_{1}

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Line broadening in a diffraction intensity profile of powdered crystalline materials due to stacking fault has been char acterized in terms of the zeroth, first, second, third, and fourth moments and the fourth cumulant. Calculations have been derived showing that the first moment causes a shift in the peak position of the profile while the third moment af fects its shape. The intensity expression has been derived on the basis of usual Cartesian coordinates and also of polar coordinates indicated by the probability of the fault and the reciprocal lattice parameter as the two axes. The expressions for the fourth cumulant have also been so derived. Here we have used three different approaches to determine methods for calculating the fourth cumulant due to stacking faults. The three forms of the equations derived here are for different coordinate systems, but will arrive at the same answers.

Perfect crystals consist of identical layers of atoms stacked one over the other. However, occasionally, due to various reasons, the position of an atom in one layer is not the same as that in the next layer. Let us call the layer containing the atom in question the A-layer, and the layer containing the same atom in another position the B-layer; then the layer arrangement in consecutive layers, instead of being AAA or BBB, may be ABAB or ABBA etc. If the defect consists of two atoms in consecutive layers being different, say, the third layer (C-layer) would enable arrangements of the type: ABCABC, etc., which further expands the level of complexity in the crystal. While in the normal lattice, A is followed by B and B is followed by C (arrangement ABCABC), in the defective crystal ABC may be followed by BCA (ABCBCA), or by BAC (ABCBAC) etc. In the first case C being followed by B instead of A constitutes a deformation stacking fault. In the second case, B instead of being followed by C is being followed by A, constitutes a Twin Fault defect. Stacking faults likely occur in hexagonal close packed metals like cobalt, tungsten and their alloys, whereas Twin Faults are more likely in the (111) planes of FCC or BCC metals and their alloys. In addition, in alloys like gold-copper which are FCC at high temperatures but simple cubic at low temperatures, are cases of three dimensional defects. In many silicate minerals, there are faults in layer arrangements including variable interlayer spacing.

The presence of the stacking fault deformation or the twin fault can be determined by X-ray diffraction line profiles, using parameters like integral width, full width at half maximum (FWHM) intensity, Fourier transforms of intensity profile etc. Additionally, for better fit, the moments [1,2] and cumulants [

The intensity scattered in the direction

where θ is the angle of scattering and λ is the wave length of the waves scattered [

and

where γ is the probability of occurrence of a stacking fault and J(t) is the tth order Fourier transform of I(s).

If I(θ) be the intensity scattered in the direction θ, the mth moment of the line profile is given by

replacings with θ in equation (1).

1) The zeroeth moment: For the zeroth moment, m = 0, and

From equation (1) we have

, where.

Hence,

Alternatively, let γ = α sinφ and, so that

.

Then,

Thus,

(4b)

by equation 268 of [

2) The first moment: For the first moment, m = 1, effectively

The first moment causes a shift in the peak position of the line profile, as observed for α-brass of composition 70 - 30 irradiated with CoKα radiation [

where γ is the stacking fault probability. They also found that,

and

where δ is the twin fault probability, α is the lattice constant and p is the particle size.

3) The third moment is given by

and affects only the shape of the line profile, as discussed by [

4) The second and the fourth moments: It has already been shown in equation (1) that the intensity of X-rays diffracted by materials with stacking fault probability γ, is given by

where

Thus, the second moment is

by equation no. 57 of [

The fourth moment is

But

by equation 2.147.3 of [

5) The Fourth Cumulant: The zeroth, first, second, and the third cumulants are the same as the corresponding moments. However, the fourth cumulant is different from the fourth moment, and will be calculated here separately: The fourth cumulant is given by:

where we have applied the formula

found in page 44 of [

Simplifying further, we have

An alternative formula to describe the second and fourth moments, and therefore the fourth cumulant, may also be derived, starting from equation (1) as follows:

and

Let us assume that γ = a sinθ and 2πs = a cosφ, so that a^{2} = γ^{2} + 4π^{2}s^{2}, and

Thus, by eqnuation no. 276 of [

and by eqnuation no. 29, p. 132 of [

(9b).

Thus, neglecting higher order sine terms,

(9c).

It has been shown in equation (2) that

So that

Hence, , , and

Now, it is known that

and

Substituting from equations (C), we have:

and as was derived by [

Thus,

There are generally three methods used for the analysis of line profiles: 1) the integral breadth of the line profile, 2) the Fourier coefficients describing the shape of the line, and 3) the second moment of the line profile about its centroid. Here we expand on that theme and derive equations for not only the second moment but also for other moments up to the fourth moment. Recently, the cumulants of the line profile has also received attention [

As stated in equation (3a), the intensity scattered at an angle θ will depend on the coefficients of Fourier transforms A_{m} and B_{m}. Since, according to [_{m} of the moment expressions, the corresponding coefficient A_{m} has been neglected.On the other hand, for the even moments, i.e. the zeroth, second and the fourth moments the portion of the equation depending on A_{m} is significant but that depending on B_{m} is negligible. On that note, Stokbro and Jacobsen had developed a simple model for the energetics of stacking faults in fcc metals [

As was earlier shown [

Multiple authors have discussed diffraction by stacking faults [4,12-16]. Therefore, in expression (1), we have not made any assumptions regarding the nature of the stacking fault, except that it is random. So the expressions for moments and cumulants derived in this work are valid for all types of stacking faults. Of course, the cases of one dimensionally disordered crystals where the faults are not random but occur preferentially on every third close packed layer [

The author expresses his deep sense of gratitude to Dr. Paramita M. Ghosh of the University of California at Davis, California, USA, for encouragement and help in the preparation of the manuscript. Thanks are also due to Mr. BishwajitHalder for secretarial help.