Anal. Chem. 1980, 52, 1300-1304
1300
(8) Mivake. S.: TOQaWa, S.: Hosoya, S. Acta Crystallogr. 1964, 17,
1083-1084. (9) Jennings, L. D. Acta Crystallogr. 1968,A24, 472-474. (IO) Suortti, P.; Jennings, L. D. Acta Crystallogr. 1977,A33, 1012-1027. (11) Mathieson, A. McL. Acta Crystallogr. 1978. A34. 404-406. . 1979,12, (12) Le Page, Y.; Gabe, E. J.: Calvert, L. D. J . ~ p p i Crysta//ogr. 25-26.
(13) Cullity, B. D. "Elements of X-ray Diffraction", 2nd ed.;Addison-Wesley: Reading, Mass., 1978: p 135. (14) Altree-Williams, S. Anal. Chem. 1978,50. 1272-1275.
RECEIYEDfor view November 13, 1979. Accepted February
29, 1980.
Powder Diffraction Analysis of a Three-Component Sample Jaroslav Fiala Central Research Institute Skoda, 3 1600 Plzefi, Czechoslovakia
A procedure is described for X-ray powder diffraction phase analysis of mixtures of components some of which are not known or are partly known substances. The procedure is based on diffractlon patterns of several secondary samples suitably prepared from the sample analyzed so that they contain various quantities of components of the same multiphase system. Correlation analysis of these diffraction patterns allows one to estimate the number of components In the mixture analyzed, to reconstruct the powder diffraction patterns of its individual components, and to determine quantitatively their volume fractions. The proposed concept is illustrated with a three-component sample problem.
The aim of powder diffraction phase analysis is to express the diffraction pattern of the mixture analyzed in terms of patterns of its components (crystalline phases). Mathematically stated (1, Z), we want to find the values c j of concentrations of the individual components that minimize
1%
k
-
Ccjyjlzsubject t o c j j=l
2 0; j = 1, 2,
. . ., k
where 2 is pattern vector ( 3 , 4 ) of the mixture analyzed and are pattern vectors of its components (see Appendix). The pattern of the unknown mixture is compared with a reference file of a large number of known patterns and the concentrations of the individual components of the mixture are determined by methods of regression analysis ( 1 , 2 , 5 , 6). But it may happen that patterns of some of the components of the mixture analyzed are not present in the reference file. In this case, the conventional methods of powder diffraction phase analysis ( I , 2, 5-12) fail. Difficulties may also arise owing to differences between the real crystal structure of components of the mixture analyzed and the corresponding reference substances. The changes of the diffraction patterns produced by real structure effects can contribute significant error to the analyses. This report describes a new method which surmounts the limitations and difficulties of the conventional methods of the powder diffraction phase analysis. In the proposed method, no reference file of powder diffraction data is used. The method is briefly the following. Starting from the sample given, several secondary samples are prepared (by sifting, magnetic separation, flotation in various liquids, etc.), in which exist a number of components (in all samples the same components) whose relative concentrations change from one sample to the next. On the powder diffraction patterns of the
gl, g2, . . ., f k
0003-2700/80/0352-1300$01 .OO/O
individual samples, the line intensities for a particular component will rise and fall together as the sample composition is changed. This allows one to reconstruct the powder diffraction patterns of the individual components by methods of correlation analysis and to determine quantitatively the composition of the mixture analyzed. The powder diffraction patterns of the individual components reconstructed by the proposed method correspond to the actual real crystal structure of the sample given. This reduces the errors due to unknown real crystal structure that appear in the conventional methods of the quantitative powder diffraction phase analysis.
METHOD Iterative Procedure. Starting from the analyzed mixture i several different mixtures (gl,i 2.,. ., 2,) are prepared (e.g., by sifting or flotation in various liquids). When the intensity data of the diffraction patterns of all these related mixtures are adjusted to the same level (e.g., by an internal standard) then it holds that k
x c i j g j + zi;i = 1, 2, . . ., p
iii =
(2)
j= 1
where cLIis the volume fraction of the component gI in the sample (mixture) 2, (i = 1, 2, . . ., p ; j = 1, 2 , . . ., k ) , so that k
CCi, = 1;i ]=1
.
