Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO
The spin Hamiltonian parameters for the two Cu²⁺ centers A1 and A2 in the calcined catalysts CuO-ZnO are theoretically investigated using the high order perturbation formulas of these parameters for a 3d⁹ ion in tetragonally elongated octahedra. In the above formulas, the tetragonal field parameters...
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| Cite this: | Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO / H.M. Zhang, S.Y. Wu, Z.H. Zhang // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23703:1-6. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1200032017-06-11T03:02:55Z Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO Zhang, H.M. Wu, S.Y. Zhang, Z.H. The spin Hamiltonian parameters for the two Cu²⁺ centers A1 and A2 in the calcined catalysts CuO-ZnO are theoretically investigated using the high order perturbation formulas of these parameters for a 3d⁹ ion in tetragonally elongated octahedra. In the above formulas, the tetragonal field parameters Ds and Dt are determined from the superposition model, by considering the relative axial elongation of the oxygen octahedron around the Cu²⁺ due to the Jahn-Teller effect. Based on the calculations, the relative elongation ratios of about 5% and 3% are obtained for the tetragonal Cu²⁺ centers A1 and A2, respectively. The theoretical spin Hamiltonian parameters are in good agreement with the observed values for both systems. The larger axial elongation in center A1 is ascribed to the more significant low symmetrical (tetragonal) distortion of the Jahn-Teller effect. The local structures characterized by the above axial elongations are discussed. Параметри спiнового гамiльтонiану для двох центрiв Cu²⁺, A1 i A2, у кальцинованих каталiзаторах CuO–ZnO дослiджуються теоретично, використовуючи формули теорiї збурень високого порядку для цих параметрiв для iона 3d⁹ у тетрагонально видовжених октаедрах. В цих формулах параметри тетрагонального поля Dsi Dt визначаються з суперпозицiйної моделi, шляхом розгляду вiдносного аксiйного видовження октаедра кисню навколо Cu²⁺ вiдповiдно до ефекту Яна-Теллера. Базуючись на цих розрахунках, отримано коефiцiєнти вiдносного видовження приблизно 5% i 3% для тетрагональних Cu²⁺ центрiв A1 i A2, вiдповiдно. Параметри теоретичного спiнового гамiльтонiану добре узгоджуються з i спостережуваними величинами для обох систем. Бiльше аксiйне видовження в центрi A1 приписується до вагомiшої низько симетричної(тетрагональної) дисторсiї ефекту Яна-Телера. Обговорюються локальнi структури, що характеризуються вище згаданим аксiйним видовженням. 2011 Article Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO / H.M. Zhang, S.Y. Wu, Z.H. Zhang // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23703:1-6. — Бібліогр.: 25 назв. — англ. 1607-324X PACS: 76.30.Fc, 75.10.Dg, 71.70.Ch DOI:10.5488/CMP.14.23703 arXiv:1107.3757 http://dspace.nbuv.gov.ua/handle/123456789/120003 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The spin Hamiltonian parameters for the two Cu²⁺ centers A1 and A2 in the calcined catalysts CuO-ZnO are theoretically investigated using the high order perturbation formulas of these parameters for a 3d⁹ ion in tetragonally elongated octahedra. In the above formulas, the tetragonal field parameters Ds and Dt are determined from the superposition model, by considering the relative axial elongation of the oxygen octahedron around the Cu²⁺ due to the Jahn-Teller effect. Based on the calculations, the relative elongation ratios of about 5% and 3% are obtained for the tetragonal Cu²⁺ centers A1 and A2, respectively. The theoretical spin Hamiltonian parameters are in good agreement with the observed values for both systems. The larger axial elongation in center A1 is ascribed to the more significant low symmetrical (tetragonal) distortion of the Jahn-Teller effect. The local structures characterized by the above axial elongations are discussed. |
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Zhang, H.M. Wu, S.Y. Zhang, Z.H. |
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Zhang, H.M. Wu, S.Y. Zhang, Z.H. Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO Condensed Matter Physics |
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Zhang, H.M. Wu, S.Y. Zhang, Z.H. |
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Zhang, H.M. |
| title |
Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO |
| title_short |
Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO |
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Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO |
| title_fullStr |
Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO |
| title_full_unstemmed |
Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO |
| title_sort |
theoretical studies of the spin hamiltonian parameters for the two tetragonal cu²⁺ centers in the calcined catalysts cuo-zno |
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Інститут фізики конденсованих систем НАН України |
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2011 |
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| citation_txt |
