BASIC ASPECTS OF X-RAY ABSORPTION In Quantitative Diffraction Analysis LEROY ALEXANDER
AND
of Powder Mixtures
HAROLD P. KLUG, Mellon institute, Pittsburgh 13, Pa.
The mathematical relationships are developed which are pertinent to the quantitative analysis of powder mixtures for the case of diffraction from the surface of a flat powder specimen. These formulas relate the diffracted intensity to the absorptive properties of the sample. Three important cases are treated: (1) Mixture of n components; absorbing power of the unknown equal to that of the matrix; concentration proportional to intensity. Direct analysis is permitted. (2) Binary mixture; absorbing power of the unknown not equal to that of the diluent; concentration not proportional to intensity. Direct analysis is possible by means of calibration curves prepared from synthetic mixtures. (3) Mixture of n components; absorbing power of the unknown not equal to that of the matrix; general case. Analysis is accomplished by the addition of an internal standard. Concentration is proportional to the ratio of the intensity of a selected reflection from the unknown to the intensity of a reflection from the internal standard.
recent years x-ray diffraction methods have been applied to problems of quantitative analysis.
t >
DURING extensively
3.2 —
X
M
Diffraction methods possess the unique advantage of detecting not only the presence of chemical elements but also their state of chemical combination. Quantitative measurements of greatly improved quality can now be made with the aid of Geigercounter tubes for receiving the diffracted energy. Finally, particular impetus has been given to the use of quantitative diffraction techniques by the development and widespread commercial distribution of the Norelco Geiger-counter x-ray spectrom-
o
P
X
sin
0
in which t is the thickness of the powder sample, p is the average density of the solid material composing the powder, p' is the density of the powder (including the interstices), 0 is the Bragg reflection angle, and m is the mean linear absorption coefficient of the solid material composing the powder.
eter.
In spite of this extensive activity in the field of diffraction analysis, no investigator to date has published a detailed statement of the simple but important mathematical relationships which relate the diffracted intensity to the absorptive properties of the sample and thereby determine the particular procedure that is suitable for the analysis of any given sample. This communication presents these mathematical considerations for the case of diffraction from the surface of a flat powder specimen, the arrangement employed in the Norelco x-ray spectrometer. The measurement of the absolute intensities of x-rays diffracted by the components of a binary powder mixture has been discussed theoretically by Brentano (4, 5), and he has applied the results to the measurement of atomic scattering factors. Glocker (8) and Schafer (10) have shown in a similar manner that the fundamental intensity formulas of Laue can be used as a basis for the quantitative diffraction analysis of binary powder mixtures and alloys. However, they did not extend their mathematical treatment to the point of evolving a systematic practical scheme of analysis. INTENSITY
DIFFRACTED
BY ONE COMPONENT
powder sample consisting of n components and incident x-ray beam of cross-sectional area A which impinges upon the sample at angle 0. Under these conditions the area of sample surface irradiated is A /sin 0. Referring to Figure 1, consider the diffraction taking place from a layer of thickness dx at depth x. The volume of this layer is Consider such
irradiated by
or, since x
a
an
—
^
s
sin
0,
OF A POWDER
dV
MIXTURE
.
