Pressure Effects on Spectra of Tunable Laser Crystal GSGG:Cr 3+ II: Energy Spectra at Normal Pressure, Low and Room Temperatures

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1 Commun. Theor. Phys. (Beijing, China) 39 (2003) pp c International Academic Publishers Vol. 39, No. 3, arch 15, 2003 Pressure Effects on Spectra of Tunable Laser Crystal GSGG:Cr 3+ II: Energy Spectra at Normal Pressure, Low and Room Temperatures ZHANG Ji-Ping 1 and A Dong-Ping 1,2, 1 Department of Applied Physics, Sichuan University, Chengdu , China 2 International Centre for aterials Physics, the Chinese Academy of Sciences, Shenyang , China (Received August 20, 2002) Abstract With the strong-field scheme and trigonal bases, the complete d 3 energy matrix in a trigonally distorted cubic-field has been constructed. By diagonalizing this matrix, the normal-pressure energy spectra and wavefunctions of GSGG:Cr 3+ 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 (R 1) and g (R 1). It is found that the contribution from EPI to R 1 line of GSGG:Cr 3+ with taking into account spin-orbit interaction (H so) and trigonal field (V trig) is much larger than the one with neglecting H so and V trig, and accordingly it is essential for the calculation of the EPI effect to take first into account H so and V trig. The admixture of base-wavefunctions, t E and t 2 2( 3 T 1)e 4 T 2, the average energy separation = Ē[t2 2( 3 T 1)e 4 T 2] Ē[t3 2 2 E] and their variations with temperature have been calculated and discussed. PACS numbers: Ch, Ej, t Key words: crystal fields, optical properties, spin-orbital interaction, electron-phonon interaction, coupling between t 2 2( 3 T 1 )e 4 T 2 and t E, tunable laser crystal 1 Introduction An important class of Cr 3+ -doped host materials is oxide garnets, and Cr 3+ -doped gallium garnets represent a new class of laser crystals for tunable continuous-wave operation at room temperature in the near infrared spectral range, such as the Cr 3+ -doped Y 3 Ga 5 O 12 (YGG), Y 3 (Sc,Ga) 2 Ga 3 O 12 (YSGG), Gd 3 Ga 5 O 12 (GGG), Gd 3 Sc 2 Ga 3 O 12 (GSGG), and (La,Lu) 5 Ga 3 O 12 (LLGG). Among them, GSGG:Cr 3+ is of special importance, since it is an intermediate crystal-field-strength system and has excellent properties. Namely, the average or effective energy separation between t 2 2( 3 T 1 )e 4 T 2 and t E states of Cr 3+ ion in GSGG:Cr 3+ is small. In this case, these states strongly couple with one another through spin-orbit interaction and electron-phonon interaction (EPI), and the lowest excited states have strong admixtures of base-wavefunctions, t E and t 2 2( 3 T 1 )e 4 T 2. Thus, the emission spectrum has characteristics of both t E and t 2 2( 3 T 1 )e 4 T 2 ; the degree of their admixture is reflected in the spectral features; unique and very attractive properties can be observed, especially under pressure. any researches have been devoted to the luminescence properties and the coupling between t 2 2( 3 T 1 )e 4 T 2 and t E states of GSGG:Cr 3+. [1 17] In order to study the luminescence properties and the degree of mixing of t E and t 2 2( 3 T 1 )e 4 T 2 basewavefunctions for GSGG:Cr 3+ and their variations with temperature and pressure, it is necessary to obtain the whole energy spectra and wavefunctions at various temperatures and pressures by means of the diagonalization of the complete d 3 energy matrix (DCE) in a trigonally distorted cubic field. In this paper, first of all, the normalpressure energy spectra and wavefunctions at 70 K and 300 K will be calculated by using DCE, which are the foundations of calculations in the forthcoming papers of this series. 2 Normal-Pressure Energy Spectra Without EPI Garnet crystals have chemical formula A 3 B 2 C 3 O 12, where A denotes the dodecahedral, B the octahedral, and C the tetrahedral lattice sites. Cr 3+ ions preferentially occupy the B sites, and possess a trigonally distorted octahedral crystal-field. In GSGG:Cr 3+, Cr 3+ substitutes for the Sc 3+ and enters a site of trigonally distorted octahedral symmetry. The Cr 3+ ion is the optical center responsible for tunable laser action at room temperature (e.g., see Refs. [5], [6], [9], and [16]). As is well known, the open shell of the Cr 3+ ion has a (3d) 3 electronic configuration. For convenience, the strong-field scheme and trigonal bases are adopted. [18] Similar to Refs. [19] and [20], we maurice@mail.sc.cninfo.net

