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Study of elastic properties of sodium, potassium and rubidium acid phthalates by brillouin scattering
J. Phyr. Printed
Chem. Solldt Vol. in Great Britain.
53. No.
4. pp.
51 l-520.
S5.00 + 0.00 0022~3697192 Pergunon Press pk
1992
STUDY OF ELASTIC PROPERTIES OF SODIUM, POTASSIUM AND RUBIDIUM ACID PHTHALATES BY BRILLOUIN SCATTERING C. ECOLIVET,~ A. MINIEWICZ$ and M. SANQUER~ tGroupe Matitre Condenske et Materiaux, URA 804, Universitt de Rennes I, Campus de Beaulieu, 35042 Rennes-Cedex, France zlnstitute of Organic and Physical Chemistry, Technical University of Wroclaw, 50370 Wroclaw, Poland (Received
13 May 1991; accepted in revisedform
24 September
1991)
Abstract-Elastic properties of three, isomorphous, orthorhombic (mm2) molecular-ionic crystals of sodium, potassium and rubidium acid phthalates have been studied employing the Brillouin scattering technique. The values of all elastic constants: stifkesses ceBand compliances sd were determined at 294 K using the measured longitudinal and transverse sound velocities for different phonon directions. Keyworcis: Brillouin scattering, elastic stiffness, elastic compliance, piezoelectric effect, acoustic phonons,
KAP, NaAP, RbAP.
Acid phthalates can be described by the general formula (C,H,O;),Me”+ .m H,O where Me+ is a cation. Some of them are widely used as monochromaters in X-ray fluorescence analysis or analysis of X-ray spectra of the Sun. We restrict our interest to three, isomorphous, polar crystals of monovalent cations, i.e. sodium acid phthalate (hereafter abbreviated as NaAP) potassium acid phthalate (KAP), and rubidium acid phthalate (RbAP). All of them crystallize in orthorhombic class C,, (mm2), thus exhibiting pyroelectric [ 1,2], piezoelectric [3,4] as well as electro-optic properties [S, 61.The latter ones when linked with the ease of obtaining large single crystals of excellent optical quality, by growth from a water solution, may be promising for a much wider application of this group of crystals. Among the three above-mentioned crystals only KAP is relatively well characterized. Its piezoelectric, dielectric and elastic properties were determined by Belyaev et al. [3], electro-optic by Shaldin er al. [5] and Miniewicz et al. [4,6], elasto-optic by Shahabuddin Khan et al. [A, pyroelectric by Poprawski et al. [l] while Raman and infrared spectra were measured by Krishnamurthy et al. [8], Ore1 et al. [9] and Miniewicz ef al.[IO]. Much less is known about NaAP and RbAP for which pyroelectric properties were measured by Poprawski et al. (21 and low-frequency Raman and infrared spectra (in NaAP) by Miniewicz et al. [ll]. The aim of this paper is to establish, by means of Brillouin scattering, the full tensors of elastic stiffnesses cZ8and elastic compliances s,,@(a, fi = l-6) for the three compounds at room temperature and com-
pare them, in the case of KAP, with the results obtained by an ultrasonic technique [3]. Studies of high-frequency (_ 10 GHz) elastic constants are important for elasto- and electro-optic applications of acid phthalates.
EXPERIMENTAL Salts of phthalic acid were obtained by dissolving stoichiometric amounts of sodium, potassium or rubidium bicarbonates and phthalic acid in water. Large, transparent crystals formed on evaporation as the major product. In any case, crystals were obtained after multiple recrystallization from water solution, by a slow evaporation of water at room temperature. Due to the presence of a perfect cleavage plane containing the C, polar axis, these crystals were easily oriented by means of a polarizing microscope and pyroelectric determination of the polar axis. Parallelepipeds of suitable size (IO x 2 x 3 mm3) were cut and cleaved from larger single crystals, with edges along the orthorhombic u-, b- and c-axes. A krypton ion laser has been used as a light source, generating a C.W.(I 50 mW), single mode, radiation at 647.1 nm. The scattered light was analyzed by a piezoelectritally scanned five-pass Fabry-Perot interferometer, which was controlled by a microprocessor-based system. This interferometer yielded a typical effective finesse of 60-70. A detailed description of the experimental system is described in [l2]. In order to measure all the necessary, and some supplementary, sound wave velocities a right angle, back- and small-angle scattering geometries were
511
512
C. ECOL~VZT etal.
employed (assuming that there is no important dispersion around the IO GHz frequency region). All the measurements were performed on samples which were mechanically free and which fulfilled electrical open circuit conditions, e.g. no electrodes were deposited onto the crystal faces.
STRUCIIJRE
Crystals of this type are molecular-ionic materials.
