Energy Spectra,g Factors and Their Pressure—Induced and ／or Thermal Shifts of SrTiO3：Cr^3＋ and SrTiO3：Mn^4＋ I：energy Spectra and g Factors at Normal Pressure

With the strong-field scheme and cubic bases,the complete d^3 energy matrix in a tetragonally distorted cubic-field has been constructed.By diagonalizing this matrix,the energy spectra of SrTiO3:Cr^3 and SrTiO3:Mn^4 at normal pressure and various temperatures have been calculated.Correspondingly,the FORTRAN program calculating the g factor of the ground state has been worked out.By using the program and the wavefunction obtained from diagonalizing the complete energy matrix,the g factors of the ground state of SrTiO3:Cr^3 and SrTiO3:Mn^4 at normal pressure and room temperature have been evaluated.The calculated results are in good agreement with the optical-spectral and EPR experimental data.The comparison and analysis of the results of two crystals have been made.It is demonstrated that the covalency of the bonding between Mn^4 and ligands(O^2-) in SrTiO3:Mn^4 is stronger than the one of the bonding between Cr^3 and ligands(O^2-)in SrTiO3:Cr^3 .It is shown that the obtained wavefunctions and values of parameters are reasonable.     

Microscopic Theoretical Calculations of R—Line Thermal Shifts and Broadenings of MgO：Cr^3＋

By taking into account all the irreducible representations and their components in the electron-phonon interaction (EPI) as well as all the levels and the admixtures of basic wavefunctions within d^3 electronic configuration,the values of all the parameters in the expressions of thermal shift (TS) and thermal broadening (TB) due to EPI for the ground level,R level and R line of MgO:Cr^3 have microscopically been evaluated;and then,TS and TB of R line and various contributions to them have uniformly been calculated.The results are in very good agreement with the experimental data.It is found that all the three terms of TS due to EPI are red shifts;the Raman term is the largest one,and the optical-branch term and neighbor-level term are important for TS;the contribution to TS from thermal expansion is blue shift,which is also important.The R-line TS of MgO:Cr^3 comes from the first-order term of EPI.The elastic Raman scattering of acoustic phonons plays a dominant role in R-line TB of MgO:Cr^3 .For both TS and TB,it is very important to take into account all the admixtures of basic wavefunctions within d^3 electronic configuration.     

Pressure—Induced Shifts of Energy Spectra of α—Al2O3：Mn^4＋

By making use of the diagonalization of the complete d^3 energy matriz in a trigonally distorted cubic-field and the theory of pressure-induced shifts (PS) of energy spectra,the whole energy spectrum of α-Al2O3:Mn^4 and PS of levels have been calculated.All the calculated results are in excdellent agreement with the experimental data.The comparison between the results of α-Al2O3:Mn^4 and ruby has been made.It is found that on one hand,R1-line and R2-line PS of α-Al2O3:Mn^4 and ruby are linear in pressure over 0-100 kbar,and their values of the principal parameter for PS are very close to each other.On the other hand,the sensitivities of R1-line and R2-line PS of α-Al2O3:Mn^4 are higher than those of ruby respectively,which comes mainly from the difference between the values of parameters at normal pressure of two crystals;moreover,the expansion of d-electron wavefunctions of α-Al2O3:Mn^4 with compression is slightly larger than the one of ruby,and the effective charge experienced by d-electrons of α-Al2O3:Mn^4 decreases with compression more rapidly than the one of ruby.In the final analysis,all these can be explained in terms of the facts that the two crystals are doped α-Al2O3 with two isoelectronic ions;the strengths of the crystal field and covalency of α-Al2O3:Mn^4 are larger than those of ruby respectively,due to the charge of Mn^4 to be larger than that of Cr^3 .     

Energy Spectrum and g Factor of MgO:Cr3+ and Their Pressure-Induced Shifts

The unified calculation of the whole energy spectrum and g factor of the ground state for MgO:Cr³⁺ at normal pressure and their pressure-induced shifts has been carried out on the basis of the theory of pressure-induced shifts and the diagonalization of the complete d³ energy matrix in a regular octahedral field. All the calculated results are in very good agreement with a lot of experimental data. For the first time, by using the wavefunctions obtained by diagonalization of the complete energy matrix, the pressure-induced shifts of g factor have satisfactorily been calculated and explained by microscopic theory. The rates of change of levels with respect to various parameters and the contributions to levels from various parameters have been calculated. The distinct differences in magnitude and/or sign of pressure-induced shifts of various levels are immediately determined by their characteristic dependencies on the parameters of interactions. The pressure-induced shifts of levels and g factor have provided important criteria for the correctness of the calculations and assignments of the energy spectrum and wavefunctions, and it is quite necessary and important to carry out the unified calculation of the whole energy spectrum and g factor at normal pressure and their pressure-induced shifts.     

