Full text of DOI:10.1002/(SICI)1521-3951(200008)220:2<941::AID-PSSB941>3.0.CO;2-J

Wen-Chen Zheng and Shao-Yi Wu: Theoretical Studies of EPR Parameters and Microstructure 941
phys. stat. sol. ꢀb) 220, 941 ꢀ2000)
Subject classification: 71.70.Ch; 76.30.Fc; S9
Theoretical Studies of EPR Parameters
and Microstructure of the Rhombic Co ±V Centre
2
+
Ag
in AgCl
Wen-Chen Zheng and Shao-Yi Wu
Department of Material Science, Sichuan University, Chengdu 610064, and International
Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110015, P. R. China
ꢀ
Received February 22, 2000; in revised from April 7, 2000)
7
The perturbation formulae of EPR g-factors gi ꢀi = x, y, z) and hyperfine constants Ai for a 3d
ꢀ
2+
Co ) ion at a rhombic octahedral site in a crystal are established from a cluster approach. In
these formulae, the contributions from the covalency effect, the configuration interaction and the
rhombic crystal field are considered and the parameters used in the calculation of EPR param-
eters, except the core polarization constant k in the calculation of Ai, can be estimated from the
optical spectra and the structure data of the studied system. From these formulae, the EPR
2
+
parameters gi and Ai for the rhombic Co ±VAg centre in AgCl are reasonably explained by
considering suitable rhombic distortion and so the microstructure of this rhombic centre is ob-
tained.
1.Introduction
Silver halides have ionic conductivity. Because of the unavoidable presence of polyva-
+
lent cation impurities which introduce some Ag vacancies ꢀVAg), ion transport at room
temperature is dominated by the concentration and mobility of free VAg, and is sensi-
tively dependent on the association of VAg with polyvalent cation impurities, forming
nearest or next-nearest neighbour bound complexes ꢀor defect centres) in these materi-
als [1]. In order to understand ionic conductivity and ionic transport phenomena in
pure and doped crystals, one needs detailed information about the microstructures of
these defect centres. The EPR spectrum is a useful tool to measure these defect centres.
2
+
For example, a rhombic Co ±VAg centre in AgCl was found and its EPR g-factors gi
i = x, y, z) and hyperfine structure constants Ai were measured from EPR experiments
ꢀ
2
+
2+
[2]. The rhombic Co ±VAg centre is identified as the substitutional Co ion associated
with one nearest VAg along the [110] axis due to charge compensation [2]. Since the
±
effective charge of VAg is negative, the two Cl ions closest to the VAg along [100] and
[
010] axes are expected to move away from the VAg by DRp and the central Co2+ ion
should be attracted towards the VAg by DRc ꢀsee Fig. 1) due to the electrostatic interac-
±
tions ꢀnote: since the distances from other Cl ions to VAg are larger, their displace-
ments should be much smaller and are neglected here). The microstructure of this
rhombic centre can be characterized by the rhombic distortions DRp and DRc. Consider-
n
ing that the EPR parameters of 3d ions are sensitive to their immediate environments,
one can estimate these microstructural parameters from the EPR data. The reliability
of the estimated results depends strongly upon the reasonableness and feasibility of the
2+
7
theoretical formulae for calculating EPR parameters. For the Co ꢀ3d ) ion in rhombic

942
Wen-Chen Zheng and Shao-Yi Wu
Fig. 1. Rhombic Co2+±VAg centre in AgCl
crystal
octahedral site, unfortunately, the previous
calculation formulae of gi and Ai are in-
complete and simplistic. For example, in
the first-order perturbation formulae of
Tinkham [3] and the second-order ones of
Robbroeck et al. [2], the configuration
interaction due to the admixture among
the ground and excited orbitals of Co2+ is
neglected; whereas in the implicit for-
mulae based on an extension of Abragam
and Pryce's theory [4] from tetragonal to
rhombic octahedra given by Osaki and
Uryu [5], the covalency effect is omitted. In addition, the treatments of the contributions
from covalency effect ꢀor configuration interaction) and rhombic crystal field are oversim-
plified. Even so, several parameters related to covalency effect ꢀor configuration interac-
tion) and rhombic crystal field used in these formulae [2 to 5] are adjustable. So, these
formulae are unable to calculate quantitatively the EPR parameters gi and Ai of the 3d7
ion at rhombic site, but they are often used to estimate some above adjustable parameters
from the observed EPR parameters. In this paper, by use of the perturbation method
7
similar to that in Ref. [4], we establish the calculation formulae of gi and Ai for 3d ion in
rhombic octahedral site from a cluster ꢀor molecular orbital) approach. In these formulae,
the contributions from configuration interaction, covalency effect and rhombic crystal
field are considered completely. The parameters in these formulae, except the core polar-
ization constant k in the formulae of Ai, can be obtained from the optical spectra and the
structural data of the studied system. Based on these formulae, the EPR parameters gi
2
+
and Ai for the rhombic Co ±VAg centre in AgCl are reasonably explained by considering
suitable rhombic distortion, and so the microstructure of this rhombic centre in AgCl is
also obtained.
