Quantitative Phase Analysis by X-Ray Diffraction R. F. KARLAK and D. S. BURNETT’ lockheed Missiles and Space Co., 3251 Hanover St., Palo Alto, Calif.
b The general equation and the associated method for quantitative analyses of mixed crystalline phases has beenderived from fundamental considerations. The results are concisely embodied in two mathematical expressions involving ordinary relative intensities, constants characteristic of pairs of substances, quantities determined from the pattern of the unknown, and the individual weight fractions. The equations apply to the analysis of mixtures of finely divided crystalline substances with or without the use of an internal standard, and regardless of the degree of reflection overlap. Three mixtures containing a-ferric oxide, nickel oxide, and ferrous ferrite were analyzed b y the new method, without an internal standard. All of the determined weight fractions were within slightly greater than one standard deviation of the actual value, as calculated on the basis of counting statistics.
M
RECEKT THEORIES of x-ray diffraction analysis of randomly oriented polycrystalline mixtures have evolved from the original work of Alexander and Klug (1). This method eliminated the need for calculating total absorption coefficients for the unknown and removed small errors probably arising from microabsorption effects. The results of this development and others will be reviewed briefly. ,411 mathematical expressions are given as in the original works, except for minor modifications in the notation to render them consistent with the present development. OST
PREVIOUS THEORY
Alexander and Klug have shown that the analytical procedure is simplified, and the results are more accurate if integrated line intensity ratios are employed from the same pattern of the unknown. The ratio of isolated reflections from two components is a linear function of the weight fraction ratio, i.e., Itj/I
kl=
(K*,pl/K~~P~)(zi/zi) = Ctl,ki(Z,/2i)
=
RGk (1)
I,, is the integrated intensity of the ith 1 Present address, College of Mechanical Engineering of the University of California, Berkeley.
reflection from the j t h component, and similarly for I k l , while REkis the ratio of these intensities as measured from the mixture. The p’s are the bulk densities; the z’s are the weight fractions. K,,and Kkl are constants characteristic, respectively, of component j and its ith reflection, and component 1 and its kth reflection. The constant C , , , k i has been introduced to indicate that no separation of the factors in the constants was suggested in the original presentation. Equation 1 is fundamental to the internal-standard method. In this case, adding a known quantity z1of substance 1 and determining the resulting intensity ratio permits a rapid determination of x, for any j once the appropriate constant CE,.kzis obtained from a calibration standard. Copeland and Bragg ( 4 ) ,extended the above method to account for the partial or complete superposition of two reflections (i = i , k ) and arrived a t the form R11
= (It,
+
Iki)/Iii
n li/In+I
=
aijzj
j=1
+
~ ( , n + l ,
i = 1, 2,
. . .,n
(3)
or, using the explicit notation as before, n Rik
= j = 1 A < j , ( k , n + i F j , [ n + i ) -k
B(&% + 1 ) ( k , n + 1) i = 1 , 2 , . . . ,k - l , k + l ,
, , . ,n + l
where k has been introduced for generality since the choice of subscript for the base reflection need not agree with that of the base component. Equations 3 represent a set of i = n equations which can be solved simultaneously to obtain the z,, the ratios of the weight fractions of the unknowns to that of the internal standard. The former are then obtained immediately since the latter is known. The n(n 1) constants appearing in Equations 3a must be determined experimentally from standards of known composition. They are of the same form as those in the previous equations. More recently, Beajak (3) considered multiphased systems and derived an expression relating the intensity ratios of isolated reflections from each of the n components present in a sample to the individual weight fractions without the use of an internal standard. The results are summarized in the following equation:
+
=
a,(Ki,pi/Kiip,)(z,/zi) 4- b k AL,,l1(2,/zJ
the standard, the following form was obtained.
