N Kirova

We consider properties of junctions for the FET geometry where molecular crystals or conducting polymers are used as semiconducting layers. In the molecular crystal Coulomb interaction of free electrons with surface polar phonons of the dielectric layer can lead to selftrapping of carriers and to the formation of a strongly coupled long-range surface polaron. The effect is further enhanced at presence of the bias electric field and strongly depends on the gate dielectric used. In conducting polymers instead of the usual band bending near the contact interface, new allowed electronic bands appear inside the band gap. As a result the bias electric field and the injected charge penetrate into the polymer via creation of the soliton lattice whose period changes with the distance from the contact surface. The current through the contact is performed via moving solitons.

Introduction

Increasing experimental activity is devoted to unconventional semiconductors: transition metal oxides and chalcogenides, molecular crystals, conjugated polymers. A new experimental dimension comes from the possibility to change carrier concentration under the applied gate at high electric field near the junction interface of the field effect transistors (FET). The main goal is to obtain the dielectric metal transition under the applied gate electric field. To avoid Wigner crystallization, one needs rather high electron concentration [1], i.e. to apply the highest possible gate voltage. And this leads us to polar dielectrics. Below in Table 1 we present the breakdown fields, dielectric susceptibilities and estimated surface charge densities for some traditional gate materials. The data are taken from Ref. 2. But with the polar gate dielectrics, we face another problem: the formation of the surface long range polaron, located near the interface between the semiconductor and the gate dielectrics [3,4]. Such an interface polaron can exist already for non-polar semiconductors, like molecular crystals, and it will be ultimately present in traditional oxides like SrTiO 3 . The polaron formation is endorsed by the bias electric field. Finally, the measured mobility of the FET is not only the property of the active semiconductor, but it intrinsically depends on the gate dielectric interface. The pronounced polaronic effects in conducting polymers [START_REF] Lu | Solitons and polarons in conducing polymers[END_REF][START_REF] Brazovskii | [END_REF] change the contact properties of these materials with respect to traditional semiconductors. Instead of the usual band bending near the contact interface, new allowed electronic bands appear inside the band gap.

The bias electric field and the injected charge penetrate into the polymer via creation of a soliton lattice, which period changes with the distance from the contact surface. Single electron carriers (polarons) are pulled to the contact area forming induced surface states. In time resolved experiments, e.g. in optically assistant junction formation, the charge injection (ii) slow kinetic process: majority carriers -polarons collide and are absorbed into the ground stateproviding one more period in the soliton lattice. The minority carriers (polarons of opposite sign) recombine with preexisting solitons reducing their number. The depletion layer is formed via reduction of soliton concentration.

Surface Long Range Polarons in Molecular Crystals.

The influence of surface phonons on the electron polaronic state inside the ionic crystal has been studied long time ago [7]. Here we address the opposite situation when the electron resides in the non-polar media (e.g. a molecular crystal) and the polaron is formed by the interaction with the surface phonons of the polar dielectrics. Since the electron is confined within the molecular crystal z>0, its wave function is distributed near the interface. The only dipole active excitations of the polar dielectrics, to which the electron is coupled, are the surface phonon modes. The electron-phonon interaction can be divided into two parts. The first one comes from the interaction with high frequency phonons which is reduced to the classical limit of the image charge potential. Another part comes from the interaction with low frequency phonons and it results in polaron formation. The total energy of such a surface polaron is W p =W 0 w, where W 0 = Ry(m/m e ):

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎥ ⎦ ⎤ - ⎢ ⎢ ⎣ ⎡ - ∆ - = - ∞ * * * * ∞ * ∫ ∫ ∫ dz e z dq a z W eE z a a dz dxdy w qz q B B B ) ( 1 2 0 0 0 2 ρ ε ψ ψ ψ ψ ε ψ ψ (11) Here e b b m m Ry W = - + = ∞ ∞ * ∞ 0 0 , 0 , 0 , ε ε ε ε ε ε (12)
m e is the bare electron mass, m is the effective band mass, ε b is the dielectric susceptibility of the molecular crystal, ε ∞ and ε 0 are the high and the low frequency dielectric susceptibilities of the gate dielectric and ρ q (z) is the in-plane Fourier component of the charge density, q=(q x ,q y ), r=(x,y) The polaron effective mass can be estimated as: So the gate electric field makes the polaron wave function distribution more two-dimensional. Notice the existence of the crossover bias electric field when the polaron energy and effective mass start drastically to increase. Experimentally it should manifest itself in a decrease of the charge carrier mobility. Moreover, at the very high gate electric fields the polaron can undergo from the long to the short range one. Correspondingly, the mobility regime changes from the band to the hopping one. The effect should be more pronounced in polar semiconducting oxides like SrTiO 3 . The polaron effective mass (14) and hence the measured mobility of the charge carriers in FET geometry strongly depends on the dielectric properties of the gate dielectrics. At Fig. 4 we present the comparison of the model predictions with the experimental data taken from Ref. [8]. Is should be mentioned that in spite we didn't take into account the variation of the surface phonon energies for various dielectrics, the agreement between theoretical curve and experimental points looks quite reasonable. In common semiconductors the band bending near the metal-semiconductor interface brings the chemical potential of the semiconductor to the metal Fermi level. But in conducting polymers the deep self-trapping of free charge carriers results in the formation of soliton (bipolaron) lattice. This originates the band branching in addition to the band bending [9,10].

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = - ∞ ∞ * ∫ ∫
The new allowed band grows inside the original gap, which also expands in its turn. The equilibrium between the metal and the polymer is defined now with respect to pairs of particles forming a lattice of solitons, rather then with respect to the single electron. The single-electron levels are not matched at the junction interface. The tunneling injection between electronic levels occurs instantaneously at a given lattice configuration, hence the activation energy will be required. To estimate the possible charge carrier concentration and the electric field near the interface we have to use the selfconsistent theory of screening by soliton (bipolaron) lattice [11]. As before, we deal with the Eqs. ( 2), but now the injected charge carriers cannot be described as a Fermi electron gas. Instead for the carriers energy density w(n) we have to use the expression derived for the periodic lattice of solitons

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - ∆ = ) ( ) ( 2 1 2 1 ) ( 2 2 0 r K r E r r n w πξ (15)
Here ∆ is the gap in the the undoped polymer, ξ 0 is the soliton size. K(r), E(r) are complete elliptic integrals of the first and the second kind [START_REF] Abramovitz | Stegun Handbook of mathematical functions[END_REF], 0<r<1. The electronic bands boundaries are given as:

2 1 , r r E r E - ∆ = ∆ = - + (16)
The parameter r is related to the soliton density n as:

) ( 1 1 ) ( 0 r rK r n ξ = (17)
The energy of the single electron excitation (polaron level) is defined as: The corresponding band structure is presented at Fig. 5. Depending on the metal and polymer, we estimate l~20-60A, C~10pF , and the built-in contact electric field E~10 5 -10 6 V/cm. The detailed analysis gives us that single-electron excitations created by charge injection or by interband absorption are attracted to the junction surface (see the polaron level at Fig. 5). The polymer may not have the surface states like those coming from dangling bonds in traditional covalent semiconductors, but the charge injection produces these states selfconsistently. Being injected, the majority carriers require an additional energy to escape into the volume. Otherwise, after pair collisions they disappear in the ground state providing one more period in the soliton (bipolaron) lattice. The minority carriers reverse their charge via recombination with preexisting solitons reducing their number. The depletion layer will be formed via reduction of the numbers of solitons as indicated at Fig. 5, left.

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′ Λ ′ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′ = r r K r r r Z r K E p , 4 ) ( 8

Conclusions

The usual choice of a gate material with a high dielectric constant brings to life the interface polaronic effects which can take place even for non-polar semiconductors. Coulomb interaction of free electrons in molecular crystals with polar phonons of the gate dielectric results in selftrapping of carriers and in formation of the surface polaron. A bias electric field drastically enhanced the effect. Even for its typical values ~10 6 eV/cm, it profoundly stabilizes the surface long range polarons. The existence of polarons will show up in enhanced effective mass, lowering of the mobility, appearance of the mid-gap states and in pseudo-gap regime in case of tunneling experiments. The mobility measured in organic FET is not only the property of specific organic molecule used, but it intrinsically depends on the organic/gate dielectric interface. Following special effects take place for the metal-polymer junction: band branching instead of usual band bending; non homogeneous soliton lattice formation; transformation of the injected minority carriers into the majority ones, the soliton current.

A c c e p t e d m a n u s c r i p t 2 goes

 2 via two steps: (i) fast dynamic process: charge injection and polaron formation;

Fig. 3 .

 3 Fig.1. Energy for the surface polaron normalized on its value at zero bias electric field Fig.2. Effective mass for the surface polaron normalized on its value at zero bias electric field

Fig. 4 .

 4 Fig.4. Single parameter fit of presented theory (solid line) for mobility as a function of the dielectric susceptibility of the gate dielectrics. The experimental points are taken from Ref.[8].

3 .

 3 Junction with conducting polymer.

  0 are the Zeta function of Jacobi and Lambda function of Heuman correspondingly[START_REF] Abramovitz | Stegun Handbook of mathematical functions[END_REF]. And we obtain in parametric form:

Fig. 5 .

 5 Fig.5. Band structure near the metal-polymer junction for major (left) and minor (right) carriers' injection. Here M and P indicate the area of metal and polymer, VB and CB are the valence and conducting bands, S(BP) denotes the new soliton band, FL-Fermi energy of the metal, PL -polaronic level (color on-line).We can determine the injected charge surface density σ, the junction area capacitance C and the penetration depth l:

Table 1 .

 1 Dielectric properties of various gate dielectrics.

	Dielectrics	10 6	Eg cm V /	ε	g	ε Eg g cm V / 10 7	13 10	n cm	2 -
	SiO 2		6	4-10	2.4-6	1.3 -3.3
	Al 2 O 3		3	10-11	4 -4.5	2.2 -2.5
	Y 2 O 3		5	9-13	4.5 -6.5	2.5-3.6
	HfO 2		50	15	75	41.4

Acknowledgments: This work was partially supported by INTAS grant 1000008-7972 and ANR grant LoMaCoQup.


