Deformation broadening and the fine structure of spectral lines in optical spectra of dielectric crystals containing rare-earth ions

The procedure of calculation of the spectral line shape in optical spectra of rare-earth ions in crystals with the inclusion of random deformations of an elastically anisotropic crystal lattice caused by point defects is developed. The distribution function of components of the random strain tensor in the case of a low defect concentration is obtained as the generalized six-dimensional Lorentz distribution. The distribution function parameters are represented by the integral functional of the strain tensor components on a sphere of unit radius containing an isotropic point defect in its center. The numerical calculations of the strain tensors induced by point defects and the parameters of the distribution functions of random strains in LiLuF$_{4}$ and LaAlO$_{3}$ crystals have been performed. The calculated envelope with the doublet structure corresponding to the $\Gamma_{2}(^{3}H_{4})\to\Gamma_{34}(^{3}H_{5})$ singlet–doublet transition in the absorption spectrum of Pr$^{3+}$ ions in the LiLuF$_4$ crystal agrees well with the data of the measurements.