Theoretical Calculations of Thermal Shifts of Ground-State Zero-Field-Splitting for Ruby

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1 Commun. Theor. Phys. (Beijing, China) 36 (2001) pp c International Academic Publishers Vol. 36, No. 3, September 15, 2001 Theoretical Calculations of Thermal Shifts of Ground-State Zero-Field-Splitting for Ruby MA Dong-Ping, 1,2 CHEN Ju-Rong 1 and MA Ning 3 1 Department of Applied Physics, Sichuan University, Chengdu , China 2 International Centre for Materials Physics, The Chinese Academy of Sciences, Shenyang , China 3 Department of Computer Science, Sichuan University, Chengdu , China (Received November 13, 2000) Abstract 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 configuration, the thermal shifts (TS) of the ground-state zero-field-splitting (GSZFS) due to EPI for ruby have microscopic-theoretically been calculated; the contribution to TS of GSZFS from thermal expansion has also been calculated. The results are in very good agreement with experiments. It is found that the contributions from the first-order perturbation of the second-order term in EPI Hamiltonian are dominant in the Raman term and optical-branch term for TS of GSZFS; the different between the TS due to EPI of t A 2 ± 1 2 e2 (G2) level and the TS due to EPI of t3 2 4 A 2 ± 3 e2 (G1) level gives rise 2 to the TS due to EPI of GSZFS, which is very small in comparison with the TS due to EPI of G 2 or G 1 level. Among various terms in TS of GSZFS, Raman term is the largest one and the signs of the Raman term and optical-branch term are opposite to the sign of the thermal-expansion term; the optical-branch term plays an important role in TS of GSZFS and increases rapidly with temperature; all various contributions to TS of GSZFS have to be taken into account, since the subtle balance among them determines the total result. The comparison between the features of TS of GSZFS and those of TS of R 1 and R 2 lines has been made. For TS of GSZFS, the contribution from thermal expansion is especially important; the neighbor-level term is insignificant. PACS numbers: Mt, Kr, Ch, Dg Key words: optical materials, crystal fields, electronic paramagnetic resonance, phonons, thermal expansion 1 Introduction The investigation on the ground state of ruby is very important, which is relevant to optical and EPR spectra of ruby as well as their high-pressure and thermal effects. Sugano et al. [1] pointed out that the lowest-order perturbations giving rise to the small splitting of the t A 2 ground state of ruby are the third-order ones involving the spin-orbit interaction twice and the trigonal field once. Many perturbation processes give comparable contributions to the ground-state zero-field-splitting (GSZFS) of ruby. Therefore, its calculation requires a very careful treatment. Especially, the calculations of the pressureinduced shift (PS) and the thermal shift (TS) of GSZFS of ruby are very interesting and difficult. For a long time, GSZFS of ruby was measured and calculated; [2 9] its PS and TS were also measured; [9,10] and the theoretical studies of TS of GSZFS for ruby and other crystals were undertaken. [10 17] However, in the previous works, [10 17] only the phenomenological fit was made and the various contributions to TS of GSZFS were incompletely taken into account. The parameters in various contributions were treated as phenomenological ones and determined by fitting the observed data. It will be seen that on one hand, the measured TS of GSZFS is very small and not so accurate; on the other hand, the sign of the contribution to TS of GSZFS from thermal expansion is opposite to the signs of two contributions to TS of GSZFS from electron-phonon interaction (involving acoustic branches and optical ones respectively), i.e., the subtle balance among them determines TS of GSZFS. Moreover, their temperature-dependencies are different. Thus, all of them have to be taken into account; and it is understandable why various phenomenological treatments by using different incomplete contributions in various previous articles gave unreasonable and contradictory results. For example, in Ref. [10], TS of GSZFS was interpreted in terms of only the contribution due to the thermal expansion; however, TS of GSZFS was considered as only the contribution due to EPI of the acoustic branches in Refs [12] [15]. In order to solve this puzzling problem, obviously, it is quite necessary to take into account all the contributions to TS of GSZFS, carry out their microscopic-theoretical calculations and reveal their physical essentials. This is exactly the purpose of the present paper. In Ref. [18], we pointed out that the TS of levels of Mailing address