M Marqués

J Contreras-García

M Flórez

J M Recio

J Manuel

On the mechanism of the zircon-reidite pressure induced transformation

Keywords: A. Oxides, C. Ab initio calculations, D. Phase transitions, D. Thermodynamic properties

We report first principles results of a detailed investigation directed to elucidate mechanistic aspects of the zircon-reidite phase transition in ZrSiO 4 . The calculated thermodynamic boundary is located around 5 GPa, and the corresponding thermal barrier, estimated from temperatures at which the transition is observed at zero and high pressure, is 133 kJ/mol. Under a martensitic perspective, we examine two different transition pathways at the thermodynamic transition pressure. First, the direct, displacive-like, tetragonal I4 1 /a energetic profile is computed using the c/a ratio as the tranformation parameter, and yields a very high activation barrier (236 kJ/mol). Second, a quasi-monoclinic unit cell allows us to characterize a transition path from zircon (β=90 • ) to reidite (β=114.51 • ) with an activation barrier of around 80 kJ/mol at β=104 • . This energy is somewhat lower than our previous estimation and supports the reconstructive nature of the transformation at the thermodynamic transition pressure.

A c c e p t e d m a n u s c r i p t A c c e p t e d m a n u s c r i p t

Efforts towards the classification of solid-solid transformations have identified two distinctive features as characteristic of displacive phase transitions: (i) parent and product phases exhibit a group-subgroup relationship, and (ii) cation nearest-neighbor coordination is mantained after the transition occurs. Under this perspective, one order parameter of the lower (subgroup) symmetry can be chosen to evolve continuosly from one phase to the other. In contraposition to this conception, reconstructive transformations are not directly symmetry connected and creation and/or breaking of bonds accompany the change of structure. In this respect, the zircon-reidite pressure induced phase transition in ZrSiO 4 presents a paradigmatic case to study the nature of the transformation mechanism since the space group of the scheelite-type (reidite, I4 1 /a) lattice is a subgroup of that of zircon (I4 1 /amd); and the four-fold and eight-fold coordinations of, respectively, Si and Zr are present in both structures. In spite of this a priori displacive or distortive character, a number of experimental results points towards a reconstructive conversion for this phase transition with a volume collapse around 10% and pressurre dependent kinetic activation barriers for the direct (around 1000 K and 300 K at 10 GPa and 20 GPa, respectively) and reverse transitions (1273 K at 0 GPa).

We have previously reported a combined experimental-theoretical work on the pressure behavior of zircon, reidite and post-scheelite phases of ZrSiO 4 with special emphasis on the equations of state and the relative thermodynamic stability of these structures [1]. Here, we focus on the characterization of the zircon-reidite phase transition mechanism, i. e., the microscopic description of the structural, energetic and bonding changes that take place across the transition pathway connecting both structures. Our investigation allows us to determine whether the cation polyhedra (SiO 4 and ZrO 8 ) involve or not the same cation-oxygen bonds in zircon and reidite, and to discriminate, based on the calculated energetic profiles, among different transition pathways.

II. Computational details and mechanistic models

Our electronic structure calculations have been performed following a standard planewavepseudopotential scheme under the local density approximation (LDA) of the density functional formalism, as implemented in the VASP code [2]. Details of the computational parameters and the comparison with the results provided by the generalized gradient approximation (GGA) have been previously discussed [1]. In general, presure (p)-volume (V ) points and all our energetic profiles have been computed at zero temperature and neglecting zero point vibrational contributions.

Nevertheless, to account for thermal effects, we have included finite temperature contributions evaluated with a Debye-like model developed in our laboratory that only needs the static bulk modulus calculated at different pressures as input parameter [3].

Assuming that the barrier height associated with the zircon→reidite transformation depends strongly on pressure but it is negligibly affected by temperature, our Debye-like model was used

A c c e p t e d m a n u s c r i p t

to evaluate thermal barriers from the observed temperatures at which the zircon-reidite transformation occurs when zircon is over-pressurized or the zircon structure is recovered from reidite at zero pressure after pressure is released. The activation energy associated with the thermal barrier is the available vibrational energy of the solid (computed in this work with our Debye model) at the temperature needed to observe the direct or reverse transformation in pressure load or unload processes. We assume a linear dependence of the activation barrier on the transition pressure.

This means that the activation volume, ∆V ‡ =V ‡ -V X , ‡ is the transition state and X is zircon or reidite, remains the same independent of the pressure at which the transition takes place. This is a reasonable assumption in analogy with the nearly constant value found at different pressures for ∆V =V zircon -V reidite (see also Ref. [4]). The ∆V ‡ =constant hypothesis allows us to evaluate thermal barriers at different pressures, in particular at the thermodynamic (equilibrium) zircon-reidite boundary.

The microscopic mechanism of the transition is described considering that the atoms throughout the crystal are displaced simultaneously and coherently from the zircon to the reidite structure following a transition pathway characterized by a specific space group. One transformation coordinate can be chosen to connect both structures in the same sense that a reaction coordinate monitors the mechanism of a chemical reaction. In spite of the obvious limitations, this martensitic modelization has shown to provide a valuable microscopic interpretation for many of the physical and chemical processes across phase transformations whether they display a displacive or reconstructive nature [5][6][7][8]. Here, we discuss a direct tetragonal (I4 1 /a) pathway originated by the group-subgroup relationship between zircon and reidite, and we explore a quasi-monoclinic path as suggested by Kusaba et al. [9].

III. Thermodynamics and thermal barriers of the zircon-reidite phase transition

The experimental facts associated with the stability of zircon and reidite polymorphs can be summarized as follows: (i) volume collapse is around 10-11% for zircon → reidite phase transition, (ii) static experiments show that this transformation is observed at pressures in the range 10-15

GPa when temperature is increased up to 1000-1500 K [10][11][12], whereas at room temperature the transition is also possible if pressure is increased up to approximately 20-23 GPa [13,14], (iii) shock wave studies found this transformation to occur very fast (microsecond time scale), and at much higher pressures (30-50 GPa) [9,15,16], (iv) zircon → reidite transformation exhibits large hysteresis, zircon is not recoverable from reidite at zero pressure unless heated up to 1273 K [15], and (v) the thermodynamic boundary presents a positive Clapeyron slope [START_REF] Finch | Zircon[END_REF].

To discuss theoretical results in relation to the above experimental data we should bear in mind the difference between our calculated equilibrium transition pressures (p t ) and the experimental ones (p exp ). That the zircon → reidite transformation is kinetically hindered is obvious from

A c c e p t e d m a n u s c r i p t

the experimental data collected so far. Thus, when the parent and product phases are close to equilibrium, the energetic barriers associated with these transformations can be overcome only at sufficient high temperatures, and then, observed transition pressures may be compared with thermodynamic transition pressures. From the high temperature data, and assuming a positive Clapeyron slope for this transformation [13,[START_REF] Finch | Zircon[END_REF], p t is expected to be below 10 GPa at room temperature. Our static value of around 5.3 GPa is in concordance with this expectation and is also in good agreement with the calculated values of Crocombette and Ghaleb [START_REF] Crocombette | [END_REF] and Farnan et al. [19]. We have also evaluated the effect of temperature in the equilibrium pressure boundary of this transformation by including vibrational energy and entropy contributions to the static results with the Debye model. We found a qualitatively satisfactory picture with Clapeyron slope slightly positive in the temperature ranges covered by our calculations since p t is computed to increase 1.9 GPa from 0 to 1800 K for zircon → reidite transformation. This means that there is a small decrease of entropy associated to this transition, supporting the analysis of Finch and Hanchar in ZrSiO 4 [START_REF] Finch | Zircon[END_REF].

Under the assumption of no dependence of the barrier height on temperature, we may estimate the thermal energy in the 1000-1500 K range as the activation barrier for the equilibrium zircon reidite transition [10,12]. The validity of this analysis is illustrated by comparing the equilibrium boundary of this transformation with the experimental zircon → reidite transition pressures of Knittle and Williams (23 GPa) [13] and van Westrenen et al. (19.7 GPa) [14] obtained at 300 K.

The overpressure needed to observe the zircon → reidite transformation at 300 K implies that the activation energy for the direct transformation has decreased with compression [13] in such a way that the thermal energy at 300 K is enough to overcome the barrier. Similarly, the reidite → zircon transformation detected at zero pressure after heating up to 1273 K informs on the barrier height for the inverse transformation at this pressure.

A good estimation of such barrier heights is obtained by coupling the Debye model implemented in the GIBBS code [3] to our static total energy calculations. At their corresponding static equilibrium unit cells, Debye's temperatures (Θ D ) are computed for zircon (Θ z D = 796 K) and reidite (Θ r D = 823 K). The available vibrational energy for zircon at 300 K and 23 GPa evaluated within our Debye model is 65 kJ/mol, whereas for reidite at p = 0 and 1273 K is 194 kJ/mol. Since we are interested in the static barriers, i.e. the difference between the static Gibbs energy of zircon (or reidite) and the static Gibbs energy of the transition state, G ‡ , we substract the corresponding zero point energies, 53 kJ/mol for zircon at 23 GPa and 46 kJ/mol for reidite at zero pressure.

Applying our linear hypothesis for ∆G ‡ (p), the thermal barrier at p t can be evaluated (see Fig. 1).

The value estimated amounts 133 kJ/mol which is equivalent (using either the vibrational energy of zircon or reidite corrected with the corresponding zero point energy) to a temperature of around 1200 K, in the range of Ono et al. experiments [10]. ), (0, 1 4 , 5 8 ), and (0,0.5653,0.1946), respectively. For the reidite structure in the I4 1 /a common cell, a=b=4.696 Å, c=10.327 Å, Zr and Si are in the same positions as in zircon in this cell, but O changes to (0.2584,0.0924,0.5456). Using the transformation matrices in Table I, all these values can be compared with the zero pressure parameters of the corresponding conventional cells collected in Table I of Ref. [1].

The transition path may be walked using either the O position or the c/a ratio as the transformation coordinate. For computational reasons, we have chosen c/a and followed several routes including linear interpolations of the unit cell volume (V lin ) or of the lattice parameter a (a lin ), with linear variations of the O coordinates in both cases, while c/a evolves from 0.9059 (zircon) to 2.1990 (reidite). These strategies yield unphysical large activation barriers of about 1540 kJ/mol (see Fig. 2). A full optimization of the unit cell lattice parameters and O coordinates (x opt ) gives an activation barrier much lower (236 kJ/mol), but still too high compared with our previous estimation based on the observed temperatures at which the transition is observed at zero and high pressure. These results allow us to rule out the consideration of the zircon → reidite as a displacive transformation and draw us to investigate further on the transition pathway.

