Structural, vibrational and thermodynamics properties of Zn-based semiconductors Y. Yu a , H.L. Han b, * , M.J. Wan a , T. Cai a , T. Gao a, ** a Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China b College of Mathematics, Sichuan University, Chengdu 610064, China article info Article history: Received 19 February 2009 Received in revised form 31 March 2009 Accepted 3 April 2009 Available online 18 April 2009 Keywords: Structural Vibrational Thermodynamics DFPT abstract We have preformed first-principle calculations for the structural, vibrational and thermodynamic prop- erties of the IIB–VIA Zn-based semiconductor compounds ZnX (X ¼ O, S, Se, Te). The phonon dispersion curves along several high-symmetry lines at the Brillouin zone together with the corresponding phonon density of states are calculated using density-functional perturbation theory. The calculated phonon frequencies at the G, X, and L points of the Brillouin zone show good agreement with the experimental values and other calculations. The thermodynamics properties including the phonon contribution to the Helmholtz free energy DF, the phonon contribution to the internal energy DE, the entropy S, and the constant-volume specific heat C V are determined within the harmonic approximation based on the calculated phonon dispersion relations. If 298 K is taken as a reference temperature, the difference values of H H 298 have been also calculated and compared with the available experimental data. Crown Copyright Ó 2009 Published by Elsevier Masson SAS. All rights reserved. 1. Introduction Wide-band-gap semiconductors are very important for appli- cations in optical devices as visual displays, high-density optical memories, transparent conductors, solid-state laser devices, photodetectors, solar cells, etc. Among the wide-band-gap semi- conductors, ZnX (X ¼ O, S, Se, Te) constitute a family of IIB–VIA compounds, crystallizing in the cubic zinc-blende structure at ambient pressure. In this work we will focus on the structural, vibrational and thermodynamic properties of ZnX. Nowadays the equilibrium structure of a large class of materials can be determined within first-principle methods based on density-functional theory (DFT) [1–3]. Furthermore, different approaches have been used to calculate thermal properties from first-principles. The thermal properties are one of the most basic properties of any material. Thermal properties of solids depend on their lattice dynamical behaviour. A simplified method for calculating the thermal properties of metals and compounds is a Debye–Gru ¨ neisen based model [4]. A more accurate approach is first-principle molecular dynamics simulation, but the ionic degrees of freedom are treated classically, so the simulations are not valid at temperatures comparable to or lower than Debye temperature. Another approach had been made possible by the achievements of density-functional perturbation theory (DFPT) [5,6], which allowed exact calculations of vibrational frequencies in each point of the Brillouin Zone (BZ). In the last decade, the implementation of perturbation theory into a density-functional framework has permitted the ab initio calculation of phonon dispersion of semiconductors [7–12] and metals [13,14]. The success of these calculations has also driven increasing interest into lattice properties that cannot be described by the harmonic approximation as e.g., the phonon lifetime [15], the thermal expansion of a solid [9,16,17] or the shift [18] of phonon frequencies when the temperature is changed. The main aim of this work is to provide a thorough DFPT investigation of the thermodynamical properties of ZnX at 0 K and high temperature. The phonon frequencies and their derivatives were computed at zero temperature and the phonon contribution to the Helmholtz free energy DF, the phonon contribution to the internal energy DE, the entropy S, and the constant-volume specific heat C V are calculated by varying only the temperature of the thermal occupation number in the corresponding formula. The knowledge of various prominent features of phonon density of states (PDOS) spectrum of solid is important for a general theo- retical understanding and detailed quantitative description of its thermal as well as optical properties. The PDOS, which includes contributions from all phonons over the entire Brillouin zone, is needed for the calculations of various thermodynamic character- istics, e.g. heat capacity and thermal conductivity. * Corresponding author. Tel./fax: þ86 28 85405234. ** Corresponding author. E-mail addresses: [email protected](H.L. Han), [email protected](T. Gao). Contents lists available at ScienceDirect Solid State Sciences journal homepage: www.elsevier.com/locate/ssscie 1293-2558/$ – see front matter Crown Copyright Ó 2009 Published by Elsevier Masson SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2009.04.007 Solid State Sciences 11 (2009) 1343–1349
