Z. Phys. B - Condensed Matter 86, 133-138 (1992)

Condensed

Zeitschrift M a t t e r for PhysikB

9 Springer-Verlag 1992

Ferroelastic phase transition in CaCl2 studied by Raman spectroscopy H.-G. Unruh, D. Miihlenberg, and Ch. Hahn* Fachbereich Physik, Universitfit des Saarlandes, W-6600 Saarbrticken, Federal Republic of Germany Received July 11, 1991; revised version August 2, 1991 CaC12 undergoes a second-order proper ferroelastic phase transition from the tetragonal ruffle type to the orthorhombic calcium-chloride type structure at T~ ~490 K. The transition is of the optical type and induced by an order parameter of B~g symmetry. An underdamped soft mode exists above and below T~, the frequency and intensity of which have been measured by Raman spectroscopy. Due to a strong coupling to strain the softening is incomplete and a transverse acoustical mode is predicted to become soft. The frequency of the optical soft mode, cos, exhibits a classical Landau behavior in a remarkably large temperature range. The unusual value of the ratio of ~(~0~)/0 Tbelow and above T~ of about - 6 . 5 can be accounted for by appropriate terms of the thermodynamical potential. The Raman active hard modes show no significant anomaly at T~.

1. Introduction At room temperature calcium chloride has an orthorhombic structure (space group P n n m, Z = 2) which may be considered a slightly deformed rutile-type lattice (space group P 4 2 / m n m , Z = 2). With increasing temperature the orthorhombic deformation decreases continuously and a phase transition to the rutile structure takes place at 491 K. This has been shown by B/irnighausen ct al. [,1] and Anselment [2] by very careful X-ray investigations. They also found that the atomic displacements with respect to the tetragonal structure essentially consist of rotations of nearly rigid Cl-octahedra around the tetragonal axis. According to Anselment [2], the phase transition is equitranslational, proper ferroelastic, and most probably of second order induced by an order parameter of Big symmetry. He also verified that the order parameter measured by the spontaneous deformation components, ~xx=-eyy, follows fairly well the classical * Present address: Dornier GmbH, Abteilung ETII, W-7990 Friedrichshafen 1, Federal Republic of Germany

Landau-type temperature dependence (T~-T) ~ with /~ =0.45 in the range 130 < T < 4 9 0 K . Furthermore, B/irnighausen et al. [,1] and Anselment [-2] showed that calcium bromide undergoes an analogous ferroelastic transition at 778 K. According to the symmetry of the order parameter a corresponding soft mode should be Raman-active above and below T~ belonging to Big- and Ag-species, respectively. Indeed, the occurrence of soft modes in the orthorhombic phases of CaC12 and CaBr2 has been described in a preliminary report by Unruh and Hahn [-3]. Independently, Raptis et al. [4] found a soft mode in the orthorhombic phase of CaBr2. The main result of these investigations is the classical temperature dependence of the soft-mode frequencies, ~0s~(T~-T) ~ with / ~ 0 . 5 , for both substances within a large temperature range covering several hundred K. However, up to now a soft mode has not been detected above T~. Raptis et al. [,4] and Raptis and McGreevy [5] argued that it is Raman inactive above T~ and assigned it to species A2~ , which is silent. On the other hand, Hahn and Unruh [6] pointed out that Landau-theory predicts a Raman active Big soft mode in the tetragonal phase in accordance with the structural investigations mentioned above, and they invoked experimental difficulties for the failure of the Raman scattering experiments. Arguments in favor of the Big symmetry of the soft mode of CaBr 2 above T~ have been also put forward by Weber et al., who studied the possibility of an analogous transition of PtO 2 [7]. Another example of this type of transition is found in stishovite, a high-pressure rutile-type phase of SiO2, which exhibits a soft B~g mode and transforms at ambient temperature at about 8 0 G P a to the CaC12-type structure [-8, 9]. In this paper we will show that a Raman active, underdamped soft mode exists in CaC12 above T~, too, and analyse its temperature dependence in the frame of a Landau-type approach. With regard to lattice dynamics of the ruffle structure [-10], the eigenvector of the Bl~-mode, which is displayed by Fig. 1, corresponds exactly to the atomic displacements of the ferroelastic

134

~4

K

689

/i~


541

":

CJ I

gg 492

Ii

,~,t ~,.

~23,1

I ~ j,. '-

/

%

Ii

j ~.

