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Raman study of SrF2 under uniaxial stress
0038-1098/86 $3.00 + .00 Pergamon Journals Ltd.
Solid State Communications, Vol. 58, No. 9, pp. 645-648, 1986. Printed in Great Britain.
RAMAN STUDY OF SrF2 UNDER UNIAXIAL STRESS A.D. Papadopoulos, Y.S. Raptis and E. Anastassakis Physics Laboratory III, National Technical University, Athens 157 73, Greece
(Received 31 January 1986 by M. Cardona) The first-order Raman scattering in SrF2 is studied at room temperature in the presence of a uniaxial stress. The phonon deformation potentials for the long-wavelength Raman-active optical phonon are determined and compared to those of other materials with the fluorite structure. 1. INTRODUCTION ALKALINE-EARTH FLUORITES, have always attracted considerable interest concerning their lattice-dynamical [ 1 - 3 ] , optical [1, 4] and transport [4, 5] properties. In addition, effects of external perturbations have been examined, theoretically and experimentally, as an independent method for testing proposed models. In this connection, the Raman spectra for all three of the most common members of this group, i.e. CaF2, BaF2 and SrF2, have been investigated as a function of temperature [6] and hydrostatic pressure [7, 8]. On the other hand, only CaF2 and partially BaF2 have been studied in the presence of a uniaxial stress [9]. It is the purpose of this Communication to report on the results of an experimental study of the firstorder Raman spectrum of SrF2 in the presence of a uniaxial stress. From such measurements the phonon deformation potentials are computed, which then are used to calculate the mode Gr~ineisen parameter. Comparison is made with similar results for CaF2 and BaF2 aiad for the infrared-active mode of the three materials. As a byproduct of the piezospectroscopic measurements we also obtained values for the elastooptical coefficients. The results agree with those in the literature and concur with the fact that there are stress inhomogeneities connected with specific crystallographic planes.
to a nearby line from a Ne calibration lamp. The latter was inserted into the optical path of the scattered radiation during the run of each spectrum. In order to obtain the position of the calibration band with improved resolution, the slits of the spectrometer were closed temporarily down to 50# shortly before the calibration line was reached. The overall uncertainty in the measured phonon frequency shifts is estimated to 0.05 cm -1 . The samples were in the form of polished rectangular bars of dimensions 1.5 x 1.5 x 15mm 3. Two crystal orientations were used, designated by (x', y', z ' ) = ([1 ]-0], [1 1 0], [00 1]) or b and (x", y", z") = ([1 1 2], [1 1 0], [1 1 1]) or c. The stressing apparatus has been described elsewhere [10]. The stress X was always applied vertically along the long axis of the crystals (z' or z"). 3. THEORETICAL BACKGROUND Fluorite-type crystals exhibit two triply-degenerate long-wavelength optical phonons with symmetries Flu and F2g (point group Oh). The former is infrared active and Raman inactive, the latter is vice versa and is the phonon we are presently concerned with. The way its frequency w shifts and its degeneracy splits in the presence of X is well understood [11]. We give next those definitions and expressions which are necessary for analyzing the data. As phonon deformation potentials we define, in general, the independent components of the fourthrank tensor
2. EXPERIMENTAL DETAILS The Raman scattering measurements were taken at room temperature in the backscattering geometry, with 300mW of 476.5 nm radiation from an Ar ÷ laser. The beam was focused inside the material (approximately in the middle of the transparent sample) with a spherical lens. The spectra were taken with a Spex double monochromator, an RCA-31034 cooled photomultiplier and a photon counting detection system. The instrumental resolution was 2.8 cm -1. Each spectrum consisted of one single band and its peak frequency was measured relative
Ki] = (a~2/~r/)i/,
(1)
or, in units of w 2, /~ij = (~ In
645
co2/~)~j,
(2)
where suppressed index notation is used and r/ is the strain produced by the applied stress. For the F2~ phonon (and, independently, for the transverse and longitudinal components of the F=u
646
RAMAN STUDY OF SrF2 UNDER UNIAXIAL STRESS
phonon) there are three such components designated by the pair of indices (11 = 22 = 33), (12 = 21 = 23 = 31) and (44 = 55 = 66). Alternative notations have been used in the literature such as p, q, r, equal to KH, K x2, K44 respectively [ 11 ], and a, b, c, equal to K1 a/2~ = K Hw/2, K12/2~, K~/2w respectively [9]. When X is applied along z'(z") the frequency co splits into a singlet co's(W's) and a doublet 6o~(6o~). Experimentally these two components are observed separately in the scattering configurations E'(y'y')x', x"(z"z")x", and E'(z'y')x', x"(z"y")x" respectively. 'fhe phonon deformation potentials and the mode Grtineisen parameter are given by the expressions K,, --/£,2 -
2(5~'/X)
(3)
~ ( S . -S~2)'
i
Vol. 58, No. 9 i
I
b
3
I
i
I
c
x//[oo q v'~
x//(.q 2
s
0 I
I
4
2
UNIAXIAL
l
0
STRESS
I
l
2
X (Kbar)
Fig. 1. Frequency shifts of the singlet (Aw = COs- w) and doublet ( A ~ = ~ a - - c o ) components of the F2¢ phonon of SrF2 as a function of the compressive stress X (absolute value) at 300 K. Solid lines are the results of least square fits.
