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On the relations between the crystal field parameter notations in the “Wybourne” notation and the conventional ones for 3dN ions in axial symmetry crystal field
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Physica B 370 (2005) 137–145 www.elsevier.com/locate/physb
On the relations between the crystal field parameter notations in the ‘‘Wybourne’’ notation and the conventional ones for 3dN ions in axial symmetry crystal field Zi-Yuan Yang, Qun Wei Department of Physics, Baoji University of Arts and Science, Baoji 721007, PR China Received 21 August 2005; received in revised form 10 September 2005; accepted 10 September 2005
Abstract The correlations of the various crystal field (CF) notations existing in the literature for 3dN ions in axial symmetry crystal field have been investigated. The confusions among these CF notations have been clarified. The correlations of the various CF notations are summarized in Table 1, which is very useful in the investigations of 3dN ions in crystal material. r 2005 Elsevier B.V. All rights reserved. PACS: 71.70.Ch; 76.30.Fc Keywords: 3dN transition-metal ions; Crystal field parameter notations; Axial symmetry crystal field; ‘‘Wybourne’’ notation
1. Introduction 3dN transition-metal (TM) ions such as Cr3+, Ni2+, Ti3+ are active ions in solid state laser materials [1,2] and non-linear optical materials [3–5]. These impurity ions in the materials play a major role because they can be responsible for the modification of optical properties. Thus, microscopic study of TM ions in various crystal Corresponding author.
E-mail address: [email protected] (Z.-Y. Yang).
materials has attracted much attention. As is well known, the crystal field (CF) parameters and the spin-Hamiltonian (SH) ones for the TM ions in crystals are very sensitive to subtle changes of the crystal structure [1–4]. Hence, the studies of these parameters can provide a great deal of microscopic insight concerning the crystal structure, structural disorder, phase transitions, pressure behavior as well as the observed magnetic and spectroscopic properties. In the phenomenological theory of CF, the CF Hamiltonian for 3dN ions in crystal materials can be written, in terms of ‘‘Wybourne’’
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.09.002
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138
CF parameter notation Bkq, as [6] " k n h i X X H CF ¼ Bk0 C ðkÞ Re Bkq C qðkÞ þ ð1Þq C ðkÞ q 0 þ q¼1
k
þ
h
i ImBkq C ðkÞ q
ð1Þ
q
C ðkÞ q
io
# ,
ð1Þ
where the CF parameters Bkq ‘‘measure the strength of interaction between the open-shell electrons of paramagnetic ions and their surrounding crystalline environment ‘‘[6,7] and hence play an important role in the CF studies. C ðkÞ q is the tensor operator defined by rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p Y kq C ðkÞ ¼ (2) q 2k þ 1 with Ykq denoting spherical harmonic functions. The CF Hamiltonian for 3dN ions at axial symmetry site can be explicitly expressed as ð4Þ q ð4Þ ð4Þ H CF ¼ B20 C ð2Þ þ B C þ B C þ ð1Þ C 40 4q q q , 0 0 (3) where q ¼ 3 for trigonal symmetry type I (C3v, D3, D3d) (zJ[1 1 1], the CF x-axis is always defined along the line joining the center of the upper oxygen trigonal to one of its vertices) and q ¼ 4 for tetragonal symmetry type I (C4v, D2d, D4, D4h) (zJ[0 0 1], xJ[1 1 0]). It can be seen from Eq. (3) that there are three independent CF parameters B20, B40, B43 for trigonal symmetry type I and B20, B40, B44 for tetragonal symmetry type I. The ‘‘Wybourne’’ CF parameter notation [8] has been widely adopted in the investigations of the TM ions of crystals. However, for the axial symmetry CF, other conventional CF parameter notations, i.e., CF parameters: (Dq, Ds, Dt) [9] and (Dq, m, d) [10] for tetragonal symmetry type I as well as (Dq, Dt, Ds) [9] and (Dq, u, u0 ) [11] have been also used by the various authors. As pointed out by Morrison [12], ‘‘the correlation of the various notations used in the analysis of the crystal-field interaction is extremely difficult’’. In fact, much confusion appears in literature. Confusions arose because the various CF parameter notations were sometimes incorrectly interpreted or understood. In this work, we will try to clarify
the relationships among the various CF parameter notations.
2. Relation of ‘‘Wybourne’’ CF notation Bkq to other CF notations in axial symmetry The axial symmetry CF can be considered as distorted from cubic one along [1 1 1] axis for trigonal symmetry CF and [0 0 1] axis for tetragonal symmetry CF. Hence, the CF can be separated into a cubic component with parameters Bcubic plus non-cubic one with parameters B0kq ; i.e. kq Bkq ¼ Bcubic þ B0kq , kq
(4)
are the cubic CF parameter, whereas where Bcubic kq B0kq measure the non-cubic trigonal CF components and vanish identically in cubic symmetry. 2.1. ‘‘Pryce & Runciman (P&R)’’ CF parameter notation in trigonal symmetry The CF Hamiltonian for trigonal symmetry can be written, in terms of the Pryce and Runciman (P&R) CF parameter notations Dq, u and u0 (see the definitions in the appendix), as [11,13] H CF ¼ H cubic ðDqÞ þ H non-cubic ðu; u0 Þ pffiffiffiffiffi 4 70 4 4 U 0 2 U 3 U 3 ¼ 10Dq 5 1 pffiffiffiffiffi 2 4 pffiffiffiffiffi 4 70U 0 þ 70U 0 þu 7 21 2 U 43 U 43 3 pffiffiffiffiffi 2 pffiffiffi 4 0 4 4 4 35 U 0 þ U 0 2 U 3 U 3 , þu 7 ð5Þ where the U kq are the irreducible tensor operators and the following relations for d electron hold: ðkÞ k 2 U , C ðkÞ (6) q ¼ 2 C q where (the reduced matrix elements) rffiffiffiffiffi rffiffiffiffiffi
ð2Þ
10 10 ð4Þ 2 C 2 ¼ ; 2 C 2 ¼ . 7 7
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By comparing Eqs. (3) and (5) for trigonal symmetry, we obtain pffiffiffi (7a) B20 ¼ u 2 2u0 ,
the Hamiltonian
pffiffiffi 4 B40 ¼ u þ 2 2u0 14Dq, 3
(7b)
pffiffiffi 7 2 u þ 2u0 þ 20Dq . B43 ¼ pffiffiffiffiffi 70 3
(7c)
where Hee, HCF, HTrees, and Hm represent, respectively, the electrostatic, the CF, Trees correction, and the magnetic interactions including the spin–spin (SS) and the spin-other-orbit (SOO) magnetic interaction besides the spin-orbit (SO) magnetic interaction [1]. The CF Hamiltonian in the Wybourne notation adopts Eq. (3) for axial symmetry crystal field. Using the optical parameters: a ¼ 70 cm1, B ¼ 645 cm1, C ¼ 3000 cm1, Dq ¼ 1815 cm1, u ¼ 800 cm1, u0 ¼ 680 cm1 for Al2O3: Cr3+(3d3), Fairbank et al. [19] obtained the energy levels in terms of the complete diagonalization method (CDM) based on the diagonalization of the CF matrix which are functions of the Racah parameters B and C, Tress parameter a, CF parameters Dq, u, and u0 . According to Eq. (7) and the CF parameters Dq, u, and u0 obtained by Fairbank et al. [22], we obtain B20 ¼ 1123.3 cm1, B40 ¼ 22420.0 cm1, B43 ¼ 31621.6 cm1 for Al2O3: Cr3+. By using these parameters as well as a ¼ 70 cm1, B ¼ 645 cm1, C ¼ 3000 cm1, we obtain the energy levels in terms of the CDM/MSH program. Our calculated results are in good agreement with ones obtained by Fairbank et al. [19] for Al2O3: Cr3+.
