- Email: [email protected]
Short range order of amorphous Mg70Zn30 investigated by means of anomalous X-ray scattering
98
Journal of Non-Crystalline Solids 130 (1991) 98 106 North-Holland
Short range order of amorphous Mgv0Zn30 investigated by means of anomalous X-ray scattering F. Paul
a, W. Press a a n d P. R a b e b
Institut fftr Experimentalphysik der Christian-Albrechts-Universtitiit Kiel, Olshausenstrasse 40, W-2300 Kiel 1, Germany b Fachhochschule Ostfriesland, Constantiaplatz 4, W-2970 Emden, Germany Received 19 October 1990 Revised manuscript received 13 February 1991
The short range order of the metallic glass Mg70Zn30 was investigated by means of anomalous dispersion of X-rays. Three characteristic X-ray photon energies of an X-ray source were used to evaluate the three partial coordination numbers of this two-component system. The values of these coordination numbers differ only slightly from those of the crystalline phase Mgs1Zn20. The similarity between the short range order of the amorphous and crystalline phase is also reflected by almost identical values of the short range order parameters. The reliability of the resulting structural parameters and the accuracy of its determination are considered in detail.
1. Introduction
The metallic glass Mg70Zn30 has already been the subject of many investigations. X-ray diffraction was carried out by Nassif et al. [1], Ito et al. [2], Rudin et al. [3] and Andonov and Chieux (they investigated Mg72Zn28 ) [4]. Neutron diffraction was done by Mizoguchi et al. [5], Andonov and Chieux [4] and EXAFS by Ito and Kawamura [6], Maurer et al. [7] and Sadoc et al. [8]. The interest in this alloy is based on its simplicity: it is one of the simplest amorphous alloys with two divalent metallic components; additionally, detailed theoretical work is available. The interatomic forces may be derived from pseudo potential theory (Hafner [9]) or using a cluster relaxation model (Heimendahl [10], Hafner [9]). The interatomic distances and coordination numbers were explicitly calculated by Hafner [9] and are available for comparison. The amorphous MgToZn30 has nearly the same composition as that of the crystalline phase Mg51Zn20 [11]. The crystalline phase therefore can serve for further comparison. Up to now, structural information on the
amorphous alloy MgToZn30 was gained by the assumption of the similiarity between the amorphous and the crystalline phase only. Here we present the first directly measured coordination numbers as well as weighted interatomic distances. The MgToZn30 alloy and other Zn-containing samples are suitable for experiments utilizing the anomalous dispersion with laboratory X-ray sources. Only 33 eV below the K absorption edge of the Zn atom, the characteristic Lo~2 X-ray of a gold target allows a variation of the atomic form factor of about 18% at q = 0 .~-1. Using two more photon energies with a sufficiently different anomalous dispersion correction term, it is possible to evaluate the partial coordination numbers.
2. Theoretical background
2.1. Structural parameters The total structure factor, S(q), of an amorphous solid can be calculated from the coherently
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
F. Paul et aL / Short range order of a-MgToZn 3o
scattered intensity, Icoh(q), using the definition of Faber and Ziman [12]
S( q) - 1 = Ic°h(q) - ( f Z ( q ) ) (f(q))Z
(1)
Here q---(4"~/X) sin O is the modulus of the wavevector transfer, with 2 0 the scattering angle and X the photon wavelength. (f(q)) and ( f ( q ) 2 ) are the mean scattering factor and the mean squared scattering factor, respectively. In the following, we restrict ourselves in the notation to the presence of just two constituents: the indices j = 1, 2 stand for Mg and Zn, respectively. Hence f = c~f~ + c2f2, where the cj are the concentrations and ~ the scattering factors of a constituents. The total structure factor S(q)-1, is the weighted sum of the three partial structure factors: 2
S(q)-l=~
2
Peaks in the R D F denote shells of the next neighbours. From the area of the first peak, the total coordination number can be calculated according to N = f~I2RDF(r) dr,
(6)
where the lower and upper integration limits, r~ and r2, refer to the minima preceding and following the first peak. The procedure is not rigorously defined, resulting in some uncertainty in the determination of N (see, for example, ref. [13]). Since p(r) is a linear combination of the three partial radial number density functions, N is a linear combination of the three partial coordination numbers, also:
(7)
cicjfi (
JY" (f)2 S , j ( q ) - l ,
(2)
where the partial structure factors, S ~ j ( q ) - 1, refer to the respective atom pairs, ij. Neglecting the only weak q-dependence of the weighting factors of eq. (2), the Fourier transform of the q-weighted total structure factor results in the reduced radial distribution function:
G(r)=4"~r[o(r)-oo] 2 =~fo q [ S ( q ) - l ] s i n q r d q ,
".).
(3)
(4)
The expression is similar to that for the total structure factor (see eq. (2)). However, the weighting factors of Oij (r) do not contain the concentration cj. In a discussion of amorphous solids it is necessary to introduce the radial distribution function (RDF):
4"~rZp(r).
The ~ j are the partial coordination numbers of the atoms of type j around the atoms of the type i obeying the relation c,NO = cjNji. By contrast with eq. (2), eq. (7) is nearly independent of q. It is advantageous, therefore, to determine coordination numbers from this relation using a variation of the weighting factors.
2.2. The anomalous dispersion
with o(r) the radial density function and 0o the average number density, p(r) is the weighted sum of three partial radial density functions, pij(r),
RDF(r) =
99
(5)
The three partial radial density functions,
Oij(r), or the corresponding coordination numbers, ~ j , can be determined from three independent experiments with varying f,. Considering the anomalous dispersion effect of the constituents, a variation of f, is possible. In the non-relativistic theory the scattering factor, f, can be written as the sum of the q-dependent, but energy-independent part, fo(q), and the complex, energy-dependent part, f'(E) + i f " ( E ) (see ref. [14]):
f(q, E) = f 0 ( q ) + f ' ( E ) + i f " ( E ) . (8) The term f'(E) represents the anomalous correction whereas the term f " ( E ) corresponds to the absorption, M E ) , of the sample according to f"(E)
=
mc
2heZpo EI~(E),
(9)
F. Paul et al. / Short range order of a-MgzoZnso
100
Table 1 The values of f ' ( E ) and f " ( E ) for the used photon energies calculated from the absorption data given by McMaster et al. [15] by means of a Kramers-Kronig integration E
Mg
(keV)
f'(E)
f"(E)
f'(E)
Zn
f"(E)
8.396 (W L . 0 9.626 (Au L~2) 17.479 (Mo K~I )
0.18
0.17
- 1.86
0.67
0.14
0.13
-5.41
0.53
0.05
0.04
0.25
1.48
and Park [18] for a finite sample, respectively. It turned out, however, that this effect is relatively small and not detectable. The density of the amorphous Mg70Zn30 alloy was measured using Archimedes' Principle. The buoyancy fluid was destilled water with a small drop of a wetting agent, as suggested by Cawthorne and Sinclair [19]. The density was determined to be 2.96 g / c m 3 with an accuracy of 0.5%. This is comparable with the results of refs. [20] and [21]. The density of the crystalline phase Mg51Zn20 is 3.0 g / c m 3 [11]. The number density, P0, is related to the macroscopic density, Pm, by
Po = Pm( A / M ), where m and e are the mass and charge, respectively, of an electron, h is Planck's constant and P0 is the number density. The absorption of the sample, M E ) , was calculated by using the compilation of X-ray cross-sections of McMaster et al. [15]. f ' ( E ) can be calculated by means of a Kramers-Kronig integration which relates the imaginary part, f " ( E ) , to the real part, f ' ( E ) . Since the EXAFS-structure on the high energy side of the absorption edge has a negligible influence on the absorption on the lower side, that structure was ignored. The calculation was carried out using the method described by Dreier et al. [16]. The values of the anomalous correction terms are listed in table 1.