/(44 = 2(86o"/X) wS~
'
(4)
"7 = -- {(RH + 2R,z)
(5a)
(8~/x) = --36o($1, + 2S,2)'
(5b)
where the components of the elastic compliances are [3] S11=9.89, $ 1 2 = - - 2 . 5 9 and $44=31.6 in units of (TPa) -1. The slopes in equations ( 3 - 5 ) are obtained directly from the measured frequencies W's,d(W'~',a) as a function of the applied compressive (negative) uniaxial stress, according to the definitions
~' X
-
d(~',
- ~o~}
dX
'
8co"
d(c~ s -- co~)
X
dX
5w X -
-
=
-
d(w', + 2w~) dX
d(.," + 2 , ~ ) dX
(6a)
(6b)
(6c) (6d)
4. RESULTS AND DISCUSSION The results of the measurements are shown in Fig. 1 for the two crystal orientations b and c. The solid lines are the results of least square fits. The four slopes in units of cm-1/GPa corresponding to equations (6a, b, c, d) are --7.4-+0.2, --2.0-+0.3, - - 5 . 6 + 0 . 3 and --7.4-+ 0.4 respectively. The values for /(11, /(12, / ( ~ and for the Griineisen parameters are listed in Table 1. One general observation is the similarity between the data in Fig. 1 and the corresponding data for CaF2 and BaF2 in [9]. That is to say, the slope for the doublet in geometry b is negative (relative to absolute X) and, further, co's >w'a contrary to Si [11], Ge [12] and
diamond [13]. In addition, in our measurements with geometry c we have experienced the same difficulties described in [9] concerning the early breaking of the BaF2 crystals in the same geometry. This is probably the reason why no value for /(44 is given in [91 for BaF2. In fact, according to our measurements in SrF2, the corresponding value of 7 " = 1.84 deviates by as much as 30% from the value of 1" = 1.40 which was measured in geometry b. Since both values are in the range of values found in the literature [4, 8, 14] we choose to use their average value of 7 = 1.62 + 0.22 for the subsequent calculation of K n and /£a2. Notice that in Fig. 1 we include all points obtained from runs with three independent samples. Within a particular run of each sample the established splitting Ws--~a was practically the same for all samples. This means that the values of Ki: which are determined from these splittings should be regarded as less uncertain than the corresponding values of "7. Besides the phonon deformation potentials for SrF2, Table 1 includes the corresponding values for CaF2 and BaF2 based on the information of [9, 7, 14]. The similarity of these values in all three materials and particularly between SrF2 and BaF2 permits a projection to be made for/(44 of BaF2, which we place at -- 0.40 -+ 0.05. Interestingly, in all three cases the F2g phonon involves the same opposing motion of the two F - sublattices; therefore one may expect a similarity in the interatomic force constants involved and also in their changes due to X (that is, the phonon deformation potentials). For completeness, we include in Table 1 the phonon deformation potentials and mode Griineisen coefficient for the transverse component of the F~u phonon. The necessary values for Kll --/~12 and K ~ are computed from the results of [ 15] where such values were obtained from piezobirefringence measurements in the infrared.