Theoretical studies of the spin Hamiltonian parameters for the two tetragonal Cu²⁺ centers in the calcined catalysts CuO-ZnO / H.M. Zhang, S.Y. Wu, Z.H. Zhang // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23703:1-6. — Бібліогр.: 25 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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2025-07-08T17:03:37Z |
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| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 2, 23703: 1–6
DOI: 10.5488/CMP.14.23703
http://www.icmp.lviv.ua/journal
Theoretical studies of the spin Hamiltonian parameters
for the two tetragonal Cu 2+ centers in the calcined
catalysts CuO–ZnO
H.M. Zhang1,2∗, S.Y. Wu1,3, Z.H. Zhang1
1 Department of Applied Physics, University of Electronic Science and Technology of China,
610054 Chengdu, China
2 Key Laboratory of Nondestructive Test, Ministry of Education, Nanchang Hangkong University,
330063 Nanchang, China
3 International Centre for Materials Physics, Chinese Academy of Science, 110016 Shenyang, China
Received October 31, 2010, in final form April 15, 2011
The spin Hamiltonian parameters for the two Cu2+ centers A1 and A2 in the calcined catalysts CuO–ZnO
are theoretically investigated using the high order perturbation formulas of these parameters for a 3d9 ion
in tetragonally elongated octahedra. In the above formulas, the tetragonal field parameters Ds and Dt are
determined from the superposition model, by considering the relative axial elongation of the oxygen octahe-
dron around the Cu2+ due to the Jahn-Teller effect. Based on the calculations, the relative elongation ratios
of about 5% and 3% are obtained for the tetragonal Cu2+ centers A1 and A2, respectively. The theoretical
spin Hamiltonian parameters are in good agreement with the observed values for both systems. The larger
axial elongation in center A1 is ascribed to the more significant low symmetrical (tetragonal) distortion of the
Jahn-Teller effect. The local structures characterized by the above axial elongations are discussed.
Key words: EPR, Cu2+, CuO–ZnO, defect structures
PACS: 76.30.Fc, 75.10.Dg, 71.70.Ch
1. Introduction
The calcined catalysts CuO–ZnO are the systems widely applied in the fields of CO ox-
idation [1–6]. The catalytic properties can be closely related to the electronic states and local
structures of Cu in ZnO, which may be conveniently investigated with the aid of electron para-
magnetic resonance (EPR) technique. Cu2+ (3d9) is usually treated as a model system with one
equivalent 3d hole, having a ground state and a single excited state under ideal octahedral crystal-
fields [7]. In CuO–ZnO catalysis, two non-equivalent Cu2+ locate on slightly tetragonally (D4h)
elongated octahedra, with four coplanar Cu2+–O2− bond lengths shorter than the axial ones [8–
11]. The properties of the above structure may play an important role in the catalytic behaviour of
these systems. That is why their local structure features are worth being further investigated. For
example, the EPR experiments were carried out for the calcined catalysts CuO–ZnO, and the spin
Hamiltonian parameters (the anisotropic g factors g‖ and g⊥ and the hyperfine structure constants
A‖ and A⊥) were also measured for the two signals A1 and A2 [12]. The signals A1 and A2 are
attributable to isolated Cu2+ ions automically dispersed in the lattice of ZnO and located in tetrag-
onally distorted octahedral cavities [12]. The observed EPR spectra reveal positive g anisotropy ∆g
(= g‖−g⊥) and a much larger A‖ in magnitude than A⊥, characteristic of a 3d9 ion under a tetrag-
onally elongated octahedron. This point is quite similar to the case of Cu2+/Al2O3 systems [13].
According to references [12, 14], the difference between signals A1 and A2 may be ascribed to
the two slightly different types of Cu2+ in different positions and unlike local environments in the
CuO–ZnO systems. Specifically, the higher g‖ and lower A‖ values for center A2 can originate from
∗E-mail: huamingzhang66@gmail.com
c© H.M. Zhang, S.Y. Wu, Z.H. Zhang, 2011 23703-1
http://dx.doi.org/10.5488/CMP.14.23703
http://www.icmp.lviv.ua/journal
H.M. Zhang, S.Y. Wu, Z.H. Zhang
the samples suffering stronger axial interaction [12, 14]. However, the above experimental results
have not been theoretically explained so far, and the data on the local structures of the Cu2+
centers have not been obtained yet. Since the analysis of the EPR spectra can provide useful data
on the electronic states and local structures for Cu2+ in the calcined catalysts CuO–ZnO, which
would be helpful in understanding the properties of the catalysts, further studies on the above EPR
experimental results are of particular scientific and practical significance. In this work, the spin
Hamiltonian parameters are theoretically investigated for the two Cu2+ centers A1 and A2 in the
calcined catalysts CuO–ZnO using the high order perturbation formulas of these parameters for a
3d9 ion in tetragonally elongated octahedra. In the above formulas, the tetragonal field parameters
Ds and Dt are determined from the superposition model, by considering the relative axial elon-
gation (in terms of the relative elongation ratio ρ of the axial Cu2+–O2− bond lengths related to
the average bond length) of the oxygen octahedron around the Cu2+ due to the Jahn-Teller effect.