Thus, from Eqs. (4) and (7) we have ¼ 0; Bcubic 20
(8a)
Bcubic ¼ 14Dq, 40
(8b)
pffiffiffiffiffi Bcubic ¼ 2 70Dq, 43
(8c)
pffiffiffi B020 ¼ B20 ¼ u 2 2u0 ,
(8d)
pffiffiffi B040 ¼ ð4=3Þ u þ 3u0 = 2 ,
(8e)
pffiffiffiffiffi pffiffiffi B043 ¼ 70=15 u þ 3u0 = 2 ,
(8f)
where B020 and B040 are the net non-cubic CF parameters and vanish identically in net cubic symmetry. The Dq(P&R) is the cubic CF parameter at a perfect cubic site [11]. Recently, a CDM/MSH program [14–17] has been developed for numerical calculations of the energy levels and eigenvectors as well as the SH parameters D, gJ, and g? for 3d3 configurations at axial symmetry crystal field including trigonal type I (C3v, D3, D3d) and tetragonal type I (C4v, D4, D2d). The origin Hamiltonian includes the Coulomb interaction Hee, the SO coupling interaction Hso, and the crystal field (CF) Hamiltonian HCF. The CDM/MSH program has recently been converted from the FORTRAN version [18] under DOS into the Visual Basic 6.0 version under Microsoft Windows environment. In the extended CDM/MSH program [15,16] two additional terms, i.e. the HSOO, and HSS, have been included in Hamiltonian. The method is based on the diagonalization of the 120 120 matrix originated by
H ¼ H ee ðB; CÞ þ H CF ðBkq Þ þ H Trees ðaÞ þ H m ðz; M 0 ; M 2 Þ,
ð9Þ
2.2. Griffith’s [10] and Yeung & Rudowicz’s(Y&R) [20] CF notations in tetragonal symmetry The CF Hamiltonian for tetragonal symmetry can be written, in terms of the Griffith’s CF parameter notation Dq, d, and m (see the definitions in the appendix), as H CF ¼ H cubic ðDqÞ þ H non-cubic ðm; dÞ " # rffiffiffiffiffi 5 ð4Þ ð4Þ ð4Þ C þ C 4 ¼ 21Dq C 0 þ 14 4 pffiffiffiffiffi 70 ð4Þ ð2Þ ð4Þ C 4 þ C ð4Þ þ d C0 C0 þ 4 10
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3 ð4Þ C þ m C ð2Þ þ 0 4 0 pffiffiffiffiffi 3 70 ð4Þ C ð4Þ þ C þ . 4 4 40
ð10Þ
By comparing Eq. (3) with Eq. (10) at tetragonal symmetry site, we have B20 ¼ d m, B40
B44
(11b)
rffiffiffiffiffi pffiffiffiffiffi 70 5 ð3m þ 4dÞ, ¼ 21 Dq þ 40 14
(11c)
where CF parameter m and d measure the tetragonal CF components since they vanish identically at a perfect cubic site, whereas Dq denotes the CF component of the perfect cubic site. It is worthwhile to point out that, by comparing the matrix elements of Griffith [10] for tetragonal symmetry, Morrison [12] also obtained the relations which are in agreement with Eq. (11) obtained by us. According to Eq. (4) and Eq. (11), we have Bcubic ¼ 0, 20
(12a)
¼ 21Dq, Bcubic 40
(12b)
rffiffiffiffiffi 5 ¼ 21 Dq, 14
(12c)
B020 ¼ d m,
(12d)
1 B040 ¼ ð3m þ 4dÞ, 4
(12e)
B044
pffiffiffiffiffi 70 ð3m þ 4dÞ. ¼ 40
(13)
Thus, Y&R obtained the following relations [21]: B020 ¼ d m,
(14a)
3 B040 ¼ ð3m þ 4dÞ. 5
(14b)
(11a)
1 ¼ 21Dq ð3m þ 4dÞ, 4
Bcubic 44
‘cubic’ one with parameter Dq. " # rffiffiffiffiffi 4 ð4Þ ð4Þ ð4Þ C þ C 4 . H cubic ¼ 21Dq C 0 þ 15 4
(12f)
In addition, Yeung and Rudowicz (Y&R) [20,21] think that the axial symmetry CF can be considered as the values of the tetragonal crystal field parameters B020 and B040 for the distorted component of the crystal field potential H Tetra ¼ ð4Þ 0 B020 C ð2Þ in addition to the dominant 0 þ B40 C 0
It can be seen from Eqs. (12) and (14) that the CF parameter B020 in Eq. (12) equal to the CF parameter B020 in Eq. (14), whereas the CF parameter B040 in Eq. (12) is different from the CF parameter B040 in Eq. (14). In fact, according to Y&R’s definition, we have B20 ¼ B020 ¼ d m,
(15a)
3 B40 ¼ Bcubic þ B040 ¼ 21Dq ð3m þ 4dÞ, 40 5
(15b)
B44 ¼
Bcubic 44
rffiffiffiffiffi 5 ¼ 21 Dq. 14
(15c)
By comparing Eq. (11) with Eq. (15), it is found that the meaning of Dq in Eq. (11) is different from that of Dq in Eq. (15). As above, Dq in Eq. (11) denotes the CF component of the perfect cubic site, i.e. it is a net cubic CF parameter. Thus, we think that Dq(Y&R) in Eq. (15) includes the tetragonal CF component besides the cubic CF one, i.e. Dq in Eqs. (15b) and (15c) correspond to a choice of ‘‘cubic’’ CF parameter incorporating effect of the axial distortion. In order to check Eq. (15), we compare our CDM results with Fairbank’s ones for MgO: Cr3+ at the tetragonal symmetry site. Fairbank et al. [22] obtained the energy levels in terms of the CDM based on the diagonalization of the CF matrix which are functions of the Racah parameters B and C, Tress parameter a, CF parameters Dq, d, and m. According to Eq. (15) and the CF parameters Dq, d, and m obtained by Fairbank et al. [22], we obtain B20 ¼ 4550 cm1, B40 ¼ 37065.0 cm1, B44 ¼ 20645 cm1 for MgO: Cr3+. By using
