Coulomb interactions and optically-active vibrations of Ionic crystals—I

1. Phys. Chw.

Solids. 1976, Vol. 37, pp. 693498.

Pergamon Press.

Printed in Great Britain

COULOMB INTERACTIONS AND OPTICALLY-ACTIVE VIBRATIONS OF IONIC CRYSTALS-I THEORY

AND APPLICATION

TO NaNOs

AKIO YAMAMOTO,t TAMAKIUTIDAand HIROMUMURATA Departmentof Chemistry,Facultyof Science, Hiroshima University, Higashisenda-machi, Hiroshima 730, Japan and YUJ~SHIRO ShinonomeBranchSchool,Facultyof Education,HiroshimaUniversity,Shinonome-3-chome, Hiroshima734,Japan (Received 5 September 1975; accepted in reuisedform

12 December 1975)

Abstract-A methodis given for polarizable-ion model calculations of optically-active vibration frequencies of ionic crystals together with an illustrative application. Frequency splittings originating from Coulomb interactions between induced dipoles are explained satisfactorily by the calculations. Important discrepancies remaining in rigid-ion treatments are considerably reduced by taking into account the electronic polarizabilities. Some effects of anisotropy of the ionic polarizabilities on crystal vibrations are discussed briefly.

INTRODUCTION In this paper, we describe the results of calculations of the optically-active vibration frequencies in calcite-type crystals. The present investigation was undertaken in order to examine the effects of Coulomb interactions on the optically-active vibrations of ionic crystals, these being characterized both by the “dynamically-effective” ionic charge and by the “dynamically-effective” electronic polarizability. Although particular attention will be given to sodium nitrate by way of illustration, the model we will use is generally applicable to other crystalline materials with different crystal structures. As will be discussed below, the model includes short-range forces which simulate the covalent-bond contributions to the potential energy, and also long-range Coulomb forces to represent those forces whose origin is in the partially ionic nature of the binding. The structure and symmetry properties of sodium nitrate, known as the mineral soda-niter, have been described on several occasions [ 11,the structure being rhombohedral with the space group 04, - R3c and 10 atoms in a unit ce11[2]. The vibrational spectra of NaNO, have been reported by many investigators[3-141, and the observed vibrational frequencies can be classified into two groups, i.e. intraand inter-molecular vibration frequencies, since there lies a frequency-gap from ca. 350cm-’ to cu. 7OOcm-‘. The force field which reproduces two groups of the observed frequencies simultaneously will be also examined in this paper. In spite of the great vogue for “shell” model calculations[lS], it should be noted that for opticallyactive vibrations of complicated crystals fairly good agreement with the experimental results can be obtained, even on the basis of the polarizable-ion model[ 161 tTo whom correspondence

should be addressed.

(abbreviated below as PI), by the use of far fewer variable Coulomb parameters than those needed for the shell model. Hence, the Coulomb interactions have been estimated on the basis of the PI model, and although the electronic polarizabilities of constituent atoms can generally be incorporated in tensor form, the off-diagonal elements will be neglected only for the sake of simplicity. Calculations have been carried out for various sets of values for the polarizabilities and it will be shown from a careful analysis of the results that the observed frequency splittings originating from the Coulomb interactions, especially in the intramolecular vibration region, can be satisfactorily accounted for with the present PI model. It has been found that agreement is poor with the rigid-ion mode1[17-191, and that the short-range force model (abbreviated as SRF) is entirely unable to account for these splittings. It will also be demonstrated that anisotropy of the electronic polarizability must be introduced in order to interpret the optical anisotropy observed in the IR- and visible-ray regions. METHOD OF CALCULATION From the Born and Huang theory of the longwavelength optical modes in ionic crystals [ 161,a dynamical matrix, D, is constructed from four constituents:

where

is the inverse-mass matrix, (Yand p = x, y or z, m and 8 are the atomic mass and the usual Kronecker delta symbol respectively, and the subscripts k and k’ specify the atoms in a unit cell. The first term of the dynamical matrix is the non-Coulomb interaction part, which is introduced

AKIO YAMAMOTO et

694

al.

to simulate the contributions from the short-range nonCoulomb interactions to the potential energy, We represent F” in terms of a function which depends upon twelve force constants, and these are left to be determined. The set of force constants used was the same as that reported previously [ 171.The second term is the Coulomb interaction part due to the undeformable-ion interactions, and we can write it in conventional matrix form:

The static dielectric constants can be estimated from the following expression[%]:

where

the subscript t identifies the transverse mode, and the summation with respect to i is extended over the IRactive modes. By eliminating the vibrational contributions, the expression for the hip-frequency dielectric constant can be obtained:

and = Qopkk’(y) - &k’%‘(&dZk)Q 0 a,?!&’

where

e”=I’+4nv?p.