2.Calculation Formulae
2
+
For a Co ion at octahedral site, the EPR spectra are typical of an effective spin
0
1
4
S ˆ , which arises from the splitting of the F ground term into six Kramers doublets
2
by the spin±orbit interaction and low symmetry crystal fields [2 to 5]. The EPR re-
sponse comes from the lowest lying doublet and yields the anisotropic factors gi and
hyperfine constants Ai . The spin Hamiltonian of this system can be given as
H ˆ H0 ‡ H0 ;
H0 ˆ HCoulꢀB; C† ‡ V ꢀDq; Ds; Dt; D ; Dh† ;
a
crys
x
0
0
0
00
0
b
crys
0
H ˆ HZꢀk; k ; a; a ; a † ‡ HSOꢀz; z † ‡ V ꢀDq; Ds; Dt; D ; Dh† ‡ HhfꢀP; P † ;
x
ꢀ1)

Theoretical Studies of EPR Parameters and Microstructure of the Rhombic Co2+±VAg Centre 943
where HCoul is the Coulomb repulsion term with Racah parameters B and C. Va and
crys
b
crys
V
are, respectively, the diagonal and off-diagonal parts of the crystal field with cubic
crystal field parameter Dq and rhombic ones Ds, Dt, Dx and Dh. HZ is the Zeeman
0
00
term with the effective Lande factors a, a and a along the axes z, x, y and the orbital
0
reduction factors k, k . HSO is the spin±orbit interaction term with the spin±orbit cou-
pling coefficients z, z and Hhf is the hyperfine interaction term with dipolar hyperfine
parameters P, P. From the cluster approach for 3d ion in octahedral cluster, the
0
0
n
LCAO molecular orbitals should be taken as the one-electron basis functions, i.e., [6, 7]
1=2
jgi ˆ Ng ꢀjdgi À lg jp i† ;
ꢀ2†
g
where g = t2g and eg denotes the irreducible representations of the Oh group. jdi is the
n
d-orbital of the 3d ion and jpi the p-orbital of the ligand. Ng and lg are, respectively,
the normalization factors and mixing coefficients. They can be obtained from a semiem-
pirical LCAO method [6, 7]. Thus, we have
0
d
2
t
0
p
z0 ˆ ꢀNtNe†1=2 ꢀz À ltlez =2† ;
0
0
z ˆ Ntꢀz ‡ l z =2† ;
d
p
2
t
k0 ˆ ꢀNtNe†1=2 ꢀ1 À ltle=2† ;
k ˆ Ntꢀ1 ‡ l =2† ;
0
1=2
P ˆ NtP0 ;
P ˆ ꢀNtNe† P0 ;
ꢀ3†
0
d
0
n
where z and z are the spin±orbit coupling coefficients of free 3d ion and free ligand
ion, respectively. P0 is the dipolar hyperfine parameter of the free 3d ion.
