are reported herewith. Table I lists a group of metal chelates in which volatilization was incomplete or otherwise unsatisfactory. Table I1 lists a group of metal chelates which are volatile and eluted from the column. Combining the information in Tables I and 11, a sharp separation of Ni(I1) and Cu(I1) from many other elements may be achieved by use of the salicylaldoximates. By the use of thenoyltrifluoroacetonates, sharp separations of many elements may be obtained from Mn(I1) and Co(I1). Solvent Action. Our first publication in this field ( 4 ) indicated that the effects on volatility achieved were caused by solvent action, not by pressure alone, because high pressures of many gases fail to achieve the volatilization obtained with CC12F2. The present study confirms this conclusion. Mn(I1) thenoyltrifluoroacetonate has a higher vapor pressure than the Fe(1II) and Al(II1) chelates ( 2 3 , yet the latter two are (27) E. W. Berg and J. T. Truemper, Anal. Chim. Acta, 32, 245
(1965).
volatilized in CCIZFZand the former is not. The Co(I1) chelate sublimes at 0.25 mm at 690 “C., the Al(II1) chelate at 125 “C,and that of Fe(II1) at 135 “C (27), yet the more volatile Co(I1) chelate is not volatilized in CClzFz while the latter two are. This is indicative of the primary solvent effect of the CClzFzduring volatilization of these compounds under pressure. ACKNOWLEDGMENT
The authors acknowledge the collaboration of Harold Hill, John Baudean, and Stanley Norem of the Perkin-Elmer Corp. in the design and construction of the hyperpressure gas chromatograph. RECEIVED for review November 29, 1967. Accepted June 24, 1968. This work was supported by the National Institutes of Health under Grant No. 2 R01 G M 11159.
Quantitative Analysis with the Aid of Calculated X-Ray Powder Patterns I. Cyrus Jahanbagloo Materials Research Laboratory, The Pennsylvania State University, University Park, Pa. 16802
Tibor Zoltai Department of Geology and Geophysics, University of Minnesota, Minneapolis, Minn.
X-RAYPOWDER DIFFRACTION provides a powerful technique for the analysis of crystalline mixtures, because each component in a mixture produces its characteristic pattern independently of the others; and, moreover, the observed intensity of each component’s pattern is proportional to the amount present provided that an absorption correction is applied. Although these features had been pointed out by Hull (1) as early as 1919, no significant work in quantitative analysis was done until 1936 when Clark and Reynolds ( 2 ) published their scheme for mine-dust analysis. Their scheme and those of others that followed were based on the use of internal standards. In the internal standard method, as described by Klug and Alexander (3), several synthetic mixtures containing known concentrations of phase A (the phase to be analyzed) and a constant concentration of a suitable standard are prepared. Then a calibration curve is established by plotting the observed intensity ratio of a line of phase A and of the standard us. the weight fraction of phase A. The sample to be analyzed is then mixed with the same concentration of the standard and the same intensity ratio is measured and, using the calibration curve, the concentration of phase A in the unknown mixture is determined. The standard should be a substance obtainable in good purity and should produce a strong diffraction line near a strong line of the unknown phase. These lines should not superpose the lines of other phases in the mixture. These conditions can not always be easily fulfilled, thus making this technique less widely applicable. The method is (1) A. W. Hull, J. Am. Chem. SOC.,41,1168 (1919). (2) G. L. Clark and D. H. Reynolds, IND.ENG.CHEM.,ANAL.ED., 8, 36 (1936). (3) H. P. Klug and L. E. Alexander, “X-Ray Diffraction Procedures,” Wiley and Sons, New York, 1954, p 410.
also very time-consuming, especially if the mixture is composed of more than two components, because for each component a new calibration curve should be prepared. In order to overcome the shortcomings of the internal standard method, a new technique is proposed that uses the absolute integrated intensities of the calculated X-ray powder patterns. In this technique, the integrated intensities of the diffraction lines of the mixture are measured on any arbitrary scale. These observed intensities are then divided by the corresponding absolute intensities of the calculated patterns. These ratios, if corrected for the absorption of the mixture, are proportional to the quantities of the components present. In the proposed technique, first an approximate composition is calculated by neglecting the absorption effect of the mixture and then both the composition and the absorption coefficient are refined in consecutive cycles until convergence is obtained. THEORETICAL
The calculation of absolute intensities of the diffraction lines of substances whose structures are well known can easily be achieved by high-speed computers (4-7). In the proposed (4) T. Zoltai and I. C. Jahanbagloo, “Encyclopedia of X-Rays and Gamma-Rays,” Reinhold, New York, 1963, p 814. (5) I. C. Jahanbagloo and T. Zoltai, “Calculated X-Ray Powder Diffraction Patterns,” Department of Geology and Geophysics, University of Minnesota, Minneapolis, Minn., 1966, pp 1-1 14. (6) D. K. Smith, “A FORTRAN Program for Calculating X-Ray Powder Diffraction Patterns,” UCRL 7196, Lawrence Radiation Laboratory, Livermore, Calif., 1963. (7) I. C. Jahanbagloo, V. Vand, and G. G. Johnson, Jr., “Calculated X-Ray Powder Patterns and Their Applications in Quantitative Interpretations,” MRL Monograph 3, Materials Research Laboratory, The Pennsylvania State University, University Park, Pa., 1968, pp 1-78. VOL. 40, NO. 1 1 , SEPTEMBER 1968
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Table I. Sample Chart Showing Sequential Calculations for Mixture NO.1 FIRST CYCLE
Quartz do
4.26 3.34 2.46 2.128 1.817 1,541 1.382 1.375
Total
Fluorite
10 40 5 5 10 5 5 5
Io 6853 35880 2547 1973 4736 3516 2153 2619
85
60277
Io
Zl,/ZI, = 0.001416 Vol = 30.4 Wt = 27.4
1.
