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Elastic constants of SrClF and BaClF crystals
Volume 67, number 2,3
CHEMICAL PHYSICSLETTERS
15 November 1979
ELASTIC CONSTANTS OF SrCIF AND BaCIF CRYSTALS K.R. BALASUBRAMANIAN, T.M. HARIDASAN and N. KRISHNAMURTHY School of Physics, Madurai Kamaraf University, Madurai-625021, India Received 10 August 1979
The six elastic constants of SrC1F and BaC1F crystals are calculated from the acoustic phonon velocities computed in four symmetry directions using the sheU model of lattice dynamics.
1. Introduction In an earlier paper [1 ] we reported the longwavelength dynamics of SrC1F and BaC1F crystals. The emphasis in that paper was on the long-wavelength optic phonons which could be compared with the experimental data from Raman scattering [2] and infrared absorption [3]. The short-range force constants entering in the dynamics were transferred from similar force constant data of the corresponding alkaline earth fluorides and alkaline earth chlorides. The calculated optic phonon frequencies in the long-wavelength limit agreed fairly well with those of the experiment, indicating that the transfer of the short-range force constants indeed is a fairly realistic representation of the short-range force field in these alkaline earth chloro fluorides. We have therefore extended the calculations on the dynamics of these crystals for small phonon wavevectors in various symmetry directions in order to obtain the velocity of propagation of the acoustic phonons, which give information on another long-wavelength property, namely the elastic constants. The results of such an investigation are reported in this note.
2. Method of calculation The details on the crystal structure and the dynamics of these crystals are given earlier [1 ] and we shall discuss here only those relevant to the present study. We have used the same shell model parameters as used 530
in ref. [1 ]. Since SrC1F and BaC1F are tetragonal crystals with space group D7h we expect six elastic constants, Cll, C33, C66, C12, C13 and C44. They can be calculated from the velocity of propagation of the acoustic phonons in four directions, namely, [100], [001], [110] and [101]. The (18 × 18) hermitean dynamical matrix was diagonalised for very small phonon wavevectors in these four directions and from the acoustic phonon frequencies and corresponding phonon wavevectors one can calculate the velocity of propagation of these acoustic phonons. The acoustic phonons in these four directions are purely longitudinal or purely transverse. The following expressions give the elastic constants in terms of the acoustic phonon velocity [4] : pv 2 [100] = Cll ,
pV21[1001 = C66 (the polarisation of VT1 is along [010]),
pdT [lOO1 = C44 (the polarisation of OT2 is along [001]); pv 2 [0011 = C33 , pv 2 [001 ] = C44
(oT is degenerate);
pv~ [I 101 = 1(Cll + C12) + C66,
Volume 67, number 2,3
pO2Tx[1101 =1~(Cll
-
CHEMICAL PHYSICS LETTERS
C12)
15 November 1979
Table 1 Velocities of the acoustic phonons (in units of 105 cm/s)
(OT1 is polarised perpendicular to [001] and is along [1TO]),
pv22 [1101 :
C44
(VT2 is polarised along [001]) ;
po211011 ='~ ( C l l
+ C33)+ C44
+ [~(Cll - C13) 2 + (C13 + C44)211/2},
,v2 [lOll -- ' ~
VL[100] oft [ 1001 OF2[ 100] oL[001 ] vy[001] vL[ 110] OFt [110] OT2[ 110] vL[101 ] OT1[101] OF2[ 101 ]
SrC1F
BaC1F
5.101 3.054 2.629 4.507 2.782 5.118 3.363 2.842 5.010 2.521 2.969
4.445 2.684 2.279 3.611 2.382 4.434 2.573 2.289 4.353 2.055 2.606
½(Cll + C33) + C44
_ [~_(Cll _ C13)2 + (C13 + C44)2] 1/2)
(VT1 is polarised along
[10T]),
OT2 [101] = 1(C66 + C44) (VT2 is polarised along [010]). One notices that C44, C12 and C13 call be obtained from more than one equation given above. Thus one can check the consistency of the results. The suffix L represents longitudinal acoustic phonons and suffix T the transverse phonons./9 is the density of the material.