p
n
By use of the perturbation method similar to that in Ref. [4], the second-order per-
7
2+
turbation formulae of gi and Ai for 3d ꢀCo ) ions in rhombic octahedra can be de-
rived from Eqs. ꢀ1) and ꢀ2) as follows:
ꢆ ꢄ
ꢂ
ꢃ
ꢅ
ꢀ
ꢁ
0
ꢀ ꢁ
2
2
a
2ka
a
12
a
gX ˆ
4
‡
‡
‡
n4X
0
00
0
a
ꢀx ‡ 2†
a
xꢀx ‡ 2†
a
),
8
n5
12n6
xꢀx ‡ 2†
a
4n7X
‡
‡
À
Z ;
2
1=2
0
00
ꢀx ‡ 2†
ꢀ
x ‡ 2†
ꢀa a †
ꢆ ꢄ
ꢂ
ꢃ
ꢅ
ꢀ
ꢁ
00
0
ꢀ
ꢁ
2
2
a
2ka
ꢀx ‡ 2†
a
12
a
gY ˆ
4
‡
‡
‡
n4Y
00
00
a
a
xꢀx ‡ 2†
a
),
8
n5
12n6
a
4n7Y
‡
‡
À
Z ;
2
1=2
xꢀx ‡ 2†
0
00
ꢀx ‡ 2†
ꢀx ‡ 2†
ꢀa a †
ꢇ
"
#
"
#
3
4
9
x2
4
gZ ˆ 2 ‡ 4ꢀka ‡ 2†
À
x2
‡
À
ꢀn1X ‡ n1Y†
2
2
ꢀ
x ‡ 2†
ꢀx ‡ 2†
)
,
ꢂ
ꢃ
aꢀn3X ‡ n3Y†
3
4
À
À
Z ;
ꢀ4†
1=2
0
00
x
x ‡ 2
ꢀ
a a †

944
Wen-Chen Zheng and Shao-Yi Wu
ꢆ
ꢄꢂ
ꢃ
ꢅ
ꢂ
ꢃ
a2
12
8ka
a2
AX ˆ P ꢀÀk=2†
‡
‡
À
WZ
0
00
0
00
a a
xꢀx ‡ 2†
x ‡ 2
a a
)
,
3
2WXY
12WX
4aWXZ
À
À
‡
Z
2
1=2
xꢀx ‡ 2†
0 00
ꢀ
x ‡ 2†
ꢀx ‡ 2† ꢀa a †
12n6
ꢇ
),
ꢀ
ꢁ
2
a
8n5
4an7X
0
‡
P
n4X ‡
‡
À
Z ;
0
2
1=2
a
xꢀx ‡ 2†
ꢅ
0 00
ꢀ
x ‡ 2†
ꢀx ‡ 2† ꢀa a †
ꢆ
ꢄꢂ
ꢃ
ꢂ
ꢃ
a2
0
12
8ka
a2
0
,
AY ˆ P ꢀÀk=2†
‡
‡
À
WZ
00
00
a a
xꢀx ‡ 2†
x ‡ 2
a a
)
3
2WXY
12WY
4aWYZ
À
À
‡
Z
2
1=2
xꢀx ‡ 2†
0 00
ꢀ
x ‡ 2†
ꢀx ‡ 2† ꢀa a †
12n6
ꢇ
),
!
ꢀ
ꢁ
2
a
8n5
4an7Y
0
‡
P
n4Y ‡
‡
À
Z ;
0
2
1=2
a
xꢀx ‡ 2†
0 00
ꢀ
x ‡ 2†
ꢀx ‡ 2† ꢀa a †
ꢇ
"
!#
3
x2
4
3
x2
),
4
AZ ˆ P ꢀÀk=2† 2 ‡ 8
À
À
‡ 4ka
À
2
2
ꢀ
x ‡ 2†
ꢀx ‡ 2†
!
ꢃ
2
9
x2
4
a WZ
‡
ꢀWX ‡ WY †
‡
Z
2
0 00
a a
ꢀ
x ‡ 2†
ꢂ
3
4
2
aꢀWXZ ‡ WYZ†
À
x
x ‡ 2
À P
1
=2
0
00
Zꢀa a †
8
ꢂ
3
ꢃ9
4
>
!