2 358 MA Dong-Ping, CHEN Ju-Rong and MA Ning Vol. 36 rare-earth or transition-metal ions in crystals includes four contributions with different characteristics. Recently, by using the diagonalizations of complete d 3 energy matrix (DCEM) at various pressures and the wavefunctions obtained from them and taking into account the variations of both radial parts and angular ones of wavefunctions with pressure, we have successfully carried out the unified theoretical-calculation of the whole energy spectrum (including GSZFS) and g factors of the ground state and their PS (including PS of GSZFS). [19] Further, by using the wavefunctions and energy spectrum obtained in Ref. [19] and 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 configuration, we have microscopically calculated all the parameters in three terms (i.e. Raman, neighbor-level and optical-branch terms) of TS due to EPI for the R 1, R 2 and ground levels of ruby; the contribution to TS from thermal expansion has also been calculated by means of the theory for PS; and then, the unified calculation of the TS of R 1 and R 2 lines and the TS of GSZFS as well as the thermal broadenings (TB) of R, R and B line-groups for ruby (including the transition probabilities for direct and Raman processes) has successfully been accomplished. According to the microscopic theoretical calculations, it will be shown (See Table 1 and the text) that the various contributions from EPI to TS of the two ground levels G 1 and G 2 are large (similar to the ones of R 1 and R 2 lines), but the various contributions from EPI to TS of GSZFS are very small differences between the ones to TS of G 2 and the ones to TS of G 1 respectively. Hence, the accurate microscopic theoretical calculations are very difficult and require a very careful treatment. Especially, the very good agreement between the theoretical results of TS of GSZFS and the experimental ones is surprising. The calculations of TS of R 1 and R 2 lines were given by the first paper (Ref. [20]) in a series of papers. In the present paper, the microscopic theoretical calculations of the TS of GSZFS are given. The microscopic theoretical calculations of TB and transition probabilities for R, R and B line-groups will be reported in the forthcoming paper. 2 Calculations of the Values of Parameters in TS due to EPI We investigate TS at constant pressure. As was mentioned in Ref. [18], the energy of a crystal-field level related to TS may be considered as a statistical average of microscopic quantity, i.e., a thermodynamic quantity. According to general thermodynamic analysis, thus, the TS of levels of transition-metal ions in crystals consists of the contribution from EPI and that from the thermal expansion. The latter will be calculated in Sec Expressions of TS due to EPI Let us consider an interaction system of the local electronic state of a transition-metal ion and lattice vibration. Its Hamiltonian is H = H latt + H ion + H int. (1) The electron-phonon interaction Hamiltonian is H int = H (1) + H (2), (2a) H (1) = ΓM C(ΓM)ɛ(ΓM), (2b) H (2) = ΓM Γ M Γ M D(ΓM; Γ M, Γ M ) ɛ(γ M )ɛ(γ M ), (2c) where only the first-order and second-order terms are considered. Γ, Γ and Γ are the irreducible representations of the point group about the central metal ion; M, M and M are their components. C(ΓM) and D(ΓM; Γ M, Γ M ) are the orbital operators of d- electrons of transition-metal ions; ɛ(γm), ɛ(γ M ) and ɛ(γ M ) are phonon operators. In H (2), Γ Γ = Γ i, i and D(ΓM; Γ M, Γ M ) transforms as the M base of the Γ representation. For convenience, the trigonal bases are adopted for ruby. [19] Taking H int as a perturbation, the wavefunction of the zero-order approximation of the i-state of the total system is i = ψ el i ; n 1, n 2,... n k,... = ψ el i n 1 n 2 n k, (3) where ψi el is the d-electronic wavefunction (local electronic state) of a transition-metal ion; n k is the occupation number of the k-th phonon state. According to the perturbation theory, the contribution of H int to energy of the i-state is δe i = (H (2) ) ii + j (H (1) ) ij 2 E i E j. (4) Obviously, (H (1) ) ii = 0; and the terms of the third-order perturbation of H (1) and the second-order perturbation of H (2) can be omitted. Taking into account all the levels and admixtures of wavefunctions within d 3 electronic configuration and all possible ΓM, the three terms of TS due to EPI of i-level were microscopically derived in Ref. [20]. The microscopic expression of Raman term is