Transcript
lable at ScienceDirect
Solid State Sciences 11 (2009) 1343–1349
Contents lists avai
Solid State Sciences
journal homepage: www.elsevier .com/locate/ssscie
Structural, vibrational and thermodynamics propertiesof Zn-based semiconductors
Y. Yu a, H.L. Han b,*, M.J. Wan a, T. Cai a, T. Gao a,**
a Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, Chinab College of Mathematics, Sichuan University, Chengdu 610064, China
a r t i c l e i n f o
Article history:Received 19 February 2009Received in revised form31 March 2009Accepted 3 April 2009Available online 18 April 2009
1293-2558/$ – see front matter Crown Copyright � 2doi:10.1016/j.solidstatesciences.2009.04.007
a b s t r a c t
We have preformed first-principle calculations for the structural, vibrational and thermodynamic prop-erties of the IIB–VIA Zn-based semiconductor compounds ZnX (X¼O, S, Se, Te). The phonon dispersioncurves along several high-symmetry lines at the Brillouin zone together with the corresponding phonondensity of states are calculated using density-functional perturbation theory. The calculated phononfrequencies at the G, X, and L points of the Brillouin zone show good agreement with the experimentalvalues and other calculations. The thermodynamics properties including the phonon contribution to theHelmholtz free energy DF, the phonon contribution to the internal energy DE, the entropy S, andthe constant-volume specific heat CV are determined within the harmonic approximation based on thecalculated phonon dispersion relations. If 298 K is taken as a reference temperature, the difference valuesof H�H298 have been also calculated and compared with the available experimental data.
Crown Copyright � 2009 Published by Elsevier Masson SAS. All rights reserved.
1. Introduction
Wide-band-gap semiconductors are very important for appli-cations in optical devices as visual displays, high-density opticalmemories, transparent conductors, solid-state laser devices,photodetectors, solar cells, etc. Among the wide-band-gap semi-conductors, ZnX (X¼O, S, Se, Te) constitute a family of IIB–VIAcompounds, crystallizing in the cubic zinc-blende structure atambient pressure. In this work we will focus on the structural,vibrational and thermodynamic properties of ZnX.
Nowadays the equilibrium structure of a large class of materialscan be determined within first-principle methods based ondensity-functional theory (DFT) [1–3]. Furthermore, differentapproaches have been used to calculate thermal properties fromfirst-principles. The thermal properties are one of the most basicproperties of any material. Thermal properties of solids depend ontheir lattice dynamical behaviour. A simplified method forcalculating the thermal properties of metals and compounds isa Debye–Gruneisen based model [4]. A more accurate approach isfirst-principle molecular dynamics simulation, but the ionicdegrees of freedom are treated classically, so the simulations arenot valid at temperatures comparable to or lower than Debye
temperature. Another approach had been made possible by theachievements of density-functional perturbation theory (DFPT)[5,6], which allowed exact calculations of vibrational frequencies ineach point of the Brillouin Zone (BZ). In the last decade, theimplementation of perturbation theory into a density-functionalframework has permitted the ab initio calculation of phonondispersion of semiconductors [7–12] and metals [13,14]. Thesuccess of these calculations has also driven increasing interest intolattice properties that cannot be described by the harmonicapproximation as e.g., the phonon lifetime [15], the thermalexpansion of a solid [9,16,17] or the shift [18] of phonon frequencieswhen the temperature is changed.
The main aim of this work is to provide a thorough DFPTinvestigation of the thermodynamical properties of ZnX at 0 K andhigh temperature. The phonon frequencies and their derivativeswere computed at zero temperature and the phonon contributionto the Helmholtz free energy DF, the phonon contribution to theinternal energy DE, the entropy S, and the constant-volume specificheat CV are calculated by varying only the temperature of thethermal occupation number in the corresponding formula. Theknowledge of various prominent features of phonon density ofstates (PDOS) spectrum of solid is important for a general theo-retical understanding and detailed quantitative description of itsthermal as well as optical properties. The PDOS, which includescontributions from all phonons over the entire Brillouin zone, isneeded for the calculations of various thermodynamic character-istics, e.g. heat capacity and thermal conductivity.
Fig. 1. Crystal structure of the zinc-blende ZnX (X¼O, S, Se, Te). The gray ball is Znatom and the yellow ball is X atom.