>,
>
<
a ꢀn3X ‡ n3Y†
À
x
>
=
9
x2
4
x ‡ 2
0
‡
P
ꢀn1X ‡ n1Y †
À
À
Z ;
2
1=2
>
0
00
>
>
:
ꢀx ‡ 2†
ꢀa a †
>
;
ꢀ
5†
0
00
2
2
where Z = ꢀa=a a † ‡ ꢀ6=x † ‡ ꢀ8=ꢀx ‡ 2† †. k is the core polarization constant. x can
4
4
4
4
be determined from the energy splittings Dꢀ= E{ B1[ T1ꢀF)]} ± E{ B3[ T1ꢀF)]}) and
4
4
4
4
4
dꢀ= E{ B2[ T1ꢀF)]} ± E{ B3[ T1ꢀF)]}) of the T1 ground state in rhombic crystal field by
the expression
ꢄ
ꢅ
0
00
za a
3
4z
za
D ˆ
‡
À 6 ꢀx ‡ 3† :
ꢀ6†
3
ax
azꢀx ‡ 2† ‡ 6d
nij are defined as
ꢄ
ꢅ
ꢄ
ꢅ
k z 15f12
0
0
2q
2
k z 15f
0
0
2
2q
2
X
1X
1Y
1Y
n1X ˆ
n3X ˆ
‡
;
n1Y ˆ
‡
ꢄ
;
3
2E1X
E2X
3
2E1Y
E2Y
ꢄ
ꢅ
ꢅ
0
0
0
0
k z 15f1X f2X
2q1X q2X
E2X
k z 15f1Yf2Y
2q1Yq2Y
E2Y
À
;
n3Y ˆ
À
;
3
2E1X
3
2EY

Theoretical Studies of EPR Parameters and Microstructure of the Rhombic Co2+±VAg Centre 945
ꢄ
ꢅ
ꢄ
ꢅ
k z 15f22
0
0
4q
2
k z 15f
0
0
2
4q
2
X
2X
2Y
2Y
n4X ˆ
‡
;
n4Y ˆ
‡
;
3
E1Y
E2Y
3
E1X
E2X
"
#
0
0
2
3
k z 15f32
0
0
2q23
2
4
k z q
2ꢀr ‡ r †
X
Y
n5 ˆ
;
n6 ˆ
‡
‡
;
3
E2Z
3
2E1Z
E2Z
E3
n7X ˆ n3X =2 ;
n7Y ˆ n3Y =2 ;
ꢀ7†
where E1X, E1Y, E1Z, E2X , E2Y , E2Z and E3 are, respectively, the energy differences
4
4
4
4
4
4
between the ground state B3[ T1ꢀF)] and the excited states B3[ T2ꢀF)], B2[ T2ꢀF)],
4
4
4
4
4
4
4
4
4
4
B1[ T2ꢀF)], B3[ T1ꢀP)], B2[ T1ꢀP)], B1[ T1ꢀP)], and A[ A2ꢀF)]. They and the split-
tings D, d can be calculated from the d±d transition energy matrices of 3d ion in rhom-
7
bic symmetry. In passing, if the rhombic field parameters is reduced to tetragonal field
0
parameters and the covalency effect is ignored ꢀi.e., lg = 0 and Ng = 1, k = k = 1,
0
0
d
z = z = z ), then gX = gY and AX = AY. Eqs. ꢀ4) and ꢀ5) become those in tetragonal sym-
metry in [4].
i
The parameters fi, qi, a and Wij in the above formulae can be determined by consid-
ering the configuration interactions among the ground and excited orbitals in rhombic
field. From the perturbation method, we obtain
p
3
15
1
2
a ˆ 2 eX eY À
ꢀeX r ‡ eYr † ‡
r r À tX tY ;
Y
X
X Y
2
p
p
a0 ˆ 3 eZeY ‡
15
eZr À tZtY ;
a00 ˆ 3 eZeX ‡
15
eZr À tZtX ;
Y
X
2
2
2
2
ꢄ
ꢂ
ꢃ
ꢃ
0
0
1
3rX
2eX
rY
2
tY t
r r
f1X ˆ ÀeY e00 1 ‡ p
À p
X
Y
X
‡
À
;
X
1
1
5
2eY
15 eY eX
eYeX
ꢄ
ꢂ
0
3rY
2eY
rX
2
tX t
rX rY
f1Y ˆ ÀeX e00 1 ‡ p
À p
Y
‡
À
;
Y
0
1
5
2eX
15 eX eY
eX eY
"
#
#
ꢂ
ꢂ
ꢃ
p ꢂ
ꢃ
ꢃ
0
0
0
1
3tX
2tY
eY
15
t
tX rY
1 t rY
0
X
X
q1X ˆ Àe eY
‡
‡
‡
À
À
À
À
;
X
0
2