do
Calcite do
IO
3.15 1.93 1.647 1,366 1,115 1,051 0,965 0,923 0.863
95 36047 100 42160 40 13070 10 4933 20 8516 5 3331 5 2909 10 4132 15 5776 Total 300 120874 ZI,/ZI, = 0.002482 Vol = 53.5 Wt = 57.8
3.86 3.035 2.495 2.095 1.912 1.875 1,604
Total
3831 27960 3914 4809 5760 6738 3271
42
56283
ZI,/ZI, = o.oO0743 Vol = 16.1 Wt = 14.8
z
z z
z z
Io
Io 2 20 3 5 5 5 2
SECOND CYCLE
w)
abs
10.43 13.32 18.27 21.22 25.07 30.00 33.87 34.07
4.87 4.82 4.72 4.66 4.56 4.43 4.33 4.32
Linear Absorption Coefficient X Radius of Sample = 0.98 IC* e( 0) abs IC* 1407 7444 540 423 1038 794 497 606
Total 12749 ZI,ZI,* = 0,006696 Newvol = 31.4 Newwt = 28.3 do = observed interplanar spacing. I, = observed relative intensity. I, = calculated absolute intensity.
z z
14.15 23.50 27.91 34.35 43.74 47.16 52.95 56.60 63.20
4.80 4.59 4.49 4.31 4.05 3.97 3.81 3.74 3.62
Total ZI,/ZI,* = 0.011047 Newvol = 52.0 Newwt % = 56.4
method, the sample must be a uniform mixture of n-components with a reasonably uniform and fine particle size, small enough to render only negligible extinction and absorption in a single particle. Equation 1 gives the expression for the integrated intensity diffracted by a powder specimen of volume u when radiated by X-rays.
z
7510 9185 2910 1145 2103 839 764 1105 1595 27156
e(
0)
abs
11.50 14.70 17.95 21.55 23.75 24.24 28.67
4.84 4.79 4.73 4.64 4.59 4.58 4.47
IC*
792 5837 827 1036 1255 1471 732
Total 11950 ZIo/ZIc* = 0.003498 New vol = 16.6 Newwt = 15.3
z
0 = Bragg angle. abs = absorption correction. I,* = I,/abs (calculated absolute intensity corrected for absorption effect of mixture).
The expression enclosed in the parentheses of Equation 1
is the absolute integrated intensity of a diffraction line per unit volume and can be calculated if the structure is well known. If this expression is abbreviated to I,, Equation 1 can be reduced to Equation 2 which is the basic relationship underlying the quantitative analysis of mixtures with the aid of calculated intensities. I, I0=K-v A
In this formula:
I,
=
N
= number of unit cells per unit volume
observed integrated intensity of a diffraction line
L and p = Lorentz and polarization factors, respectively, M = multiplicity factor F = structure factor corrected for thermal vibration of constituent atoms A = absorption correction K = a factor which includes the intensity of the incident beam, the charge and mass of electron, the velocity of light, the wavelength of x-radiation, and the geometry of the apparatus. If the experimental conditions are kept constant, this f2,ctor will remain constant and therefore will be referred to as constant of proportionality. 1740
ANALYTICAL CHEMISTRY
If Equation 2 is applied to a diffraction line of component i in the mixture, it will result in Equation 3.
(3) In this equation (Z& can be measured, (I& can be calculated, A is the absorption correction term of the mixture whose value is not known in advance, ut is the volume of component i which is to be determined, and K is a constant whose value is not known. Similar equations can be written for a number of well resolved diffraction lines of all the components. These equations should be self-consistent and simultaneous solutions of them should give the volume percentages of the components. This set of equations can be solved by successive approximations-Le., neglecting the absorption effect at the first stage for obtaining the approximate composition, and then refining
both the composition and the absorption coefficient in consecutive Cycles until convergence iS obtained. If the mixture does not contain substances with great differences in their absorption coefficients, the convergence can be reached after two cycles. If the absorption effect is neglected at the first stage, Equation 3 will be reduced to Equation 4 where K‘ is a new constant of proportionality.