RESULTS
AND
classified
as
model
DISCUSSION
The Brillouin shift (Av) for a birefringent crystal is given by
DESCRIPTION
Av= f~(n:+nj-2n,n,cos8)‘.‘, crystal structures of NaAP, KAP and RbAP were reported by several authors to belong to the orthorhombic class tnrn2. However, al1 of the compounds were originally described in a non-standard setting 2mm. In Table I the exact chemical formula, space group symmetry, cell occupation factor 2, lattice parameters and densities are listed. In fact NaAP is a sodium acid phthaiate hemihydrate. The B2,ab space group of NaAP is a non-standard setting of the A&2, one and the P2,ab group of KAP and RbAP is a non-standard setting of the Pca2, space group. In order to keep a standard (mm2) setting for all the crystals we adopted here an assignment of the Cartesian x, y, z axes system after Belyaev et al. [3], i.e. .Y= h, y = c and z = a of the originally reported structures [13, 14, 151, where the z-axis is a polar one and the cleavage plane within this system is (010). All these structures are similar, having a perfect xz cleavage plane due to double layers of ionic and hydrogen bonded phthahc acid ions with alkali metal ions, interacting only by weak van der Waals forces along the y-axis. The structures of KAP and RbAP are almost identical, whereas the structure of NaAP [13] differs slightly due to the presence of water of crystallization which occupies a position of site symmetry 2 and is hydrogen bonded to two atoms of ionized carboxyl groups with the G-H. . . 0 distance being 2.80 A. The strong H-bonds between oxygen of the non-ionized carboxyl group of the phthalate ion and the oxygen of another ion were found in all the considered compounds, in KAP the shortest H-bond amounts to 2.546A [14]. Alkali metal cations are surrounded by six oxygen atoms. The average values of the distance of the six nearest oxygens to the central cation amount to 2.45 A, 2.80 A and 2.95 A for NaAP, KAP and RbAP, respectively, thus reflecting the increase in the ionic radius of the cation.
(I)
The
where & is the incident laser wavelength, V the phase velocity of the acoustic phonon, ni and n, are the indices of refraction for the incident and scattered light beams, respectively, and 0 is the scattering angle. Refractive indices necessary to calculate sound velocities were measured at 6’74.1nm by a modified prism method and confirmed by Brillouin scattering at different angles. For crystals belonging to an orthorhombic system the tensor of elastic stiffness coefficients contains only nine non-zero, independent components cJ (in conventional matrix notation): c,,, cZZ,cX3,c,, C,,, C,, C,*, C,s and C,,. The final values of c,# the elastic stiffness tensor components, measured in adiabatic conditions, for any of the crystals studied were calculated on the basis of over 20 sound velocity measurements in various directions. In order to obtain their best average values the least squares fitting procedure was employed. The results of these calculations are gathered in Table 2. The errors of determination of the c, (Z = l-6) components are estimated not to exceed +2%, whereas the off-diagonal components c,*, c,~ and c23are determined with an error &7%. Inversion of the c,,, matrix yields directly the elastic compliance tensor elements s,~. In order to visualize the anisotropy of sound propagation in the materials studied we calculated sound velocity diagrams. Such diagrams for any of the three principal planes are shown in Figs 1-3. Despite the presence of weak van der Waals bonds parallel to the y-axis the longitudinal sound wave velocities do not show any remarkable anisotropy, in any of the three compounds. One can suppose that the lack of anisotropy in this series of crystals must be due to the effect of mutual compensation of weak van der Waals interactions between double-layers by
Table 1. Crystal data of NaAP, KAP and RbAP Formula
Space group
2
Na(C~H~O,).~H~O KC&&O, RM=,H,O,
B2,nb P2,ab P2,ab
8 6.75 4 6.46 4 6.561
a
b 9.31 9.60 10.064
c (4
PC&
26.60 13.85 13.068
1.566 1.572 1.579 1.64 1.929 1.94
Densities: pnk (calculated) and P,,, (measured) are expressed in gem
Pm
Ref. 1131 (141 [IS]
’ units.
Elastic properties of sodium, potassium and rubidium acid phthalates
513
Table 2. Elastic stiffnesses Coe and elastic compliances SDafor NaAP, KAP and RbAP crystals obtained from a Brillouin scattering experiment at 294K. The densities p, (in gcne3) and the indices of refraction at
1 = 674.1nm are also ouoted in 10mgN m-2 C, at _ 15 GHz NaAP (p, = 1.572) n, = 1.64 nv = 1.65 n = 1.48 KAP (p,= 1.64) n,= 1.656 nu= 1.654 nz= 1.491 RbAP (p, = 1.94) n, = 1.67 nY= 1.66 n, = 1.49 S in lo-“mZN-’ a?- 15 GHz NaAP KAP RbAP
Cn
c44
C,,
18.37 15.73
7.95
8.06 10.56
18.40 13.65 19.32
5.11
6.82
19.75 14.30 15.46
4.95
7.82
22.9
c22
c,,
C,
C,,
c,,
C,,
12.94
5.5
10.19
6.4
8.26
7.66
12.45
6.03
9.06
6.73
12.12
%I s 33 su s,, & 42 s22 9.13 9.05 8.93 12.58 12.41 9.47 -5.21 10.71 10.47 9.56 19.57 14.66 15.62 -3.35 11.12 10.03 12.68 20.20 12.79 16.58 -3.70
relatively strong ionic and H-bonded interactions within a double layer. Comparison of the elastic properties of NaAP to the two other crystals KAP and RbAP reflects the difference between their respective structures. Presence of water molecules in the proximity of Na+ ions produces two additional H-bonds and adds one coordination oxygen which strengthens the bonding within double-layers of the phthalate ions. This makes the elastic stiffness of NaAP generally higher than those of KAP and RbAP. We believe that additional H-bonds present in the NaAP structure are responsible for the longitudinal phonon velocities to be the highest along the (110) direction, which is not the case for KAP and RbAP. In all the crystals the smallest anisotropy for both longitudinal or transverse sound velocities was observed within the (100) plane. As one can expect, the overall sound
s23
&3
0.21 -1.99 -1.46
- 4.09 -5.57 -7.11
are the largest for NaAP, smaller for KAP and still smaller for RbAP, in accordance with the respective density increase. In Table 3 we compare the results of elastic stiffness determinations by the two measurement methods: Brillouin scattering (this work) and an ultrasonic technique (Belyaev et ai. [3]). The elastic from Brillouin scattering stiffnesses obtained experiments are consistently higher than those obtained from ultrasound, except for the cSs component. The most important difference (-34%) is observed for the cz3 component. This discrepancy can be understood if one takes into account the fact that all off-diagonal coefficients cannot be measured directly by either of the methods, being only calculated as an average of some independent measurements of sound velocities where they are involved. velocities
Fig. l(1)