eX
2
e
eX eY
2 e eY
X
X
"
ꢃ
p ꢂ
0
0
1
2
3tY
eY
2tX
eX
15
2
t
tY rX
1 t rX
0
Y
Y
q1Y ˆ Àe eX
‡
;
Y
0
0
e
eY eX
2 e eX
Y
Y
ꢄ
ꢅ
ꢅ
0
3
rY
2
tZ t
f2X ˆ ÀeZe00 1 À p
‡ p
Y
0
;
Y
1
3
5 eY
15 eZ eY
ꢄ
0
rX
2
tZ t
f2Y ˆ ÀeZe00 1 À p
‡ p
X
;
X
0
1
5 eX
15 eZ eX
"
#
p
0
0
3
tY
tZ
15 t
0
Y
q2X ˆ Àe eZ
‡ eZ
À
À
;
Y
2
eY
2
e
Y
"
#
p
0
0
3
tX
eX
tZ
15 t
0
X
q2Y ˆ Àe eZ
‡ eZ
;
X
2
2
e
X

946
Wen-Chen Zheng and Shao-Yi Wu
1
2
1
f3 ˆ
ꢀeX ‡ eY† ‡ p ꢀr ‡ r † ;
X
Y
2
15
ꢇ"
"
#)
p
1
2
tZ
eZ
3
2
15
q3 ˆ e0
ꢀtX ‡ tY † ‡
ꢀeX ‡ eY † ‡
ꢀr ‡ r †
;
Z
X
Y
2
p
2
2
2
WX ˆ 1 ‰2e ‡ 12eX tX À 7t À 2 15 r ꢀeX À 2tX †Š ;
05
X
X
X
p
2
05
2
Y
2
Y
WY ˆ 1 ‰2e ‡ 12eYtY À 7t À 2 15 r ꢀeY À 2tY†Š ;
Y
4
05
2
Z
2
Z
WZ ˆ 1 ‰2e ‡ 12eZtZ À 7t Š ;
p
1
1
WXZ ˆ 7 ‰eZeX ‡ 8ꢀeX tZ ‡ eZtX † ˆ ‡ 3 15 r ꢀeZ ‡ 8tZ†Š ;
0
X
p
1
1
WYZ ˆ 7 ‰eZeY ‡ 8ꢀeYtZ ‡ eZtY † ˆ ‡ 3 15 r ꢀeZ ‡ 8tZ†Š ;
0
Y
ꢇ
p
1
15
WXY ˆ
eX eY ‡ 8ꢀeX tY ‡ eYtX † ‡ 14tX tY À
7
0
3
)
 ‰r ꢀeY ‡ 8tY† ‡ r ꢀeX ‡ 8tX † À 5r r Š
;
ꢀ8†
X
Y
X Y
j
j
where e , t and r are the mixing coefficients of configuration interaction. Considering
i
i
i
the linear combinations of orbitals in the above configuration interactions, we have the
normalization relationships
2
2
2
0
2
0
0
i
2
e ‡ t ‡ r ˆ 1 ;
e ‰1 ‡ ꢀti=ei† ‡ ꢀt =e † Š ˆ 1 ;
ꢀi ˆ X; Y; Z†
ꢀ9†
i
i
i
i
i
0
0
2
2
e ‰1 ‡ ꢀr =ei† ‡ ꢀti=ei† ˆ 1 ;
ꢀi ˆ X; Y† :
ꢀ10†
i
i
From the perturbation procedure and the d±d transition energy matrices, we obtain
5
6
12
5
tX
eX
À4Dq À Dt ‡ Ds À
Dx À Dh
39
2
5
ˆ
ˆ
;
;
9
15
15B À 6Dq ‡ D À D ‡ D ‡ 6Dh
s
t
x
5
4
10
5
2
6
5
12
5
tY
eY
À4Dq À Dt ‡ Ds ‡
Dx ‡ Dh
9
5
15
4
39
10
15B À 6Dq ‡ D À D À D À 6Dh
s
t
x
p
1
5
À
ꢀ5Dt ‡ 4Ds À 2D À 16Dh†
x
rX
20
ˆ
ˆ
ˆ
;
eX
2
15
9
À8Dq ‡
p
Ds À
Dt ‡
D ‡ 8Dh
x
5
2
5
1
5
À
ꢀ5Dt ‡ 4Ds ‡ 2D ‡ 16Dh†
x
rY
20
;
eY
2
15
9
À8Dq ‡
Ds À
Dt ‡
Dx À 8Dh
5
2
5
p
1
5
0
t
ꢀ5Dt ‡ 4Ds ‡ 8Dx À 6Dh†
X
10
;
0
e
7
15
57
X
À15B À 2Dq À
Ds À
Dt À
D ‡ 2Dh
x
5
4
10

Theoretical Studies of EPR Parameters and Microstructure of the Rhombic Co2+±VAg Centre 947
p
1
5
0
ꢀ5Dt ‡ 4Ds À 8Dx ‡ 6Dh†
t
e
Y
20
ˆ
ˆ
;
0
7
15
57
Y
À15B À 2Dq À Ds À
Dt ‡
Dx À 2Dh
p
5
4
10
1
2
4
15
À4Dq À
Ds
0
À
ꢀD ‡ Dh†
x
tZ
eZ
t
5
Z
5
;
ˆ
;
ꢀ11†
0
18
e
14
Z
1
5B À 6Dq ‡ 5 Ds
À15B À 12Dq À 5Dt ‡
Ds
5
So, from the optical spectrum parameters B, C, Dq, Ds, Dt, Dx and Dh of the studied
system ꢀnote: the rhombic field parameters Ds, Dt, Dx and Dh are often calculated from