3. Experimental 3.1. The sample The sample was prepared by the melt-spinning technique. It had a typical" size of 10 mm width and 40 ~m thickness. A frame was used as sample holder, with a size sufficient to avoid parasitic scattering from the sample holder. In order to realize a semi-infinite sample - at least for longer wavelengths - a stack of five ribbons aligned parallel was mounted on the frame. With this geometry it is principally possible to use the correction algorithm for multiple scattering corrections according to Warren and Mozzi [17] for a semi-infinite sample and according to Dwiggins
(10)
where A is the Avogadro constant and M is the mass of one mole. The knowledge of the number density, P0, is required for the calculation of the R D F and the coordination numbers.
3.2. X-ray diffraction To obtain a high flux of photons, a Rigaku RU 200 PL rotating anode was used as photon source. A molybdenum anode with Mo K~I (17.479 keV) and a gold anode with Au L,2 (9.626 keV) were used. Because of a tungsten contamination of the anode caused by evaporation of the cathode also W L~1 (8.396 keV) was available for the experiment with the MgZn alloy. In order to profit from the focussing geometry, a bent LiF crystal was used as an incident beam monochromator. The scattered intensity was detected by a Si(Li) solid state detector. To reduce the contribution of fluorescent radiation in the diffraction pattern especially for the Au L~2 line in the neighbourhood of the Zn-K-absorption edge, the height of the pulses was suitably discriminated with help of a multichannel analyzer. The data were read into the memory of a personal computer. The scattered intensity was measured in reflection geometry. The scattering angles covered the range from 2 ° up to 130 ° corresponding to 0.34 , ~ - 1 < q < 8.91 ~ - 1 for Au L~2, 0.62 , ~ - 1 < q < 16.06 ~ - 1 for Mo K ~ and 0 . 7 8 , ~ - ~ _ < q _ < 7 . 8 , ~ 1 for W L~I. The entire range of angles was scanned in steps of 0.2 ° in 6).
lq Paul et al. / Short range order of a-MgroZnso
4. Data analysis The data were taken at T = 2 0 ° C in the Bragg-Brentano reflection geometry. Additional measurements with the sample removed were carried out to determine the parasitic scattering from the air and the holder. Half of the latter scattering intensity was subtracted from the uncorrected intensity data. These background corrected data were multiplied by a factor that considers the double scattering of X-rays in a semi-infinite or a finite sample. Tabulated values of a correction term considering the scattering angle 20, the mean squared scattering factor and the thickness of the sample are given by Warren and Mozzi [17] or Dwiggins and Park [18], respectively, from which the double scattering was calculated by a procedure described by the same authors. These corrected data were multiplied by the usual polarization factor
P(a,
O)=
1 + cos22a + cos220 cos22oe ,
(11)
where a is the monochromator angle and 2 0 the scattering angle (see, for example, ref. [22]). In case of a symmetric reflection geometry, the absorption correction is independent of the direction of the incident beam and the corresponding correction factor is A = 1/2/,, with ~ the absorption of the sample. Finally, the corrected data were normalized by the method of Krogh-Moe [23] and Norman [24]. The q-dependent scattering factor was calculated from the corresponding expressions in the International Tables [25]. The scattered intensity also contains Compton scattering. This incoherent part of the scattering was determined from an analytic expression given by Hajdu [26] for light elements (Z_< 36). Also the Breit-Dirac recoil factor was taken into account. A Fourier transform up to r = 2 ,~ proved that the measured density leads to reliable results: the initial slope of G(r) must be proportional to the number density P0- The total structure factor which was determined according to eq. (1) was weighted by q and then a fast Fourier transform was performed. An additional damping factor exp( - otq 2) with c~ = 0.005 ,~2 was used in the Fourier transform to reduce termina-
101
tion ripples in the transformed data. The reduced radial distribution function, G(r), as well as the R D F ( r ) were obtained by applying eqs. (3) and (5).
5. Results Figure 1 shows the three corrected and normalized diffraction patterns of the amorphous MgZn alloy for the three different photon energies. The diffraction patterns are typical for amorphous alloys with an additional prepeak at q = 1.60 ~ - 1 . When extrapolating from qmin to q = 0 ,~-~, the measured intensity goes linearly to zero. The influence of a non-vanishing intensity at q = 0 A-1 due to the compressibility (see ref. [27]) of the solid could be neglected. The three diffraction patterns differ only slightly for the different Xrays. However, due to the varying anomalous dispersion correction term of eq. (8) the scattering power is significantly different for the measurements with Au L~2. This can best be seen in fig. 1 in the case of the first maximum at q = 2.70 A-1. The normalized intensity (Au) is much lower than 600 eu (electron units), whereas the other intensities (Mo, W) are more than 600 eu.
,..., 6O0 I_ I1)
400 -
;
\ 200 W
Au 0
~
0
I
5
i
I
t
q (x')
I
i
10
t
1
I
I
15
Fig. 1. Corrected and normalized intensities, lnorm(q) (in electron units, eu), of the amorphous Mg70Zn30 alloy for the W L~I (8.396 keV), the Au L,2 (9.626 keV) and the Mo K~l lines (17.479 keV).
F. Paul et aL/ Short range orderof a-MgToZn3o
102 3 I v
09 1
W 0
.
.
.
.
.
.
.
Au 0
Mo
0 -1 0
5 q (A-t) 10
15
Fig. 2. Structure factors, S(q)- 1, of the amorphous Mg70Zn3o alloy for the W L.a (8.396 keV), the Au L.2 (9.626 keV) and the Mo K,I lines (17.479 keV) calculated according to Faber and Ziman [12].
T h e range of c h e m i c a l short range o r d e r of a b o u t 8 - 1 0 ,~ a n d o f a t o p o l o g i c a l short range o r d e r of a b o u t 15 A is in a g r e e m e n t with o t h e r p u b l i s h e d results (see, for e x a m p l e , ref. [1] a n d ref. [4]). These two f u r t h e r sets o f i n f o r m a t i o n are listed in table 2, t o g e t h e r with the structural p a r a m e t e r s , the p o s i t i o n a n d the w i d t h of (i) the p r e p e a k a n d (ii) m a i n p e a k in S ( q ) which are q u o t e d together with the results o f o t h e r a u t h o r s [1,4]. T h e occurrence of the p r e p e a k which is caused b y a chemical short range o r d e r is discussed in detail b y N a s s i f et al. [1] a n d A n d o n o v a n d Chieux [4] a n d is n o t a subject of discussion in this paper.
6. D i s c u s s i o n b
F r o m the c o r r e c t e d a n d n o r m a l i z e d intensity d a t a the structure factor was c a l c u l a t e d a c c o r d i n g to eq. (2) (fig. 2). W i t h this p r e s e n t a t i o n , the limiting qmax a n d increasing noise with increasing q is better visible t h a n in fig. 1. F r o m the values of the full width at half m a x i m u m ( F W H M ) Aq of the p r e p e a k a n d the first peak, the respective correlation lengths due to a chemical short range o r d e r a n d a topological short range o r d e r were d e t e r m i n e d using the Scherrer f o r m u l a X = 2 ~r/Aq.