C. ECOLIWTet al.
514
nap2
Fig. l(2) nap3
x B)
Fig. l(3)
Fig. 1. Sound velocity diagrams in NaAP single crystal at 294 K as viewed along the three principal crystallographic directions. QL-Quasilongitudinal acoustic phonon, QT-quasitransverse acoustic phonon, T-transverse acoustic phonon.
Other, less pronounced, differences, seen in Table 3, between the values of cGPobtained by us and by Belyaev et af. [3] are, in our opinion, uniquely connetted with the way a given coefficient was determined by either of the methods, i.e. directly from the measurements or indirectly from calculations. In our experiment, the only indirectly obtained elastic stiff-
nesses are: c,, cn, cl2 and cz3. However, looking for the source of possible errors, one must take into account that the cctsfmatrix used for comparison purposes was obtained through the inversion of the matrix s,~ quoted by Belyaev et al. 131,and thus the values of .rERmeasured with an error propagate this error towards values of the cclamatrix. Taking into
Elastic properties of sodium, potas! Gum and rubidium acid phthalates
all the above considerations, one must admit that the differences between the c,~ values determined for KAP by the two methods are acceptable. One should also mention that some differences may occur due to the dispersion effect of the elastic stiffness coefficients Q(V). Brillouin scattering establishes elastic stiffness components at frequency (w 15 GHz) whereas ultrasonic measurements were done at N 1 MHz. Usually, for non-ferroelectric piezoelectrics, if there is no important relaxation in account
515
the 1 MHz-15 GHz frequency range, the dispersion accounts for no more than 1%. As all the crystals studied in this work belong to the class of nonferroelectric piezoelectrics (no phase transition in any of these compounds has been reported to our knowledge), one should not expect important relaxation processes in the above-mentioned frequency range. There are some indications that this is true at least for KAP single crystal. Namely, its dielectric constants measured at 1 MHz are relatively low (6: = 6.00,
kapl
Fig. 2(A)l kap2
Fig. 2(A)2
516
C. ECOLIVET et al.
Fig. 2(A)3
Fig. 2(B)l
cs=3gl and es2 - 4.34 [9) and according to our Y . estimations based on recent far-infrared reflectivity spectra in KAP crystal [ 161the low-frequency dielectric constants are in great part explained by contribution of the lattice and internal modes and pure electronic polarizabilities. It is well known, that the propagation velocity of sound waves in a piezoelectric solid is different from that in a non-piezoelectric solid due to the coupling of acoustic phonons with polarization
components. The values of the elastic constants established for a piezoelectric can be different depending on electrical boundary conditions fulfilled in the course of the measurements [17]. In the case when an electric field E can be considered as constant one deals with CL elastic stiffness tensor components and when an electric displacement vector D can be considered as constant one speaks of elastic stiffness tensor components C
D
iikl
Elastic properties of sodium, potassium and rubidium acid phthalates
517
kapb2
Fig. 2(B)2 kapb3
Fig. 2(B)3
Fig. 2. Sound velocity diagrams in KAP single crystal at 294K as viewed along the three principal crystallographic directions (A) Brillouin, (B) Ultrasonics [3].
Due to the open electrical circuit conditions used in our experiment (no electrodes at the crystal faces), the values of the elastic constants determined by us cannot be treated as either pure c$’ or co ones. Assuming that the optical polarization components are coupled to the phonons propagating at frequenties around IS GHz (as in our Brillouin experiment)
one can suggest that the elastic constants measured by us are rather closer to the c$’ values than the c$ ones. Nevertheless one can estimate the possible influence of a piezoelectric effect on phonon propagation velocities in different directions by solving the so called modified Christoffel equation which properly
C. Fkouvm
518
describes the acoustic propagation solid [18]: det II’, + y
in a piezoelectric
- p Y2Si,J= 0,
tz)
where V is the sound velocity, p is the density and S,, is the Kronecker symbol, r, = c&n,nn. where nj is a component of the unitary propagation vector, yr = ekqnjnk where e,, is piezoelectric tensor component and finally t = c$rj~~ with E; a dielectric
et al.
tensor component under constant strain. From the solutions of the above equation for an mm2 class piezoelectric it follows that sound velocities for certain transverse or iongitudina~ phonons propagating in some principal directions are not affected by the piezoelectric effect. They are affected, however, in any arbitrarily chosen direction. The estimation of piezoelectric influence on sound velocity can however be done only for the KAP crystal for which all the necessary data can be found.
