Electrostatic field in a pseudo-cubic ionic structure. Application to all BaTio3 structure-types

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Electrostatic field in a pseudo-cubic ionic structure.

Application to all BaTio3 structure-types

C. Tsallis, R. Machet, J. Bouillot

To cite this version:

C. Tsallis, R. Machet, J. Bouillot. Electrostatic field in a pseudo-cubic ionic structure. Ap- plication to all BaTio3 structure-types. Journal de Physique, 1971, 32 (2-3), pp.171-175.

�10.1051/jphys:01971003202-3017100�. �jpa-00207039�

ELECTROSTATIC FIELD IN A PSEUDO-CUBIC IONIC STRUCTURE.

APPLICATION TO ALL

BaTiO3

by ^{C. TSALLIS} (*)

Laboratoire de Magnétisme ^et Physique des Solides C. N. R. S. Bellevue/Meudon

R. MACHET and J. BOUILLOT

Laboratoire de Dielectriques-Ferroelectriques, Faculté des Sciences, Dijon (Reçu ^{le 17} septembre 1970)

Résumé. 2014 Description d’une méthode générale ^{de calcul du} champ électrostatique ^{au voisi-}

nage de chaque point ^de symétrie d’un sous-réseau cristallin pseudo-cubique quelconque ^{dont les}

sites sont occupés par des charges identiques. ^Une superposition de sous-réseaux de ce type peut être, pour certains calculs statiques ^et dynamiques (en particulier dans le cadre du modèle de

couches), ^une image utile d’un cristal ionique. ^Le champ ^est développé ^{au 3e ordre en fonction}

du déplacement (par rapport ^au point ^de symétrie voisin) ^{et de la} déformation du réseau (par rap- port ^{au réseau} cubique associé). ^Comme application, les coefficients du développement ^{sont cal-}

culés pour ^toutes les structures du type de celles du BaTiO3 (quadratique, orthorhombique ^et rhomboédrique).

Abstract. ²⁰¹⁴ A general ^{method is} given ^for calculating ^the electrostatic field in the neighborhood

of every symmetry point ^of any pseudo-cubic crystalline sublattice the sites of which are occupied by ^identical charges. ^A superposition ^{of such} sublattices _may be, ^{for some} statical and dynamical

calculations (particularly ^{in a} shell-model framework), ^{an useful} image ^{of an ionic} crystal. ^The

field is developed ^to the 3rd order in displacement (respect ^{to the} neighbouring symmetry point)

and lattice strain (respect ^to the associated cubic lattice). ^{As an} application, ^the development

coefficients are evaluated for all the BaTiO3 structure-types (tetragonal, orthorhombic and rhom-

bohedral).

LE JOURNAL PHYSIQUE TOME 32, FÉVRIER-MARS 1971,

Classification

physics ^Abstracts

16.35

1. Introduction. ^- Since Slatter’s paper [1] ^{it has}

been recorded [2 ^to 19] ^a regain of interest on

calculations of local electrostatic field because of their usefulness in quantitative ^{statical or} dynamical ^consi-

derations on ferroelectricity, piezoelectricity, ^electro- optics, ^{etc..., in} crystals having ^an important ^ionic

behaviour. Particularly ⁱⁿ any shell-model theory, ^the

local electrostatic field should be handled with preci-

sion. Present paper is ^an attempt ^to improve ^and

extend existing calculations by : a) taking ^{into account}

the lattice-strain contribution ; b) considering ^the

contribution of terms (in Taylor’s development ^{of the} field) higher ^{than those} corresponding ^{to a set of} point dipoles ^{located at the symmetry} points ^{of the} crystal ; c) considering ^{also other than} tetragonal phases. ^Point (a) ^has already ^been payed ^{much atten-}

tion [7 ^to 18] ^and recently [19] ^{we made a} calculation of tetragonal BaTi03 ^and PbTi03 birefringences

where importance ^of point (a) ^was confirmed and

importance ^of point (b) ^{recorded, even for a} simple

ionic model. Attention payed ^{here to} point (c) ^{is due}

(*) ^{Now at} the Service de Physique des Solides et de Réso-

nance Magnétique, ^{C. E. N.} Saclay, 91, Gif-sur-Yvette, ^France.