= 1, 2 , . . ., p
(3)
The residuals (errors) 7,, i = 1, 2, . . ., p appear in consequence of the deviations from an ideal powder specimen and of the instrumental aberrations. In the n-dimensional Euclidean model ( I ) , the diffraction pattern of the ith mixture takes the form i ,= ( x L 1x, L 2 ,, . ., x,,,), the diffraction pattern of the j t h component is SI = Cyll,y l z , . . ., y,,,), and the ith residual is 7, = (ql,t12, . . ., tin), so that the Equation 2 may be written as a matrix equation
Y kn
(4) J
The problem of the powder diffraction phase analysis now reads as follows: find the numbers yll, . . ., yk,, (identification) Q 1980 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 52, NO. 8 , JULY 1980
1301
and cll, . . ., Cpk (quantitative powder diffraction phase analysis) that minimize k
P
P
k
n
where
A = Yii/YlJ subject to the constraints ci, 2 0; i = 1, 2 , . . . , p ; j = 1, 2 , .
. ., k yj1 2 0; j = 1, 2 , . . ., k ; 1 = 1, 2 , . . ., n
does not depend on r. In other words
(6) (7)
The solution of this problem is obtainable by iteration for t = 1, 2, 3, . . .: (a) For given cj;) (i = 1, 2, . . ., p; j = 1, 2, . . ., k ) find nonnegative numbers yji = y$) (J' = 1, 2, . , ., k ; 1 = I, 2, . . ., n) that minimize
Le., the column vectors
(b) Using the intermediate result y$' (j = 1, 2, . . ., k ; 1 = 1, 2, . . ., n) find nonnegative numbers cij = c$+l) (i = 1, 2, . . ., p ; j = 1, 2, . . ., k ) that minimize
and
P
k
n
C C(xii -
i=ll=l
CCijy$))*
(10)
;=l
subject to the constraints k
X c i ; = 1; i = 1, 2, . . ., p j=l
(11)
This procedure can be repeated until the differences
fall below some chosen limit. The optimization problems in the individual steps of the iteration process can be solved by standard methods for minimization of quadratic functions subject to linear constraints (13-15). Clearly the rate of convergence of the iterative algorithm depends on the starting (i = 1, 2, . . .,p ; j = 1, 2, . . ., k ) . In fact the crucial point part of the method proposed is a procedure providing sufficiently good first approximations c!,?). Initial Approximation. The initial approximations c:,? (i = 1, 2, . . .,p ; j = 1 , 2 , . . ., k ) will be calculated using isolated lines or almost isolated lines, Le., diffraction lines contributed (mostly) by a single component of the mixture analyzed. Consider the intensity data matrix
ct)
Xi1
[x]= [ i : "pi
!: !: I::
Xin
Xpz
xpn
Xi,
X13
Xp3
' ' '
' ' '
c,lyl,;
r = 1, 2,
. . ., p
For j = 1, 2,
. . ., k , let us denote
dx?;+
..
*
+ x;j
so that
cII = e,,b,; i = 1, 2,
. . ., p ; j
= 1, 2,
. . ., k
(26)
(14)
Substitution of clJinto Equation 3 yields the following system of p equations in k unknowns bl, b p , . . ., b k :
(15)
Cellbl = 1; i = 1, 2, . . ., p
and
k
xrJ =
so that
of the matrix [XIthat correspond to the ith and jth diffraction line will be parallel. Applying methods of cluster analysis (16, 17), we can classify the n column vectors of the matrix [XI according to their orientation into several groups (clusters) of mutually parallel or almost parallel vectors. Groups containing at least two vectors correspond to isolated lines of the particular components. Now suppose without detriment of' universality that the first k lines on the diffraction patterns of our mixtures are isolated lines of the individual ( k ) components, i.e. x r j = cr;yjj;r = 1, 2, . . ., p ; j = 1, 2, . . ., k (21)
(13)
where xlJis intensity value of the j t h line on the diffraction pattern of the ith mixture (i = 1, 2, . . ., p ; j = 1, 2, . . ., n ) . When the diffraction lines i and j are contributed by a single component, say 91, then the intensity values of these lines will rise and fall simultaneously on the diffraction patterns of individual mixtures. Referring to Equation 4 and neglecting instrumental aberrations and real crystal structure effects, we can write x,, =
Plil
c,lyl,;
r = 1, 2,
. . ., p
j=l
The least squares estimates b1('), bz('), .
