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Tb2(Mo O4)3 spontaneous shear strain measurements and free energy expression
P.M. Bastie, J. Bornarel
To cite this version:
P.M. Bastie, J. Bornarel. Tb2(Mo O4)3 spontaneous shear strain measurements and free energy expression. Journal de Physique, 1982, 43 (5), pp.795-800. �10.1051/jphys:01982004305079500�. �jpa- 00209453�
Tb2(Mo O4)3
Spontaneous shear strain measurements and free energy expression (*)
P. M. Bastie and J. Bornarel
Université Scientifique et Médicale de Grenoble, Laboratoire de Spectrométrie Physique (**),
Boîte postale n° 53 X, 38041 Grenoble Cedex, France
(Rep le 10 juillet 1981, révisé le 28 décembre, accepté le 18 janvier 1982)
Résumé. 2014 Nous avons mesuré la déformation spontanée uxy en fonction de la température entre 300 et 430 K.
Cette transition ferroélectrique-ferroélastique est du premier ordre avec une discontinuité pour uxy d’environ 100"
à la transition. Nos résultats sont en très bon accord avec des descriptions phénoménologiques qui correspondent
à des expressions de l’énergie libre comprenant quatre paramètres ajustables.
Abstract 2014 The spontaneous shear strain uxy versus the temperature is measured by the y ray diffractometry technique in the (300 K-430 K) range. The first-order type of the ferroelectric-ferroelastic transition is clearly
demonstrated (discontinuity of some 100" for uxy at the transition temperature Ttr) : our data are in accordance with phenomenological descriptions corresponding to the free energy expressions with four adjustable parameters.
Classification
Physics Abstracts
64.70K - 77.80 - 62.20
1. Introduction. - Tb2(Mo 04)3 (TMO) is a ferro-
electric-ferroelastic crystal known since 1965 [1].
It is a good example of an improper ferroelectric [2, 3] :
an inelastic neutron scattering study [4] has shown
that the antiferroelectric static displacements consti-
tute the order parameter. The shear strain u.y and the polarization produced by piezoelectric coupling are only coupled with the order parameter at the second order.
Numerous experiments have been performed with
this crystal especially in the two last years : heat capacity, entropy and magnetic moment measure-
ments at very low temperatures [5], Raman spectrum [6], pyroelectric figure of merit [7], light scattering from phonons [8], second harmonic gene- ration [9] and acoustic measurements [10]. Although polarization [11] and birefringence [12] measurements exist on TMO, knowledge of the spontaneous shear strain is fair : the lattice parameters have been mea- sured [11,13] at room temperature but the spontaneous shear strain uxy versus temperature has been deduced with the help of thermal expansivity measurements [14].
In this paper, we report on the first direct measure- ment in TMO of the shear strain uxy versus tempe-
rature employing the y ray diffractometry tech- nique [15], using samples on which optical studies
of domain textures had been previously made, then
we compare the data with predictions deduced from
a free energy expression [16].
2. Experimental. - The samples were cut from a poled boule grown by the Czochralski process, in
a parallelepiped shape with faces perpendicular to tetragonal crystallographic axes. Dimensions were
ai = 6 mm, a2 = 2 mm, c = 7 mm for the sample
for which the shear strain results reported herein
were measured. Optical observations of the ferro- electric domain texture and the phase front at the
transition near 160 OC have been made with the aid of a polarizing microscope. The influence of the cooling
rate on the domain structure formed during the phase
transition in the crystal is evident : in figures la and 1 b, are shown similar parts of a sample (the width
of the picture corresponds to 2 mm on the sample).
In figure la, the cooling rate is a few degrees per
minute; different domain families with perpendi-
cular domain walls can be observed; there are many domain tips; the boundaries between two domain families are not in tetragonal crystallographic planes;
sometimes two perpendicular domains families inter- penetrate ; the volumes of + or - domains are not
always equal and some latent polarization can exist.