to the increasing ^{amount of} experimental information

on other than tetragonal phases. ^{In section 2 we}

describe a method for calculating ^the electrostatic field in the nieghborhood ^of every symmetry point ^{of any} pseudo-cubic crystalline sublattice whose sites are

occupied by ^identical point charges. This method fur- nish immediately, by adding ^{the fields on a} given

sublattice due to all the others, ^{the total} local field on

the given sublattice of _any pseudo-cubic ^ionic crystal decomposable ⁱⁿ sublattices located eachone near a

symmetry position of every single ^{one of} the others.

It is worth while to clear up that for evaluating ^each

sublattice contribution (to ^{the total field on the} given sublattice) its effective charge q ^{and actual} displace-

ment b ( * *) (respect ^to the associated actual symmetry point ^{of the} given sublattice) ^{must be used. In this}

section we describe also general characteristics of a

3rd order (in lattice strain and sublattice displacement development ^{of the field.}

In section 3 we apply ^{this method to all the} BaTi03

structure-types ^and evaluate all the corresponding.

development coefficients.

(* *) Actually 8 used in this paper is the dimensionless reduced displacement (expressed ^{in units of the} crystalline parameters)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01971003202-3017100

172

2. Field caIcuMon method. ^- 2. 1 CUBIC SUBLAT- TICE. ^- Let us consider a simple ^cubic sublattice with a

parameter ^{a* and a} charge q ^{on each site. The} posi-

tions of charges, ^{respect to a} given point ^{in the} neighborhood ^{of a} symmetry point (cube ^{corner, cube}

center, edge ^{center and face} center), ^are given by

where (l, ^m, n) ^are integral ^or half-integral ^numbers (this depends ^{on the} neighboring symmetry point), (i, j, k) ^{are the} unitary ^{vectors of the unit} cell, ^and (ôi, £52, d3) ^{are small relative to} unity. ^{So the field’s} z-component ^{at the} given point ^{is :}

where (l, ^m, n) ^{run over all the} sublattice sites (term

1 = m = n ⁼ 0 must obviously be excluded from this and similar sums). Because of symmetry, Ez ^vanishes

at every symmetry point (ô ⁼ 0). ^If Ez ^is developed ^to

the 3rd order in displacement, ^a dipole ^term (in ^the

form qb3) ^and octopole ^terms (in ^{the form} qb3 Ôk bk’

where k and k take in general ^{the values} 1, 2, 3) ^are

obtained. To calculate the development coefficient,

we divide, following Lorentz, the sublattice in two

regions by the condition 12 + m2 + n 2 N 2

(already

used in

[19])

is Na*, ^{where N is an} integer. ^The inside-dipoles

contribution Em z is treated microscopically ^{and the} outside-dipoles contribution

Et

it holds Ez = EmZ ⁺

Emz.

2.2 PSEUDO-CUBIC SUBLATTICE. ^- Let us assume now that the sublattice has ^a pseudo-cubic ^deforma-

tion. The new unit cell vectors (i’, j’, k’) ^{are related to}

the old ones by

where tij ^{« 1. The} positions ^of charges ^{are still} given by (1) ^where (i, j, k) ^{must be} replaced by (i’, j’, k’).

If we develop Ez ^{to the} third order in displacement ^and

strain we obtain

where the A, B, C, ^D are sums over 1, m, ^n. Ex ^and Ey

may be obtained from E , by cyclic permutation. ^The microscopic contributions A’", Bm‘, C’", ^{D’" to} precedent

coefficients _may be machine-computed, respecting

the condition l’ + m2 + n2 N2 (in present ^work N ⁼ 50 have been used for the tabulation), ^which

determines ^{now a} pseudo-spherical ellipsoïd. ^To

obtain the macroscopic contributions AM, BM, CM, ^DM

we must calculate the field at the center of an ellipsoïd

hollowed in a polarized homogeneous dielectric, ^the uniform polarization being ^{P =} qa* ôlv’, ^{where v’}

is the volume of the deformed unit cell and ô is given by ô, ^{i’ +} Ô2 j’ ⁺ b3 ^{k’. If} (R) is ^{the rotation matrix}

which diagonalizes ^the ellipsoid, ^the macroscopic

contribution to the field is given by ^a generalization

of the Lorentz relation :

where the elements n ii i of the diagonal ^matrix ( A )

are easily ^{obtained from} [19].

