The lattice dynamics of alkaline earth oxides

The lattice dynamics of alkaline earth oxides

1 Phya. Chum. Solids. 1975. Vol. 36. pp. X-297. Pcrganwn Press. Pnotcd in Great Britain THE LAmICE DYNAMICS OF ALKALINE EARTH OXIDES K. S. UPADHY...

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1 Phya. Chum. Solids. 1975. Vol. 36. pp. X-297.

Pcrganwn Press.

Pnotcd in Great Britain



K. S. UPADHYAYA* ~p~rnent of Physics,BanarasHinduUniversity, Varanasi-5,India and R. K. SINGH Department of Physics,N.R.E.C. College, Khurja, India (Received

9 May 1974)

Abstract-A theoreticalinvestigationof the latticevibrationsin alkalineearth oxides (CaO and SrO) has been carried out by means of a three-body force shell model which considers the ionic charge Z as an independent parameter and both the ions polarizable. A satisfactory description of the dispersion of phonons, De.byetemperature variations and second-order Raman and i.r. spectra is obtained. Analysis of these descriptions has revealed that the alkaline earth oxides are ionic crystals with simple interionic interactions.

In two earlier papers [ 1,2], we have reported a thorough investigation of the lattice dynamics of transition metal oxides [ l] (7’MO) and magnesium oxide [2] which belongs to the alkaline earth oxide (AEO) family. These investigations have been carried out by means of one-ion-poiarizable three-body force shell model [2] (TSM) developed by us for the divalent ionic crystals on the basis of TSM of Verma and Sir&[31 devised already for the monovalent ionic crystals. The interesting feature notable from these reports is that the TSM, which is reasonably appropriate for the simple ionic solids [4,5], has described the lattice dynamics of MgO much better than those of the transition metal oxides. This Ieads to an obvious conclusion that MgO is more ionic in nature and possesses simpler interionic interactions as compared to the TMO. The complexities in the nature of TM0 may be mainly due to the magnetic interactions whose existence in them has been well established by Sakurai et ai. [6] and Haywood and Collinsf71. Thus, in order to gene&se the distinctive features between the AEO and the TM0 systems, it seems desirable to investigate the complete lattice dynamics of the remaining members of the AEO family (i.e. CaO, SrO, BaO). The basic aim of the present paper outlined above has been achieved by applying both-ions-poiarizable TSM to compute the phonon dispersion relations, the Debye temperature variations, and the second-order Raman and i.r. spectra for CaO and SrO. These computed results have been found to present an excellent interpretation of *Present Address: Department of Physics, K.N. Govt. P.G. College, Gyanpur, Varanasi-221304,India.

the observed data (wherever available) on them. A detailed discussion of these results given in Section 2 has been presented in the Section 3. However, the lattice dynamics of BaO could not be studied at present due to lack of sufficient experimental data required for the calculation of model parameters.

2. coMPurA~orusANDRESULTS A both-ions-polarizable TSM used in the present study contains 9 parameters: the two short-range nearest Neiman force constants A and B, the two parameters f0 = f(rO) and r&, = (r d~(r)~d~~,-~ arising from threebody interactions, the ionic charge 2 f=2), the shell charges (Y,, YZ)and the mechanical polarizabilities (d,, d& Without giving the details of the strategy (which can be obtained by extending the one-ion-polarizable TSM [2]) of the parameter dete~ination we report the values of these parameters in the Table 1. The input data used in the calculation are also given in the same Table together with their relevant references. The complete phonon spectra for CaO and SrO has been calculated using the parameters listed in Table 1I The curves representing the dispersion of phonons along the principal symme~ d~ections for CaO have been shown in Fig. 1 and compared with those obtained from the neutron scattering technique applied by Vijayaraghvan and Iyengar[l3], and Saunderson and Peckham[ 141. However, such a comparison was not possible for SrO in Fig. 2 due to unavailability of the measured data. The Debye temperature variations, which are sensitive to the lower range of phonon spectra, have been computed as a function of temperature T and plotted in



K. %bie





data and model parameters


SINGH for CaO and SrO

Input data Physical


C, 1(IO” dyn

cm ‘f CL2 (IO” dyn cm- ‘) C,(lO”dyncm ‘) v,.(lO’“sec ‘) V, (10” set-‘1 Q, (10~“cm’) a,(lO~“cm’) r,, (IO-” cm) e,


CaO values


22.330 5.930 8.100 17.100 8,850

8 8 x 9 9

Model Para-

SrO values

References meters

17.330 4.530 5,600 14.400 6,810

8 8 8 9 9

A ! r& Y,



I ,1os*





I ,800 2,406 2.330


2.09s* ?..s80 3460






Fig. 3 showing the experimental points[ E-171 for visual comparison. In order to test the correctness of higher range of phonon spectra, the second-order Raman and i.r. features have been studied following the combined density of states (CDS) approach[[email protected] and the critical point analysisI 191.The CDS curves showing theoretical and experimental features have been given in Figs. 4 and 5. The assignments obtained from the criticaf point analysis and compared with the observed [9,20,2 11ones have been given in Tables 2 and 3. This analysis has been carried out to interpret the fine structure of the i.r. and Raman spectra.




7.8766 0.9846 0.0258 Wo676 5.5952 3.3910 0.0713 [email protected] 2X000

74874 - 09oO9 . . 0.0378 - oG442 - 4.6333 - 3.5454 0.1229 0.5299 [email protected]



Fig. I. Phonon dispersion curves for CaO. Theoretical curves: ----_-_RSM. Experimental Points: The points with circles and triangles correspond, Vijayaraghvan[l3]andSaundersonandPeckham[ 14](.li,Clongitudinal


d: Z

parameters CaO SrO

3. DISCUSSIONAND CONCLUSION clear from Fig. I that the phonon dispersion

9Both-Ions-Polarizable respectively. IO andA,Otransverse).