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these parameters as well as a ¼ 70 cm1, B ¼ 570 cm1, C ¼ 3165 cm1, we obtain the energy levels in terms of the CDM/MSH program. Our calculated results are in good agreement with ones obtained by Fairbank et al. [22] for MgO: Cr3+(3d3). It is worthwhile to point out in Fairbank’s work [22] that Dq for tetragonal symmetry site is chosen as 1640 cm1 including the non-cubic component besides the cubic CF one, whereas Dq at a perfect cubic site is chosen as 1590 cm1 including only the cubic CF component. 2.3. Ballhausen’s CF parameter notation in axial symmetry In order to investigate the 3dN ions in axial symmetry crystal field, Ballhausen designates a set of CF parameters: Dq, Ds, and Dt for trigonal symmetry and Dq, Ds, and Dt for tetragonal symmetry. Ballhausen’s CF parameter notation has been widely used in the investigations of 3dN ions in crystals [23–25]. By comparing the matrix elements, one finds that the relationships between the ‘‘Ballhausen’’ CF parameter notation and the ‘‘Wybourne’’ parameters can be written as [12] B20 ¼ 7Ds,
(16a)
B40 ¼ 14Dq 21Dt,
(16b)
pffiffiffiffiffi B43 ¼ 2 70Dq,
(16c)
where the definitions for the CF parameters Ds, and Dt are given in the appendix. It can be seen from Eqs. (7) and (16) that the parameter Dq(P&R) in Eq. (7) is not equal to the parameter Dq(Ballhausen) in Eq. (16). In fact, the physical meanings of the Dq(P&R) is different from those of Dq(Ballhausen). The parameter Dq(P&R) denotes the CF parameter of the perfect cubic site, whereas the parameter Dq(Ballhausen) contains the trigonal CF component besides the cubic component. We obtain the following relationships: 1 3 0 p ffiffi ffi DqðBallhausenÞ ¼ DqðP&RÞ þ uþ u 30 2 7 ð17Þ ¼ DqðP&RÞ Dt. 18
141
For tetragonal CF, Ballhausen [9] and Febbraro [26,27] obtained the following relations: B20 ¼ 7Ds,
(18a)
B40 ¼ 21Dq 21Dt,
(18b)
rffiffiffiffiffi 5 ¼ 21 Dq, 14
(18c)
B44
where the definitions for the CF parameters Ds, and Dt are given in the appendix. By comparing Eqs. (18) with (11) and Eq. (15), we find that the meaning of Dq(Y&R) in Eq. (15) is equivalent to that of Dq(Ballhausen) in Eq. (18), whereas the meaning of Dq(Griffith) in Eq. (11) is different from that of Dq(Ballhausen) in Eq. (18) or Dq(Y&R) in Eq. (15). The parameter Dq(Griffith) denotes the CF parameter of the perfect cubic site, whereas the CF parameter Dq(Ballhausen) in Eq. (18) or Dq(Y&R) in Eq. (15) contains the tetragonal CF component besides the cubic component. We obtain the following relationships: DqðY&RÞ ¼ DqðBallhausenÞ 1 ð3m þ 4dÞ 60 7 ¼ DqðGriffithÞ þ Dt. 12 ¼ DqðGriffithÞ þ
ð19Þ
2.4. Sharma [28] and Yu’s [29] (S&Y) notation in axial symmetry For the perfect cubic symmetry CF, The CF Hamiltonian can be written as ð4Þ ð4Þ z½1 1 1 , H CF ¼ B40C C ð4Þ þ B C C 43C 0 3 3 (20a) H CF
ð4Þ ð4Þ ¼ B40C C ð4Þ þ B C þ C 44C 0 4 4
z½0 0 1 .
(20b) It is well known that there exists a constant ratio between B40C and B43C(B44C). These rations can be expressed as pffiffiffiffiffiffiffiffiffiffi B40C ¼ 7=10B43C , (21a) B40C ¼
pffiffiffiffiffiffiffiffiffiffi 14=5B44C .
(21b)
Sharma and Yu’s
DqðS&YÞ ¼ DqðP&RÞ
1 B40 21
rffiffiffiffiffi ! 7 B43 10
1 3 7 u þ pffiffiffi u0 ¼ DqðP&RÞ Dt 30 18 2
Ds ¼
1 4 u pffiffiffi u0 7 2 3 3 u þ pffiffiffi u0 Dt ¼ 35 2
1 Dq ¼ pffiffiffiffiffi B43 2 70
Dt ¼
1 Ds ¼ B20 7
13 3 3 u þ pffiffiffi u0 ¼ DqðBallhausenÞ þ Dt 420 4 2
DqðBallhausenÞ ¼ DqðP&RÞ þ
u ¼ 3Ds
20 Dt 3 ffiffiffi p pffiffiffi 5 2 0 u ¼ 2Ds Dt 3
pffiffiffiffiffi 1 70 B40 B43 54 189
pffiffiffi pffiffiffi pffiffiffiffiffi 2 5 2 35 B43 B20 þ B40 63 7 63
Dq ¼
u0 ¼
pffiffiffiffiffi 3 20 2 70 u ¼ B20 þ B40 B43 7 63 63
B43
B40 ¼ 21Dq 21Dt pffiffiffiffiffi B43 ¼ 2 70Dq
B20 ¼ 7Ds
Dq ¼
B040
1 B40 þ 28
rffiffiffiffiffi ! 7 B43 10 pffiffiffiffiffi 1 70 B40 þ B44 36 252
DqðS&YÞ ¼ DqðGriffithÞ þ
1 B40 21
rffiffiffiffiffi ! 14 B44 5
1 7 ð3m þ 4dÞ ¼ DqðGriffithÞ þ Dt 60 12
1 ðm dÞ 7 1 ð3m þ 4dÞ Dt ¼ 35 Ds ¼
2 Dq ¼ pffiffiffiffiffi B44 3 70
Dt ¼
1 1 ð3m þ 4dÞ ¼ DqðBallhausenÞ Dt 420 2
DqðBallhausenÞ ¼ DqðGriffithÞ þ
m ¼ 4Ds þ 5Dt d ¼ 3Ds þ 5Dt
Dq ¼
3 5 10 d ¼ B20 B40 þ pffiffiffiffiffi B44 7 21 3 70
1 Ds ¼ B20 7
B40 ¼ 21Dq 21Dt rffiffiffiffiffi 5 Dq B44 ¼ 21 14
1 B40 ¼ 21Dq ð3m þ 4dÞ rffiffiffiffiffi 4 pffiffiffiffiffi 5 70 Dq þ ð3m þ 4dÞ B44 ¼ 21 14 40 4 5 10 m ¼ B20 B40 þ pffiffiffiffiffi B44 7 21 3 70
B20 ¼ 7Ds
B20 ¼ d m
Ballhausen’s
Dq ¼
rffiffiffiffiffi ! 14 B44 5
rffiffiffiffiffi 14 B44 5
1 B40 þ 42
B040 ¼ B40
Sharma and Yu’s
142
B40
B20
rffiffiffiffiffi 7 B43 ¼ B40 10
Griffith’s
Ballhausen’s
Pryce and Runciman’s
pffiffiffi ¼ u 2 2u0 pffiffiffi 4 ¼ u þ 2 2u0 14Dq 3 pffiffiffi 7 2 u þ 2u0 þ 20Dq ¼ pffiffiffiffiffi 70 3
Tetragonal symmetry type I( C4v, D2d, D4, D4h )
Trigonal symmetry type I(C3v, D3d, D3)
Table 1 Relations among the various CF parameter notations
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Z.-Y. Yang, Q. Wei / Physica B 370 (2005) 137–145
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can been seen from p Eq. (21) that the B40C pIt ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi 7=10B43C and B40C 14=5B44C atpffiffiffiffiffiffiffiffiffiffi a perfect cubic site vanish. However, the 7=10B43C 40C pB ffiffiffiffiffiffiffiffiffiffi at trigonal site and B40C 14=5B44C at tetragonal site is no longer zero. Thus Sharma [28] and Yu [29,30] denote the difference by B040 (S&Y) pffiffiffiffiffiffiffiffiffiffi B040 ðS&YÞ ¼ B40 7=10B43 , for trigonal site, ð22aÞ pffiffiffiffiffiffiffiffiffiffi B040 ðS&YÞ ¼ B40 14=5B44 , for tetragonal site.