Energy Spectra of the Tunable Laser Crystal Gd3Ga5O12:Cr3+ and Their Pressure-Induced Shifts

By means of both the theory for pressure-induced shifts (PS) of energy spectra and the theory for shifts of energy spectra due to electron-phonon interaction (EPI), at 300 K, the `pure electronic' contributions and the contributions from EPI to R₁ line, R₂ line, and U band of GGG:Cr³⁺ as well as their PS have been calculated, respectively. The total calculated results are in good agreement with all the experimental data. Their physical origins have been explained. It is found that the mixing-degree of |t₂²(³T₁)e ⁴T₂> and |t₂³²E> base-wavefunctions in the wavefunctions of R₁ level of GGG:Cr³⁺ is considerable under normal pressure, and the mixing-degree rapidly decreases with increasing pressure. The change of the mixing-degree with pressure plays a key role for PS of R₁ line or R₂ line. At 300 K, both the temperature-independent contribution to R₁ line (or R₂ line or U band) from EPI and the temperature-dependent one are important. The remarkable difference between pressure-dependent behaviors of PS of R₁ lines of GGG:Cr³⁺ and GSGG:Cr³⁺ results from the differences of their microscopic properties. The features of emission spectra of GGG:Cr³⁺ at various pressures have satisfactorily been explained.     

Energy Spectra, g Factors and Their Pressure-Induced and/or Thermal Shifts of SrTiO3:Cr3 and SrTiO3:Mn4 Ⅲ: R-Line Thermal Shifts of SrTiO3:Mn4

By taking into account all the irreducible representations and their components in the electron-phonon interaction (EPI) as well as all the levels and the admixtures of basic wavefunctions within d3 electronic configuration,the values of the parameters in the expressions of thermal shift (TS) from EPI for the ground level, R level and R line of SrTiO3:Mn4 have been evaluated; the R-line TS and various contributions to it have been calculated in the low-temperature region. It is found that all the three terms of R-line TS from EPI relevant to the lattice vibration are red shifts. The Raman term is the largest, the neighbor-level term is the second, and the optical-branch term is very small over the range of T ≤ 80 K. The contribution to R-line TS from thermal expansion has been approximately neglected in this work. The very strong EPI relevant to its lattice vibration for SrTiO3:Mn4 causes its R-line TS to be an unusually large red-shift. Only by taking into account the strong softening of the low-frequency acoustic modes of the lattice vibration at low temperatures, can we successfully explain the variation of R-line TS of SrTiO3:Mn4 with temperature.``     

Theoretical Calculation of Energy Spectrum of YGG：Cr^3＋ and Thermal Shifts of Its R-Lines

With the strong-field scheme and trigonal bases, the complete d3 energy matrix in a trigonally distorted cubic-field has been constructed. By diagonalizing this matrix, the energy spectrum of YGG：Cr^3＋ at normal pressure and low temperature has been calculated. The g factor of the ground-state has been evaluated in terms of the energy spectrum. At the same time, by using the wavefunctions obtained from diagonalizing the complete d^3 energy matrix and Thermal Shifts theory, we calculate the thermal shifts of the sharp lines of YGG：Cr^3＋ and determine the relevant parameters. The calculated results are all in good agreement with the optical-spectrum and EPR experimental data. It is demonstrated that the obtained wavefunctions and the values of parameters are reasonable.     