+
=
B&l,ll
(2)
where R,l is used since I,, and I k l are treated as a single reflection-i.e., i = k. The notation here is the same as used in Equation 1 with the exception of a, which will be shown to be the ratio of the relative intensities of the ith reflection of component j, to the first reflection of component 1 and the constant b k , which can be shown to be the ratio of the ith relative intensity to the first for component 1. As in Equation 1, all constants but b k are combined into a single coefficient As,,ll. It should be noted that although the individual K’s depend on the geometry of the apparatus, this factor cancels in the ratios thereby leaving the constants on the right of Equation 2 as functions of specific lines of the components only. These authors also considered the case of composite intensities with contributions from each of n phases present, and derived an expression for calculating the weight fractions by adding an internal standard which contributes to no other intensity. By dividing selected intensities by the intensity of the selected line from the standard, designated by subscript i = n 1, and dividing the terms involving the weight fractions by the corresponding term for
+
for j
=
1, 2, . , ,,s,
. , ., n (4)
Explicitly,
j = 1 , 2 , . . . , s , . . . ,n (4a) The constant 1 has been put in the summation by including m = s. The notation in Equations 4a is the same as used previously and the summation index has been changed from i to m for clarity in comparing Equations 4a with subsequent equat.ions. The subscript order has been reversed from the original order for consistency-i.e., here, VOL 38, NO. 12, NOVEMBER 1966
1741
the first subscript refers to the numerator of a ratio and the second to the denominator. The A’s appearing in Equations 4a have four subscripts to indicate that they are not characteristic of the components alone but arise from the arbitrarily selected sth and mth reflection of the sth and mth component, respectively. A single subscript is associated with each Z since there is no overlap and the designation for the reflections and the components agree (i j). Although the foregoing provide means of analyzing the composition of mixtures under specific conditions of superposition and use of an internal standard, none is completely general, that is, none includes all possible cases. GENERALIZED THEORY
In certain processes, such as reactionsintering at elevated temperatures, it may not be possible to analyze the rate of product formation with an internal standard; also overlapping reflections may exist. None of the previous methods are then applicable. Such a situation requires an approach accounting for absorption effects, as previously, and for contributions from all possible components. It also would be convenient to employ accurate, previously tabulated data such as relative intensities or constants characteristic of each component, or pairs of components, only. By starting with the expression, from Alexander and Klug, which was the basis for the previous theories, it is possible to derive a unifying equation with general applicability and fulfilling the requirements stated in the preceding paragraph. For generality, each intensity is assumed to contain a reflection from each component. The contribution from component j to the arbitrarily numbered ith intensity of the diffraction pattern from the mixture, can be written -?$I
=
aZ]-?OJ
(5)
where a,, is the relative intensity and I , , is the intensity of the major reflection, and can be expressed as
I,,
= KoJX,/UI*PI
(6)
This is the basic expression derived by Alexander and Klug which led to Equation 1. The factor K O ,depends on the component j and geometrical characteristics of the apparatus, x , is the weight fraction of component j, ut* is the mass absorption coefficient of the mixture, and p , the bulk density of component j . Combining Equations 5 and 6 and summing to include contributions to the ith intensity from each of the n components leads to
1742
ANALYTICAL CHEMISTRY
where C,, = Kol/u,*p,,and I , is the intensity of the composite ith reflection from the mixture of n components. Multiplying Equation 7 by C,/C, letting C,,/C,, = C,,, and x l / x n = x j n results in n
1%=
C o n ~ n
(8)
abiCjnxjn
3=1
The generalized form is obtained by dividing Equation 8 by a similar expression for the kth intensity, defining R L k = I c / z k as before, and by combining terms involving like weight fraction ratios to produce the following for each of the i equations:
i = l , 2 , . . . ,k - l , k + l ,
. . . , n (9)
When j = n, C,, = z,, = 1, from the definition of these quantities, and the resulting term is a constant which can be transposed to the left. The expanded form of Equations 9 is
- U k 1 R l k ) X l n -k C,,(a1, - U & i k ) X 2 n . . -k Cn - I , n ( U l . n - 1 - a k , n - 1 R l k ) X n - 1 , n
aknR1.k
0f
- aln
aknRkk Cin(aki
=
Cln(U11
- Ukn -
akiRkk)xin
f
+
.