3 No. 3 Theoretical Calculations of Thermal Shifts of Ground-State Zero-Field-Splitting for Ruby 359 δe (1) i = α i T 4 TD /T α i = π Cρ( 3 vl vt 5 + j i 0 x 3 e x 1 dx, )( kb h ) 4 { M ψ el i D(T 2 M) ψ el M ψel i C(T 2M) ψ el j M ψel i C(EM) ψel E el i E el j i j 2 (5a) }, (5b) where k B is the Boltzmann constant; h = h/2π, and h is the Planck constant; T D = hω D /k B is Debye temperature of acoustic branches. C is the velocity of light in the vacuum; ρ is the crystal density; v l and v t are the longitudinal and transverse velocities of sound respectively; ψ el and E el denote the d-electronic wavefunction and d-electronic energy level respectively. For convenience, the definition of α i differs from that in Ref. [18]; and we can write α i = αi c + αd i, where superscript c indicates the contributions of C(T 2 M) and C(EM); d indicates those of D(T 2 M) [The detailed calculation shows that the contributions of D(EM) can be neglected]. The so-called Raman term is the contributions to TS of i-level from the phonons of acoustic branches and all the j-levels which satisfy Ei el Ej el hω D (Noticing that it also includes the contribution of M ψel i D(T 2M) ψi el due to phonons of acoustic branches). The neighbor-level term (i.e., the so-called direct-process term) represents the contributions to TS of i-level from the phonons of acoustic branches and the j-levels which satisfy Ei el Eel j < hω D or hω D. Its microscopic expression is δe (2) i = j i β ij ( T T ij ) 2P TD /T 0 x 3 ( e x 1)[x 2 (T ij /T ) 2 dx, (6a) ] β ij = T 3 ijd ij = T 3 jid ij, k 3 B ( 1 D ij = 60π 3 Cρ h 4 vl v 5 t )[ M ψ el i C(T 2 M) ψ el j M ψ el i C(EM) ψ el j 2], (6b) (6c) where T ij = (Ei el Ej el)/k B; P denotes the principal value of the integral (when T ij < T D, there is a singular point in the integrand). The optical-branch term represents the contributions to TS of i-level from the phonons of optical branches and all the j-levels (Ei el Ej el), including the contribution of M ψel i D(T 2M) ψi el due to phonons of optical branches. Its microscopic expression is δe (3) i = γ i (TD/T 5 o )( e hω eff /k B T 1) 1 = γ i (TD/T 5 o )( e To/T 1) 1, (7a) { M γ i = ψel i C(T 2M) ψj el M ψel i C(EM) ψel j 2 (E el j i i Ej el (E el )2 ( hω eff ) 2 i Ej el ) } + M ψ el i D(T 2 M) ψ el i 1 120π 3 Cρv 5 s [ kb h ] 4, (7b) where the approximation of single-frequency model [18,20] was made; ω eff is the phonon frequency of optical branches for the single-frequency model, and T o hω eff /k B ; v s is defined as 3/vs 3 = 2/vt 3 + 1/vl 3. Similar to α i = αi c + αd i, we can write γ i = γi c + γd i. The T D and v s in Eq. (7) come from the sum k k2 = V ωd/(10π 5 2 vs) 5 (k is the wave vector; V is the crystal volume). [20] This is obtained by using acoustic branches and Debye model, because the values of k are independent of whether optical branches or acoustic ones. α i and γ i are in unit K 4 cm 1 ; D ij is in unit K 3 cm 1, δe (1) i, δe (2) i and δe (3) i are in unit cm Theoretical Calculations of α i, γ i and D ij As is well known, the combined action of trigonal-field and spin-orbit interaction gives rise to the splitting of the ground term t A 2 for ruby. [1,4] Namely, it is split into two levels [t A 2 ± 3 2 e 2 (i.e., t A 2 2Ā) G 1 and t A 2 ± 1 2 e 2 (i.e., t A 2 Ē) G 2 ]; the former is lower than the latter by δ = 2D = 2 D (D is the axial-symmetry ZFS constant in the spin [or effective] Hamiltonian, see Refs [3] [5], [9]