m@kk’(O);

where z is the effective charge, Q the Coulomb coefficient matrix representing the long-range electrostatic interactions, and y the wavevector. The third term is the Coulomb interaction part resulting from the induceddipole interactions through the electronic polarizability:

This expression can be used to check the chosen polarizability parameters. Actual calculations were carried out on the TOSBAC 3400 TOPS 14 electronic computer at the Computing Center of Hiroshima University. RESULTS AND DISCUSSION

F’ = - ZQRQZ, Electronic

where R = (I - AQ)--‘A, I,8tr. = Srr,.&, and A ir@t’--6

kk~~,r@k,

being the electronic polarizability tensor of the k th atom. The last term is the macroscopic field part:

cyk

FM = ZSYSZ, where S=ItQR,

I) is the unit cell volume, and a tilde denotes the transposed matrix. In order to determine the normal mode of vibration and the co~es~nding vibration frequency we must solve the equation of motion for the atomic displacements, X: DX=AX, where A is the eigenvalue. In the case of the opticallyactive vibration, the magnitude of the wavevector is set to be zero.

polarizability

From the crystal symmetry it follows that the electronic pola~zability of the constituent atoms is (rilli = trz2k(designated below as o;(k))+ 033k (designated as all(k)). Unlike the formal ion charge approximation for the effective charge, there seems to be no well-established criterion for the first-approximation values of the polarizability. Therefore, the various polarizability sets of values were chosen so as to reproduce the observed refractive indices[‘tO], nL = 1.587 and nt = 1.336, it being assumed that the electronic polarizability of the sodium atom is isotropic and fixed at 0.292 A3, since that of the atom is considered to be much smaller than that of the nitrate group from the physicochemica1 point of view. The resulting electronic ~la~zabilities of the oxygen and nitrogen atoms are shown in Fig. 1, where all of the sets of values on the curves, {al(O), al(N)} and {all(O), (YII(N)], were able to account for the high-frequency dielectric constants corresponding to n, and nil respectively. The determination of the electronic polarizabilities must therefore be based on a fit of the calculated to the observed frequencies. In the present investigation, off-diagonal components of the polarizabilities were neglected as a first approximation, The validity of the simplifying assumption is unclear at present, although it does predict correct orders of magnitude for the Coulomb frequency splittings. It seems likely that the influence of the components, if any, would be appreciable in the case of atoms having lower site symmetry, such as oxygen atoms in the calcite lattices. Therefore, further work is required to clarify the significance of the off-diagonal components. ~~t~cu~~y-ucti~e

ui~r~tio~

On the basis of the PI model, the frequency splittings in

Optically-active

vibrations of ionic crystals-I

695

in Fig. 1, the E,-E. splitting in the NO stretching modes was calculated to be comparable with the observed width, while that in the ON0 deformation modes was seriously overestimated. In other words, a small value of al(O) is required to explain the NO-stretching splitting; however, the value cannot be too small because of the narrow width 4 of the splitting observed for the ON0 deformation modes. Decius et al.[23-261 have discussed the effects of the 4 3 / dipolar coupling upon the TO-LO and E,-Eu splittings of intramolecular vibration modes. For crystals with the calcite structure, they have employed a model with (~~(0)= (u11(0) = 0.0 A’. In the present study, not only have the effects of the Coulomb interactions on the splittings of the other modes been ascertained, but also it has become apparent that with their model fairly good agreement with the observed splitting for the NO stretching modes can be obtained, but that, however, the agreement is less satisfactory for the other modes. The observed E,-E. splittings [27-301 of calcite-type 0 0.5 1.0 15 crystals, such as calcite, magnesite or dolomite, show the a(O), 2 following inequality relations: vS1> P+,,and vszs vG2.The Fig. 1. The electronic polarizabilities of the nitrogen and oxygen calculations for NaN03 have shown that the relations are atoms. All of the sets of values on the curves,{a,(O), ar(N)} and {a,,(O) and q(N)}, are able to account for the high-frequency satisfied only with a small value of al(O). In this work, dielectricconstants, l,- and ~(7,respectively.n,, perpendicular therefore, we have attempted to find the set of polarizabilcomponent;ai,,parallelcomponent. ity values which reproduces the observed width, v~,-v~~, as well as possible, and at the same time which gives the the intramolecular vibrations, e.g. the TO-LO and EC-E, inequality relation, us2s v62.In certain cases, the origin of splittings[23-261, are expected to be sensitive to the the E,-E. splittings can be qualitatively understood electronic polarizabilities of the oxygen and nitrogen through the difference in the Coulomb interactions beatoms, and also to the charge distribution within the tween the Eg and E, symmetric modes. In the NO nitrate groups. The effects of the Coulomb terms, which stretching modes, for example, a large dipole moment is considered to be vibrationally induced on each nitrate mainly affect the intramolecular modes, will be discussed group, since wide TO-LO splittings are observed experiin the following two subsections. mentally. Arrangements of the induced-dipole arrays are schematically shown in Fig. 2 for the Es and E, modes; E8 and E, symmetric modes The present calculations suggest that the E,-E,, split- the origin of the Es-E. splitting can be attributed satisfactings in the NO stretching and ON0 angle deformation torily to the differences in the Coulomb interactions of the modes are dependent on the electronic polarizabilities of induced dipoles. Furthermore, small differences in the the nitrogen and oxygen atoms rather than on the charge non-Coulomb interactions may also play an important role distribution. The calculated frequencies are given in Table in the splittings when there is only a small difference in the 1 for three typical polarizability sets of values, showing Coulomb interactions as in the case of the ON0 angle that the electronic polarizability has considerable influ- deformation modes. The small difference in the Coulomb ence upon the intramolecular vibrations. In the case of the interactions is likely to be cancelled out by the nonlargest value of al(N), i.e. al(N) = 4.7912, as is shown Coulomb contribution, and the Coulomb contribution may I