1/

I

.":'-J" Y!.i Ca

Cl

Fig. 1. Structure of ruffle and atomic displacements of the B~g optical mode after Traylor et al. [10]

phase of CaClz and CaBr2 as concluded from X-ray diffraction measurements [1, 2].

0

-60

-30 30 frequency [cm-1]

60

Fig. 2. Raman spectra of the soft mode of polycrystalline CaC12 at various temperatures above and below the ferroelastic transition temperature at about 490 K. The intensities of the spectra are not to scale. The peak marked by 9 is an emission line of the laser plasma. Instrumental width: 1.5 cm- 1

2. Experimental results 3000

It is known that CaCI2 crystals grown from the melt are sensitive to mechanical stress and may undergo a transformation to the a-PbO2-type structure, which is metastable at room temperature [2]. Therefore, and because of the extreme hygroscopicity of this substance we used polycrystalline superdry CaC12, ampouled under argon and consisting mainly of individual spheres of 0.4 to 0.8 mm diameter (Johnson Matthey G m b H , Alfa Products, Karlsruhe, FRG). F r o m this material samples of about 1 cm 3 were loosely poured under nitrogen into a glass cell, which was sealed. The cell was put into a thermostat, the sample chamber of which was filled with helium gas to improve the heat exchange between the cell and the thermostat. The temperature of the sample was measured by a thermocouple fixed to the outside of the cell. In addition, some Raman experiments have been done on pellets pressed from coarse grained CaC12 (E. Merck, Darmstadt, FRG). The pellets were handled similarly in nitrogen-filled glass cells. A conventional R a m a n spectrometer with photon counting equipment and laser excitation of 400 m W at a wavelength of 514 nm was used throughout. An iodine cell served as an additional filter of the unshifted laser light when measuring the low frequency part of the Raman spectra above room temperature. The recorded spectra were corrected for the absorption of the iodine cell. 2.1. Soft mode behavior Figure 2 displays some spectra of the optical mode with the lowest frequency in the temperature range between

500 \\

?"o/

~ N

'E

2000

o~ ~ x

1

I

\

o4

1000

%

~J c_

o% %

o

-'-

-250

o~ o

O.O

c ell

%

q% %

......

~"'

0

I-emperafure [K]

Fig. 3. Square of the eigenfrequency of the soft mode of CaCI2 versus temperature. Central portion displayed also on a larger scale. The characteristic temperatures according to (2a, b) are Tp= 244 K, T~=491 K, T/=528 K

20 and 700 K. A substantial softening is noticed with a minimum of the frequency of ee~g14 cm -1 at about 490 K. The mode parameters were evaluated by a least squares fit of the data to the response function of a damped harmonic oscillator,

S (co) ~ (eeL-- COo)4 (n (co) + 1) I0 F ee/((ee2 _ 0)2)2 +

Fz

c02), (1)

where n ( e e ) = l / ( e x p ( h c o c / k T ) - l ) , coL the frequency of the incident laser light, COo the eigenfrequency of the oscillator, F its damping constant, and I o an intensity parameter. Additionally the fit function allowed for an ap-

135 propriate background, which results mainly from the fluorescence of powdery portions of the samples. The temperature dependence of the square of the soft mode eigenfrequency is depicted in Fig. 3. Apart from saturation effects below about 50 K, a classical behavior is observed above as well as below T~,

o~z~=~p(T-- Tp),

T>=T~,

(2a)

c ~ = e l ( T - - TS),

T_<_T~,

(2b)

within experimental errors. The enlarged portions of the data and of the fitted curves in Fig. 3 demonstrate that a sharp minimum exists at T~. Further remarkable results are the large difference between T~ and Tv, the temperature of instability of the (bare) optic soft mode according to (2a), T~-Tp~247 K, and the unusually large value of the ratio of the slopes, %~/~p~ -6.5, as compared to the value of - 2 expected from Landau theory. The results on the temperature dependence of the integrated intensity, I(r)~SS(co, r)/(n(co, r ) + l ) d o and of the damping constant, F(T), of the soft mode are given in Fig. 4. Due to the not compact, polycrystalline samples the measured intensities of the modes and of the background depend sensitively on the spot where the laser impinges on the sample. Therefore, care was taken to include into Fig. 4 only those data of I(T) which are considered to be comparable. We deduce from Fig. 4 that a substantial decrease of the Raman cross section occurs when the temperature approaches T~ from below, whereas it stays roughly constant above the transition temperature. The accuracy of the full width of the soft mode, F(T), is indicated by bars in Fig. 4. These data have been evaluated in consideration of the instrumental bandwidth of 1.5 cm -1 by convoluting the response function (1) with the instrumental profile numerically in the fitting procedure. There is no anomaly of F(T) around T~ apart from a slight steplike decrease.