(23)
with DqC ¼
1 B40C 14
for trigonal site,
(24a)
pffiffiffiffiffiffiffiffiffiffi DqðS&YÞ ¼ B40 þ 7=10B43 =28 for trigonal site,
DqC ¼
1 B40C 21
ð24bÞ
for tetragonal site,
(24c)
pffiffiffiffiffiffiffiffiffiffi DqðS&YÞ ¼ B40 þ 14=5B44 =42 for tetragonal site.
13 3 u þ pffiffiffi u0 420 2 3 ¼ DqðBallhausenÞ þ Dt 4 for trigonal site,
ð26aÞ
1 ð3m þ 4dÞ 420 1 ¼ DqðBallhausenÞ Dt 2 for tetragonal site.
ð26bÞ
DqðS&YÞ ¼ DqðP&RÞ
DqðS&YÞ ¼ DqðGriffithÞ þ
ð22bÞ
In fact, Yu [30] defines the three parameters B20 (S&Y) (or B020 (S&Y)), B040 (S&Y), Dq0 (S&Y) to measure the net trigonal or tetragonal CF component since they vanish identically in perfect cubic site. Dq0 is defined by [30] Dq0 ¼ DqðS&YÞ DqC .
143
ð24dÞ
where DqC denotes the CF parameter of the perfect cubic site. We find that B20 (S&Y) equal to B20 (or B020 ) in Eq. (8d), and DqC for trigonal site in Eq. (24a) and DqC for tetragonal site in Eq. (24c) equal to Dq (P&R) in Eq. (7) and Dq(Griffith) in Eq. (11) respectively, whereas B040 (S&Y) in Eq. (22a) is different from B040 in Eq. (8e). We obtain: 9 3 0 0 u þ pffiffiffi u ¼ 21Dt, B40 ðS&YÞ ¼ (25) 5 2
3. Summary The relationships among the various crystal field notations for 3dN ions in axial symmetry crystal field have been studied. The confusions among these CF notations have been clarified. The following conclusions are obtained: (i) For trigonal symmetry CF, Pryce and Runciman (P&R) CF parameter notation Dq(P&R) in Eq. (7) is not equal to the parameter Dq(Ballhausen) in Eq. (16). The physical meanings of the Dq(P&R) is different from those of Dq(Ballhausen). The parameter Dq(P&R) denotes the CF parameter of the perfect cubic site, whereas the parameter Dq(Ballhausen) contains the trigonal CF component besides the cubic component. (ii) For tetragonal symmetry CF, the meaning of Dq(Y&R) in Eq. (15) is equivalent to that of Dq(Ballhausen) in Eq. (18), whereas the meaning of Dq(Griffith) in Eq. (11) is different from that of Dq(Ballhausen) in Eq (18) or Dq(Y&R) in Eq. (15). The parameter Dq(Griffith) denotes the CF parameter of the perfect cubic site, whereas the parameter Dq(Ballhausen) in Eq. (18) or Dq(Y&R) in Eq. (15) contains the tetragonal CF component besides the cubic component. (iii) The B20 (S&Y) is equal to B20 (or B020 ) in Eq. (8d), and DqC (S&Y) for trigonal site in Eq. (24a) and DqC (S&Y) for tetragonal site in Eq. (24c) equal to Dq (P&R) in Eq. (7) and Dq(Griffith) in Eq. (11) respectively, whereas B040 (S&Y) in Eq. (22a) is different from B040 in Eq. (8e). The correlations of the various crystal field notations are summarized in Table 1, which is very useful in the investigations of 3dN ions in crystal material.
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Acknowledgments One of the authors (Z.Y. Yang) would like to thank Prof. C. Rudowicz and Dr. Y.Y. Yeung for their helpful discussions at City University of Hong Kong. This work was supported by the Education Committee Natural Science Foundation of Shaanxi Province (Grant No. 05JK139) and a Baoji University of Arts and Sciences Key Research Grant.
(2) Pryce and Runciman’s CF parameter notation for trigonal symmetry [9,11,13]
1 (A.3a) tþ V trig tþ ¼ t V trig t ¼ u, 3
2 t0 V trig t0 ¼ u, 3
(A.3b)
tþ V trig eþ ¼ t V trig e ¼ u0 .
(A.3c)
(3) Griffith’s CF parameter notation for tetragonal [10]
Appendix We present here the definitions of the various CF parameters notations. They will be convenient for reader to consult.
2 2 EðzÞ ¼ Eðd xy Þ ¼ D d, 5 3
(A.4a)
(1) Ballhausen’s CF parameter notation (a) Trigonal symmetry [9] D
E
V trig t ¼ Ds þ 2 Dt, t 2g 2g 3
2 1 Eðx; ZÞ ¼ Eðd yz ; d zx Þ ¼ D þ d, 5 3
(A.4b)
3 1 EðyÞ ¼ E ðd z2 Þ ¼ D m, 5 2
(A.4c)
(A.1a)
D
E t02g V trig t02g ¼ 2Ds 6Dt,
(A.1b)
pffiffiffi D
E pffiffiffi 5 2
Dt, t2g V trig eg ¼ 2Ds 3
(A.1c)
3 1 EðeÞ ¼ E d x2 y2 ¼ D þ m, 5 2 where D ¼ 10 Dq.
(A.4d)
References D
E
V trig e ¼ 7 Ds. e g g 3 (b) Tetragonal symmetry [9]
d x2 y2 jV tetra jd x2 y2 ¼ 2Ds Dt,
(A.1d)
(A.2a)
d z2 jV tetra jd z2 ¼ 2Ds 6Dt,
(A.2b)
d xy jV tetra jd xy ¼ 2Ds Dt,
(A.2c)
hd xz jV tetra jd xz i ¼ Ds þ 4Dt.
(A.2d)
[1] Z.Y. Yang, Y. Hao, C. Rudowicz, Y.Y. Yeung, J. Phys.: Condens. Matter. 16 (2004) 3481. [2] Z.Y. Yang, J. Magn. Magn. Matter. 238 (2000) 200. [3] L.H. Xie, P. Hu, P. Huang, J. Phys. Chem. Solids 66 (2004) 918. [4] W.C. Zheng, Physica B 212 (1995) 420. [5] H. Ebisu, M. Arakawa, H. Takeuchi, J. Phys.: Condens. Matter. 17 (2005) 4653. [6] C. Rudowicz, Magn. Reson. Rev. 13 (1987) 1. [7] D.J. Newman, B. Ng, Rep. Prog. Phys. 52 (1989) 699. [8] B.G. Wybourne, Spectroscopic Properties of Rare Earth, Wiley, New York, 1965. [9] C.J. Ballhausen, Introduction to Ligand Field Theory, McGraw-Hill, New York, 1962. [10] J.S. Griffith, Theory of Transition Metal Ions, Cambridge University Press, London, 1961. [11] M.H.L. Pryce, W.A. Runciman, Discuss. Faraday Soc. 26 (1958) 34.
