BASIC ASPECTS OF X-RAY ABSORPTION
In Quantitative Difraction Analysis of Powder Mixtures
I,EKOY ALEX4XDF:K
AND
HAROLD P. KLI'G, Mellon Institute, Pittshrrrgh 13, Pa.
..1he mathetiidtical relatiorishipsare deteloped 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 intensitj
of a selected reflection from the unknown to tbe intensity of a reflection from the
internal Standard.
D
URING recent years x-lag diffraction methods have been
t
extensively applied t o problems of quantitative analysis
Diffraction methods possess the unique advantage of detecting
not only the presence of chemical elements but also their state
'If 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 spectrometer.
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
Ehich 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
liffraction 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 , 6),and he has applied the results
io the measurement of atomic scattering factors. Glocker (8)
and Schafer (IO) have shown in a similar manner that the fundamental intensity formulas of Laue can be used as a basis for the
quantitative d8raction analysis of binary powder mixture5
and alloys. However, they did not extend their mathematical
treatment to the point of evolving a systematic practical wheme
if analysis.
3.2
P
x
P
x
sine
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 der,.
sity of the powder (including theinterstices), eis the Bragg reflertion angle, and ,u is the mean linear absorption coefficient of the
solid material composing the powder.
' am
Figure 1.
Diffraction from a La)er of Powder
a t Depth x
Consider such a powder sample consisting of n Components and
irradiated by an incident x-ray beam of cross-sectional area A
which impinges upon the sample a t angle 0. Under these condi:ions the area of sample surface irradiated is A/sin 0. Referring
ro Figure 1, consider the diffraction taking place from a layer cd
thickness dx at depth 2. The volume of this layer is
dV
or. since z =
INTENSITY DIFFRACTED BY ONE COMPONENT O F A POWDER
MIXTURE
1
-
2
s sin
=
Adz
sin 0
e,
1
dV = HAda
I t will be initially assumed that the sample is a uniform mixture of 1c 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.
-4 satisfactory criterion for the latter condition is that ~8 2 6.4,
L being the linear absorption coefficient of the sample and s being
che maximum path length traversed by the x-rays through the
sample (1.8). Applied to the geometrical arrangement of the
Geiger-counter x-ray spectrometer, this criterion takeq the form
(1)
Let us now define (Zo)i as theintensity diffracted by unit volume
of pure ith component a t the angle 28 to the primary beam and
under conditions of nonabsorption, and let fi be the volume fraction occupied by the i t h component, neglecting the interstices.
The intensity diffracted from element dV by component i of the
mixture is then
and, integrating between the limits 3 = 0 and m, we obtain for tht
total intensity of the diffracted beam from the zth component
886
887
V O L U M E 20, NO. 10, O C T O B E R 1 9 4 8
the coniponcnt to be analyzed for, component 1, and the sum of
the other components, which we may call the matrix and refer t o
by the subscript M. The weight fraction of the matrix is
XX = 1
1
K , = -2 ( I o ) , A , n-hich is a funcion of the nature of the component L and of the gwmetrv of the
annaratus Equation 2 can then be nntten
LT e may now dehm a constant,
I<
-
Kc
(t)
-51
=
w.u/w
?;ow the weight fraction of the i t h component in the matrix is
3:
=-
In ordei t u apply kquation 3 to quantirarive aiialydis, it is desirable to express I ; as a function of the xveight fraction of the ith
icomponent. Let H’and T- be, respectively, the sample weight and
,iolume (neglecting interstices), a-hile,wi, 51, pE. and pi represent,
respectively, the weight, weight fraction, density, and linear abgorption coefficient of the i t h component. It, can then be seen
-hatfi, thevolumr fraction of the ith component, is given by
I
.C,
‘8
- XI
The mass absorption coefficient of the matrix is given by
ILZ
= d(22)u
+
+
p:(53)M
PXcdM
+....
which by reference to Equation 8 may be nritten
I n terms of rhe component to he anal1 z ~ dtor, component 1,
Equation 7 becomes
Uultiplication of the numerator and denoniiriator h p 11W gives
KlXl
= --
fi=+--
c
!X%’PI
1
T--
PI
/*.“J,
,
- -
>ow thr. mas* absorption roefficirnt, p / p , of a powder mixture i4
iefined b j the expresion
slu;xi
__ ! ! E l - n
+
--
P 3 ,
Pi
2
ix-hich in view of Equation 9 becomes
01
(10)
Equatioii 10 i> the basic relationship underlying quantitative
diffraction analj,ia with the x-ray spectrometer. I t ielates the
intensity of anv given diffraction maximum of the unknown component to i t \ concentration in the sample and to thr relative
mass alxxption coefficients of the unknown and matrix.
a
THREE IMPORTAYT ANALYTICAL CASES
iccordiriy to the number of components and the equalitj or
nonequalitv of p: and p: in Equation 10, three importanr
cases of quantitative analysis arise, each permitting or requiring
a particular procedure.