3. Results and discussion
The values of the acoustic phonon velocities obtained are entered in table 1. Table 2 gives the values of the elastic constants of SrC1F and BaC1F obtained from these acoustic phonon velocities. The values of C12 obtained from the above expressions gave a spread of roughly +0.4 X 1011 dyne/cm 2 from the mean value. The spread in the values of C13 obtained is also of the same order, whereas for C44 the spread obtained is much smaller, namely +0.2 X 1011 dyne/ cm 2. Accordingly the uncertainty in the estimated values would be more for C12 than for the others. Tabulated values of C12, C13 and C44 are the means of those obtained from the above expressions. There is no experimental data on the elastic con-
stants of these materials for a direct comparison. However, since the force field in SrC1F is obtained from similar data from SrF 2 and SrC12, a comparison of the present results with those of cubic SrF 2 and SrC12 is in order. One finds that the C 11, C44 and C 12 values of SrC1F lie in between the experimental data of SrF 2 [5] and of SrC12 [6,7], as is to be expected. The values are closer towards those of SrF 2. Similar trend, namely the elastic constants of BaCIF being closer towards those of BaF2, is also seen. However a comparison with the elastic constants of BaC12 is not possible due to the lack of experimental data on crystalline BaC12. It would be highly useful to obtain the elastic constants of these materials experimentally, either from ultrasonic pulse method and/or by Brillouin scattering technique. Such experimental data would confirm the validity or otherwise of the force-field employed in the present investigation. The experimental results would also be useful to work out more direct and realTable 2 Elastic constants of SrC1F and BaC1F (in units of 1011 dyne/ cm2)
C11 C12 C13 C44 Css C66
SrC1F
BaCIF
10.61 3.32 4.58 3.06 8.29 3.79
9.08 2.67 4.16 2.43 6.00 3.32 531
Volume 67, number 2,3
CHEMICAL PHYSICS LETTERS
istic lattice dynamical models. However, it is hoped that the present results would be helpful to the experimentalists to know the region o f the acoustic velocities that may come up in their measurements. Work is in progress on the detailed investigation o f the phonon dispersion in these crystals.
Acknowledgement One of us (KRB) wishes to thank the University Grants Commission for the award of a teacher fellowship.
532
15 November 1979
References [1] K.R. Balasubramanian, T.M. Haridasan and N. Krishnamurthy, Solid State Commun. (1979), to be published. [2] J.F. Scott, J. Chem. Phys. 49 (1968) 2766. [3] H.L. Bhat, M.R. Srinivasan, S.R. Girisha, A.H. Rama Rao and P.S. Narayanan, Indian J. Pure Appl. Phys. 15 (1977) 74. [4] E.W. Kammer, L.C. Cardinal, C.L. Vold and M.E. Glicksman, J. Phys. Chem. Solids 33 (1972) 1891. [5] D. Gerlich, Phys. Rev. A135 (1964) 1331; A136 (1964) 1336. [6] H.V. Lauer, K.A. Solberg, D.H. Kuhner and W.E. Bron, Phys. Letters A35 (1971) 219. [7] C. An, Phys. Star. Sol. A43 (1977) K69.
CHEMICAL PHYSICSLETTERS
15 November 1979
ELASTIC CONSTANTS OF SrCIF AND BaCIF CRYSTALS K.R. BALASUBRAMANIAN, T.M. HARIDASAN and N. KRISHNAMURTHY School of Physics, Madurai Kamaraf University, Madurai-625021, India Received 10 August 1979
The six elastic constants of SrC1F and BaC1F crystals are calculated from the acoustic phonon velocities computed in four symmetry directions using the sheU model of lattice dynamics.
1. Introduction In an earlier paper [1 ] we reported the longwavelength dynamics of SrC1F and BaC1F crystals. The emphasis in that paper was on the long-wavelength optic phonons which could be compared with the experimental data from Raman scattering [2] and infrared absorption [3]. The short-range force constants entering in the dynamics were transferred from similar force constant data of the corresponding alkaline earth fluorides and alkaline earth chlorides. The calculated optic phonon frequencies in the long-wavelength limit agreed fairly well with those of the experiment, indicating that the transfer of the short-range force constants indeed is a fairly realistic representation of the short-range force field in these alkaline earth chloro fluorides. We have therefore extended the calculations on the dynamics of these crystals for small phonon wavevectors in various symmetry directions in order to obtain the velocity of propagation of the acoustic phonons, which give information on another long-wavelength property, namely the elastic constants. The results of such an investigation are reported in this note.