Energy Spectrum,g Factors,Strain-Induced Level-Splittingsand R-Line Thermal Shift of MgO:V2+

Traditional ligand-field theory has to be improved by taking into account both pure electronic contribution and electron-phonon interaction one (including lattice-vibrational relaxation energy). By means of improved ligand-field theory, the R line, t₂^{3 2}T₁ and t₂^{3 2}T₂ lines, t₂² (³T₁)e⁴T₂, t₂² (³T₁)e⁴T₁ and t₂ e²(⁴A₂)⁴T₁ bands, g factors of t₂^{3 4}A₂ and t₂^{3 2}E, four strain-induced level-splittings and R-line thermal shift of MgO:V₂₊ have been calculated. The results are in very good agreement with the experimental data. It is found that for MgO:V₂₊, the contributions due to electron-phonon interaction (EPI) come from the first-order term; the contributions from the second-order and higher terms are insignificant. In thermal shift of R line of MgO:V₂₊, the temperature-dependent contribution due to EPI is dominant. The results obtained in this work may be used in theoretical calculations of other effects of EPI.     

Thermal Shifts and Electron—Phonon Interactions of ^4T2 and ^4T1 Broad Bands for Ruby

For the first time,by taking into account all the irreducible representations and their components in the electron-phonon interaction (EPI) as well as all the levels and the admixtures of wavefunctions within d^3 electronic electron-phonon interaction (EPI) as well as all the levels and the admixtures of wavefunctions within d^3 electronic configuration,the values of parameters in expressions of Raman and optical-branch terms of thermal shifts (TS) due to EPI for three levels,^4T2 band and ^4T1 band of ruby have been evaluated;the contributions to TS of ^4T2 and ^4T1 broad bands from thermal expansion have also been calculated;and then,the TS of the peak energies of ^4T2 and ^4T1 broad bands have been calculated.The results are in satisfactory agreement with observed data.The values of single-electron reduced matrix elements representing the strengths of EPI of ^4T2 and ^4T1 bands have respectively been determined.For TS of the peak energies of ^4T2 and ^4T1 bands,it is found that the contribution to TS from the second-order term in EPI Hamiltonian is dominant;TS due to EPI of acoustic branches are over two times as much as those of optical branches,and both of them increase rapidly with temperature;the neighbor-level term is insignificant;the contribution to TS from thermal expansion is specially important,and all the three terms of TS of ^4T2 or ^4T1 band are red shifts.     

Improved Ligand-Field Calculation of Energy Spectrum and R-Line Thermal Shift of MgO：Cr^3＋

Traditional ligand-field theory has to be improved by taking into account both pure electronic contribution and electron-phonon interaction one （including lattice-vibrational relaxation energy）. By means of improved ligand-field theory, the R-line, t^3 2^2 T1 lines, t^2 2（^3 T1）e^4 T2, and t^2 2（^3T1）e^4T1 bands, ground-state g factor, four strain-induced level- splittings, and R-line thermal shift of MgO：Cr^3＋ have been calculated. The results are in very good agreement with the experimental data. It is found that for MgO：Cr^3＋, the contributions due to electron-phonon interaction （EPI） come from the first-order term. In thermal shift of R-line of MgO：Cr^3＋, the temperature-dependent contribution due to EPI is dominant.     

Microscopic—Theoretical Calculations of R—Line Thermal Shifts and Broadenings of MgO：V^2＋

A great improvement on a previous work (PHYS.Rev.B48 (1993) 14067) has been made.By taking into account all the irreducible representations and their components in the electron-phonon interaction (EPI) as well as all the levels and the admixtures of basic wavefunctions within d^3 electronic configuration,the values of all the parameters in the expressions of thermal shift(TS) and thermal broadening (TB) from EPI for the ground level,R level and R line of MgO:V^2 have microscopically been evaluated;and then,both the TS and TB of R line and various contributions to them have uniformly been calculated.The results are in very good agreement with the experimental data.It is found that all the three terms of TS from EPI are red shifts;the term of the contribution to TS from thermal expansion is blue shift.The Raman term is the largest,and the other terms are also important for TS.The R-line TS of MgO:V^2 comes from the first-order term of EPI.The elastic Raman scattering of acoustic phonons plays a dominant role in R-line TB of MgO:V^2 .For calculations of both the TS and TB,it is very important to take into account all the admixtures of wavefunctions.     

Energy Spectra and g Factors of α-A1203:Cr3+ and α-A1203:Mn4+

By diagonalizing the complete d³ energy matrix in a trigonally distorted cubicfield and using the wavefunctions from it, unified calculations of the whole energy spectrum as well as the g factors of the ground state and t₂^{3 2} E excited states for α-A1₂0₃:Cr³⁺ and α-A1₂0₃:Mn⁴⁺ have been carried out respectively. A11 the calculated results are in very good agreement with the experimental data. The comparison between the results of the two crystals has been made, which demonstrates that the covalency of α-A1₂0₃:Mn⁴⁺ is stronger than the one of α-A1₂0₃:Cr³⁺. For the zero-field splittings of the ground state and t₂^{3 2} E , their physical origins are revealed; the comparison and analysis of their values of the two crystals have been made.     