Vol. 58, No. 9
RAMAN STUDY OF SrF2 UNDER UNIAXIAL STRESS
647
Table 1. Phonon deformation potentials, mode Grfineisen parameters and frequencies (in cm -~) for the Ramanactive (F2g) and infrared-active (Fl u) long-wavelength transverse optical (TO) phonons in alkaline-earth fluorites CaF2
SrF2
BaF2
F~'g
F, u(TO) b
F2g
F, u(TO)
F~'g
F,.(TO) b
289 1.8 d
285 1.4e 1.84 f 1.5 i 1.6 e
251 b
247 2.0 e 1.8 d
213 0.8 a
7
327 1.9e 1.9 h
/~11 --/~12 /~'H &'12 /~
--6.21 --7.94 -- 1.73 -- 0.63
1.17 --2.82 -- 4.00 -- 0.57
--4.16-+0.12 --6.00-+0.40 -- 1.90 -+ 0.40 -- 0.44 -+ 0.07
1.36 --1.10 -- 2.45 -- 0.43 b
--4.08 --6.52 -- 2.44 -- 0.40 g -+ 0.05
1.63 --0.51 -- 2.14 -- 0.47
co
1.0 g
aReference [9], .bReference [15], CReference [7], aReference [16], eX II [0 0 1], f x II [1 1 1], gSuggested, hReference [14], 1Reference [8]. The corresponding mode Griineisen parameters have been measured in [16] from reflectivity and transmission experiments in the infrared. Notice however that substantial differences exist between the Flu phonon frequencies (TO) stated in [15, 16]. Since the results of [15] are heavily based on the frequency values mentioned there, we choose to use the same ones (Table 1). Again the overall similarity for SrF2 and BaF2 (both are less ionic than CaF2) allows us to project a value of 3' = 1.0 -+ 0.2 for SrF2 (not measured in [16] ) and proceed with the calculation of/~ll and K12 for the Flu (TO) phonon of SrF2. Finally, a comparison between the results for the F2g and Flu (TO) phonons leads to the following remarks. For all three materials, the differences Kll--/~12 are larger (in absolute value)for the F2g phonon and of opposite sign than those of the Flu (TO) phonon. Secondly, in all cases the values and signs of K44 are alike. As already mentioned, stressing SrF2 along [1 1 1] created particular technical difficulties, as the samples often broke at low stress levels or cleaved in planes containing the stress axis. In order to assess the state of the crystal while under stress, independent piezobirefringence measurements were taken with both propagation directions in both samples b and c. The static method was employed with 632.8nm, as described in [17, 18]. Since crystals c broke before the applied stress reached the necessary value for inducing piezobirefringence of 180 °, the measurements were taken with a Babinet-Soleil compensator at much lower stresses. From the linear relationship between the observed phase shifts and the corresponding stresses [18] the following values were reached from the elastooptical coefficients, Pit -- P~2 = -- 0.170 from sample b and either direction of propagation, and P44 = 0.0185 (0.022) for propagation along x"(y"). The values given in the literature for 632.8nm are [19] Pll--Pl2 =
--0.189 and P44 = 0.0185. Therefore, it appears that for propagation along y", the light beam probes areas where residual and/or induced strain inhomogeneities are present and are associated with specific crystallographic planes. Similar observations have been made and commented on in [20] in connection with piezoRaman work in Si. A systematic investigation of such residual strains using X-ray techniques seems worthwhile.
Acknowledgements - This work was supported by the Ministry of Industry, Energy and Technology, Greece. Thanks are due to Mr. G. Vangelakis of Ioannina University for providing the SrF2 material and to the crystal preparation group of Max Planck Institute, Stuttgart, for preparing the samples used in these experiments.
REFERENCES 1.
