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A NEW METHOD FOR STUDYING THE SIZE AND DEFORMATION OF CRYSTAL GRAINS
As was shown in the preceding abstract, there is an error-free procedure by means of which the true shape of the diffraction line, represented as a Fourier series, can be obtained from experimental data.
It can be shown that the coefficients of this series give direct information on the distribution of stresses in the specimen; namely, the following formula can be derived:
\[ P(2\vartheta)=KN\sum_{-\infty}^{+\infty}\left(A_n\cos 2\pi n h_3+B_n\sin 2\pi n h_3\right), \]
where
\[ A_n=\cos 2\pi h_3 Z_n \quad \text{and} \quad B_n=-\sin 2\pi h_3 Z_n, \]
and \(h_3\), for a given value of \(2\vartheta\), is calculated from the formula
\[ h_3=2a_3\frac{\sin\vartheta}{\lambda}. \]
(In the expression for \(A_n\) and \(B_n\), instead of \(h_3\) one may substitute its maximum value \(l\)—the order of reflection from the plane perpendicular to the edge \(a_3\) of the lattice.) Here \(N\) is the total number of cells in the crystal,
\[ K=\frac{MjR\lambda^3F^2}{16\pi v^3 |b_3|\sin^3\theta}, \]
\(M\) is the number of crystallites in the specimen, \(j\) is the multiplicity factor, \(\lambda\) is the wavelength, \(R\) is the distance from the object to the film, \(v\) is the volume of a cell, \(b_3\) is the edge of the reciprocal lattice corresponding in direction to \(a_3\), in the case of a rhombic lattice, \(2\theta\) is the scattering angle, and \(F\) is the structure factor.
Thus it is shown that, in the Fourier series representing the “true” diffraction curve \(P(2\vartheta)\), the expansion coefficients are determined by the deformation in the direction perpendicular to the reflecting plane.
If the coefficients \(B_n\) are not equal to zero, then the positions of the maxima for unstressed and undeformed material will be different.
Thus, in the method described, such distortions as lead to broadening or contraction of the lattice are also automatically taken into account.
The coefficients \(A_n\) and \(B_n\) are calculated by the method discussed in the preceding abstract.
It is clear that
\[ A_n=\frac{F^{\text{act}}}{KN} \quad \text{and} \quad B_n=\frac{F^{\text{imag}}}{KN}. \]
The result of the calculation can be represented in the form of curves \(A_n\) and \(B_n\) as functions of \(n\). If the relative displacements \(Z_n\) are small or follow a Gaussian distribution, then from the coefficients \(A_n\) one can pass to the mean-square displacements and represent \(A_n\) in the form
\[ A_n \sim e^{-2\pi^2 n^2 \overline{Z_n^2}}. \]
Thus, experiment makes it possible to construct curves of deformation as a function of the length of the deformed column. If the column contains
If we have \(n\) cells and length \(L = n a_3\), then the relative deformation corresponding to \(Z_n\) is, by definition, equal to \(\Delta L = a_3 Z_n\).
Having put \(\Delta L\) as a function of \(L\), we would have to obtain a straight line in the case of homogeneous deformation.
In Fig. 1 such curves are shown for cold-rolled \(\alpha\)-brass. Two reflections, (400) and (331), were measured, and the measurement results were processed by the method described.
As is evident from the figure, the experiment leads to a curve, not to a straight line. Portions of the curves can be approximated by a straight line over an interval of no more than 20 Å. This, consequently, is the size of regions within which the deformation may be considered homogeneous.
Judgments about material of this character can be obtained only by the new method described above.
A very significant advantage of the method is the fact that, as a result of processing measurements of a single diffraction line, it makes it possible to distinguish line broadening caused by stresses from line broadening whose cause is the smallness of the grain, and also to characterize the distribution of crystalline grains by size.\(^{2}\)
Fig. 1.
The grain size, when considering crystals along the normal to the reflecting plane, can be characterized by the number \(i\) of cells lying in this direction. Let \(n_i\) be the number of crystals of thickness \(i\), so that
\[ N=\sum_{i=1}^{\infty} i n_i \]
is the total number of cells in the specimen.
The formula for the intensity curve remains valid with the simplification that, because of the equality \(Z_n=0\),
\[ P(2\vartheta)=K N \sum_{n=-\infty}^{+\infty} A_n \cos 2\pi n h_3 \]
\[ A_n=\frac{1}{N}\sum_{i=|n|+1}^{\infty}(i-|n|)n_i \]
or, if one introduces the distribution function with respect to thicknesses,
\[ A_n=\frac{1}{N}\int_{i=|n|+1}^{\infty}(i-|n|)\,p(i)\,di, \]
where \(p(i)\,di\) is the number of crystallites with thickness between \(i\) and \(i+di\).
It is not difficult to see that, having obtained from experiment the curve \(A_n\) as a function of \(n\), we can find all the essential data concerning the size of the crystalline grains. Indeed,
\[ \frac{dA_n}{dn}=-\frac{1}{N}\int_{i=|n|+1}^{\infty}p(i)\,di, \]
where
\[ \int_{|n|+1}^{\infty}p(i)\,di \]
is the number of crystallites with thickness greater than \(n\). The value of the first derivative at \(n=0\) is the number of all crystallites divided by \(N\), i.e., in other words, the mean size of a crystallite in the direction normal to the reflecting plane.
If we differentiate once more, we obtain directly the curve of the distribution of particles by size, since
\[ \frac{d^2A_n}{dn^2}=\frac{1}{N}p(n). \]
The presence of an initial slope in the curve \(A_n\) as a function of \(n\) is a distinctive indication that the broadening of the diffraction curve has occurred because of the small size of the particles.
In the case when the deformation tends to zero or the crystallite size becomes large, all the coefficients \(A_n\) become equal to 1.
Fig. 2.
The form of the curve \(A_n\) as a function of \(n\) makes it possible to study experimentally thermal vibrations.
Figure 2 shows three typical curves \(A_n(n)\). Curve \(a\) is characteristic of the stressed state of a material, curve \(b\) of a substance consisting of small grains, and, finally, curve \(c\), which passes into a horizontal line, characterizes the influence of thermal expansion.
In the absence of diffuse thermal scattering, the influence of an increase in temperature would have had to reduce merely to a decrease in the ordinates of the function \(P(2\vartheta)\), and correspondingly the coefficients \(A_n\) would have had to lie on a horizontal straight line, which in fact arises at large \(n\). The values of \(A_n\) lying to the left of this transition should be divided into two parts. Everything that lies above the horizontal straight line is responsible for thermal diffuse scattering.
There is no doubt that the method we have described for studying the shape of diffraction lines on radiographs of a polycrystal has a great future.
A. I. Kitaigorodsky
References
- B. E. Warren and B. L. Averbach, J. Appl. Phys. 21, 595 (1950).
- M. F. Bertant, Comptes Rendus 228, 492 (1949).