Previously we derived equations determining line broadening in ax-ray diffraction profile due to stacking faults. Here, we will consider line broadening due to particle size and strain which are the other factors affecting line broadening in a diffraction profile. When line broadening in a diffraction profile is due to particle size and strain, the theoretical model of the sample under study is either a Gaussian or a Cauchy function or a combination of these functions, e.g. Voigt and Pseudovoigt functions. Although the overall nature of these functions can be determined by Mitra’s <i>R</i>(x) test and the Pearson and Hartley <i>x</i> test, details of a predicted model will be lacking. Development of a mathematical model to predict various parameters before embarking upon the actual experiment would enable correction of significant sources of error prior to calculations. Therefore, in this study, predictors of integral width, Fourier Transform, Second and Fourth Moment and Fourth Cumulant of samples represented by Gauss, Cauchy, Voigt and Pseudovoigt functions have been worked out. An additional parameter, the coefficient of excess, which is the ratio of the Fourth Moment to three times the square of the Second Moment, has been proposed. For a Gaussian profile the coefficient of excess is one, whereas for Cauchy distributions, it is a function of the lattice variable. This parameter can also be used for determining the type of distribution present in aggregates of distorted crystallites. Programs used to define the crystal structure of materials need to take this parameter into consideration.
Diffraction line profiles due to polycrystalline materials have long been recognized to be caused by physical parameters like sizes of and defects in constituent powder particles. For this purpose, parameters like full width at half maximum (FWHM) intensity, integral width of line profile, Fourier transforms of intensity profile etc. have been utilized. Additionally, the moments [
The author previously showed that line broadening in a diffraction intensity profile of powdered crystalline materials due to stacking fault can be characterized in terms of the Zeroth, the First, the Second, the Third, and the Fourth Moment and the Fourth Cumulant [
The author has in the past described a method of differentiating between Cauchy and Gauss distributions in diffraction line profiles [
1) Gauss distribution:
2) Cauchy distribution:
where f(x) is the distribution for the variable x. The author also showed that the cumulative distribution
For Gauss distribution:
For Cauchy distribution:
Therefore, comparison of a graph of
with the calculated values of
would determine the nature of the distribution, i.e., whether it is of a Cauchy or Gauss type. Using these criteria, a plot of
for a sample of Kaolinite, to test the formula
yielded the value of μ, the physical parameter describing the diffraction pattern, and revealed that the distribution in this case was of a Cauchy type [
In actual practice, the distribution may be neither pure Gaussian nor pure Cauchy—but a combination of both. Several options are possible—1) a sum of partly Gaussian or partly Cauchy, 2) the convolution of partly Gaussian or partly Cauchy functions, 3) or even the same proportion of Gauss and Cauchy functions. These combinations may be purely additive or purely convolutional. The convolution may be of purely Gaussian and purely Cauchy (Voigt profile) [
Convolution of two functions is by definition the inverse Fourier Transform of the product of the Fourier Transforms of the two functions i.e.
where Fg is of the Fourier Transform of the function g and Fc is the Fourier Transform of the function c (F−1 is the inverse Fourier Transform of the product).
The Fourier Transform of
Since Fourier Transform carries a function in x space into t space while inverse Fourier Transform takes it back into x space,
Thus for convolutions representing the Voigt function:
Pseudovoigt functions may be additive like η Cauchy + (1 − η) Gauss or of convolution type like
where η is called the mixing fraction, and can be used to denote the proportion of intensity of the Cauchy type.
Although peak position, peak height, half intensity width etc. have been used to describe an intensity distribution in a diffraction profile, the first satisfactory parameter has been the integral width
and for the Pseudovoigt function of the additive type, this can be described in terms of
Whereas for the Pseudovoigt function of the convolution type,
Langford introduced the Voigt function through the integral width [
Fourier Transform of
Thus
then
Integral width
Ida [
where
As mentioned in Section 2 above, Fourier Transform of
For Voigt function the Fourier Transform is
While the Fourier Transform for Pseudovoigt function of additive type is
Intensity of X-ray diffracted in the direction
where τ is the limit where A(t) vanishes. By inverse Fourier Transform
And so,
By differentiating Equation (8) with respect to t over and over again and then assuming t = 0, we obtain the values of
Wilson has shown that [
where σ1 and σ2 are limits of the integral (8) instead of 0 to τ where A(t) vanishes. Very often σ1 = σ2.
The author [
In Equations (11) and (12), there are other terms involving
ation (12). But these terms are negligible compared to the remaining terms. The size of the particle comprising the powder is given by the 0th Moment of I(s) i.e.