Fig. 3(l)
Fig. 3(2)
Elastic properties of sodium, potassium and rubidium acid phthalates
519
rap3
L.-.-
Fig. 3(3) Fig. 3. Sound velocity diagrams in RbAP single crystal at 294 K as viewed along the three principal c~stallographic directions.
One can calculate its piezoelectric e,, coefficients using the relation [I81 e, = dibciz,
(3)
and employing the dippiezoelectric coefficients quoted in [3] and respective values of CL. The eh piezoelectric coefficients of KAP obtained in this way are: e31= -0.147, ejz=0.0315, e3) = -0.0212, e,, = -0.0544 and ez4= 0.02089 Cm-‘. Then taking the values of dielectric ~~ittivities of KAP at 1 MHz: t,, = 5.312, 622= 3.427 and ~~~= 3.846 in IO-” Fm-’ units [3] one can calculate effective elastic constants Cr,i, (for a given type of a phonon and a given direction of its propagation in the crystal). Similar calculations can be performed without taking account of the piezoelectric effect, just by solving a normal Christoffel equation without the 7,7,/c term present in eqn (2). In this way one obtains the respective effective constant C. Introducing the quantity 100 x (C,,, - C)/C one can express, in per cent, the expected modification of the effective elastic constant by the presence of piezoelectricity in KAP.
Results of such calculations for some phonons propagating along important directions in KAP are shown in Table 4. We have found that for any of the three principal directions (LOO),(010) and (001) only one among the three phonons is affected by the piezoelectric coupling (a detailed description of phonon polarization and type is given in Table 4). In the (011) and (101) directions two phonons are influenced by the piezoelectric effect: a quasi-longitudinal (QL) and a quasi-transverse (QT) one along the (110) only one of a pure transverse type (T). The most important change for phonon propagation velocity in KAP, due to piezoelecttic coupling, occurs for the QL phonon propagating along the (101) direction. The relative change in the effective elastic constant in this case amounts to 2.34%. For other directions and other phonons the effect is smaller, usually much less than I %. One should mention that the calculations were performed with t values in the MHz range, however, due to the small dispersion expected for the dielectric permittivity of KAP between 1 MHz and 15 GHz these results will also apply
Table 3. Comparison of elastic stiffnesses obtained in KAP by Brillouin scattering technique at - I5 GHz with those determined by an ultrasonic method at - I MHz by Belyaev er al. [3] C,, in 10%N m-’ Brillouin ( - 15 GHz) Ultrasonics (- 1 MHz) C$ - C$
-* Cf*
100%
C,,
c22
G,
G
18.40 13.65 19.32 5.11 17.56 13.52 16.97 4.86 4.6
2.9
12.2
4.9
cs,
6.82 7.59 -11.3
G&
c12
c23
6.4 6.23
8.26 7.39
7.66 5.05
12.45 10.38
G,
2.6
10.5
34.1
16.6%
C. ECoLivETet al.
520
Table 4. Estimated effect of piezoelectticity on phonon propagation velocities in KAP Phonon Wave-vector direction
Effective elastic constant C,,,,, =pv2
Polarization type
e?, 611
100 x CC,,,, - C)/C (%) 0.74%
-f@f)l>
c&t-
(010)
T (ml>
c”+& 44
LWI)
c:,+--
QL
+Cz=“‘*
0.07%
QT
c%-
0.30%
QL
c%-
2.34%
QT
GYve
0.36%
T
Cl> __ + CM : (e,5+e24)* 2 4 e33
0.12%
<110)
0.27% 622 43
0.07%
c13
t The complicated formulae for nonprincipal directions of phonon propagation are omitted
to the conditions of Brillouin scattering. Similar, rather negligible, influence of piezoelectricity on the elastic properties of two other crystals studied, NaAP and RbAP, can safely be predicted. Acknow/e&emenrs-Gne of the authors (A.M) is grateful to the Groupe de Physique Cristalline of Universite de Rennes I for their hospitality and financial support during his stay at Rennes. The help in preparations of the figures by Dr Krzysztof Wolinski (IRISA, Rennes) is acknowledged. REFERENCES
Poprawski R., Matyjasik S. and Shaldin Yu. V., S&d Stare Commun. 62, 257 (1987).
Poprawski R., Matyjasik S. and Shaldin Yu. V., Mut. Sci. 13, 203 (1987). Belyaev L. M., Behkova G. S., Gilvarg A. B. and Silvestrova 1. M., Sou. P&r. Crysrailogr. 14,544 (1970). Miniewicz A., Samoc M., Dziedzic J. and Poprawski R., in Proc. 11th Mol. Cryst. Symposium, p. 198. Lugano, Switzerland (1985).
5. Shaldin Yu. V. and Okhrimenko T. M., Sov. Phys. Crystallop. 20, 750 (1975). 6. Miniewin A., Samoc A. and Sworakowski J., J. Mol.