Energy Spectrum of YAG:Cr3+ and Thermal Shifts of Its R Lines

Traditional ligand-field theory has to be improved by taking into account both “pure electronic” contribution and electron-phonon interaction one (including lattice-vibrational relaxation energy). By means of improved ligand-field theory, R₁, R₂, R'₃, R'₂, and R'₁ lines, U band, ground-state zero-field-splitting (GSZFS) and ground-state g factors as well as thermal shifts of R₁ line and R₂ line of YAG:Cr³⁺ have been calculated. The results are in very good agreement with the experimental data. In contrast with ruby, the octahedron of ligand oxygen ions surrounding the central Cr³⁺ ion in YAG:Cr³⁺ is compressed along the [111] direction. Thus, for YAG:Cr³⁺ and ruby, the splitting of t₂³⁴A₂ (or t₂³²E) has opposite order, and the trigonal-field parameters of the two crystals have opposite signs. In thermal shifts of R₁ and R₂ lines of YAG:Cr³⁺, the temperature-dependent contributions due to EPI are dominant.     

Energy Spectrum and g Factors of α-A12O3:Ni2+ and Their Pressure-Induced Shifts

A unified calculation of the whole energy spectrum and g factors of the ground stateat normal pressure and their pressure-induced shifts for α-A1₂O₃:Ni²⁺ has been carried outon the basis of the theory of pressure-induced shifts and the diagonalization of the completed⁸ energy matrix adopting C_3υ symmetry. The calculated results are in very good agreementwith all the experimental data. The rates of change of all the levels with respect to variousparameters and the contributions to typical levels or splittings from various parameters havebeen calculated. The distinct differences of pressure-induced shifts of various levels are immediatelydetermined by their characteristic dependencies on -the parameters of interactions,and the pressure-induced shifts (especially those of t₂⁶e^{2 1}EÊ) have provided important or,crucial criteria for the correctness of the calculation and assignment of the energy spectrum of α-A1₂O₃:Ni²⁺.     

Energy Spectrum and Its Pressure-Induced Shift and g Factor of ZnS:Mn2+

By using strong-field scheme, the complete d⁵ energy matrix with D_2d symmetry has been constructed. Then, by diagonalization of this matrix at normal and various pressures,the whole energy spectrum [including the ground-state zero-field-splitting (GSZFS)], its PS and the g factor of the ground state for zns:Mn²⁺ have uniformly been calculated. According to the eigenfunctions and PS, the new assignments of five absorption bands have been given.The variation of tetragonal field with pressure makes a main contribution to the pressureinduced shift (PS) of GSZFS of zns:Mn²⁺, which supports the existence of tetragonal Jahn-Teller distortion in zns:Mn²⁺. It is found that when P≥62 kbar, t₂⁴(³T₁)e⁴T₁ merges with t₂e^{4 2}T₂, which has to be taken into account in the calculation of PS of the fifth band in the range of 1 bar ~ 95 kbar. It is demonstrated that the Mn²⁺ ions in ZnS:Mn²⁺ have tetrahedral coordination, and the difference between ζ and ζ' caused by the covalency effect is very important for GSZFS. The physical essentials of typical levels, GSZFS and their PS have been revealed. By taking into account the influence of covalency on t₂³(⁴A₂)e²(³A₂)⁴A₁ and t₂³(²E)e²(³A₂)⁴E, the positon of the third absorption band at normal pressure has been estimated.     

Pressure-Induced Shift of R-Line of MgO：Cr^3＋ with Electron-Phonon Interaction Effect

By means of improved ligand-field theory, the ＂pure electronic＂ pressure-induced shift （PS） and the PS due to electron-phonon interaction （EPI） of R-line of MgO：Cr^3＋ have been calculated, respectively. The calculated results are in very good agreement with the experimental data. The behaviors of the pure electronic PS of R-line of MgO：Cr^3＋ and the PS of its R-line due to EPI are different. It is the combined effect of them that gives rise to the total PS of R-line, which has satisfactorily explained the experimental results. The comparison between the feature of R-line PS of MgO：Cr^3＋ and that of R1-line PS of ruby has been made.     

