only if the first variations of the functional in hi and h 2 are equal to zero. This leads to a system of linear algebraic equations which when represented in matrix form is
d,*2g, +
]
----- - d l ~ d l l + ~d2~';~21 + B1 ,.~1~11 q-- ~.5J2~.5~/21
"5~1~J~12 q- ~JJ2~ J~22 .5J1"2,-~12 "Jr-~..~/2"2J~/22 + B 2
×
. (21)
h2
Here .-~u and ad22 are matrices with elements a,j which are determined by formulas (18) and (19) for all i,j except for aq+lj = Wmj~--£- The symbol (*) denotes transposition. The vector G = [G,, U/:, .... ~,, 0]. The Pm × P,,, matrix B., has the form
I
0 -~,,,
Bm =
g~m + 2fim
-.- 0
--fl,,,
0
--~m
am
0
0
0
+ 2fi~
"'" .-.
0 0
•••
ffm + 2tim
(22)
10
Yu.A. Babanov et al. / A new method for determining RDFs: I
where
~,, = ( amAr ) / a k , ~',, = flm/( ArAk ), ~,, = Vm/ak.
(23)
The matrix involved in eq. (21) is of the dimension (P1 q ' P 2 ) X ( P l + P 2 ) " It is nondegenerate due to the presence of matrices B,, which contain small positive parameters ~,, and /9,,. As has already been stated above, standard methods may be used to solve the system of linear algebraic equations (21). In our calculations, for example, the square root method was applied. For solution refinement the iteration procedure according to formula (9) was employed. What was chosen as an initial approximation in the iteration process was the solution to system (21) specially transformed to meet requirements (~) and (~), i.e. h,,(r) >__ - 1 for any r, hm(r )--) 0 when r --* oo. To ascertain the efficiency of the algorithm described above numerical experiments were carried out. As model partial functions of Fe-Fe, Fe-B, and B-B we selected functions qualitatively similar to those calculated according to the relaxed dense random packing (DRP) model for metallic glass Fe85B15 [2] (fig. 4, curves la, 2a, and 3a). For these functions we calculated the normalized oscillating part x ( k ) according to eq. (3) and the structure factor I(s) according to eq. (2). The plots for the normalized oscillating part x ( k ) and structure factor I(s) are presented in fig. 2 on a single scale (s = 2k). Structure factor I(s) oscillations at low s, as is known, arose from model RDF cutoffs. Two peculiarities of diffraction and EXAFS data must be noted. (1) The X-ray scattering signal is approximately two orders of magnitude greater than the EXAFS signal. That is why we introduce the parameter ~ into functional (17).
I
~il, 3'
0 t_..._.z
¸1102
2-
10
0
)0
,,('A")
s(#')
r5
~3
r(~)
Fig. 2. Model interference function l ( s ) and normalized oscillating part of X-ray absorption x(k). Fig. 3. Partial radial distributions functions g,j(r) for amorphous state model (a) and result of solving a system of two integral equations for model l ( s ) without taking account of B-B contribution and model x(k) (b).