+ +
Cn--l,n(an,.-1
-
a k , ~ - & ) ~ ~ - i , ”
(9%)
The n equations in Equation 8 are reduced in number by one when the ratio of i to k is formed, hence the kth equation (Equations 9a) is identically zero. Subscript k is used instead of n in Equations 9 since the ratios might possibly be formed more conveniently with an intensity that is not the last in serial order on the diffraction pattern. Since there is no natural ordering of the components, any may be assigned the terminal subscript n. As in the previous developments, the C’s involve ratios of the K’s and the densities. However, in this case, each refers to the major lines of the components j and n. Because the geometric dependence of the K’s appears as a multiplicative factor, it cancels in the ratios and thus the C’s can be regarded as dependent on pairs of components only. Cjn
=
K o i ~ n / K o n ~= t
constant dependent on j and n only (10)
+
A very simple expression results from substituting into Equation 12 the solutions to Equations 9 expressed as
D,/D, j = 1, 2, . . ., (n - 1) (13) where D is the determinant of the coefficients, [C,,(uL9[ where the i’s refer to rows and the j’s to columns, and D , is the determinant resulting from the replacement of the j t h column of D by the column of constants appearing on the left side of Equations 9a. Finally, by substitution of x,,, = x , / x , we obtain 2,. =
- CCkkRkk)Xkn . Cn-I,n(ak,n-l - Uk,n-LR*k)Xn-l,n U k n R n k - a n n = C l n ( a n 1 - U k 1 R n k ) X l n -I-
+
CJr = CJn/CLn = 1/ci] (11) This is permissible because the C’s depend on the major intensity of the components only and not on any of the various reflections from them. The individual 2,’s are calculated from Equations 9a by applying Cramer’s rule and by using the fact that the sum of the weight fractions is unity-Le.,
+
=
Ckn(Ukk
(n - 1)) - C,,,’s. If the relative intensities are available from previous measurements, ASTM file cards, or if they may be calculated from fundamental considerations, the equations may be solved for the weight fractions after determining the CJn’s with data obtained from the diffraction pattern of a single standard specimen containing all of the components in known amounts. Since Equations 9 are symmetrical with respect to the C’s and the x ’ s , the method of solution is identical in each case, as will be shown. Knowledge of C,, and C I , permits the calculation of C , Iand C L as , follows.
There are n2 n - 1 constants in Equations 9 and 9a: n2 - a,,’s, with both j and i ranging from 1 to n, and
x j
=
Di
n-l
,
j
=
1,2,
...,n
D+ C D j j=1
(14) Thus, computation required for an analysis is reduced to three steps: calculating the coefficients of the x j , in Equations 9; evaluating the determinants; and evaluating the x i s by inserting the appropriate D’s into Equations 14. AGREEMENT OF THEORIES
Since Equations 9 were derived for the most general case, they should reduce to the equations of the previous theories when the same restrictions are applied. When there are two significant components-one unknown j and one standard, n = 1 in Equations 9-and there is no overlap, athention is focused on two intensities (i = i and i = k ) . Letting the ith intensity correspond to the component j and the kth intensity to
__
~~
~~
Table 1.
~~
Relative Intensity Notation
~
for Reducing Equations 9 to Equations 3 n
. . .
. . . k k + l
0
ak
+
ak
component I , and expressing Equations 9 for this case, lead to Cji(aij
- akjRik)Zji f Cii(aii
-
(15)
aeiRik)Zii
But a k j = 0 and ail = 0 because component j has no line a t position i = k and component I has no reflection at i = i . Also (711 = Z ~ = I 1, by definition. With these conditions, Equation 15 reduces to (16)
Rik = C j i ( a i j / a k r ) Z j i
which is the compact form of Equation 1. Using the definition of x j l and C j l as well as t'hat of C,j from Equation 7 in Equation 16 results in Rik
=
(17)
( a i j K , j p i / a k i K , I P j ) (Zj/ZI)
which is Equation 1 with Ki, = airKO,, and K k i = aklK,1, as was to be shown. The method of Copeland and Bragg ( 4 ) expressed in Equation 2 can be shown to exist in Equations 9 by considering a superposition of reflections from components j = j and j = n = 1 at. position i = i, (aii # 0 # air), and a single reflection from component I , the standard, at i = k = 1, ( a l j = 0 # all). With these substitutions, Equations 9 reduce to
0=
Cji(ai,)zji
+
Cii(aii
- allRil)xll (1%
Which, by noting that C I I = = 1, as before, and expanding the C's and the Z'S, can be brought to the form
Ril
+
1 al I
= - [aij(K~jpi/K~zpj)(xj/xi)
ail]
(19)
This is Equation 2, exactly, with the coefficient and constant term expressed in t.erms of relative intensities or constant characteristic of components only-Le. , A ij,li
0
. . .
+ 282
ak
ak,n
+ 1.n
0
-
m=l I
which is the required equation with I,,,/Ia = R,,, as previously defined, and A,,,,, = Cam(aae/amm), each with m = 1,2, . . . ,j.,s It has been demonstrated that the present development accounts for all the previously considered special cases. Furthermore, the adopted approach reduces the number of calibration samples required when reliable relative intensities are available. When all the pertinent u,,'s are known, only a single standard sample of known composition need be analyzed to evaluate the C's. This is then accomplished by solving the (n - 1) equations in Equation 9, using the procedure implicit in Equations 13. In this case, the determinants, D' and D,',are evaluated with known weight fractions in the coefficients instead of CIS and the left side of Equations 13 becomes C,,-i.e., C,, = D,'/D'.
an + 1-1
n-i-1
0=
0 0
azn
. . .