Y. Yu et al. / Solid State Sciences 11 (2009) 1343–13491344
Several calculations have been performed to study thedynamical and thermodynamical properties of ZnX. Agrawal et al.[19] used the full-potential self-consistent linearized muffin-tinorbital (LMTO) method for calculating the dynamical properties ofZn-based semiconductors. Serrano et al. [8] have calculated thelattice dynamics for ZnO using first-principles based on density-functional theory (DFT) and the local-density approximation(LDA) for the exchange-correlation (XC) potential. Hamdi andAouissi [9] have performed first-principle calculations based onDFPT to study the vibrational and thermal properties of ZnSe, atzero and high pressures. The values of Debye temperature and thespecific heat capacity at different pressures and differenttemperatures are calculated using the plane-wave pseudopoten-tial DFT by Hu et al. [20]. Even though a number of calculationsfor the dynamical and thermodynamical properties of ZnX usingdifferent methods exist, few DFPT calculations exist for thesecompounds. Our calculations are performed by using a pseudo-potential plane-wave approach (PPPW) and the generalizedgradient approximation (GGA) for the exchange–correlationpotential. The thermodynamical properties are obtained withinthe harmonic approximation.
This paper is organized as follows. In Section 2, a short outlineof the used theoretical approach and computational details isgiven. In Section 3, our results are presented and discussed incomparison with the available experimental data and othertheoretical results. Finally, the main results and conclusions aresummarized in Section 4.
Table 1Calculated and experimental lattice parameters a (in Å) and volumes V (Å3) for ZnX(X¼O, S, Se, Te).
Our calculations reported here were performed with the ABINITcode [21,22], which is based on first-principle PPPW in the frame-work of the DFT. It relies on an efficient fast Fourier transformalgorithm [23] for the conversion of wave functions between realand reciprocal spaces, on an adaptation to a fixed potential of theband-by-band conjugate-gradient algorithm [24], and on a poten-tial-based conjugate-gradient method [25] for the determination ofthe self-consistent potential. The exchange–correlation term hasbeen determined within the generalized gradient approximation(GGA) parameterized by Perdew–Burke–Ernzerhof [26]. Weemployed norm-conserving pseudopotentials to treat the interac-tion between valence electrons and ions. The pseudopotentialswere generated by adopting the scheme described by Troullier andMartins [27]. The wave functions are expanded in plane waves up toa kinetic energy cutoff of 30 hartree. Integrals over the Brillouinzone (BZ) are replaced by a sum on a Monkhorst-pack grid of4� 4� 4 special k-point. Convergence tests show that the BZsampling and the kinetic energy cutoff are sufficient to guaranteean excellent convergence within 1 cm�1 for the calculated phononfrequencies.
The phonon frequencies were subsequently obtained using thelinear-response method, which avoids the use of supercells andallows the calculation of the dynamical matrix at arbitrary qvectors. Since ZnX (X¼O, S, Se, Te) are polar materials, themacroscopic electric field, caused by the long-rang character of theCoulomb forces, contributes to the longitudinal optical phonons inthe long-wavelength (q / 0) limit. This effect is included bycalculating the nonanalytical part of the force constants. The forceconstants were extracted from a Fourier transform of the dynamicalmatrix obtained for uniform grid in the first BZ. These were lateremployed to obtain the phonon frequencies at arbitrary points inreciprocal space and the phonon dispersion relations. Phonondensities of states for ZnX are plotted. The thermodynamic func-tions of ZnX can be determined by the entire phonon spectrum.
3. Results and discussion
3.1. Structural properties
The zinc-blende ZnX (X¼O, S, Se, Te) have a cubic symmetry asshown in Fig. 1 and belong to the space group F-43M. The primitivecell consists of two atoms. The Zn atom is located in the 4a Wyckoffsite (0, 0, 0) and the X atom occupies the 4c (0.25, 0.25, 0.25) site.We first determine the equilibrium volume of the ground state ofthe zinc-blende phase of ZnX by calculating the total energy perprimitive unit cell as a function of V. Because these compoundshave very high symmetry, we have only optimized the volumekeeping the angles fixed and not optimized the atomic position. TheMurnaghan’s equation of state [35] is then used to fit the calculatedenergy-volume data.