Mixture of n Components; p: = p;; direct analyais The
absorbing power nf the unknown equals that of the niatrn
Eqiiaticm 10 rediiri- ‘ ( 1
6J
Substituting Equations 4 and 6 in 3, and reprearriting the r n a s
absorption coefficient of the Ith component bv p.* = ut / p l . wo arFive a t the relationship
Equation 7 can be put into a very useful form by regarding the
rnixture of n components as if it consisted of just two component<.
which shon-s that the diffracted intensity ik directly pruportioiial
to the concentration, allon-ing direct linear analysis. I n pracricc
3uch cases are relatively rare, the most important being mixture
of the polymorphic crystalline fornis of a given compound. Morf
frequently it happens that p: and p: are approximately but not
precisely equal. In this event the degree of accuracy desired
will determine whether or riot the linear relationship of Equatiort
11 can be assumed to apply.
I n Figure 2 the validity of Equation 11 is illustrated by mean.
of diffraction data from mixtures of quartz and cristobalite.
polymorphs of silica, for which p* = 34.9 for VuKu radiation.
Three synthetic mixtures were prepared containing 25, 50, arid
75% micronized quartzite admixed with Johns Manville’s <‘elite.
888
ANALYTICAL CHEMISTRY
Type I. Celite is a commercial thermal insulating powder which
diffraction analysis showed to consist very largely of finely divided a-cristobalite. I t 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 Sorelco
Geiger-counter x-ray spectrometer. The intensity of the 3.33 A.
quartz maximum was measured by manual scanning, taking sufficient counts a t each point to keep Geiger-counter statistical errors small. The recorded counts were corrected for the resolving
time of the counter. I t is seen that the four experimental points
lie very rlose to a straight line as predicted.
standard. This method has been applied to emission spectrographic analysis for some years (fl), 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 theoretiral
basis.
Consider, then, the addition of an internal standard, component s, to the sample in knonm amount. The unknown and internal standard components then occupy the volume fractions f I J
and f,, the prime being used to distinguish the volume fraction of
the unknown componpnt after addition of the internal standard
from its fraction in the original sample,fl.
From Equation 3 n-e have for these two components,
I1 = Kljl’/p and I. = K6.fB/p
Dividing I1 by I , and substituting for fi’ and fa from Equation
4, we obtain
5
-
RlPSZll
I,
KaW1
and solution of this equation for xi’ gives
provided xI is kept constant. But the weight fraction of coniponent 1 in original sample is related to xl’ in the following manner:
Khen this is combined with Equation 14 we find that
-0
0.2
0.4
0.6
0.8
WEIGHT FRACTION OF QUARTZ, XI
I .o
Figure 2. Comparison of Theoretical IntensityConcentration Curves (Solid Lines) with Experimental Measurements (Open Circles) for Several Binary
Mixtures
Mixture of 2 Components; p: # p t . 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 firstcomponent
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 11/18.
Figure 3 shows the applicability of Equation 15 to the
analysis of quartz in powder mixtures using fluorite as internal
standard.
Ki
(Ilh = PIP?
while for a mixture containing a weight fraction x1 of the first
component
Division of the second equation by the first gives
In Figure 2 the upper and lower curves shoFv the agreement between the theoretical values (solid lines) as calculated with Equation 12 and experimental points for mixtures of quartz and beryllium oxide (p,” > &) and mixtures of quartz and potassium chloride ( p : < p ; ) . The values of the mass absorption coefficients for
CuKa radiation were calculated from the absorption data given in
the “Internationale Tabellen zur Bestimmung von Kristallstrukturen” and found to be: B e 0 8.6, Si02 34.9, and KC1 124.
The intensity measurements v ere made in the manner described
above.