V. Quasi-monoclinic transition pathway

Under the low symmetry P 1 unit cell with four formula units defined in Table I, we can study general transition paths including the two step mechanism proposed by Kusaba et al. [9]. According to these authors, the volume is reduced to the value of the reidite phase in the first step, whereas in the second step slight atomic displacements provide the final high pressure structure; and both steps may be carried out simultaneously. Using the corresponding values in their conventional unit cells, lattice parameters, cell angles and Zr, Si and O atomic coordinates of zircon and reidite in the P 1 unit cell are easily derived using the I4 1 /amd (I) → P 1 and I4 1 /a (II) → P 1 transformation matrices of Table I and noting that there are not origins shifts. In the P 1 cell, the positions of Zr and Si, respectively, are the same in zircon and reidite whereas the sixteen oxygens have different coordinates in the two structures.

The calculation of the Gibbs energy profile, and of the structural and bonding changes across the P 1 pathway, is performed using the angle β as the transformation coordinate. After a full optimization of unit cell lattice parameters and atomic coordinates, we found that α and γ remains with the pathways studied in this work. I and II stand for zircon and reidite conventional unit cells.

I4 1 /amd (I) → I4 1 /a I4 1 /a (II) → I4 1 /a I4 1 /amd (I) → P 1 I4 1 /a (II) → P 1      1 0 0 0 1 0 0 0 1           1 0 0 0 1 0 0 0 1           0 0 1 1 0 0 0 1 0           1 2 1 1 2 -1 2 1 -1 2 -1 2 0 1 2      (0, 1 2 ,0) (0,0, 1
2 ) (0,0,0) (0,0,0)

a=b=a I a=b=a II a=c=a I a=c= 1 2 a 2 II + 1 4 c 2 II c=c I c=c II b=c I b= √ 2a II β=90 o β=arccos ( 1 2 a 2 II -1 4 c 2 II / 1 2 a 2 II + 1 4 c 2 II )
close to 90 • while β evolves from 90 In summary, our investigation reveals that the thermal barrier associated with zircon → reidite phase transition is around 130 kJ/mol at the thermodynamic transition pressure. A direct, displacive-like, transition pathway produces an unrealistic description of the transformation mechanism. If we use a low symmetry triclinic unit cell, we find a quasi-monoclinic transition path with a reasonable activation energy that explains the structural and bonding changes that take place in this phase transition and allows us to characterize it as reconstructive at least at the thermodynamic transition pressure.

FIG. 1 :

 1 FIG.1: Pressure dependence of the Gibbs energy (kJ/mol) of reidite relative to zircon according to present LDA calculations. Thermal barriers at 0 (reidite → zircon), p t , and 23 GPa (zircon → reidite) denoted by arrows and calculated using also our Debye model.

  transition pathwaySince the space group of reidite (I4 1 /a) is a subgroup of that of zircon (I4 1 /amd), we have first chosen a I4 1 /a common unit cell to describe the direct displacive-like mechanism of the zircon → reidite phase transition at the thermodynamic transition pressure. According to the transformation matrices shown in TableI, the lattice parameters of zircon in the I4 1 /a common cell keep the same values as in the I4 1 /amd unit cell (a=b=6.545 Å and c=5.9296 Å), but Zr, Si, and O display new positions at (0,1 4 , 1 8

FIG. 2 :

 2 FIG.2: Energy profile of the zircon → reidite I4 1 /a transition path. V lin , a lin , and x opt stand for different levels of calculation (see text).

A c c e p t e d m a n u s c r i p t

  • (zircon) to 114.51 • (reidite). Changes in the lattice parameters are smooth up to the neighborhood of the transition state where abrupt jumps occur (a and b axes are interchanged), though volume follows a continuous and softer decreasing across the transition path. Concerning atomic coordinates, we found that the changes in Si and Zr are negligible, whereas for the oxygen crystallographic coordinates two well differentiated β ranges of a rather constant behavior appear in the transition path enabling the location of the transition state. This behavior can be explained as due to the phase inertia associated with the high mechanical stability of the two structures, i.e., the lattice parameters and the oxygen coordinates of intermediate P 1 quasimonoclinic structures only suffer substantial deviations from the values in the zircon and reidite phases after β achieves values in the middle of the transition pathway. The transition state is situated at β 104 • , being the calculated value of the activation barrier around 80 kJ/mol. Given the approximations involved in our martensitic description of the mechanism, this value below the estimation of 133 kJ/mol suggests the relevance of surface and/or diffusion barriers to produce the reidite structure in the laboratory, and allows us to conclude that this transition has a reconstructive character at least at the thermodynamic transition pressure. Further support to the above conclusion is provided following the changes in the chemical bonding across the P 1 pathway. In the zircon structure, ZrO 8 and SiO 4 polyhedra alternate to form chains through edge sharing, whereas in reidite ZrO 8 and SiO 4 are linked by corners. Two Zr-O bonds break and two new Zr-O bonds appear in the mechanism to keep the same cation coordination in both phases. At p t , this reorganization takes place in the neighborhood of the transition state and can be associated with the increase of energy in the Gibbs energy profile. Thus, the tilting of the SiO 4 tetrahedra to pass from the sharing with the ZrO 8 bidisphenoid of a O-O edge in zircon to a O corner in reidite involves an energy barrier high enough to hinder the phase transformation at

A c c e p t e d m a n u s c r i p t ambient temperature.

TABLE I :

 I Transformation matrices of the lattice vectors (first row) and origin shifts (second row) associated

Acknowledgments

Finantial support from the Spanish MEC and FEDER program under project MAT2006-13548-C02-02 is gratefully acknowledged. JCG thanks the Spanish MEC for a FPU postgraduate grant.

Part of this work was performed during the stay of JMR at the Laboratoire de Chimie Théorique (Université Pierre et Marie Curie, Paris) under a HPC-EUROPA grant.


A Szytuła

D Kaczorowski
email: d.kaczorowski@int.pan.wroc.pl

M Kalychak

B Penc

Yu Tyvanchuk

A Winiarski

M Smoluchowski

Electronic structure and magnetic properties of the compound CeCuIn

Keywords: Cerium intermetallic compounds, Magnetic properties, Transport properties, Electronic structure

Magnetization, magnetic susceptibility, electrical resistivity, thermoelectric power and X-ray photoemission measurements were performed on a polycrystalline sample of CeCuIn. This compound crystallizes in a hexagonal structure of the ZrNiAl type. The magnetic data indicate that CeCuIn remains paramagnetic down to 1.9 K with a paramagnetic Curie temperature of -13 K and an effective magnetic moment equal to 2.5 µ B . The electrical resistivity has metallic character, yet in the entire temperature range studied here, it is a strongly nonlinear function of temperature. The temperature dependence of the thermoelectric power is dominated by a small positive maximum near 76 K and a deep negative minimum at about 16 K. Above 150 K the thermopower exhibits a Mott's type behavior. The positive sign of the Seebeck coefficient in this temperature region indicates that the holes are dominant charge and heat carriers. The structure of Ce 3d 5/2 and Ce 3d 3/2 XPS spectra has been interpreted in terms of the Gunnarsson-Schönhammer theory. Three finalstate contributions f 0 , f 1 and f 2 are clearly observed, which exhibit a spin-orbit splitting ∆ SO ≈ 18.7 eV. The appearance of the 3d 9 f 0 component is a clear evidence of the intermediate valence behavior of Ce. From the intensity ratio I(f 0 )/[I(f 0 ) + I(f 1 ) + I(f 2 )] the 4f-occupation number is estimated to be 0.95. In turn, the ratio I(f 2 )/[I(f 1 ) + I(f 2 )] = 0.08 yields a measure of the hybridization energy that is equal to 45 meV.

A c c e p t e d m a n u s c r i p t 1. Introduction

Cerium ternary intermetallic compounds have attracted considerable attention due to their anomalous properties, such as valence fluctuation phenomena, Kondo effect, heavy fermion behavior, superconductivity etc. [START_REF] Maple | Handbook on the Physics and Chemistry of Rare Earths[END_REF][START_REF] Wohlleben | [END_REF][START_REF] Smith | Theoretical and Experimental Aspects of Valence Fluctuations and Heavy Fermions[END_REF]. These anomalous properties are associated with the instability of the 4f states. The magnetic properties result from a competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions, crystal electric field (CEF) and Kondo interactions [START_REF] Steglich | [END_REF]. The long-range exchange RKKY interactions mediated by conduction electrons favor magnetic ordering. The Kondo interactions tend to prevent the formation of local magnetic moments and lead to a variety of correlated electronic ground states observed in cerium compounds.

CeTX compounds, where T is an nd-electron element and X a p-electron element, form in a few different crystal structures. Our recent investigations concentrate on the CeTIn phases that crystallize in the hexagonal ZrNiAl type, and exhibit various properties [5][6][7][8], e.g. CeNiIn and

CeRhIn are valence fluctuating systems, CePtIn is a non-magnetic heavy fermion compound, while

CePdIn and CeAuIn order antiferromagnetically at low temperatures.

In this paper we present the results of magnetization, electrical resistivity, thermoelectric power and XPS measurements performed on a polycrystalline sample of CeCuIn. This compound was reported in the literature to crystallize either in a ZrNiAl-type structure [START_REF] Dwight | Proc. Rare Earth Res. Conf. 12 th[END_REF][START_REF] Baranyah | [END_REF] or a CaIn 2 -type structure [11]. Published magnetic and resistivity data indicate that it is paramagnetic down to 4.2 K [11].

Experimental details

Polycrystalline samples of CeCuIn and LuCuIn were obtained by arc melting stoichiometric amounts of the constituents (Merck AG; purity: Ce and Lu 99.9 wt.%, Cu and In 99.99 wt.%) under high-purity argon atmosphere on a water-cooled copper hearth, using a titanium-zirconium alloy as a getter. Subsequently, the buttons were annealed in evacuated quartz tubes at 800°C for a week.

The quality of the products was checked by X-ray powder diffraction at room temperature. The Xray data were analyzed by the program FULLPROF [12].

The magnetic properties were studied in the temperature range 1.9-300 K and in external magnetic fields up to 5 T using a commercial SQUID magnetometer (Quantum Design MPMS-5).