Yu.A. Babanov et al. / A new method for determining RDFs. I
11
(2) The structure factor I(s) has the main contribution at 2 A-1 < s < 15 ~, i whereas x(k) has the main contribution in the interval 3.5 , - 1 < k < 15 A -~ (7 ,~ i < s < 30 A - l ) . It is therefore natural to regard these two experiments as mutually complementary. For the calculated functions x(k) and I(s) the quasibinary problem (21) was solved with the following parameters: the number of points for l(s) ql = 400, the number of points for x(k) q2 = 241, which corresponds to the intervals [Q, d~] = [0.05 ~k-1; 20 A -l] and [c 2, d 2 ] = [3.5 ,~-1; 15.5 A - I [ with the step Ak = 0.05 A - l ; the number of points in r space p~ =P2 = 120 with the step Ar = 0.08 ,A corresponds to the intervals in which the solutions [a 1, b l ] = [2.15 ,~; 11.75 A], [a2, b2]=[1.75 A; 11.35 A] are sought; the mean atomic density for Fe85B15 P0 = 0.09594 a t . / A 3. In the first version of the problem solution the contribution of two pairs, F e - F e and Fe-B, was taken into account in I(s). But since the EXAFS equation (10) involves only these same pairs the inverse quasibinary problem is exact in this respect. The results are presented as plots in fig. 3 and in table 1. Comparing curves l a and l b we see that the solution gve-Fe practically coincides with the model function over the entire interval prescribed. As far as the function gvc-~ is concerned, the contribution of the F e - B pair to both the diffuse scattering intensity I(s) and x(k) is appreciabloy smaller. Therefore the solution gw-B (fig- 3, curve 2b) is distorted at r > 5 A because of the errors introduced by the calculation procedure. Here we must note that the EXAFS data interval was restricted to k > 3.5 A-1. The information contained in the first peak of the R D F is evident at high k, but the information contained in the tail is present only for small k, and hence is lost. For a quantitative comparison of the model and of the solution, the following major characteristics of the first peak of the function gij(r) have been selected: the maximal value of g, peak coordinate r, coordination number Nq, first-peak asymmetry index x. The quantities g and r are determined as the value and position of the vertex of the parabola obtained from three points in the vicinity of the peak maximum. The number of nearest/-species neighbors is estimated according to the formula e
Nij = 4~rpocj f " r2ggj( r ) dr,
(24)
where a is determined from condition (7), e corresponds to the first minimum behind the first function gij(r) peak. The asymmetry parameter x is found as the ratio of the right-hand and left-hand first-peak halfwidths measured on the half-height. Comparison of the major characteristics of the model and of the solution (see table 1) testifies that the regularization solution of the inverse problem is practically exact. The choice of regularization parameters and the stability of regularized solutions were discussed in [10]. In the solution of a quasibinary problem the contribution of the B-B pair to the intensity I(s) may be regarded as a perturbation of input data. In the second version of the model calculations the contribution of this pair was
5 × 10 -8
5 × 10- 7
10 -7
103
104
1
2.56
2.56
2.56
3.63
3.66
3.66
3.64
relaxed D R P model X-ray diffraction Neutron diffraction X-ray diffraction Neutron diffraction with n~t Fesol~ B20 Neutron diffraction with 57Fe8o 11B2° EXAFS X-ray diffraction EXAFS 2.57 2.55 2.57 + 0.02
12.4 8.2 12.0 h) +- 0.3
10.7
2.56
10 9
2)<10 - 9
5 × 10 - 7
2.14 2.06 2.07 +_0.02
2.57
2.08
r (A)
12.2
r (,~) 2.56
Fe-B
10 -8
2 × 10- 7
5 × 10 -9
F e - Fe N (at.)
1.80
1.79
1.77
1.78
a) This value has been obtained from the relation CBNB_Fe = cveNve_ e where NB_ w = 6.9 at. [2]. b) Integration interval [2.15 ,~, 3.20 A]. ~) Integration interval [1.65 A, 2.55 A.].
Fes0 B2o Fes0 B20
Fes0132o
FessB15 Fes3 B17
13.82
14.02
13.95
14.00
(at.)
b) Integration interval [1.75 ik, 2.55 A].
2 × 1 0 -8
10- v
10 -8
2.56
Table 2 Structure parameters for oearest neighbors in amorphous F e - B alloys
a) Integration interval [2.15 ,~, 3.45 A].
Model Result (without B-B) Result (with B-B) Result (with B-B)
(A)
N ~
Fe-B g 0' 2
r
al
Pl
Fe-Fe
Table 1 Interatomic correlations of nearest neighbors - results of model numerical experiments
2.2 2.2 1.6 c) +_0.2
1 . 4 ~)
1.5
N (at.)
2.08
2.08
2.08
2.08
(A)
r
4.63
4.94
4.96
4.91
this paper
[3] [51
[21
[2]
Ref.
1.38
1.44
1.54
1,56
(at.)