Table 1. Observed and calculated vibrational frequencies Cbserved’

syn

[Electronic ul(0)=a,,(O)=O.O

ut

*lb?

VI

Av

+

1068

*2u

938

E g

1385

Eu

1357

843

5

TO-LO

815

862

47

-

62

u1

al(o)=u,,(o)=l.a

Au+

822

vt

"1

A"+

lOGi 915

93

1373

735 1419

vt

1069

1385

* Refs. 11, 12 and 27. t

v1

A”+

modes

polarizabiIity15

ni(0)=o,,(O)=0.4

1068

724

727

Yt

(cm-‘) of the intramolecular

e39

EL2

3

:3?6

727

730

1362

1436

74

1367

1428

61

1377

1414

37

691

e08

117

726

735

9

730

730

0

vt, tran~"erse node frequency; ul, Icngitudiml

node

frequency.

-

splitting in cm -1

5 T112 electronic polarizability In

;;3.

6%

AKIO YAMAMOTO

_qgE$:

active lattice vibrations by using a shell model, regarding the nitrate group as one large ion. The observed latticemode frequencies are well accounted for by the use of eleven parameters, four short-range force parameters and the rest shell model parameters. Plihal[35] have calculated the phonon dispersion branches and the lattice specific heat by using the same model. An applicability of their shell model parameters to the intramolecular vibrations remains obscure, although the parameters are satisfactory for the explanation of low-frequency intermolecu-

s;

;.$SLL+

(?&$+

.

. EU

5

et al.

Fig. 2. The vibration modes and the induced dipoles in NO stretching vibrations. o, sodium; l, nitrogen; 0, oxygen.

lar vibrations. Recently, Trevino et al. [36] have measured the lattice-mode frequencies by means of neutron inelastic scattering, but there seems to be a certain ambiguity in

not always be unambigously reflected in the experimental ES-E, splittings in such a case. On the other hand, the TO-LO splittings of the NO

the assignment of the observed AZg phonon peaks. As is shown in Table 2, the results of the present study offer strong support for their group theoretical assignment[37].

stretching modes were found to depend appreciably on the charge distribution, and the determination of the distribution parameter based on the PI model therefore becomes more meaningful than that based on the RI model[l7]. Careful analysis of the results leads to the conclusion that the best fit to the observed splittings can be obtained by using an optimized set, al(O) = 0.40 A’ and clJN) = 0.682 w”. The effective charges are also determined to be ZNa= 0.77, ZN = 0.15 and Z0 = - 0.31, all in electron units; the values are slightly smaller than those obtained with the RI model[l7]. The calculated vibrational frequencies are given in Table 2. The calculated frequencies of the intermolecular vibrations agree fairly well with the observed frequencies. Normal coordinate calculations have been made on the basis of the SRF model by several investigators [ 11,3 l331. Plihal and Schaack[34] have analyzed the optically-

Azu symmetric modes The TO-LO splitting of the out-of-plane angle defor-

mation modes tends to be overestimated for small values of (Eli (see Table 1). The result that the observed TO-LO splitting is relatively small can best be explained by assuming the polarizabilities of the oxygen and nitrogen atoms to be anisotropic; the experimental estimate [ 121that the “A2p-A2.” splitting of the modes is at most small offers additional evidence in support of this assumption. The AZ,-A*, splitting of the modes has been calculated to be fairly sensitive to the electronic polarizabilities, but this additional confirmation of the chosen polarizability sets has not been verified because of the lack of the experimental AZ, data. The calculated vibrational frequencies are given in Table 2 for the best-fit set: (Y,,(O) = 0.95 A’ and cyil(N)= 0.899 A’.