j
j
the structural data because it is difficult to measure them), the parameters e , t and r
i
i
i
can be calculated from Eqs. ꢀ9) to ꢀ11) and so the parameters in the formulae of gi and
Ai, except the core polarization constant k in the formulae of Ai, can be obtained.
3
.Application to the Rhombic Co 2+±VAg Centre in AgCl
2+
The above formulae are now used to the studies of the rhombic Co ±VAg centre in
AgCl. The LCAO coefficients Ng and lg are estimated from a semiempirical LCAO
method, which yields the approximate relation [6, 7]
2
g
2
2
fg ˆ N ‰1 ‡ l S ꢀg† À 2lgSdpꢀg†Š ;
ꢀ12†
g dp
and the normalization correlation
2
g
Ngꢀ1 À 2lgSdpꢀg† ‡ l † ˆ 1 ;
ꢀ13†
n
where fg [ꢁ ꢀB/B0 ‡ C/C0)/2] is the ratio of Racah parameters for a 3d ion in a crystal
to those of the free ion. Sdpꢀg) are the group overlap integrals which can be calculated
2
+
±
2+
from the Slater-type SCF functions [8, 9] and the Co ±Cl distance R0 in AgCl:Co .
2+
ꢀ
ꢀ
Since the ionic radius ri ꢀꢁ 0.72 A [10]) of Co differs from the radius rh ꢀꢁ 1.26 A
+
[
[
10]) of the replaced Ag ion, we estimate the distance R0 from the empirical formula
ꢀ
11, 12]: R0 ꢁ RH ‡ ꢀri À rh)/2, where RH ꢀꢁ 2.775 A [10]) is the metal±ligand distance
ꢀ
in host AgCl crystal. So, we have R0 ꢁ 2.505 A and hence Sdpꢀt2g) ꢁ 0.00855 and
2
+
Sdpꢀeg) ꢁ 0.02847. From the optical spectra of AgCl:Co [13] , we have
Dq ꢁ À680 cmÀ1
;
B ꢁ 810 cmÀ1
;
C ꢁ 3390 cmÀ1
:
ꢀ14†
For the free Co2+ ion [14], B0 ꢁ 1115 cm and C0 ꢁ 4366 cm . Then, we have fg ꢁ
±1
±1
0
.7515 and hence Nt ꢁ 0.8698, Ne ꢁ 0.8770, lt ꢁ 0.3955 and le ꢁ 0.4041 from Eqs. ꢀ12)
2
+
0
d
±1
±4
±1
and ꢀ13). For the free Co ion, z ꢁ 533 cm [14], P0 ꢁ 254 Â 10 cm [15] and for
±
0
p
±1
free Cl ion, z ꢁ 587 cm [16], from Eq. ꢀ3), we obtain
z ꢁ 504 cmÀ1
k ꢁ 0:9378 ;
P ꢁ 221 Â 10 cm
;
z ꢁ 425 cm
0
À1
;
0
k ꢁ 0:8036 ;
À4
À1
0
À4
À1
:
;
P ꢁ 222 Â 10 cm
ꢀ15†

948
Wen-Chen Zheng and Shao-Yi Wu
4
±
According to the superposition model [17] and the geometry relation of the ꢀCoCl6)
cluster in the rhombic Co ±VAg centre ꢀsee Fig. 1), we have
2
+
ꢄꢂ
ꢄ
ꢃ
ꢂ
ꢃ
ꢂ
ꢃ ꢅ
t2
t2
t2
2
R0
R1
R0
R2
R0
R3
ꢁ
2
Ds ˆ À A2ꢀR0†
‡
À ꢀ3cos Q À 1†
;
7
ꢂ
ꢃ
ꢂ
ꢃ
ꢂ
ꢃ ꢅ
t2
t2
t2
2
R0
R1
R0
R2
R0
R3
2
ꢁ
D ˆ À A2ꢀR0† cos 2F1
À cos 2F2
‡ sin Q
;
x
7
ꢄ
ꢂ
ꢃ
ꢂ
ꢃ ꢅ
t4
t4
2
R0
R0
R2
ꢁ
Dt ˆ
A4ꢀR0† ꢀ7 cos 4F1 ‡ 3†