TSM; results


relations predicted by 9-parameter TSM present an excellent interpretation of the data measured extensively by Vijayaraghvan and Iyengar [ 131,and Saunderson and Peckham[l4] using the most powerful technique of coherent inelastic scattering of slow neutrons. The rigid shell model 1221(RSM) with 9- and I l-parameters used by Saunderson and Peckham [ 141 provides comparable agreement but yields the values of the elastic constants and the optical constants that possess deviations of about 5-20 per cent. In order to interpret their measured data, Vijayaraghvan and Iyengar[l3] have used a S-parameter RSM and obtained a very poor description of the phonon dispersion. In addition, these authors [ 131have reported the value of the ionic charge Z = 1.70 which will lead to a considerable decrease in binding energy of the crystal. In


The lattice dynamics of alkalineearth oxides


r 90

VI o




0.2 OL

cx 1 I




I III I I 0.2 I IV 0.6 0.8 1.0 0.8 06 0.~ q-_q


I 0.2

L 1

' o/s I


Fig. 2. Phonondispersion curves for SrO.

our model, these deviations are never more than 5-10 per cent. The phonon dispersion relation for SrO is shown in Fig. 2. At present this has only academic interest but it might be helpful in analysing the complex data measured in future. Theoretical dispersion curves for this solid have coo


Fig. 3. Debye temperature variation for CaO and SO. 0 Gmelin [1S],AParksandKelly[l6].OAnderson[17].

another report1231 they have used a Wparameter RSM which describes the dispersion of phonons fairly well, indeed slightly better than ours; nevertheless the model contains a large number of parameters, some of which have physically unrealistic values. Prior to the reports of measured data, Mon [241 applied ‘I-parameter breathing shell model [25] (ESM), which is an important extension of RSM, to obtain the dispersion curves of CaO. The results obtained by Mon. on comparison with experimental data, show a deviation of about 15-20 per cent for TA modes and 20-25 per cent for LO and TO modes with higher wave-vectors along [qoo] and [qqq] directions. For

JPCS Vol. 36. No. 4-F

Fig. 4. Combined density of states curve for CaO. The continuous and broken arrows indicate Raman featores[20] and i.r. features .191. ..resoectivelv. . ~,


K. S.




also been reported by Mon[24] using ‘I-parameter BSM. A comparison between the two curves, not shown in Fig. 2, has revealed some difference particularly along [qoo) and [qqo] directions while curves along [qqq] direction are almost identical. The difference for some branches may be due to the low value of ionic charge (~2) appearing in BSM used by Mon. The other discussions must, however, be deferred until the experimental results are reported. It is clearly seen from the Fig. 3 that the agreement between the measured[H-171 and derived values of Debye temperatures 80 is excellent. The small deviations at higher temperatures may be mainly due to the anharmonicity of vibrations, the effect of which is ignored in TSA4.The suitability of the model for SrO is evident from the good agreement obtained for Debye temperature variation. An inspection of the CDS curves for CaO and SrO shown, respectively, in Figs. 4 and 5, demonstrates that i 0



I 6


I 15






, 21


\I 2L

Fig. 5. Combined density of states curve for SrO. The continuous and broken arrows indicate, respectively, Rarnan[21] and i.r.[9] features.

theoretical and experimental peaks compare favourably but that a satisfactory interpretation of fine structure is not achieved. This may he mainly due to the coarseness

involved in the division of the Brillouin zone and the restrictions imposed by the selection rules. Thus, in order to interpret the measured data satisfactorily we have obtained assignments in terms of the combinations and overtones of the phonon frequencies at the critical points

Table 2. Assignments of Raman spectra of CaO and SrO CaO Present study Value Assignments (cm-‘)

Observed [20] peaks (cm-‘) 400 -

2 TA(L) TA(X) LA + TA(L) 2 TO(T) 2 LA(L) 2 Lo(r)


580 750 1160

389 413 580 593 773 1147

Observed[Zl] peaks (cm-‘) 3G 450 600 750 960

SIO Present study Value (cm-‘) Assignments LA + TA(L) TO t TA(X) 2 TO(f) LO + LA(X) 2 LA(L) r 2 LO(X) I 2 LO(L) 2 LO(T)


353 453 600 870 970

Table 3. Assignments of i.r. spectra of CaO and SrO CaO Present study Observed 191 Values peaks (cm-‘) Assignments (cm-‘) 475 504 590 629 668 739

LA + TA(A) TO + TA(A) TO + LA (A) LO + TA(A) LO + TA(L) TO + LA(L) LO + LA(A) LO + TO(A)

Observed[9] peaks (cm-‘)

450 492 575 612 670

387 470 503 510 537 -

700 742


SrO Present study Values Assignments (cm-‘)

TO + LA(A) LO + TA(A) To+ LA(L) LO + LA(L) LO+TA(L) LO + TO(A) -

397 460 517 530 540 583 -

The lattice dynamics of alkaline earth oxides

(r, X, L, A). It is interesting to note from Tables 2 and 3 that these assignments are in fair agreement with the observed i.r. [9] and Raman[20,21] peaks. These successful results provide a fairly good test of the correctness of the higher range of phonon spectra. Thus, in view of the results reported here and earlier[l, 21 ones it is evident that the oxides of alkaline earths are more simple ionic solids than those of transition metals.

Acknowkdgements-The authors express their gratitude to Prof. B. Dayal and Dr. R. S. Srivastava for the encouragement, and principal Dr. P. C. Gupta for providing facilities. One of us (K.S.U.) is thankful to the C.S.I.R. (India) for the financial


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