4 360 MA Dong-Ping, CHEN Ju-Rong and MA Ning Vol. 36 and [10]). This is the GSZFS of ruby. For the two ground levels and GSZFS of ruby, the values of α c, α d, α, γ c, γ d, γ and D ij are microscopictheoretically calculated as follows. From Eqs (5) (7), α i, D ij and γ i depend on the elastic stiffness constants and density of crystal, the electronic wavefunctions and energy spectrum and the 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 ), Z d ( t 2 D(T 2 ) e ), by which the reduced matrix elements of C(T 2 ), C(E) and D(T 2 ) for three electrons in Eqs (5) (7) are expressed respectively. [20] Besides, γ i also depends slightly on T o. The electronic energy spectrum and wavefunctions were obtained by means of diagonalization of the complete energy matrix (DCEM) of the d 3 electronic configuration in a trigonally distorted cubic-field in Ref. [19]. The elastic stiffness constants [21,22] C 11 = dyn/cm 2, C 44 = dyn/cm 2 and the density ρ = g/cm 3 [23] (They determine the velocities of sound) were adopted in Ref. [20] and this work. Furthermore, in Ref. [20], by using the observed results of the stress-induced changes in splittings of t A 2, t E, t T 1 and t T 2, we obtained Y c = cm 1, Z c = cm 1, P c = cm 1, Q c = cm 1 ; by least-square fit to the observed data of TS of R 1 and R 2 lines for ruby, Y d = cm 1, Z d = cm 1, T D = 780 K and T o = 822 K were determined; and accordingly the calculated results of TS of R 1 and R 2 lines are in very good agreement with the observed ones. Besides, the values of T D and T o are also in agreement with the observed peaks of acoustic and optical branches. [20,24] In the present paper, with all the values of Y c, Z c, P c, Q c, Y d, Z d, T D and T o obtained in Ref. [20], the TS of GSZFS due to EPI for ruby and various contributions to it are microscopic-theoretically calculated. However, in order to get necessary digits of the values of α i, D ij and γ i (See Table 1), it should be assumed that the values of Y c, Z c, P c, Q c, Y d, Z d, T D and T o have more digits. Namely, by the unified calculation of the TS of R 1 line, R 2 line and GSZFS, the uncertainties of these parameters are much smaller than the ones obtained by the calculations of only the TS of R 1 and R 2 lines. Table 1 The values of α c, α d, α, γ c, γ d and γ of two levels and GSZFS (in unit K 4 cm 1 ). α c α d α γ c γ d γ G Level G GSZFS G 1 and G 2 represent t A 2 2Ā and t3 2 4 A 2 Ē respectively. D 2 Ā Ē between G 1 and G 2 is equal to K 3 cm 1. By using these values of Y c, Z c, P c, Q c, Y d, Z d and T o (T D will be employed in calculation of δe (1), δe (2) and δe (3) ), the aforementioned wavefunctions and energy spectrum given in Ref. [19], the microscopic expressions of α i, D ij and γ i Eqs (5) (7) and the appropriate FORTRAN programs, the values of α c, α d, α, γ c, γ d, γ and D ij of G 1 and G 2 levels and GSZFS have been microscopic-theoretically evaluated and shown in Table 1. The α c (or α d,..., γ) of GSZFS is α c (G 2 level) α c (G 1 level), etc. It is noteworthy that due to (H (1) ) ii = 0, we have to consider j [ (H(1) ) ij 2 /(E i E j )] in the derivations of Eqs (5) (7). Obviously, (H (2) ) ii has to be simultaneously taken into account according to the perturbation theory. It has already been included in Eq. (5) (for acoustic branches) or Eq. (7) (for optical branches). From Table 1, we can find the very important feature: α d α c and γ d γ c for GSZFS [On the contrary, we have α c > α d and γ c > γ d for G 1 and G 2 levels. It can be seen that α d (or γ d ) of GSZFS is the small difference between α d (or γ d ) of G 2 and α d (or γ d ) of G 1 ; however, α c (or γ c ) of GSZFS is the especially small difference between α c (or γ c ) of G 2 and α c (or γ c ) of G 1. This causes α d α c and γ d γ c for GSZFS]. Namely, the contribution of D(T 2 M) (the one of the