In figure 1 b, the cooling rate is less than 0.01 K/min.;
there is mainly one domain family with large domain widths; the volume of + domains is approximately equal to the volume of - domains; the domain walls
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004305079500
796
Fig. 1. - Domain structure observed in a section perpen- dicular to the c ferroelectric axis for two different cooling
rates across the transition : a) A few degrees per minute.
b) Less than 0.01 K per minute (the picture width corres- ponds in a and b to 2 mm on the sample).
of this equilibrium domain structure move easily
under applied stress or electrical field.
To obtain cooling rate and temperature conditions
as in figure lb, the uxy measurements have been made
using an oven of cylindrical shape where the tempe-
rature was regulated on the axis by an Artronix temperature regulator which provided a 10- 2 K stability. The sample, inside the oven, is placed in a
small aluminium cylinder whose temperature homo- geneity is 0.01 K, and in which the temperature is measured near the sample with a Rosemount plati-
num resistor and a Dana 5000 digital multimeter
readout. The relative precision of two measure-
ments was of order 0.01 K but the absolute accuracy
was only several tenths of a degree. All the uxy measure-
ments were made in figure lb conditions (cooling
rate lower than 0.01 K/min.). The samples were elec- trically isolated (no electrodes), but in the GMO
family crystals, the conductivity at these temperatures
seems sufficient to avoid any difference in properties
between short-circuited and isolated samples. Such
is not the case in the situation observed at low tempe-
rature with KH2P04(KDP) [17].
The preceding arrangement, which allows good
temperature measurement, is possible because the y radiation used (412 keV, A = 0.03 A) has a very weak interaction with matter (mean free path for Al
is about 5 cm) and the diffraction measurement is
possible even in the oven. A detailed description of the
y ray diffractometer has been given previously [15].
Due to the small Bragg angle and the chosen radiation,
this apparatus is very well adapted for lattice tilt measurements [15]. This technique has been there- fore used to study twinned textures [18] and ferro-
elastic phase transitions [19]. Let us only recall that the rocking curve obtained on a sample such as
TMO with the reflexion (2. 0. 0) shows one peak
in the paraelectric phase (Bragg reflexion on tetra- gonal a, planes for example). In figure lb conditions,
this peak can split at the transition into two peaks (one corresponds to domains +, the other to
domains - ). The difference in rotation angle between
these two peaks is 2 Uxy. It is easily understandable that the evolution of the rocking curves with the temperature T gives precise information on uxy versus T
as well as on the transition order [20] (accuracy on U,y measurements of a few seconds of arc).
3. Spontaneous shear strain uxy,. - The results for uxy versus T are shown in hgure 2 and given in detail
in table I. Each experimental point corresponds to
Fig. 2. - a) Spontaneous shear strain uxy versus the tempe- rature, experimental data (x) and fit obtained with the relation (9). b) The deviation Au (magnified 25 times)
between our experimental points and the curve of the fit.
Table I. - Data of ’ spontaneous shear strain Uxy in seconds of arc (uncertainty ± 5") versus the tempe-
rature T (uncertainty ± 0.02 K).
a rocking curve in which the angular variation
between the barycentres of the two Bragg peaks is
considered equal to 2 uxy (by taking into account the divergence of the y ray incident beam (30") and the
mechanics of the apparatus, we appreciate the uncer- tainty on the u.y data to ± 5" of arc). During the rocking curve drawing, the temperature was defined with a 0.02 K in precision and stability. The first
order nature of the transition is demonstrated without
ambiguity : we measure for the jump at the transition
103" of arc and consider that the uncertainty should
be larger near the lower values (estimated range for the uxy jump : 95"-107" of arc). A small thermal
hysteresis (less than a tenth of a degree) has been
observed at each transition crossing.
Fig. 3. - Linear dependence of the spontaneous bire- fringence An, [12] with the spontaneous shear strain uxy for the same T - Ttr values.