It will be initially assumed that the sample is a uniform mixof n components, that the particle size is very small so that extinction and so-called microabsorption effects (6) are negligible, and that the thickness of the sample is sufficient to give maximum diffracted intensities. A satisfactory criterion for the latter condition is that ms > 6.4, a being the linear absorption coefficient of the sample and s being the maximum path length traversed by the x-rays through the sample (12). Applied to the geometrical arrangement of the Geiger-counter x-ray spectrometer, this criterion takes the form
=
^
Ads
(1)
Let us now define (/„)< as the intensity diffracted by unit volume of pure ith component at the angle 20 to the primary beam and under conditions of nonabsorption, and let/< be the volume fraction occupied by the ith component, neglecting the interstices. The intensity diffracted from element dV by component i of the mixture is then
ture
dli
=
i
(I0)if>Ae"'Psds
and, integrating between the limits s = 0 and , we obtain for the total intensity of the diffracted beam from the ith component 886
VOLUME
2 0,
NO, 10, OCTOBER
1948
887
the component to be analyzed for, component 1, and the sum of the other components, which we may call the matrix and refer to by the subscript M. The weight fraction of the matrix is
(DifiA 2
u
xu
which is a funcconstant, Ki ^ (70)q4, tion of the nature of the component i and of the geometry of the apparatus, Equation 2 can then be written We may
now
define
a
=
'•-'•'•(
=
1
Xi
—
wufW
=
Now the weight fraction of the ith component in the matrix is
(xi)u
=
Wxí Xi) W{1
Wi/u’M
-
=»
Xi
In order to apply Equation 3 to quantitative analysis, it is desirable to express 7; as a function of the weight fraction of the ith component. Let W and T be, respectively, the sample weight and volume (neglecting interstices), while m, Xi, pq and µ; represent, respectively, the weight, weight fraction, density, and linear absorption coefficient of the ith component. It can then be seen •hat/,·, the volume fraction of the ith component, is given by
1
The
—
18’
Xi
absorption coefficient of the matrix is given by
mass Pm
=
P*{Xi)u +
M
3
+ p*(xt)u +.....
(m3) m
which by reference to Equation 8 may be written
>·
Wi
(9:
2___
1 l
1
Xi
—
In terms of the component to be analyzed for, component Equation 7 becomes
Pi
1,
Multiplication of the numerator and denominator by l/W gives /,
Xi /Pi
=
µ. v,
(4)
{xi'p
7)
1
Equation 7 can be put into a very useful form by regarding the mixture of n components as if it consisted of just two components.
which shows that the diffracted intensity is directly proportional to the concentration, allowing direct linear analysis. In practice such eases are relatively rare, the most important being mixtures of the polymorphic crystalline forms of a given compound. More frequently it happens that p* and µ* are approximately but not precisely equal. In this event the degree of accuracy desired will determine whether or not the linear relationship of Equation 11 can be assumed to apply. In Figure 2 the validity of Equation 11 is illustrated by means of diffraction data from mixtures of quartz and cristobalite, 34.9 for CuKa radiation. polymorphs of silica, for which µ* =
Three synthetic mixtures were prepared containing 25, 50, and 75% micronized quartzite admixed with Johns Manville’s Celite.
ANALYTICAL
888
Type I. Celite is a commercial thermal insulating powder which diffraction analysis showed to consist very largely of finely divided or-cristobalite. It also contains a very small amount of clay and possibly a little amorphous silica, but the absorbing powers of these impurities differ but little from that of cristobalite. Each analysis for quartz was performed in triplicate using a Norelco Geiger-counter x-ray spectrometer. The intensity of the 3.33 Á. quartz maximum was measured by manual scanning, taking sufficient counts at each point to keep Geiger-counter statistical errors small. The recorded counts were corrected for the resolving time of the counter. It is seen that the four experimental points lie very close to a straight line as predicted.
CHEMISTRY
standard. This method has been applied to emission spectrographic analysis for some years (11), while more recently it has been utilized in diffraction analysis (1, 2, 3, 7, 9). The latter application has seemed valid on more or less intuitive grounds, but to date no one appears to have investigated its theoretical basis.
Consider, then, the addition of an internal standard, component s, to the sample in knorvn amount. The unknown and internal standard components then occupy the volume fractions// and/,, the prime being used to distinguish the volume fraction of the unknown component after addition of the internal standard
from its fraction in the original sample,/i. From Equation 3 we have for these two components,
Dividing 11 4, we
h by I,
=
Krfi'/µ
and I,
and substituting
obtain
h
,/,/µ
=
for// and/,
from Equation
Dp,z/
_
Ksx,pi
7,
and solution of this equation for z/ gives
z/
=
K.P1X.