W. Hayes & A.M. Stoneham, Crystals with the Oxford, Clarendon Press, (1974). S. Dasgupta, A.N. Basu & A. Basu, Phys. Rev. B30, 7255 (1984); M.O. Manasreh & D.O. Pederson,Phys. Rev. B31, 3960 (1985). S. Alterovitz & D. Gerlich, Phys. Rev. B1, 2718 (1970). R.J. Elliot, W. Hayes, W.G. Kleppmann, A.J. Rushworth & J.F. Ryan, Proc. R. Soc. Lond. A360, 317 (1978). W. Hayes, Contemp. Phys. 19, 469 (1978). D.G. Mead & G.R. Wilkinson, J. Phys. C. Solid StatePhys. 10, 1063 (1977). J.R. Kessler, E. Monberg & M. Nicol, J. Chem. Phys. 60, 5057 (1974). G. Kourouklis & E. Anastassakis, Phys. Rev. to appear. S. Venugopalan & A.K. Ramdas, Phys. Rev. B8, 717(1973). E. Anastassakis, Appl. Opt. 13, 1971 (1974).
Fluorite Structure,
2. 3. 4. 5. 6. 7. 8. 9. 10.
648 11. 12. 13. 14. 15.
RAMAN STUDY OF SrF2 UNDER UNIAXIAL STRESS E. Anastassakis, A. Pinczuk, E. Burstein, F.H. Pollak & M. Cardona, Solid State Commun. 8, 133 (1970). F. Cerdeira, C.J. Buchenauer, F.H. Pollak & M. Cardona, Phys. Rev. BS, 580 (1972). M.H. Grimsditch, E. Anastassakis & M. Cardona, Phys. Rev. BI8, 901 (1978). S.S. Mitra, (unpublished data cited in [7]). A. Feldman & R.M. Waxler, Phys. Rev. Lett. 45, 126 (1980).
16. 17. 18. 19. 20.
Vol. 58, No. 9
J.R. Ferraro, H. Horan & A. Quattrochi, J. Chem. Phys. 55,664 (1971). A.D. Papadopoulos & E. Anastassakis, unpublished. M.H. Grimsditch, E. Anastassakis & M. Cardona, Phys. Rev. BI9, 3240 (1979). O.V. Shakin, M.F. Bryzhina & V.V. Lemanov, Soy. Phys. Solid State 13, 3141 (1972); C. Sanchez & M. Cardona,Phys. Status Solidi ( b ) 50, 293 (1972). V.V. Baptizmanskii, I.I. Novak & Y.F. Titovets, Soy. Phys. Solid State 21, 1915 (1979).
Solid State Communications, Vol. 58, No. 9, pp. 645-648, 1986. Printed in Great Britain.
RAMAN STUDY OF SrF2 UNDER UNIAXIAL STRESS A.D. Papadopoulos, Y.S. Raptis and E. Anastassakis Physics Laboratory III, National Technical University, Athens 157 73, Greece
(Received 31 January 1986 by M. Cardona) The first-order Raman scattering in SrF2 is studied at room temperature in the presence of a uniaxial stress. The phonon deformation potentials for the long-wavelength Raman-active optical phonon are determined and compared to those of other materials with the fluorite structure. 1. INTRODUCTION ALKALINE-EARTH FLUORITES, have always attracted considerable interest concerning their lattice-dynamical [ 1 - 3 ] , optical [1, 4] and transport [4, 5] properties. In addition, effects of external perturbations have been examined, theoretically and experimentally, as an independent method for testing proposed models. In this connection, the Raman spectra for all three of the most common members of this group, i.e. CaF2, BaF2 and SrF2, have been investigated as a function of temperature [6] and hydrostatic pressure [7, 8]. On the other hand, only CaF2 and partially BaF2 have been studied in the presence of a uniaxial stress [9]. It is the purpose of this Communication to report on the results of an experimental study of the firstorder Raman spectrum of SrF2 in the presence of a uniaxial stress. From such measurements the phonon deformation potentials are computed, which then are used to calculate the mode Gr~ineisen parameter. Comparison is made with similar results for CaF2 and BaF2 aiad for the infrared-active mode of the three materials. As a byproduct of the piezospectroscopic measurements we also obtained values for the elastooptical coefficients. The results agree with those in the literature and concur with the fact that there are stress inhomogeneities connected with specific crystallographic planes.