The electrical resistivity was measured over the temperature interval 4.2-280 K employing a standard dc four-probe technique. The current contacts were made using silver paste and the voltage contacts were made by ultrasonic welding. The Seebeck coefficient was measured in the temperature range 5-300 K by a differential method with copper as reference material. eV) and AlK α (hν = 1486.6 eV) radiation. The total energy resolution of the spectrometer with a hemispherical energy analyzer was about 0.3 eV for the PHI 5700/660 spectrometer and 0.75 eV for the LHS10 spectrometer for Ag 3d. Binding energies were referred to the Fermi level (E F = 0). The spectrometer was calibrated using the Cu 2p 3/2 (932.5 eV), Ag 3d 5/2 (368.1 eV) and Au 4f 7/2 (84.0 eV) core-level photoemission spectra. The surface of the sample was mechanically cleaned by scraping with a diamond file in a preparation chamber under high vacuum (10 -9 mbar) and the sample was then immediately moved into the analysis chamber. This procedure was repeated until the C 1s and O 1s core-level peaks were negligibly small or were not changed after further scrapings. Such a cleaning procedure was performed before each XPS measurement. The Shirley method [13] was used to subtract the background. The experimental spectra were numerically fitted using an 80% Gaussian and 20% Lorentzian model.

Results

The powder X-ray diffraction data revealed that the investigated sample of CeCuIn crystallizes in a hexagonal ZrNiAl-type structure (space group No 189), in agreement with [START_REF] Dwight | Proc. Rare Earth Res. Conf. 12 th[END_REF][START_REF] Baranyah | [END_REF]. The refined lattice parameters are equal to a = 7.4915(6) Å and c = 4.2452(5) Å and are in agreement with earlier results [START_REF] Dwight | Proc. Rare Earth Res. Conf. 12 th[END_REF][START_REF] Baranyah | [END_REF]. In the unit cell the atoms are located as follows:

Ce at 3(g) site:

x Ce , 0, ½ with x Ce = 0.5832(4);

Cu1 at 1(b) site: 0, 0, ½;

Cu2 at 2(c) site: ⅓, ⅔, 0;

In at 3(f) site:

x In , 0, 0 with x In = 0.2486 (5).

As shown in Fig. 1, above about 40 K the reciprocal magnetic susceptibility of CeCuIn follows the Curie-Weiss law with an effective magnetic moment µ eff of 2.50 µ B (i.e. close to µ eff = 2.54 µ B expected for a free Ce 3+ ion) and a strongly negative paramagnetic Curie temperature θ p being equal to -15 K. Some deviation from the Curie-Weiss behavior is observed below 40 K, presumably due to crystal field effects. These results are in good agreement with those reported in [11]. The cerium moments do not order magnetically down to 1.9 K. The magnetization curve recorded at this temperature has a paramagnetic character (see the inset in Fig. 1). The Ce magnetic moment measured at 1.9 K in a field of 5 T amounts to 0.66 µ B .

The electrical resistivity of CeCuIn is presented in Fig. 2(a). The ρ(T) dependence is strongly non-linear in the entire temperature range, yet the resistivity increases with rising temperature, as

A c c e p t e d m a n u s c r i p t

4 expected for a metal. This resistivity curve differs from that reported in [11] because the latter shows a straight-line behavior from room temperature down to 100 K, followed by some curvature at lower temperatures only.

The value of the residual resistivity of our sample equals 36.7 µΩ cm, i.e. it is smaller than that observed in [11] (56.2 µΩ cm), which indicates that our sample is of better quality. The resistivity of CeCuIn is also quite different from the ρ(T) behavior of LuCuIn (see Fig. Assuming that the electron-phonon scattering in CeCuIn is similar to that in LuCuIn the magnetic contribution to the resistivity of the Ce compound may be derived:

]} [ ] )[ ( { ]} [ ] )[ ( { ] )[ ( 0 0 LuCuIn LuCuIn T CeCuIn CeCuIn T CeCuInT T m ρ ρ ρ ρ ρ - - - = .
The thick solid line in Fig. 2 At 76 K a broad maximum is observed, which may be due to a phonon drag effect [START_REF] Blatt | Thermoelectric Power of Metals[END_REF] or caused by inelastic scattering of carriers by acoustic phonons [START_REF] Durczewski | [END_REF]. The positive sign of the Seebeck coefficient at ambient temperatures possibly indicates that holes are dominant charge and heat carriers [START_REF] Durczewski | [END_REF].

Fig. 3 shows the XPS spectrum of CeCuIn collected in a wide 0-1200 eV binding energy range.

The binding energies are referred to the Fermi level (E F = 0 eV). The XPS valence band spectrum of CeCuIn, extending from the Fermi level energy to a binding energy of about 12.5 eV, is presented in Fig. 4. Near the Fermi level there is a peak at 1.9 eV that corresponds to the Ce 4f and Ce 5d6s 2 states. At 4.0 eV a peak occurs due to the Cu 3d state, and then at 6.3 and 8.9 eV the peaks of In 5s and In 5p states are observed. In turn, the In 4d 5/2 and In 4d 3/2 states manifest themselves as

A c c e p t e d m a n u s c r i p t

5 peaks occurring at 16.85 and 17.75 eV, respectively. Their spin-orbit splitting amounts to 0.9 eV, while the literature value is 0.8 eV [START_REF] Hüfner | Photoemission Spectroscopy[END_REF].

The Ce 3d spectrum is displayed in Fig. 5. The Ce 3d 5/2 and Ce 3d 3/2 states spin-split of ∆ SO ≈ 18.7 eV are clearly seen. Both peaks have three final state contributions, f 0 , f 1 and f 2 , which can be analyzed in terms of the Gunnarsson-Schönhammer theory [START_REF] Gunnarsson | [END_REF]. For this purpose the spectrum was separated into particular components using an approach based on the Doniach-Šunjić theory [18].

The appearance of the f 0 contribution is a clear evidence of a mixed valence state. The intensity ratio r 1 = I(f 0 )/[I(f 0 ) + I(f 1 ) + I(f 2 )] provides an estimate for the occupation of the final f states n XPS = 1-r 1 being equal to 0.95. The f 2 components located at the low-binding energy side of the f 1 components in the Ce XPS spectrum may be attributed within the Gunnarsson-Schönhammer model to the hybridization between the f states and the conduction band states. The hybridization energy ∆ is defined as πV 2 ρ max , where ρ max is the maximum of the density of states of the conduction band and V is the hybridization matrix element. Since the intensity ratio r 2 = I(f 2 )/[I(f 1 ) + I(f 2 )] had been calculated for Ce in [19] as a function of ∆, it is possible to determine the strength of the hybridization interactions. From the calculated value r 2 = 0.08, the hybridization energy in CeCuIn is estimated to be 45 meV. This value is considerably lower than that reported for the isostructural compound CeNiIn (140 meV) [START_REF] Hüfner | Photoemission Spectroscopy[END_REF].

The Cu 2p 3/2 and Cu 2p 1/2 peaks are located at 933.1 and 952.9 eV, respectively, which yields the spin-orbit splitting ∆ SO = 19.1 eV. Similarly, the In 3d 5/2 and In 3d 3/2 states, which occur at 443.8 and 451.4 eV, respectively (see Fig. 3), are spin-split by ∆ SO = 7.6 eV. The two latter values of ∆ SO are in good agreement with the literature data [START_REF] Hüfner | Photoemission Spectroscopy[END_REF].

Discussion

The reported investigations aimed at determining the magnetic and electrical transport properties as well as the electronic structure of CeCuIn. The obtained magnetic data seem to indicate that the cerium magnetic moments are fairly well localized. In the paramagnetic state the effective magnetic moment is close to the free Ce 3+ ion value, and at the lowest temperatures some manifestation of crystalline electrical field is observed. However, the Ce moments do not order magnetically down to 1.9 K, and nevertheless the paramagnetic Curie temperature is strongly negative. From this behavior one might infer that Kondo interactions play an important role. Such a presumption is, however, not compatible with the electrical behavior of CeCuIn, which does not provide any hint at the latter effect.

Actually, in the isostructural series CeTIn, where T = Ni, Rh, Pd and Au, Kondo screening interactions have been found to govern the ground state properties, which range from valence [20]. In this model the key parameter is the coupling constant J cf , which represents the interaction between 4f and conduction electrons. The indirect magnetic exchange between local moments via conduction electrons polarization, the so-called RKKY interaction, is characterized by a temperature defined as T RKKY ~ J cf 2 N(E F ), where N(E F ) is the density of states at the Fermi level, whereas the Kondo effect, which tends to compensate for the local moment and leads to the formation of a nonmagnetic ground state, is given by the temperature

T K ~ exp ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ cf F J ) E ( N 1
. The ground-state behavior of a given compound results from a competition/interplay of these two mechanisms. Fig. 6 shows the Kondo and Néel temperatures of the CeTIn compounds, plotted as a function of the unit cell volumes. Apparently, the RKKY interactions take over in the compounds having large unit cell volumes, so that antiferromagnetic ordering is found, as for CeAuIn and CePdIn. In contrast, for small unit cell volumes the Kondo effect is the dominant interaction and hence no long-range order is observed.

A c c e p t e d m a n u s c r i p t 3 The

 3 XPS spectra were obtained at room temperature in a vacuum of about 10 -9 mbar using a Multipurpose Electron Spectrometer PHI 5700/660 with AlK α monochromatized radiation (hν = 1486.6 eV) and a Leybold LHS10 electron photoemission spectrometer with MgK α (hν = 1253.6

  2(a)), which is a nonmagnetic isostructural counterpart to the Ce-based compound studied here. As shown in the figure by a thin solid line, the resistivity of the latter can be well described by the standard Blochscattering the conduction electrons on lattice imperfections and phonons. The fit parameters found by the least-squares method are the residual resistivity ρ 0 = 25.1 µΩcm, the BG coefficient R = 0.105 µΩ⋅cm/K and the Debye temperature Θ D = 183 K.

  Fig.2(b) shows the temperature dependence of the thermoelectric power of CeCuIn. At low temperatures a deep negative minimum occurs at about 16 K with a sign reversal in the vicinity of 25 K. At 76 K a broad maximum is observed, which may be due to a phonon drag effect[START_REF] Blatt | Thermoelectric Power of Metals[END_REF] or

A c c e p t e d m a n u s c r i p t 6 fluctuations

 6 to long-range magnetic order. Qualitatively these properties can be explained based on the Doniach phase diagram

A c c e p t e d m a n u s c r i p t 8 Figure captions Fig. 1 . 9 K

 8captions19 Figure captions

Fig. 2 .