5 No. 3 Theoretical Calculations of Thermal Shifts of Ground-State Zero-Field-Splitting for Ruby 361 first-order perturbation of the second-order term in EPI Hamiltonian H (2) ) is dominant in both the Raman term and the optical-branch term for TS of GSZFS. It should be emphasized that similar results were also obtained for TS of R 1 and R 2 lines in Ref. [20]. Furthermore, it is noteworthy that all the α c, α d,..., γ of G 1 and G 2 levels are negative; however, those of GSZFS are positive (because α c, α d,..., γ of G 1 are larger than those of G 2 respectively), which have exactly the same signs as those of the experimental TS of GSZFS (See Table 2). From Table 2 of Ref. [20] and Table 1 of this paper, we can see that the value of α c of G 1 or G 2 level is close to the one of R 1 or R 2 level; the value of α d (or γ d ) of G 1 or G 2 level is one order of magnitude smaller than the one of R 1 or R 2 level. However, the value of α c of GSZFS is five orders of magnitude smaller than the value of α c of G 1 or G 2 level, and the value of α d of GSZFS is two orders of magnitude smaller than the value of α d of G 1 or G 2 (which causes α d α c for GSZFS); the value of γ c of GSZFS is four orders of magnitude smaller than the value of γ c of G 1 or G 2, and the value of γ d of GSZFS is three orders of magnitude smaller than the value of γ d of G 1 or G 2 (which causes γ d γ c for GSZFS). Thus, the accurate microscopic theoretical calculation of TS of GSZFS is very difficult and requires a very careful treatment. It is quite necessary to take into account the contributions of all the levels and admixtures of wavefunctions within d 3 electronic configuration and all the ΓM in the EPI in detail. Table 2 P eff, thermal shift of GSZFS and various contributions to it. T (K) P eff (kbar) δ te (10 4 cm 1 ) δ (1) (10 4 cm 1 ) δ (3) (10 4 cm 1 ) δ calc (10 4 cm 1 ) δ exp (10 4 cm 1 ) From Ref. [10], see the text. 3 Calculation of TS due to Thermal Expansion As was mentioned in Refs [18] and [20], the value of the shift due to the thermal expansion with χ T ( R T /R n, i.e., from normal temperature T n (= 298 K) to T and at normal pressure P n ) is equal to the value of the pressureinduced shift (PS) with the same change of interionic distance χ p ( R p /R n, i.e., from normal pressure P n to P and at normal temperature T n ) = χ T ; this is because they possess the same microscopic mechanism, i.e., the reduction of the interionic distance R and the corresponding expansion of the electronic wavefunction cause the level shift. The latter is called the effective (or equivalent) PS. R n is the interionic distance at normal P n and T n ; R T is the one at T and P n ; R P is the one at P and T n. According to this equivalency, the shift caused by thermal expansion can be evaluated by using the thermal expansion coefficient and the effective PS. White and Roberts measured the linear expansion coefficients α and α of α-al 2O 3 (in the range K, their anisotropy is only 12%), and gave average αav from 20 K to 2000 K at normal pressure. [25] Similar to Refs [18] and [20], the average values of χ T (= l T /l 298 = R T /R 298, where l T and R T are the length of specimen and interionic distance at T respectively; l 298 and R 298 [ R n ] are the quantities at 298 K) at various temperatures can be evaluated by applying αav; further, as an approximation, the shift due to thermal expansion can be calculated by using χ T (the average) and the effective PS with χ P = χ T (See the discussion in Sec. 5). The P χ P dependence is calculated from the three-parameter Birch equation P = 3B 0 2 χ 5 3 b i (χ 2 1) i. (8) i=1 For ruby B 0 = kbar, b 1 = , b 2 = 0 and

6 362 MA Dong-Ping, CHEN Ju-Rong and MA Ning Vol. 36 b 3 = [19,26] From Eq. (8) and χ P = χ T, we have evaluated the values of P, which correspond to χ T = χ P. It should be emphasized that the TS at constant pressure are studied in this work, and the TS due to thermal expansion can be calculated by employing the aforementioned equivalent PS. [27] In order to avoid confusion, therefore, the obtained values of P are called the effective (or equivalent) pressures P eff. From the definition of χ T, χ T and corresponding P eff are relative to 298 K. For convenience, they have been converted into the P eff relative to 0 K. The values of P eff at various temperatures have been listed in Table 2. It seems that the measured results of dδ/dp in Ref. [9] are more accurate than those in Refs [7] and [8]. According to Nelson et al., [9] the GSZFS δ varies linearly with pressure up to 75 kbar, and dδ/dp = (6.0 ± 0.4) 10 4 cm 1 /kbar. Then, the contribution to TS of GSZFS from thermal expansion (relative to 0 K) δ te has been evaluated and shown in Table 2. It is noteworthy that with increasing temperature, the values of P eff and δ te are negative (The thermal expansion corresponds to the negative compression), which are opposite to the signs of δ (1), δ (2) and δ exp (See Table 2). 