T h e F o u r i e r t r a n s f o r m of the total structure factors, that is the r e d u c e d r a d i a l d i s t r i b u t i o n functions G(r), are shown in fig. 3. F r o m the initial slope in the s m a l l - r region (see ref. [13]) which is - 4 v p 0 , the average n u m b e r density, 00, can be derived. T h e r e are strong oscillations in this region of distances. A m o r e reliable c o n f i r m a tion o f the c u r r e n t analysis can be m a d e . The values of n u m b e r densities, P0, for all three det e r m i n e d values of G ( r ) agree within errors of
Table 2 Structural parameters derived from the structure factor S(q)-1. qpp, q~ and q2 axe the positions of the prepeak and the first and second maximum, respectively. Aqpp, Aql denote the FWHM of the respective maxima, from which the correlation lengths are calculated. The results of Nassif et al. [1] and Andonov and Chieux [4] are also listed for comparison. In the case of the neutron measurement, no prepeak was found by Andonov and Chieux E (keV) 8.396 (W L~l) 9.626 (Au L~2) 17.479 (Mo K,,1) 17.479 (ref. [1]) (h = 0.942 ,~) (neutron) 22.159 (Ag K,,1) (ref. [4])
qpp
Aqpp
Aqa
Xpp
Xl
( m - 1)
( m - 1)
(m)
(m)
4.50
0.75
0.40
8.4
15.7
2.70
4.50
0.625
0.40
10.1
15.7
1.60
2.70
4.55
0.70
0.40
9.0
15.7
1.54
2.65
-
0.78
0.44
8.0
14.2
-
2.65
4.50
-
0.40
-
15.7
1.5
2.70
4.5
0.63
0.4,.'.
ql
q2
( m - 1)
(m
1.65
2.70
1.50
(A -
1)
1)
10.0
14.5
60 v
F. Paul et al. / Short range order of a-Mg7oZn3o
2
103
~., 40
0
2
0
~
0
0
G I0
r (A)
0 20
Fig. 3. The reduced radial distribution function, G(r), of amorphous Mg7oZn3o alloy for the W L~I (8.396 keV), the L~2 (9.626 keV) and the Mo K,~1 lines (17.479 keV). From slope of the line between 0 A and approximately 2 ,~, atomic number density can be calculated.
the Au the the
measurement with the measured density and indicate a correct normalization. Usually interatomic distances are derived from the reduced radial distribution function G(r) because a further mathematical treatment which yields the RDF shifts the maxima to higher values in r [28]. To obtain values of partial G(r), a matrix inversion of eq. (2) with the appropriate weighting factors is necessary. Because of the noise at high q, the different cut-off values, qmax, and especially the very ill-conditioned matrix such a matrix inversion does not lead to reasonable results, unfortunately. The RDFs based on eq. (5) are displayed in fig. 4. The areas of the first peak denoted by NAue,2, NWL,, and NMoK., are also listed in table 3. Only in the case of the measurement with the W L,~ line, the area of the first peak must be corrected. The oscillations for r < r~ are too strong and modify the slope considerably. The left side of this peak must be fitted by a Gaussian to evaluate the correct area. The three partial radial distribution functions of both constituents are expected to contribute to the first peak. The fact that only one peak is observed indicates that the Goldschmidt radii [29] of both constituents (1.62 ,~ for Mg and 1.33 ,~ for Zn) are either of comparable size or the vari-
5
- (I)
iO
Fig. 4. The radial distribution function, RDF(r), of the amorphous MgToZn30 alloy for the W L~I (8.396 keV), the Au L,2 (9.626 keV) and the Mo K m lines (17.479 keV).
ance of the distribution is large compared with these distances. Otherwise the shells of the first neighbours must be seen separately. However, due to the weighting of the scattering power according to the anomalous dispersion, a shift of the maximum was detected (see table 3). In case of the measurement with the Au L~2 line, the lower scattering power of the Zn atom must be accounted for. This means that the influence of this contribution must also be lower. Indeed the radial distance
Table 3 Radial distances of the next neighbours and the areas of the first maximum in the MgToZn3o alloy. The distances were derived from the reduced radial distribution function G(r) whereas the areas are obtained from the RDF(r) E (keV)
rl (A)
N1
rz (A)
C~ (A)
8.396 (W L~:) 9.626 (Au L~2) 17.479 (Mo K~l)
2.99
10.8
4.88
5.71
2.92
11.0
4.94
5.77
2.85
10.5
4.86
5.77
17.479 (Mo Kca ) (ref. [1]) 17.749 (Mo K,d) Mg 7oZn 3o-meh (ref. [32])
2.88
10.7
2.92
11.4
--
5.70
-
104
F. Paul et a L / Short range order of a-MgzoZn 3o
increases due to the bigger Goldschmidt radius of Mg. According to eq. (7), the following system of linear equations containing the three partial coordination numbers is obtained:
O M#70Zn30. thin work anomalous X-ray scattertn i
• MgsIZn20 (crystalline) (Higashi et aL, 19el)
t-a
Mg?OZn3O - models : <> cluster relaxaUon delxH random bard sphere packl~ i
z
mAuL,2 = 0.448NMgMg + 0.702NZnMg
(Hafuer, 1980)
+ 0.652Nznz,, NWL,2 ----0.340NM~Mg + 0.714NZnMg + 0.875mznzn,
Zq170Zn30 . EXAFS (Maurer et aL. 1963) X MiToZn30, EXAFS
10
(Sadoc et al., 1985)
02)
o g~2ZnZ6, X-ray, neutron (Andonov st al., 1987) + a statinUcal distribution
NMoK~ ' = 0.252NMgMg + 0.692Nz.Mg
+ 1.131Nznzn.
5
Althofigh this is an intrinsically ill-conditioned system, a solution of the system of linear equations in eq. (12) was obtained: NMgMg =
0
2.4,
NM~n = 8.1, 3.5,
NZnMg
=
NznMg
= 8.4.
0 (13)
The value of NMaz, was determined from NznMg considering the relation cz,Nz.r,,tg = CMgNM~n. We have compared these results with those of Sadoc et al. [8], Maurer et al. [7], Andonov and Chieux [4], and theoretical calculations by Hafner [9] as well as the data of the crystalline phase (Higashi et al. [11]). EXAFS experiments at the Z n - K absorption edge yield structural parameters of the ZnZn and ZnMg correlation. Therefore we eliminated NmgM~ from the system of linear equations in eq. (12) by subtraction of each equation from the others. With an error of 0.1 in determination of the area of the first peak in the RDF, the linear equations reads
NMgZ. = 13.2 _ 1.0 - 2.1Nz,zn, NM# n = 14.4 _ 0.5 -- 2.6Nznzn,
(14)
NM# ~ = 15.3 + 1.0 -- 3.0Nz,zn. These linear relations are drawn in fig. 5. Such a representation of our results allows also an estimation of the reliability of structural data that were determined in eq. (13) via anomalous X-ray scattering. It is obvious that the system (12) is
5 N ZnZn 10
Fig. 5. The solution of the system of linear equations containing the partial coordination numbers, Nznmg and Nznzn. The bars indicate an error in the determination of the area N of about 0.1 [4,7-9,11].
