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.     

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.     

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.     

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.     

Improved Ligand-Field Theory with Effect of Electron-Phonon Interaction

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, R₃', R₂', and R₁' lines, U band, ground-state zero-field-splitting (GSZFS), and ground-state g factors of ruby and/or GSGG:Cr³⁺ as well as thermal shifts of GSZFS, R₁ line and R₂ line of ruby have been calculated. The results are in very good agreement with the experimental data. Moreover, it is found that the value of cubic-field parameter given by traditional ligand-field theory is inappropriately large. For thermal shifts of GSZFS, R₁ line and R₂ line of ruby, several conclusions have also been obtained.     

Energy Spectrum,g Factors,Strain-Induced Level-Splittings and R-Line Thermal Shift of MgO：V^2＋＋

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^322T1 and t^322T2 lines, t^22（^3T1）e^4T2, t^22（^3T1）e^4T1 and t2e^2（^4A2）4T1 bands, g factors of t^32 ^4A2 and t32E, four strain-induced level-splittings and R-line thermal shift of MgO：V^2＋ have been calculated. The results are in very good agreement with the experimental data. It is found that for MgO：V^2＋, the contributions due to electronphonon 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：V^2＋, 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.     

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 Shifts of R1 and R2 Lines of YAG:Cr3+

By means of improved ligand-field theory, the "pure electronic" pressure-induced shifts (PS's) and the PS's due to electron-phonon interaction (EPI) of R1 line and R2 line of YAG:Cr3 have been calculated, respectively.The calculated results are in very good agreement with the experimental data. It is demonstrated that the admixture of |t22(3T1)e4T2> and |t322E> bases in the wavefunction of R1 level of YAG:Cr3 and its change with pressure play a key role for the PS of R1 line. The behaviors of the "pure electronic" PS of R1 line and the PS of R1 line due to EPI are different. It is the combined effect of them that gives rise to the total PS of R1 line, which has satisfactorily explained the experimental results. The systematic analyses and comparisons between the feature of R1-line PS of YAG:Cr3 and the ones of three laser crystals (GSGG:Cr3 , GGG:Cr3 and ruby) have been made, and the origin of the difference between them has been revealed.     

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 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.     

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.     

Energy Spectrum of La3Lu2Ga3O12:Cr3+ 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 andβ centers of Cr3 ions for LLGG:Cr3 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:Cr3 , the pressureinduced low-high crystal-field transition and the reversal of R1-line PS take place. The pressure-dependent variation of Relmix (2E- 4T2) [mixing-degree of |t22(3T1)e4T2) and |t322E) 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:Cr3 and LLGG:Cr3 has been made. It is found that a peak of R1-line PS appears at Relmix (2E - 4T2) ≈ 0.08.     

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.     

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 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.     

BeAl2O4:Cr3+的能谱和R1线压力移位的理论计算

采用强场方案和三角基,通过对角化三角畸变立方场的d3电子组态完全能量矩阵,拟合得到BeAl2O4:Cr3+的能谱和波函数.利用获得的波函数,计算出了基态g因子和R1 、R2线能谱压致移位值,计算值和实验结果相符合.各参数对能级压力移位率的影响也被定量计算,揭示了R1和R2线压力移位的物理起源.     

Improved Ligand-Field Theoretical Calculation of R-Line Pressure-Induced Shift of MgO:V2+

By means of an improved ligand-field theory, the “pure electronic” PS and the PS due to EPI of R line of MgO:V²⁺ 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:V²⁺ 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 mixing-degree of |t₂²(³T₁)e⁴T₂〉and |t₂^{3 2}E〉 in the wavefunction of R level and its variation with pressure have been calculated and analyzed. The comparison between the feature of R-line PS of MgO:V²⁺ and that of MgO:Cr³⁺ has been made.     

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.     

LiNbO_3:Ni~(2+)的常压能谱和g因子

采用强场方案,通过将d8完全能量矩阵对角化,统一计算了LiNbO3:Ni2+的常压能谱和g因子,计算结果与大量实验数据符合很好.给出了各个能级对不同参量的变化率.研究表明,利用对角化完全能量矩阵获得的波函数对g因子所作的计算为整个理论计算及波函数的归属提供了重要判据,充分体现了将能谱和g因子作统一计算的重要性和必要性. 关键词： 晶场 能谱 g因子     

Theoretical Studies of Energy Spectra and g Factors of Cr3+-Doped YAl3(BO3)4

With the strong-field scheme and trigonal bases, by diagonalizing the complete d³ energy matrix in a trigonally distorted cubic-field, the energy spectra and wavefunctions of YAl₃(BO₃)₄:Cr³⁺ have been calculated. The rates of change of levels with respect to various parameters and the contributions to levels from various parameters are calculated, and the physical origins of various levels or splittings have been clearly and quantitatively shown. By using the wavefunctions obtained from diagonalizing the complete energy matrix, the g factors of the ground state of YAl₃(BO₃)₄:Cr³⁺ have been evaluated. The calculated results are in good agreement with the optical-spectral and EPR experimental data. It is demonstrated that the bonding between Cr³⁺ and ligands (O^2-) is ionic.