2 350 ZHANG Ji-Ping and A Dong-Ping Vol. 39 have calculated all the matrix elements of the trigonal field and spin-orbit interaction within d 3 electronic configuration. The matrix elements of both the cubic field and Coulomb interaction between d-electrons were given in Ref. [18]. Then, with all the matrix elements of the cubic-field, trigonal-field, Coulomb and spin-orbit interactions, we have constructed the complete energy matrix of d 3 electronic configuration in a trigonally distorted cubic-field. Among the parameters including in this matrix, B 0 and C 0 are Racah parameters; Dq 0 is the cubic-field parameter; ζ 0 and ζ 0 are the spin-orbit coupling parameters; K 0 and K 0 are the trigonal-field parameters. In order to reduce the number of parameters, the approximations ζ 0 = ζ 0 and K 0 = K 0 can be made. The subscript 0 indicates the quantities at normal pressure. According to Refs. [5], [7], and [9], the GSGG:Cr 3+ exhibits a degree of disorder in the arrangement of cations, so that Ga 3+ ions can occupy Sc 3+ sites while Sc 3+ ions can occupy Ga 3+ sites. The presence of such disorder in the vicinity of the Cr 3+ ions leads to perturbed sites of higher and of lower crystal field strength, respectively. Thus, in GSGG:Cr 3+, there are four non-equivalent Cr 3+ sites (A, B, C and D). From observed fluorescence line narrowing (FLN) spectra at 4.2 K and normal pressure (see Fig. 4 in Ref. [5]), the R 1 -line transition energies of the four sites were approximately cm 1, cm 1, cm 1 and cm 1 ; their R 1 R 2 separations were 20 cm 1, 30 cm 1, 36 cm 1 and 40 cm 1, respectively. The measured ratio for the peak intensities of these R 1 -lines 1:0.65:0.16:0.06 was obtained. [5] From these data, we get that the weight-mean of these R 1 -line transition energies is cm 1 ; the weight-mean of these R 1 R 2 separations is 25.5 cm 1. It is noteworthy that the observed R 1 -line transition energy at 70 K and normal pressure in Refs. [15] and [16] was also cm 1. Then, this value is taken as experimental result to determine the values of parameters B 0 and C 0 at low temperature and normal pressure, since the R 1 -line transition energy depends mainly on B 0 and C 0. The value of ζ 0 = 170 cm 1 is taken from Ref. [14]. The R 1 R 2 separation 25.5 cm 1 is taken as experimental result to determine the value of the parameter K 0 at low temperature and normal pressure, because the R 1 R 2 separation depends mainly on K 0 and ζ 0. According to Tables 1 and 2 in Ref. [2], from 30 K to room temperature, the R 1 -line red shifts of YGG:Cr 3+, GGG:Cr 3+ and YSGG:Cr 3+ were 50 cm 1. Thus, from 70 K to room temperature, the R 1 -line red shift of GSGG:Cr 3+ may be estimated as 42 cm 1, and accordingly its R 1 -line transition energy at room temperature and normal pressure should be cm 1. According to Fig. 4. (b) in Ref. [16], at room temperature, the observed R 1 R 2 separation of GSGG:Cr 3+ at 50 kbar was 27 cm 1, and its R 1 R 2 separation at 100 kbar was 25 cm 1 (notice that its R 2 line cannot be observed at room temperature and 45 kbar). From the extrapolation, its R 1 R 2 separation at room temperature and normal pressure is 29 cm 1. We employ these results to determine the values of B 0, C 0 and K 0 at room temperature and normal pressure (We consider ζ 0 approximately unchanged with temperature). According to Ref. [16], at room temperature and normal pressure, the peak position of U emission band (t 2 2( 3 T 1 )e 4 T 2 t A 2 transition) was 740 nm ( cm 1 ). From Ref. [2], the peak position of U absorption band at room temperature and normal pressure was cm 1. These results are employed to determine the value of parameter Dq 0. By using aforementioned results and by means of DCE, after taking into account the contributions to energy spectra form EPI (see the later text), we have determined the values of parameters at 300 K and normal pressure Dq 0 = 1472 cm 1, B 0 = cm 1, C 0 = 3251 cm 1, ζ 0 = 170 cm 1, K 0 = 87 cm 1, (1) the whole energy spectrum and wavefunctions have also been obtained. For simplicity, only part of the levels is shown in Table 1. Table 1 The parts of normal-pressure energy spectra of GSGG:Cr 3+ at 70 K and 300 K without EPI, in unit cm 1. α 2S+1 Γ S Calculated at 300 K Calculated at 70 K t A 2 ± 1 2 e t A 2 ± 3 2 e t E ± 1 2 u t E ± 1 2 u ± t 2 2 (3 T 1 )e 4 T 2 ± 3 2 x t 2 2 (3 T 1 )e 4 T 2 ± 1 2 x t 2 2 (3 T 1 )e 4 T 2 ± 1 2 x ± t 2 2 (3 T 1 )e 4 T 2 ± 3 2 x ± t 2 2 (3 T 1 )e 4 T 2 ± 1 2 x t 2 2 (3 T 1 )e 4 T 2 ± 3 2 x t 3 2 2T 1 ± 1 2 a t 3 2 2T 1 ± 1 2 a ± t 3 2 2T 1 ± 1 2 a t 2 2 (3 T 1 )e 4 T 1 ± 3 2 a ± t 2 2 (3 T 1 )e 4 T 1 ± 1 2 a ± t 2 2 (3 T 1 )e 4 T 1 ± 1 2 a t 2 2 (3 T 1 )e 4 T 1 ± 3 2 a t 2 2 (3 T 1 )e 4 T 1 ± 1 2 a t 2 2 (3 T 1 )e 4 T 1 ± 3 2 a At 300 K, the average energy of the six electronic levels of t 2 2 (3 T 1 )e 4 T 2 (without EPI) is cm 1, and the energy of the ground electronic level (without EPI) is taken as zero.

