Unified research of the optical and EPR spectral data for the trigonal Cr3+ centers in emerald crystals

Optik 127 (2016) 4131–4133

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Unified research of the optical and EPR spectral data for the trigonal Cr3+ centers in emerald crystals Yang Mei a , Ren-Ming Peng a , Wen-Chen Zheng b,∗ a b

School of Physics & Electronic Engineering, Mianyang Normal University, Mianyang 621000, PR China Department of Material Science, Sichuan University, Chengdu 610064, PR China

a r t i c l e

a b s t r a c t

i n f o

Article history: Received 18 November 2015 Accepted 11 January 2016 Keywords: Optical spectrum Electron paramagnetic resonance Crystal- and ligand-field theory Cr3+ Emerald

The complete diagonalization (of energy matrix) method founded on the two-spin–orbit-parameter model (where the contributions from both the spin–orbit parameter of central dn ion and that of ligand ions are contained) is adopted to calculate together the optical and EPR spectral data for the trigonal Cr3+ centers in emerald crystal. The computed 18 optical and EPR spectral data (15 optical band positions and three spin-Hamiltonian parameters) with only five adjustable parameters are in rational accordance with the experimental values. The calculations indicate that the above diagonalization method is able to explain unifiedly the optical and EPR spectral data for d3 ions in crystals. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction Emerald is beryl (Be3 Al2 Si6 O18 ) doped with Cr3+ ions. It can be used as gemstone, and more importantly, is a famous tunable laser crystal [1–3]. So, its spectroscopic properties have aroused considerable research interest [4–12]. The optical spectra and electron paramagnetic resonance (EPR) spectra of emerald were investigated by many researchers [10–12]. From these studies, 15 optical band positions and three spin-Hamiltonian parameters (g factors g// , g⊥ and zero-field splitting D) were reported. These spectroscopic data are attributed to Cr3+ ion located in the trigonal octahedral Al3+ site in beryl crystal [10–12]. In the conversional crystal-field theory, the optical and EPR (i.e., spin-Hamiltonian parameters) data are calculated unifieldly by using the complete diagonalization (of energy matrix) method (CDM) founded on the one-spin–orbitparameter model [13] (in which only the contributions to the spectroscopic data from the spin–orbit parameter of central dn ion are considered). In the present paper, we compute these optical and EPR spectral data together through the CDM founded on the two-spin–orbit-parameter model [14,15]. The advantages of the CDM founded on the present model are as follows: (i) The method is more reasonable because the contributions to the spectroscopic (in particular, EPR) data due not only to the spin–orbit parameter of central dn ion, but also to that of ligand ions via covalence effect are contained. (ii) The number of adjustable parameter is fewer

∗ Corresponding author. Fax: +86 28 85416050. E-mail address: [email protected] (W.-C. Zheng). http://dx.doi.org/10.1016/j.ijleo.2016.01.084 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

because the spin–orbit parameters and orbit reduction factors (note: they are often treated as the adjustable parameters in the one-spin–orbit-parameter model) can be calculated from the Racah parameters (in fact, in the system studied, only five adjustable parameters are needed). The results are discussed. 2. Calculation In the two-spin–orbit-parameter model, the one-electron basis functions change from the d orbitals |d  of central dn ion in the one-spin–orbit-parameter model to the molecular orbitals (MO) consisting of both the |d  of dn ion and the p orbitals |p  of ligand ion [14,15]. Thus, there are two spin–orbit parameters ,   and two orbit reduction factors k, k in this model, i.e., [14,15]  = Nt



d0 +



1 2 0   , 2 t p



k = Nt 1 − 2t Sdp (t2g ) +





  = (Nt Ne )1/2 d0 −



1 t e p0 2



1 2  , 2 t

k = (Nt Ne )1/2 1 − 2t Sdp (t2g ) − e Sdp (eg ) −

1 t e 2

 (1)

where d0 and p0 denote the spin–orbit parameters of free dn ion and free ligand ion, respectively. In the emerald crystal studied, we have d0 (Cr3+ ) ≈ 273 cm−1 [16] and p0 (O2− ) ≈ 150 cm−1 [17]. Sdp () ( = t or e, the irreducible representation of Oh group) stand for the group overlap integrals. We yield Sdp (t) ≈ 0.03490 and Sdp (e) ≈ 0.09627 for the (CrO6 )9− cluster in emerald from the Slater-type SCF functions [18,19] with the Cr3+ –O2− distance R ≈ 1.975 A˚ obtained from the

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Y. Mei et al. / Optik 127 (2016) 4131–4133

Table 1 The crystal field energy levels (or optical band positions, in cm−1 ) for Cr3+ ion in emerald crystal. Energy level D3

Calc.