2. Method of calculation The details on the crystal structure and the dynamics of these crystals are given earlier [1 ] and we shall discuss here only those relevant to the present study. We have used the same shell model parameters as used 530
in ref. [1 ]. Since SrC1F and BaC1F are tetragonal crystals with space group D7h we expect six elastic constants, Cll, C33, C66, C12, C13 and C44. They can be calculated from the velocity of propagation of the acoustic phonons in four directions, namely, [100], [001], [110] and [101]. The (18 × 18) hermitean dynamical matrix was diagonalised for very small phonon wavevectors in these four directions and from the acoustic phonon frequencies and corresponding phonon wavevectors one can calculate the velocity of propagation of these acoustic phonons. The acoustic phonons in these four directions are purely longitudinal or purely transverse. The following expressions give the elastic constants in terms of the acoustic phonon velocity [4] : pv 2 [100] = Cll ,
pV21[1001 = C66 (the polarisation of VT1 is along [010]),
pdT [lOO1 = C44 (the polarisation of OT2 is along [001]); pv 2 [0011 = C33 , pv 2 [001 ] = C44
(oT is degenerate);
pv~ [I 101 = 1(Cll + C12) + C66,
Volume 67, number 2,3
pO2Tx[1101 =1~(Cll
-
CHEMICAL PHYSICS LETTERS
C12)
15 November 1979
Table 1 Velocities of the acoustic phonons (in units of 105 cm/s)
(OT1 is polarised perpendicular to [001] and is along [1TO]),
pv22 [1101 :
C44
(VT2 is polarised along [001]) ;
po211011 ='~ ( C l l
+ C33)+ C44
+ [~(Cll - C13) 2 + (C13 + C44)211/2},
,v2 [lOll -- ' ~
VL[100] oft [ 1001 OF2[ 100] oL[001 ] vy[001] vL[ 110] OFt [110] OT2[ 110] vL[101 ] OT1[101] OF2[ 101 ]
SrC1F
BaC1F
5.101 3.054 2.629 4.507 2.782 5.118 3.363 2.842 5.010 2.521 2.969
4.445 2.684 2.279 3.611 2.382 4.434 2.573 2.289 4.353 2.055 2.606
½(Cll + C33) + C44
_ [~_(Cll _ C13)2 + (C13 + C44)2] 1/2)
(VT1 is polarised along
[10T]),
OT2 [101] = 1(C66 + C44) (VT2 is polarised along [010]). One notices that C44, C12 and C13 call be obtained from more than one equation given above. Thus one can check the consistency of the results. The suffix L represents longitudinal acoustic phonons and suffix T the transverse phonons./9 is the density of the material.
3. Results and discussion
The values of the acoustic phonon velocities obtained are entered in table 1. Table 2 gives the values of the elastic constants of SrC1F and BaC1F obtained from these acoustic phonon velocities. The values of C12 obtained from the above expressions gave a spread of roughly +0.4 X 1011 dyne/cm 2 from the mean value. The spread in the values of C13 obtained is also of the same order, whereas for C44 the spread obtained is much smaller, namely +0.2 X 1011 dyne/ cm 2. Accordingly the uncertainty in the estimated values would be more for C12 than for the others. Tabulated values of C12, C13 and C44 are the means of those obtained from the above expressions. There is no experimental data on the elastic con-
stants of these materials for a direct comparison. However, since the force field in SrC1F is obtained from similar data from SrF 2 and SrC12, a comparison of the present results with those of cubic SrF 2 and SrC12 is in order. One finds that the C 11, C44 and C 12 values of SrC1F lie in between the experimental data of SrF 2 [5] and of SrC12 [6,7], as is to be expected. The values are closer towards those of SrF 2. Similar trend, namely the elastic constants of BaCIF being closer towards those of BaF2, is also seen. However a comparison with the elastic constants of BaC12 is not possible due to the lack of experimental data on crystalline BaC12. It would be highly useful to obtain the elastic constants of these materials experimentally, either from ultrasonic pulse method and/or by Brillouin scattering technique. Such experimental data would confirm the validity or otherwise of the force-field employed in the present investigation. The experimental results would also be useful to work out more direct and realTable 2 Elastic constants of SrC1F and BaC1F (in units of 1011 dyne/ cm2)
C11 C12 C13 C44 Css C66
SrC1F
BaCIF
10.61 3.32 4.58 3.06 8.29 3.79
9.08 2.67 4.16 2.43 6.00 3.32 531
Volume 67, number 2,3
CHEMICAL PHYSICS LETTERS
istic lattice dynamical models. However, it is hoped that the present results would be helpful to the experimentalists to know the region o f the acoustic velocities that may come up in their measurements. Work is in progress on the detailed investigation o f the phonon dispersion in these crystals.
Acknowledgement One of us (KRB) wishes to thank the University Grants Commission for the award of a teacher fellowship.
532
15 November 1979
References [1] K.R. Balasubramanian, T.M. Haridasan and N. Krishnamurthy, Solid State Commun. (1979), to be published. [2] J.F. Scott, J. Chem. Phys. 49 (1968) 2766. [3] H.L. Bhat, M.R. Srinivasan, S.R. Girisha, A.H. Rama Rao and P.S. Narayanan, Indian J. Pure Appl. Phys. 15 (1977) 74. [4] E.W. Kammer, L.C. Cardinal, C.L. Vold and M.E. Glicksman, J. Phys. Chem. Solids 33 (1972) 1891. [5] D. Gerlich, Phys. Rev. A135 (1964) 1331; A136 (1964) 1336. [6] H.V. Lauer, K.A. Solberg, D.H. Kuhner and W.E. Bron, Phys. Letters A35 (1971) 219. [7] C. An, Phys. Star. Sol. A43 (1977) K69.