The calculation results and the local structures of the Cu2+ centers are discussed.
2. Calculation
In the calcined catalysts CuO–ZnO, the original Cu2+ in the form of CuO may be located in
the center of oxygen octahedra. As a Jahn-Teller ion, Cu2+ will suffer the Jahn-Teller effect via
stretching the two Cu2+–O2− bonds along the C4 axis and contracting the other four bonds in the
perpendicular plane, which reduces the local symmetry to tetragonal (an elongated octahedron).
This point has been illustrated in various studies on Cu2+ in oxygen octahedra [15]. Thus, the
local structures of the Cu2+ centers can be characterized by the relative tetragonal elongation
ratio (labeled as ρ) in the [CuO6]
10− clusters. For a Cu2+ (3d9) ion in tetragonally elongated
octahedra, the lower 2Eg irreducible representation may be separated into two orbital singlets
2B1g and 2A1g , with the former lying lowest [7, 15]. Meanwhile, the upper 2T2g representation
would split into an orbital singlet 2B2g and a doublet 2Eg [7, 15]. The perturbation formulas of
the spin Hamiltonian parameters for a tetragonally elongated 3d9 cluster can be expressed as [16]:
g‖ = gs + 8kζd/E1 + kζ2d/E
2
2 + 4kζ2d/(E1E2) + gsζ
2
d
[
1/E2
1 − 1/
(
2E2
2
)]
+kζ3d
[
4/
(
E1E
2
2
)
− 1/E3
2
]
− 2kζ3d
[
2/
(
E2
1E2
)
− 1/
(
E1E
2
2
)]
,
g⊥ = gs + 2kζd/E2 + kζ2d
[
2/ (E1E2)− 1/E2
2 − 4/(E1E2)
]
+2gsζ
2
d/E
2
1 + kζ3d
(
4/E2
1 − 1/E2
2
)
/(2E2),
A‖ = P
[
(−κ− 4N/7) + (g‖ − gs) + 3(g⊥ − gs)/7
]
,
A⊥ = P [(−κ+ 2N/7)− (g⊥ − gs)/7] (1)
where gs ≈ 2.0023 is the spin-only value. Parameter k is the orbital reduction factor, which is
equivalent to the covalency factor N. Parameter κ is the core polarization constant. ζd and P are,
respectively, the spin-orbit coupling coefficient and the dipolar hyperfine structure parameter for
the 3d9 ion in crystals. They can be written in terms of the corresponding free-ion values, i.e.,
ζd ≈ Nζ0d and P ≈ NP0 . E1 and E2 are the energy separations between the excited 2B2g and 2Eg
and the ground 2B1g states [16]:
E1 = 10Dq,
E2 = 10Dq − 3Ds+ 5Dt. (2)
Here Dq is the cubic field parameter, and Ds and Dt are the tetragonal field parameters.
For the Jahn-Teller elongated [CuO6]
10− clusters, the parallel and perpendicular bond lengths
can be expressed in terms of the relative tetragonal elongation ratio ρ and the reference distance
R as: R‖ ≈ R(1 + 2ρ) and R⊥ ≈ R(1 − ρ). Thus, the cubic and tetragonal field parameters are
determined from the superposition model [16] and the geometrical relationship of the Cu2+ centers:
Dq ≈ (3/4)Ā4 (1− ρ)−t4 ,
Ds ≈ (2/7)Ā2
[
(1− ρ)−t2
− (1 + 2ρ)−t2
]
,
Dt ≈ (16/21)Ā4
[
(1− ρ)−t4
− (1 + 2ρ)−t4
]
. (3)
23703-2
Theoretical studies of the spin Hamiltonian parameters
Here t2 ≈ 3 and t4 ≈ 5 are the power-law exponents in view of the ionic nature of the bonds [17].