Pressure-Induced Shift of R-Line of MgO:Cr3+ with Electron-Phonon Interaction Effect

By means of improved ligand-field theory, the "pure electronic" pressure-induced shift (PS) and the PS due to electron-phonon interaction (EPI) of R-line of MgO:Cr3 have been calculated, respectively. The calculated results are in very good agreement with the experimental data. The behaviors of the pure electronic PS of R-line of MgO:Cr3 and the PS of its R-line due to EPI are different. It is the combined effect of them that gives rise to the total PS of R-line, which has satisfactorily explained the experimental results. The comparison between the feature of R-line PS of MgO:Cr3 and that of R1-line PS of ruby has been made.     

Energy Spectrum of YAG:Cr3+ and Thermal Shifts of Its R Lines

Traditional ligand-field theory has to be improved by taking into account both “pure electronic“ contribution and electron-phonon interaction one (including lattice-vibrational relaxation energy). By means of improved ligand-field theory, R1, R2, R‘3, R′2, and Ri lines, U band, ground-state zero-field-splitting (GSZFS) and ground-state g factors as well as thermal shifts of R1 line and R2 line of YAG:Cr3 have been calculated. The results are in very good agreement with the experimental data. In contrast with ruby, the octahedron of ligand oxygen ions surrounding the central Cr3 ion in YAG:Cr3 is compressed along the [111] direction. Thus, for YAG:Cr3 and ruby, the splitting of t23 4A2 (or t23 2E) has opposite order, and the trigonal-field parameters of the two crystals have opposite signs. In thermal shifts of R1 and R2 lines of YAG:Cr3 , the temperature-dependent contributions due to EPI are dominant.     

Pressure Effects on Spectra of Tunable Laser Crystal GSGG:Cr3+Ⅱ:Energy Spectra at Normal Pressure, Low and Room Temperatures

With the strong-field scheme and trigonal bases, the complete d3 energy matrix in a trigonally distorted cubic-field has been constructed. By diagonalizing this matrix, the normal-pressure energy spectra and wavefunctions of GSGG:Cr3+ at 70 K and 300 K have been calculated without the electron-phonon interaction (EPI), respectively. Further, the contributions to energy spectra from EPI at two temperatures have also been calculated, where temperatureindependent terms of EPI are found to be dominant. The sum of aforementioned two parts gives rise to the total energy spectrum. The calculated results are in good agreement with all the optical-spectral experimental data and the experimental results of g||(R1) and g⊥(R1). It is found that the contribution from EPI to R1 line of GSGG:Cr3+ with taking into account spin-orbit interaction (Hso) and trigonal field (Vtrig) is much larger than the one with neglecting Hso and Vtrig, and accordingly it is essential for the calculation of the EPI effect to take first into account Hso and Vtrig. The admixture of base-wavefunctions, |t32 2E) and |t22(3T1)e4T2 ), the average energy separation △＝ E[t22 (3T1)e4T2]-E[t32 2E] and their variations with temperature have been calculated and discussed.     

Energy Spectrum of La3Lu2Ga3O12：Cr^3＋ and Its Pressure-Induced R-Line-Shift Reversal

By means of both the theory for pressure-induced Shifts （PS） of energy spectra and the theory for shifts of energy spectra due to electron-phonon interaction （EPI）, the normal-pressure energy spectra of α and β centers of Cr^3＋ ions for LLGG：Cr^3＋ and the PS＇s of R1 lines and U band of these centers have been calculated at 10 K, respectively. The total calculated results are in very good agreement with the experimental data. For LLGG：Cr^3＋, the pressureinduced low-high crystal-field transition and the reversal of R1-line PS take place. The pressure-dependent variation of Rmix^ei （2E - 4T2） [mixing-degree of （t2^2 （^3T1）e^4T2） and （t2^3 E） base-wavefunctions in the wavefunction of R1 state without EPI] plays a key role for the reversal of R1-line PS. The behavior of the pure electronic PS of R1 line is quite different from that of the PS of R1 line due to EPI. It is the combined effect of them that gives rise to the total PS of R1 line. The comparison between R1-line PS＇s of GSGG：Cr^3＋ and LLGG：Cr^3＋ has been made. It is found that a peak of R1-line PS appears at Rmix^ei （^2E - ^4T2） ≈ 0.08.