4 Numerical Calculation and Comparison with Experimental Data From Eqs (5) (7), the TS due to EPI of i-level is δe i = δe (1) i + δe (2) i + δe (3) i, (9) where i can be taken as G 1 or G 2 level. The TS of GSZFS is the difference between TS of G 2 level and TS of G 1 level. So, the TS of GSZFS due to EPI can be written as δ e p = δ (1) + δ (2) + δ (3), (10) where δ (k) = δe (k) (G 2 level) δe (k) (G 1 level) [k = 1, 2, 3]. From Eqs (5) (7), (10) and Table 1, the values of various contributions to TS of GSZFS from EPI at ten temperatures have been microscopic-theoretically evaluated and shown in Table 2. It is found that the value of neighbor-level term δ (2) is insignificant and can be omitted. So, the TS of GSZFS has finally been obtained by δ calc = δ te + δ (1) + δ (3). (11) Klein et al. measured TS of GSZFS for ruby in the range of K. [10] By using the experimental points in Fig. 1 of Ref. [10], we draw a smooth D (= δ/2) T curve. From the experimental curve, then, the data of δ exp at ten temperatures are obtained and listed in Table 2. It can be seen from Table 2 that the theoretically calculated results of TS of GSZFS are in very good agreement with the experimental data within the measured error (by taking into account the measured error of dδ/dp in Ref. [9] and the one of D in Ref. [10], the total measured error should be larger than 15%). 5 Discussions and Conclusions In the previous theoretical work for the thermal shifts of GSZFS, [10 17] only the phenomenological fit was made, i.e., the α and γ of GSZFS were treated as phenomenological parameters and determined by fitting the observed data. Moreover, the various contributions to TS of GSZFS were incompletely taken into account. In the present paper, for the first time, by taking into account the contributions of all the levels and admixtures of wavefunctions within d 3 electronic configuration and all the ΓM in EPI, the results of α, D ij, γ and the TS of GSZFS due to EPI have microscopic-theoretically been calculated. Because the microscopic mechanism of the level shift caused by the compression under pressure is the same as the one of the level shift caused by the negative thermal expansion, [18,20] on the basis of the theory of PS, [19,20] the TS of GSZFS due to thermal expansion has been calculated by applying the observed αav and dδ/dp. In this way, without fit procedure, the TS of GSZFS and all the various contributions to it have theoretically been evaluated. In view of the difficulties in these calculations, it is a great success that the theoretically predicted results are in very good agreement with experiments. Wybourne emphasized that it is not enough to be able to just fit energy levels. A correct theory should also yield eigenfunctions that can be used to calculate other physical observables with a comparable precision. [28] In Ref. [19], by means of DCEM, we calculated the energy spectra and eigenfunctions of ruby at normal and various pressures. The results of the energy spectra are in very good agreement with observed optical spectra, GSZFS and their PS. The obtained wavefunctions were used to calculate g factors and their PS, which are also in very good agreement with experiments. Further, these wavefunctions and energy spectrum have been used to calculate the TS of R 1 line, R 2 line (in Ref. [20]) and the TS of GSZFS (in this paper) as well as the TB and transition probabilities of R, R and B line-groups (in the forthcoming paper). All the results are in very good agreement with a lot of experimental data. For the main features of TS of GSZFS and the comparison between them and those of TS of R 1 and R 2 lines, the following conclusions can be drawn from this work. (i) The microscopic theoretical calculations have demonstrated the remarkable feature: α d α c and