to a nearby line from a Ne calibration lamp. The latter was inserted into the optical path of the scattered radiation during the run of each spectrum. In order to obtain the position of the calibration band with improved resolution, the slits of the spectrometer were closed temporarily down to 50# shortly before the calibration line was reached. The overall uncertainty in the measured phonon frequency shifts is estimated to 0.05 cm -1 . The samples were in the form of polished rectangular bars of dimensions 1.5 x 1.5 x 15mm 3. Two crystal orientations were used, designated by (x', y', z ' ) = ([1 ]-0], [1 1 0], [00 1]) or b and (x", y", z") = ([1 1 2], [1 1 0], [1 1 1]) or c. The stressing apparatus has been described elsewhere [10]. The stress X was always applied vertically along the long axis of the crystals (z' or z"). 3. THEORETICAL BACKGROUND Fluorite-type crystals exhibit two triply-degenerate long-wavelength optical phonons with symmetries Flu and F2g (point group Oh). The former is infrared active and Raman inactive, the latter is vice versa and is the phonon we are presently concerned with. The way its frequency w shifts and its degeneracy splits in the presence of X is well understood [11]. We give next those definitions and expressions which are necessary for analyzing the data. As phonon deformation potentials we define, in general, the independent components of the fourthrank tensor
2. EXPERIMENTAL DETAILS The Raman scattering measurements were taken at room temperature in the backscattering geometry, with 300mW of 476.5 nm radiation from an Ar ÷ laser. The beam was focused inside the material (approximately in the middle of the transparent sample) with a spherical lens. The spectra were taken with a Spex double monochromator, an RCA-31034 cooled photomultiplier and a photon counting detection system. The instrumental resolution was 2.8 cm -1. Each spectrum consisted of one single band and its peak frequency was measured relative
Ki] = (a~2/~r/)i/,
(1)
or, in units of w 2, /~ij = (~ In
645
co2/~)~j,
(2)
where suppressed index notation is used and r/ is the strain produced by the applied stress. For the F2~ phonon (and, independently, for the transverse and longitudinal components of the F=u
646
RAMAN STUDY OF SrF2 UNDER UNIAXIAL STRESS
phonon) there are three such components designated by the pair of indices (11 = 22 = 33), (12 = 21 = 23 = 31) and (44 = 55 = 66). Alternative notations have been used in the literature such as p, q, r, equal to KH, K x2, K44 respectively [ 11 ], and a, b, c, equal to K1 a/2~ = K Hw/2, K12/2~, K~/2w respectively [9]. When X is applied along z'(z") the frequency co splits into a singlet co's(W's) and a doublet 6o~(6o~). Experimentally these two components are observed separately in the scattering configurations E'(y'y')x', x"(z"z")x", and E'(z'y')x', x"(z"y")x" respectively. 'fhe phonon deformation potentials and the mode Grtineisen parameter are given by the expressions K,, --/£,2 -
2(5~'/X)
(3)
~ ( S . -S~2)'
i
Vol. 58, No. 9 i
I
b
3
I
i
I
c
x//[oo q v'~
x//(.q 2
s
0 I
I
4
2
UNIAXIAL
l
0
STRESS
I
l
2
X (Kbar)
Fig. 1. Frequency shifts of the singlet (Aw = COs- w) and doublet ( A ~ = ~ a - - c o ) components of the F2¢ phonon of SrF2 as a function of the compressive stress X (absolute value) at 300 K. Solid lines are the results of least square fits.