Etecrronics 4, 25 (1988). 7. Shahabuddin Khan M. D. and Narasimh~urty T. S., J. Mat. Sci. Lett. 1, 268 (1982). 8. Krishnamurthy N. and Soots V., J. Rammt Spectr. 6,
221 (1977). 9. Ore1 B., Hadzi D. and Cabassi F., Spectrochimka Acta 31A, 169 (1975). IO. Miniewicz A., Ecolivet C. and Marqueton Y., Mat. Sci.
13, 169 (1987). 11. Miniewicz A., Lefebvre J., Fontaine H. and Marqueton Y., J. Raman Spectr. 21, 177 (1990). 12. Ecolivet C., Sanquer M., Pellegrin J. and Dewitte J., J. Chem. Phys. 78, 6317 (1983).
13. 14. IS. 16. 17.
Smith R. A., Acrcl Crysl. 831, 2345 (1975). Okaya Y., Acla Cryst. 19, 879 (1965). Smith R. A., AC& Crysl. 871, 2347 (1975). Miniewicz A. and Marqueton Y., unpublish~ results. Mason W. P., Phys. Rev. 69, 173 (1946); Brody E. M. and Cummins H. 2.. Phvs. Rev. B9, 179 ( 1974).
18. Dieulesaint E. and Royer D., On&s Elastiques abns les So/ides. Mason et Cie., Paris (1974).
Chem. Solldt Vol. in Great Britain.
53. No.
4. pp.
51 l-520.
S5.00 + 0.00 0022~3697192 Pergunon Press pk
1992
STUDY OF ELASTIC PROPERTIES OF SODIUM, POTASSIUM AND RUBIDIUM ACID PHTHALATES BY BRILLOUIN SCATTERING C. ECOLIVET,~ A. MINIEWICZ$ and M. SANQUER~ tGroupe Matitre Condenske et Materiaux, URA 804, Universitt de Rennes I, Campus de Beaulieu, 35042 Rennes-Cedex, France zlnstitute of Organic and Physical Chemistry, Technical University of Wroclaw, 50370 Wroclaw, Poland (Received
13 May 1991; accepted in revisedform
24 September
1991)
Abstract-Elastic properties of three, isomorphous, orthorhombic (mm2) molecular-ionic crystals of sodium, potassium and rubidium acid phthalates have been studied employing the Brillouin scattering technique. The values of all elastic constants: stifkesses ceBand compliances sd were determined at 294 K using the measured longitudinal and transverse sound velocities for different phonon directions. Keyworcis: Brillouin scattering, elastic stiffness, elastic compliance, piezoelectric effect, acoustic phonons,
KAP, NaAP, RbAP.
Acid phthalates can be described by the general formula (C,H,O;),Me”+ .m H,O where Me+ is a cation. Some of them are widely used as monochromaters in X-ray fluorescence analysis or analysis of X-ray spectra of the Sun. We restrict our interest to three, isomorphous, polar crystals of monovalent cations, i.e. sodium acid phthalate (hereafter abbreviated as NaAP) potassium acid phthalate (KAP), and rubidium acid phthalate (RbAP). All of them crystallize in orthorhombic class C,, (mm2), thus exhibiting pyroelectric [ 1,2], piezoelectric [3,4] as well as electro-optic properties [S, 61.The latter ones when linked with the ease of obtaining large single crystals of excellent optical quality, by growth from a water solution, may be promising for a much wider application of this group of crystals. Among the three above-mentioned crystals only KAP is relatively well characterized. Its piezoelectric, dielectric and elastic properties were determined by Belyaev et al. [3], electro-optic by Shaldin er al. [5] and Miniewicz et al. [4,6], elasto-optic by Shahabuddin Khan et al. [A, pyroelectric by Poprawski et al. [l] while Raman and infrared spectra were measured by Krishnamurthy et al. [8], Ore1 et al. [9] and Miniewicz ef al.[IO]. Much less is known about NaAP and RbAP for which pyroelectric properties were measured by Poprawski et al. (21 and low-frequency Raman and infrared spectra (in NaAP) by Miniewicz et al. [ll]. The aim of this paper is to establish, by means of Brillouin scattering, the full tensors of elastic stiffnesses cZ8and elastic compliances s,,@(a, fi = l-6) for the three compounds at room temperature and com-
pare them, in the case of KAP, with the results obtained by an ultrasonic technique [3]. Studies of high-frequency (_ 10 GHz) elastic constants are important for elasto- and electro-optic applications of acid phthalates.
EXPERIMENTAL Salts of phthalic acid were obtained by dissolving stoichiometric amounts of sodium, potassium or rubidium bicarbonates and phthalic acid in water. Large, transparent crystals formed on evaporation as the major product. In any case, crystals were obtained after multiple recrystallization from water solution, by a slow evaporation of water at room temperature. Due to the presence of a perfect cleavage plane containing the C, polar axis, these crystals were easily oriented by means of a polarizing microscope and pyroelectric determination of the polar axis. Parallelepipeds of suitable size (IO x 2 x 3 mm3) were cut and cleaved from larger single crystals, with edges along the orthorhombic u-, b- and c-axes. A krypton ion laser has been used as a light source, generating a C.W.(I 50 mW), single mode, radiation at 647.1 nm. The scattered light was analyzed by a piezoelectritally scanned five-pass Fabry-Perot interferometer, which was controlled by a microprocessor-based system. This interferometer yielded a typical effective finesse of 60-70. A detailed description of the experimental system is described in [l2]. In order to measure all the necessary, and some supplementary, sound wave velocities a right angle, back- and small-angle scattering geometries were
511
512
C. ECOL~VZT etal.
employed (assuming that there is no important dispersion around the IO GHz frequency region). All the measurements were performed on samples which were mechanically free and which fulfilled electrical open circuit conditions, e.g. no electrodes were deposited onto the crystal faces.
