1. Introduction

Multiple scattering is generally not considered in X-ray or neutron diffraction from polycrystals, due to the negligibly low intensity expected from this effect; however, measurements of pole figures show significant differences in many cases when density maxima from first and second orders are compared to each other, indicating that primary or secondary extinction could be present^{1}^{-}^{3}. If secondary extinction is dominant, a way to verify it could be to test if the curve of crystallite orientations for the second diffraction coincides with high populated zones of the Orientation Distribution Function (ODF) of the textured sample. As well known of diffraction from polycrystals, the intensity at a point of a Debye-Scherrer ring hkl, or the pole density at a point on the pole figure, is caused by the superposition of the intensities diffracted by many crystallites, all of which satisfy the Bragg condition for the planes (hkl). This crystallites have in common the normal direction of planes (hkl), but differ otherwise in orientation so that their orientations lie along a curve in the Eulerian space, i.e. the pole figure is a projection of pole densities along a path through the ODF corresponding to a 2 ^{5}. This happens also for the second diffraction process, being the incident vector orientation the only difference, so that the diffracting crystallites lie also along a curve in the Eulerian space. So the aim of this work is to evaluate the orientation curves for the main reflections for a secondary diffraction, and for a general sample orientation given by the conventional angles (^{1}, however, the mathematical method used allowed only one solution of the main equation expressing the Laue condition for a second reflection (Eq. (7) of ^{1}), and no modifications were made for especial cases where no solution (for example reflections where

2. The method

Let

To determine possible secondary diffraction processes this wave can produce in the polycrystalline sample, the orientation curve of crystallites contributing to the intensity registered at a general point of a Debye-Scherrer ring produced by diffraction from plane (hkl) is to be calculated. Figure 1 shows the primary diffraction layout, and Fig. 2 shows the sequence of both diffraction processes.

It is convenient to rotate the reference system an angle

And expressed in the sample reference system, it is

If

with

where the superindex 0 indicates that the sample is in its initial orientation (0,0).

Q_{0} in terms of the pole figure angles (

As in a measure of a pole figure, let the sample be rotated an angle

Crystallites for which Q satisfies Laue conditions

Let hkl be a particular reflection, and C a crystallite such that the vector Q satisfies Laue conditions for a certain reciprocal vector

where ^{6} (Eq. (2.50)):

Then

where

where **í**, **j** and

After applying matrix

where

*Some properties of coefficients A, B, C and*

From (11a) and (11c) it can be readily be seen that

Multiplying (10a) by

and similarly for 𝐵

It follows then

Also, from (11b) and (11c)

And with (11d)

Since rotations do not change the magnitude of vectors

from which, using (5), (8), and (9)

Using (12e), this equation gives

and using (12d),

Solution of Equations (10)

Equations (10) are not independent, and the way adopted here to solve them is as follows:

From (11b) and (11c), B(

From (10a) and (10b)

To solve (10c) the following method is proposed for a general equation with coefficients a, b and c:

Let

Equation (13) is then:

Multiplying by

This is a quadratic equation in

from which

Applying this to (10c) with

From (12f) these equations become

Let

i.e

This means

Equations (10a) and (10b) form a system of two linear equations whose solution is

Using (12d), expressing

with the condition

Especial case

If

And from (12a)

Which, using (18a) can be written

Which means that for this case there is only one value for

Equation (18a) has no solution if

Total of solutions

Except for the especial case

These are two curves if

3. The computer program

A program in Dev C

Primary data are given: Pole figure affected, i.e. (

The especial cases

4. Results

Figures 5 and 6 show some results for a silver sample using copper radiation. Curves are usually closed but they appear fragmentary when some angle reaches the ^{6}, Eq. (2.4)).

5. Conclusions

A method has been developed to determine the orientation curves of crystallites able to produce a second diffraction of an incident photon. Calculations require only modest computer capabilities, and they could be applied in future work to determine the integrated intensity of the whole Debye-Scherrer rings, as a measure to evaluate the intensity loss due to a possible effect of secondary extinction.

The aim of this work is thus fulfilled; however, to know if these curves pass through high pole density zones, further calculations will be necessary since on the one hand, due to symmetric properties of the samples, only a part of this cube of side length 2𝜋 will be sufficient, and on the other hand, every curve should be first subdivided in segments of equal length. Both aspects require careful operations, and for the moment they are beyond the scope of this work.