3 No. 3 Pressure Effects on Spectra of Tunable Laser Crystal GSGG:Cr 3+ II 351 Similarly, the values of parameters at 70 K and normal pressure have been determined, Dq 0 = 1476 cm 1, B 0 = cm 1, C 0 = 3253 cm 1, ζ 0 = 170 cm 1, K 0 = 90 cm 1. (2) Correspondingly, the whole energy spectrum and wavefunctions have also been obtained, and only part of the levels are also shown in Table 1. The differences between Eqs. (1) and (2) are mainly the increases of Dq 0 and K 0 with decreasing temperature, which means the increase of crystal-field strength (corresponding to the decrease of interionic distance) with decreasing temperature. So, in view of the thermalexpansion, this is reasonable. Because Ē[t2 2( 3 T 1 )e 4 T 2 ] depends principally on Dq 0 and increases with Dq 0, while Ē[t3 2 2 E] depends only slightly on Dq 0, the average energy separation = Ē[t 2 2( 3 T 1 )e 4 T 2 ] Ē[t3 2 2 E] increases with temperature decreasing. When the trigonal field and spin-orbit interaction are omitted, by using DCE as well as the values of Dq 0, B 0 and C 0 in Eq. (1) and ζ 0 = K 0 = 0, we get the value of to be cm 1 at 300 K and normal pressure; similarly, from the values of Dq 0, B 0, and C 0 in Eq. (2), we get the value of to be cm 1 at 70 K and normal pressure. Therefore, we have δ /δt = 0.12 cm 1 /K, which is close to the value 0.15 cm 1 /K given by Ref. [13]. Now, we study admixture of t E and t 2 2( 3 T 1 )e 4 T 2 base-wavefunctions. At 70 K or 300 K, from the normalpressure wavefunctions corresponding to R 1 (t E ± 1 2 u ) level obtained by DCE (Generally, it is the admixture of various bases, and the maximum mixing-coefficient is one of mixing-coefficients of t E bases), we get the sum of square mixing-coefficients of all t 2 2( 3 T 1 )e 4 T 2 bases, the sum of square mixing-coefficients of all t E ± 1 2 u bases, and their ratio. The ratio may represent the degree of the mixing of t E and t 2 2( 3 T 1 )e 4 T 2 base-wavefunctions. It can be seen from Table 2 that it increases with temperature increasing, which results mainly from the decrease of with temperature increasing. Here, we take into account only the mixing of electronic wavefunctions, and the nuclear-vibrational wavefunctions and EPI are not considered. EPI will lead to extra wavefunction-mixing. With decreasing temperature, the degree of mixing of t E and t 2 2( 3 T 1 )e 4 T 2 electronic base-wavefunctions decreases, and the contribution to R 1 level from EPI (and accordingly the degree of the wavefunction-mixing due to EPI) decreases, too. oreover, as will be shown in forthcoming papers, both the degree of mixing of electronic wavefunctions and the contribution to R 1 level from EPI (and accordingly the degree of the wavefunction-mixing due to EPI) decreases rapidly with the increasing of pressure. Hence, for the sake of simplification and convenience, we may approximately represent the relative variation of the degree of the total wavefunction-mixing by use of the relative variation of electronic wavefunction-mixing without EPI. Table 2 Admixtures of t E ± 1 u and 2 t2 2( 3 T 1)e 4 T 2 bases in the normal-pressure wavefunctions corresponding to R 1 level at 300 K and 70 K without EPI. T 300 K 70 K The sum of square mixing-coefficients of t 2 2 (3 T 1 )e 4 T 2 bases The sum of square mixing-coefficients of t E ± 1 2 u bases Their ratio From Ref. [10], when Cr 3+ ions in GSGG:Cr 3+ are optically pumped to t E and t 2 2( 3 T 1 )e 4 T 2 states, a very fast nonradiative relaxation occurs between t E and t 2 2( 3 T 1 )e 4 T 2 states (relaxation rate s 1 ), which is much faster than the radiative decay rate out of these states. Thus, the populations in these states are in thermal equilibrium during the luminescence process. According to the Boltzmann distribution, obviously the populations in t 2 2( 3 T 1 )e 4 T 2 levels at 70 K are much smaller than those in t 2 2( 3 T 1 )e 4 T 2 levels at 300 K. This is mainly due to the difference in temperature; besides, as was mentioned, the average separation between t 2 2( 3 T 1 )e 4 T 2 and t E at 70 K is larger than the one at 300 K. It is noteworthy that the radiative decay rate between t 2 2( 3 T 1 )e 4 T 2 and t A 2 is s 1, which is spin-allowed; however, the radiative decay rate between t E and t A 2 is only 10 3 s 1, which is spin-forbidden, but become weakly allowed due to the admixture of t 2 2( 3 T 1 )e 4 T 2 and t E bases. oreover, the ratio between the degeneracy of t 2 2( 3 T 1 )e 4 T 2 and the one of t E is equal to three. Thus, the higher the temperature, the more important the t 2 2( 3 T 1 )e 4 T 2 emission. In fact, the emission spectrum of GSGG:Cr 3+ at 70 K and normal pressure consists of overlapping contributions of the R 1 -lines and their sidebands from t E emission and the broad band from t 2 2( 3 T 1 )e 4 T 2 emission. [15] However, at 300 K and normal pressure, only the t 2 2( 3 T 1 )e 4 T 2 emission band is observed, and it appears

4 352 ZHANG Ji-Ping and A Dong-Ping Vol. 39 that the R-line emission is too broad and flat to be observed, due to broadenings caused by EPI and the admixture of t 2 2( 3 T 1 )e 4 T 2 and t E bases. 3 Contributions to Energy Spectra from EPI According to Ref. [21], the contribution to i-state energy from EPI due to acoustic branches (the so-called acousticbranch term, it is in cm 1 unit) is δe ac (i) = δe ac,t (i) + δe ac,0 (i), δe ac,t (i) = j i D ij T ij T 2 P δe ac,0 (i) = 1 D ij P 2 D ij = F ii = j i k 3 B 60π 3 cρ 4 ( 1 v 5 l k 4 B 60π 3 cρ 4 ( 1 v 5 l + 3 2v 5 t + 3 2v 5 t TD 0 TD /T 0 )[ ) x 3 TD /T ( e x 1)[x 2 (T ij /T ) 2 ] dx + F iit 4 0 (3) x 3 e x dx, (4) 1 y 3 y T ij dy F iit 4 D, (5) ψ el i C(T 2 ) ψ el j ψ el i C(E) ψ el j 2], (6) ψi el D(T 2 ) ψi el, (7) where k B is the Boltzmann constant; = h/2π, h is the Planck constant, T D = ω D /k B is Debye temperature of acoustic branches, T ij = (Ei el Ej el)/k B, P indicates the principal value of an integral (when T ij < T D, there is a singular point in the first integral, there is a singular point in the second integral, when 0 < T ij < T D ), c is the velocity of light in the vacuum, ρ is the crystal density, ψ el and E el denote the d-electronic wavefunction and energy obtained from DCE, respectively. The summation