0
111
n+l
aln
= (aij/air)(K.jP,/Kolpj)
and
+
each of these. Substance j = n 1 is assumed to contribute a single reflection located a t i = k , and each of the remaining substances cont'ributes to each of the i # k positions. The pertinent relative intensities are shown in Table I. Substitution of the appropriate relative inbensities into ith equation of Equations 9 and recalling that C ~ = I X ~= I 1 for j = 1 results in 0
=
C1,n+1(ail)Z1,n+1
C2.n+l(ai2)r2.,+1
aii/aii
The more complex Equations 3 are contained in Equations 9 as can be shown by considering the ith equation from
+ ... +
which, when solved for Rik
ai.n.1
+
ak,n
+1Rik
(20)
Rik
yields
+
GENERAL DISCUSSION
= Ci,n+~(ai~/ak,n+~)X~,.+i
C2,, + l(aia/aken+ 1)22,, + I ai,,, + l / a k , , + I
+ ... +
=
(21)
This is exactly the ith equation of Equations 3a with the coefficients of the weight fraction ratios and the constant expressed in terms of the fundamental relative intensities and component ratios -i.e., A,,.irc,n+~)= C , , n + l a r i / a k , n + l for j = 1,2,. . . , n , a n d B ( r . n + l ) ( k , n + l ) - a,.* + 1 / a k q n + I . Also Equation 20 reduces to Equation 19 for the twocomponent case. The general expression, Equations 9, also reduces to the form presented by Berjak, Equations 4, as a special case involving no complex peaks and no internal standard. With these conditions, each selected diffraction peak of the unknown corresponds to only one component-Le., i = j, so that a,, = 0 for i # j. Also n, k = s and the determinant of Equations 9 is diagonal. These conditions when applied to Equations 9 and substituted into Equations 14 for the j t h component, yield 2,
Bii.li =
+
D,
=
D
+
5' D ,
(22)
m=l
Dividing numerator and denominator of Equation 22 by D and rearranging gives
Application of Equations 9 and 14 to a specific analysis involves identification of the possible components, selection of appropriate reflections, calibration, and measurement of integrated intensities from the diffraction pattern of the unknown, calculation of coefficients, the determinants, and finally the concentrations. It is conveneint to tabulate pertinent data using a consistent indexing scheme, particularly when many components are present. Since the degree of overlap will seldom be as complete as assumed for the derivation, many factors in Equation 9 will be zero. Identification of all the components in a sample is important when no internal standard is used. This may be accomplished by examining the diffraction pattern from the unknown in conjunction with a detailed understanding of the chemical reactions involved in the sample formation. When a component that is not present is included in the analysis, no accuracy is lost since the corresponding determinant in Equations 14 will be null. The effect of neglecting a component depends on whether an internal standard is used, providing no reflections from the component occur in the analysis intensities. With no internal standard, an error will occur because Equation 12 will no longer apply and, furthermore, the unknown may affect the measured intensity ratios. In this case, all possible components should be included. The strongest reflection from each component should be selected for
VOL. 38, NO. 12, NOVEMBER 1966
1743
analysis since the accuracy depends on the number of counts obtained for each integrated intensity. This is one criterion that forces the selection of complex diffraction intensities4.e. , containing reflections from more than one component. Designation of the base intensity 11,is completely arbitrary although it is convenient to make the selection such that the ratios (R,k’s) are manageable numbers. Calibration (determination of the constants in Equations 9) may require a t 1) samples if none of the least (n needed relative intensities are known. There would be n samples of the pure substances for ascertaining the a,,’s, and a single mixture for the C I n k The latter would be a well-blended mixture containing all of the components, preferably in amounts to produce nearly equal intensities for the major lines of each. For each component with known relative intensities the number of calibration specimens is reduced by one. Furthermore, some of the CjnJs may be obtainable from previous calibrations and by Equation 11. When this is the case only the remaining substance should be mixed in the standard to improve the statistical accuracy by increasing the number of counts in each integrated intensity measurement. In any case, it is worthwhile to obtain the maximum accuracy for calibration since all subsequent analyses depend on these values. When more than three components are present, the C’s may be determined with greater accuracy from binary mixtures of each of the j # n substances with the nth. If no relative intensities are known, a total of 2n - 1 calibration samples will be required. If an internal standard is to be used and the analysis is made for fewer than all components, only those of interest are included in Equations 9. The standard component is then designated as n, the denominator component in the weight fraction ratios. Because the total absorption coefficient cancels in the derivation of Equations 9, there is no loss of applicability. However, since Equation 12 no longer applies, the solutions are obtained directly from Equations 13 and the definition of x,,-i.e., xln =x,/ xn,where xn is the known weight fraction of the internal standard. The method of specimen preparation for the spectrometric powder technique and the thoretical discussions given by Klug and Alexander (6) apply to this method of analysis.