Ā2 and Ā4 are the rank-2 and rank-4 intrinsic parameters, respectively. For octahedral 3dn clusters,
the relationships Ā4 ≈ (3/4)Dq and Ā2 ≈ 10.8Ā4 [17–19] are proved valid in many crystals and are
reasonably applied here. Thus, the g factors, especially the anisotropy ∆g(= g‖ − g⊥) is connected
with the tetragonal field parameters and hence with the local structure (i.e., the relative tetragonal
elongation ratio ρ) of the systems studied.
According to the optical spectra for Cu2+ in some oxides [20, 21], the values Ā4 ≈ 800 cm−1
and N ≈ 0.82 can be obtained. The spin-orbit coupling coefficient ζd and the dipolar hyperfine
structure parameter are acquired for the systems studied using the free-ion data ζ0d ≈ 829 cm−1 [22]
and P0 ≈ 416× 10−4 cm−1 [23]. The core polarization constant is taken as the expectation value
0.3 [22] for 3dn ions in crystals. Thus, only the relative tetragonal elongation ratio ρ is unknown in
the formulas of the spin Hamiltonian parameters. Substituting these values into equation (1) and
fitting the calculated results to the experimental data, one can obtain
ρ ≈ 5% and 3% (4)
for the two centers A1 and A2, respectively. The corresponding theoretical results are shown in
table 1.
Table 1. The g factors and the hyperfine structure constants (in 10
−4 cm−1) for the two
tetragonal Cu2+ centers in the calcined catalysts CuO–ZnO.
g‖ g⊥ A‖ A⊥
A1 Cal. 2.329 2.070 −142.5 18.8
Expt. [12] 2.325 2.075 −140.1 18.7
A2 Cal. 2.367 2.083 −139.4 17.3
Expt. [12] 2.366 2.085 −130.7 18.7
3. Discussion
Table 1 reveals that the calculated spin Hamiltonian parameters for the two centers A1 and A2
in the calcined catalysts CuO–ZnO based on the relative tetragonal elongation ratios in equation (4)
are in good agreement with the observed values. Thus, the experimental EPR spectra [12] for both
systems are satisfactorily interpreted in this work, and the local structure information is obtained.
1. The EPR spectra for the Cu2+ centers can be characterized by the positive anisotropy ∆g.
The anisotropy largely depends upon the tetragonal distortion (i.e., the tetragonal field pa-
rameters Ds and Dt) arising from the relative tetragonal elongation ratio ρ (≈ 3% − 5%).
Further, the tetragonal elongations are of the Jahn-Teller nature via relaxation of the parallel
bond lengths and contraction of the perpendicular ones. Similar tetragonal elongation of the
oxygen octahedra was also reported for Cu2+ on the Al3+ site in LaSrAlO4 [15]. The relative
elongation ratio ρ in the A1 center larger than in the A2 center may be attributed to the more
significant Jahn-Teller distortion in the former. Thus, one can expect that the two centers
A1 and A2 may be distorted in different ways. The difference in the Jahn-Teller distortion
strength is likely due to the [CuO6]
10− clusters at different octahedral cavities and hence dif-
fer from local environments in the CuO–ZnO systems. Similar difference in the Jahn-Teller
distortion was also reported for the two trigonal Ti3+ (with the same spin S = 1/2) centers
at non-equivalent octahedral Al3+ sites in LaMgAl11O19 corresponding to two different sets
of EPR spectra [24].
2. The hyperfine structure constants for the Cu2+ centers are close to each other, which can be
illustrated by the similar spectral parameters and the core polarization constant. However,
the parallel component (A‖) is smaller in A2 center than that in A1 center. From equation (1)
23703-3
H.M. Zhang, S.Y. Wu, Z.H. Zhang
it is seen that A‖ is effected by g‖− gs , while A⊥ remains almost the same because g⊥ shows
the similar values for both centers. Thus, the larger g‖ in A2 center may somewhat cancel
the negative isotropic term (proportional to the core polarization constant) and then leads
to the lower magnitude of A‖ . The signs of the experimental hyperfine structure constants
were not given in [12]. Nevertheless, the theoretical studies in this work indicated negative
signs of hyperfine structure constants A‖ .