4

A2g

4

A2g

Eg (G)

2

Eg

T1g (G)

2

A1g Eg

0 1.7864 14,380 14,450 14,830 15,467 15,514 15,677 15,694 16,362 16,429 16,474 16,506 21,274 21,290 22,835 22,982 23,023 23,077 23,146 24,885 24,889 28,179 29,989 30,195 30,250 31,139 31,776 31,827 33,220 33,278 33,623 33,634 36,345 36,420 36,707 38,449 38,459 38,467 38,479 41,773 41,820 42,211

2

2

2

4

2

T2g (F)

T2g (F)

4

A1g

4

Eg

2

2 4

2 2

4

T1g

A1g (G) T2g (H)

Eg A2g Eg

4

A2g

2

A1g Eg

2

2 2

A2g Eg

2

T1g (H)

2

Eg (H)

2

2

T1 (H)

2

Eg

2 4

T1g (P)

A2g A1g

2

2

a

T2g (D)

H = Hf (B, C) + HSO (,   ) + HCF (Dq, ,  )

Expt.

Oh

4

A1g Eg

4

Eg

2

Eg

2

A2g

[11]

[10]

1.7912(6)a 14,655 14,717

14,624 14,686 15,140

15,700

15,860

16,800

16,750

 

1 E 2

g// = 2 22,800

23,000

g⊥ = 24,000

24,000

30,700 31,650





4

A2 , ±

4A , 2

4A , 2

3 2

37,730

39,900 39,900 41,800

41,320

−E



4

A2 , ±

1 2



1 1 |(k, k )LZ + ge SZ |4 A2 , − 2 2

 (5)



These formulas are acquired by means of the approximate equivalence between the physical Hamiltonian and the effective spin-Hamiltonian. Thus, the optical and EPR spectral data can be computed together by diagonalzating the energy matrix. In the above energy matrix and formulas, there are five unfixed parameters B, C, Dq, v and v . We treat them as the adjustable ones. By fitting the experimental optical and EPR spectral data of the emerald crystal with the complete diagonalization method, one can find C ≈ 2950 cm−1 ,

v ≈ − 2092 cm−1 ,

37,400



1 1 |(k, k )LX + ge SX |4 A2 , − 2 2

B ≈ 800 cm−1 , 36,300

(4)

where the three terms are, respectively, the Coulomb, spin–orbit and crystal-field (with the cubic field parameter Dq and the trigonal field parameters v, v’) interaction terms. The 120 × 120 complete energy matrix of the above Hamiltonian is constructed by the application of strong field basis functions [21]. The crystal field energy levels and hence the optical band positions are corresponding to the eigenvalues of the energy matrix, and the spin-Hamiltonian parameters can be calculated from the configuration eigenfunctions |4 A2 , Ms  and eigenvalues E (4 A2 , Ms ) of the ground state 4 A2 by using the formulas D=

Dq ≈ 1659 cm−1 ,

v ≈ 1950 cm−1 .

(6)

The computed optical and EPR spectral data are contrasted with the experimental values in Tables 1 and 2, respectively. 3. Discussion

Obtained from EPR spectra in Ref. [12].

EXAFS measurement [20]. N (the normalization coefficient) and  (the orbital mixing coefficient) are the MO coefficients. They follow the normalization connections [14,15] N [1 − 2 Sdp () + 2 ] = 1

(2)

and the approximate relationships [14,15] f =

The Hamiltonian in the two-spin–orbit parameter model for a d3 ion in trigonal crystal field takes the form

1 2 (B/B0 + C/C0 ) = N2 [1 − 2 Sdp () + 2 Sdp ()] 2

(3)

in which f is the covalence reduction factor. B and C are the Racah parameters of dn ions in crystals, and B0 and C0 are the corresponding parameters of free dn ions. Here B0 ≈ 1030 cm−1 and C0 ≈ 3850 cm−1 [16] for the free Cr3+ ion.

There are only five adjustable parameters in our calculations. Even so, the 18 computed optical and EPR spectral data in emerald crystal (15 optical band positions and three spin-Hamiltonian parameters) are in reasonable accordance with the experimental values (see Tables 1 and 2). Specially, the calculated ground state splitting |2D| (≈1.7864 cm−1 ), first excited state splitting E(2 E) (≈70 cm−1 ) and g anisotropy g (=|g// − g⊥ | ≈ 0.0077) are close to the observed |2D| ≈ 1.7912(6) cm−1 , E(2 E) ≈ 62 cm−1 and the g ≈ 0.0052(6). The large values of |2D|, E(2 E) and g are due to the large trigonal distortion and hence to the large trigonal field parameters |v| and |v | (see Eq. (6)) of Cr3+ center in emerald. There are small disparities of spectral data between calculation and experiment (see Tables 1 and 2). One of the reasons for the small disparities may be due to the neglect of vibrational contribution (i.e., electron–phonon interaction) to the optical and EPR spectral data. So, the complete diagonalization method based on the twospin-parameter model is effective in the unified studies of optical and EPR data for Cr3+ ion in crystals.

Table 2 Spin-Hamiltonian parameters of Cr3+ ion in emerald crystal. g//

D (cm−1 )

g⊥

Calc.

Expt. [12]

Calc.

Expt. [12]

Calc.

Expt. [12]

1.9717

1.9740(3)

1.9794

1.9792(3)

−0.8932

−0.8956(3)

Y. Mei et al. / Optik 127 (2016) 4131–4133

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