Diffraction Analysis of Solid Solutions Ludo K. Frevel Chemical Physics Research Laboratory, The Dow Chemical Company, Midland, Mich. 48640 Quantitative identification of substitutional solid solutions is made possible through precision X-ray diffractometry and an extension of Vegard’s rule. Two cases are considered; namely, the case for which literature data on lattice parameter vs. composition are known, and the other for which only the structure of the pure phase is known. Formulas are developed for calculating the degree of substitution in a homogeneous solid solution. The detection of a distribution of solid solutions is also discussed. THE INCREASED USE of precision diffraction data within the past five years has revealed that many solids submitted for phase identification are found to be dilute solid solutions rather than mixtures of pure phases. Detection and appropriate analysis of these solid solutions augment the utility of the powder diffraction method in a modern analytical laboratory. Of the three types of solid solutions (substitutional, interstitial, and subtractional) the substitutional variety is commonly encountered in all types of solids, be they ionic, covalent, or metallic. Interstitial solutions are often present in metallic phases containing such elements as H, B, C, N , or 0 in nonstoichiometric ratios to the host element. Vacancy solid solutions are defect structures in which some of the atoms are “subtracted” from the ideal stoichiometric structure, e.g., Fel-, S based on the B81 structure (1). It is possible t o have two types of solid solutions exhibited by one phase such as (Fe, Ni)l-,(32S, “S), where the italicized elements are present in major concentrations. The work reported in this paper will be restricted to the identification of substitutional solid solutions. Two cases will be considered; namely, the case for which literature data o n lattice parameter us. composition are known, and the other for which only the structure of the pure phase is known. Case 1. UNIT CELL CONSTANTSus. COMPOSITIONAL CHANGES.X-ray diffraction is the preferred method of detecting solid solutions, especially in multiphase mixtures. After having established cell constants throughout a homogeneous phase field, one may determine the composition of a particular solid solution from its measured unit cell, provided the changes of the lattice constants with composition are measurably large. The metallurgists have practiced this technique with great proficiency and have compiled useful data on several thousand alloys ( 2 , 3). The types of curves empirically established for binary alloys fall into five classifications illustrated in Figure 1 . Curve 1 typifies a substitutional alloy such as the Ag-Pd system ( 4 ) ; the straight line 2 holds for those cases which follow Vegard’s rule (5, 6) of solid solution; curve 3 corresponds to a binary alloy with vacant sites as in Af-Zn (7); curve 4 exhibits a minimum cell edge such as the alloy 60 Ag-40 Au (partial ordering of Au (1) H. Haroldsen, 2. Anorg. Chem., 246, 169 (1941). (2) W. B. Pearson, “A Handbook of Lattice Spacings and Structure of Metals and Alloys,” Vol. I, Pergamon Press, New York, N. Y., 1958. ( 3 ) Ibid., Vol. 11, 1967. (4) B. R. Coles, J. Inst. Metals, 84, 346 (1956). ( 5 ) L. Vegard and H. Schjelderup, Phys. Z., 18,93 (1917). (6) L. Vegard, 2.Phys., 5 , 17 (1921). (7) E. C. Ellwood, J. Inst. Metals, 80,217 (1952).