extremely ill-conditioned, consequently also eqs. (14). The slopes of the eqs. (14) differ only weakly. This fact means that a linear combination of both coordination numbers can be obtained with high confidence, whereas a single value of the coordination numbers must be taken with care. However, it is interesting to note that all coordination numbers lie within a small area of the NMgZnNznzn plane. The values of coordination numbers which result from the measurements of Sadoc et al. [8], Higashi et al. [11], and Andonov and Chieux [4] are in agreement with our results considering the error bars. There also seems to be a weak tendency of a short range ordering in case of the investigations of Mauer et al. [7] and Hafner [9]. Compared with a purely statistical distribution, these values are shifted to somewhat smaller values. Another quantity that describes the short range order of a solid is the short range order parameter, first introduced by Cowley [30]. From several dif-
F. Paul et aL / Short range order of a-Mg7oZn3o
ferent definitions published since then, we use the definition of Cargill and Spaepen [31]:
NAB
ct--=] --
CB[CA(NAA -~- NAB ) -I1-CB(NBA "~ NBB)] " (15) The subscripts A and B denote the atoms of type A and B, respectively. The expression in the squared brackets denotes a mean total coordination n u m b e r and its p r o d u c t with %, that part of the B-type atoms on this mean total coordination number. Thus the fraction 1 - a in eq. (15) describes the probability of finding A - t y p e atoms a r o u n d the B-type atoms related to a mean total coordination number. A value ct < 0 means a tendency of forming pairs of unlike atoms, ct = 0 means a totally r a n d o m distribution of the constituents and a > 0 means a tendency of forming a pair of like atoms or clustering. According to the definition of a, we obtained ct = - 0 , 0 3 for the short range order, thus a very weak tendency for a chemical short range order. In order to allow a judgement, whether this departure from ct = 0 is significant, it is important to consider the error of a due to the errors in determination of the areas below the mainpeak in the R D F . Since all partial coordination numbers are linearly dependent a can also expressed by
ct = 1
b - tuNas "
(16)
Considering the coefficients of eq. (12), b = 3.595 and rn = 0.058 follows. The error of a is Act
b (b - tuNAs) 2 ANAB"
(17)
Assuming a reasonable error of about 0.5 atoms in NAB, one obtains a = - 0 . 0 3 _+ 0.15. Generally the very small value of a = - 0 . 0 3 supports the results of former investigations. C o m p a r i n g the a = - 0 . 0 4 for the crystalline phase, the short range order of the crystalline and a m o r p h o u s phase is of comparable size and signals a r a n d o m distribution of the constituents rather than a tendency of pair formation. The departure of a from 0 is not significant.
105
7. Conclusions F r o m the total reduced distribution function, weighted radial distances were derived. It was possible to determine partial coordination numbers f r o m a matrix inversion of that system of linear equations which contains these partial coordination numbers. A n illustration of this system of linear equations demonstrates the accuracy and also the limits of the method. Considering this accuracy it was shown, without any additional assumption about the structure of the a m o r p h o u s alloy, that the short range order in the a m o r p h o u s phase is almost identical to that in the complex crystalline phase MgslZn20. The tendency of ordering is described by the short range order parameter a. The values of ct which were determined for the a m o r p h o u s and crystalline phase indicate an ordering that is comparable to that of a pure r a n d o m distribution. Earlier assumptions about the similarity of the a m o r p h o u s and the crystalline phase seem to be justified. The authors would like thank H.J. Giinterrodt and P, Reimann, Institut fiir Physik, Basel, for the preparation of the samples.
References [1] E. Nassif, P. Lamparter, W. Sperl and S. Steeb, Z. Naturforsch. 38a (1983) 142. [2] M. Ito, H. Iwaska, N. Shiotani, H. Narumi, T. Mizoguchi and T. Kawamura, J. Non.-Cryst. Solids 62&63 (1984) 303. [3] H. Rudin, S. Jost and H.J. Giintherodt, J. Non-Cryst. Solids 62&63 (1984) 291. [4] P. Andonov and P. Chieux, J. Non.-Cryst. Solids 93 (1987) 331. [5] T. Mizoguchi, H. Narumi, N. Ahutsu, N. Watanabe, N. Shiotani and M. Ito, J. Non-Cryst. Solids 62&63 (1984) 285. [6] M. Ito and T. Kawamura, Philos. Mag. 49 (1984) L9. [7] M. Maurer, J.M. Friedt and G. Krill, J. Phys. F: Met. Phys. 13 (1983) 2369. [8] A. Sadoc, R, Krishnan and P. Rougier, J. Phys. F: Met. Phys. 15 (1985) 241. [9] J. Hafner, Phys. Rev. B21 (1980) 406. [10] L. Heimendahl, J. Phys. F: Met. Phys. 19 (1979) 161. [11] J. Higashi, N. Shiotani, M. Uda, T. Mizoguchi, and H. Katoh, J. Solid State Chem. 36 (1981) 225.
106
F. Paul et al. / Short range order of a-Mg7oZn3o
[12] T.E. Faber and J.M. Ziman, Philos. Mag. 11 (1965) 153. [13] Y. Waseda, The Structure of Non-Crystalline Materials, Liquids and Amorphous Solids (McGraw-Hill, New York, 1980). [14] R.W. James, The Optical Principals of the Diffraction of X-rays (Ox Bow Press, Woodbridge, CT, 1982). [15] W.H. McMaster, N. Kerr del Grande, J.H. Mallet and J.H. Hubbel, Lawrence Livermore Lab. report UCRL50174-SEC2-R1 (1969). [16] P. Dreier, P. Rabe, W. Malzfeldt and W. Niemann, J. Phys. C17 (1984) 3123. [17] B.E. Warren and R.L. Mozzi, Acta Crystallogr. 21 (1966) 459. [18] C.W. Dwiggins and D.A. Park, Acta Crystallogr. A27 (1971) 264. [19] C. Cawthorne and W.D. Sinclair, J. Scient. Instr. 2 (1972) 531. [20] M. Matsuda and U. Mizutani, in: Proc. 4th Int. Conf. on Rapidly Quenched Metals (Japanese Institute of Metals, Sendai, 1981) p. 1315.
[21] U. Mizutani and T. Mizuguchi, J. Phys. F l l (1981) 1385. [22] B.E. Warren, X-Ray Diffraction, (Addison-Wesley, Reading, MA, 1969). [23] J. Krogh-Moe, Acta Crystallogr. 9 (1956) 951. [24] N. Norman, Acta Crystallogr. 10 (1957) 370. [25] J.A. Ibers and W.C. Hamilton, eds., International Tables for Crystallography, Vol. 4 (Kynoch Press, Birmingham, 1974). [26] F. Hajdu, Acta Crystallogr. A28 (1972) 250. [27] G.L. Squires, Introduction to the Theory of Thermal Neutron Scattering, (Cambridge University Press, Cambridge, 1978). [28] C.N.J. Wagner, J. Non-Cryst. Solids 31 (1978) 1. [29] V.M. Goldschmidt, Skr. Nor. Videns.-Akad. Oslo 1 1(2) (1926) 1. [30] J.M. Cowley, J. Appl. Phys. 21 (1950) 24. [31] G.S. Cargill, F. Spaepen, J. Non-Cryst. Solids 43 (1981) 91. [32] E. Btihler, P. Lamparter and S. Steeb, Z. Naturforsch. 42a (1987) 507.
