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ANALYTICAL CHEMISTRY, VOL. 50, NO. 9, AUGUST 1978
Calculated X-ray Diffraction Data and Quantitative X-ray Diffractometry Stephen Altree-Williams Division of Occupational Health and Radiation Control, Health Commission of New South Wales, P.O. Box 163, Lidcambe, Australia 2141
An equation is derived that relates the quantitation constant for direct quantitative x-ray powder diffractometry, k,,, to the calculated x-ray powder diffraction data parameters, ASF, and Idrd. The derived equation is tested experimentally and the results confirm the potential usefulness of both the derived equation and, more generally, calculated x-ray diffraction data. A diffractometer constant, ti, is introduced that should allow experimentally determined kUvalues from different laboratories to be directly compared.
As a quantitative technique, x-ray powder diffractometry (XRD) attempts to relate the quantity of an analyte phase to the intensity of one of its diffraction lines. For the conventional Bragg-Brentano parafocusing powder diffractometer ( I ) and the general case of a flat sample of depth less than t h e penetration depth of the x-ray beam, the quantitation equation is ( 2 )
For a sample “infinitely” thick to the x-ray beam, Equation 1 becomes
For a thin-layer sample supported on a diffracting underlay, Equation 1 becomes (3,4 )
where IIJ = integrated intensity of diffraction line z nf phase = the quantitation constant, its value depending on the diffractometry conditions and on diffraction line I of phase J; W J = weight fraction of phase J in the sample; p = mass absorption coefficient of the sample; M = weight of sample per unit area; M j = weight of phase J per unit area; Od = angle of incidence of the x-ray beam to the sample for diffraction line z of phase J; On = angle of incidence of the x-ray beam to the sample for a chosen diffraction line from the thin-layer support material; L = In (Pn/ID);PD,I D = integrated intensity of the chosen diffraction line from the material supporting the thin-layer before and after loading, respectively. The quantitation constant, k d , is the same constant in each of Equations 1-3. To date this constant has had t o be determined by diffractometry on samples of pure phase J . However, the availability of calculated XRD data (5-8) allows the determination of kLJfrom such data alone. Consider the conventional powder diffractometer with a constant divergence slit, without monochromator, and with a n “infinitely” thick sample of pure phase assumed t o he a t theoretical density. Let the crystallites of phase J be randomly oriented and let them have a particle size such as t o eliminate extinction and microabsorption effects wit horit introducing any significant line broadening Then the In-
J; kd
0003-2700/78/0350-1272$01.00/0
tegrated intensity of diffraction line i of phase J is theoretically given by (9- 12)
I,
pi3 0:
-
Rr
. ~-e4 . -1 + cos* 20,J . PlJIFlJI**m2c4 sin2 O,,
COS
1
V J 2F*
O1,J
(4)
where ZIJ, OIJ are as previously defined; Io, A, Q, r = constants for a given diffractometer run under constant conditions, being the intensity of the primary x-rays used for diffraction, their wavelength, the scan rate used for counting the integrated intensity, and the radius of the goniometer circle, respectively; e, m, c = physical constants, being the charge and rest mass of the electron and the velocity of light, respectively; pJ,lFJ = the multiplicity. and the amplitude of the structure factor (including thermal effects) for line i (Miller index, h k l ) of phase J , respectively; VJ = volume of the unit cell of phase J; p* = linear absorption coefficient of the sample, in this case, phase J at theoretical density. Equation 4 can be transposed into a more convenient form by considering the following. For a given diffractometer run under constant conditions, the diffractometer constants, the physical constants, and the constant of proportionality can he incorporated into a single constant, K . For the specific wavelength used in the diffractometer, the combined relative effect of the Lorentz-polarization term, the multiplicity factor, and the structure factor term on the intensity of the diffraction lines of phase J is given by the calculated relative intensities, IfJrel, for the lines of phase J . T h e most intense diffraction line of phase J is given a n Il,y’value of 100. The absolute effect of the Lorentz-polarization, multiplicity, and structure factor terms on the diffraction intensities of phase J is a property of phase J and A. The factor for this absolute effect adjusts the relative intensity data to the absolute scale and is incorporated with 1/ V j 2to give the absolute scale factor for phase J , designated ASFJ (6). Equation 4 can thus be written
Equation 5 refers to a sample of pure phase J a t theoretical density. For real samples containing phase J (including pure powdered phase J as packed in a diffractometer), Equations 4 and 5 can be modified by including a term for the volume fraction of phase J in the sample, with p* referring to the linear absorption coefficient of the sample. Expressing p * in terms of the mass absorption coefficient of the sample and expressing the volume fraction of phase J in terms of the weight fraction of phase J gives W.