 2 Fig. 2. Temperature dependencies of (a) electrical resistivity of CeCuIn and LuCuIn and (b) thermoelectric power of CeCuIn.

Fig. 3 .

 3 Fig. 3. XPS spectrum of CeCuIn collected for the 0-1100 eV binding energy range.

Fig. 4 .

 4 Fig. 4. XPS valence band spectrum of CeCuIn of its deconvolution into components.

Fig. 5 .

 5 Fig. 5. Ce 3d XPS spectrum of CeCuIn of its deconvolution into components.

Fig. 6 .

 6 Fig. 6. Kondo (○) and Néel (∆) temperatures vs. the unit cell volume for several CeTIn compounds.

A c c e p t e d m a n u s c r i p t 9 Fig. 1

 91 Fig. 1

A c c e p t e d m a n u s c r i p t 10 Fig. 2a

 10 Fig. 2a

A c c e p t e d m a n u s c r i p t 11 Fig. 2b

 11 Fig. 2b

A c c e p t e d m a n u s c r i p t 12 Fig. 3

 123 Fig. 3

A c c e p t e d m a n u s c r i p t 13 Fig. 4

 134 Fig. 4

A c c e p t e d m a n u s c r i p t 14 Fig. 5 A c c e p t e d m a n u s c r i p t 15 Fig. 6

 145156 Fig. 5

  Here, the prominent examples are the valence fluctuation systems CeNiIn and CeRhIn. The compound CeCuIn, investigated in the present work, lies in the intermediate region between these two regions, and is somewhat similar to the heavy fermion paramagnet CePtIn.More advanced discussion is possible based on the experimentally determined electronic structures of the CeTIn compounds. From the XPS data obtained for CeNiIn[21] it is clear that Ce 4f and Ni 3d are mutually hybridized, which brings about the non-magnetic state. On the contrary, the analysis of the XPS valence band spectra of CeAuIn[21] confirms the well localized character of the Ce 4f state. In the case of CeCuIn the XPS spectra of Ce 3d states hint at little hybridization of the 4f states with the conduction band states. It is remarkable that the hybridization strength in CeCuIn derived from the XPS data (45 meV) is much lower that that reported for CeNiIn (140 meV).

Acknowledgements

This work was partially supported by the Polish Ministry of Science and Information Society Technologies with Grant No. 1P03B 111 29.

A c c e p t e d m a n u s c r i p t


M G Brik
email: brik@fi.tartu.ee

Yau-Yuen Yeung

model and crystal field parameters for Co 2+ ions using the exchange charge model

Keywords: A. Optical materials, D. Crystal fields, D. Optical properties

This paper attempts for the first time to establish a reliable linkage between the two wellknown and independent models of crystal field (CF), namely the exchange charge (ECM) and superposition models (SM). Our approach aims to show that the SM parameters can be reliably extracted from the distance dependence of the CF invariants for Co 2+ as derived from the ECM through some semi-ab initio calculations which involved a single fitting parameter and a set of newly constructed procedures. Complete sets of the numerical values of SM parameters k B and t k for Co 2+ in its own host lattices of Li 2 Co 3 (SeO 3 ) 4 , CoSO 4 •H 2 O, CoSeO 4 •H 2 O, and Co(OH ) 2 are obtained and they are found to be around 13,000 -16,000 cm - 1 for 2 B , 4,100 -5,700 cm -1 for 4 B , 4.1 -5.0 for t 2 and 6.2 -6.5 for t 4 . The present results generally agree with but should be much better than those incomplete sets of results found by previous researchers using the conventional fitting approach. Plausible explanations for some noticeable discrepancies are also discussed together with the effects of different CF contributions on values of the SM parameters.

Introduction

It is a well established fact that the energy levels of any ion with an unfilled electron shell are split by the crystal field, if this ion is placed into a crystal or a glass (Refs. 1-6, and references therein). Since the pioneering paper by Bethe 1 , several models of crystal field have been Very often the k p B parameters are treated as the fitting parameters defined by matching the calculated and observed energy level schemes. In this case, initial assumptions about the point symmetry of the cluster should be introduced before fitting. Though such an approach allows for getting very good agreement between the calculated and observed energy levels, employment of a great number of fitting parameters (which can be as great as 14 for d-ions and 27 for f-ions) makes the fitting procedure rather complicated.

However, there are several models of crystal field which allows for direct calculation or estimation of the numerical values of CFP's. In the next section two such models used in the present paper -superposition model (SM) 7 and exchange charge model (ECM) [START_REF] Malkin | Spectroscopy of solids containing rare-earth ions[END_REF] will be described briefly. The main goal of the present study is to establish a feasible way to extract the values of the SM intrinsic parameters for Co 2+ using the values of CFP obtained within the framework of ECM and to re-examine the plausible form of the distance dependence of the SM intrinsic parameters. The present approach is very different from the traditional approach which fits the SM parameters to the empirical CFP. The latter approach is often found to be unfeasible in many physical systems because of rather limited or an incomplete set of CFP derived from the experimental energy levels. This is especially true for the spectra of 3d ions, when the number of the non-zero CFP in the low-symmetry centers may be greater than the number of the well-detected energy levels.

Superposition model

Since the SM was introduced by Newman and his co-workers 7,[START_REF]Crystal Field Handbook[END_REF] to successfully analyse the CFP of lanthanide ions in various garnets and other crystal systems [START_REF] Yeung | [END_REF] , its equation and formalism had been widely extended to model the spin-Hamiltonian parameters (SHP) for the zero-field splittings of the ground 8 S 7/2 state of the 4f 7 (Gd 3+ and Eu 2+ ) ions or ground 6 S 5/2 state of 3d 5 (Fe 3+ and Mn 2+ ) ions [11][12][13] , even though the contributing mechanisms to SHP are substantially different from those of CFP. For analyzing the crystal field splittings of the 3d transition-metal ions, the examples of SM application were relatively fewer [14][15][16] because of the rather limited number or incomplete set of CFP derived from some experimental energy levels of which their states are often poorly determined or identified. Since the successful demonstration 17 of the effective use of SM for modeling the impurity-induced local distortion of Eu 2+ doped in various alkali halides, another major application of SM is related to the study of local distortion for the doping of transition-metal ions in various crystal lattices. 12,16 However, it must be noted that a number of researchers have held or overlooked a common fallacy in this area of study because of (i) their adoption of an oversimplified local distortion model and (ii) fitting too many (i.e. over fitting) distortion parameters to the limited number of experimental data via the SM parameterization.

For analyzing the CFP of 3d ions, we shall adopt the approximate assumptions of SM that the major interactions between the transition-metal ion and its neighbouring ligands are cylindrically symmetric (i.e. characterized by the C v point symmetry group) and independent.

Hence, the CFP can be written as linear sum of the crystal field contributions from individual ligand L located at the polar position

) , , R ( L L L
with respect to a chosen coordinate frame:

L L L p , k L k k p ) , ( G ) R ( B B , (2)
where the coordination factors

) , ( G L L p , k are explicitly given in Table 5.1 of Ref. 9. For the SM intrinsic parameters ) R ( B k
, their distance dependence is often assumed to follow a power-law form: , but this was commented by Newman and Ng 9 to be phenomenologically no better or worse than using the conventional SM parameterization which involves just two parameters ) R ( B k 0 and t k for each rank of crystal field. Newman and Ng 7 have remarked that the power-law expression (3), in fact, has no special merit and because of the narrow range of R L for the experimental data, it is empirically indistinguishable from the exponential form:

k t k k ) R / R ( ) R ( B ) R ( B 0 0 , (3)
R k k k e ) R ( B ) R ( B 0 , (4)
While there had been many extensive applications of the SM in lanthanide (4f n ) crystal field 7 , rather few papers were published on the "proper" use of SM in the crystal field of transition metal (3d n ) ions. The most noticeable work was performed by Yeung and Newman 19 on the first successful use of the SM to simultaneously analyze strain-induced zero field splittings and optical spectra of Cr 3+ :Al 2 O 3 system which clarified the past serious "misuse" of SM by various researchers. The approach was further extended to study other oxide systems (especially some ions located at very low site-symmetry) by Yeung and his co-workers 16,[20][21][22] , yielding some empirical estimates of SM parameters for the Cr 3+ ion. On the other hand, Wildner, Giester and Andrut [23][START_REF] Wildner | Neues Jahrbuch fuer Mineralogie[END_REF][25][26] had started from late 1990s to make a systematic extraction of the SM parameters for Co 2+ ion from the electronic absorption spectra of several oxide crystals containing Co 2+ ions in their original host lattices. However, there still lacks some reliable ab initio calculations for the CFP of Co 2+ ion in the corresponding oxide crystals for triangulation with the empirical results. Therefore, it would be scientifically meaningful to make numerical calculations through the exchange charge model (as described in the next section) to (a) predict the values of the intrinsic parameters ) R ( B k 0 for Co 2+ located in some oxygen-ligand crystals and to (b) critically examine and determine the distance dependence of the intrinsic parameters for the Co 2+ ion located in those physical systems.

Exchange charge model

In the exchange charge model (ECM) framework, the energy levels of a 3d impurity ion (

The first contribution is due to the electrostatic interaction between the valence electrons of an impurity ion and crystal lattice ions (which, at this point, are treated as the point charges only, without taking into account their electronic structure), and the second one is proportional to the overlap of the wave functions of an impurity ion and ligands. This term includes all effects of the covalent bond formation between an impurity ion and ligands. Inclusion of these effects significantly improves agreement between the calculated and experimentally observed energy levels. Analytical expressions for calculating both contributions in the case of 3d-ion are as follows [START_REF] Malkin | Spectroscopy of solids containing rare-earth ions[END_REF] :

i p i i i k p i p k p k q p R V q r e K B 1 2 , , , (6)
i i i i k p i p i i s k p k S p R V S G S G s S G p e K B , ) ( ) ( ) ( 5 ) 1 2 ( 2 2 2 2 2 , . (7)
The sums are carried out over lattice ions numbered by symbol i with charges q i ;

i i i R , ,
are the spherical coordinates of the i-th ion of crystal lattice in the system of reference centered at the impurity ion. As reported in one of our previous paper (see Ref. [35]), our detailed analysis showed that the rank 2 (B 2q ) and rank 4 (B 4q ) CF parameters of Ni 2+ : MgAl 2 O 4 would require about 1500 and 500 ions to get their values stabilized, respectively. For our present 4 crystals, we have a much larger size of cluster which ranges from 24,616 to 53,385 ions with ions truncated at a given radial distance (of around 44.99 to 50.00 Å) to preserve the proper local site-symmetry of the Co 2+ ion which is placed at the centre of this cluster. A large cluster is especially essential for the accurate calculation of the long-range electrostatic contribution as given by Eq. ( 6) while it is much less crucial for the short-range exchange contribution given by Eq.( 7).