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[21] Y.Y. Yeung, J. Phys.: Condens. Matter. 2 (1990) 2461. [22] W.M. Fairbank, G.K. Klauminzer, Phys. Rev. B 7 (1973) 500. [23] E. Konig, R. Schnakig, Chem. Phys. Lett. 32 (1975) 553. [24] H. Riesen, E. Krausz, L. Dubicki, J. Lumin. 44 (1989) 97. [25] Y. Lei, X.T. Zu, M.G. Zhao, Physica B 364 (2005) 122. [26] S. Febbraro, J. Phys. C: Solid State Phys. 20 (1987) 5367. [27] S. Febbraro, J. Phys. C: Solid State Phys. 21 (1988) 2577. [28] R.R. Sharma, T.P. Das, Phys. Rev. 149 (1966) 257. [29] W.L. Yu, M.G. Zhao, Phys. Rev. B 37 (1988) 9254. [30] W.L. Yu, J. Phys.: Condens. Matter. 6 (1994) 5105.
Physica B 370 (2005) 137–145 www.elsevier.com/locate/physb
On the relations between the crystal field parameter notations in the ‘‘Wybourne’’ notation and the conventional ones for 3dN ions in axial symmetry crystal field Zi-Yuan Yang, Qun Wei Department of Physics, Baoji University of Arts and Science, Baoji 721007, PR China Received 21 August 2005; received in revised form 10 September 2005; accepted 10 September 2005
Abstract The correlations of the various crystal field (CF) notations existing in the literature for 3dN ions in axial symmetry crystal field have been investigated. The confusions among these CF notations have been clarified. The correlations of the various CF notations are summarized in Table 1, which is very useful in the investigations of 3dN ions in crystal material. r 2005 Elsevier B.V. All rights reserved. PACS: 71.70.Ch; 76.30.Fc Keywords: 3dN transition-metal ions; Crystal field parameter notations; Axial symmetry crystal field; ‘‘Wybourne’’ notation
1. Introduction 3dN transition-metal (TM) ions such as Cr3+, Ni2+, Ti3+ are active ions in solid state laser materials [1,2] and non-linear optical materials [3–5]. These impurity ions in the materials play a major role because they can be responsible for the modification of optical properties. Thus, microscopic study of TM ions in various crystal Corresponding author.
E-mail address: [email protected] (Z.-Y. Yang).
materials has attracted much attention. As is well known, the crystal field (CF) parameters and the spin-Hamiltonian (SH) ones for the TM ions in crystals are very sensitive to subtle changes of the crystal structure [1–4]. Hence, the studies of these parameters can provide a great deal of microscopic insight concerning the crystal structure, structural disorder, phase transitions, pressure behavior as well as the observed magnetic and spectroscopic properties. In the phenomenological theory of CF, the CF Hamiltonian for 3dN ions in crystal materials can be written, in terms of ‘‘Wybourne’’
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.09.002
ARTICLE IN PRESS Z.-Y. Yang, Q. Wei / Physica B 370 (2005) 137–145
138
CF parameter notation Bkq, as [6] " k n h i X X H CF ¼ Bk0 C ðkÞ Re Bkq C qðkÞ þ ð1Þq C ðkÞ q 0 þ q¼1
k
þ
h
i ImBkq C ðkÞ q
ð1Þ
q
C ðkÞ q
io
# ,
ð1Þ
where the CF parameters Bkq ‘‘measure the strength of interaction between the open-shell electrons of paramagnetic ions and their surrounding crystalline environment ‘‘[6,7] and hence play an important role in the CF studies. C ðkÞ q is the tensor operator defined by rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4p Y kq C ðkÞ ¼ (2) q 2k þ 1 with Ykq denoting spherical harmonic functions. The CF Hamiltonian for 3dN ions at axial symmetry site can be explicitly expressed as ð4Þ q ð4Þ ð4Þ H CF ¼ B20 C ð2Þ þ B C þ B C þ ð1Þ C 40 4q q q , 0 0 (3) where q ¼ 3 for trigonal symmetry type I (C3v, D3, D3d) (zJ[1 1 1], the CF x-axis is always defined along the line joining the center of the upper oxygen trigonal to one of its vertices) and q ¼ 4 for tetragonal symmetry type I (C4v, D2d, D4, D4h) (zJ[0 0 1], xJ[1 1 0]). It can be seen from Eq. (3) that there are three independent CF parameters B20, B40, B43 for trigonal symmetry type I and B20, B40, B44 for tetragonal symmetry type I. The ‘‘Wybourne’’ CF parameter notation [8] has been widely adopted in the investigations of the TM ions of crystals. However, for the axial symmetry CF, other conventional CF parameter notations, i.e., CF parameters: (Dq, Ds, Dt) [9] and (Dq, m, d) [10] for tetragonal symmetry type I as well as (Dq, Dt, Ds) [9] and (Dq, u, u0 ) [11] have been also used by the various authors. As pointed out by Morrison [12], ‘‘the correlation of the various notations used in the analysis of the crystal-field interaction is extremely difficult’’. In fact, much confusion appears in literature. Confusions arose because the various CF parameter notations were sometimes incorrectly interpreted or understood. In this work, we will try to clarify
the relationships among the various CF parameter notations.
2. Relation of ‘‘Wybourne’’ CF notation Bkq to other CF notations in axial symmetry The axial symmetry CF can be considered as distorted from cubic one along [1 1 1] axis for trigonal symmetry CF and [0 0 1] axis for tetragonal symmetry CF. Hence, the CF can be separated into a cubic component with parameters Bcubic plus non-cubic one with parameters B0kq ; i.e. kq Bkq ¼ Bcubic þ B0kq , kq
(4)
are the cubic CF parameter, whereas where Bcubic kq B0kq measure the non-cubic trigonal CF components and vanish identically in cubic symmetry. 2.1. ‘‘Pryce & Runciman (P&R)’’ CF parameter notation in trigonal symmetry The CF Hamiltonian for trigonal symmetry can be written, in terms of the Pryce and Runciman (P&R) CF parameter notations Dq, u and u0 (see the definitions in the appendix), as [11,13] H CF ¼ H cubic ðDqÞ þ H non-cubic ðu; u0 Þ pffiffiffiffiffi 4 70 4 4 U 0 2 U 3 U 3 ¼ 10Dq 5 1 pffiffiffiffiffi 2 4 pffiffiffiffiffi 4 70U 0 þ 70U 0 þu 7 21 2 U 43 U 43 3 pffiffiffiffiffi 2 pffiffiffi 4 0 4 4 4 35 U 0 þ U 0 2 U 3 U 3 , þu 7 ð5Þ where the U kq are the irreducible tensor operators and the following relations for d electron hold: ðkÞ k 2 U , C ðkÞ (6) q ¼ 2 C q where (the reduced matrix elements) rffiffiffiffiffi rffiffiffiffiffi
ð2Þ
10 10 ð4Þ 2 C 2 ¼ ; 2 C 2 ¼ . 7 7
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139
By comparing Eqs. (3) and (5) for trigonal symmetry, we obtain pffiffiffi (7a) B20 ¼ u 2 2u0 ,
the Hamiltonian
pffiffiffi 4 B40 ¼ u þ 2 2u0 14Dq, 3
(7b)
pffiffiffi 7 2 u þ 2u0 þ 20Dq . B43 ¼ pffiffiffiffiffi 70 3
(7c)
where Hee, HCF, HTrees, and Hm represent, respectively, the electrostatic, the CF, Trees correction, and the magnetic interactions including the spin–spin (SS) and the spin-other-orbit (SOO) magnetic interaction besides the spin-orbit (SO) magnetic interaction [1]. The CF Hamiltonian in the Wybourne notation adopts Eq. (3) for axial symmetry crystal field. Using the optical parameters: a ¼ 70 cm1, B ¼ 645 cm1, C ¼ 3000 cm1, Dq ¼ 1815 cm1, u ¼ 800 cm1, u0 ¼ 680 cm1 for Al2O3: Cr3+(3d3), Fairbank et al. [19] obtained the energy levels in terms of the complete diagonalization method (CDM) based on the diagonalization of the CF matrix which are functions of the Racah parameters B and C, Tress parameter a, CF parameters Dq, u, and u0 . According to Eq. (7) and the CF parameters Dq, u, and u0 obtained by Fairbank et al. [22], we obtain B20 ¼ 1123.3 cm1, B40 ¼ 22420.0 cm1, B43 ¼ 31621.6 cm1 for Al2O3: Cr3+. By using these parameters as well as a ¼ 70 cm1, B ¼ 645 cm1, C ¼ 3000 cm1, we obtain the energy levels in terms of the CDM/MSH program. Our calculated results are in good agreement with ones obtained by Fairbank et al. [19] for Al2O3: Cr3+.