‡ ꢀ7 cos 4F2 ‡ 3†
21
R1
ꢂ
ꢃ
t4
2
R0
4
4
2
ꢁ
‡ 2
A4ꢀR0† ꢀ7 sin Q ‡ 35 cos Q À 30 cos Q ‡ 3†
;
1
R3
ꢄ
ꢂ
ꢃ
ꢂ
ꢃ
t4
t4
1
2
0
R0
R1
R0
R2
ꢁ
Dh ˆ À
A4ꢀR0† cos 2F1
Àcos 2F2
1
ꢂ
ꢃ ꢅ
t4
R0
2
2
‡
sin Q ꢀ7 cos Q À 1†
;
ꢀ16†
R3
with
p
p
2
2 1=2
R1 ˆ ‰ꢀR0 À DRc= 2† ‡ ꢀDRp ‡ DRc= 2† Š
;
p
p
À1
F1 ˆ p=4 ‡ tg ‰ꢀ 2 DRp ‡ DRc†=ꢀ 2 R0 À DRc†Š ;
p
2
2 1=2
R2 ˆ ‰DR =2 ‡ ꢀR0 ‡ DRc= 2† Š
;
c
p
À1
F2 ˆ p À tg ‰R0=ꢀR0 ‡ 2 DRc†Š ;
2
2 1=2
Q ˆ tgÀ1ꢀDRc=R0† ;
R3 ˆ ꢀDR ‡ R †
;
ꢀ17†
c
0
where the power-law exponents take the values t2 ꢁ 3and t4 ꢁ 5 because of the ionic
n
nature of the bonds [17, 18]. For the 3d ion in octahedral clusters, the intrinsic param-
ꢁ
ꢁ
ꢁ
eter A4ꢀR0) ꢁ ꢀ3/4) Dq [18 to 20] and the ratio A2ꢀR0)/A4ꢀR0) is in the range of 9 to 12
ꢁ
ꢁ
[
18 to 20]. We take A2ꢀR0) ꢁ 12A4ꢀR0) here. Thus, there are only two unknown param-
eters DRp and DRc in the formulae of gi. We obtain
ꢀ
ꢀ
DRp ꢁ 0:112A ;
DRc ꢁ 0:122A
ꢀ18†
by fitting the calculated gi to the observed values. The comparisons between the calcu-
lated and observed gi are shown in Table 1. It should be pointed out that due to the
use of approximated values for parameters and simplifying assumptions ꢀe.g., the slight
±
displacements of other four Cl ions in CoCl6 octahedron are not considered), there
may be some errors in the magnitudes of DRp and DRc. Even so, since DRp > 0 and
±
2+
DRc > 0, the displacement directions of Cl and Co ions are consistent with the
expectation based on the electrostatic interaction ꢀsee Fig. 1), suggesting that the esti-
mated microstructural parameters are reasonable in physics. Substituting the above
parameters ꢀincluding the rhombic distortion parameters DRp and DRc) into Eq.ꢀ5) and
assuming the core polarization constant k ꢁ 0.26 ꢀnote: the value of k may be com-
2
+
2+
pared with the value 0.325ꢀ10) in Co ion [4] and 0.25 to 0.3in Cu
ion [21] in
Tutton salts), the constants Ai are calculated. They also show good agreement with the

Theoretical Studies of EPR Parameters and Microstructure of the Rhombic Co2+±VAg Centre 949
Ta ble 1
EPR g-factors and hyperfine structure constants Ai ꢀin units of 10±4 cm±1) for the rhom-
bic Co2+±VAg centre in AgCl
0
0
0
0
0
0
A
Z
g
g
g
A
A
X
Y
Z
X
Y
calculation
4.46
3.40
5.08
170
51
180
experiment [2]
4.70 ꢀ5)
3.39 ꢀ3)
5.06 ꢀ5)
177 ꢀ9)
43 ꢀ27) 190 ꢀ9)
ꢁ