7 No. 3 Theoretical Calculations of Thermal Shifts of Ground-State Zero-Field-Splitting for Ruby 363 γ d γ c for GSZFS of ruby, which means that the contributions from the first-order perturbation of the secondorder term in EPI Hamiltonian ( M ψel i D(T 2M) ψi el ) are dominant in both the Raman term and the opticalbranch term of TS of GSZFS. The similar results were also obtained for TS of R 1 and R 2 lines in Ref. [20]. Only from the microscopic theoretical calculation using the wavefunctions and energy spectrum, obviously, can we separate the contribution of the first-order perturbation of the second-order term in EPI (H (2) ) from that of the second-order perturbation of the first-order term in EPI (H (1) ) and find the former to be dominant. (ii) According to the microscopic calculation, the values of all the α c, α d, α,... and γ of G 1 and G 2 levels are negative, but those of GSZFS are positive. This is because α c (or α d,..., γ) of GSZFS has a very small difference between α c (or α d,..., γ) of G 2 and the one of G 1 and so on; and α c, α d,..., γ of G 1 are larger than those of G 2 respectively. Therefore, the contribution to TS of GSZFS from EPI δ e p (= δ (1) + δ (3) ) is positive. Its sign is opposite to the signs of the contributions to TS of R 1 and R 2 lines from EPI E e p (See Ref. [20]). Since dδ/dp > 0 (i.e., the blue shift of δ takes place with increasing pressure) [9,19] and the thermal expansion corresponds to negative compression (P eff < 0), the contribution to TS of GSZFS from thermal expansion δ te is negative. Its sign is also opposite to the signs of the contributions to TS of R 1 and R 2 lines from thermal expansion E te (See Ref. [20], noticing that the PSs of R 1 and R 2 lines are red shifts [19] ). For both TS of GSZFS and TS of R 1 or R 2 line, the contribution to TS from EPI is dominant. Thus, the total TS of GSZFS δ exp is positive, and the total TS of R 1 or R 2 line E exp is negative. (iii) From the detailed calculation, it is found that the neighbor-level term due to EPI of acoustic branches can be neglected for TS of GSZFS. On the contrary, the neighbor-level term is very important for TS of R 1 and R 2 lines. This is because there is only a couple of neighbor levels G 1 and G 2 for the ground state; the energy gap between them is very small (at normal temperature and pressure, it is only cm 1[19] ); and the ratio D ij /α of GSZFS is much smaller than the one of R 1 and R 2 lines. (iv) The temperature dependencies of various contributions to TS of GSZFS are different. Especially, the contribution to TS of GSZFS from EPI of optical branches (i.e. the optical-branch term) δ (3) increases rapidly with temperature. From Table 2, the Raman term due to EPI of acoustic branches δ (1) is the largest one in various contributions to TS of GSZFS; δ te (the contribution due to thermal expansion) only is slightly smaller than δ (1) ; they have opposite signs and cancel almost each other. Thus, the contribution to TS of GSZFS from EPI of optical branches is important. For example, δ (3) / δ exp are 62.0%, 76.1% and 82.9% at 250 K, 400 K and 550 K, respectively. Then, it can be seen that the subtle balance among various contributions to TS of GSZFS determines the total TS of GSZFS. Hence, all the various contributions to TS of GSZFS have to be completely taken into account, otherwise incorrect results may be given. In Ref. [10], for example, the contribution to TS of GSZFS from EPI was only taken as C[ctgh( hω/2k B T ) 1] = 2C[exp( hω/k B T ) 1] 1, which is exactly the form of the optical-branch term Eq. (7a). By using only C[ctgh( hω/2k B T ) 1], Klein et al. argued the temperature dependence of the contribution due to EPI not to exist and accordingly concluded TS of GSZFS to be only the contribution due to thermal expansion. [10] According to this paper, however, the contribution to TS of GSZFS from EPI includes both the Raman term of EPI of acoustic branches Eq. (5) and the optical-branch term Eq. (7); their temperature dependencies are different and the former is dominant. Thus, the aforementioned argument and conclusion given in Ref. [10] are incorrect. In fact, the signs of the contributions due to thermal expansion δ te are opposite to the signs of the experimental findings δ exp ; and the contribution due to EPI δ e p (= δ (1) + δ (3) ) is larger than the absolute value of the contribution due to thermal expansion δ te (See Table 2). Moreover, in Ref. [10], the maximum phonon frequency was taken as ω = s 1, i.e., hω = erg = 122 cm 1, which is not in agreement with the phonon spectrum of ruby. (v) In comparison with the results of TS of R 1 and R 2 lines, the contribution from thermal expansion to TS of GSZFS is more important than that to TS of R 1 and R 2 lines. For example, at 500 K, and for TS of GSZFS; and δ te / δ calc e p = 66.3% δ te / δ exp = 1.92 E te (R 1 )/ E calc e p(r 1 ) = 18.1% E te (R 1 )/ E exp (R 1 ) = 0.22 for TS of R 1 line. This can be explained in terms of the theory of PS (See Ref. [19]) as follows. The PSs of GSZFS of ruby are blue shifts (i.e., dδ/dp > 0, which causes δ te < 0) and depend mainly on trigonal field (Noticing