Thus, from Eqs. (4) and (7) we have ¼ 0; Bcubic 20
(8a)
Bcubic ¼ 14Dq, 40
(8b)
pffiffiffiffiffi Bcubic ¼ 2 70Dq, 43
(8c)
pffiffiffi B020 ¼ B20 ¼ u 2 2u0 ,
(8d)
pffiffiffi B040 ¼ ð4=3Þ u þ 3u0 = 2 ,
(8e)
pffiffiffiffiffi pffiffiffi B043 ¼ 70=15 u þ 3u0 = 2 ,
(8f)
where B020 and B040 are the net non-cubic CF parameters and vanish identically in net cubic symmetry. The Dq(P&R) is the cubic CF parameter at a perfect cubic site [11]. Recently, a CDM/MSH program [14–17] has been developed for numerical calculations of the energy levels and eigenvectors as well as the SH parameters D, gJ, and g? for 3d3 configurations at axial symmetry crystal field including trigonal type I (C3v, D3, D3d) and tetragonal type I (C4v, D4, D2d). The origin Hamiltonian includes the Coulomb interaction Hee, the SO coupling interaction Hso, and the crystal field (CF) Hamiltonian HCF. The CDM/MSH program has recently been converted from the FORTRAN version [18] under DOS into the Visual Basic 6.0 version under Microsoft Windows environment. In the extended CDM/MSH program [15,16] two additional terms, i.e. the HSOO, and HSS, have been included in Hamiltonian. The method is based on the diagonalization of the 120 120 matrix originated by
H ¼ H ee ðB; CÞ þ H CF ðBkq Þ þ H Trees ðaÞ þ H m ðz; M 0 ; M 2 Þ,
ð9Þ
2.2. Griffith’s [10] and Yeung & Rudowicz’s(Y&R) [20] CF notations in tetragonal symmetry The CF Hamiltonian for tetragonal symmetry can be written, in terms of the Griffith’s CF parameter notation Dq, d, and m (see the definitions in the appendix), as H CF ¼ H cubic ðDqÞ þ H non-cubic ðm; dÞ " # rffiffiffiffiffi 5 ð4Þ ð4Þ ð4Þ C þ C 4 ¼ 21Dq C 0 þ 14 4 pffiffiffiffiffi 70 ð4Þ ð2Þ ð4Þ C 4 þ C ð4Þ þ d C0 C0 þ 4 10
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140
3 ð4Þ C þ m C ð2Þ þ 0 4 0 pffiffiffiffiffi 3 70 ð4Þ C ð4Þ þ C þ . 4 4 40
ð10Þ
By comparing Eq. (3) with Eq. (10) at tetragonal symmetry site, we have B20 ¼ d m, B40
B44
(11b)
rffiffiffiffiffi pffiffiffiffiffi 70 5 ð3m þ 4dÞ, ¼ 21 Dq þ 40 14
(11c)
where CF parameter m and d measure the tetragonal CF components since they vanish identically at a perfect cubic site, whereas Dq denotes the CF component of the perfect cubic site. It is worthwhile to point out that, by comparing the matrix elements of Griffith [10] for tetragonal symmetry, Morrison [12] also obtained the relations which are in agreement with Eq. (11) obtained by us. According to Eq. (4) and Eq. (11), we have Bcubic ¼ 0, 20
(12a)
¼ 21Dq, Bcubic 40
(12b)
rffiffiffiffiffi 5 ¼ 21 Dq, 14
(12c)
B020 ¼ d m,
(12d)
1 B040 ¼ ð3m þ 4dÞ, 4
(12e)
B044
pffiffiffiffiffi 70 ð3m þ 4dÞ. ¼ 40
(13)
Thus, Y&R obtained the following relations [21]: B020 ¼ d m,
(14a)
3 B040 ¼ ð3m þ 4dÞ. 5
(14b)
(11a)
1 ¼ 21Dq ð3m þ 4dÞ, 4
Bcubic 44
‘cubic’ one with parameter Dq. " # rffiffiffiffiffi 4 ð4Þ ð4Þ ð4Þ C þ C 4 . H cubic ¼ 21Dq C 0 þ 15 4
(12f)
In addition, Yeung and Rudowicz (Y&R) [20,21] think that the axial symmetry CF can be considered as the values of the tetragonal crystal field parameters B020 and B040 for the distorted component of the crystal field potential H Tetra ¼ ð4Þ 0 B020 C ð2Þ in addition to the dominant 0 þ B40 C 0
It can be seen from Eqs. (12) and (14) that the CF parameter B020 in Eq. (12) equal to the CF parameter B020 in Eq. (14), whereas the CF parameter B040 in Eq. (12) is different from the CF parameter B040 in Eq. (14). In fact, according to Y&R’s definition, we have B20 ¼ B020 ¼ d m,
(15a)
3 B40 ¼ Bcubic þ B040 ¼ 21Dq ð3m þ 4dÞ, 40 5
(15b)
B44 ¼
Bcubic 44
rffiffiffiffiffi 5 ¼ 21 Dq. 14
(15c)
By comparing Eq. (11) with Eq. (15), it is found that the meaning of Dq in Eq. (11) is different from that of Dq in Eq. (15). As above, Dq in Eq. (11) denotes the CF component of the perfect cubic site, i.e. it is a net cubic CF parameter. Thus, we think that Dq(Y&R) in Eq. (15) includes the tetragonal CF component besides the cubic CF one, i.e. Dq in Eqs. (15b) and (15c) correspond to a choice of ‘‘cubic’’ CF parameter incorporating effect of the axial distortion. In order to check Eq. (15), we compare our CDM results with Fairbank’s ones for MgO: Cr3+ at the tetragonal symmetry site. Fairbank et al. [22] obtained the energy levels in terms of the CDM based on the diagonalization of the CF matrix which are functions of the Racah parameters B and C, Tress parameter a, CF parameters Dq, d, and m. According to Eq. (15) and the CF parameters Dq, d, and m obtained by Fairbank et al. [22], we obtain B20 ¼ 4550 cm1, B40 ¼ 37065.0 cm1, B44 ¼ 20645 cm1 for MgO: Cr3+. By using
ARTICLE IN PRESS Z.-Y. Yang, Q. Wei / Physica B 370 (2005) 137–145