In the calculation, the axes are taken as X k [110], Y k [110] and Z k [001], while the
0
0
ꢁ
0
EPR experiment coordination axes are X k [001], Y k [110] and Z k [110]. Therefore,
we need a rotation of the axes of the calculation coordination so that the theoretical
0
values can be compared with the experimental results [2]. Thus, we have g = gZ,
X
0
0 0 0
= gX , g = gY , A = AZ, A = AX , A = AY .
Z X Y
0
g
Y
Z
observed values ꢀsee Table 1). It appears that from the above formulae based on the
2+
cluster approach, the EPR parameters gi and Ai for the rhombic Co ±VAg centre in
AgCl can be reasonably explained by considering the suitable rhombic distortion, and
so the microstructures of this rhombic centre can be obtained. These formulae can also
2
+
be applied to other rhombic Co octahedral clusters in crystals.
References
[
[
[
[
[
[
[
[
[
1] R. J. Lieb and L. M. Slifkin, J. Phys. Chem. Solids 57, 101 ꢀ1996).
2] L. Van Robbroeck, E. Goovaerts, and D. Schoemaker, phys. stat. sol. ꢀb) 132, 179 ꢀ1985).
3] M. Tinkham, Proc. Roy. Soc. ꢀLondon) A 236, 549 ꢀ1956).
4] A. Abragam and M. H. I. Pryce, Prog. Roy. Soc. ꢀLondon) A 206, 173ꢀ1951).
5] K. Osaki and N. Uryu, J. Phys. Soc. Jpn. 40, 1575 ꢀ1976).
6] M. L. Du and C. Rudowicz, Phys. Rev. B 46, 8974 ꢀ1992).
7] W. C. Zheng and S. Y. Wu, Physica B 252, 173ꢀ1998).
8] E. Clementi and D. L. Raimondi, J. Chem. Phys. 38 , 2686 ꢀ1963).
9] E. Clementi, D. L. Raimondi, and W. P. Reinhardt, J. Chem. Phys. 47, 1300 ꢀ1967).
[
10] R. C. Weast, CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton 1989 ꢀpp.
F187, B196).
[
[
[
[
11] W. C. Zheng, Physica B 215, 255 ꢀ1995).
12] Z. M. Li and W. L. Shuen, J. Phys. Chem. Solids 57, 1073ꢀ1996).
13] T. R. Sliker, Phys. Rev. 130, 1749 ꢀ1963).
14] J. S. Griffith, The Theory of Transition-Metal Ions, Cambridge University Press, London
1
964.
[
[
[
[
[
[
[
15] B. R. McGarvey, J. Phys. Chem. 71, 51 ꢀ1967).
16] G. L. McPerson, R. C. Kach, and G. D. Stucky, J. Chem. Phys. 60, 1424 ꢀ1974).
17] D. J. Newman and B. Ng, Rep. Prog. Phys. 52, 699 ꢀ1989).
18] W. L. Yu, X. M. Zhang, L. X. Yang, and B. Q. Zen, Phys. Rev. B 50, 6756 ꢀ1994).
19] J. D. Newman, D. C. Pryce, and W. A. Runciman, Amer. Mineral. 63, 1278 ꢀ1978).
20] A. Edgar, J. Phys. C 9, 4304 ꢀ1976).
21] A. Abragam and M. H. I. Pryce, Prog. Roy. Soc. ꢀLondon) A 206, 164 ꢀ1951).