over j includes all the levels except for level i. δe ac,t (i) is temperature-dependent, and δe ac,0 (i) is temperature-independent. The contribution to i-state energy from EPI due to optical branches (the so-called optical-branch term, in cm 1 unit) is δe op (i) = δe op,t (i)] + δe op,0 (i), δe op,t (i) = γ T (i)(td/t 5 op )( e Top/T 1) 1 and δe op,0 (i) = γ 0 (i)(td/t 5 op ), (9) [ k 4 [ ψi el C(T 2) ψj el ψi el C(E) ψel j 2 ] B γ T (i) = 120π 3 cρvs 5 4 (Ei el Ej el (E el )2 ( ω eff ) 2 i Ej el ) + ] ψi el D(T 2 ) ψi el, γ 0 (i) = k 4 B 240π 3 cρv 5 s 4 j i [ [ j i ψi el C(T 2) ψj el ψi el C(E) ψel j 2 ] E el i E el j ω eff + ψ el i D(T 2 ) ψ el i ] (8), (10) where v s is defined as 3/vs 3 = 2/vt 3 +1/vl 3, T op = ω eff /k B, ω eff is the effective frequency of the single-frequency model for optical branches, δe op,t (i) is temperature-dependent and δe op,0 (i) is temperature-independent. The detailed calculations show that the behavior of the variation of the optical-branch term with temperature is different from the one of the acoustic-branch term. However, at a constant temperature, the behavior of the variation of the optical-branch term with pressure is very close to the one of the acoustic-branch term. Namely, the ratio of the optical-branch term to the acoustic-branch term is almost unchanged with pressure. oreover, it is found that the contribution of the optical-branch term is obviously smaller than the one of the acoustic-branch term. Thus, for the calculation at a constant temperature, we may approximately take the sum of the acoustic-branch term and the optical-branch term as the acoustic-branch term multiplying a factor C a (C a > 1). On one hand, this approximation simplifies greatly calculations; on the other hand, it can avoid the shortcoming of the single-frequency model [i.e., if (Ei el Eel j ) ω eff, the singularity will make the calculation invalid]. In order to calculate all the ψi el C(T 2) ψj el, ψel i C(E) ψj el, and ψel i D(T 2) ψi el, the reduced matrix elements of C(T 2 ), C(E), and D(T 2 ) for three electrons are necessary. According to Ref. [22], all of them were virtually expressed in terms of six reduced matrix elements for a single electron Y c ( t 2 C(T 2 ) t 2 ), Z c ( t 2 C(T 2 ) e ), P c ( t 2 C(E) t 2 ), Q c ( e C(E) e ), Y d ( t 2 D(T 2 ) t 2 ) and Z d ( t 2 D(T 2 ) e ). The values of Y c, Z c, P c, Q c, Y d, and Z d can be determined from

5 No. 3 Pressure Effects on Spectra of Tunable Laser Crystal GSGG:Cr 3+ II 353 the observed data of the stress-induced changes in splittings and the thermal shifts of optical spectra. [22] At the present time, however, there are no such observed data for GSGG:Cr 3+. Hence, we have to take approximately the values of Y c, Z c, P c, Q c, Y d, and Z d of GSGG:Cr 3+ as corresponding values of ruby multiplying an adjustable factor C b, which takes already into account the differences between properties (such as the strength of EPI, elastic properties and density) of GSGG:Cr 3+ and ruby. So, in the calculation, the value of (1/ρ)(1/vl 5 + 3/2v5 t ) for ruby is also adopted. By combining C a and C b, only an adjustable parameter C m = C a C b is necessary, which will be determined by fitting the experimental data of the optical spectra of GSGG:Cr 3+ and their pressure-induced shifts (PS). We fail to find the datum of T D of GSGG:Cr 3+. According to Ref. [23], T D of Y 3 Al 5 O 12 (YAG):Cr 3+ is K, and T D of ruby is K. These results were obtained by fitting the specific heat, and all phonon branches contribute to the specific heat. However, according to Ref. [22], T D of acoustic branches of ruby is 780 K. Therefore, T D of acoustic branches of YAG:Cr 3+ should approximately be (750/1030) 780 K = 568 K. Because the distance between Cr 3+ and ligand ions of GSGG:Cr 3+ is larger than the one of YAG:Cr 3+ and accordingly the crystal field and force constants of GSGG:Cr 3+ are smaller than those of YAG:Cr 3+, T D of acoustic branches of GSGG:Cr 3+ should be smaller than T D of acoustic branches of YAG:Cr 3+. As a reasonable approximation, we can take T D 500 K for acoustic branches of GSGG:Cr 3+. In calculations of two principal values of integrals and I 2 ( T ij, T ) = P I 1 (T ij ) = P TD /T 0 TD 0 y 3 y T ij dy (11) x 3 ( e x 1)[x 2 (T ij /T ) 2 dx, (12) ] it is necessary to make some corrections. Firstly, from Figs. 1 3, we can see that they show singularly large values, when T ij [for I 1 (T ij )] or T ij [for I 2 ( T ij, T )] is in the vicinity of T D. Obviously, this is caused by the Debye approximation (i.e., the Debye distribution function g(ω) is cut off at ω D k B T D / ). In fact, according to the observed phonon sideband, the real phonon distribution is different from the Debye distribution. For example, when ω > ω D, g(ω) 0 and it usually has a long tail; the peak of Debye distribution should be flattened and shifted to the left. Note that the integrand has opposite signs at two sides of the T ij point. Thus, as a correction, the fictitious singular peaks in Figs. 1 3 have to be appropriately cut off. Secondly, it is seen from Figs. 1 3 that I 1 (T ij ) [or I 2 ( T ij, T )] is equal to zero at a turning point Tij turn (or Tij turn ) in the range of T ij [or T ij ] from 0 K to 500 K, and it has opposite signs at two sides of the turning point. It is readily seen that the substitution of the real phonon distribution for Debye distribution leads to a decrease of T turn ij (or T turn ij ). Fig. 1 I 1(T ij) as a function of T ij (T D = 500 K). Fig. 2 The variation of I 2( T ij, T ) (Eq. (12)) with T ij (T D = 500 K, T = 300 K). According to aforementioned argument, we adopt the following corrections. For I 1 (T ij ), it is calculated in four various ranges, respectively. (i) In the range of T ij 600 K, the correction is not made. (ii) In the range of 500 K T ij 600 K, I 1 (T ij ) is taken as I 1 (600 K). Namely, we adopt the same value as its value at 600 K. In other words, the singular peak has been appropriately cut off. (iii) In the range of 0 K T ij 430 K, we take Eq. (11) and TD C is reasonably taken as 450 K. The correction leads to a decrease of Tij turn. (iv) In the range of 430 K T ij 500 K, the linear interpolation is adopted.