(6) where all terms have been previously defined except pJ, the density of phase d. Comparison of Equation 6 with its equivalent experimental equation, Equation 2, gives the equation for determining k,,, from calculated XRD data
‘C 1978 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 50, NO. 9, AUGUST 1978
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Table I. Comparison of h i , Values Determined by Calculation' and by Experiment k i J , net counts cmz f i g - ' phase, J line, i, hkl ASE'j P J , g cm-3 calculated experimentalC corundum (e-Al,O,) 113 0.120 3.987 b 11.5 i 0.1 a-quartz (SiO,) 101 0.321 2.65 46.3 45.3 t 0.3 calcite (CaCO,) 0.250 104 47 t 1 2.71 35.2 sylvite (KC1) 0.329 1.987 200 65 t 1 63.3 zincite (ZnO) 0.912 5.68 61.3 101 53 i 2 anatase (TiO,) 101 47 I 1 0.497 3.89 48.8 eskolaite (Cr,O,) 104 22 T 1 0.361 5.21 26.5 hematite (e-Fe,O,) 104 0.56 5.26 40.7 30.2 t 0.2 111 5.33 10.5 122 c 1 194 Ag a Values for A S F J and p J are from Hubbard, Evans, and Smith ( 7 ) except for hematite (JCPDS card 24-72A) and are corrected for the use of a graphite crystal monochromator with Cu Kcu radiation. The diffractometer constant, K , was determined as (3.82 * 0.03) X l o 6 net counts. ASF" cm-' by equating the experimental measurements and calculated data for corundum. Average value of four sample preparations I mean deviation.
'
~ - _ -
Equation 7 indicates that k I J can be determined from calculated XRD data (ASFJand IIJre')for all phases except for the one chosen to experimentally determine h for the diffractometer. With this one limitation, x-ray powder diffractometry can be considered an absolute instrument technique where the quantitation of phase J can be performed without recourse either to a calibration standard of phase J or to a n internal standard for each sample. Two further deductions of practical interest can be made from Equation 7. Since K , ASFJ and pJ are constant for a given diffractometer, wavelength, and phase, then the quantitation constant for each diffraction line of a given phase is directly proportional to IdE'. Again, since for any given diffraction line of any given phase ASFJ, Idre'and pJ are constant for a given wavelength and independent of the diffractometer, then the quantitation constants for a set of phases determined on one diffractometer will be directly proportional to the quantitation constants for t h a t same set of phases determined on any other diffractometer, subject t o the same wavelength being used. Note that Equation 4 refers to the use of a conventional diffractometer without monochromator. When a monochromator is used (either on the incident beam to the sample or on the diffracted beam) the polarization term, 1 cos2 28,,,,, in Equation 4 needs t o be changed to 1 + cos' 2 8 , ~cos2 28,, where Om is the Bragg angle for the monochromator crystal reflection and wavelength used (13). This change affects the intensity of all lines depending on their BLJ and on 8,. It necessitates adjustment of the calculated values of ASFJ and Zdrel, which are routinely determined assuming a polarization factor of 1 + cos2 2/ILJ(8). T h e validity of Equation 7 for determining hIJ has been investigated experimentally, as has the deduction that for lines of the one phase h,J is proportional to I l j e l . T h e results are considered below.
+
EXPERIMENTAL Phases. All phases used were laboratory chemicals except for corundum (Linde A, Union Carbide Corporation), a-quartz (natural crystal, Kingsgate, New South Wales), and silver (Selas Flotronics silver membrane filter, 0.2-gm pore size). Diffractometry. A Philips (Eindhoven) x-ray diffractometer was used comprising vertical goniometer (PW 1050/70),diffracted beam monochromator with graphite crystal, sample spinner, xenon filled proportional counter, and copper anode broad focus x-ray tube (PW 2253/20). The quantitation constant was determined, by the use of Equation 3, from measurements obtained from deposits of