Mixture of n Components; p: # P:; general
absorbing power of the unknown is different from
matrix, the latter being unknown as a general rule.
the analysis can be performed by the addition of
case. The
that of the
I n this case
an internal
Figure 3. Linearity of Typical Calibration Curve
for Quartz Analysis Using Fluorite as Internal
Standard
Three synthetic mixtures of micronized quartzite and calduxn
carbonate were prepared containing 30, 60, and 100% quartz.
Fluorite was then added to each in the constant proportion x. =
0.20. The intensity ratio of the 3.33 d. quartz and 3.16 A. fluorite
maxima was then measured with the Norelco spectrometer in the
manner described earlier. Each plotted point is the average of 10
determinations. I t is seen that the experimental results substantiate the linearity between I J I , and 21 which is predictrtf hy
Equation 15.
V O L U M E 20, NO. 10, O C T O B E R 1 9 4 8
ACKNOWLEDGMENT
The authors are indebted to Elizabeth Kummer for part of
the experimental measurements.
LITERATURE CITED
(1) Ballard, J. \I7., Oshry, H. I.,and Schrenk, H. H., J . Optical Soc.
Am., 33, 667-75 (1943).
(2) Ballard, J. W., Oshry, H. I., and Schrenk, H. H., U . S. Bur
Mines. Rept. Invest. 3520 (June 1940).
(3) Ballard, J. W., and Schrenk, H. H., I b i d . , 3888 (June 1946).
(4) Brentano, J. C. XI.,Phil. -'dug., 6, 178-91 (1928).
(5) Brentano, J. C. &I., Proc. P h y s . SOC.,47, 932-47 (1935).
(6) Brindlry, G. 'S., Phil. Mag., 36, 347-69 (1945).
889
( 7 ) Clark, G. L., and Reynolds. D. H., IND.ENG.CHEM.,A s + i .
ED.,8, 36-40 (1936).
( 8 ) Glocker, R., Metallwirtsch., 12, 599-602 (1933).
(9) Gross, S. T., and Mart,in, D. E., IND.EKG.CHEM.,AKAL.R I B . ,
16, 95-8 (1944).
(10) Schafer, K., 2 . Krist., 99, 142-52 (1938).
(11) Scheibe, G., "Chemische Spektralanalyse, physikalische Methoden der analytischen Chemie," F'ol. I, p. 108, Leipsig. Aksdemische T'erlagsgesellschaft, 1933.
(12) Taylor, A , , P h i l . Mug.,35, 632-9 (1944).
RECEIVED
May 3, 1948. Presented before the First Congress of the International <-nion of Crystallography, Cambridge, Mass.. July 28 to Aumrt 3
1948.
POLAROGRAPHY
Some Factors ARecting Drop Time
F. L. ENGLISH
Chambers F o r k s , E. I . clu Pont de J e m o u r s & Co., Deepwater Point, V. J .
Data 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.
I
N 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, hoTyever, depends
upon several factors besides the concentration, as shown in the
fundamental equation for the dropping mercury electrode, first
devised by IlkoviE (1):
id
= 60jnD1 JzCmZ'at1
18
in which n = number of faradays of electricity required per
mole
D = diffusion coefficient, sq. cm. per second
C = concentration of substance sought, millimoles
per liter
m = mass of mercury flowing, mg. per second
t = drop time, seconds
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.
EXPERlMENTAL
h model XX visible recording polarograph, manufactured
by the E. H. Sargent Co., Chicago, Ill., was used in this work.
The time for fulblength 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 CORstant temperature bath maintained a t 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. Under
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 oscillatione
per drop, usually for five chart divisions, representing an elapsed
time of 322.5 secondj. The oscillations were then counted and
the average time per drop waq obtained. In some few instances.
the average drop time over a definite voltage range waq determined by a similar procedure.
Mercury Head. The effect of the mercury pressure wab not
investigated, as it is fully covered in the literature ( 2 , 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 n-hich
it is dropping.
Vibration. The polarograph i t d f , containing no galvanometer.
is vibration-proof. The effect of vibration on the mercury drops
has been noted (9, 10) but no drtailed 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 TI ater bath and capillary support
were mounted did not show enough vibration to interfere with
the proper functioning of an analvtical balance of 0.05-mg
sensitivity.
A stand was made for the bath assembly consisting of a 35-cm
(14-inch) square of 5 / 8 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 S potassium chloride containing
1 ml. each of 1 &Vi' cadmium chloride and 1% gelatin per 100 ml.,
a t -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:
RUN1. The boiler plate stand was set directly on the t)mcb