atoms about Ag atoms and uice versa); and curve 5 displays a maximum cell edge as seen in the y-phase of Fe-Ni (8) in the region of 45 at. Ni. Various combinations of these solid solution effects may be realized in a binary or ternary phase diagram. We shall restrict our analysis to those substitutional solid solutions which follow curves 1, 2, 3, as well as the terminal solid solutions illustrated by curves 4 and 5 . Two determinations of a binary solid solution will be cited to illustrate the type of analysis a diffractionist can perform on known systems. The first case involves the analysis of a small spherical bead of Pd-Ag alloy. Conventional powder data from the filings of this sample yielded a cell constant of 3.916 f 0.005 A. Using the precise data of Coles ( 4 ) , one can express the increment in lattice constant with compositional change by Equation 1
where a = lattice constant of the cubic solid solution Pd-Ag, a, = lattice constant of the pure matrix phase Pd, ax = lattice constant of the pure substituting element Ag, and x is the atom fraction of Ag. The exponents a and p are empirically evaluated t o give the best fit with the published data ( 4 ) . With the aid of a Wang 360 electronic calculator, the averaged values of a and B for the interval o 6 x 6 0.50 were computed as 1.032 i 0.021 and 1.052 i 0.002, respectively. Substituting the value of a = 3.916A, u, = 3.8898 A (9) and ax = 4.08626 A (10) in the above formula, one obtains the composition of the solid solution alloy as Pdo.ss Ago,,,. X-ray fluorescence analysis on filings of the bead gave the composition as Pd0.833 Ago,,,,. It should be mentioned that for the Ag-rich phases where the small Pd atom is substituting in the Ag matrix, the appropriate values for the exponents a and (? are 0.912 f 0.042 and 0.973 + 0.003, respectively. The second example pertains to a n annealed powder of Pd contaminated by a small amount of Ag. To detect the presence of a dilute solid solution, precision powder data were required for the sample and for a specimen of pure Pd taken in the same camera and preferably on the same film. This procedure was followed by employing an AEG double cylinder Guinier camera (1 14.7-mm diameter) with C u K a l radiation and an internal standard (11). The cell constants thus obtained for the dilute solid solution Pd-Ag and for pure Pd powder are 3.8919 i 0.0012 A and 3.8890 f 0.0007 A, respectively. From Equation 1, the composition of the second alloy was computed as Pd0.9sAgo.op. However, the broadness of the diffraction lines and the actual Ag content of 4.1 at. (X-ray fluorescence analysis), suggest that the alloy is not homogeneous in composition. Case 2. UNKNOWNSOLID SOLUTIONHAVINGKNOWN CRYSTAL STRUCTURE.The availability of accurate powder diffraction standards from the U S . National Bureau of (8) E. A. Owen, E. L. Yato, and A. H. Sully, Proc. Phys. SOC.,49,
315 (1937). (9) H. E. Swanson and E. Tatge, Nut. Bur. Stand. ( U . S.) Circ., 539, Vol. 1, 21 (1953). (10) H. E. Swanson, M. C. Morris, and E. H. Evans, Nat. Bur. Stand. (U.S.)Monogr. 25, Section 4, 4 (1966). (11) L. K. Frevel, ANAL.CHEM., 38, 1914 (1966).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
1583
atom B a t ( U B , CB, WE). Designating the translation from the center of atom A to the center of B by the vector ,: we may write
4;
-
= (UB
O:
UA) a 0
+
(GB
-
CA)
$0
+
-
(WE
WA)
2.0
(3)
where ?io, &, and t oare the unit cell translations for the pure phase. Now if we substitute x mole fraction of B by “impurity” atom X and thus alter the unit cell translations to ax,bx, &, then the corresponding closest interatomic distance is given by Equation 4
4
2X = (UX
0
-
uA’)ZX
+
(OX
-
f (WX
L’A’)bX
-
WA’)zX
(4)
For moderately dilute solid solutions the differences in the corresponding atomic coordinates remain essentially unchanged, Le., UX
-
UA’ z u g
-
UA, 6X L’B
- L‘A’ - L’A, W X -
WA’
WB
- WA
(5)
From the measured unit cell dimensions of the two respective phases and from the known atom-coordinates of the pure phase, we obtain the increment in the closest interatomic distance as given by Equation 6 sx
At. %SOLUTE
Figure 1. Representative curves expressing change in lattice parameter as function of alloy composition Standards in combination with precision diffractometry has made possible the detection of dilute solid solutions where the “contaminating” element(s) may range between 0.2 and 2 atom per cent. Usually an elemental analysis by X-ray fluorescence, atomic absorption, or chemical analysis reveals the presence of the impurity element(s). In most instances, no data are extant on the compositional variation of the unit cell constants; hence, the diffractionist merely states the qualitative composition of the solid solution. However, if the structure type of the pure phase is known, it is possible t o calculate the extent of substitution. THEORY
The dominant factor determining solid-solution formation is the atomic size factor, generally treated in terms of the ratio of ionic radii, covalent radii, or metallic radii. For example the usual limit imposed on binary alloy formation is given by Expression 2 0.87
< rl/r2 6 1.15
(2)
where ri and r2 are the metallic radii of the two alloying elements corrected for the appropriate coordination number (12, 13). Based on the approximate validity of the additivity of atomic radii (ionic, covalent, or metallic), the following deductions can be established. Let us consider the closest interatomic distance in a known crystal structure between atom A located in the unit cell a t ( U A , L’A, W A ) and another
- so = & . S,)1/* - ,(: . : a ) ’ / 2
zz
x(rx -
YE)
(6)
where rx and rB are the respective appropriate radii of atoms X and B based on the same applicable coordination number (12, 13). If the ratio rx/rB approaches or exceeds the limit (1,15)*l or if the electronegativity of element X differs markedly from that of element B, then the linear additivity postulate has t o be modified by empirically introducing exponents a and ,8 in the right-hand portion of Equation 6 ; namely, xa / Y X - rB Equation 6 obviously can be solved for x provided only one random substitution obtains. If two different elements are randomly substituted for B, then the right-hand portion of Equation 6 becomes XI (YX, - YB) X ~ ( Y X~ rB). Moreover, if the ratio x2/x1can be ascertained from X-ray fluorescence or some other method, then the solution for XI and x2 can be realized.