G G G G s
(in this case only, the first absorption band is required to determine the value of G , while employing three fitting parameters would require at least three absorption bands detected precisely in the experimental spectra). We restrict ourselves by this simplified model. As will be shown later, even one fitting parameter is sufficient to get reasonable agreement between positions of the calculated and experimentally detected energy levels.

Previously, the ECM has been successfully applied for calculations of the energy levels of both rare earth 8, 28-32 and transition metal ions [33][34][35][36] in different hosts, and in analysis of the electron-phonon coupling. [37][38][39][40] 1.

Short description of the studied crystals

In all these hosts, Co 2+ ions occupy octahedral positions formed by oxygen ions. Their absorption spectra are typical for 6-fold coordinated Co 2+ ion and have been studied in details by Wildner and Andrut 23,25,26 using the Superposition Model and other empirical approaches.

For the Li 2 Co 3 (SeO 3 ) 4 system, the cluster for the summation over i in Eq.( 6) of our ECM calculations consists of 8,173 ions (1,183 Co 2+ , 769 Li + , 4,667 O 2-and 1,554 Se 4+ ions at the cut-off radial distance of 28.995Å) and 8,177 ions (1,185 Co 2+ , 775 Li + , 4,665 O 2-and 1,552

Se 4+ ions at the cut-off radial distance of 28.999Å) for site I and site II, respectively. For the two kieserite systems, the cluster consists of 53,385 ions (5,

Results of CFP and energy levels calculations for all crystals

Calculations of CFP's were performed using the ECM equations from Section 2.2. The overlap integrals between Co 2+ and O 2-ions, which are needed to evaluate the exchange charge contribution, were calculated using the radial parts of the Co 2+ and O 2-wave functions given in Ref. 43. For modeling the CFP's distance behavior, the interionic distances in all studied crystals were varied from 0.90R 0 to 1.10R 0 with a step of 0.01 (R 0 stands for an equilibrium interionic distance at normal conditions). Such an approach allows for taking into account pressure effects and effects of thermal expansion. So, the 21 set of CFP's was obtained and used for a further analysis. The fitting parameter G was determined by matching the calculated and experimentally observed Co 2+ absorption bands, when the interionic separation is R 0 ). The results of the energy calculations (for the equilibrium interionic distances in all considered crystals), which include the free-ion Hamiltonian 2-4, 20 (with Calc. 25 Exp. 25 Calc. (this work) Calc. 25 Exp. 25 4 T 1g ( together with the corresponding calculations by Wildner and Andrut 23,25,26 for direct comparison. The results of the SM fitting for the present sets of CFP are also given in those tables but the detailed method of fitting will be described in the next section.

A c c e p t e d m a n u s c r i p t

11

Extraction of SM parameters from CFP

To extract the SM intrinsic parameters k B and power-law exponents t k for Co 2+ ion, the traditional approach is to carry out a least-squares fit to the sets of calculated CFP using Eqs.

(2) and (3). The fitting process is notoriously difficult for low-symmetry CFP because it is a kind of non-linear fitting which may encounter many local minimas, and probably lead to divergence. Therefore, we put forward a new method for the present work consisting of the following two key steps:

Step (1) -Use the ECM to calculate the CFP for a range of uniform expansion (or contraction) in the lattice parameters of a scaling factor f which gives rise to an identical fractional change (f-1) in all ionic distances. Then plot the graph (see Figures 1234)

for the rotational invariants s k against the scaling factor f in which s k are related to the crystal field parameters B kp through the following equations:

, B ) k ( s p kp k 2 2 1 2 1 ( 8
)
When Eq.( 2) and ( 3) are substituted into (8), Yeung and Newman [START_REF] Yeung | [END_REF] found that

ij ij , k t j i k k ) ( G R R R R B k s k 0 2 0 2 0 2 1 2 1 (9)

A c c e p t e d m a n u s c r i p t

14 where ij are the angles subtended by the ligands i, j is at the origin (rather than heir angles referred to a specific coordinate system as noted in Eq.( 2)). When the whole crystal is uniformly expanded (or contracted) by the same scaling factor f, the new ionic distance R i ' for ion i is related to its original host lattice distance R i by the same factor f = R i '/ R i but all kinds of angles remain unchanged. Hence, we can easily derive from Eq.( 9) that the rotational invariant s k also follows the same power-law dependence on f (like that of the intrinsic SM parameter k B on the ionic distance R i ), i.e. k t k f s (10) Consequently, the graph of rotational invariants against the expansion factor f can help us easily determine the numerical value of the power-law exponent. It is remarked that this simple approach for the power-law formula (3) does not work for the exponential expression (4) as the term containing the scaling factor f could not be completely separated from individual ionic distance R i .

Step 2 -Once the numerical values of t k and the polar coordinates of the ligands are known (as tabulated in Table 8(a)-(c) using the relevant crystal structure data already given in Table 1 and references therein), we could easily employ Eq.( 9) to uniquely determine the numerical value of the SM intrinsic parameters k B without involving any non-linear least-squares in the fitting process. The results have also been presented in Tables 567.

Discussion

From a quick glimpse of Tables 2-4, Wildner and Andrut's previous calculations 23,25,26 of energy levels for Co 2+ in the 4 crystal lattices seem to match better with the experimental results than compared with the results calculated from this work. However, one must note that Wildner and Andrut used a much greater number of the fitting parameters in their calculations, and so it would normally lead to an apparently better agreement. Our present ECM results

were obtained from almost first-principle calculations which involved only one dimensionless adjustable parameter G, as derived from the first absorption band. Much better results will likely be obtained if the approximation

G G G G s
is relaxed but at least three absorption bands must first be precisely detected in the experimental spectra so that the 3 dimensionless adjustable parameters G s , G , and G of the ECM model could be uniquely determined to enhance the accuracy of our calculated results. Anyway, the energy levels From Table 5, we see that our presently calculated set (shown in columns (a)) of rank 4 CFP for the Co 2+ located at sites I and II in Li 2 Co 3 (SeO 3 ) 4 are generally in quite good agreement with those given by Wildner and Andrut 23 as the difference between each pair of corresponding components is often less than 20%, or just a few hundred cm to the rank 2 CF as predicted by the SM, so the value of this kind of calculations for the rank 2 CF will again be strongly dependent on the accurate value of the power-law exponent t 2 because of the cancellation between contributions from individual ligands.

Table 9 reports a summary of the SM parameters k B (all at the same reference ligand distance of R o = 2.1115Å) and t k for Co 2+ located in 6 different host systems which all contain oxygen as their key ligands. Corresponding values for the short-range exchange (including covalency and overlapping) contributions (as calculated through the use of Eq.( 7)) are also reported in brackets for revealing its importance as relative to the overall values. They are obtained when B kp,s in Eq. ( 7) are put into the LHS of Eq. ( 2) and re-label the term k B as

S , k

B

with the suffix S being added to remind us that it comes from the short-range exchange contribution. as proposed by Levin and Cherepanov 18 . However, we have attempted some fittings using the alternative exponential form (given in Eq. ( 4)) of distance-dependence for the SM intrinsic parameters. Even though this alternative form also works quite well for a short range of distances (±10% of the mean ligand distance), some noticeable discrepancies are found at the two extremes of each fitted curve.

Conclusion

For the first time, two independent models of crystal field -the exchange charge and superposition models -have been linked together. The basic idea of our approach was to use the explicit expressions for the CFP in the ECM framework for numerical calculations of the CFP values and CF invariants for different inter-ionic distances. The main aim of such an approach is to employ the calculated sets of CFP to extract the SM parameters from the distance dependence of the calculated CF invariants, using a set of newly constructed procedures which are very simple and independent of the coordinate frames used in the calculations of the CFP's. This new method of calculating the SM parameters from the distance dependence of the CF invariants (calculated using ECM and crystal structure data) can be easily applied in a straightforward manner to other crystals, both doped and/or selfactivated. The traditional way of fitting the SM parameters to the experimental CF parameters does not work for the 3d ion in low-symmetry sites because there are usually insufficient number of energy levels observed for determining all the CF parameters and the free-ion parameters. The ECM calculations enable us to determine all the SM parameters with very few spectroscopic data and so the present approach has its definite advantage. purposes such as prediction of the spectroscopic and EPR spectra 20-22, 23, 26 , proper analysis of the strain-induced zero field splittings 19 and estimation of the local distortions induced by the doping of the 3d n ion in certain host lattices 12,16 .

BB

  developed and are now actively being used worldwide by many researchers. All of them have a common starting point, namely the Hamiltonian of the ion under consideration, (as a rule, with d or f open shell; in what follows this ion will be referred to as a central ion) can be has dimensions of energy and stands for the crystal field parameters (CFP), and l is the orbital momentum quantum number of the unfilled shell electrons. The k p O operators act upon the angular parts of the central ion wave functions. The k p parameters contain all information about the crystal properties, such as geometry of the surrounding ions, chemical bond lengths and angles. They are also related to the electron density of the central ion. The structure of the Hamiltonian (1) or, in other words, the number of non-zero CFPs depends on the symmetry of the considered complex and increases by lowering the symmetry.

A c c e p t e d m a n u s c r i p t 3 2.