Dp,
v X
h
7,,
_
7,
w
X
7i 7,
(14)
provided z, is kept constant. But the weight fraction of component 1 in original sample is related to z/ in the following manner: Xi
Xi' 1
x,
—
When this is combined with Equation 14 xi
Figure 2. Comparison of Theoretical IntensityConcentration Curves (Solid Lines) with Experimental Measurements (Open Circles) for Several Binary Mixtures Mixture of 2 Components; µ ^ µ*. The absorbing powers of the unknown and diluent are unequal, so that the intensityconcentration curve is not linear. However, direct analysis is possible by reference to a calibration curve prepared from synthetic mixtures of the two components. An equation giving the theoretical form and position of the analysis curve can be obtained from Equation 10. For the pure first component
while for a mixture containing component
_
XTt
=
find that
kXT,
do)
Equation 15 states that when the internal standard is added in a constant proportion, z„ the concentration of the unknown component is proportional to the intensity ratio Ii/I,. Figure 3 shows the applicability of Equation 15 to the analysis of quartz in powder mixtures using fluorite as internal standard.
weight fraction xi of the first
a
__
PiIDmÍ
=
tve
—
µ) + µ]
Division of the second equation by the first gives 7i
(Do
_
_ µ_ (µ
-
µ*) +
µ*2
(12)
In Figure 2 the upper and lower curves show the agreement between the theoretical values (solid lines) as calculated with Equation 12 and experimental points for mixtures of quartz and beryllium oxide (µ > µ2) and mixtures of quartz and potassium chloride (µ* < µ|). The values of the mass absorption coefficients for CuKct radiation w'ere calculated from the absorption data given in the “Internationale Tabellen zur Bestimmung von Kristallstrukturen” and found to be: BeO 8.6, Si02 34.9, and KC1 124. The intensity measurements were made in the manner described above.
Mixture of n Components; µ ^ µ%; general case. The absorbing power of the unknown is different from that of the matrix, the latter being unknown as a general rule. In this case the analysis can be performed by the addition of an internal
Figure 3. Linearity of Typical Calibration Curve for Quartz Analysis Using Fluorite as Internal Standard Three synthetic mixtures of micromzed quartzite and calcium carbonate were prepared containing 30, 60, and 100% quartz. Fluorite was then added to each in the constant proportion z, 0.20. The intensity ratio of the 3.33 Á. quartz and 3.16 Á. fluorite maxima was then measured with the Norelco spectrometer in the manner described earlier. Each plotted point is the average of 10 determinations. It is seen that the experimental results substantiate the linearity between Ii/I, and xi which is predicted by =
Equation
15.
VOLUME
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NO. 10, OCTOBER
1948
889
ACKNOW LEDGM ENT
The authors are indebted to Elizabeth Kummer for part of the experimental measurements. LITERATURE
CITED
(1) Ballard, J. W., Oshry, . I., and Schrenk, . H., J. Optical Soc· Am., 33, 667-75 (1943). (2) Ballard, J. W., Oshry, . I., and Schrenk, . H., U. S. Bur. Mines, Rept. Invest. 3520 (June 1940). (3) Ballard, J. W., and Schrenk, . H., Ibid., 3888 (June 1946). (4) Brentano, J. C. M., Phil. Mag., 6, 178-91 (1928). (5) Brentano, J. C. M., Proc. Phys. Soc., 47, 932-47 (1935). (6) Brindley, G. W., Phil. Mag., 36, 347-69 (1945).
(7) Clark, G. L., and Reynolds, D. H., Ind. Eng. Chem., Anm. Ed., 8, 36-40 (1936). (8) Glocker, R., Metallwirtsch., 12, 599-602 (1933). .. (9) Gross, S. T., and Martin, D. ., Ind. Eng. Chem., Anal. 16, 95-8 (1944). (10) Schafer, K„ Z. Krist., 99, 142-52 (1938). (11) Scheibe, G., “Chemische Spektralanalyse, physikalische Methoden der analytischen Chemie,” Vol. I, p. 108, Leipzig. Akademische Verlagsgesellschaft, 1933. (12) Taylor, A., Phil. Mag., 35, 632-9 (1944). Received May 3, 1948. Presented before the First Congress of the International Union of Crystallography, Cambridge, Mass.. July 28 to August 3 1948.