3. The studied Cu2+ centers in the calcined catalysts CuO–ZnO are ascribed to the octahedral
[CuO6]
10− clusters embedded in the ZnO host. Although ZnO exhibits some covalency, the
Cu2+ centers are mainly ionic, characterized by the covalency factor N (≈ 0.82). In addition,
the spin-orbit coupling coefficient (≈ 151 cm−1) [25] of the ligand oxygen is much smaller than
that (≈ 829 cm−1) [18] of the central ion Cu2+. Thus, contributions to the spin Hamiltonian
parameters from the ligand orbital and spin-orbit coupling interactions can be regarded as
very small and negligible for simplicity.
4. The calculation errors of the present work can be discussed as follows. First, the approxima-
tion of the theoretical model (i.e., crystal-field model) and formulas may lead to some errors.
Second, the spectral parameters Dq (or Ā4) and N obtained from those for Cu2+ in some
oxides can effect the final results. As Dq varies by 10% , the relative elongations ratio ρ and
the spin Hamiltonian parameters would be modified by about 0.5% , because the tetragonal
distortion (or ρ) and the resultant ∆g [see equation (3)] have a little relation to Dq. While
N changes by 10% , the final ρ and spin Hamiltonian parameters deviate by only 0.4% ,
suggesting that covalency affects mainly the average of the g factors and brings forward a
few effects on ρ and ∆g. Third, the errors can also be introduced from the approximation
of the relationship Ā2 ≈ 10.8Ā4 [17–19], which would slightly modify the tetragonal field
parameters. The errors are estimated to be no more than 1% for ρ and the spin Hamiltonian
parameters when the ratio Ā2/Ā4 varies within the widely accepted range of 9–12. Finally,
the uncertainty of the core polarization constant κ used in the present calculations of the
hyperfine structure constants may lead to some errors. Since this value is the expectation
result for transition-metal ions in crystals, the effects of the above errors on the hyperfine
structure constants may be regarded as very small or negligible.
4. Summary
The spin Hamiltonian parameters and the local structures of the tetragonal Cu2+ centers A1
and A2 in the calcined catalysts CuO–ZnO are theoretically investigated from the perturbation
formulas for a 3d9 ion in tetragonally elongated octahedra. The oxygen octahedra around Cu2+ are
found to suffer the relative tetragonal elongation ratios of about 3% and 5% due to the Jahn-Teller
effect for centers A1 and A2, respectively. The axial elongation for center A1 larger than for center
A2 can be attributed to the more significant tetragonal Jahn-Teller distortion in the former.
Acknowledgements
This work was financially supported by the Support Program for Academic Excellence of
UESTC.
23703-4
Theoretical studies of the spin Hamiltonian parameters
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H.M. Zhang, S.Y. Wu, Z.H. Zhang
Теоретичнi дослiдження параметрiв спiнового гамiльтонiану
для двох тетрагональних центрiв Cu2+ в кальцинованих
каталiзаторах CuO–ZnO
Г.M. Жанг1,2, С.Й. Ву1,3, Ж.Г. Жанг1
1 Унiверситет електронiки i технологiї Китаю, 610054 Ченду, Китай
2 Провiдна лабораторiя неруйнiвних випробувань, Мiнiстерство освiти, Унiверситет Нанчанг Ганконг,
330063 Нанчанг, Китай
3 Мiжнародний центр фiзики матерiалiв, Китайська академiя наук, 110016 Шеньянг, Китай
Параметри спiнового гамiльтонiану для двох центрiв Cu2+, A1 i A2, у кальцинованих каталiзаторах
CuO–ZnO дослiджуються теоретично, використовуючи формули теорiї збурень високого порядку
для цих параметрiв для iона 3d9 у тетрагонально видовжених октаедрах. В цих формулах параме-
три тетрагонального поля Ds i Dt визначаються з суперпозицiйної моделi, шляхом розгляду вiдно-
сного аксiйного видовження октаедра кисню навколо Cu2+ вiдповiдно до ефекту Яна-Теллера. Ба-
зуючись на цих розрахунках, отримано коефiцiєнти вiдносного видовження приблизно 5% i 3% для
тетрагональних Cu2+ центрiв A1 i A2, вiдповiдно. Параметри теоретичного спiнового гамiльтонiану
добре узгоджуються зi спостережуваними величинами для обох систем. Бiльше аксiйне видовже-
ння в центрi A1 приписується до вагомiшої низько симетричної (тетрагональної) дисторсiї ефекту
Яна-Телера. Обговорюються локальнi структури, що характеризуються вище згаданим аксiйним ви-
довженням.
Ключовi слова: EPR, Cu2+, CuO–ZnO, дефектнi структури
23703-6
Introduction
Calculation
Discussion
Summary
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