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Manuscript Title: Computer calculation of the Van Vleck second moment for materials with internal rotation of spin groups.
Authors: Roman Goc
Program title: m2rc3
Catalogue identifier: ADUC_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 162(2004)102
Programming language: Fortran 90.
Computer: Cray SV1, Cray T3E-900, PCs.
Operating system: UNICOS version on Cray SV1, UNICOS/mk on Cray T3E-900, Windows 98 and Windows XP.
Keywords: NMR, Van Vleck second moment, Computer simulation, Internal rotation.
PACS: 82.56.Ub, 61.43.Bn, 33.15.Hp.
Classification: 16.13.

Nature of problem:
The NMR second moment reflects the strength of the nuclear magnetic dipole-dipole interaction in solids. This value can be extracted from the proper experiment and can be calculated on the basis of the Van Vleck formula. The internal rotation of molecules or their parts averages this interaction decreasing the measured value of the NMR second moment. The analysis of the internal dynamics based on the NMR second moment measurements is as follows. The second moment is measured at different temperatures. On the other hand it is also calculated for different models and frequencies of this motion. Comparison of experimental and calculated values permits the building of the most probable model of internal dynamics in the studied material. The program described in this paper calculates the second moment for solids with rotation of different groups of spins with C3 symmetry.

Solution method:
The rotation of molecules or their parts, for example CH3 groups, is simulated as a random walk process by rotating each individual group of spins about its symmetry axis by an angle allowed by the type of symmetry. It is not a continuous rotation, but is in the form of jumps between consecutive positions allowed by symmetry of rotating group. Such a model of rotation fulfils assumptions on which theoretical equations used in NMR are derived. The value of Van Vleck's second moment averaged by this rotation is evaluated. The degree of averaging depends on the number of rotational jumps simulated during calculation. This number is then recalculated into the frequency of rotation and finally into the temperature. As a result we are getting simulated values of the NMR second moment as a function of temperature.

The only restriction is the number of spins for which calculations can be performed in a reasonable amount of CPU time. This restriction is therefore a combination of the number of spins in the unit cell, number of unit cells taken for calculation and speed of the computer used. The tested version of the program was compiled for a maximum number of 6250 spins, arranged in 125 unit cells. There are 15 axes of rotation allowed per unit cell. Any of these restrictions can be overcome by increasing the dimensions of the proper arrays in the program. The dimensions given in the program are sufficient for analysis of most NMR data one can find in scientific literature. This is due to the fact that magnetic dipole-dipole interaction decreases with the third power of distance between spins and calculations including spins up to the distance of about 2.0 nm give a final accuracy of the second moment equal to about 1%, while experimental values are determined with 5% accuracy or even worse. The program was designed to handle any combination of complex rotations, but only about C3 axes. Overcoming this restriction by introducing the possibility of C6 or C4 rotations would require some changes in the program. They may be quite easily introduced by an experienced programmer.