The present results of u.y versus T have been compared to the similar results of birefringence An,
for a light propagating along the c axis by Konak
et al. [12]. In figure 3 each point has for coordinates the values of uxy and Anc measured for a temperature corresponding to a same T - T tr value (where 1;r
is the transition temperature). For all the temperature
range studied by Konak et al. (between 90 OC and
160 OC), the linear variation can be formulated by the
relation An, = (0.126 5 ± 0.001 5) uxy. This is consis- tent with the same linear dependence between sponta-
neous birefringence and polarization reported by
these authors [12]. Let us note that, at the transition temperature, the smallest uxy value it has been possible
to measure with the y ray diffractometry technique
is relatively smaller than the corresponding value
of the measured birefringence Anc.
4. Discussion. - Our results will now be compared
to theoretical laws which can be found by adopting
classical free energy expressions. Following
V. Dvorak [16], let us take a free energy expansion
in terms of phase transition parameters q, and q2,
polarization along the c axis and strain components.
After employing zero external field conditions, it is
convenient to introduce the « polar » coordinates r
and 0 as :
Then, with the same sense for the different parameters
as in Dvorak’s paper, it is possible to express the free energy F :
where a = a(T - To) a>O; ç is constant and posi- tive ; fi is a more complicated parameter; it contains terms depending on 0, and is negative for a first
order transition.
The spontaneous value of the order parameter r(T)
is calculated as usual and given by :
By writing that the transition temperature Ttr corres- ponds to F( Ttr) = Fo, it follows that :
and Dvorak [16] shows that the spontaneous shear strain u.y can be deduced by a relation
798
where K is in particular a function of 0 which could be temperature dependent. But, as all the other authors Dorner et al. [4] suppose that 0 is temperature inde- pendent for two reasons :
1. Several superlattice intensities were observed
to have nearly identical relative temperature depen-
dences.
2. No very precise determination of r(T) behaviour
was made.
For a first hypothesis, let us suppose like these authors that 0 and fl are temperature independent.
Using the relations (2) to (5), we get :
or
/MfT 2 2
Figure 4 shows the function x equal to
u T ) 3
(u(T) 3versus T, using our experimental data,
especia ly
u(T,,) = 103" and Tr = 430.16 K. A linear decrease of x(T) is not perfectly verified in the studied range of temperature (295 K-430 K) but the discrepancy is not
too bad : a least squares fit over the whole range leads to a To value of 428.66 ± 0.02 K and a root mean square deviation of about 16 seconds of arc. This is three times greater than the experimental uncertainty
and gives the limit of the model. Strictly speaking, a
Landau expansion is valid only near Ttr ; so the tempe-
rature ranges (Ttr - 10 K, Ttr) and (Tlr - 5 K, Ttr)
have been studied. In these two temperature ranges, the
Q value is about equal to the experimental uncertainty
and the calculated value of To is the same : 429.1 K (we shall keep to this value from here onwards).
Using the value of f32 laç obtained by this fit in these
Fig. 4. 2013 The function x =
( u(T) - 2 2
versus the tempe-rature with M(TJ equal to 103".
Fig. 5. - Variation of B = P(T)IP(Ttr) with the tempe-
rature.
temperature ranges, it is possible to calculate TM - To
where TM is the upper existence temperature of the ferroelectric phase, To being the lower; we find TM - To = 1.4 K. For example for quartz (Ttr = 846 K), TM - To = 9.5 K [21] but the situation, is different because the phenomenological description with a, fl and ç temperature independent gives a better result :
u less the experimental uncertainty in a temperature
range of 550°C [21]. In the present case, with TMO
samples, in order to extend the validity of the descrip-
tion used to the whole studied temperature range, it is necessary to increase the number of adjustable para- meters in expansion (1).
We cannot exclude the eventual existence of the critical fluctuations near the transition [22]. But as
the TMO transition is a first order one, the chosen
hypothesis is that a classical description can be used.
