March-Dollase model

In eonneetion with the implementation in the Rietveld codes, the Dollase March model and the spherical harmonics approach, for pole distributions determination, is developed in the next two parts. The problem of pole figure inversion is outside the scope of this chapter. 

The Dollase approach is based on the observation that, if the distribution of a given pole ho is known, the distribution of any other pole h can be derived. Let us denote by a the angle between h and hg. For the crystallites that have h parallel to y, the pole ho lies on the surface of a cone of axis y and angle 2a. On the other hand, the pole h is a member of a family of Wh equivalents making different angles a with ho. Then we can write  

Note that ho is also a member of a family of equivalents. If the pattern indexation is changed into an equivalent indexation the terms in the sum in Equation (13) permute but the sum remains unchanged. 

The March distribution fulfills the normalization condition Equation (12) and is monotonic in the limits 1/0 for P = 0 and 0 for P = njl. The maximum is 

If the sample symmetry axis y3 is defined by the standard polar and azimuthal angles (T, y ), then the angle can be generally written as follows  

---

The Dollase March model to describe the texture in the Rietveld method has become very attractive due to the small number of refinable parameters. Nevertheless, frequently it produces an incomplete texture correction, even if complex variants like Equation (22) are used. In fact the condition of disc-shaped crystallites or needles is not fulfilled in general and the specimen does not have cylindrical sample symmetry. In principle, by spinning the specimen, an apparent cylindrical texture is obtained, but in this case there is no prominent plane (/zq o/q) as required by the Dollase approach. Any diffraction plane can be used as a placeholder, but it is highly improbable that the corresponding pole ho has a monotonic distribution in the range 0 < < njl, which is described by the March formula Equation (16). 

Dollase WA. 1986. Correction of intensities for preferred orientation in powder diffracto-metry application of the March model. J. Appl. Cryst. 19 267-272. 

W. A. Dollase, Correction of Intensities for Preferred Orientation in Powder Difffactometry Application of the March Model,/. Appl. Crystallogr., 19, 267-272 (1986)... 

The preferred orientation correction was accounted for in two ways during the refinement. First, the March-Dollase approach with one texture axis [001] resulted in x = 1.247(2) and correction coefficients ranging from 0.52 to 1.39, which gives the preferred orientation magnitude of 2.70. Second, the 8 -order spherical harmonics expansion, which corresponds in this crystal system to six adjustable parameters (200, 400, 600, 606, 800, and 806) was attempted with the March-Dollase preferred orientation correction (i) left as is but fixed (i.e. the spherical harmonics were in addition to the March-Dollase model), or (ii) eliminated. Both ways result in practically an identical result except for the magnitudes of the coefficients. In the second case, the correction coefficients ranged from 0.61 to 1.54, which corresponds to the preferred orientation magnitude of 2.52. 

---