Table 1 displays the obtained lattice parameters compared toother calculations and experimental data reported in the literature.For ZnO, the energy minimum appears at a value of latticeparameter (a) equal to 4.627, which is very close to the experi-mental value of 4.62 Å. On the other hand, for ZnS, ZnSe and ZnTe,
Fig. 2. Phonon frequencies of ZnX (X¼O, S, Se, Te), at 0 K and zero pressures.
Y. Yu et al. / Solid State Sciences 11 (2009) 1343–1349 1345
the minimum energy appears at a¼ 5.395, 5.738 and 6.144 Å,which are also very near to the experimental values of 5.409, 5.662(5.668) and 6.101 Å, respectively. Our calculations overestimate theequilibrium lattice parameter with the maximal error of 1.34% withrespect to experimental values, a normal agreement by GGA stan-dards. This is largely sufficient to allow the further study ofdynamical and thermodynamic properties.
3.2. Vibrational properties
ZnX semiconductors belong to the cubic system, with pointgroup Td and space group F-43M (symmorphic group). According
to lattice vibration theory [36], vibration frequency u is a func-tion of wave vector q, which is described with the dispersionrelation: u¼uj(q). The subscript j is the branch index. A crystallattice with n atoms per unit cell has 3n branches, three of whichare acoustic and the remainder optical. The dispersion curveexhibits symmetry properties in q-space, which enables us torestrict consideration to the first Brillouin zone only. The latticevibration mode with q z 0 plays a dominant role for Ramanscattering and infrared absorption. For this reason, the vibrationfrequency with q¼ 0, i.e. at the center G point of the first Bril-louin zone, is called as normal vibration mode. There are 2 atomsin the primitive unit cell of ZnX crystal, so there are 6 dispersion
Fig. 3. Phonon density of states of ZnX (X¼O, S, Se, Te), at 0 K and zero pressures.
Y. Yu et al. / Solid State Sciences 11 (2009) 1343–13491346
curves, which means 6 normal vibration modes at the center G
point. Based on the factor group theory, the reducible represen-tations for the space group F-43M at the G point can be reducedas follows:
Wyckoff Position 4a : 0A1 þ 0A2 þ 0E þ 0T1 þ T2;
Wyckoff Position 4c : 0A1 þ 0A2 þ 0E þ 0T1 þ T2:
The irreducible representations of the lattice vibration in ZnX arethe following: 2T2.
Group theoretical analysis predicts the following irreduciblerepresentation for acoustical and optical zone center modes:Gaco¼T2 and Gopt¼ T2. T2 symmetry mode is triply degenerate, andis Raman active and also IR active. In particular, the dipole–dipolecontribution is found to be responsible for the splitting betweenthe longitudinal and transverse optic (LO and TO, respectively)modes T2.
The results for the phonon dispersion curves of ZnX are dis-played in Fig. 2 along several high-symmetry lines and the corre-sponding phonon density of states (DOS) are shown in Fig. 3.Despite the similarity, there are also several distinctions in thephonon dispersion curves of ZnX due to the different strengths ofthe elastic forces and the degree of the ionicity. For ZnSe, there is nogap between the acoustical and optical phonon branches in g(u),since there is a considerable overlap between the TO and longitu-dinal-acoustical (LA) phonon branches. The overlap is caused by thealmost identical masses of Zn and Se atoms. One of the two TOphonon dispersion curves shows a considerable flatness along all
high-symmetry directions which leads to a sharp peak in g(u). ForZnO, ZnS, and ZnTe, gaps exist between the acoustical and opticalphonon branches. As a result not only the acoustic and optical areseparated by a gap from 258 to 379 cm�1 for ZnO (from 213 to278 cm�1 for ZnS), but also the LO- and TO-phonon branches bya gap from 498 to 506 cm�1 for ZnO (from 318 to 325 cm�1 for ZnS).The acoustic and optical are separated by a gap from 138 to159 cm�1 for ZnTe, but there is not a considerable overlap betweenLO- and TO-phonon branches. The results for the phononfrequencies along several high-symmetry points (at G, X, and Lpoints) of ZnX are compared with the experimental values andother calculations in Table 2. For ZnO, to the best of our knowledge,the experimental values of phonon frequencies are not available inthe literatures, while the phonon frequencies of high-symmetrypoints (at G, X, and L points) are calculated by Serrano et al. [8]. Onthe whole, they are in perfect agreement. The calculated frequen-cies underestimate that of Serrano et al. by 4.7%. Differentexchange–correlation interactions were used may caused thedifferent results. For ZnS, the calculated frequencies uO
G, uTOX , uLA
X
and uTAX equal to 278, 315, 213 and 84 cm�1, respectively, which are
very near to the experimental values of 276, 316, 211, 90 cm�1 [40].For ZnTe, the calculated frequencies are also close to the experi-mental values from Reference [40] obtained by Neutron scattering.For ZnSe, our calculated phonon frequencies at zero P are in
excellent agreement with the experimental data, obtained using
INS [42]. The zone center LO phonon frequency uLOG is slightly
smaller than the Raman spectroscopy data [43]. However, the zone
center TO phonon frequency uTOG is in excellent agreement with the
Table 2Phonon frequencies (in cm�1) at certain high-symmetry point (G, X, and L points) of the Brillouin zone of ZnX (X¼O, S, Se, Te).