Journal of Non-Crystalline Solids 130 (1991) 98 106 North-Holland
Short range order of amorphous Mgv0Zn30 investigated by means of anomalous X-ray scattering F. Paul
a, W. Press a a n d P. R a b e b
Institut fftr Experimentalphysik der Christian-Albrechts-Universtitiit Kiel, Olshausenstrasse 40, W-2300 Kiel 1, Germany b Fachhochschule Ostfriesland, Constantiaplatz 4, W-2970 Emden, Germany Received 19 October 1990 Revised manuscript received 13 February 1991
The short range order of the metallic glass Mg70Zn30 was investigated by means of anomalous dispersion of X-rays. Three characteristic X-ray photon energies of an X-ray source were used to evaluate the three partial coordination numbers of this two-component system. The values of these coordination numbers differ only slightly from those of the crystalline phase Mgs1Zn20. The similarity between the short range order of the amorphous and crystalline phase is also reflected by almost identical values of the short range order parameters. The reliability of the resulting structural parameters and the accuracy of its determination are considered in detail.
1. Introduction
The metallic glass Mg70Zn30 has already been the subject of many investigations. X-ray diffraction was carried out by Nassif et al. [1], Ito et al. [2], Rudin et al. [3] and Andonov and Chieux (they investigated Mg72Zn28 ) [4]. Neutron diffraction was done by Mizoguchi et al. [5], Andonov and Chieux [4] and EXAFS by Ito and Kawamura [6], Maurer et al. [7] and Sadoc et al. [8]. The interest in this alloy is based on its simplicity: it is one of the simplest amorphous alloys with two divalent metallic components; additionally, detailed theoretical work is available. The interatomic forces may be derived from pseudo potential theory (Hafner [9]) or using a cluster relaxation model (Heimendahl [10], Hafner [9]). The interatomic distances and coordination numbers were explicitly calculated by Hafner [9] and are available for comparison. The amorphous MgToZn30 has nearly the same composition as that of the crystalline phase Mg51Zn20 [11]. The crystalline phase therefore can serve for further comparison. Up to now, structural information on the
amorphous alloy MgToZn30 was gained by the assumption of the similiarity between the amorphous and the crystalline phase only. Here we present the first directly measured coordination numbers as well as weighted interatomic distances. The MgToZn30 alloy and other Zn-containing samples are suitable for experiments utilizing the anomalous dispersion with laboratory X-ray sources. Only 33 eV below the K absorption edge of the Zn atom, the characteristic Lo~2 X-ray of a gold target allows a variation of the atomic form factor of about 18% at q = 0 .~-1. Using two more photon energies with a sufficiently different anomalous dispersion correction term, it is possible to evaluate the partial coordination numbers.
2. Theoretical background
2.1. Structural parameters The total structure factor, S(q), of an amorphous solid can be calculated from the coherently
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
F. Paul et aL / Short range order of a-MgToZn 3o
scattered intensity, Icoh(q), using the definition of Faber and Ziman [12]
S( q) - 1 = Ic°h(q) - ( f Z ( q ) ) (f(q))Z
(1)
Here q---(4"~/X) sin O is the modulus of the wavevector transfer, with 2 0 the scattering angle and X the photon wavelength. (f(q)) and ( f ( q ) 2 ) are the mean scattering factor and the mean squared scattering factor, respectively. In the following, we restrict ourselves in the notation to the presence of just two constituents: the indices j = 1, 2 stand for Mg and Zn, respectively. Hence f = c~f~ + c2f2, where the cj are the concentrations and ~ the scattering factors of a constituents. The total structure factor S(q)-1, is the weighted sum of the three partial structure factors: 2
S(q)-l=~
2
Peaks in the R D F denote shells of the next neighbours. From the area of the first peak, the total coordination number can be calculated according to N = f~I2RDF(r) dr,
(6)
where the lower and upper integration limits, r~ and r2, refer to the minima preceding and following the first peak. The procedure is not rigorously defined, resulting in some uncertainty in the determination of N (see, for example, ref. [13]). Since p(r) is a linear combination of the three partial radial number density functions, N is a linear combination of the three partial coordination numbers, also:
(7)
cicjfi (
JY" (f)2 S , j ( q ) - l ,
(2)
where the partial structure factors, S ~ j ( q ) - 1, refer to the respective atom pairs, ij. Neglecting the only weak q-dependence of the weighting factors of eq. (2), the Fourier transform of the q-weighted total structure factor results in the reduced radial distribution function:
G(r)=4"~r[o(r)-oo] 2 =~fo q [ S ( q ) - l ] s i n q r d q ,
".).
(3)
(4)
The expression is similar to that for the total structure factor (see eq. (2)). However, the weighting factors of Oij (r) do not contain the concentration cj. In a discussion of amorphous solids it is necessary to introduce the radial distribution function (RDF):
4"~rZp(r).
The ~ j are the partial coordination numbers of the atoms of type j around the atoms of the type i obeying the relation c,NO = cjNji. By contrast with eq. (2), eq. (7) is nearly independent of q. It is advantageous, therefore, to determine coordination numbers from this relation using a variation of the weighting factors.
2.2. The anomalous dispersion
with o(r) the radial density function and 0o the average number density, p(r) is the weighted sum of three partial radial density functions, pij(r),
RDF(r) =
99
(5)
The three partial radial density functions,
Oij(r), or the corresponding coordination numbers, ~ j , can be determined from three independent experiments with varying f,. Considering the anomalous dispersion effect of the constituents, a variation of f, is possible. In the non-relativistic theory the scattering factor, f, can be written as the sum of the q-dependent, but energy-independent part, fo(q), and the complex, energy-dependent part, f'(E) + i f " ( E ) (see ref. [14]):
f(q, E) = f 0 ( q ) + f ' ( E ) + i f " ( E ) . (8) The term f'(E) represents the anomalous correction whereas the term f " ( E ) corresponds to the absorption, M E ) , of the sample according to f"(E)
=
mc
2heZpo EI~(E),
(9)
F. Paul et al. / Short range order of a-MgzoZnso
100
Table 1 The values of f ' ( E ) and f " ( E ) for the used photon energies calculated from the absorption data given by McMaster et al. [15] by means of a Kramers-Kronig integration E
Mg
(keV)
f'(E)
f"(E)
f'(E)
Zn
f"(E)
8.396 (W L . 0 9.626 (Au L~2) 17.479 (Mo K~I )
0.18
0.17
- 1.86
0.67
0.14
0.13
-5.41
0.53
0.05
0.04
0.25
1.48
and Park [18] for a finite sample, respectively. It turned out, however, that this effect is relatively small and not detectable. The density of the amorphous Mg70Zn30 alloy was measured using Archimedes' Principle. The buoyancy fluid was destilled water with a small drop of a wetting agent, as suggested by Cawthorne and Sinclair [19]. The density was determined to be 2.96 g / c m 3 with an accuracy of 0.5%. This is comparable with the results of refs. [20] and [21]. The density of the crystalline phase Mg51Zn20 is 3.0 g / c m 3 [11]. The number density, P0, is related to the macroscopic density, Pm, by
Po = Pm( A / M ), where m and e are the mass and charge, respectively, of an electron, h is Planck's constant and P0 is the number density. The absorption of the sample, M E ) , was calculated by using the compilation of X-ray cross-sections of McMaster et al. [15]. f ' ( E ) can be calculated by means of a Kramers-Kronig integration which relates the imaginary part, f " ( E ) , to the real part, f ' ( E ) . Since the EXAFS-structure on the high energy side of the absorption edge has a negligible influence on the absorption on the lower side, that structure was ignored. The calculation was carried out using the method described by Dreier et al. [16]. The values of the anomalous correction terms are listed in table 1.