Table 2. Observed and calculated vibrational frequencies (cm-‘)

*1tz

V11

A

u21

1”

*2g

1068

1069 201

V31

846

i 13 2

170

i79

124

117

w33

** 175 ** 120

vlrl

a38

V42

e43

5

838

205

251

46

204

51

89

38

56

7

837

844

7

261

57

198

267

69

58

2

53

75

22

845

u51

1335

1373

1376

Us2

774

725

726

186

177

185

98

111

V53 u5'i EU

LO70

202

Vu3 E g

iC6E

“22 V32

A2u

LO69

u61

1351

v62

127

1419

62

-

-

104

1367

1428

61

1376

1380

4

725

735

10

726

726

0 71

Us3

204

271

67

209

255

46

196

267

us4

175

185

10

163

178

15

179

178

3

\65

87

97

10

86

90

4

85

95

10

* F.ers.11, 12 longitudinal

and 27.

mode

Vt, transverse mode

frequency.

B.Ref. 17. “” Ref.

36.

: TO-LO

see text.

frequency;

splitting

ir cn

ul,

-1

Optically-active

697

vibrations of ionic crystals--I

Table3. Forceconstants(mdyn/A)and Coulombparameters F.C.*

CPII

De”.+

0.000

ml5

[SRFI’

K

5.254

HS

0.626

5.680

5.656

0.581

0.581

FS

2.753

a

0.687

0.017

1.645 0.469

1.645

P "S

-1.442 0.71955

0.122

0.017 55 0.150

fl

0.039

0.003

0.016

0.055

f2

0.222

0.020

0.210

0.118

f3S

0.002

0.007

0.000

f4

0.047

0.003

0.052

0.054

*5

-0.001

0.000

-0.012

0.050

f6S

-0.001

-0.008

0.036

0.485 -0.036 0.14555

** (Caulomb parameters) ZNa

0.77

0.87

zN

0.15

0.17

zO

-0.31

-0.35

o(Na)

0.292

a1 (N)

0.682

a,,(N)

0.899

a1 (0)

0.400

a,,(0)

0.950

* The notation is the same as in Ref. 17. + Standard deviation. 5 Ref. 17. ** The effective charges and the electronic polarizabilities are in electron units and In i3 respectively. 0 % Fixed force constant. SB Force constant n in ndyn.ii.

Comparison of the PI model with the RI model

It is apparent by comparing the results of this study with those based on the RI model[l7] that the PI model yields much better agreement with the observed frequencies of the intramolecular vibrations than the RI model. The apparent discrepancies between calculated and observed frequencies for some lattice modes can be attributed to originate from the optimization procedure used to obtain the parameters, and hence not necessarily to defects in the model itself. Thus, it can be concluded that the weak interactions between moderately separated groups can be approximated quite well by the RI model, but that the contribution from the distortion of the electronic distribution cannot be interpreted without the use of a model where the influence of the electronic polarizability can be appropriately estimated. If attention is paid only to the intermolecular modes, it is noteworthy that the optically-active vibrations can be interpreted satisfactorily on the basis of the RI model with considerably fewer variable Coulomb parameters. There seems to be no reasonable way in the RI model to introduce optical anisotropy effects than by conventional charge anisotropy, while in the PI model this can be accomplished through the anisotropy of the electronic polarizabilities of the constituent atoms. Short -range force constants

The force constants were refined by the use of leastsquares techniques, and the values obtained are listed in

Table 3, together with those of the RI and SRF models [ 171.It is seen that for the intermolecular parameters, f, - f6, there are no significant differences between the PI model and the RI and SRF models, but there are differences in the values of intramolecular force constants, H, F, p and n, which are appreciably changed from those of the other models by the introduction of the electronic polarizability. This seems to be in accordance with the generally recognized concept that the Coulomb interactions through the electronic polarizability are more short-range than those between rigid ions. The force constant F’ of the Urey-Bradley force field has been assumed to be F’ = - 0.1F as in ordinary molecules, since the Coulomb contributions are separately taken into account. Acknowledgement-The authors wish to express their sincere thanks to Mr. Masaru Ohsaku of Hiroshima University for his kind and helpful suggestions.

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698

AK~O YAMAMOTOet al.

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