6 354 ZHANG Ji-Ping and A Dong-Ping Vol. 39 For I 2 ( T ij, T ) at T = 300 K, similar corrections are adopted. (i) In the range of T ij 600 K, the correction is not made. (ii) In the range of 500 K T ij 600 K, I 2 ( T ij, T ) is taken as I 2 ( T ij, 600 K). (iii) In the range of 0 K T ij 430 K, we take Eq. (12) and TD C is taken as 480 K. (iv) In the range of 430 K T ij 500 K, the linear interpolation is adopted. For I 2 ( T ij, T ) at T = 70 K, the correction is very simple. From Fig. 3, it is sufficient to make a correction of linear interpolation between 485 K and 515 K. Fig. 3 The variation of I 2( T ij, T ) (Eq. (12)) with T ij (T D = 500 K, T = 70 K). We have found that the effect of aforementioned corrections on the calculation of EPI of R-lines is small; however, its effect on the calculation of EPI of U-band is sometimes obvious. By using the aforementioned method and results as well as the electronic wavefunctions and energy spectrum obtained by DCE, with fitting the observed results of optical spectra of GSGG:Cr 3+ and their PS, [2,5,15,16] we have determined C m = Namely, from this and Ref. [22], we have Y c = cm 1, Z c = cm 1, P c = cm 1, Q c = cm 1, Y d = cm 1 and Z d = cm 1. Then, the contributions to normal-pressure energy spectra from EPI have been evaluated. At 300 K, the contributions to normal-pressure R 1 -line, R 2 -line and U-band from EPI are cm 1, cm 1 and cm 1, respectively; at 70 K, the contributions to normal-pressure R 1 -line, R 2 -line and U-band from EPI are cm 1, cm 1 and cm 1, respectively. Note that the contribution to normal-pressure ground level from EPI has been deducted (e.g., the contribution to R 1 -line is the one to R 1 [t E ± 1 2 u ] level minus the one to ground level). The contribution to normal-pressure U-band from EPI is the average of the contributions to normal-pressure six levels of t 2 2( 3 T 1 )e 4 T 2 from EPI minus the contribution to normal-pressure ground level from EPI. From the aforementioned results, obviously, the contribution from EPI to U-band is much larger than the one to R 1 - or R 2 -line. This is in accordance with the fact that EPI of t 2 2( 3 T 1 )e 4 T 2 is much larger than EPI of t E. Furthermore, at 300 K or 70 K, the ratio of the contribution from EPI between R 2 (t E ± 1 2 u ±) levels and t 2 2( 3 T 1 )e 4 T 2 levels to the one between R 2 and all the levels other than R 2 is about 74%. There is similar result for R 1 at 300 K or 70 K. These results indicate also that t 2 2( 3 T 1 )e 4 T 2 plays a dominant role in the effect of EPI. The contributions from EPI consist of temperatureindependent contribution and temperature-dependent one. It is found that the temperature-independent contribution is dominant, and the temperature-dependent contribution is also considerable at 300 K. For example, at 300 K, the contribution to R 1 line from the temperature-independent terms of EPI is cm 1, and the one from the temperature-dependent terms of EPI is cm 1 ; the temperature-independent contribution to R 1 R 2 separation is 4.96 cm 1, and the temperature-dependent one is 2.64 cm 1 ; there are similar results for R 2 -line and U-band. However, at 70 K, it is found that the temperature-dependent contribution is much smaller than the temperature-independent contribution. For example, the contribution to R 1 -line from the temperature-independent terms of EPI is cm 1, and the one from the temperature-dependent terms of EPI is only 1.10 cm 1 ; the temperature-independent contribution to R 1 R 2 separation is 4.70 cm 1, and the temperature-dependent one is only 0.01 cm 1. In Ref. [24], from the experiment and theoretical calculation, it is demonstrated that the temperatureindependent contribution of EPI (i.e., zero-temperature one) is very important, which gives rise to the isotope shifts in the R-lines of Cr 3+ in ruby and go:cr 3+ (because the zero-temperature contribution depends on the mass of the dopant ion). In Ref. [25], it was pointed out that the temperature-independent contribution of EPI may be called as zero field contribution. It is the phonon zero point (zero-point vibration of the lattice) contribution similar to the Lamb shift due to the interaction of the atomic systems with the zero electromagnetic field. In this paper, we have found the important effect of the temperature-independent contribution of EPI on the normal-pressure R 1 -line, R 2 -line and U-band of GSGG:Cr 3+, especially at low temperatures. Furthermore, because the temperature-independent contribution of EPI depends on pressure, the very important effect of the temperature-independent contribution of EPI on the

7 No. 3 Pressure Effects on Spectra of Tunable Laser Crystal GSGG:Cr 3+ II 355 pressure-induced shifts of R 1 -line, R 2 -line and U-band of GSGG:Cr 3+ will be shown in the forthcoming papers. As for the temperature-dependent terms, from their derivation, their dependencies on temperature are mainly from the phonon occupation number n kλ (see Ref. [21]). Thus, their effects increase with T. However, the dependence on T of acoustic-branch term is different from the one of optical-branch term, since n kλ is substituted by ( e ω eff /k B T 1) 1 and is taken out of summation (or integral) for the latter (single-frequency model). 