3. Experimental 3.1. The sample The sample was prepared by the melt-spinning technique. It had a typical" size of 10 mm width and 40 ~m thickness. A frame was used as sample holder, with a size sufficient to avoid parasitic scattering from the sample holder. In order to realize a semi-infinite sample - at least for longer wavelengths - a stack of five ribbons aligned parallel was mounted on the frame. With this geometry it is principally possible to use the correction algorithm for multiple scattering corrections according to Warren and Mozzi [17] for a semi-infinite sample and according to Dwiggins
(10)
where A is the Avogadro constant and M is the mass of one mole. The knowledge of the number density, P0, is required for the calculation of the R D F and the coordination numbers.
3.2. X-ray diffraction To obtain a high flux of photons, a Rigaku RU 200 PL rotating anode was used as photon source. A molybdenum anode with Mo K~I (17.479 keV) and a gold anode with Au L,2 (9.626 keV) were used. Because of a tungsten contamination of the anode caused by evaporation of the cathode also W L~1 (8.396 keV) was available for the experiment with the MgZn alloy. In order to profit from the focussing geometry, a bent LiF crystal was used as an incident beam monochromator. The scattered intensity was detected by a Si(Li) solid state detector. To reduce the contribution of fluorescent radiation in the diffraction pattern especially for the Au L~2 line in the neighbourhood of the Zn-K-absorption edge, the height of the pulses was suitably discriminated with help of a multichannel analyzer. The data were read into the memory of a personal computer. The scattered intensity was measured in reflection geometry. The scattering angles covered the range from 2 ° up to 130 ° corresponding to 0.34 , ~ - 1 < q < 8.91 ~ - 1 for Au L~2, 0.62 , ~ - 1 < q < 16.06 ~ - 1 for Mo K ~ and 0 . 7 8 , ~ - ~ _ < q _ < 7 . 8 , ~ 1 for W L~I. The entire range of angles was scanned in steps of 0.2 ° in 6).
lq Paul et al. / Short range order of a-MgroZnso
4. Data analysis The data were taken at T = 2 0 ° C in the Bragg-Brentano reflection geometry. Additional measurements with the sample removed were carried out to determine the parasitic scattering from the air and the holder. Half of the latter scattering intensity was subtracted from the uncorrected intensity data. These background corrected data were multiplied by a factor that considers the double scattering of X-rays in a semi-infinite or a finite sample. Tabulated values of a correction term considering the scattering angle 20, the mean squared scattering factor and the thickness of the sample are given by Warren and Mozzi [17] or Dwiggins and Park [18], respectively, from which the double scattering was calculated by a procedure described by the same authors. These corrected data were multiplied by the usual polarization factor
P(a,
O)=
1 + cos22a + cos220 cos22oe ,
(11)
where a is the monochromator angle and 2 0 the scattering angle (see, for example, ref. [22]). In case of a symmetric reflection geometry, the absorption correction is independent of the direction of the incident beam and the corresponding correction factor is A = 1/2/,, with ~ the absorption of the sample. Finally, the corrected data were normalized by the method of Krogh-Moe [23] and Norman [24]. The q-dependent scattering factor was calculated from the corresponding expressions in the International Tables [25]. The scattered intensity also contains Compton scattering. This incoherent part of the scattering was determined from an analytic expression given by Hajdu [26] for light elements (Z_< 36). Also the Breit-Dirac recoil factor was taken into account. A Fourier transform up to r = 2 ,~ proved that the measured density leads to reliable results: the initial slope of G(r) must be proportional to the number density P0- The total structure factor which was determined according to eq. (1) was weighted by q and then a fast Fourier transform was performed. An additional damping factor exp( - otq 2) with c~ = 0.005 ,~2 was used in the Fourier transform to reduce termina-
101
tion ripples in the transformed data. The reduced radial distribution function, G(r), as well as the R D F ( r ) were obtained by applying eqs. (3) and (5).
5. Results Figure 1 shows the three corrected and normalized diffraction patterns of the amorphous MgZn alloy for the three different photon energies. The diffraction patterns are typical for amorphous alloys with an additional prepeak at q = 1.60 ~ - 1 . When extrapolating from qmin to q = 0 ,~-~, the measured intensity goes linearly to zero. The influence of a non-vanishing intensity at q = 0 A-1 due to the compressibility (see ref. [27]) of the solid could be neglected. The three diffraction patterns differ only slightly for the different Xrays. However, due to the varying anomalous dispersion correction term of eq. (8) the scattering power is significantly different for the measurements with Au L~2. This can best be seen in fig. 1 in the case of the first maximum at q = 2.70 A-1. The normalized intensity (Au) is much lower than 600 eu (electron units), whereas the other intensities (Mo, W) are more than 600 eu.
,..., 6O0 I_ I1)
400 -
;
\ 200 W
Au 0
~
0
I
5
i
I
t
q (x')
I
i
10
t
1
I
I
15
Fig. 1. Corrected and normalized intensities, lnorm(q) (in electron units, eu), of the amorphous Mg70Zn30 alloy for the W L~I (8.396 keV), the Au L,2 (9.626 keV) and the Mo K~l lines (17.479 keV).
F. Paul et aL/ Short range orderof a-MgToZn3o
102 3 I v
09 1
W 0
.
.
.
.
.
.
.
Au 0
Mo
0 -1 0
5 q (A-t) 10
15
Fig. 2. Structure factors, S(q)- 1, of the amorphous Mg70Zn3o alloy for the W L.a (8.396 keV), the Au L.2 (9.626 keV) and the Mo K,I lines (17.479 keV) calculated according to Faber and Ziman [12].
T h e range of c h e m i c a l short range o r d e r of a b o u t 8 - 1 0 ,~ a n d o f a t o p o l o g i c a l short range o r d e r of a b o u t 15 A is in a g r e e m e n t with o t h e r p u b l i s h e d results (see, for e x a m p l e , ref. [1] a n d ref. [4]). These two f u r t h e r sets o f i n f o r m a t i o n are listed in table 2, t o g e t h e r with the structural p a r a m e t e r s , the p o s i t i o n a n d the w i d t h of (i) the p r e p e a k a n d (ii) m a i n p e a k in S ( q ) which are q u o t e d together with the results o f o t h e r a u t h o r s [1,4]. T h e occurrence of the p r e p e a k which is caused b y a chemical short range o r d e r is discussed in detail b y N a s s i f et al. [1] a n d A n d o n o v a n d Chieux [4] a n d is n o t a subject of discussion in this paper.
6. D i s c u s s i o n b
F r o m the c o r r e c t e d a n d n o r m a l i z e d intensity d a t a the structure factor was c a l c u l a t e d a c c o r d i n g to eq. (2) (fig. 2). W i t h this p r e s e n t a t i o n , the limiting qmax a n d increasing noise with increasing q is better visible t h a n in fig. 1. F r o m the values of the full width at half m a x i m u m ( F W H M ) Aq of the p r e p e a k a n d the first peak, the respective correlation lengths due to a chemical short range o r d e r a n d a topological short range o r d e r were d e t e r m i n e d using the Scherrer f o r m u l a X = 2 ~r/Aq.