these parameters as well as a ¼ 70 cm1, B ¼ 570 cm1, C ¼ 3165 cm1, we obtain the energy levels in terms of the CDM/MSH program. Our calculated results are in good agreement with ones obtained by Fairbank et al. [22] for MgO: Cr3+(3d3). It is worthwhile to point out in Fairbank’s work [22] that Dq for tetragonal symmetry site is chosen as 1640 cm1 including the non-cubic component besides the cubic CF one, whereas Dq at a perfect cubic site is chosen as 1590 cm1 including only the cubic CF component. 2.3. Ballhausen’s CF parameter notation in axial symmetry In order to investigate the 3dN ions in axial symmetry crystal field, Ballhausen designates a set of CF parameters: Dq, Ds, and Dt for trigonal symmetry and Dq, Ds, and Dt for tetragonal symmetry. Ballhausen’s CF parameter notation has been widely used in the investigations of 3dN ions in crystals [23–25]. By comparing the matrix elements, one finds that the relationships between the ‘‘Ballhausen’’ CF parameter notation and the ‘‘Wybourne’’ parameters can be written as [12] B20 ¼ 7Ds,
(16a)
B40 ¼ 14Dq 21Dt,
(16b)
pffiffiffiffiffi B43 ¼ 2 70Dq,
(16c)
where the definitions for the CF parameters Ds, and Dt are given in the appendix. It can be seen from Eqs. (7) and (16) that the parameter Dq(P&R) in Eq. (7) is not equal to the parameter Dq(Ballhausen) in Eq. (16). In fact, the physical meanings of the Dq(P&R) is different from those of Dq(Ballhausen). The parameter Dq(P&R) denotes the CF parameter of the perfect cubic site, whereas the parameter Dq(Ballhausen) contains the trigonal CF component besides the cubic component. We obtain the following relationships: 1 3 0 p ffiffi ffi DqðBallhausenÞ ¼ DqðP&RÞ þ uþ u 30 2 7 ð17Þ ¼ DqðP&RÞ Dt. 18
141
For tetragonal CF, Ballhausen [9] and Febbraro [26,27] obtained the following relations: B20 ¼ 7Ds,
(18a)
B40 ¼ 21Dq 21Dt,
(18b)
rffiffiffiffiffi 5 ¼ 21 Dq, 14
(18c)
B44
where the definitions for the CF parameters Ds, and Dt are given in the appendix. By comparing Eqs. (18) with (11) and Eq. (15), we find that the meaning of Dq(Y&R) in Eq. (15) is equivalent to that of Dq(Ballhausen) in Eq. (18), whereas the meaning of Dq(Griffith) in Eq. (11) is different from that of Dq(Ballhausen) in Eq. (18) or Dq(Y&R) in Eq. (15). The parameter Dq(Griffith) denotes the CF parameter of the perfect cubic site, whereas the CF parameter Dq(Ballhausen) in Eq. (18) or Dq(Y&R) in Eq. (15) contains the tetragonal CF component besides the cubic component. We obtain the following relationships: DqðY&RÞ ¼ DqðBallhausenÞ 1 ð3m þ 4dÞ 60 7 ¼ DqðGriffithÞ þ Dt. 12 ¼ DqðGriffithÞ þ
ð19Þ
2.4. Sharma [28] and Yu’s [29] (S&Y) notation in axial symmetry For the perfect cubic symmetry CF, The CF Hamiltonian can be written as ð4Þ ð4Þ z½1 1 1 , H CF ¼ B40C C ð4Þ þ B C C 43C 0 3 3 (20a) H CF
ð4Þ ð4Þ ¼ B40C C ð4Þ þ B C þ C 44C 0 4 4
z½0 0 1 .
(20b) It is well known that there exists a constant ratio between B40C and B43C(B44C). These rations can be expressed as pffiffiffiffiffiffiffiffiffiffi B40C ¼ 7=10B43C , (21a) B40C ¼
pffiffiffiffiffiffiffiffiffiffi 14=5B44C .
(21b)
Sharma and Yu’s
DqðS&YÞ ¼ DqðP&RÞ
1 B40 21
rffiffiffiffiffi ! 7 B43 10
1 3 7 u þ pffiffiffi u0 ¼ DqðP&RÞ Dt 30 18 2
Ds ¼
1 4 u pffiffiffi u0 7 2 3 3 u þ pffiffiffi u0 Dt ¼ 35 2
1 Dq ¼ pffiffiffiffiffi B43 2 70
Dt ¼
1 Ds ¼ B20 7
13 3 3 u þ pffiffiffi u0 ¼ DqðBallhausenÞ þ Dt 420 4 2
DqðBallhausenÞ ¼ DqðP&RÞ þ
u ¼ 3Ds
20 Dt 3 ffiffiffi p pffiffiffi 5 2 0 u ¼ 2Ds Dt 3
pffiffiffiffiffi 1 70 B40 B43 54 189
pffiffiffi pffiffiffi pffiffiffiffiffi 2 5 2 35 B43 B20 þ B40 63 7 63
Dq ¼
u0 ¼
pffiffiffiffiffi 3 20 2 70 u ¼ B20 þ B40 B43 7 63 63
B43
B40 ¼ 21Dq 21Dt pffiffiffiffiffi B43 ¼ 2 70Dq
B20 ¼ 7Ds
Dq ¼
B040
1 B40 þ 28
rffiffiffiffiffi ! 7 B43 10 pffiffiffiffiffi 1 70 B40 þ B44 36 252
DqðS&YÞ ¼ DqðGriffithÞ þ
1 B40 21
rffiffiffiffiffi ! 14 B44 5
1 7 ð3m þ 4dÞ ¼ DqðGriffithÞ þ Dt 60 12
1 ðm dÞ 7 1 ð3m þ 4dÞ Dt ¼ 35 Ds ¼
2 Dq ¼ pffiffiffiffiffi B44 3 70
Dt ¼
1 1 ð3m þ 4dÞ ¼ DqðBallhausenÞ Dt 420 2
DqðBallhausenÞ ¼ DqðGriffithÞ þ
m ¼ 4Ds þ 5Dt d ¼ 3Ds þ 5Dt
Dq ¼
3 5 10 d ¼ B20 B40 þ pffiffiffiffiffi B44 7 21 3 70
1 Ds ¼ B20 7
B40 ¼ 21Dq 21Dt rffiffiffiffiffi 5 Dq B44 ¼ 21 14
1 B40 ¼ 21Dq ð3m þ 4dÞ rffiffiffiffiffi 4 pffiffiffiffiffi 5 70 Dq þ ð3m þ 4dÞ B44 ¼ 21 14 40 4 5 10 m ¼ B20 B40 þ pffiffiffiffiffi B44 7 21 3 70
B20 ¼ 7Ds
B20 ¼ d m
Ballhausen’s
Dq ¼
rffiffiffiffiffi ! 14 B44 5
rffiffiffiffiffi 14 B44 5
1 B40 þ 42
B040 ¼ B40
Sharma and Yu’s
142
B40
B20
rffiffiffiffiffi 7 B43 ¼ B40 10
Griffith’s
Ballhausen’s
Pryce and Runciman’s
pffiffiffi ¼ u 2 2u0 pffiffiffi 4 ¼ u þ 2 2u0 14Dq 3 pffiffiffi 7 2 u þ 2u0 þ 20Dq ¼ pffiffiffiffiffi 70 3
Tetragonal symmetry type I( C4v, D2d, D4, D4h )
Trigonal symmetry type I(C3v, D3d, D3)
Table 1 Relations among the various CF parameter notations
ARTICLE IN PRESS
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ARTICLE IN PRESS Z.-Y. Yang, Q. Wei / Physica B 370 (2005) 137–145
can been seen from p Eq. (21) that the B40C pIt ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi 7=10B43C and B40C 14=5B44C atpffiffiffiffiffiffiffiffiffiffi a perfect cubic site vanish. However, the 7=10B43C 40C pB ffiffiffiffiffiffiffiffiffiffi at trigonal site and B40C 14=5B44C at tetragonal site is no longer zero. Thus Sharma [28] and Yu [29,30] denote the difference by B040 (S&Y) pffiffiffiffiffiffiffiffiffiffi B040 ðS&YÞ ¼ B40 7=10B43 , for trigonal site, ð22aÞ pffiffiffiffiffiffiffiffiffiffi B040 ðS&YÞ ¼ B40 14=5B44 , for tetragonal site.