4 Comparison Between Theoretical and Experimental Results In Sec. 2, by means of DCE, the normal-pressure energy spectra without EPI have been obtained. Further, the contributions to normal-pressure energy spectra from EPI have also been obtained in Sec. 3. The sum of the two parts gives rise to the total energy spectra. Therefore, at 300 K, we have obtained that the calculated R 1 -line transition energy is cm cm 1 = cm 1 ; the R 1 R 2 separation is 21.5 cm cm 1 = 29.1 cm 1. They are in good agreement with aforementioned experimental data in Sec. 2. The average energy of the six electronic levels of t 2 2( 3 T 1 )e 4 T 2 minus the energy of the ground electronic level gives rise to the zero-phonon-line energy of U emission band (of course, the contributions to electronic levels from the effect of EPI should be taken into account). From Table 1, without EPI, the average energy of the six electronic levels of t 2 2( 3 T 1 )e 4 T 2 minus the energy of the ground electronic level is cm 1. As for the contribution of EPI, from Sec. 3, the average of the contributions from EPI to the six levels of t 2 2( 3 T 1 ) 4 T 2 minus the contribution from EPI to the ground level is cm 1. Thus, their algebraic sum cm 1 gives rise to the zero-phononline energy of U emission band. Now, the physical meaning of this result is explained as follows. According to Franck Condon principle, for the t 2 2( 3 T 1 )e 4 T 2 (it may be briefly called as 4 T 2 ) absorption band, the vertical transition from the relaxed ground state t A 2 of Cr 3+ ion to the unrelaxed 4 T 2 state (electronicvibrational state) occurs. Then, the Cr 3+ ion reaches the relaxed 4 T2 state (zero-vibrational electronic state) from the 4 T 2 state via lattice vibrational relaxation (the energy variation is half of the Stokes shift). For the emission band, the vertical transition from the 4 T2 state to the unrelaxed ground state t A 2 of Cr 3+ ion occurs. And then, the Cr 3+ ion reaches the relaxed t A 2 ground state from the unrelaxed ground state (the energy variation is also half of the Stokes shift). [8,9] For the emission band considered in this paper, obviously, we should consider 4 T2 state (including the interaction between it and t E. Note that t E t A 2 transition has no Stokes shift; t E depends only slightly on Dq). Thus, aforementioned result cm 1 represents the energy separation between 4 T2 and t A 2 electronic levels including the effect of EPI (i.e., the energy of the zero-phonon line). According to Ref. [2], the Stokes shift of 4 T 2 broad band of GSGG:Cr 3+ is 2290 cm 1. Therefore, the transition energy of the 4 T 2 absorption band is cm cm 1 /2 = cm 1, which is in good agreement with the observed result of Ref. [2] cm 1 within experimental error. oreover, the transition energy of the 4 T 2 emission band is cm cm 1 /2 = cm 1, which is also in good agreement with the observed result of Ref. [16] cm 1. It should be pointed out that the average energy separation between 4 T2 and t A 2 levels (14554 cm 1 ) consists of the average of six levels of t 2 2( 3 T 1 )e 4 T 2 without EPI ( cm 1 ) and the average of the contributions due to EPI of the six levels ( cm 1 ). The former is obtained from Eq. (1), in which Dq = 1472 cm 1. This is different from the value of Dq in Table 1 of Ref. [2], which was given by 10Dq = cm 1 (It is the position of the peak of 4 T 2 absorption band). oreover, in Ref. [2], the value of the energy separation between t 2 2( 3 T 1 )e 4 T 2 and t E levels (noticing that the splitting of t 2 2( 3 T 1 )e 4 T 2 or t E was neglected in Ref. [2]) was given as 50 cm 1, which was determined by the difference between the zero-phonon line position E 0 ( 4 T 2 ) = cm 1 and the one E 0 ( 2 E) = cm 1. However, these values were inaccurate. For example, according to this work, E 0 ( 4 T 2 ) = cm 1, E 0 ( 2 E) = cm 1 (It is the average of E 0 (R 1 ) = cm cm 1 = cm 1 and E 0 (R 2 ) = cm cm 1 = cm 1 ), and accordingly the value of is 212 cm 1. In a word, the evaluation and physical meaning of Dq, B and in this work are different from those in Ref. [2]. In Ref. [11], it was pointed out that Ref. [2] fails to explain the energies of perturbed states, especially the ones with small quartet-doublet separation (i.e., 4 T 2 2 E energy gap); the 4 T 2 2 E energy gap versus the 4 T 2 zerophonon energetic position E 0 ( 4 T 2 ) of various Cr 3+ -doped garnets (Fig. 1 in Ref. [2]) is apparent lack of anticrossing behavior. As is well known, by taking into account spin-orbit interaction and EPI, the anticrossing behavior of t E and t 2 2( 3 T 1 )e 4 T 2 levels takes place. [11,15] oreover, the values of Dq and B in Ref. [2] were obtained from observed peaks of absorption bands (i.e., E a [t 2 2( 3 T 1 )e 4 T 2 ] and E a [t 2 2( 3 T 1 )e 4 T 1 ]) and approximate formulas (Eqs. (1) and (2) in Ref. [2]), where spin-orbit interaction and trigonal field were neglected. At low temperatures and normal pressures, there are only the observed data of the transition energy of R 1 - line and the R 1 R 2 separation (see Sec. 2). Here, thus,

8 356 ZHANG Ji-Ping and A Dong-Ping Vol. 39 only their calculated results are given at 70 K. We have evaluated that the R 1 -line transition energy without EPI is cm 1, the contribution to the R 1 -line transition energy from EPI is cm 1, and their algebraic sum gives rise to the total R 1 -line transition energy cm 1, which is in very good agreement with the observed datum. oreover, the calculated R 1 R 2 separation is cm 1, which is also in good agreement with the observed datum (see Sec. 2). In the calculated result of R 1 R 2 separation at 300 K, the contribution without EPI is cm 1, and the contribution from EPI is 7.60 cm 1. Similarly, in the calculated result of R 1 R 2 separation at 70 K, the contribution without EPI is cm 1, and the contribution from EPI is 4.71 cm 1. Therefore, the contributions from EPI are considerable. 