POLAROGRAPHY Some Factors
Affecting Drop Time
F. L. ENGLISH Chambers Works, E. I. du Pont de Nemours & Co., Deepwater Point, N. J. are presented showing the influence of capillary vibration, applied voltage, magnitude of current flowing through the cell, capillary active agents, purity of mercury (including certain amalgams), and stray currents in the constant temperature bath upon the drop time.
Data
POLAROGRAPHIC analysis, the concentration of the material sought is determined by the magnitude of the diffusion current produced under the prevailing conditions and the precision is indicated by the degree of reproducibility of the diffusion current in repeated runs. The magnitude of the current, however, depends upon several factors besides the concentration, as shown in the fundamental equation for the dropping mercury electrode, first devised by Ilkovic (1):
IN
eOónD1,2Cm2 1/6 number of faradays of electricity required per mole diffusion coefficient, sq. cm. per second concentration of substance sought, millimoles per liter mass of mercury flowing, mg. per second drop time, seconds id
in which
=
D
=
C
=
m
=
t
=
=
Of these factors, n and D are inherent in the nature of the material being determined and hence beyond the control of the analyst under the conditions of any given analysis. The mass of mercury is substantially constant for a given capillary operating under a fixed head of mercury and at constant temperature. Variations in diffusion current from one run to another, under otherwise identical conditions, will, therefore, be attributable primarily to irregularities in the drop rate and although time enters the equation to the extent of only the sixth root, a considerable degree of consistency is required if high accuracy is to be attained. Thus, excluding all other errors, a variation of 3% in drop time from one run to another will cause an error of 0.5% in the determination; to attain 1.0% accuracy, the variation in time may not exceed 6.2%. The investigation described herein was, therefore, undertaken to identify the factors primarily responsible for variation in drop time. EXPERIMENTAL
A model XX visible recording polarograph, manufactured by the E. H. Sargent Co., Chicago, 111., was used in this work. The time for full-length chart travel (410 mm.) was 645 seconds. The polarographic cell was the H-shaped type recommended by Kolthoff and Lingane (4), with saturated calomel cell anode,
supported in a 30 X 30 cm. (12 X 12 inch) cylindrical Pyrex constant temperature bath maintained at 25° ± 0.05° C. by a LoLag thermostatically controlled immersion heater. The neoprene tube connecting the capillary with the mercury reservoir was provided with a branch leading to a fixed buret for convenient maintenance of the 443-mm. head used throughout the work, binder this pressure, the capillary passed 1.366 mg. per second. The determinations of drop time were made by establishing the desired conditions and permitting the instrument to run at sensitivity and damping settings to give easily legible pen oscillations per drop, usually for five chart divisions, representing an elapsed time of 322.5 seconds. The oscillations were then counted and the average time per drop was obtained. In some few instances, the average drop time over a definite voltage range was determined by a similar procedure.
Mercury Head. The effect of the mercury pressure was not investigated, as it is fully covered in the literature (3, 9). It has been shown that drop time is an inverse linear function of the head if proper correction is made for the back pressure due to the interfacial tension between the mercury and the solution in which it is dropping. Vibration. The polarograph itself, containing no galvanometer, is vibration-proof. The effect of vibration on the mercury drops has been noted (9, 10) but no detailed study of it appears to have been made. At the start of the work, variations of
as much as 10% were noted under apparently identical conditions. Vibrations seemed the most likely explanation, although the steel and stone construction bench upon which the water bath and capillary support were mounted did not show enough vibration to interfere with the proper functioning of an analytical balance of 0.05-mg.
sensitivity. A stand was made for the bath assembly consisting of a 35-cm, (14-inch) square of 5/s inch thick boiler plate, drilled and tapped at the back, left side, and right rear corner for 0.5-inch diameter rods for the support of the capillary and other accessories. Three runs were then made upon 0.1 N potassium chloride containing 1 ml. each of 1 M cadmium chloride and 1% gelatin per 100 ml., at —0.9 volt, a sensitivity of 2-50 (33.9 mm. per microampere) and No. 1 damping. These conditions were chosen to give wide amplitude to the pen oscillations. Varying degrees of vibration insulation were used: Run 1. The boiler plate stand was set directly on the bench