The Second and the Fourth Moment of the line profile yield the shape of the particles comprising the powdered sample. Of course, they and other moments yield information regarding crystal deformations due to strain, stacking faults, nature and the extent of dislocations etc. In Equations (11)-(13) we have used equations for particle size only, disregarding other factors like strain, dislocation density etc.
The author previously also showed that for a powder particle of size
where
where α, β and γ are angles between the axes
Thus
Therefore,
Hence
For h00, 0k0 and 00l reflection, only one dimension of the particle will be obtained, with hk0, 0kl and h0l reflections only two dimensions will be obtained. For a cylindrical crystal, the above equations will be modified in terms of equations derived by Langford and Louër [
It is known from Equation 509 [
Putting n = 1 and a = 1/2, we have the expression for the Second Moment of a Gaussian function,
And putting n = 2 and a = 1/2, we have the expression for the Fourth Moment of a Gaussian function
Equations (15) and (16) represent the Second and the Fourth Moment of a Gaussian profile. Cernansky has derived the equations [
and
For a Cauchy profile
and the Fourth Moment, from Equation 2.147.3 [
The Second Moment of the Voigt profile is given by
where
For large positive value of x
Similarly the Fourth Moment of the Voigt profile is given by
For the Pseudovoigt profile of the convolution type the Second and Fourth Moment will be given by Equations
(19) and (20) multiplied by
1) Second Moment
2) Fourth Moment
Since the Second Moment and Second Cumulant are equal and we have already studied the Second Moment, we shall be examining only the Fourth Cumulant. All cumulants along with the Fourth Cumulant have the additive property namely
x1, x2, x3 being specific properties like size, strain, stacking fault, probability etc. The Fourth Cumulant C4 is given by
where μ4 and μ2 are the Fourth and Second Moment respectively which have been described in this paper. The Fourth Cumulant is given as shown below.
Fourth Cumulant
according to Cernansky
By Equation (18)
By Equation (17)
Thus
The Fourth Cumulant can be found similarly from Equations (21) and (22). Now, to distinguish between Voigt and Pseudovoigt functions, an additional parameter, the coefficient of excess, is introduced here. It is the ratio of the Fourth Moment and three times the square of the Second Moment [
Hence this study proposes that Equations (19) and (21) may be used for identifying Voigt and Pseudovoigt distributions, respectively.
Pearson and Hartley [
For Voigt and Pseudovoigt distribution Equations (20) and (19), (22) and (21) should be used.
That more than one parameter can be studied from one line in a Debye Scherrer pattern was noticed by Williamson & Hall who assumed that particle size and strain broadenings had Cauchy like distribution [
or
where t is the apparent particle size and e is apparent strain, θ is the angle of scattering and λ is the wave length scattered.
Thus
Plotting
curve. Equation (25) is the well-known Williamson-Hall relation much used by many investigators [
Warren and Averbach showed that the Fourier Transform of a line profile yield significant information regarding particle size, particle strain, probability of faulting and many other crystal defects [
where v(t) is the Fourier Transform of the particle size, (st) stands for strain. For determining particle size and strain only, Warren and Averbach has developed a simple method [
Now
For small value of l and n, the logarithm of the measured Fourier coefficient is given by
By plotting lnA(n) against l2 we obtain at l = 0, the value of
In general, Equation (26) can be written in a summation form
where
where p is the stalking fault probability. Also
And
where γ is the twin fault probability, and a is the lattice constant.
The application of the above formulae depends on the recognition of the proper mathematical model. To achieve this, the best way is the
be obtained by comparing with
voigt of convolution type. This can probably be resolved by examining additional peaks. Similarly, determining particle sizes by trying the relevant formulae may be helpful. The advantage of this procedure is that overlapping of lines will not interfere with the determination of the parameters. It is important to note that the Fourth Cumulant is an important parameter—since it is the only parameter which can decompose the observed line profile into profiles for different reasons of line shape. By a reverse analysis of the shape of the diffraction profile, it should be possible to identify the nature of the line profile—Gauss, Cauchy, Voigt, Pseudovoigt and hence, of the mixing parameter. From these—it should be possible to determine the parameters of line broadening. It is expected to be a good test for pattern decomposition. In this connection, another test—the χ test due to Pearson and
Hartley may be mentioned [
voigt functions using Formulae (15), (17) and (18). Pearson and Hartley have given numerical values to identify Pearson type IV and Pearson type VII curves as well [
Sincere and heartfelt thanks are due to Dr. Paramita M Ghosh of the University of California at Davis, California, USA for encouragement and helping in many ways and to Mr. Bishwajit Halder for secretarial help. The author was previously Professor and Head of the Department of Physics, Indian Institute of Technology, Kharagpur, West Bengal, INDIA.