 3 Superposition model and exchange charge model of crystal field.

A c c e p t e d m a n u s c r i p t 4 Levin

 4 where the power-law exponents t k > 0 (k = 2, 4 for 3d ions and 2, 4, 6 for 4f ions) are treated as adjustable parameters and their values are usually greater than the values of k + 1 (as predicted by the point charge electrostatic model) because of the dominant (short-range) overlap and covalency contribution to the crystal field in the transition-metal ions. However, and Cherepanov 18 proposed an alternative two-power law formula for the distance

BA c c e p t e d m a n u s c r i p t 5 are

 5 in a crystal field (CF) of an arbitrary symmetry are represented by the eigenvalues of crystal field Hamiltonian from Eq. (1). 8 k p O are the suitably chosen linear combinations of the irreducible tensor operators acting on the angular parts of the 3d ion wave functions, and k p the crystal field parameters (CFP) containing all information about geometrical structure of an impurity center. The basic assumption of the ECM is that these parameters can be written as a sum of two terms 8 :

The averaged values p rA c c e p t e d m a n u s c r i p t 6 found in Ref. 8 .

 p68 of p-th power of the radial coordinate of electrons from the impurity ion unfilled shell are given in Ref. 27, for example, or can be easily calculated numerically using the corresponding wave functions. The values of the numerical factors p k p K , , expressions for the polynomials k p V and definitions of the operators k p O can be The overlap integrals between the d-functions of the central ion and pand sfunctions of the ligands are denoted by ) s , G , and G are dimensionless adjustable parameters of the model, which are determined from the positions of the first three absorption bands. For the sake of simplicity, they can be assumed to be equal to each other:

  All of the four crystals considered in the present work (Li 2 Co 3 (SeO 3 ) 4 , CoSO 4 •H 2 O, CoSeO 4 •H 2 O, Co(OH) 2 ) contain Co 2+ ion as a part of crystal lattice. Such a choice of crystals eliminates all phenomena related to the lattice distortions after doping and allows for a direct use of the crystal structure data without introducing any corrections to the ionic positions. Three crystals (Li 2 Co 3 (SeO 3 ) 4 , CoSO 4 •H 2 O, CoSeO 4 •H 2 O) have monoclinic crystal structures, whereas the remaining one (Co(OH) 2 )) has a trigonal structure. A brief summary of crystal structure data is given below in Table

A c c e p t e d m a n u s c r i p t 9 Racah

 9 schemes are available from the authors upon request). Since the reported absorption bands are broad, we have fitted positions of the calculated barycenters of groups of energy levels arising from the orbital triplets to the barycenters of the observed bands.

  calculations based on the exchange charge model. (b) Wildner's calculations 25 refer to the superposition model analysis with R o = 2.1115Å. (c) Superposition model fitting of present calculated values of CFP as given in column (a).

A c c e p t e d m a n u s c r i p t 13

 13

A c c e p t e d m a n u s c r i p t 17 calculated

 17 from the present ECM approach for most systems are more or less consistent with those of Wildner and Andrut's fitted results.

- 1 .A c c e p t e d m a n u s c r i p t 18 significant

 118 The results in column (c) reveal that the SM fitting to the presently calculated rank 4 CFP in column (a) through the use of Eqs.(9-10), and Figures1-2yielded two sets of highly consistent values of the SM parameters 4 B and t 4 for the two sites I and II (with less than 1% discrepancy). The SM fitted values of individual B 4p parameters in column (c) also match well with those given in column (a), verifying the goodness of the superposition model in describing the angular distribution of the CFP. Although the SM intrinsic parameters 4 B from Wildner and Andrut's work (columns (b)) do not differ much for our present SM fitted values in columns (c), yet our fitted value of the power-law exponent t 4 is twice of their value for each of the two sites. On the other hand, there are much greater discrepancies found for the rank 2 CFP when we compare the results across different methods of calculations and across the two sites. The most probable reason is that the various contributions to the rank 2 CFP tend to cancel each other and they are very sensitive to the changes in the polar coordinates of the nearest neighbour ligands. Our fitted values of the rank 2 SM intrinsic parameter 2 B is about twice of Wildner and Andrut's work, and this difference is very significant, so we shall later make justification for the reliability of our values. For the CFP of Co 2+ located in the two kieserite-type compounds called CoSO 4 •H 2 O and CoSeO 4 •H 2 O, Table 6 shows that the SM fits (columns (c)) to the presently calculated CFP work quite well (especially for the rank 4 parameters). The fitted values of the rank 2 and 4 SM parameters match very well between the two kinds of crystal and it implies that the SM parameters are transferable between these two similar crystal systems. Since Wildner's 25 CF analysis had employed only three tetragonal CFP, namely Dq, Dt and Ds which are equivalent to Wybourne's B 20 , B 40 and ReB 44 , his results should be considered as primitive findings from a pseudotetragonal approximation 25 . However, the overall CF strength of his rank 4 CFP (as measured by the rotational invariant s 4 ) agrees very well with our present calculations (with about 5% discrepancy) while that for the rank 2 CFP (measured by s 2 ) indicates a very difference of about 4 times. The most likely reason is because the 6 nearest oxygen ligands form a CoO 6 complex with an approximate octahedral site-symmetry which gives rise mostly to the rank 4 CF. The remaining rank 2 CF, which arises from the pseudotetragonal elongation of the CoO 6 octahedra, is very sensitive to the value of the power-law exponent t 2 because of the cancellation between contributions from individual ligands.

A c c e p t e d m a n u s c r i p t 20 Finally,

 20 Figures 1-4 show the distance dependencies of the rotational invariants calculated in the present paper (marked by various symbols). The curves, which are calculated from the aforementioned fitting procedures through the use of Eqs. (9) and(10), clearly reveal that both the short-range exchange contribution S , k B and the overall SM intrinsic parameters k B do closely follow the power-law form of distance dependence. Even though both the short-range exchange contribution and the long-range electrostatic contribution follow the power-law form of distance dependence (with different values for t k ), the goodness of single power-law fits shown in Figures 1 and 2 confirms Newman and Ng's 7 comment of no obvious phenomenological advantage to apply the alternative two-power law formula for the distance dependence of ) R ( B k

A c c e p t e d m a n u s c r i p t 21 The

 21 numerical values of SM parameters k B and t k for Co 2+ in the six oxide crystals of Li 2 Co 3 (SeO 3 ) 4 , CoSO 4 •H 2 O, CoSeO 4 •H 2 O, and Co(OH) 2 have been found to be around 13,000 -16,000 cm -1 for 2 B , 4,100 -5,700 cm -1 for 4 B , 4.1 -5.0 for t 2 and 6.2 -6.5 for t 4 . While the values of the rank 4 SM intrinsic parameter 4 B are in agreement with those found in the literature, a substantial difference is found for the rank 2 one. Our present results should be more reliable because the power-law exponents t k have been properly determined (instead of using the electrostatic value of k+1), and there is additional no uncertainty about the ligand polar positions as arisen from the doping of a transition-metal ion in a charge mismatch host which will then induce local distortions around that impurity ion. In summary, the significance of the present work is three-folded: (1) The SM is confirmed by the ECM calculations to be very good for empirical modeling of the CF parameters in the 4 oxide crystal lattices. (2) A simple method has been devised to determine the SM intrinsic parameters and power-law exponents from the ECM calculations in low-symmetry CF where many coordinated dependent CF parameters (up to 14 in total for each 3d ion) are notoriously difficult for processing and for comparison across different systems. (3) It is the first time that a reliable range of values for SM intrinsic parameters and power-law exponents are determined for Co 2+ in pure undistorted host lattice of 4 different oxide crystals. Our present ranges of values for the SM parameters k B and t k will provide some reliable information and very useful initial guidance in future use of the superposition model for many different

Figure 1 :A c c e p t e d m a n u s c r i p t 24 Figure 2 :

 1242 Figure 1: distance dependence of the rank 2 rotational invariants s 2 for Co 2+ in 6 oxide crystal systems.

Figure 3 :

 3 Figure 3: distance dependence of the short-range exchange component of the rank 2 rotational invariants for Co 2+ in 6 oxide crystal systems.

Figure 4 :A c c e p t e d m a n u s c r i p t 25 Figure 1

 4251 Figure 4: distance dependence of the short-range exchange component of the rank 4 rotational invariants for Co 2+ in 6 oxide crystal systems.

Table 1 .

 1 923 Co 2+ , 11,856 H -, 29,670 O 2- and 5,936 S 6+ ions at the cut-off radial distance of 49.999Å) and 49,109 ions (5,461 Co 2+ , , 27,288 O 2-and 5,470 Se 6+ ions at the cut-off radial distance of 49.999Å) for the CoSO 4 •H 2 O and CoSeO 4 •H 2 O, respectively. For the brucite system, the cluster consists of 46,561 ions (9,271 Co 2+ , 18,642 H -and 18,648 O 2-ions at the cut-off radial distance of 44,993Å) and 64,803 ions (12,957 Co 2+ , 25,932 H -and 25,914 O 2-ions at the cut-off radial distance of 49.995Å) for the temperatures 270K and 90K, respectively. There are more ions included in the cluster for the temperature of 90K system than that of the 270K system because the cut-off radial distances for those two cases are different in order to maintain the site-symmetry of the CF experienced by the central Co 2+ ion. On the other hand, the summation over i in Eq.(7) for the short-range exchange contribution merely runs through the 6 nearest oxygen ligands in all of those oxide crystal lattices in this study because the overlap integrals with the ions from the second and further coordination spheres are negligible. Crystal structure data for Li 2 Co 3 (SeO 3 ) 4 , CoSO 4 •H 2 O, CoSeO 4 •H 2 O, Co(OH) 2

	10,890 H
	A c c e p t e d	m a n u s c r i p t
		7

-

[*] X-ray data from private communication with Prof. Manfred

Wildner (2007)

.

Table 2 .

 2 Calculated energy levels (in cm -1 ) of Co 2+ in sites I and II of Li 2 Co 3 (SeO 3 ) 4 with the Racah parameters B = 805 cm -1 and C = 3630 cm -1 .

		Site I			Site II
		Calc. (this work)	Calc. 23	Calc. (this work)	Calc. 23	Exp. 23
		0	0	0	0
		641	520	1595	857	0
		1454	788	2020	1657
	4 T 2g	5942 6940 7925	6982 7600 7778	4513 7055 7775	6605 7201 7617	5850 7550
	4 A 2g	13372	14818	12058	13848	13800
		16926	17768	15819	16025	16030
	4 T 1g ( 4 P)	18553	18676	17950	18536	18700
		21952	20297	22460	20905	20750

Table 3 .

 3 Calculated energy levels (in cm -1 ) of Co 2+ in CoSO 4 •H 2 O (with B = 856 cm -1 and C = 3580 cm -1 ) and CoSeO 4 •H 2 O (with B = 841 cm -1 and C = 3490 cm -1 )

	CoSO 4 •H 2 O	CoSeO 4 •H 2 O
	Calc. (this
	work)

Table 4 .