N b}
1.16
1.22
1.24
1.31
Yu.A. Babanov et al. / A new method for determining RDFs. 1
13
~4
0 5
5
g+ ~3
c~
6 r(~)
8
1'0
Fig. 4. Partial radial distribution functions g u ( r ) for amorphous state model (a) and results of solving a system of two integral equations for model l ( s ) taking account of B - B contribution and model x(k) at ~ = 10 4 (b) and ~ = 1 (c).
taken into account and the effect of the above perturbation on the solution estimated. The solutions gFe-Fe and gve_ u in this case are given as plots in fig. 4. The label b pertains to the curves corresponding to the solution at ~ = 1 0 4, and the label c to the solution at ~ = 1. The parameter ~ is responsible for the weight of the EXAFS data in functional (17). Comparison of these curves as well as of the major characteristics of the first peak of partial functions with the model (see table 1) suggests that enhancing the role of an exact equation when a problem is solved approximately is a fairly efficient technique. The weight of the EXAFS data in eq. (17) should be increased for two more reasons: (1) the intensity of oscillations in x ( k ) is approximately two orders of magnitude smaller than that in the structure factor I(s), (2) the relative contribution of the F e - B pair to x ( k ) is considerably larger than the contribution in the case of X-ray scattering.
4. Atomic structure of metallic glass FesoB2o The X-ray scattering experiments were performed in reflection geometry using an automated diffractometer [22]. Monochromatized radiation (Mo Kc~, 2~= 0.7107 ,~) was produced using pyrolytic graphite on the reflected beam. The time needed to gain 10 4 counts was measured. The recording was performed stepwise with a step As = 0+05 ,~-1 uniform in s space over the interval 0.8 ,~-1 < s < 16.6 ,~-t. This interval corresponds to the angular
14
Yu.A. Babanov et al. / A new method for determining RDFs. 1
interval 5 ° < 20 < 140 °. The specimen for diffraction experiments was composed of amorphous FesoB20 alloy ribbons 1.5 mrn wide produced by melt spinning. The thickness of the specimen was such that the beam attenuation coefficient at normal incidence was equal t o - 100. The preliminary experimental-data processing was effected following a program in which the ideas advanced in [23] were used. The structure factor I(s) for metallic glass FesoB20 is presented in fig. 5. In the interval [0 to 2 sk-t] the structure factor was extrapolated to zero. The same figure gives the normalized oscillating part of the X-ray absorption coefficient, x ( k ) . The Fe absorption K spectrum for a specimen in the form of a 12/~m thick ribbon was registered using the synchrotron radiation of a VEPP-3 storage ring (Novosibirsk). The experimental conditions and the preliminary processing are described in [24]. The preliminary processing was done using optimal parameters for pure iron. The result of solving the system of algebraic equations (21) was obtained with the following regularization parameters: al = 10-3,/31 = 10-3, a2 = 1 0 - 1 , /32 = 2 × 10 2, "/a = "72= 10-l, n = 3. The value of the mean atomic density for FesoB20 was 00 = 0.09515 at./,~ 3. The experimental values were prescribed in the limits: [ca, da] = [0.05 k -1, 16.6 j - i ] for l(s) and [c 2, d2] = [3.5 k -1, 15.5 j - 1 ] for x(k) with the step Ak = As = 0.05 A-~. The solutions were tried in
p. :.
gtO~ - ".
/':
t,.
/:7-
::
•
.
".-all'/
~'
..
• "
o¥
.-
V
"
-:
'.."
I
,
L
,
!
,~
:.
I
,
lo
I~
I
(l")
I
I
~5
#,,. •
:~, ; \.f"---"-%1/°
J |
0
i
i
i
i
|
i
i
,0
I
i
I.
2o
i
s (#i-')
Fig. 5. Experimental interference function l(s) and normalized oscillating part of X-ray absorp-
lion x(k) for amorphous alloy Feso-B2o.