+
EXPERIMENTAL VERIFICATION
The new method implicit in Equations 9 was tested by analyzing three ternary mixtures containing NiO, aFez 03,and Fe304without an internal standard. The diffraction pattern of Fe304 is very similar to that of NiFezO4. Consequently, the analysis would re1744
ANALYTICAL CHEMISTRY
Table It.
Relative Intensities Contributing to Diffraction Lines Used for Analyzing Mixtures of NiO, aFezO3, and FesO4
Components
Brw
Intensity, i 1
2 = k 3
angle, degree 42 45 55
j = l all a21 a31
-
= 1 = 0.705
0
semble that for determining the progress of the following high-temperature solidstate reaction. NiO
+ Fe203
+ NiFe204
j = 2
aFe203
(24)
Sample Preparation. Submicron reagent grade powders were thoroughly mixed by repeated screening through an 80-mesh sieve (20 times) followed by hand grinding with an agate mortar and pestle for 30 minutes to break up agglomerates of the individual components. The progress of grinding was checked periodically by smearing a small quantity of powder on a microscope slide and examining the surface a t lOOx for evidence of the colored streaks which occur when single-component agglomerates are broken. Grinding was terminated when colored streaks were no longer detected. The uniformity of mixing is very important, particularly for materials with high absorption coefficients, since the volume of diffracting materia1 is extremely small. Precautions should be exercised to ensure the random alignment of powders which might tend to orient in preferred crystallographic directions because of the particle shape. Apparatus. The analysis was made with a G. E. XRD-5 diffractometer, filtered F e K a radiation, an 0.1-degree receiving slit, a 1-degree medium resolution beam slit, and a proportional counter. The specimen was packed into an aperture cut into a microscope slide. The dimensions were such that the entire beam was intercepted by the specimen for the full range of pertinent angles. Experimental Details. The diffraction intensities for the analyses were selected to include the major reflections from each component. These are listed in Table I1 with the relative intensities of their components. The relative intensities were obtained from diffraction patterns of the pure components using scan speeds of 0.2 degree/minute. This provided about 40,000 counts above background for the principal reflections. Calibration and Data Reduction. Calibration with a standard containing nearly equal weight fractions of each component provided the C,, values required for analyzing the unknowns. Primed quantities are used here to designate calibration values. The calculations from Equations 9 are outlined below. For this case, with the subscript scheme shown in Table I,
j = 3
NiO
Fe304
alz = 0 a22 = 0 a32 = 1
aI3 = 0 a13 = 1 aa3 = 0.245
k
= 2 (261 = 45 degrees) and n = 3 (Fea04), Equations 9 become 3
0
=
C,3(Uij
j= 1
- a?,R’iZ)dj3,
i = 1, 2 (25)
which, when expanded, yield a23R’lZ
- ai3 = Z’i3(aii
a2&’3?
-
- aziR‘i?)C13 +
- ~zzR’iz)Cz3 - aZlR’3?)c13 + X ’ Z ~ ( Q Z - a&’dCz3 (26)
X’Z~(~IZ
=
a33
X113(a31
Making use of a12= a13 = aZ2= reduces Equations 26 to
a31
=
0
+ (0)Czs az3R’a2 - aS = - X ‘ I ~ ~ Z I R+ ’~~C~~
a23R‘lZ
=
6’13(~11
- aziR’iJCi3
x ’ z ~ ~ ~ z(27) CZ~ The determinants required in Equations 13 are formed as follows.
