Analysis of Nuclear Magnetic Resonance Line Shapes by Lattice Harmonics

Abstract

The second ($M_2$) and fourth ($M_4$) moments of magnetic resonance absorption lines of nuclei in crystals resulting from dipolar and exchange interactions have been given by Van Vleck. The dependence of $M_2$ and $M_4$ on the orientation of the magnetic field in the crystal coordinate system may be rewritten in terms of lattice harmonics of the crystal point group. Only lattice harmonics belonging to the identity representation occur. The number of such functions, and hence the number of independent quantities needed to specify $M_2$ and $M_4$ have been determined for all 32 point groups. These numbers vary from 15 and 45 for triclinic $C_1$ symmetry to 2 and 4 for cubic $O_h$ symmetry. $M_2$ and $M_4$ are given as a finite series of lattice harmonics of the crystal orientation, the coefficients of which are expressed as irreducible lattice sums. Application is made to available data on the resonance of ${\mathrm{F}}^{19}$ in Ca${\mathrm{F}}_2$, ${\mathrm{Al}}^{27}$ in aluminum metal and ${\mathrm{H}}^1$ in urea, CO${\left(\mathrm{N}{\mathrm{H}}_2\right)}_2$; the effect of lattice vibrations on the moments of Ca${\mathrm{F}}_2$ and Al are examined. The influence of an applied electric field of NMR moments and the use of lattice harmonics in other spectroscopics of the solid state are considered.