5 Calculations of g (R 1 ) and g (R 1 ) In order to justify the wavefunctions and the values of parameters obtained in Sec. 2, it is necessary to calculate the values of g (R 1 ) and g (R 1 ) by using these wavefunctions. As is well known, due to the combined effect of the trigonal field and spin-orbit interaction, t E is split into two Kraners doublets t E± 1 2 u [i.e., t E(Ē)] and t3 2 2 E± 1 2 u ± [i.e. t E(2Ā)]. The transition t3 2 2 E(Ē) t3 2 4 A 2 corresponds to R 1 line. According to Ref. [26], the g values of R 1 (t E ± 1 2 u ) excited state can be calculated by using g = t3 2 2 E 1 2 u + (L z + g s S z ) t E 1 2 u + 1 2, 1 2 S z 1 2, 1 2 = 2 t E 1 2 u + (L z + g s S z ) t E u, (13) g = t3 2 2 E 1 2 u (L x + g s S x ) t E 1 2 u + 1 2, 1 2 S x 1 2, 1 2 = 2 t E 1 2 u (L x + g s S x ) t E u = 2 t E 1 2 u ( (1) L 1 + g s 2 2 S ) t E 1 2 u +, (14) where g s = is the free-spin (or spin-only) g value. It is more convenient to adopt the operators S = S x is y and L (1) 1 = 1 2 (L x il y ). In Refs. [26] and [27], all the matrix elements of L z and L (1) 1 were calculated. They are expressed in terms of the orbital-angular-momentum reduction factors k and k ; and k = (1 ε) 1/2 k for octahedral fields (ε is the covalency parameter). In Eqs. (13) and (14), the prime indicates that all the admixtures of base-wavefunctions within d 3 electronic configuration have been taken into account. In Ref. [28], it was pointed out that only the transition from the component t E 1 2 u + was observed for t 3 2 2E(Ē) at liquid-helium temperature. For simplification, the electronic wavefunctions of t E 1 2 u + and t E 1 2 u states obtained by DCE without EPI in Sec. 2 are adopted; the values of k and ε are reasonably taken as the ones for ruby, i.e., k = 0.63 and ε = [26] In this way, without fitting procedure, we have theoretically evaluated out g calc (R 1 ) = 2.62 and g calc(r 1) = 0.03 for the R 1 [t E(Ē)] excited state of GSGG:Cr 3+, which are in good agreement with the experimental results g exp (R 1 ) = 2.6 ± 0.05 and g exp (R 1) < 0.1. [29] In Ref. [29], the value of g exp (R 1 ) was obtained by fitting the experimental data of the magnetic-field dependence of the magnetic-circular-polarization emissionspectrum intensity for the R 1 -line of GSGG:Cr 3+ at 1.6 K. It is noteworthy that g exp (R 1 ) = ± and g exp (R 1) = ± for ruby, [26,30,31] which are close to the values of g exp (R 1 ) and g exp (R 1) for GSGG:Cr 3+, respectively. The agreement between the calculated and experimental results of g (R 1 ) and g (R 1 ) have once more demonstrated the reasonableness of the wavefunctions obtained by DCE in Sec. 2 and the values of parameters resulting in these wavefunctions. 6 Discussion and Conclusion In this work, by means of DCE, the normal-pressure energy spectra and wavefunctions of GSGG:Cr 3+ without EPI have been obtained at 300 K and 70 K, respectively. Further, the contributions to normal-pressure energy spectra from EPI have also been obtained by using the electronic wavefunctions and energy spectra obtained by DCE, where the temperature-independent and temperature-dependent terms, all levels, all admixtures of electronic wavefunctions, all phonon-branches, all irreducible representations and their components have been taken into account. And then, the sum of the aforementioned two parts gives rise to the calculated result of the total energy spectrum. It is shown that the calculated results are in good agreement with all the optical-spectral experimental data and the experimental results of g (R 1 ) and g (R 1 ).

9 No. 3 Pressure Effects on Spectra of Tunable Laser Crystal GSGG:Cr 3+ II 357 On the basis of the normal-pressure results in this paper, all the experimental results of pressure-induced shifts (PS) of R 1 -line of GSGG:Cr 3+ at 70 K as well as PS of its R 1 -line, R 2 -line and U-band at 300 K can be successfully reproduced and explained by means of both the theory for PS of energy spectra and the theory for shifts of levels caused due to EPI, which will be reported in the forthcoming papers. Further, it is found that the contribution from temperature-independent terms of EPI is dominant. Especially, at low temperatures, the contribution from temperature-dependent terms of EPI is insignificant. oreover, although the energy spectrum without EPI is predominant, the contribution to energy spectrum from EPI is considerable. In the effect of EPI, t 2 2( 3 T 1 )e 4 T 2 plays a dominant role. By means of DCE, the eigenfunctions and accordingly the admixtures of t E and t 2 2( 3 T 1 )e 4 T 2 bases in the wavefunctions corresponding to R 1 (t E ± 1 2 u ) level have also been obtained at 70 K and 300 K, respectively. We have found that the degree of mixing of t E and t 2 2( 3 T 1 )e 4 T 2 base-wavefunctions decreases with decreasing temperature, and the average energy separation = Ē[t2 2( 3 T 1 )e 4 T 2 ] Ē[t3 2 2 E] increases with decreasing temperature. Considering this as well as the Boltzmann populations and relevant properties of t E and t 2 2( 3 T 1 )e 4 T 2 levels, the dramatic difference between the emission spectrum of GSGG:Cr 3+ at room temperature and the one at very low temperature (such as 4.2 K) can be explained. We should emphasize the importance of the sequence of calculations adopted by this work. For the first step, with taking into account spin-orbit interaction (H so ) and trigonal field (V trig ), the energy spectrum and wavefunctions are calculated by DCE without EPI. For the second step, on the basis of the aforementioned energy spectrum and wavefunctions, the contribution to energy spectrum from EPI is calculated. It is found that the contribution to energy spectrum from EPI obtained by this approach is much larger than the one obtained by neglecting H so and V trig. At 300 K and normal pressure, for example, with ζ 0 = 170 cm 1 and K 0 = 87 cm 1, the contribution from EPI to R 1 -level of GSGG:Cr 3+ is cm 1, and the one to the ground level is 2.94 cm 1 ; however, with ζ 0 = K 0 = 0, the