Theory of spectral thermal shift and R-line thermal-shift calculations of MgO:Cr3+

Abstract

The thermal shift of the crystal-field spectra of 3d or 4f ions includes contributions of different characters. In this paper we develop an approach to calculating all those contributions in detail. The thermal shift caused by thermal expansion is interpreted and calculated by use of the theory of pressure-induced spectral shift. Contributions of phonons of optical branches are given using a single frequency model. Taking into account contributions of all the \ensuremath{\Gamma}\ensuremath{\gamma} in the electron-phonon interaction, two terms coming from phonons of acoustic branches are derived in detail, and theoretical formulas for the parameters in the most important term are given. By adequately taking into account all the contributions, we calculated the R-line thermal shift of MgO:${\mathrm{Cr}}^{3+}$ by fitting the experimental data. The results show good agreement with thermal-shift experimental data and a series of other experimental data of MgO:${\mathrm{Cr}}^{3+}$. Values of the most important parameters are calculated theoretically with wave functions obtained from diagonalization of complete ${\mathit{d}}^{3}$ energy matrices. They are in good agreement with those from fitting the experimental data. We found that it is the mixing of wave functions caused by the Coulomb interaction and/or the spin-orbital interaction that makes D(${\mathrm{\ensuremath{\Gamma}}}_{6}$) and D(${\mathrm{\ensuremath{\Gamma}}}_{8}$) nonzero. Our investigation shows that among all the contributions to thermal shifts, the ${\mathit{t}}_{2}^{32}$${\mathit{T}}_{1}$ term (resulting in redshift) is the most important one; the Raman term (resulting in blueshift) eliminates part of the ${\mathit{t}}_{2}^{32}$${\mathit{T}}_{1}$ term; their algebraic sum gives the contribution of electron-phonon interaction of the acoustic branches; contributions of optical branches (resulting in redshift) increases rapidly with temperature, and exceeds that of the acoustic branches when T\ensuremath{\ge}175 K; contribution by thermal expansion (resulting in blueshift) holds about 30% of total thermal shift in absolute value.