2.2. Hard modes

~

52K

o

IOO 200 frequency [cm-I]

300

Fig. 5. Raman spectra of the hard modes of C a C I 2 at various temperatures above and below T~. The inset shows the splitting of the band at about 160 cm i that occurs at low temperatures. Peaks

marked by * are laser plasma lines, the frequencies of which have been used for calibration purposes

250

--200

~

o

o

o o c o o

o

o ~ o o o

oooo

o

~,o

o

oo-

~

o

o

o

~o

o

o

oooo

o

%e

o

ooj

~

o

o

o o ~ o

o

Dq)

o

~e

o

ooe

o

o

o

o

o

o

e o o o

E t.a

~150 C =o o- 100GJ

o

~o

ooo

~ooo

ooo

~

oooo

c-

50-

0

~

~

o~

o ooi

z60

~

K00 ~emperafure [K]

Spectra of the hard modes at several selected temperatures during a cooling run are displayed in Fig. 5. Sam-

660

860

Fig. 6. Temperature dependence of the frequencies of the Raman active modes of CaC12. Full circles refer to results obtained on pellets pressed from coarse grained CaC12

10 % u

%\o 4---

8 a

~

\o

:~

l+ L_ o~

8 oB~

(I)

o~o

2

\

o

28o

4oo

600

I-emperature [K]

Fig. 4. Integrated intensity, I(T), and damping constant, F(T), of the soft mode of CaC1z versus temperature. The curves serve as guide lines only

ples prepared as described above exhibit at ambient temperature several Raman lines more than when heated to about 650 K before the measurements. However, the extra lines reappear generally with varying intensities when the temperature is below about 300 K as shown in Fig. 5. We suppose that the additional lines have to be attributed to another phase of CaC12, possibly to the c~-PbO/-type structure which is known to result from mechanical stresses at ambient temperatures [2]. Such stresses may originate from the polydomain structure of the particles and will increase in the ferroelastic phase with increasing spontaneous strain on cooling. Nevertheless, besides the soft mode three Raman active hard modes were identified within the range of the tetragonal phase and five hard modes below T~: The mode at about

136 115 cm-1 is obviously silent above T~, and a very small splitting of the line at about 160 cm -1 occurs at the lowest temperatures as depicted in the inset at the upper left of Fig. 5. The temperature dependence of the hard modes shows a small but definite influence of the phase transition at 490 K as can be seen in Fig. 6. Full symbols represent some results from another series of measurements on pellets pressed from coarse grained CaC12. The data agree one with another within the experimental errors, including the soft mode. However, due to the particularly high intensity of light scattered elastically at the surface of the pellets, it was not possible to measure the soft mode above about 400 K in this case. Nevertheless, the correspondence between the results of these differently processed samples indicates that the method of preparation is not essential.

F = Fo + (a/2) x~ - dxl Q1 + (ovz/2) Q~,

(3)

where c@=C~p(T-Tp). The deformation component of Big symmetry, which couples bilinearly with the order parameter, Q~, is Xl = exx-eyy. Terms which do not contain Q, or Xl are included in Fo. It is well known that (3) predicts a proper ferroelastic transition of the mechanically free crystal at

T~= Tp + d2/(C~pa).

3. Discussion

Landau-theory predicts that an equitranslational structural transition between space groups P4z/mnm and Pnnm is induced by an order parameter of Bag symmetry in the tetragonal phase. This result can be read from existing tables in the literature [11, 12] and has been used successfully in case of the analogous transition of CaBr2 [6]. Thus the soft mode must belong to species Big (point group 4/mmm) above T~ and to species Ag(mmm) below T~and it is Raman active in both phases. This kind of transition has been classified by Aizu as a ferroelastic transition of the optical type [13]. The factor group analysis shows that the representation structure of zone-center Raman-modes of the tetragonal cell is Alg@A2gq-Blg+B2g-bEg, which changes for the equitranslational orthorhombic cell to 2Ag + 2B~g +B2g+B3g. The complete correlation diagram of Raman active modes of the high- and low-temperature phase is shown in Table 1 together with mode-frequencies determined at 300 and 600 K. The assignment of the hard modes was made by comparison with CaBr2 [6] taking into account the eigenvectors of the modes [10] and the ratio of masses of C1 and Br. In particular, the line at about 115 em -1 has to be assigned to B2[ because it vanishes above T~, the line at about 160 cm to species Eg above and to B2g+B3g below T~ as only this band shows a splitting in the low temperature phase. In order to describe the phase transition and the temperature dependence of the soft mode, we start from a Landau-type expansion of the free energy. We designate the normal coordinates of the zone-center optical