. ,
(27) bk(l) for the
1302
ANALYTICAL CHEMISTRY, VOL. 52, NO. 8, JULY 1980
quantities b,, b2, . . ., b k are obtained by solving the system of normal equations P
k P
1=1
i=ll=l
x e L j= CCel,eljbi;j = 1, 2, . . ., k
(28)
Thus from Equation 26 we find the initial approximations of the values of ciJ
c(L) 11 = eilbjl);i = 1, 2 ,
. . ., p ; j
= 1, 2 ,
. . ,, h
(29)
N u m b e r of Components. The number of components of the mixture analyzed ( k ) is determined from the rank of the matrix
[A] =
[
( Z l Z 1 KSz) ’ . . st,) [X][X]’ = ( 7 7 1 ) .)27!( ”.
(751
(30)
‘-f
(%J>Zl) (xp,xz) . . . Gp3p)
then the sample corresponding to g p contains a new (or modified) substance, absent in the first (p - 1)samples, and, therefore, this sample should not be employed in the analysis. Significant statistical error in measurement of the diffraction spectra increases the rank of the set of vectors a,, fiz, . . ., 5ip, this effect being equivalent to introducing a new (or modified) substance. The described matrix rank test allows one to recognize the lower quality of such patterns and to exclude them from the analysis. In the case of complex mixtures, it may happen that some subsystems
(18). In fact
rank[A] = rank[X] Ik and it can be shown that rank[X]
(31) of the analyzed mixture
(33)
i-k+l}
(21, 22).
Other statistical criteria for the determination of the rank of high order matrices have been developed and published elsewhere (23). DISCUSSION Conditions which may lead to incorrect results are: (1)the real crystal structure of some of the components of the original mixture are changed when preparing particular secondary samples; or (2) new substances arise in the samples as a result of inconvenient preparation techniques. These conditions can be avoided by careful attention to the number of components in various groups of the samples. It has been shown that the number of components can be determined from the rank of the matrix [A] (Equation 30). Let 2, be the pattern vector of the original mixture of k components and 5i2, 5i3, . . ., 2, ( p > K ) be the pattern vectors of ( p - 1) secondary samples prepared from the original mixture. If the rank of the matrix
of the components samples
gl, Y z , . . ., 3 k
remain in the individual
k ai
= xcijgj;; = 1, 2 , j= 1
. . ., p
(36)
invariable, so that we obtain instead of Equation 2 the relations 1
2i =
Cih?h h=l
f ?i;
i = 1, 2 ,
a
,, p
(37)
. . ., k
(38)
where C.,= LJ
1
CihDhj; h=l
i = 1, 2,
. . ., p ; j
= 1, 2,
One may expect that the subsystems q h , h = 1 , 2 , . . ., 1 will be simpler than the original system ? (some of the values &, being equal to zero). In this case, the procedure described does not provide directly the composition of the sample 2 but reduces the problem of the phase analysis of this complex mixture to, an- easier problem of analyzing 1 less complex mixtures Y1, Yz, . . ., YL.
SAMPLE PROBLEM Starting from a given sample, five secondary samples were derived (a model system simulated by means of a computer), the powder diffraction patterns of which are given in Table I. The intensity data J of all samples are adjusted to the same reference level. Following (I),we shall assign vectors g1,a,, , . ,, jz5 to the powder diffraction patterns of the individual samples. It is (fil,?il) = 57958.8; (a,,?,)= 63881.6; (a,,&) = 56873.9; (%&) = 49176.2; (21,aS)= 40928.9; (a&) = 75545.9; (2z,2,)= 66844.1; (%,,a,)= 53046.3; (a,,%,)= 38726.3; (%3,?3) = 61355.3; (a&) = 49985.5; ( 3 3 3 5 ) = 38081.3; (%4,24) = 45258.6; (a,,?,)= 39959.7; (a&,) = 41257.5, so that the matrix
1
(ZlFl) (st,,%) . . . (zl,zp) (%%) (%,%) . . ( L Z p ) ’
. . .