Further simplifications can be made by summing Equation 4 for a number of well resolved diffraction lines of component i. This will result in Equation 5. Z(1o)i = K’0tVZc)t
Table 11. Comparison between Calculated and Prepared ComDosition for Mixture No. 1
(5)
~
K’Z(1c)t
The constant K’ can also be obtained from Equation 7. 100%
(7)
The weight percentages of the components can then be calculated from their volume percentages. Once the approximate composition is obtained, the absorption coefficient of the mixture can be calculated and the corresponding absorption corrections can be applied in subsequent cycles. It should be noted that in diffractometer technique, absorption is independent of the Bragg angle. Therefore, if diffractometer data are used, one cycle of calculations will give the final results. The reliability of the technique is a function of the quality and resolution of the observed patterns. It increases if more lines are considered for each component, and if proper weights are assigned to different reflection types. The size of the crystallites composing the powder and the method of sample preparation are important factors in the reproducibility of the intensity measurements, and consequently in the reliability of the technique. For satisfactory results, it was found that the crystallites should be smaller than 45 microns in diameter. At that size, the effect of preferred orientation is reduced considerably. EXPERIMENTAL VERIFICATION
The authors have used this technique in the analysis of several synthetic mixtures and have obtained encouraging results. In these examples, the calculated intensities have been obtained from the manual of the “Calculated X-ray Powder Patterns” prepared by Jahanbagloo and Zoltai (5). In order to demonstrate the calculations in sequential steps, the analysis of a mixture composed of three components has been shown in Table I. In this example, film technique and Cu-radiation were used. This table is self-explanatory and shows two cycles of calculations. In the first cycle, the approximate composition was calculated by neglecting the absorption effect of the mixture. In the second cycle, the absorption coefficient of the mixture (having the composition obtained from the first cycle) was calculated. Then the corresponding absorption correction for each diffraction line was evaluated and the calculated intensities were modified by dividing them by the absorption corrections. The new composition was then calculated after this modification. The final results of this analysis have been shown in Table 11. Notice that absorption corrections have improved the results.
1st Cycle
Analyzed 2nd Cycle
Quartz Fluorite Calcite
29.2 54.3 16.5
27.4 57.8 14.8
28.3 56.4 15.3
Table 111. Comparison between Calculated and Prepared Compositions for Mixtures Nos. 2 to 5
WLJi
ZVi =
Prepared
z
This equation can be solved for ui as shown in Equation 6. vi =
Minerals, wt.
Examples Mixture No. 2
Minerals Halite Fluorite Rutile
Mixture” No. 3 Mixture” No. 4 Mixtureb No. 5
Pyrite Calcite Hematite Niter Sylvite
Quartz
Prepared, wt % 50.0
30.0 20.0 66.0 34.0 65.0
35.0 90.0 10.0
Analyzed, .wt 48.7 28.8 22.5 69.2 30.8 68.2 31.8 92.4 7.6
z
One component is a weak and the other is a strong absorber. b One is a minor component. Q
Table I11 shows the results of the analysis for four additional mixtures with components having varying degrees of concentrations and absorption powers and demonstrates the good agreements between the calculated and prepared compositions. LIMITATIONS
There are certain limitations to this technique. Only those substances whose crystal structures are well known can be analyzed. Similar restrictions apply to mixtures containing amorphous phases. In such mixtures, only the relative proportion of the crystalline phases may be determined. Further limitation is imposed by samples in which the diffraction lines of the components superimpose one another. In such cases, the use of longer wavelength radiations or larger camera diameters can improve the resolution of the patterns and, consequently, the reliability of the technique. Substances which exhibit strong preferred orientations require special sample preparation techniques, such as the one described by Bloss, Frenzel, and Robinson (8). Special care should also be exercised in samples which consist of constituents varying in brittleness and hardness. Such heterogeneous samples should be ground and screened in several cycles until all the material passes through the screen. It should also be pointed aut that the reliability of the technique is reduced in samples consisting of components with crystal imperfections or in which there are great variations in particle size.
RECEIVED for review May 6, 1968. Accepted June 13, 1968. Part of this work was supported by The Joint Committee on Powder Diffraction Standards (Grant Nos. 2212-2112). Presented at the annual meeting of The Geological Society of America in New Orleans, La., Nov. 20-22, 1967.
(8) F. D. Bloss, G. Frenzel, and P. D. Robinson, Amer. Mineral., 52, 1243 (1967). VOL. 40, NO. 1 1 , SEPTEMBER 1968
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