frequencies at zero P are also in excellent agreement with other
calculations using different methods in Table 2.
3.3. Thermal properties
The knowledge of the entire phonon spectrum granted by DFPTmakes possible the calculation of several important thermody-namical properties as functions of temperature T. The thermody-namical properties of a system are usually determined by theappropriate thermodynamic potential relevant to the givenensemble. In the ensemble where the sample volume (V) and T areindependent variables, the relevant potential is the Helmholtz freeenergy (F). In the adiabatic BO approximation, F of a semiconductorcan be written as
F ¼ E þ DF ¼ E þ DE � TS; (1)
where E is the static contribution to the internal energy, DF is the
Fig. 4. The phonon contribution to the Helmholtz free energies DF of ZnX (X¼O, S, Se, Te).
phonon contribution to the Helmholtz free energy, DE and S are thecontribution of the lattice vibration to the internal energy and entropy,respectively. The electronic entropy contribution to S, vanishes iden-tically for insulators, and thus, it is not included in Eq. (1). Even formetals this contribution is usually neglected, although it is easy tocalculate. Thus, the key quantity to calculate in order to have access tothe thermal properties and to phase stability is DF.
The DF is usually calculated within harmonic approximation:[45]
DF ¼ 3nNkBTZumax
0
ln�
2sin hZu
2kBT
�gðuÞdu; (2)
where kB is the Boltzmann constant, n is the number of atoms perunit cell, N is the number of unit cells, u is the phonon frequencies,umax is the largest phonon frequency, and g(u) is the normalizedphonon density of states with
Rumax0 gðuÞdu ¼ 1. In Figs. 4 and 5, we
have displayed the relations of the DF and DE as a function of the
Fig. 5. The phonon contribution to the internal energies DE of ZnX (X¼O, S, Se, Te).
Fig. 6. The constant-volume specific heats of ZnX (X¼O, S, Se, Te).
Table 4Thermodynamic functions for ZnS.
T (K) Calculated results Experimental resultsa
S (J/mol K) (H�H298) (kJ/mol) S (J/mol K) (H�H298) (kJ/mol)
Y. Yu et al. / Solid State Sciences 11 (2009) 1343–13491348
temperature T from 0 to 1000 K. DF and DE at zero temperaturerepresent the zero-point motion, which can be calculated from theexpression as DF0 ¼ DE0 ¼ 3nN
Rumax0 ðZu=2ÞgðuÞdu. For ZnO, the
calculated DF0¼DE0¼11.1 kJ/mol. On the other hand, the value is7.7 for ZnS, 5.3 for ZnSe, and 4.4 for ZnTe. The zero-point motioncontribution to the thermodynamic functions of ZnO is moreimportant than that of ZnS, ZnSe and ZnTe. The temperature-dependent DF and DE are higher for ZnO than ZnS, ZnSe and ZnTesince ZnO has much higher average phonon frequencies and lowerentropy.
Within harmonic approximation, CV is given as [45]
CV ¼ 3nNkB
Zumax
0
�Zu
2kBT
�2
csch2�
Zu
2kBT
�gðuÞdu (3)
The calculated heat capacity CV of ZnX, as a function of T is shown inFig. 6. ZnO is found to have lower CV than the other three
Table 3Thermodynamic functions for ZnO.