.
.
. . .
.
has eigenvalues 260191.9; 18645.4; 2532.0; 3.7; 3.1. Let the
ANALYTICAL CHEMISTRY, VOL. 52, NO. 8, JlJLY 1980
Table 111. Powder Diffraction Patterns of the Components ( First Approximations)
Table I. Diffraction Data
no. of line d , A 1
2 3
4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19
20
2.38 2.37 2.25 2.20 2.17 2.11 2.06 2.02 1.96 1.87 1.85 1.80 1.78 1.76 1.74 1.68 1.66 1.64 1.61 1.58
J no. of sample
J
component
1
2
3
4
5
32.0 22.4 38.2 12.1 22.7 79.6 39.2 200.4 42.1 41.7 39.8 20.4
21.4 46.1 23.8 7.8 44.8 48.4 26.2 246.4 27.3 50.2 26.0 40.3 23.2
5.7 51.3 40.0 2.1 52.0 70.0 7 .O 208.9 23.6 41.8 20.1 44.9 24.8 2.2 7 .O 6.7 12.6
12.2 28.8 60 .O 4.6 30.4 103.6 15.0 154.3 38.9 30.1 32.7 26.2 15.1 4.8 9.3 7.3 17.6 2.2 8.4 4.2
19.3 8.5 77.8 7.3 8.6 136.1 24.6 97.4 53.1 20.4 43.5 7.7 4.4 7.6
11.8 13.1 3.8
13.5 7.3 5.9 6.3 11.5
8.8
2.4 12.2 4.3 4.1 12.5 7.7
1.0
13.9 1.9
11.8
7.6 22.2 3.5 2.4 6.7
Table 11. First Ap roximations of the Concentrations cij
8)
no. of line
p)
d, A
2.38 2.37 2.25 2.20 2.17 2.11 2.06 2.02 1.96 1.87 1.85 1.80 1.78 1.76 1.74 1.68 1.66 1.64 1.61 1.58
1
2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20
54.0074 0.1634 29.7471 20.2103 0.0671 75.8862 66.2580 196.1290 53.9481 41.9933 54.3545 1.3889 1.6615 21.7816 0.3623 19.3966 0.8165 9.9624 0.4549 19.0979
T2(U 0
89.5565 0 0
90.3746 0 0
330.3375 0 65.0027 0
78.3114 4.3.6931 0
0.0157 8.4316 0 0
24.3917
0 0 121.1036 0 0 197.9180 0.5384 0 60.2701 0 45.1838 0 0 0 21.4560 0 39.9324 0 0
0
0
Table IV. Second Approximations of the Concentrations c i j ( 2 )
j 1
1
2
3
1
0.587 0.399 0.104 0.2 29 0.359
0.247 0.495 0.574 0.339 0.098
0.166 0.106 0.322 0.432 0.543
2 3 4 5
Since rank [A] = 3 (2532.0> 1038.5 > 3.7), we infer that there are 3 components (say gl, g2,and y3) in the sample analyzed ( 2 ) . Thus *, 5
1
1
2
3
1
0.590 0.399 0.106 0.229 0.353
0.249 0.508 0.574 0.331 0.089
0.161 0.093 0.320 0.440 0.558
2 3 4 5
estimated error of the intensity measurements be 1%. Then the standard deviation of an element of the matrix [A] is given by 0025 5 S = -E E (Zi,Zj) = 1038.5 25 i = y = 1
(41)
Now, applying the described iterative procedure, we determine the powder diffraction patterns of the components f l , gz,y3 and their concentrations cLI (i = 1, 2, . . ., 5; j = 1, 2, 3) in the , i, is). , individual mixtures (gl,i,g3, The correlation analysis of the matrix of measured intensity data yields the following optimized choice of three unoverlapped diffraction lines
Table V.