T h e F o u r i e r t r a n s f o r m of the total structure factors, that is the r e d u c e d r a d i a l d i s t r i b u t i o n functions G(r), are shown in fig. 3. F r o m the initial slope in the s m a l l - r region (see ref. [13]) which is - 4 v p 0 , the average n u m b e r density, 00, can be derived. T h e r e are strong oscillations in this region of distances. A m o r e reliable c o n f i r m a tion o f the c u r r e n t analysis can be m a d e . The values of n u m b e r densities, P0, for all three det e r m i n e d values of G ( r ) agree within errors of
Table 2 Structural parameters derived from the structure factor S(q)-1. qpp, q~ and q2 axe the positions of the prepeak and the first and second maximum, respectively. Aqpp, Aql denote the FWHM of the respective maxima, from which the correlation lengths are calculated. The results of Nassif et al. [1] and Andonov and Chieux [4] are also listed for comparison. In the case of the neutron measurement, no prepeak was found by Andonov and Chieux E (keV) 8.396 (W L~l) 9.626 (Au L~2) 17.479 (Mo K,,1) 17.479 (ref. [1]) (h = 0.942 ,~) (neutron) 22.159 (Ag K,,1) (ref. [4])
qpp
Aqpp
Aqa
Xpp
Xl
( m - 1)
( m - 1)
(m)
(m)
4.50
0.75
0.40
8.4
15.7
2.70
4.50
0.625
0.40
10.1
15.7
1.60
2.70
4.55
0.70
0.40
9.0
15.7
1.54
2.65
-
0.78
0.44
8.0
14.2
-
2.65
4.50
-
0.40
-
15.7
1.5
2.70
4.5
0.63
0.4,.'.
ql
q2
( m - 1)
(m
1.65
2.70
1.50
(A -
1)
1)
10.0
14.5
60 v
F. Paul et al. / Short range order of a-Mg7oZn3o
2
103
~., 40
0
2
0
~
0
0
G I0
r (A)
0 20
Fig. 3. The reduced radial distribution function, G(r), of amorphous Mg7oZn3o alloy for the W L~I (8.396 keV), the L~2 (9.626 keV) and the Mo K,~1 lines (17.479 keV). From slope of the line between 0 A and approximately 2 ,~, atomic number density can be calculated.
the Au the the
measurement with the measured density and indicate a correct normalization. Usually interatomic distances are derived from the reduced radial distribution function G(r) because a further mathematical treatment which yields the RDF shifts the maxima to higher values in r [28]. To obtain values of partial G(r), a matrix inversion of eq. (2) with the appropriate weighting factors is necessary. Because of the noise at high q, the different cut-off values, qmax, and especially the very ill-conditioned matrix such a matrix inversion does not lead to reasonable results, unfortunately. The RDFs based on eq. (5) are displayed in fig. 4. The areas of the first peak denoted by NAue,2, NWL,, and NMoK., are also listed in table 3. Only in the case of the measurement with the W L,~ line, the area of the first peak must be corrected. The oscillations for r < r~ are too strong and modify the slope considerably. The left side of this peak must be fitted by a Gaussian to evaluate the correct area. The three partial radial distribution functions of both constituents are expected to contribute to the first peak. The fact that only one peak is observed indicates that the Goldschmidt radii [29] of both constituents (1.62 ,~ for Mg and 1.33 ,~ for Zn) are either of comparable size or the vari-
5
- (I)
iO
Fig. 4. The radial distribution function, RDF(r), of the amorphous MgToZn30 alloy for the W L~I (8.396 keV), the Au L,2 (9.626 keV) and the Mo K m lines (17.479 keV).
ance of the distribution is large compared with these distances. Otherwise the shells of the first neighbours must be seen separately. However, due to the weighting of the scattering power according to the anomalous dispersion, a shift of the maximum was detected (see table 3). In case of the measurement with the Au L~2 line, the lower scattering power of the Zn atom must be accounted for. This means that the influence of this contribution must also be lower. Indeed the radial distance
Table 3 Radial distances of the next neighbours and the areas of the first maximum in the MgToZn3o alloy. The distances were derived from the reduced radial distribution function G(r) whereas the areas are obtained from the RDF(r) E (keV)
rl (A)
N1
rz (A)
C~ (A)
8.396 (W L~:) 9.626 (Au L~2) 17.479 (Mo K~l)
2.99
10.8
4.88
5.71
2.92
11.0
4.94
5.77
2.85
10.5
4.86
5.77
17.479 (Mo Kca ) (ref. [1]) 17.749 (Mo K,d) Mg 7oZn 3o-meh (ref. [32])
2.88
10.7
2.92
11.4
--
5.70
-
104
F. Paul et a L / Short range order of a-MgzoZn 3o
increases due to the bigger Goldschmidt radius of Mg. According to eq. (7), the following system of linear equations containing the three partial coordination numbers is obtained:
O M#70Zn30. thin work anomalous X-ray scattertn i
• MgsIZn20 (crystalline) (Higashi et aL, 19el)
t-a
Mg?OZn3O - models : <> cluster relaxaUon delxH random bard sphere packl~ i
z
mAuL,2 = 0.448NMgMg + 0.702NZnMg
(Hafuer, 1980)
+ 0.652Nznz,, NWL,2 ----0.340NM~Mg + 0.714NZnMg + 0.875mznzn,
Zq170Zn30 . EXAFS (Maurer et aL. 1963) X MiToZn30, EXAFS
10
(Sadoc et al., 1985)
02)
o g~2ZnZ6, X-ray, neutron (Andonov st al., 1987) + a statinUcal distribution
NMoK~ ' = 0.252NMgMg + 0.692Nz.Mg
+ 1.131Nznzn.
5
Althofigh this is an intrinsically ill-conditioned system, a solution of the system of linear equations in eq. (12) was obtained: NMgMg =
0
2.4,
NM~n = 8.1, 3.5,
NZnMg
=
NznMg
= 8.4.
0 (13)
The value of NMaz, was determined from NznMg considering the relation cz,Nz.r,,tg = CMgNM~n. We have compared these results with those of Sadoc et al. [8], Maurer et al. [7], Andonov and Chieux [4], and theoretical calculations by Hafner [9] as well as the data of the crystalline phase (Higashi et al. [11]). EXAFS experiments at the Z n - K absorption edge yield structural parameters of the ZnZn and ZnMg correlation. Therefore we eliminated NmgM~ from the system of linear equations in eq. (12) by subtraction of each equation from the others. With an error of 0.1 in determination of the area of the first peak in the RDF, the linear equations reads
NMgZ. = 13.2 _ 1.0 - 2.1Nz,zn, NM# n = 14.4 _ 0.5 -- 2.6Nznzn,
(14)
NM# ~ = 15.3 + 1.0 -- 3.0Nz,zn. These linear relations are drawn in fig. 5. Such a representation of our results allows also an estimation of the reliability of structural data that were determined in eq. (13) via anomalous X-ray scattering. It is obvious that the system (12) is
5 N ZnZn 10
Fig. 5. The solution of the system of linear equations containing the partial coordination numbers, Nznmg and Nznzn. The bars indicate an error in the determination of the area N of about 0.1 [4,7-9,11].