T (K) Calculated results Experimental resultsa
S (J/mol K) (H�H298) (kJ/mol) S (J/mol K) (H�H298) (kJ/mol)
semiconductors over the entire temperature range, due to itsdensity of states for low-frequency modes. In the low-temperaturelimit, the four specific heats exhibit the T3 power-law behaviour,and they all approach at high temperature the classical asymptoticlimit of CV¼ 3nkB¼ 50 J/mol K. Due to the thermal expansioncaused by anharmonicity effects CP is different from CV. The relationbetween CP and CV is determined by [46].
CP � CV ¼ a2V ðTÞB0VT; (4)
where aV is the volume thermal expansion coefficient, B0 is the bulkmodulus, V is the volume and T is absolute temperature. In order todeduce the theoretical Cp, the determination of thermal expansioncoefficients is necessary, which is in progress.
Entropy and enthalpy are two important conceptions in ther-modynamics. The entropy S is an extensive state function thataccounts for the effects of irreversibility in thermodynamicsystems. The change of entropy has often been defined as change toa more disorder state at a molecular level. Quantitatively, theentropy is defined by the formulas [45].
S¼3nNkB
Zumax
0
�Zu
2kBTcoth
Zu
2kBT�ln
�2sinh
Zu
2kBT
���gðuÞdu: (5)
Table 5Thermodynamic functions for ZnSe.
T (K) Calculated results Experimental resultsa
S (J/mol K) (H�H298) (kJ/mol) S (J/mol K) (H�H298) (kJ/mol)
Y. Yu et al. / Solid State Sciences 11 (2009) 1343–1349 1349
On the other hand, the enthalpy H is a description of potential ofa system, which can be used to calculate the ‘‘useful’’ workobtainable from a thermodynamic system under constant pressure.For a simple system, with a constant number of particles, thedifference in enthalpy is the maximum amount of thermal energyderivable from a thermodynamic process with a constant pressure.The calculated results S and H�H298 are displayed in Tables 3–6together with the corresponding experimental data [47]. Theoryand experiment show satisfactory agreement within the limitationof the ABINIT program and the harmonic approximation. From this,we have computed the entropy S and H�H298, over a wide range oftemperatures T from 298.15 to 2248 K for ZnO (from 298.15 to1995 K for ZnS, from 298.15 to 1300 K for ZnSe and ZnTe). ZnO isfound to have lower entropy than the other three semiconductorsover the entire temperature range; it is also due to its density ofstates for low-frequency modes. The enthalpy can be written as:
H ¼ E0 þ pV ¼ E þ DE þ pV : (6)
If 298 K is taken as a reference temperature, the difference values ofH�H298 can be calculated with the interval 100 K. For lowtemperatures, our harmonic approximation works best, and alsothe quality of the experimentally obtained difference values ofH�H298 is quite good and can thus be compared to our calculatedresults. Of course, the discrepancies between the calculated andexperimental specific heats become larger at high temperatures asthe lattice undergoes thermal expansion due to anharmonicinteractions.
4. Conclusions
To summarize, the lattice dynamics and thermodynamic prop-erties of ZnX (X¼O, S, Se, Te) using DFPT and pseudopotentialmethods are performed. For all semiconductors, the relaxed latticeparameters are found to be in good agreement with experimentalones with the errors less than 1.34%. This is largely sufficient toallow the further study of dynamical and thermodynamicproperties.
Based on the factor group theory, the reducible representationsfor the space group F-43M at the G point can be reduced. Theirreducible representation for acoustical and optical zone centermodes: Gaco¼T2 and Gopt¼ T2. We obtain the phonon frequenciesat the Brillouin zone center (G point), as well as X and L points, andthe phonon dispersion curves, as well as corresponding density ofstates. The calculated phonon frequencies are generally speaking invery good agreement with experimental data and other similartheoretical calculations.
The thermodynamics properties including the phonon contri-bution to the Helmholtz free energy DF, the phonon contribution tothe internal energy DE, the entropy S, and the constant-volumespecific heat CV are determined within the harmonic approxima-tion based on the calculated phonon dispersion relations. Theirdifference among the four semiconductor compounds can beunderstood in terms of the phonon density of states. If 298 K istaken as a reference temperature, the values of H�H298 have beencalculated and compared with the available experimental data.
Acknowledgments
The authors wish to thank Dr Y.N. Zhao and W.H. Xue forstimulating discussions and for their keen interest in this work. Ourthanks are also due to Dr J.J. Zhou for computation-practical help.
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