Table 1. Correlation diagram and assignment of Raman-modes of the high- and low-temperature phase of CaC12

modes of the ruffle structure by Q~ and the homogeneous deformations, which transform according to the irreducible representations (IR's) of the point group 4/mmm, by xi. Taking i= 1 for quantities of the symmetry of the order parameter, Big, and considering that only one Big-mode exists in the paraelastic tetragonal phase of CaC12, the free energy can be expressed to second order in the form [13]

(4)

In order to stabilize the ferroelastic phase, (3) is usually supplemented by a single fourth order term, (b/4)Q4, with b>0. This leads also to a renormalization of the frequency of the soft mode in the ferroelastic phase. Taking into account that the crystal is mechanically clamped at optical frequencies one yields [13, 14],

o)}=-2co2 + 3d2/a with co}(T~)--c@(T~)=d2/a.

Fitting (2a, b) to the data of Fig. 3 one obtains T~= (491+1.5) K and ey/ccp=-6.45_0.15. The value of T~ is in excellent agreement with the transition temperature of 491 K, which has been determined by X-ray measurements [2]. Comparison of (5) with (2a, b) and use of (4) shows that

c@c~v = - 2 and

Tf = Tp+ 3 d2/(2 c~pa) = (3 T~- Tp)/2, (6)

respectively. These relations are not verified experimentally in the case of CaC12 as mentioned earlier, although (2 a, b) hold in large ranges of temperature (Fig. 3). A complete set of invariants of the expansion of F may be found by inspection of the representation structures of the symmetrized powers, [F"], and of the direct products of those IR's which are relevant to the quantities of interest, i.e. the normal modes, Qi, and the deformations, xi. Looking for monomials up to the fourth order, which are invariant with respect to point group 4/mmm and which contain the order parameter, one has to consider [15]:

Dzh(mmrn)

D4a(4/mrnm)

co[cm- 1] at 300 K

co[era- 1] at 600 K

34.9, 208.8 113.5, 250.5

Alg A2g

156.0

Ag ~ B~g ~

~B2g ~B3g ~

(5)

~

Big Bzg G

203.0 silent mode 17.1, soft mode 246.7 15o.o

137 [B2g]=Alg, [B~g3=Bxg, [B4gl=Alg, [E 2] = [E23 = A1 g + B1 g+ B2g, (7) AagxBag=Blg, AI, XBI,=Blg, A2~xB2,=Blg. For simplicity we will not discuss here the invariants which renormalize some non-critical coefficients of the Landau potential, e.g. the squares of the eigenfrequencies of the hard modes. Furthermore, we neglect terms of higher than second order if they contain variables of higher than first order different from the order parameter, i.e. invariants like x~ Q 21, x 31 Q 1. In this case the equations of state, ~F/O xi = Xi,

(8 a)

or/~Q;= 0,

(8 b)

are linear in all the independent variables except the order parameter. Thereby one takes into account that nonlinear effects are due primarily to the order parameter whereas other quantities, even if coupled bilinearly to it, generally do not leave the range of linear behavior. Under the assumptions of the last paragraph the free energy of the tetragonal phase of CaC12 takes the form F = Fo +(a/2) x Z - d x ~ Ot +(co2/2)Q~-gxl O 3 + (b/4) Q4.