There are two possibilities.
1. To introduce a higher order term in the free energy expression ;
2. To assume a temperature dependence of f3 (across a ø variation with T or for another reason).
The first hypothesis, i.e. the introduction in the free energy expression of an eighth order term, allows an
improvement in the fit within the experimental uncer- tainty. Let us also try to explore the second hypothesis :
if we suppose that fl is a function of T, it is possible by using the equations (2) and (5), to write the relation :
which allows the calculation of, at each temperature, with our u values, the corresponding B value. Using
the experimental data u(Ttr)=103" and T,, = 430.16 K,
and for To the value (429.1 K) obtained by the fit near
the transition, we obtain for B the temperature depen-
dence shown in figure 5 which is surprisingly a linear
variation. This result is consistent with a free energy
expansion of the following form :
which has already been used with success to describe
the order parameter variation versus T observed in
crystals near the tricritical point [23, 24, 25, 26]. To be
consistent we have fitted our data to a u(T) function
limited to the (8) expansion :
The agreement is very good : the root mean square deviation is only two seconds of arc and no syste- matic deviation occurs, as it can be seen in figure 2b.
The values of the adjustable parameters, for u in radians and T in degree Kelvin are M = -105 ± 6;
N = (14.5 + 0.4). 106 K ; To = 427.7 + 0.2 K;
T 1 = 313 ± 12 K. The value of To - T1 1 equal to
about 115 K is very large compared to the one
obtained for example in KDP (about 1 K) [24].
This proves that TMO at atmospheric pressure is not very near a tricritical point. But let us compare
our results to those obtained in KDP : in this crystal,
when an eighth order term is introduced in the free energy expression, a good description can be obtained in a lower temperature range (within 1.5 K of the
transition [27]) than if one supposes a fourth order term linearly temperature dependent (a good descrip-
tion in a temperature range of 15 K or more [24]).
A free energy expansion such as equation (8) has
been established to describe the vicinity of a tri-
critical point, and the fact that it works, even in the
KDP family on a large temperature range, is not inconsistent with the previous microscopic model.
This point will be discussed in the next paper of
our group. But the present result, on TMO, is more surprising because first it does not seem so near a
tricritical state and the second, two descriptions (eighth order term, or b(T - T 1) r4 term) are suitable
for the whole temperature range (295 K-430 K).
Perhaps an extension of the studied temperature
range could permit a choice between these two pheno- menological descriptions.
5. Conclusion. - The presented data of ux)’ versus
the temperature in TMO samples are to a first approxi- mation, in agreement with a free energy expression,
as proposed by Dvorak [16], in which the order parameter is the antiferroelectric displacement of
some atoms r. But the accuracy of our measurements demonstrates that a free energy expression with three
terms of which only the first one is temperature depen- dent, leads to an imperfect description : the intro-
duction of a supplementary adjustable parameter is necessary. It could be the introduction of an eighth
order term, or the supposition of temperature varia- tion of the second term in the free energy expression.
In order to choose between these descriptions it
would be interesting to perform simultaneously bire- fringence, electrical and y ray diffractometry mea-
surements in TMO and other crystals of the GMO family until the temperature is near 0 K. It would also be interesting to apply hydrostatic pressure p to a TMO crystal, to study the transition versus p as in ’ KDP [24] although TMO is not as near a tri-
critical state as the latter.
Acknowledgments. - The authors thank Dr. Gri- mouille of the C.N.E.T. (Bagneux-France) ’who pro- vided a very good sample; they thank Pr. C. Mal- grange, Dr. J. Baruchel and A. Freund for cooperation
and fruitful discussions. They are grateful to the
Institut Laüe Langevin for the allocation of facilities and they wish to acknowledge the assistance of its staff. Thanks are also due to Pr. J. Lajzerowicz for stimulating discussions.
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