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EXPERIMENTAL DETERMINATION OF THE TRUE SHAPE OF A DIFFRACTION LINE IN A POLYCRYSTAL X-RAY DIFFRACTION PATTERN
A diffraction line in the X-ray diffraction pattern of a polycrystal has a certain width, which depends both on the conditions of the experiment (the size of the specimen, the divergence of the beam, the imperfect monochromaticity of the beam and the presence of a doublet in the \(K_{\alpha}\)-radiation, etc.) and on the crystalline structure, namely, on the size of the crystalline grains and the character of the lattice distortion. Investigation of these two features of the crystalline structure by measuring the broadening of diffraction lines has been the subject of a very considerable number of works. Many methods have been proposed for estimating and calculating the “true” width of a diffraction line. All of them suffer from essential defects, since they contain a number of arbitrary assumptions.
Even if some of these methods are considered satisfactory for estimating the half-width of a line, they all nevertheless have the drawback that they do not give us information about the form of the curve, i.e., about the distribution of the intensity \(P(2\theta)\) in the diffraction line. Whereas judging the properties of a specimen by a single parameter is very imperfect, an exhaustive description of the properties of the specimen may be obtained if the function \(P(2\theta)\) is known to us.
In a recently completed work \({}^{1}\) it was shown that there exists a direct, objective method for reproducing the form of the diffraction curve from X-ray measurement data.
Let us denote by \(f(x)\) the intensity of the diffracted ray at the point \(x\) (where \(x\) is measured along the film from some definite origin) for the case when the line broadening is due only to the properties of the specimen. \(f(x)\) will be the desired function, immediately convertible into \(P(2\theta)\). Let us denote by \(g(x)\) the experimental curve obtained from a specimen with sufficiently large and unstrained grains, and by \(h(x)\) the curve of the specimen of interest to us.
Thus, \(g(x)\) is the curve broadened by the experimental conditions, while \(h(x)\) is the curve broadened by the experimental conditions and by the properties of the specimen.
The three functions introduced are related by the equality \({}^{2}\)
\[ h(x)=\int_{-\infty}^{+\infty} f(y)\,g(x-y)\,dy. \]
In practice we measure \(g\) and \(h\) through discrete intervals. Therefore the integral may be replaced by a sum
\[ h(x)=\sum f(y)\,g(x-y)\,\delta y, \]
where \(\delta y=0.1\) mm (or of this order). We obtain from 20 to 50 such equalities from experiment.
To determine \(f(x)\), let us expand all three functions in a Fourier series over the interval from \(-\dfrac{a}{2}\) to \(+\dfrac{a}{2}\), chosen so that \(g(x)\) and \(h(x)\) vanish at the ends of the interval. Then
\[ f(x)=\sum_{t=-\infty}^{+\infty} F(t)e^{-2\pi i t \frac{x}{a}}, \]
\[ g(x)=\sum_{t=-\infty}^{+\infty} G(t)e^{-2\pi i t \frac{x}{a}}, \]
\[ h(x)=\sum_{t=-\infty}^{+\infty} H(t)e^{-2\pi i t \frac{x}{a}}, \]
where \(t\) is an integer—the summation index—and \(F, G, H\) are complex amplitudes.
The calculation of the coefficients \(G(t)\) and \(H(t)\) is carried out with the aid of tables of sines and cosines available in many x-ray laboratories, after which the real and imaginary parts of the desired coefficients \(F\) are determined from the formulas
\[ F^{\text{real}}= \frac{H^{\text{real}}G^{\text{real}}+H^{\text{imag}}G^{\text{imag}}} {(G^{\text{real}})^2+(G^{\text{imag}})^2}; \]
\[ F^{\text{imag}}= \frac{H^{\text{imag}}G^{\text{real}}-H^{\text{real}}G^{\text{imag}}} {(G^{\text{real}})^2+(G^{\text{imag}})^2}. \]
It remains to synthesize the required function; for this, two more summations are performed (with the aid of tables) according to the formula
\[ f(x)=\sum_t F^{\text{real}}(t)\cos 2\pi t\,\frac{x}{48} +\sum_t F^{\text{imag}}(t)\sin 2\pi t\,\frac{x}{48}. \]
The entire procedure takes no more than 4–5 hours.
As an example, all three curves are given in the figure. As is seen from the curve \(g(x)\), the \(K_\alpha\)-doublet is strongly pronounced; nevertheless, in the method described no special measures are taken to account for its presence. The influence of the doublet, as well as of all the other experimental conditions, is taken into account automatically.
The condition under which integration can be replaced by summation is as follows: the coefficients \(H(t)\) and \(G(t)\) must become equal to zero for values of \(t\) smaller than \(\frac{a}{2}\). If \(a = 48\), then, consequently, the quantities \(H\) (and, analogously, \(G\)), beginning approximately with \(H_{20}\), must be close to zero. If this is not so, then dividing the interval into 48 parts is insufficient. Experience shows, on the contrary, that division into 48 parts is quite sufficient.
Of course, the accuracy of the intensity distribution obtained depends on photometric errors. It can be shown that these errors will not play a noticeable role, especially in the case when the line \(g(x)\) is sufficiently narrow in comparison with \(h(x)\).
A. I. Kitaigorodskii
REFERENCES CITED
- A. R. Stokes. Proc. Phys. Soc. 61, 382 (1948).
- A. I. Kitaigorodskii, X-ray Structure Analysis, 1950, p. 592.