ID.
+
EXAMPLES
A sample of slaked dolomitic lime was exposed to CuKal radiation in a n AEG Guinier camera with A1 and cubic Asz03 as internal standards. Two phases were identified: slightly expanded Mg(OH)* and slightly contracted Ca(OH)*. The cell constants for the brucite phase were determined by the axial ratio method (14) yielding a = 3.160 i 0.004 A and c = 4.784 i 0.002 A. Pure Mg(OH)2 crystallizes in the C19 structure with a, = 3.147 A and c, = 4.769 A (15). The shortest interatomic distance between a magnesium ion and its nearest oxygen neighbor is given by (aO2/3 u 2 c 0 2 ) 1 ~with z, u = 0.222. Substituting the appropriate constants in Equation 6, one obtains
+
SX
- so = 2.110 - 2.1029
=
x(rcaz+ - rarr2+) = x (1.06
- 0.78)
(7)
The value of the mole fraction x computed from Equation 7 is 0.029 giving the composition of the solid solution as Mgo.9 Cao.os(OH)*. A comparable calculation for the Ca(OH)* phase (16) with a = 3.5855 + 0.0010 A and c = 4.901
*
(12) Linus Pauling, “The Nature of the Chemical Bond,” 3rd ed., Cornel1 University Press, Ithaca, N. Y.,1966, p 154. (13) W. H. Zachariasen, cited in “Introduction to Solid State Physics,” Charles Kittel, 2nd ed., John Wiley & Sons, Inc., London, 1961, p 81. 1584
(14) L. K . Frevel, Acta CrystaNogr., 17,907(1964). (15) H. E. Swanson, N. T. Gilfrich, and M. I. Cook, Nat. Bur. Stand. (U.S.) Circ. 539,Vol. 6, 32 (1956). (16) H. E. Swanson and E. Tatge, ibid., Vol. 1, 58 (1953).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
30
-
'v *)
P
.40
-
.50
-
.60
-
t
P 0
0 0
I
f .70 -
N
a
-
.eo
.so 1.00
I
I
I
I
I
,
I
I
I
I
I
I
I
I
-
0
s
*40
N
f
p!
Figure 2. Slow scan of diffraction patterns in region of 2.00 to 2.10 (0.5 rnmimin) 0.002 A yielded the formula Cao.9gMg0.01 (OH)'. An elemental analysis of the above sample revealed that less than 0.01 C1 was present, thus eliminating the possible substitution of C1- for OH-. A second set of examples pertains to four solid solutions having the barite structure. The first specimen was prepared by adding 0.99 gram of concd H & 0 4 to a hot agitated aqueous solution (-20 ml) containing 2.620 grams of Ba(NO& and 0.2117 gram of Sr(NO&. The fine precipitate was filtered and washed three times with 20-ml portions of hot distilled water. After drying, the white powder was heated to 500 "C for 3 hours yielding crystallites approximately 1 p i n diameter; and, subsequently, to 800 "C for an additional 5 hours to give -3-p crystallites (as measured by optical microscopy). The respective cell constants for the two heat-treated samples were found to be u1 = 8.8213 0.0068 A,bl = 5.4529 i 0.0018 A, ci = 7.1305 i 0.0021A; and ar = 8.8157 + 0.0041 02 = 5.4470 f 0.0011 A, c2 = 7.1191 i 0.0034 A. This shrinkage of the unit cell with the second heat-treatment and the unsymmetrically broad diffraction maxima (see Figure 2) from the 3-p crystallites points conclusively to a n asymmetric distribution of solid solutions. The asymmetric (410) reflection did not sharpen materially after the second heat treatment which tripled the size of the crystallites. Contraction of the barite unit cell on substituting Sr2+for Ba2+ involves a shortening of the average interatomic distance between the cation and the oxygens of the surrounding sulfate ions. I n the BaS04 structure, each Ba2+is surrounded by 12 oxygen atoms a t distances ranging from 2.75 A to 3.28 A. The "ORTEP"
A,
program (17) is convenient for computing these interatomic distances. The separation between a barium atom at ( u . 0 , w ) and a n oxygen atom a t ( x , y , z ) is given by Equation 8 [(x
- U ) ~ U , * + ( y - ~)'h,,'+ ( Z
- I~')~c,.']''?