Residuals Ri
-100
i 1
2 3 4 5
Ri
240.7 274.8 247.7 212.7 203.1
4.2 1.7 3.2 1.8
2.4
IZi I
1.7 0.6 1.3 0.8 1.2
with the patterns of the file of the Joint Committee on Powder Diffraction Standards (JCPDS), the components 1, 2, and 3 can be readily identified with Fe3C (JCPDS-6-688),see also ( 2 4 ) , Cr2& (JCPDS-9-122), and Fe3W3C (JCPDS-3-980). $I, and gi'' into Equation 10, we obtain the Substituting second approximations of the concentrations c:;) (i = 1, 2, . . ., 5; j = 1, 2, 3) (see Table IV). The residuals
26.2 7.7 which we shall assign to the components gl, g2,and y3 From this, we shall calculate the initial approximations cj!) (i = 1, 2 , . . ,,5; j = 1, 2,3) of these components in mixtures 1 , 2 , . . ., 5 (see Table 11). Substituting the values of c$) into Equations 9 we obtain the first approximations of the powder diffraction patterns of the components 96') (see Table 111). By and 96'' comparing the powder diffraction patterns $I),
si1),
1303
interplanar spacing intensity [ll]
[A] dl, d z , . . ., d n 11,
12, . ., I n
1304
Anal. Chem. 1980, 52, 1304-1308
the j t h of which being equal to Iiwhen 0.99 + j 0.01 5 d, < 1 + j 0.01; i = 1, 2, . . ., n; j = 1, 2, . . ., 900. Thus, e.g., the diffraction pattern interplanar spacing intensity[l]
[AI
I 2.115 1.972 1.224 I
80
100
60
LITERATURE CITED
corresponds to a string of 900 numbers, the 23rd of which being equal to 60, the 98th of which being equal to 80, the 112th of which being equal to 100 and all others being equal to zero: [0, 0 , 0, . . ., 0, 60, 0 , . . ., 0 , 8 0 , 0 , . . ., 0 , 100, 1. 2. 3.
22. 23. 24.
97. 98. 99.
0,
...)
113.
111.112.
0,
0,
01
898. 899. 900.
In this way, a one-to-one correspondence is defined between the set of all diffraction patterns and the class M of all ntuplets of nonnegative numbers ( n = 900). Define further addition, scalar multiplication, and inner product in the class M as follows:
fi + 5 = (hl, hz, . ', h900) *
+ (g1, gz, . . *, g900) = + g1, h2 + g2, ' * .?h900 + g900) ., hgO0)= ( a h l , a h 2 , . . ., ahgo0); (hl
+
ah = a ( h l , hZ,. .
diffraction phase analysis (Equation l),the overlapping of the diffraction lines of individual components of the mixture analyzed corresponds simply to the summation of the corresponding coordinates of their pattern vectors according to the rule of vector addition.