contribution from EPI to R 1 - level is only 1.45 cm 1 (it results mainly from the EPI between R 1 -level and very high levels), and the one to the ground level is 2.80 cm 1 (notice that the separations of the ground level and the other levels are larger than cm 1, and accordingly the effects of H so and V trig are small). Hence, the contribution from EPI to R 1 level with ζ 0 = K 0 = 0 is much smaller than that with ζ 0 = 170 cm 1 and K 0 = 87 cm 1. This shows that it is essential for the calculation of the EPI effect to take first into account H so and V trig. In Ref. [11], it was pointed out that EPI may contribute to the mixing of the 2 E and 4 T 2 states, but only after some mixing is introduced through H so. It is this point that causes these authors to reverse the conventional order in which perturbations (H so and EPI) are included. Namely, in their new model, H so is first included. In Ref. [11], it was found that the results obtained by the new model (in which H so is first included) are much better than the ones obtained by conventional models. As to the average energy separation = Ē[t2 2( 3 T 1 )e 4 T 2 ] Ē[t3 2 2 E], various references gave different rough estimations and its values were scattering; various definitions were also proposed (e.g., see Refs. [9] and [13]). In many references, was defined by omitting spin-orbit interaction and trigonal field. By using DCE and setting ζ 0 = K 0 = 0, without EPI, we have obtained the value of to be cm 1 at 300 K and cm 1 at 70 K (see Sec. 2). However, as was mentioned in Sec. 4, when ζ 0 = 170 cm 1, K 0 = 87 cm 1 and EPI are taken into account, we have obtained = 212 cm 1 at 300 K. Furthermore, several aspects should be noticed. Firstly, in GSGG:Cr 3+, there are four non-equivalent Cr 3+ sites with different strengths of crystal fields, which causes the values of their to be very different. [4,5,7,9] oreover, the crystal imperfections also cause some distribution of the strengths of crystal field, which results in a distribution of. [13] Secondly, t 2 2( 3 T 1 )e 4 T 2 of GSGG:Cr 3+ includes six levels and its t E includes two levels, and the energy separations between them are remarkably different. In fact, these energy separations are comparable with splittings of t 2 2( 3 T 1 )e 4 T 2 for GSGG:Cr 3+. In view of the aforementioned two aspects, we can only speak of an effective or average energy separation between t 2 2( 3 T 1 )e 4 T 2 and t E. [9,13] Thirdly, according to Refs. [2] and [9], the larger plays a more important role for R lines, and the smaller is more important for U-band. oreover, the average energy separation responsible for the transition energy of R-line or the one of 4 T 2 band or their emission line shapes and lifetimes may be different. In view of the importance of the average energy separation between t 2 2( 3 T 1 )e 4 T 2 and t E levels, we need to consider more accurately it and its effect. For this purpose, the aforementioned several aspects should be taken into account in detail.

10 358 ZHANG Ji-Ping and A Dong-Ping Vol. 39 References [1] B. Struve, G. Huber, V.V. Laptev, I.A. Shcherbacov, and E.V. Zharikov, Apll. Phys. B30 (1983) 117. [2] B. Struve and G. Huber, Apll. Phys. B36 (1985) 195. [3] A. Suchocki and R.C. Powell, Chem. Phys. 128 (1988) 59. [4] B. Henderson, A. arshall,. Yamaga, K.P. O Donnell, and B. Cockayne, J. Phys. C: Solid State Phys. 21 (1988) [5] S.. Healy, C.J. Donnelly, T.J. Glynn, G.F. Imbusch, and G.P. organ, J. Lumin. 44 (1989) 65. [6]. Yamaga, B. Henderson, and K.P. O Donnell, J. Phys.: Condens. atter 1 (1989) [7] A. onteil, W. Nie, C. adej, and G. Boulon, Optical and Quantum Electronics 22 (1990) S247. [8] K. P. O Donnell, A. arshall,. Yamaga, B. Henderson, and B. Cockayne, J. Lumin. 42 (1989) 365. [9] S.. Healy, C.J. Donnelly, T.J. Glynn, G.P. Imbusch, and G.P. organ, J. Lumin. 46 (1990) 1; G.P. Imbusch, T.J. Glynn, and G.P. organ, J. Lumin. 45 (1990) 63. [10] C.J. Donnelly, T.J. Glynn, G.P. organ, and G.P. Imbusch, J. Lumin. 48 & 49 (1991) 283. [11] A.J. Wojtowicz,. Grinberg, and A. Lempicki, J. Lumin. 50 (1991) 231. [12]. Yamaga, B. Henderson, and K.P. O Donnell, Phys. Rev. B46 (1992) [13]. Grinberg, J. Lumin. 54 (1993) 369. [14] J.B. Gruber,.E. Hills, C.A. orrison, G.A. Turner, and.r. Kokta, Phys. Rev. B37 (1988) [15] U. Hommerich and K.L. Bray, Phys. Rev. B51 (1995) [16] U. Hommerich and K.L. Bray, Phys. Rev. B51 (1995) [17] Hong Hua, S. irov, and Y.K. Vohra, Phys. Rev. B54 (1996) [18] S. Sugano, Y. Tanabe, and H. Kamimura, ultiplets of Transition-etal Ions in Crystals, Academic Press, New York (1970). [19] Y.Y. Liu, D.P. a, D.C. Wang, and J.R. Chen, Commun. Theor. Phys. (Beijing, China) 27 (1997) 285. [20] D.P. a, H.. Zhang, Y.Y. Liu, J.R. Chen, and N. a, J. Phys. Chem. Solids 60 (1999) 463. [21] D.P. a and Zhang Ji-Ping, Commun. Theor. Phys. (Beijing, China) 39 (2003) 249. [22] D.P. a, Y.Y. Liu, N. a, and J.R. Chen, J. Phys. Chem. Solids 61 (2000) 799. [23] A.A. Kaminskii, Laser Crystals (Their Physics and Properties), Splinger-Verlag, Berlin (1981) p [24] G.F. Imbusch, W.. Yen, A.L. Schawlow, G.E. Devlin, and J.P. Remeika, Phys. Rev. 136 (1964) A481. [25] B. Di Bartolo, Optical Interactions in Solids, Wiley, New York (1968) pp. 373, [26] D.P. a, N. a, J.R. Chen, and Q. Lin, Commun. Theor. Phys. (Beijing, China) 33 (2000) 167. [27] D.P. a, H.. Zhang, J.R. Chen, and Y.Y. Liu, Commun. Theor. Phys. (Beijing, China) 30 (1998) 491. [28].H. Crozier, Phys. Lett. 18 (1965) 219. [29]. Yamaga, B. Henderson, A. arshall, and K.P. O Donnell, J. Lumin. 43 (1989) 139. [30] S. Geschwind, R.J. Collins, and A.L. Schawlow, Phys. Rev. Lett. 3 (1959) 545. [31] T. uramoto, T. Fukuda, and T. Hashi, J. Phys. Soc. Japan 26 (1969) 1551.