15
Yu.A. Babanov et al. / A new method for determining RDFs. 1
~4
~2 1
0 4
1
o
t (4)
Fig. 6. Partial radial distributionfunctionsof amorphous alloy F%o-B2o.
the intervals: [a,, ba] = [2.15 ,~, 16.55 ,~] for g v e - w and [a2, b2]= [1.65 A, 16.05 A] for g v e - B with the step Ar = 0.12 A. The partial radial distribution functions g v e - V e and gve B are presented as plots in fig. 6. The curve g w - v e is similar to the total radial distribution function. However, one feature merits special consideration. The first coordination sphere is asymmetric in form (the asymmetry factor x = 1.6) and is separated from the other sphere by a minimum typical for coordination spheres of crystalline materials. As might be expected proceeding from model calculations, the function gFe-B is most distorted in the region of higher coordination spheres. It is next to impossible to single out the signal from these spheres from among random noise. As was shown earlier [5,9], the intensity of the EXAFS spectrum for amorphous systems is determined chiefly by the contribution of the nearest neighbors. By increasing the relative weight of the exact EXAFS equation in the quasibinary problem (21) to 104 we increase the relative weight of the EXAFS contribution by the Fe-B pair. Therefore it is hoped that trustworthier information may be obtained concerning the first coordination sphere gF~-B. The high narrow first peak is symmetric in form (K = 1). The total width of this peak on the halfheight is equal to 0.4 A, which is close to the value obtained in [3]. However, the other peak characteristics determined in the present paper differ from the data obtained in [3]. In table 2 the values of interatomic distances and coordination numbers for F e - F e and Fe-B pairs are given, obtained in Fujiwara's theoretical model [2] and from the data of different experiments [2,3,5]. First and foremost, it should be noted that the stability of the experimental results for the F e - F e pair is comparatively high. An exception is the coordination number NF~_Fe 8.2 at. in ref. [5] where only EXAFS data were analyzed, Our results are the closest to the data obtained by Nold et al. [3] and give a fairly good fit to the results of the theoretical relaxed DRP structure model [2]. ~---
16
Yu.A. Babanoo et al. / A new method for determining RDFs. 1
The largest spread in data occurs for gFe-~. The interatomic distance rFe-B = 2.07 A determined by us coincides within the experimental error with the data obtained by Haensel et al. [5] and with the result of theoretical calculations [2]. The difference in coordination numbers (12.0 at. and 8.2 at. for F e - F e ) (1.6 at. and 2.2 at. for F e - B ) apparently stems from the assumption, adopted by the authors of [5], as to the asymmetry of the first peak of the gFe-B function. As is well known, in the processing of EXAFS spectra a strong correlation arises between the parameters varied, specifically between the coordination number and the asymmetry parameter. A similar conclusion may be arrived at if one analyzes the results of EXAFS studies of metallic glass Fe~oB20 presented in [25]. In refs. [2,3] systems of two and three linear algebraic equations were solved. The matrix of those systems is ill-conditioned. In this case it may be expected that errors in initial data will lead to large distortions of the solution, especially for the function gFe-B whose contribution to the total intensity is comparatively small. The function g~e-B is particularly essential to our knowledge about the atomic structure of T M - M metallic glasses. Modern theoretical T M - M structure models [2,11,12,26] rest on the assumption: the interaction between metallic and metalloid atoms is several times stronger than that between metallic atoms. Therefore the metallic atoms surrounding the metalloid atom form a perfect configuration. This peculiar molecular unit is a major characteristic of the atomic structure of an alloy. It persists, as shown in [11,27], when the concentration of the components is varied. In the Boudreaux model each boron atom is surrounded by 6.6 Fe atoms. The coordination number obtained in the present paper, NBFe = 6.4 at., is close to the result of Boudreaux and to the experimental value NBFe = 6.9 at. of ref. [2]. In the Fujiwara relaxed D R P model [2] the value N B v e = 7.7 at. differs rather significantly from the data obtained by Nold et al. ( N B v e = 8.7 at.) [3] and Haensel et al. [5]. The symmetric form of the g w - B curve (see fig. 6) is also testimony to the presence of a perfect configuration of Fe atoms. On the strength of the experimental evidence for one alloy it is difficult to draw final conclusions concerning the atomic structure of amorphous T M - M alloys. The results of further investigations into the atomic structure of the F e - B system will be published elsewhere. We wish to thank Dr A.V. Serebryakov who kindly supplied us with a specimen for analysis.