STRUCIIJRE
Crystals of this type are molecular-ionic materials.
RESULTS
AND
classified
as
model
DISCUSSION
The Brillouin shift (Av) for a birefringent crystal is given by
DESCRIPTION
Av= f~(n:+nj-2n,n,cos8)‘.‘, crystal structures of NaAP, KAP and RbAP were reported by several authors to belong to the orthorhombic class tnrn2. However, al1 of the compounds were originally described in a non-standard setting 2mm. In Table I the exact chemical formula, space group symmetry, cell occupation factor 2, lattice parameters and densities are listed. In fact NaAP is a sodium acid phthaiate hemihydrate. The B2,ab space group of NaAP is a non-standard setting of the A&2, one and the P2,ab group of KAP and RbAP is a non-standard setting of the Pca2, space group. In order to keep a standard (mm2) setting for all the crystals we adopted here an assignment of the Cartesian x, y, z axes system after Belyaev et al. [3], i.e. .Y= h, y = c and z = a of the originally reported structures [13, 14, 151, where the z-axis is a polar one and the cleavage plane within this system is (010). All these structures are similar, having a perfect xz cleavage plane due to double layers of ionic and hydrogen bonded phthahc acid ions with alkali metal ions, interacting only by weak van der Waals forces along the y-axis. The structures of KAP and RbAP are almost identical, whereas the structure of NaAP [13] differs slightly due to the presence of water of crystallization which occupies a position of site symmetry 2 and is hydrogen bonded to two atoms of ionized carboxyl groups with the G-H. . . 0 distance being 2.80 A. The strong H-bonds between oxygen of the non-ionized carboxyl group of the phthalate ion and the oxygen of another ion were found in all the considered compounds, in KAP the shortest H-bond amounts to 2.546A [14]. Alkali metal cations are surrounded by six oxygen atoms. The average values of the distance of the six nearest oxygens to the central cation amount to 2.45 A, 2.80 A and 2.95 A for NaAP, KAP and RbAP, respectively, thus reflecting the increase in the ionic radius of the cation.
(I)
The
where & is the incident laser wavelength, V the phase velocity of the acoustic phonon, ni and n, are the indices of refraction for the incident and scattered light beams, respectively, and 0 is the scattering angle. Refractive indices necessary to calculate sound velocities were measured at 6’74.1nm by a modified prism method and confirmed by Brillouin scattering at different angles. For crystals belonging to an orthorhombic system the tensor of elastic stiffness coefficients contains only nine non-zero, independent components cJ (in conventional matrix notation): c,,, cZZ,cX3,c,, C,,, C,, C,*, C,s and C,,. The final values of c,# the elastic stiffness tensor components, measured in adiabatic conditions, for any of the crystals studied were calculated on the basis of over 20 sound velocity measurements in various directions. In order to obtain their best average values the least squares fitting procedure was employed. The results of these calculations are gathered in Table 2. The errors of determination of the c, (Z = l-6) components are estimated not to exceed +2%, whereas the off-diagonal components c,*, c,~ and c23are determined with an error &7%. Inversion of the c,,, matrix yields directly the elastic compliance tensor elements s,~. In order to visualize the anisotropy of sound propagation in the materials studied we calculated sound velocity diagrams. Such diagrams for any of the three principal planes are shown in Figs 1-3. Despite the presence of weak van der Waals bonds parallel to the y-axis the longitudinal sound wave velocities do not show any remarkable anisotropy, in any of the three compounds. One can suppose that the lack of anisotropy in this series of crystals must be due to the effect of mutual compensation of weak van der Waals interactions between double-layers by
Table 1. Crystal data of NaAP, KAP and RbAP Formula
Space group
2
Na(C~H~O,).~H~O KC&&O, RM=,H,O,
B2,nb P2,ab P2,ab
8 6.75 4 6.46 4 6.561
a
b 9.31 9.60 10.064
c (4
PC&
26.60 13.85 13.068
1.566 1.572 1.579 1.64 1.929 1.94
Densities: pnk (calculated) and P,,, (measured) are expressed in gem
Pm
Ref. 1131 (141 [IS]
’ units.