extremely ill-conditioned, consequently also eqs. (14). The slopes of the eqs. (14) differ only weakly. This fact means that a linear combination of both coordination numbers can be obtained with high confidence, whereas a single value of the coordination numbers must be taken with care. However, it is interesting to note that all coordination numbers lie within a small area of the NMgZnNznzn plane. The values of coordination numbers which result from the measurements of Sadoc et al. [8], Higashi et al. [11], and Andonov and Chieux [4] are in agreement with our results considering the error bars. There also seems to be a weak tendency of a short range ordering in case of the investigations of Mauer et al. [7] and Hafner [9]. Compared with a purely statistical distribution, these values are shifted to somewhat smaller values. Another quantity that describes the short range order of a solid is the short range order parameter, first introduced by Cowley [30]. From several dif-
F. Paul et aL / Short range order of a-Mg7oZn3o
ferent definitions published since then, we use the definition of Cargill and Spaepen [31]:
NAB
ct--=] --
CB[CA(NAA -~- NAB ) -I1-CB(NBA "~ NBB)] " (15) The subscripts A and B denote the atoms of type A and B, respectively. The expression in the squared brackets denotes a mean total coordination n u m b e r and its p r o d u c t with %, that part of the B-type atoms on this mean total coordination number. Thus the fraction 1 - a in eq. (15) describes the probability of finding A - t y p e atoms a r o u n d the B-type atoms related to a mean total coordination number. A value ct < 0 means a tendency of forming pairs of unlike atoms, ct = 0 means a totally r a n d o m distribution of the constituents and a > 0 means a tendency of forming a pair of like atoms or clustering. According to the definition of a, we obtained ct = - 0 , 0 3 for the short range order, thus a very weak tendency for a chemical short range order. In order to allow a judgement, whether this departure from ct = 0 is significant, it is important to consider the error of a due to the errors in determination of the areas below the mainpeak in the R D F . Since all partial coordination numbers are linearly dependent a can also expressed by
ct = 1
b - tuNas "
(16)
Considering the coefficients of eq. (12), b = 3.595 and rn = 0.058 follows. The error of a is Act
b (b - tuNAs) 2 ANAB"
(17)
Assuming a reasonable error of about 0.5 atoms in NAB, one obtains a = - 0 . 0 3 _+ 0.15. Generally the very small value of a = - 0 . 0 3 supports the results of former investigations. C o m p a r i n g the a = - 0 . 0 4 for the crystalline phase, the short range order of the crystalline and a m o r p h o u s phase is of comparable size and signals a r a n d o m distribution of the constituents rather than a tendency of pair formation. The departure of a from 0 is not significant.
105
7. Conclusions F r o m the total reduced distribution function, weighted radial distances were derived. It was possible to determine partial coordination numbers f r o m a matrix inversion of that system of linear equations which contains these partial coordination numbers. A n illustration of this system of linear equations demonstrates the accuracy and also the limits of the method. Considering this accuracy it was shown, without any additional assumption about the structure of the a m o r p h o u s alloy, that the short range order in the a m o r p h o u s phase is almost identical to that in the complex crystalline phase MgslZn20. The tendency of ordering is described by the short range order parameter a. The values of ct which were determined for the a m o r p h o u s and crystalline phase indicate an ordering that is comparable to that of a pure r a n d o m distribution. Earlier assumptions about the similarity of the a m o r p h o u s and the crystalline phase seem to be justified. The authors would like thank H.J. Giinterrodt and P, Reimann, Institut fiir Physik, Basel, for the preparation of the samples.
References [1] E. Nassif, P. Lamparter, W. Sperl and S. Steeb, Z. Naturforsch. 38a (1983) 142. [2] M. Ito, H. Iwaska, N. Shiotani, H. Narumi, T. Mizoguchi and T. Kawamura, J. Non.-Cryst. Solids 62&63 (1984) 303. [3] H. Rudin, S. Jost and H.J. Giintherodt, J. Non-Cryst. Solids 62&63 (1984) 291. [4] P. Andonov and P. Chieux, J. Non.-Cryst. Solids 93 (1987) 331. [5] T. Mizoguchi, H. Narumi, N. Ahutsu, N. Watanabe, N. Shiotani and M. Ito, J. Non-Cryst. Solids 62&63 (1984) 285. [6] M. Ito and T. Kawamura, Philos. Mag. 49 (1984) L9. [7] M. Maurer, J.M. Friedt and G. Krill, J. Phys. F: Met. Phys. 13 (1983) 2369. [8] A. Sadoc, R, Krishnan and P. Rougier, J. Phys. F: Met. Phys. 15 (1985) 241. [9] J. Hafner, Phys. Rev. B21 (1980) 406. [10] L. Heimendahl, J. Phys. F: Met. Phys. 19 (1979) 161. [11] J. Higashi, N. Shiotani, M. Uda, T. Mizoguchi, and H. Katoh, J. Solid State Chem. 36 (1981) 225.
106
F. Paul et al. / Short range order of a-Mg7oZn3o
[12] T.E. Faber and J.M. Ziman, Philos. Mag. 11 (1965) 153. [13] Y. Waseda, The Structure of Non-Crystalline Materials, Liquids and Amorphous Solids (McGraw-Hill, New York, 1980). [14] R.W. James, The Optical Principals of the Diffraction of X-rays (Ox Bow Press, Woodbridge, CT, 1982). [15] W.H. McMaster, N. Kerr del Grande, J.H. Mallet and J.H. Hubbel, Lawrence Livermore Lab. report UCRL50174-SEC2-R1 (1969). [16] P. Dreier, P. Rabe, W. Malzfeldt and W. Niemann, J. Phys. C17 (1984) 3123. [17] B.E. Warren and R.L. Mozzi, Acta Crystallogr. 21 (1966) 459. [18] C.W. Dwiggins and D.A. Park, Acta Crystallogr. A27 (1971) 264. [19] C. Cawthorne and W.D. Sinclair, J. Scient. Instr. 2 (1972) 531. [20] M. Matsuda and U. Mizutani, in: Proc. 4th Int. Conf. on Rapidly Quenched Metals (Japanese Institute of Metals, Sendai, 1981) p. 1315.
[21] U. Mizutani and T. Mizuguchi, J. Phys. F l l (1981) 1385. [22] B.E. Warren, X-Ray Diffraction, (Addison-Wesley, Reading, MA, 1969). [23] J. Krogh-Moe, Acta Crystallogr. 9 (1956) 951. [24] N. Norman, Acta Crystallogr. 10 (1957) 370. [25] J.A. Ibers and W.C. Hamilton, eds., International Tables for Crystallography, Vol. 4 (Kynoch Press, Birmingham, 1974). [26] F. Hajdu, Acta Crystallogr. A28 (1972) 250. [27] G.L. Squires, Introduction to the Theory of Thermal Neutron Scattering, (Cambridge University Press, Cambridge, 1978). [28] C.N.J. Wagner, J. Non-Cryst. Solids 31 (1978) 1. [29] V.M. Goldschmidt, Skr. Nor. Videns.-Akad. Oslo 1 1(2) (1926) 1. [30] J.M. Cowley, J. Appl. Phys. 21 (1950) 24. [31] G.S. Cargill, F. Spaepen, J. Non-Cryst. Solids 43 (1981) 91. [32] E. Btihler, P. Lamparter and S. Steeb, Z. Naturforsch. 42a (1987) 507.