3. Application ^{to the} BaTio3 structure-types. ^-

3.1 TETRAGONAL. - Let us suppose the following

lattice strain

Then we obtain for the microscopic contribution

where the T are dimensionless sums over (l, ^m, n)

and are easily ^obtained from the K of [19]. ^Their

evaluations _{appear in} Table 1

((X,

TABLE 1. ^- Microscopic contribution’s coefficients (tetragonal case)

to the associated symmetry point coordinates in the unit

cell).

The x-and y-components ^{of Em and} EM may be obtained from Emz ^and

Et

structure we must obviously impose ti ⁼ t2. ^The strain parameters ^{are then related to the usual tetra-} gonal crystalline parameters ^{a and c} by

tl = t2 ⁼ a/a* - ^{1 and} t3 ⁼ c/a* - 1, where a* is the parameter ^{of an} arbitrary associated cubic lattice.

We may choose a* to be the parameter (obtained by ^thermal extrapolation) ^{the structure would have}

if it was still cubic at the temperature of interest.

Another possible choice is a* ⁼ a, hence ti ⁼ t2 ^{= 0} and t3 ⁼ c/a - 1. To visualize the _way of using ^the

results of this paper let ^us consider the tetragonal phase ^of BaTi03 ^{at room} temperature [20] : ^its crystalline parameters ^{are a = 3.992} A and ^{c =} 4.036 A (which ^allows for t1 = t2 ^{= 0 and} t3 ⁼ 0.011), ^and

the atomic positions ^are given (the Ba-sublattice being

the reference system) by ^Ba (0, 0, 0), ^Ti (t, 1/2,1/2 ⁺ Ô31i), 0I (1/2, 1/2, d3ÔI.), On (t, 0, 1/2+d30II) ^and OIII (0, 1/2, 2+b3oII)

where d3Ti = 0.013,

Replacing ^{these values in} the above expressions ^for Emz ^and

Et

ionic charge associated to each sublattice) the contri- bution (to ^{the field at the Ba} positions) of each sublat- tice.

3.2 ORTHORHOMBIC. - Let us suppose the follow-

ing lattice strain

Then we obtain for the microscopic contribution

Comparing ^with tetragonal ^{case, the} following

relations are obtained :

The evaluations of the rest of the 0 appear in Table II. The macroscopic contribution is given by

TABLE Il

Microscopic contribution’s coefficients (orthorhombic case)

174

TABLE III

Microscopic contribution’s coefficients (rhombohedral case)

Eÿ

Em

^Et

The strain parameters ^{are related to the usual} orthorhombic crystalline parameters a, b ^{and c} by t1 ^- (a ⁺ b)/2

/2

2

and t3 ⁼ c/a* - ^{1. We} may again extrapolate (as

indicated for the tetragonal case) ^{a value for} a*, ^or

we may adopt ^an arbitrary value, ^for example ^{a * = c}

which implies ^a vanishing t3.

3.3 RHOMBOHEDRAL. - Let us suppose the follow-

ing lattice strain :

Then, ^{we obtain for the} microscopic contribution

Comparing ^{with the} tetragonal ^{case, the} following

relations are obtained :

The evaluations of the rest of the R appear in Table III. The macroscopic contribution is given by

The x- and y-components of E’" and EM _may be obtained from Emz ^and

Et

The strain parameters ^{are related to the usual} rhombohedral crystalline parameters a ^{and oc}

(in

our notation a ⁼ a* i’ i and ^{a is the} angle (i’,

j’))

An useful choice for a* _may be a* = a

,12

which simplifies ^{the above} expressions :

Acknowledgements. ^{- We are} greatly ^{indebted to}

Professor L. Godefroy ^for stimulating this work and

making ^{comments on the} manuscript.

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