Elastic properties of sodium, potassium and rubidium acid phthalates
513
Table 2. Elastic stiffnesses Coe and elastic compliances SDafor NaAP, KAP and RbAP crystals obtained from a Brillouin scattering experiment at 294K. The densities p, (in gcne3) and the indices of refraction at
1 = 674.1nm are also ouoted in 10mgN m-2 C, at _ 15 GHz NaAP (p, = 1.572) n, = 1.64 nv = 1.65 n = 1.48 KAP (p,= 1.64) n,= 1.656 nu= 1.654 nz= 1.491 RbAP (p, = 1.94) n, = 1.67 nY= 1.66 n, = 1.49 S in lo-“mZN-’ a?- 15 GHz NaAP KAP RbAP
Cn
c44
C,,
18.37 15.73
7.95
8.06 10.56
18.40 13.65 19.32
5.11
6.82
19.75 14.30 15.46
4.95
7.82
22.9
c22
c,,
C,
C,,
c,,
C,,
12.94
5.5
10.19
6.4
8.26
7.66
12.45
6.03
9.06
6.73
12.12
%I s 33 su s,, & 42 s22 9.13 9.05 8.93 12.58 12.41 9.47 -5.21 10.71 10.47 9.56 19.57 14.66 15.62 -3.35 11.12 10.03 12.68 20.20 12.79 16.58 -3.70
relatively strong ionic and H-bonded interactions within a double layer. Comparison of the elastic properties of NaAP to the two other crystals KAP and RbAP reflects the difference between their respective structures. Presence of water molecules in the proximity of Na+ ions produces two additional H-bonds and adds one coordination oxygen which strengthens the bonding within double-layers of the phthalate ions. This makes the elastic stiffness of NaAP generally higher than those of KAP and RbAP. We believe that additional H-bonds present in the NaAP structure are responsible for the longitudinal phonon velocities to be the highest along the (110) direction, which is not the case for KAP and RbAP. In all the crystals the smallest anisotropy for both longitudinal or transverse sound velocities was observed within the (100) plane. As one can expect, the overall sound
s23
&3
0.21 -1.99 -1.46
- 4.09 -5.57 -7.11
are the largest for NaAP, smaller for KAP and still smaller for RbAP, in accordance with the respective density increase. In Table 3 we compare the results of elastic stiffness determinations by the two measurement methods: Brillouin scattering (this work) and an ultrasonic technique (Belyaev et ai. [3]). The elastic from Brillouin scattering stiffnesses obtained experiments are consistently higher than those obtained from ultrasound, except for the cSs component. The most important difference (-34%) is observed for the cz3 component. This discrepancy can be understood if one takes into account the fact that all off-diagonal coefficients cannot be measured directly by either of the methods, being only calculated as an average of some independent measurements of sound velocities where they are involved. velocities
Fig. l(1)
C. ECOLIWTet al.
514
nap2
Fig. l(2) nap3
x B)
Fig. l(3)
Fig. 1. Sound velocity diagrams in NaAP single crystal at 294 K as viewed along the three principal crystallographic directions. QL-Quasilongitudinal acoustic phonon, QT-quasitransverse acoustic phonon, T-transverse acoustic phonon.
Other, less pronounced, differences, seen in Table 3, between the values of cGPobtained by us and by Belyaev et af. [3] are, in our opinion, uniquely connetted with the way a given coefficient was determined by either of the methods, i.e. directly from the measurements or indirectly from calculations. In our experiment, the only indirectly obtained elastic stiff-
nesses are: c,, cn, cl2 and cz3. However, looking for the source of possible errors, one must take into account that the cctsfmatrix used for comparison purposes was obtained through the inversion of the matrix s,~ quoted by Belyaev et al. 131,and thus the values of .rERmeasured with an error propagate this error towards values of the cclamatrix. Taking into
Elastic properties of sodium, potas! Gum and rubidium acid phthalates
all the above considerations, one must admit that the differences between the c,~ values determined for KAP by the two methods are acceptable. One should also mention that some differences may occur due to the dispersion effect of the elastic stiffness coefficients Q(V). Brillouin scattering establishes elastic stiffness components at frequency (w 15 GHz) whereas ultrasonic measurements were done at N 1 MHz. Usually, for non-ferroelectric piezoelectrics, if there is no important relaxation in account
515
the 1 MHz-15 GHz frequency range, the dispersion accounts for no more than 1%. As all the crystals studied in this work belong to the class of nonferroelectric piezoelectrics (no phase transition in any of these compounds has been reported to our knowledge), one should not expect important relaxation processes in the above-mentioned frequency range. There are some indications that this is true at least for KAP single crystal. Namely, its dielectric constants measured at 1 MHz are relatively low (6: = 6.00,
kapl
Fig. 2(A)l kap2
Fig. 2(A)2
516
C. ECOLIVET et al.
Fig. 2(A)3
Fig. 2(B)l
cs=3gl and es2 - 4.34 [9) and according to our Y . estimations based on recent far-infrared reflectivity spectra in KAP crystal [ 161the low-frequency dielectric constants are in great part explained by contribution of the lattice and internal modes and pure electronic polarizabilities. It is well known, that the propagation velocity of sound waves in a piezoelectric solid is different from that in a non-piezoelectric solid due to the coupling of acoustic phonons with polarization
components. The values of the elastic constants established for a piezoelectric can be different depending on electrical boundary conditions fulfilled in the course of the measurements [17]. In the case when an electric field E can be considered as constant one deals with CL elastic stiffness tensor components and when an electric displacement vector D can be considered as constant one speaks of elastic stiffness tensor components C
D
iikl
Elastic properties of sodium, potassium and rubidium acid phthalates
517
kapb2
Fig. 2(B)2 kapb3
Fig. 2(B)3
Fig. 2. Sound velocity diagrams in KAP single crystal at 294K as viewed along the three principal crystallographic directions (A) Brillouin, (B) Ultrasonics [3].