- ~ i R ‘ i z )= 0.709
D’ =
~‘i3~’*3~az(~ii
D‘i
=
~ ‘ 2 3 ~ 2 3 ~ 3 z R ’= iz
D’z
= 2’13(aii
+
-
0.420
~ziR’u)(a&’az
X’i3CL?iaz3R’izR13? =
- US)
1.038 (28)
These were evaluated by substitution of the relative intensities from Table I1 and the experimentally measured ratios (R’i? = 0.420, R‘az = 1.204, 2 ’ 1 3 = 1.006, and 2 ’ 2 3 = 1.001). The CInJsare then obtained by taking the ratios as in Equations 13, to give Ci3
=
D’i/D’
=
0.593
Cas = D’z/D’ = 1.465
It is sometimes convenient to evaluate the coefficients in Equations 9 directly because the subsequent evaluation of the determinants is simplified. If the algebra is retained in forming the determinants, it may be possible to simplify the numerical calculations by factoring. In forming the ratio D’l/D’, for example, x’23a23 disappears from numerator and denominator. From the two caliulated Cjn values four others may be evaluated:
c31= Ci3-l
=
1.686,
c3z = Cz3-l Ciz =
C13/C23
=
=
0.683
0.405, C21
=
Ciz-’ = 2.469
Any of the six constants can be applied directly to the analyses of any mixture containing the appropriate subscriptpair components. Sample Analysis and Results. The Cjn’s determined above were used to analyze three ternary mixtures each containing one of the three compounds as the major component (90 weight %) and the remaining two as minor components (5 weight % ’ each). Integrated intensities were measured as previously, by planimeter, from charts made a t 1 inch per minute with a 0.2 degree per minute scan speed to obtain maximum accuracy. The ratios obtained from these measurements were used to evaluate the weight fractions by following a procedure similar to that outlined in Equations 25 through 28. For this calculation, however, the D’s were as in Equations 28 but with the substitution of the calculated Cj3’s for the corresponding X ’ , ~ ’ S . Equations 14 were then used to obtain the x,’s. The analytical results are listed in Table I11 with the observed relative errors and the relative theoretical standard deviations calculated from Equations 9 and 14 on the basis of counting statistics by assuming that, for the large number of events employed, the normally Poisson distribution becomes essentially Gaussian (2) and ( 5 ) . DISCUSSION
The applicability of the method, and its accuracy in the range of compositions studied, is evident. Greater accuracy occurs for the components with the greater weight fraction as would be expected from the statistical nature of the measurements. This situation could be improved by increasing the effective number of counts for those peaks containing reflections from the less abundant components. The only limita-
Table 111.
Sample
Quantitative X-Ray Diffraction Analyses of Test Samples
Component
3
dev.
Exptl. rel. error -10.00
rei.
90.39 5.11 4.50
zk0.46 rt2.63 f9.91
a-Fe203
5.00 90.00 5.00
5.03 90.08 4.89
zt3.47 f0.24 53.27
a-Fe208
5.00 5.00 90.00
4.89 4.83 90.28
rt4.64 15.68 rt0.35
NiO
Fe304
2
Theoretical
90.00 5.00 5.00
a-F&Oa
1
Composition (weight %) True Calcd.
NiO Fe304 NiO
Fe304
tion on the accuracy is the time required to obtain the calibration and analysis measurements. Since the theoretical standard deviation is inversely proportional to the square root of the counts, the time required to obtain a negligible error may be very large. To obtain a measure of the number of counts required for each reflection or peak, to provide a given accuracy, involves a complete statistical analysis wherein the resulting equations for the fractional standard deviations are expressed in terms of constants pertinent to the mixture and actual scaler counts, or effective counts determined from the area of reflections together with settings of the integrating circuitry and chart speeds. These calculations, performed after a preliminary scan of the unknown, provide an accurate means for ascertaining the extent to which each datum contributes to the final error. Hence, any degree of accuracy within the limits imposed by the apparatus, may be subsequently obtained by repeating the most
++ 20.43 .20
++ 00.09 .60 - 2.20 -
2.20
- 3.40
+ 0.31
sensitive measurements with the calculated accuracy. ACKNOWLEDGMENT
The authors are grateful to R. H. Bragg and J. C. Robinson for many helpful discussions during the performance of this work. LITERATURE CITED
(1) Alexandei CHEM.20, E (2) Beers, Y.
197 (4) c
CHEM.30,lYti
(1Y58).
(5) Cullity, B. D., “Elements of X-Ray Diffraction,” p. 202, Addison-Wesley, Reading, Mass., 1956. (6) Klug, H. P., Alexander, L. E., “X-Ray Diffraction Procedures,” Chap. 5, Wiley, New York, 1954.
RECEIVEDfor review April 20, 1966. Accepted August 24, 1966.
VOL 38, NO. 12, NOVEMBER 1966
1745