 4 Calculated energy levels (in cm -1 ) of Co 2+ in Co(OH) 2 at the temperature T of 290 K and 90 K. See Table7for values of the free-ion parameters B, C and used in all calculations.The calculated values of CFP and the rotational invariant[START_REF] Yeung | [END_REF] s k in Wybourne convention B kp for the Co 2+ ion in the aforementioned four oxide crystals are presented in Tables5, 6 and 7

	4 F)	0 1148 1407	0 618	-	0 1511 1744	0 730	-
	4 T 2g	7029 8397 8739	7182 7497	7225 7750	6952 8233 9195	7140 7477	7200 7850
	4 A 2g	15301	15444	15300	14681	15311	15100
	4 T 1g ( 4 P)	19116 20591 24350	18656 20439	18450 18625 19000 20500	19655 21185 25416	18338 20406	18220 18240 18690 20270
			T = 290 K			T = 90 K
		Calc. (this work)	Calc. 26	Exp. 26 Calc. (this work)	Calc. 26	Exp. 26
	4 T 1g ( 4 F)	0 218	0	0	0 28	0	0
	4 T 2g	7852 9767	8124 8518	8190	7791 9499	8425 8797	8350
	4 A 2g	15991	15965	15850	15817	16623	16900
	4 T 1g ( 4 P)	18822 25540	18720 21454	18800 21450	18680 25099	19057 21562	21600

Table 5 :

 5 Comparison of calculated crystal field parameters (in Wybourne notation) and SM parameters for the Co 2+ located at sites I and II in Li 2 Co 3 (SeO 3 ) 4 . All parameters except t 2 and t 4 (which are dimensionless) are expressed in units of cm -1 .

	Parameter	(a)	Site I (b)	(c)	(a)	Site II (b)	(c)
	ImB 22	-478.47	484.4	763.0	-4957.8	-3851.52	-6128.2
	ImB 21	5523.60	2435.9	4246.8	-3513.3	-2071.61	-3136.8
	B 20	1914.00	2900.2	5735.1	3133.6	1171.1	2188.0
	ReB 21	2860.19	666.3	2004.1	76.8	283.45	-8.5
	ReB 22	879.37	846.1	1562.4	2819.9	-185.85	41.4
	s 2	4075.52	2147.90	4075	4462.7	2823.22	4463
	2 B		7,000	14,650		7,000	13,440
	t 2		5.5	4.07		5.5	4.37
	ImB 44	2727.99	2929.3	2542.2	3170.6	4419.35	3604.1
	ImB 43	3947.88	4516.9	4217.7	-3773.3	-4071.18	-3572.9
	ImB 42	-9294.57	-10311.6	-9229.2	-4747.1	-5496.25	-4538.9
	ImB 41	1479.38	1066.9	1025.2	6973.6	8058.03	7030.2
	B 40	-3848.00	-4647.4	-3995.2	-597.6	403.39	-273.4
	ReB 41	-5832.56	-6479.4	-5940.3	-2247.3	-2281	-2209.1
	ReB 42	-342.79	-791	-571.5	-3754.3	-4392.62	-3652.6
	ReB 43	-3675.41	-4253.2	-3508.4	-3082.8	-3030.21	-3188.7
	ReB 44	-5228.41	-5909	-5256.4	-4231.5	-4909.03	-4240.5
	s 4	6565.73	7346.41	6566	5621.2	6482.56	5621
	4 B		4740	4,092		4,740	4,060
	t 4		3.1	6.20		3.1	6.21
	Remark:
	(a) Present calculations based on the exchange charge model.
	(b) Wildner & Andrut's calculations 23 refer to the superposition model analysis with R o =
	2.1115Å and their CFP have been transformed into the present coordinate frame.
	(c) Superposition model fitting of present calculated values of CFP as given in column (a).

Table 6 :

 6 Comparison of calculated crystal field parameters (in Wybourne notation) and SM parameters for the Co 2+ located in CoSO 4 •H 2 O and CoSeO 4 •H 2 O. All parameters except t 2 and t 4 (which are dimensionless) are expressed in units of cm -1 .

	Parameter	(a)	CoSO 4 •H 2 O (b)	(c)	(a)	CoSeO 4 •H 2 O (b)	(c)
	ImB 22	3804	[0]	2401	5111	[0]	3655
	ImB 21	-4268	[0]	-4210	-4405	[0]	-4349
	B 20	-4349	-2450	-5709	-6395	-2842	-8378
	ReB 21	-2465	[0]	-3465	-2396	[0]	-3128
	ReB 22	2197	[0]	1120	3471	[0]	2574
	s 2	4606	1096	4606	5789	1271	5789
	2 B			15357			15679
	t 2			4.54			4.49
	ImB 44	-4105	[0]	-3951	-5556	[0]	-5336
	ImB 43	8708	[0]	8807	8022	[0]	8241
	ImB 42	4744	[0]	4598	5292	[0]	5143
	ImB 41	4524	[0]	4513	4449	[0]	4505
	B 40	-4856	16506	-5438	-3266	16233	-3899
	ReB 41	6278	[0]	6117	6266	[0]	6132
	ReB 42	-4010	[0]	-4095	-4654	[0]	-4743
	ReB 43	-4458	[0]	-4894	-2690	[0]	-3148
	ReB 44	6595	10366	6220	6465	10253	6130
	s 4	7693	7359	7693	7577	7255	7577
	4 B			4719			4764
	t 4			6.35			6.35
	Remark:

Table 7 :

 7 Comparison of calculated crystal field parameters (in Wybourne notation) and SM parameters for the Co 2+ located at the trigonally compressed octahedral site of Co(OH) 2 with temperatures of 290K and 90K. The SM parameters refer to the reference ligand distance R o = 2.1115Å. All parameters except t 2 and t 4 (which are dimensionless) are expressed in units of cm -1 .

		Andrut & Wildner's SM analysis 26	Present calculations
	Parameter	Temp=290K	Temp=90K	Temp=290K	Temp=90K
	B	825	830	825	830
	C	3550	3500	3550	3500
	B 20	-3931	-3274	-11673	-11650
	B 40	-8739	-8860	-9433	-9808
	B 43	15010	15512	16234	17296
	2 B	4920	3900	14650	13127
	t 2	[3]	[3]	4.79	4.98
	4 B	5260	5320	5704	5531
	t 4	[5]	[5]	6.51	6.51

Table 8 (

 8 a): Polar coordinates for the 6 nearest oxygen ligands of Co 2+ as located at the distorted octahedral sites I and II in Li 2 Co 3 (SeO 3 ) 4

		Site I (C i symmetry)			Site II (C 1 symmetry)
	R/Å	/°	/°	R/Å	/°

Table 9 :

 9 Comparison of the superposition model intrinsic parameters k B (in cm -1 ) and the power-law exponents t k for the Co 2+ located at different oxide crystals. All the SM intrinsic parameters refer to the same reference ligand distance R o = 2.1115Å. Values for the shortrange exchange component of

				k B	S ,	and t k,S are given in brackets.
	SM	Li 2 Co 3 (SeO 3 ) 4		Kieserite	Co(OH) 2
	Parameters	Site I	Site II		CoSO 4 •H 2 O CoSeO 4 •H 2 O Temp=290 K Temp=90 K
	2 B	14,650	13,440	15,360	15,680	14,650	13,130
	B 2	S ,	(4,530)	(5,710)	(6,790)	(6,440)	(5,820)	(5,780)
	t 2	4.07	4.37		4.54	4.49	4.79	4.98
	t 2,S	(7.02)	(6.72)		(6.79)	(6.88)	(7.51)	(7.48)
	4 B	4,090	4,060		4,720	4,760	5,700	5,530
	B	S ,

Table 7

 7 reports the values of free ion parameters B, C and used in all our present calculation of the energy levels for Co 2+ in Co(OH) 2 crystal as given in Tables 2. Besides, there are calculated values of CFP and fitted values of SM parameters from our present work and Andrut and Wildner's 26 work for Co 2+ located at the trigonally compressed octahedral site of Co(OH) 2 with temperatures of 290K and 90K. While Andrut and Wildner fixed the powerlaw exponent t k at their electrostatic values of 3 and 5 for rank 2 and 4 crystal field, respectively, our present findings of t 2 = 4.79 to 4.98 and t 4 = 6.41 indicate that the electrostatic model is not a very good approximation. On the other hand, the values of the rank

	4 SM intrinsic parameter 4 B from their work and our present work do agree fairly well (with
	about 10% discrepancy), but the rank 2 SM intrinsic parameter 2 B is very significantly
	different, by nearly 3 times. From Table 8(c), we note that both the radial distance R and
	azimuthal angle of all the 6 ligands follow the perfect octahedral symmetry, but the polar
	angle deviates slightly from the perfect octahedral values of	cos 1	1 (	/	3	)	= 54.74 o and
	125.26 o . Since the non-cubic configuration of the nearest neighbour oxygen ions will give rise

T 1g ( 4 F)

Acknowledgement

Prof. Manfred Wildner is thanked for providing some of his unpublished X-ray data for Co(OH) 2 at 90K.

A c c e p t e d m a n u s c r i p t 19

This Table 9 shows that while the short-range exchange contribution S , k B (described by the second term in Eq. ( 5)) account for about 60% of the rank 4 CF, they are significantly less important in the rank 2 CF, as the latter is predominated by the electrostatic contribution q , k B which can be taken as the difference between k B and S , k B . This way of analysis is also consistent with findings given in the last two rows of Table 7 B also closely follow the power-law form of distance dependence as shown in Figures 3 and4.

The corresponding values of the power-law exponent are lying between 6.7 to 7.5 for both rank 2 and rank 4 CF as given in Table 9. Furthermore, our present sets of k B values for Co 2+ actually are highly consistent with the values of 2 B = 12,600 ± 1,600 cm -1 (assuming t 2 = 3.5 ± 0.5) and 4 B = 4,770 cm -1 as obtained by Newman et al 14 in their SM analysis of Fe 2+ in a garnet host with oxygen ligands. Our present results also match well with the value of 2 B = 14,100 cm -1 (with t 2 = 3) but are significantly lower than the value of 4 B = 7,900 cm -1 (with t 4 = 8.1) of which they were derived by Chang et al 22 from the direct fitting of the SM intrinsic parameters to the energy level splittings for the two low-lying states (i.e. [START_REF] Wybourne | Spectroscopic Properties of Rare Earth Ions[END_REF] A 2 and 2 E) of Cr 3+ ions for Nb 5+ ions.