(8)
For the shortest interatomic distance the appropriate atomic coordinates are u , L', w = 0.1844, 1/4,0.1587; x , y , z = 0.187, 1/4, 0.543; and the cell constants for pure BaSOr are u, =
A,
8.8741 ==I 0.0040 A,0, = 5.4517 i- 0.0017 c, = 7.1514 i 0.0007 A (18, 19). The only additional datum required is
the difference in the ionic radii of Ba2+and Sr?+which difference can be derived from the cell constants of the isomorphous oxides ( 2 0 , 2 / )or sulfides ( 2 2 ) : 2(rl+n2+- r s r ? + )=
UB,O
- as,o = 0.363
=
seas - nsrs
=
0.366
A
(9)
Substituting the appropriate values in Equation 6. one computes the mole fraction x for the doubly heat-treated (17) C. K. Johnson, Rep! ORNL-3794, Oak Ridge National Laboratory, Oak Ridge, Tenn., June 1965. (18) H. E. Swanson, R . K . Fuyat. and 0.M. Ugriiiic, Nut. Bur. Sfuiid. ( U . S.)Circ., 539, Vol. 3, 65 (1954). (19) G . Walton and G . H . Walden, Jr., J . Anicr. Chern. SOC.,68, 1750 ( 1 946). (20) H. Huber and S. Wagner, Z . Tech. Phys., 23, 1 (1942). (21) H. E. Swanson, V. T. Gilfrich, and G . M. Ugrinic. N u / . Bur. Sfand. ( U . S.) Circ.,'539, Vol. 5 , 68 (1955). (22) H. E. Swanson, N. T. Gilfricli, and M. I. Cook, Nat. 8ur. Stand. ( U . S.) Circ. 539, Vol. 7 , pp 8 and 52 (1957).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
1585
Table I. Change of Lattice Constanta of KIl-=Br2 as Function of x Change of Nominal lattice cons,tant Calculated composition, 102x - Aa, A composition, 102x 1,39 0.0052 1.12 2.77 0,0103 2.21 6.84 0.0275 5.90 100.00 0,4656 100.00 a i i ~(23 “C) = 7.0618 i 0.0013 A. Bal-, Sr, SO4 as 0.069. Utilizing the atomic coordinates for the second shortest Ba-0 distance (2.77 A) in Equation 6, one calculates x = 0.085. Likewise for the third closest oxygen, of which there are two, the value of x is equal to 0.074. If the atom coordinates for the crystallographically nonequivalent oxygens were known more accurately, then the three computed values of x would agree more closely. Nonetheless, the mean value of x (0.076 i 0.006) is considerably lower than the nominal value of 0.091 and actually pertains to the mode (peak) of the distribution of solid solutions. A similar example was brought to the author’s attention by Laverne Ruhberg who identified in an unknown solid a major phase with the barite structure. X-ray fluorescence analysis revealed the presence of Sr, Ba, and S. Accordingly, he prepared a precipitate by adding an aqueous solution of N a ? S 0 4 to one containing 0.10 gram of SrCI? and 0.02 gram of BaCl?. The washed and dried precipitate was examined by powder diffraction and found to contain a celestite phase Srl-u Ba,S04 and a barite phase Bal-,Sr,S04 with the following respective cells: ul = 8.3727 A, bl = 5.3566 A, c1 = 6.8764 and u2 = 8.519 A,bz = 5.407 A,c? = 6.981 From Equation 6, the calculated value for y is 0.022 and for x , 0.242. The fourth example illustrates the substitution of a sulfate ion by a selenate ion. A precipitate of Ba(S04)1-,(Se04)zwas prepared by adding a hot aqueous solution of 2.614 grams of Ba(NO& to a hot solution of 1.568 grams of K 2 S 0 4and 0.2232 gram of K2Se04. After thorough washing, the white precipitate was centrifuged, dried, and heated to 300 “C for 16 hours. Precision powder diffraction data yielded the cell dimensions u = 8.8921 i 0.0022 A, b = 5.4734 f 0.0005 A, I t is noteworthy to mention that c = 7.1698 i 0.0011 the diffraction maxima were not broadened in comparison to those of well crystallized BaS04, thus indicating a homogeneous solid solution. The expansion of the barite unit cell on substituting Se6’ for S6+(i.e., S e 0 4 z - for SO4+) results from an effective lengthening of the average interatomic distance in the tetrahedral radical ion. For the sulfate ion in BaS04,the computed S-0 distances vary between 1.476 and 1.515 with atomic coordinates for sulfur at (0.064, 1/4, 0.690) and oxygens at (0.079, 0.034, 0.813), (-0.092, 114, 0.612) and (0.187, 1/4,0.543). Since there is no reliable information on the radius of hexavalent selenium, it is of interest t o calculate (rg,n+-r86+) from Equation 6 for the known nominal value of z = 0.100. For the shortest S-0 distance (1.476 A) we have
A;
A.