References [1] Y. Waseda and S. Tamaki, Z. Phys. B23 (1976) 315. [2] T. Fujiwara, H.S. Chen and Y. Waseda, J. Phys. Fll (1981) 1327. [3] E. Nold, P. Lamparter, H. Olbrich, G. Rainer-Harbach and S. Steeb, Z. Naturf. 36a (1981) 1032.
Yu.A. Babanov et al. / A new method for determining RDFs. 1
17
[4] 3. Hafner, in: Liquid and Amorphous Metals, eds., E. Luscher and H. Coufal, Proc. Nato Adv. Study Inst. (Sijthoff & Noordhoff Int. B.V., Alphen aan den Rijn, The Netherlands, 1980) p. 183 ]5] R. Haensel, R. Rabe, G. Tolkiehn and A. Werner, ibid, ref. 4, p. 459. [6] P.A. Lee, P.H. Citrin, P. Eisenberger and B.M. Kincaid, Rev. Mod. Phys. 53 (1981) 769. [71 P.H. Gaskell, Proc. of the study Weekend 13-14 Nov 1982 (DL/SCI/RI9, 1983) p. 28. [8] Yu.A. Babanov, V.V. Vasin. A.L. Ageev and N.V. Ershov, phys. stat. sol. (b)105 (1981) 747. [9] A.L. Ageev, Yu.A. Babanov, V.V. Vasin, N.V. Ershov and A.V. Serikov, phys. stat. sol. (b)l 17 (1983) 345. [10] N.V. Ershov, A.L. Ageev, A.V. Serikov, Yu.A. Babanov and V.V. Vasin, phys. star. sol. (b)121 (1984) 451. [11] D.S. Boudreaux, Phys. Rev. B18 (1978) 4039. [12] D.S, Boudreaux and H.J. Frost, Phys. Rev. B23 (1981) 1506. [13] C.N.J. Wagner, J. Non-Cryst. Solids 42 (1980) 3. [14] A.N. Tikhonov and V.Ya. Arsenin, Solution of ill-posed problems, translated from the Russian. Preface by translation, ed., Filtz John Scripta Series in Mathematics, (Winston. Washington DC; Wiley, New York, Toronto, London, 1977). [15] V.K. Ivanov, V.V, Vasin and V.P. Tanana, Teoriya lineynykh nekorrekmych zadach i eyo prilozheniya (Izd. Nauka, Moscow, 1978). [16] A.N. Tikhonov, Dokl. Akad. Nauk SSSR 151 (1963) 501. [17] V.V. Vasin, Dokl. Akad. Nauk SSSR, 258 (1981) 271. [18] V.V. Vasin, Proksimalnyi algoritm s proektirovaniem v zadachakh vypuklogo programmirovaniya (Preprint: Institute of Mathematics, Acad. Sci. USSR, Ural Sci. Res. Center, SverdIovsk, 1981). [19] P.A. Doyle and P.S. Turner, Acta Cryst. A24 11968) 390. [20] B.K. Teo and P.A. Lee, J. Amer. Chem. Soc. 101 (1979) 2815. [21] A.L. Ageev, Yu.A. Babanov, V.V. Vasin and N.V. Ershov, in: Chislennye i analiticheskie metody resheniya zadach mekhaniki sploshnoi sredy (Ural Sci. Res. Center, Acad. Sci. USSR. Sverdlovsk, 1981) p. 3. [22] E.Yu. Medvedev, V.A. Boshegov, Yu.A. Deryabin, Yu.A. Babanov, N.V. Ershov and A.V. Serikov, in: Apparatura i metody rentgenovskogo analiza 29 (Izd. Mashinostroenie, Leningrad. 1983) p.79. [23] S. Ergun, J. Bayer and W. van Buren, J. Appl. Phys. 38 (1967) 3540. [24] N.V. Ershov, Yu.A. Babanov and V.R. Galakhov, phys. stat. sol. (b)l17 (1983) 749. [25] M. de Crescenzi, A. Balzarotti, F. Comin, L. Incoccia, S. Mobilio and N. Motta, Solid. St. Commun. 37 (1981) 921. [26] P.H. Gaskell, J. Non-Cryst. Solids 32 (1979) 207. [27] T. Fujiwara and Y. Ishii, J. Phys. F10 (1980) 1901.