/(44 = 2(86o"/X) wS~
'
(4)
"7 = -- {(RH + 2R,z)
(5a)
(8~/x) = --36o($1, + 2S,2)'
(5b)
where the components of the elastic compliances are [3] S11=9.89, $ 1 2 = - - 2 . 5 9 and $44=31.6 in units of (TPa) -1. The slopes in equations ( 3 - 5 ) are obtained directly from the measured frequencies W's,d(W'~',a) as a function of the applied compressive (negative) uniaxial stress, according to the definitions
~' X
-
d(~',
- ~o~}
dX
'
8co"
d(c~ s -- co~)
X
dX
5w X -
-
=
-
d(w', + 2w~) dX
d(.," + 2 , ~ ) dX
(6a)
(6b)
(6c) (6d)
4. RESULTS AND DISCUSSION The results of the measurements are shown in Fig. 1 for the two crystal orientations b and c. The solid lines are the results of least square fits. The four slopes in units of cm-1/GPa corresponding to equations (6a, b, c, d) are --7.4-+0.2, --2.0-+0.3, - - 5 . 6 + 0 . 3 and --7.4-+ 0.4 respectively. The values for /(11, /(12, / ( ~ and for the Griineisen parameters are listed in Table 1. One general observation is the similarity between the data in Fig. 1 and the corresponding data for CaF2 and BaF2 in [9]. That is to say, the slope for the doublet in geometry b is negative (relative to absolute X) and, further, co's >w'a contrary to Si [11], Ge [12] and
diamond [13]. In addition, in our measurements with geometry c we have experienced the same difficulties described in [9] concerning the early breaking of the BaF2 crystals in the same geometry. This is probably the reason why no value for /(44 is given in [91 for BaF2. In fact, according to our measurements in SrF2, the corresponding value of 7 " = 1.84 deviates by as much as 30% from the value of 1" = 1.40 which was measured in geometry b. Since both values are in the range of values found in the literature [4, 8, 14] we choose to use their average value of 7 = 1.62 + 0.22 for the subsequent calculation of K n and /£a2. Notice that in Fig. 1 we include all points obtained from runs with three independent samples. Within a particular run of each sample the established splitting Ws--~a was practically the same for all samples. This means that the values of Ki: which are determined from these splittings should be regarded as less uncertain than the corresponding values of "7. Besides the phonon deformation potentials for SrF2, Table 1 includes the corresponding values for CaF2 and BaF2 based on the information of [9, 7, 14]. The similarity of these values in all three materials and particularly between SrF2 and BaF2 permits a projection to be made for/(44 of BaF2, which we place at -- 0.40 -+ 0.05. Interestingly, in all three cases the F2g phonon involves the same opposing motion of the two F - sublattices; therefore one may expect a similarity in the interatomic force constants involved and also in their changes due to X (that is, the phonon deformation potentials). For completeness, we include in Table 1 the phonon deformation potentials and mode Griineisen coefficient for the transverse component of the F~u phonon. The necessary values for Kll --/~12 and K ~ are computed from the results of [ 15] where such values were obtained from piezobirefringence measurements in the infrared.
Vol. 58, No. 9
RAMAN STUDY OF SrF2 UNDER UNIAXIAL STRESS
647
Table 1. Phonon deformation potentials, mode Grfineisen parameters and frequencies (in cm -~) for the Ramanactive (F2g) and infrared-active (Fl u) long-wavelength transverse optical (TO) phonons in alkaline-earth fluorites CaF2
SrF2
BaF2
F~'g
F, u(TO) b
F2g
F, u(TO)
F~'g
F,.(TO) b
289 1.8 d
285 1.4e 1.84 f 1.5 i 1.6 e
251 b
247 2.0 e 1.8 d
213 0.8 a
7
327 1.9e 1.9 h
/~11 --/~12 /~'H &'12 /~
--6.21 --7.94 -- 1.73 -- 0.63
1.17 --2.82 -- 4.00 -- 0.57
--4.16-+0.12 --6.00-+0.40 -- 1.90 -+ 0.40 -- 0.44 -+ 0.07
1.36 --1.10 -- 2.45 -- 0.43 b
--4.08 --6.52 -- 2.44 -- 0.40 g -+ 0.05
1.63 --0.51 -- 2.14 -- 0.47
co
1.0 g