A.
A
sx - so = 0.0052 = 0.100 (rses+ -
TSG-)
(10)
The mean value of ( r S e 6 + - rss+) computed from the three S-0 distances is 0.040 f 0.008 A and can be substituted in Equation 6 for ascertaining z when selenate is substituted for sulfate. It should be pointed out that the infrared mull technique is 1586
* ANALYTICAL CHEMISTRY, VOL.
very useful and sensitive in detecting the presence of selenate ions in the presence of sulfate ions, but is practically incapable of distinguishing between a physical mixture and a solid solution. Virginia B. Carter of this laboratory examined the three solid solutions of barite by the mull technique and found that the sulfate bands are little affected by the solid solution phases. Slight frequency shifts occur in the asymmetric S-0 stretching region ( < l o cm-l for 1080, 1120, and 1170 cm-l bands). The other infrared active bands are not shifted; bands in the Raman spectra are not shifted from the BaS04 frequencies. The Se04?- ion has bands similar to the SO4*ion, with frequencies about 200 cm-I lower than the respective S 0 4 z - band in both the infrared and Raman spectra. The Ba(S04)o.g(SeO~)o,l sample, prepared as described above, showed also a minor impurity band, probably from adsorbed nitrate ions. Additional techniques such as X-ray fluorescence, atomic absorption, or infrared absorption are required to confirm the quantitative interpretation of the diffraction data. The accuracy one may expect from Equation 6 is illustrated in the following two sets of known solid solutions. The first illustration pertains to three dilute solid solutions of KBr in K I . Carefully weighed portions of reagent grade K I and KBr were dissolved in distilled water to form a concentrated solution which was then evaporated to dryness in an oven heated to 120 “C. The dried mixture was ground in an agate mortar and subsequently fused in a gold boat over Nz. A clear crystalline mass was obtained which upon pulverization yielded a sharp powder pattern. For precise comparison the solid solution and the K I standard were exposed simultaneously in the AEG Guinier camera. Table I contains the pertinent diffraction data on three such solid solutions. The composition of the solid solution was calculated from Equation l l
where ( T I - - n ~ ) ‘ = / * ( ~ K I - U K B ~ ) = 0.233 A. On the average the computed composition is -22% too low. If one uses the nonlinear Equation 1 with cy = 1.04 =t0.04 and p = 1.07 f 0.03, the calculated composition is accurate within 5 % . For an unknown solid solution, at least two homogeneous solid solutions of known composition would have to be prepared to evaluate cy and p. In case the calculated composition differs more than 40 % from the elemental determination, one should look for inhomogeneity in the solid solution. To simulate a distribution of solid solutions, the following 5-phase mixture was prepared: 25 mg of KI, 100 mg of KIo.9d3ro.ol4, 250 mg of KI0.972Bro.028, 112.5 mg of KI0.9azBro.osB,and 12.5 mg of KBr. The five highly crystalline powders were thoroughly mixed in a n agate mortar and gently ground together. A precision powder pattern of this mixture looked like a single solid-solution with asymmetrically “broadened” diffraction lines. A short portion of the photometer trace is reproduced in Figure 3. For the above homogeneous KIl-,Br, phases (Table I), the diffraction lines were symmetrical and had the same halfwidth as the aluminum reference-reflection (31 1). From a change of the KI-lattice constant of -0.0098 A, the composition of the “single” phase was calculated as K10.979Br0.021 corresponding to 1.51 % Br by weight whereas the actual value is 3.24% Br for the total mixture. On careful inspection of the powder pattern and its print-out { d,, I, it is noted that the strongest line of KBr was recorded as a sharp line at 3.2995 with intensity 6.5 relative to 388 for the (200) reflection of the K(1,Br) phase. How-
42, NO. 13, NOVEMBER 1970
1,
A
*401 .50
.600
z a
E .70$ z
2I-
.eo.so 1.001
I
I
I
I
I
I
I
I
I
I
2rO
1
I
P
Imm+