6,5) = ((h, hz, . . ., h g d , (gl, g2, . . ., g
d ) = hlgl + hzgz + ' * + h900g900
Then M is a euclidean 900-dimensional space. Of course, the choice of the number and distribution of the coordinate hyperplanes of this space ( 3 ) can be suitably adjusted to the individual problem in question. Another, more general model was formulated in (2),where powder diffraction patterns were represented as vectors in a real Hilbert space (infinite-dimensional vector space). It is an important benefit of the vector representation of powder diffraction patterns that it removes problems connected with line overlap. In the vector model of powder
J. Fiala, J . Phys. D, 5 , 1874-76 (1972). J. Fiala, J . Appl. Crysta//ogr.,9, 429-32 (1976). P. C. Jurs and T. L. Isenhour, Chemical Applications of Pattern Recognition", John Wiley & Sons, New York, 1975, Chapter 2. P. C. Jurs, Anal. Chem., 43, 1812-15 (1971). J. Fiala, Hutn. Llsty, 32, 435-37 (1977). E. M. Burova, N. P. Zidkov. A. G. Zilberman, V. V. Zubenko, N. S. Nabutovsklj, M. M. Umanskij, and B. M. Scedrin, Krlstallogr., 2 2 , 1182-90 (1977). G. G.Johnson, "Fortran IV Programs (Version XII) for the Identification of Muhiphase Powder Diffraction Patterns", Joint Committee on Powder Diffraction Standards, Philadelphia, Pa., 1970. R. W. Schliephake, News Jahrb. Mineral., 112, 302-19 (1970). L. K. Frevel, Anal. Chem., 37, 471-82 (1965). G.G. Johnson and V. Vand, Ind. Eng. Chem., 59, 19-31 (1967). L. K. Frevel, C. E. Adams, and L. R. Ruhberg, J . Appl. Ctysta//ogr.,9, 199-204 (1976). E. H. O'Connor and F. Bagliani, J . Appl. Crystalbgr., 9, 419-23 (1976). P. Whittle, "Optimization Under Constraints", J. Wiley, New York. 1971. S. L. S. Jacoby, J. S. Kowalik, and J. T. Pizzo. "Iterative Methods for Nonlinear Optimization Problems", Prentice-Hall, Englewocd Cliffs, N.J., 1972. J. L. Kuester and J. H. Mize, "Optlmization Techniques with Fortran", McGraw-Hill Book Company, New York, 1973. R. C. Tryon, and D. E. Bailey, Cluster Analysis", McGraw-Hill Book Company, New York, 1970. J. A. Hartigan, "Clustering Algorithms", John Wiley & Sons, New York, 1975. G. LT Ritter, S. R. Lowry, T. L. Isenhour, and C. L. Wilkins, Anal. Chem., 48, 591-95 (1976). R. M. Wallace and S. M. Katz, J . Phys. Chem., 66, 3890-92 (1964). H. Margenau and G. M. Murphy, "The Mathematics of Physics and Chemistry", Van Nostrand, Princeton, N.J., 1956. R. W. Rozett and E. M. Petersen, Anal. Chem., 47, 1301-08 (1975). J. J. Kankare, Anal. Chem., 42, 1322-26 (1970). D. L. Duewer, B. R. Kowalski, and J. L. Fasching, Anal. Chem., 48, 2002-20 11976); J. Fiala. Cesk. Cas. Fys. A , 20, 1-4 (1970).
RECEIVED for review April 12, 1979. Resubmitted February 7, 1980. Accepted March 10, 1980.
Flow-Rate Independent Component of the Steady-State Current in Tubular Electrodes P. Lawrence Meschi and Dennis C. Johnson* Department of Chemistty, Iowa State University, Ames,
Iowa 500 1 1
The limiting steady-state response of tubular electrodes is predicted by the equation I , = nFAK, Vl"Cb, where a is Practical electrodes frequently respond according to the equation but with slight deviations of a from I/,. ine ear leastmSquares fits of of vs. v 1 ~ / 3 have a intercept when a # 'I3.The Intercept has been interpreted as resulting from the contribution of "end diffusion" to the total flux in the electrodes even though both positive and negative intercepts have been observed. The use of this evidence for end diffusion is brought into question.
I,, = nFAK,VfaCb
In Equation 1: Cb is the bulk concentration of analyte (mol Cm-3), vf is the fluid flow rate (cm3 S-'), cy is a constant dependent on electrode design and the nature of the fluid dynamics, Kl is the limiting mass transfer coefficient (cm'-& sr') which is a function of electrode design, A is the electrode area (cm2), and and have their usual electrochemical signif.under conditions of fluid cance. For dynamics, (y is predicted to be 1,3 and the product AK, is given by Equation 2 (1-4)
AK1 = 5.43D2/3X213 The limiting steady-state response, I,,, of flow-through electrodes is described by Equation 1. 0003-2700/80/0352-1304$01 .OO/O
(1)
(2)
where X is the length of the tubular electrode (cm) and D is the diffusion coefficient (cm2s-l). The validity of Equations @ 1980 American Chemical Society