aReference [9], .bReference [15], CReference [7], aReference [16], eX II [0 0 1], f x II [1 1 1], gSuggested, hReference [14], 1Reference [8]. The corresponding mode Griineisen parameters have been measured in [16] from reflectivity and transmission experiments in the infrared. Notice however that substantial differences exist between the Flu phonon frequencies (TO) stated in [15, 16]. Since the results of [15] are heavily based on the frequency values mentioned there, we choose to use the same ones (Table 1). Again the overall similarity for SrF2 and BaF2 (both are less ionic than CaF2) allows us to project a value of 3' = 1.0 -+ 0.2 for SrF2 (not measured in [16] ) and proceed with the calculation of/~ll and K12 for the Flu (TO) phonon of SrF2. Finally, a comparison between the results for the F2g and Flu (TO) phonons leads to the following remarks. For all three materials, the differences Kll--/~12 are larger (in absolute value)for the F2g phonon and of opposite sign than those of the Flu (TO) phonon. Secondly, in all cases the values and signs of K44 are alike. As already mentioned, stressing SrF2 along [1 1 1] created particular technical difficulties, as the samples often broke at low stress levels or cleaved in planes containing the stress axis. In order to assess the state of the crystal while under stress, independent piezobirefringence measurements were taken with both propagation directions in both samples b and c. The static method was employed with 632.8nm, as described in [17, 18]. Since crystals c broke before the applied stress reached the necessary value for inducing piezobirefringence of 180 °, the measurements were taken with a Babinet-Soleil compensator at much lower stresses. From the linear relationship between the observed phase shifts and the corresponding stresses [18] the following values were reached from the elastooptical coefficients, Pit -- P~2 = -- 0.170 from sample b and either direction of propagation, and P44 = 0.0185 (0.022) for propagation along x"(y"). The values given in the literature for 632.8nm are [19] Pll--Pl2 =
--0.189 and P44 = 0.0185. Therefore, it appears that for propagation along y", the light beam probes areas where residual and/or induced strain inhomogeneities are present and are associated with specific crystallographic planes. Similar observations have been made and commented on in [20] in connection with piezoRaman work in Si. A systematic investigation of such residual strains using X-ray techniques seems worthwhile.
Acknowledgements - This work was supported by the Ministry of Industry, Energy and Technology, Greece. Thanks are due to Mr. G. Vangelakis of Ioannina University for providing the SrF2 material and to the crystal preparation group of Max Planck Institute, Stuttgart, for preparing the samples used in these experiments.
REFERENCES 1.
W. Hayes & A.M. Stoneham, Crystals with the Oxford, Clarendon Press, (1974). S. Dasgupta, A.N. Basu & A. Basu, Phys. Rev. B30, 7255 (1984); M.O. Manasreh & D.O. Pederson,Phys. Rev. B31, 3960 (1985). S. Alterovitz & D. Gerlich, Phys. Rev. B1, 2718 (1970). R.J. Elliot, W. Hayes, W.G. Kleppmann, A.J. Rushworth & J.F. Ryan, Proc. R. Soc. Lond. A360, 317 (1978). W. Hayes, Contemp. Phys. 19, 469 (1978). D.G. Mead & G.R. Wilkinson, J. Phys. C. Solid StatePhys. 10, 1063 (1977). J.R. Kessler, E. Monberg & M. Nicol, J. Chem. Phys. 60, 5057 (1974). G. Kourouklis & E. Anastassakis, Phys. Rev. to appear. S. Venugopalan & A.K. Ramdas, Phys. Rev. B8, 717(1973). E. Anastassakis, Appl. Opt. 13, 1971 (1974).
Fluorite Structure,
2. 3. 4. 5. 6. 7. 8. 9. 10.
648 11. 12. 13. 14. 15.
RAMAN STUDY OF SrF2 UNDER UNIAXIAL STRESS E. Anastassakis, A. Pinczuk, E. Burstein, F.H. Pollak & M. Cardona, Solid State Commun. 8, 133 (1970). F. Cerdeira, C.J. Buchenauer, F.H. Pollak & M. Cardona, Phys. Rev. BS, 580 (1972). M.H. Grimsditch, E. Anastassakis & M. Cardona, Phys. Rev. BI8, 901 (1978). S.S. Mitra, (unpublished data cited in [7]). A. Feldman & R.M. Waxler, Phys. Rev. Lett. 45, 126 (1980).
16. 17. 18. 19. 20.
Vol. 58, No. 9
J.R. Ferraro, H. Horan & A. Quattrochi, J. Chem. Phys. 55,664 (1971). A.D. Papadopoulos & E. Anastassakis, unpublished. M.H. Grimsditch, E. Anastassakis & M. Cardona, Phys. Rev. BI9, 3240 (1979). O.V. Shakin, M.F. Bryzhina & V.V. Lemanov, Soy. Phys. Solid State 13, 3141 (1972); C. Sanchez & M. Cardona,Phys. Status Solidi ( b ) 50, 293 (1972). V.V. Baptizmanskii, I.I. Novak & Y.F. Titovets, Soy. Phys. Solid State 21, 1915 (1979).