Figure 3. Slow scan of diffraction pattern of 5-phase mixture in region between 1.19 and 1.25 (1 mm/min) ever, the second strongest line (2.333 A) was missed because of superposition with the 2.337Y-A reflection of the aluminum reference line (111). From the known relative concentration of the phases present in the 5-phase mixture, it is evident that the “single” phase determined from the diffraction data corresponds to the peak (mode) of the distribution; namely, the KIo.y72Br0.0.14 phase. Extensive overlap of its diffraction lines phase accounts for the slightly with those of the KIO.DBGBrO lower Br content calculated for the mode of the distribution. Asymmetry of the broadened reflection was caused by the KT0.932Br0.088 phase constituting 22.5 % by weight of the mixture. Line broadening from crystallite size is symmetrical and was not significant in this case. The nature of the crystallites in an unknown solid sample should be ascertained from optical microscopy or electron microscopy to assist in the meaningful interpretation of the diffraction data. Although Equation 6 was derived for homogeneous solid solutions, it may profitably be applied to inhomogeneous solid solutions in order to ascertain the approximate composition of the peak of the distribution of solid solutions and also to obtain a qualitative measure of the skewness of the distribution from the shape of the dilfraction lines. The second illustration on known homogeneous solid solutions applies to calcium magnesitini carbonates with the calcite structure. Goldsmith and Graf (23) prepared magnesian calcite by reacting pclletized mixtures of basic magnesium carbonate and Johnson and Mathey’s “Specpure” CaCO:, in cold-seal pressure vessels. The runs were held one or two days a t tcmpcratures of 800 to 860 “C under COz pressures of bctween 20,000 and 24,000 Ib/im2 The temperatures chosen for individual runs werc sufficiently high to ensure complete solid solution of the mixture. Precise lattice-constants of four solid solutions Ca,-,Mg,CO;~, synthetic calcite, and synthetic magnesite were determined on the General Electric XRD-3 X-ray dilrractometer. Taking their averaged cell constants for nine scparate runs and the atomic coordinates for oxygen (0.25706,0, 1 /4) of Chessin et crl. ( 2 4 ) , one can calculate the closest interatomic distance between CaZi-and its six nearest oxygen neighbors as 2.359 A . Table 11 lists the (23) J. R . Goldsmith and D. L. Graf, Awwr. Miiioro/. 43, 84 (1958). (24) H. Chessin, W. C. Hamilton, and B. Post, Acto Crys/u//ogr. 18, 689 (1965).
Table 11. Cell Constants of Cal-,Mg,COB Nominal Closest composition interatomic Calculated (synthesis) distanc? composition 102x
n, A
c, A
0.00
4.9899 4.9686 4.9436 4.9218 4,9022
17.063 16.953 16.851 16.738 16.637
4.94 9.89 14.83 19.78
Ca-0, A 2.359 2.347 2.334 2.322 2.311
102x 0.00 4.3 8.9 13.2 17.1
cell constants for the four solid solutions, the closest interatomic distance between the cation and its nearest oxygens, and the composition of the solid solutions calculated from Equation 6. The calculated values are about 13% too low. The linear approximation of Equation 6 can be improved by adjusting the parameters Q and in the dimensionless Equation 12
Equation 12 can be transformed into the nonlinear form of Equation 6 mentioned above. As reliable data on homogeneous solid solutions accumulate, it will be possible to develop empirical values for exponents Q and /3 applicable to various ionic, covalent, metallic, or molecular crystals. Many examples of dilute solid solutions have been encoiintered by H. W. Rinn and the author in the phase identification of unknown solids. On the basis of our experience, it is reasonable to predict that precision powder diffraction methods coupled with the techniques described in this paper can enhance the quality and utility of phase identification in the fields of chemistry, mineralogy, metallurgy, and ceramics. JICKNOWLEDGMENT
The author is grateful to June W. Turley for assisting in some of the crystallographic computations and for carefully reviewing the manuscript. RECEIVED for review February 6, 1970. Accepted August 17,1970.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
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