Due to the open electrical circuit conditions used in our experiment (no electrodes at the crystal faces), the values of the elastic constants determined by us cannot be treated as either pure c$’ or co ones. Assuming that the optical polarization components are coupled to the phonons propagating at frequenties around IS GHz (as in our Brillouin experiment)
one can suggest that the elastic constants measured by us are rather closer to the c$’ values than the c$ ones. Nevertheless one can estimate the possible influence of a piezoelectric effect on phonon propagation velocities in different directions by solving the so called modified Christoffel equation which properly
C. Fkouvm
518
describes the acoustic propagation solid [18]: det II’, + y
in a piezoelectric
- p Y2Si,J= 0,
tz)
where V is the sound velocity, p is the density and S,, is the Kronecker symbol, r, = c&n,nn. where nj is a component of the unitary propagation vector, yr = ekqnjnk where e,, is piezoelectric tensor component and finally t = c$rj~~ with E; a dielectric
et al.
tensor component under constant strain. From the solutions of the above equation for an mm2 class piezoelectric it follows that sound velocities for certain transverse or iongitudina~ phonons propagating in some principal directions are not affected by the piezoelectric effect. They are affected, however, in any arbitrarily chosen direction. The estimation of piezoelectric influence on sound velocity can however be done only for the KAP crystal for which all the necessary data can be found.
Fig. 3(l)
Fig. 3(2)
Elastic properties of sodium, potassium and rubidium acid phthalates
519
rap3
L.-.-
Fig. 3(3) Fig. 3. Sound velocity diagrams in RbAP single crystal at 294 K as viewed along the three principal c~stallographic directions.
One can calculate its piezoelectric e,, coefficients using the relation [I81 e, = dibciz,
(3)
and employing the dippiezoelectric coefficients quoted in [3] and respective values of CL. The eh piezoelectric coefficients of KAP obtained in this way are: e31= -0.147, ejz=0.0315, e3) = -0.0212, e,, = -0.0544 and ez4= 0.02089 Cm-‘. Then taking the values of dielectric ~~ittivities of KAP at 1 MHz: t,, = 5.312, 622= 3.427 and ~~~= 3.846 in IO-” Fm-’ units [3] one can calculate effective elastic constants Cr,i, (for a given type of a phonon and a given direction of its propagation in the crystal). Similar calculations can be performed without taking account of the piezoelectric effect, just by solving a normal Christoffel equation without the 7,7,/c term present in eqn (2). In this way one obtains the respective effective constant C. Introducing the quantity 100 x (C,,, - C)/C one can express, in per cent, the expected modification of the effective elastic constant by the presence of piezoelectricity in KAP.
Results of such calculations for some phonons propagating along important directions in KAP are shown in Table 4. We have found that for any of the three principal directions (LOO),(010) and (001) only one among the three phonons is affected by the piezoelectric coupling (a detailed description of phonon polarization and type is given in Table 4). In the (011) and (101) directions two phonons are influenced by the piezoelectric effect: a quasi-longitudinal (QL) and a quasi-transverse (QT) one along the (110) only one of a pure transverse type (T). The most important change for phonon propagation velocity in KAP, due to piezoelecttic coupling, occurs for the QL phonon propagating along the (101) direction. The relative change in the effective elastic constant in this case amounts to 2.34%. For other directions and other phonons the effect is smaller, usually much less than I %. One should mention that the calculations were performed with t values in the MHz range, however, due to the small dispersion expected for the dielectric permittivity of KAP between 1 MHz and 15 GHz these results will also apply
Table 3. Comparison of elastic stiffnesses obtained in KAP by Brillouin scattering technique at - I5 GHz with those determined by an ultrasonic method at - I MHz by Belyaev er al. [3] C,, in 10%N m-’ Brillouin ( - 15 GHz) Ultrasonics (- 1 MHz) C$ - C$
-* Cf*
100%
C,,
c22
G,
G
18.40 13.65 19.32 5.11 17.56 13.52 16.97 4.86 4.6
2.9
12.2
4.9
cs,
6.82 7.59 -11.3
G&
c12
c23
6.4 6.23
8.26 7.39
7.66 5.05
12.45 10.38
G,
2.6
10.5
34.1
16.6%
C. ECoLivETet al.
520
Table 4. Estimated effect of piezoelectticity on phonon propagation velocities in KAP Phonon Wave-vector direction
Effective elastic constant C,,,,, =pv2
Polarization type
e?, 611
100 x CC,,,, - C)/C (%) 0.74%
-f@f)l>
c&t-
(010)
T (ml>
c”+& 44
LWI)
c:,+--
QL
+Cz=“‘*
0.07%
QT
c%-
0.30%
QL
c%-
2.34%
QT
GYve
0.36%
T
Cl> __ + CM : (e,5+e24)* 2 4 e33
0.12%
<110)
0.27% 622 43
0.07%
c13
t The complicated formulae for nonprincipal directions of phonon propagation are omitted
to the conditions of Brillouin scattering. Similar, rather negligible, influence of piezoelectricity on the elastic properties of two other crystals studied, NaAP and RbAP, can safely be predicted. Acknow/e&emenrs-Gne of the authors (A.M) is grateful to the Groupe de Physique Cristalline of Universite de Rennes I for their hospitality and financial support during his stay at Rennes. The help in preparations of the figures by Dr Krzysztof Wolinski (IRISA, Rennes) is acknowledged. REFERENCES
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