8 364 MA Dong-Ping, CHEN Ju-Rong and MA Ning Vol. 36 that the trigonal-field parameters are very sensitive to the angular variations of the crystal lattice and wavefunctions with pressure, which makes the dominant contribution to PS of GZFZS of ruby). [19] However, the PSs of R 1 line are red shifts (consequently E te (R 1 ) > 0) and depend mainly on Coulomb interaction between d-electrons. And the variation of trigonal field with pressure is obviously larger than that of Coulomb interaction (See Ref. [19]). Therefore, it is understandable that the contribution due to thermal expansion is more important for the TS of GSZFS. According to this work, it is a good approximation to calculate δ te by using the average thermal-expansion coefficient αav measured by White et al. [25] This is understandable. Firstly, the anisotropy of thermal expansion is small (only 12% in the range of K [25] ). Secondly, as we have mentioned in Refs [29] and [30], at very high pressure, the local compression around Cr 3+ is slightly smaller than the bulk compression for ruby or MgO:Cr 3+. When P < 100 kbar, however, this effect can be ignored. In the range of K, the variation of P eff is smaller than 18 kbar (See Table 2). So, the difference between the local compression (or expansion) around Cr 3+ and the bulk one can be neglected; and accordingly the measured bulk data from Ref. [25] can be employed. The similar conclusion was obtained in Ref. [10]. References [1] S. Sugano, Y. Tanabe and H. Kamimura, Multiplets of Transition-Metal Ions in Crystals, Academic, New York (1970). [2] A.A. Manenkov and A.M. Prokhorov, J. Exptl. Theoret. Phys. (U.S.S.R) 28 (1955) 762; Soviet Phys.-JETP 1 (1955) 611. [3] R.M. Macfarlane, J. Chem. Phys. 42 (1965) 442. [4] R.M. Macfarlane, J. Chem. Phys. 47 (1967) [5] R.M. Macfarlane, Phys. Rev. B1 (1970) 989. [6] S. Sugano and M. Peter, Phys. Rev. 122 (1961) 381. [7] J.O. Artman and J.C. Murphy, Paramagnetic Resonance, ed. W. Low, Academic, New York (1963) Vol. 2, p [8] T.I. Alaeva, L.F. Vereshchagin and E.N. Yakovlev, Fiz. Tverd. Tela (Leningrad) 11 (1969) 502 [Sov. Phys. Solid State 11 (1969) 398]. [9] H.M. Nelson, D.B. Larson and J.H. Gardner, J. Chem. Phys. 47 (1967) [10] H. Klein, V. Scherz, M. Schulz, H. Setyono and K. Wisznewski, Z. Phys. B28 (1977) 149. [11] W.M. Walsh, Jr., Phys. Rev. 114 (1959) [12] K.N. Shrivastava, Phys. Rep. 20 (1975) 137. [13] K.N. Shrivastava, G.V. Rubenacker, S.L. Hutton, J.E. Drumheller and R.S. Rubins, J. Chem. Phys. 88 (1988) 634. [14] K.N. Shrivastava, Proc. R. Soc. London A419 (1988) 287. [15] K.N. Shrivastava, Chem. Phys. Lett. 20 (1973) 106. [16] R.R. Sharma, Phys. Rev. B2 (1970) [17] W.C. Zheng and S.Y. Wu, Phys. Rev. B54 (1996) [18] D.P. Ma, X.Y. Huang, J.R. Chen, J.P. Zhang and Z.G. Zhang, Phys. Rev. B48 (1993) [19] D.P. Ma, H.M. Zhang, Y.Y. Liu, J.R. Chen and N. Ma, J. Phys. Chem. Solids 60 (1999) 463. [20] D.P. Ma, Y.Y. Liu, N. Ma and J.R. Chen, J. Phys. Chem. Solids 61 (2000) 799. [21] R.V.G. Sundara Rao, Proc. Indian Acad. Sci. A26 (1949) 352. [22] J.B. Wachtman Jr., W.E. Tefft, D.G. Lam Jr. and R.P. Stinchfild, J. Res. Nat. Bar. Stand. 64A (1960) 213. [23] Handbook of Chemistry & Physics, 66th ed., eds R.C. Weast, M.I. Astle and W.H. Beyer, CRC, Boca Raton, FL (1985) p [24] T. Kushida and M. Kikuchi, J. Phys. Jpn. 23 (1967) [25] G.K. White and R.B. Roberts, High Temperature-High Pressure 15 (1983) 321. [26] D.P. Ma, X.T. Zheng, Y.S. Xu and Z.G. Zhang, Phys. Lett. A115 (1986) 245. [27] D.P. MA, J.R. CHEN and Y.Y. LIU, Commun. Theor. Phys. (Beijing, China) 29 (1998) 13. [28] B.G. Wybourne, Spectroscopic Properties of Rare Earths, John Wiley & Sons, New York (1965) p. 40. [29] D.P. Ma and D.E. Ellis, J. Lumines. 71 (1997) 329. [30] D.P. Ma, D.E. Ellis, Y.Y. Liu and J.R. Chen, Commun. Theor. Phys. (Beijing, China) 28 (1997) 265.