(23)
with DqC ¼
1 B40C 14
for trigonal site,
(24a)
pffiffiffiffiffiffiffiffiffiffi DqðS&YÞ ¼ B40 þ 7=10B43 =28 for trigonal site,
DqC ¼
1 B40C 21
ð24bÞ
for tetragonal site,
(24c)
pffiffiffiffiffiffiffiffiffiffi DqðS&YÞ ¼ B40 þ 14=5B44 =42 for tetragonal site.
13 3 u þ pffiffiffi u0 420 2 3 ¼ DqðBallhausenÞ þ Dt 4 for trigonal site,
ð26aÞ
1 ð3m þ 4dÞ 420 1 ¼ DqðBallhausenÞ Dt 2 for tetragonal site.
ð26bÞ
DqðS&YÞ ¼ DqðP&RÞ
DqðS&YÞ ¼ DqðGriffithÞ þ
ð22bÞ
In fact, Yu [30] defines the three parameters B20 (S&Y) (or B020 (S&Y)), B040 (S&Y), Dq0 (S&Y) to measure the net trigonal or tetragonal CF component since they vanish identically in perfect cubic site. Dq0 is defined by [30] Dq0 ¼ DqðS&YÞ DqC .
143
ð24dÞ
where DqC denotes the CF parameter of the perfect cubic site. We find that B20 (S&Y) equal to B20 (or B020 ) in Eq. (8d), and DqC for trigonal site in Eq. (24a) and DqC for tetragonal site in Eq. (24c) equal to Dq (P&R) in Eq. (7) and Dq(Griffith) in Eq. (11) respectively, whereas B040 (S&Y) in Eq. (22a) is different from B040 in Eq. (8e). We obtain: 9 3 0 0 u þ pffiffiffi u ¼ 21Dt, B40 ðS&YÞ ¼ (25) 5 2
3. Summary The relationships among the various crystal field notations for 3dN ions in axial symmetry crystal field have been studied. The confusions among these CF notations have been clarified. The following conclusions are obtained: (i) For trigonal symmetry CF, Pryce and Runciman (P&R) CF parameter notation Dq(P&R) in Eq. (7) is not equal to the parameter Dq(Ballhausen) in Eq. (16). The physical meanings of the Dq(P&R) is different from those of Dq(Ballhausen). The parameter Dq(P&R) denotes the CF parameter of the perfect cubic site, whereas the parameter Dq(Ballhausen) contains the trigonal CF component besides the cubic component. (ii) For tetragonal symmetry CF, the meaning of Dq(Y&R) in Eq. (15) is equivalent to that of Dq(Ballhausen) in Eq. (18), whereas the meaning of Dq(Griffith) in Eq. (11) is different from that of Dq(Ballhausen) in Eq (18) or Dq(Y&R) in Eq. (15). The parameter Dq(Griffith) denotes the CF parameter of the perfect cubic site, whereas the parameter Dq(Ballhausen) in Eq. (18) or Dq(Y&R) in Eq. (15) contains the tetragonal CF component besides the cubic component. (iii) The B20 (S&Y) is equal to B20 (or B020 ) in Eq. (8d), and DqC (S&Y) for trigonal site in Eq. (24a) and DqC (S&Y) for tetragonal site in Eq. (24c) equal to Dq (P&R) in Eq. (7) and Dq(Griffith) in Eq. (11) respectively, whereas B040 (S&Y) in Eq. (22a) is different from B040 in Eq. (8e). The correlations of the various crystal field notations are summarized in Table 1, which is very useful in the investigations of 3dN ions in crystal material.
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Acknowledgments One of the authors (Z.Y. Yang) would like to thank Prof. C. Rudowicz and Dr. Y.Y. Yeung for their helpful discussions at City University of Hong Kong. This work was supported by the Education Committee Natural Science Foundation of Shaanxi Province (Grant No. 05JK139) and a Baoji University of Arts and Sciences Key Research Grant.
(2) Pryce and Runciman’s CF parameter notation for trigonal symmetry [9,11,13]
1 (A.3a) tþ V trig tþ ¼ t V trig t ¼ u, 3
2 t0 V trig t0 ¼ u, 3
(A.3b)
tþ V trig eþ ¼ t V trig e ¼ u0 .
(A.3c)
(3) Griffith’s CF parameter notation for tetragonal [10]
Appendix We present here the definitions of the various CF parameters notations. They will be convenient for reader to consult.
2 2 EðzÞ ¼ Eðd xy Þ ¼ D d, 5 3
(A.4a)
(1) Ballhausen’s CF parameter notation (a) Trigonal symmetry [9] D
E
V trig t ¼ Ds þ 2 Dt, t 2g 2g 3
2 1 Eðx; ZÞ ¼ Eðd yz ; d zx Þ ¼ D þ d, 5 3
(A.4b)
3 1 EðyÞ ¼ E ðd z2 Þ ¼ D m, 5 2
(A.4c)
(A.1a)
D
E t02g V trig t02g ¼ 2Ds 6Dt,
(A.1b)
pffiffiffi D
E pffiffiffi 5 2
Dt, t2g V trig eg ¼ 2Ds 3
(A.1c)
3 1 EðeÞ ¼ E d x2 y2 ¼ D þ m, 5 2 where D ¼ 10 Dq.
(A.4d)
References D
E
V trig e ¼ 7 Ds. e g g 3 (b) Tetragonal symmetry [9]
d x2 y2 jV tetra jd x2 y2 ¼ 2Ds Dt,
(A.1d)
(A.2a)
d z2 jV tetra jd z2 ¼ 2Ds 6Dt,
(A.2b)
d xy jV tetra jd xy ¼ 2Ds Dt,
(A.2c)
hd xz jV tetra jd xz i ¼ Ds þ 4Dt.
(A.2d)
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