Profval: Functions to calculate powder-pattern peak profiles with axial-divergence asymmetry

Abstract

The crystallographic problem: In a powder diffractometer, the curvature of the Debye-Scherrer rings causes an asymmetry in the peak profiles as the Bragg angle approaches either 0 or 180°. For accurate Rietveld refinement, this asymmetry, which is known as axial divergence, must be modeled correctly. Traditionally, empirical ad hoc functions such as a split-Pearson-VII function or multiple overlapping Gaussian peaks have been employed. Finger et al. (1994) extended the method of van Laar & Yellon (1984) for convolution of the ring shape with an intrinsic peak profile - a procedure that describes extremely asymmetric profiles and involves only physically meaningful parameters. To date, however, only the program GSAS (Larson & von Dreele, 1990) has incorporated this method. This contribution makes available tested code for the convolution of the asymmetry function with the intrinsic line function in the hope that more Rietveld codes will incorporate this method. Method of solution: As described by Finger et al. (1994), the convolution is performed using a Gauss-Legendre quadrature procedure, with the weights and abscissae generated with routine GAULEG of Press et al. (1992). For speed, these values have been previously calculated and are stored in large initialization statements in the Fortran and C versions of the code. In the present versions, the overall profile is evaluated as a convolution of the asymmetry function with a pseudo-Voigt function; however, this choice is arbitrary, and any other functional shape can be substituted. Although this method is computationally intensive for very asymmetric peaks, the amount of extra computation drops rapidly as the Bragg angle of the peak deviates from 0 or 180°. Experience has shown that the calculation of an entire profile with this algorithm is as fast, or faster, than using the multiple-Gaussian method. The routines provide not only the value of the convolved function, but also analytical partial derivatives of each of the parameters, so that the routines may be used for least-squares minimization. Software environment: The code consists of functions written in Fortran, C and Pascal that implement the calculation; a user may select their language of choice. Each of the functions includes a test driver showing the method of calling the routine. The routines are written in generic dialects of these languages and should compile and run under any software environment. They were written and tested on a PC running Windows 95 using Microsoft PowerStation Fortran V4.0, Microsoft C V5.1 and Borland Turbo Pascal V6.0. Hardware environment: These routines will run on any computer that has any of the three languages implemented. Documentation: The source codes for the convolution function are heavily commented, with each section referring to the appropriate equation in Finger et al. (1994). In addition, an output file of the results for each language is included with the software distribution. Availability: The source code for these routines is available by anonymous ftp from cryst.ciw.edu in directory anonymous.profile as files profile.for, profile.c and profile.pas. The output files are named o_for.out, o_c.out and o_pas.out. © 1998 International Union of Crystallography.