(9)

Solving the equilibrium condition of the mechanically free crystal, i.e. (8a) with X t = 0 , for xl, eliminating Xl in (9) and neglecting terms of powers higher than four, one yields QI~, the spontaneous value of the order parameter, by applying (8 b), Q ~ = % ( T ~ - T)/(b - k),

(10)

with k = 4 d g / a and cop z = % ( r - T ~ ) + de/a. Using (10) the spontaneous value of the strain, xl~, results from OF/~ xl = O, %(T~- T) ( g%(T~- T)) 2" x ~ s - ( b _ k ) a 2 d-~ b-k

(11)

Now the frequency of the soft mode in the ferroelastic phase may be obtained from co} = (02 F/OQ~)xl =xl~- The nearly linear dependence of x2s on temperature [2] justifies the linearization of (11) and this approximation leads to 2 2 k-4b (6b-3k)d2/a coy = cop 2 (b - k~ ~ 2 (b - k)

(12)

Equation (12) shows in comparison with (5) that the linear dependence of co} on temperature is maintained, but the ratio of the slopes of the squared frequencies now becomes c@%=(k-4b)/(2(b-k)),

(13)

which is smaller than - 2, if 0 < k < b. From our experimental result of c~i/c~p =- - 6 . 5 follows Ic/b = 0.75. This implies that the coefficients d and g must have the same sign and it would be tempting to evaluate the measure-

ments of the spontaneous strain by Anselment [21 for the coupling coefficients d and g on the basis of (11). However, the values of the coefficients a and b in (9) are not yet known and the temperature dependence of the 'normal' coefficients must be also taken into account, when the whole temperature range of the ferroelastic phase is considered. We believe that a similar explanation of deviations from a behavior according to (6) may apply to other ferroelastic transitions, too. From measurements of Pinczuk et al. [141, for example, one gets a J % ~ - 3 . 5 in the case of BiVO4, a crystal which has a soft mode of Bg symmetry in its paraelastic phase of point group 4/m. The linear dependence of co} on temperature holds to about 50 K as seen in Fig. 3. The saturation effects below that temperature may be treated by the low-temperature extension of Landau theory, which has been worked out recently by Salje et al. [16]. According to this theory CaC12 may be regarded as a system being close to the displacive limit with quantum effects becoming essential for T < O ~ 40 K. The comparatively incomplete softening of the soft mode of CaC12 to only 14 cm- 1 at T~ indicates a strong coupling between the order parameter and the strain component xl. The elastic stiffness, a, in (9) is assumed to be practically temperature independent and becomes renormalized by this coupling. Determining Q1 by means of the equilibrium condition of (8b) yields the elastic stiffness in the paraelastic phase, ap, with the help of (8 a) and (4), ap = a - d2/co~ = a ( T - T~)/(T- Tp).

(14)

Equation (14) shows that an elastic instability occurs at T~ due to the vanishing of the static elastic stiffness and a Curie-Weiss-type behavior is expected in the vicinity of T~. The stiffness ap may be expressed by the corresponding tensorial components of the tetragonal phase as ap = (c 11- C12)/2" The solution of the acoustical wave equation shows that a purely transverse sound wave propagates in the x - y plane at an angle of +45 ~ with respect to the axes with a velocity of v=(%/p) 1/2, where p is the density of the crystal. It is this sound wave which is predicted to soften according to (14) on approaching T~from above. The analysis shows that the transformation of CaC12 may be considered a clear example of a proper ferroelastic transition of the optical type, and we infer that the same holds true for CaBr z from its analogous behavior.

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138 7. Weber, W.H., Graham, G.W., McBride, J.R.: Phys. Rev. B42, 10969 (1990) 8. Henley, R.J.: In: High-pressure research in mineral physics. Y. Sonyo, Manghnani, M.H. (eds.), p. 347. Tokyo: Terra Scientific 1987 9. Tsuchida, Y., Yagi, T.: Nature 340, 217 (1989) 10. Traylor, J.G., Smith, H.G., Nicklow, R.M., Wilkinson, M.K.: Phys. Rev. B3, 3457 (1971) 11. Boyle, L.L., Lawrenson, J.E.: Acta Crystallogr. A28, 485 (1972)

12. Janovec, V., Dvorak, V., Petzelt, J.: Czech. J. Phys. B25, 1362 (1975) 13. Aizu, K.: J. Phys. Chem. Solids 32, 1959 (1971) 14. Pinczuk, A., Burns, G., Dacol, F.H.: Solid State Commun. 24, 163 (1977) 15. Ljubarski, G.J.: Anwendungen der Gruppentheorie in der Physik. Berlin: Deutscher Verlag der Wissenschaften 1962 16. Salje, E.K.H., Wruck, B., Thomas, H. : Z. Phys. B - Condensed Matter 82, 399 (1991)