Abstract. The tetragonalarthorhombic phase transition in arsenic pentoxide has been studied by x-ray powder diffraction and is found to be a proper ferroelastic transition. The spontaneous strain behaves as the order parameter following Landau theory with E, CL IT, -TIs, 6 = 1, T, = 578 K. The order parameter susceptibility has been determined from line broadening and follows IT, -TI-7, y = 1. No deviation from Landau behaviour has been observed experimentally

E Salje

S A T Redfern

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J .  Phys. C :  Solid State Phys. 21 (1988) 277-285. Printed in the UK 
Spontaneous strain and the ferroelastic phase transition 
in As,O, 
S A T Redfern and E Salje 
Department of Earth Sciences, University of Cambridge, Downing Street, 
Cambridge CB2 3EQ, UK 
Received 11 June 1987, in final form 12 August 1987 
Abstract. The tetragonalarthorhombic phase transition in arsenic pentoxide has been 
studied by x-ray powder diffraction and is found to be a proper ferroelastic transition. 
The spontaneous strain behaves as the order parameter following Landau theory with 
E ,  CL IT, - TIs, 6 = 1, T,  = 578 K.  The order parameter susceptibility has been determined 
from line broadening and follows IT, - TI-7, y = 1. No deviation from Landau behaviour 
has been observed experimentally. 
1. Introduction 
Arsenic pentoxide, As205, is a binary oxide comprising As06  octahedral chains linked 
by A s 0 4  tetrahedra (Jansen 1978,1979). The room-temperature orthorhombic form, 
space group P212121 (Schonflies symbol Di), transforms to the tetragonal space group 
P4,2,2 (Schonflies symbol D:) above 578 K. Bismayer er a1 (1986) describe the driving 
free energy for the transformation in terms of a Landau expansion of the order 
parameter, the elastic energy, and direct coupling between the order parameter and the 
spontaneous strain. The thermodynamic potential is 
U b 
2 4 1.2 
A F  = - Q2 + - Q4 + . . . + h Q ( ~ 2  - ~ 1 )  + 4 C i k ~ i ~ k ,  
If the behaviour at the transition is that of a pure proper ferroelastic, the structural 
instability would traditionally be described by critical behaviour of the strain components 
E ~ .  To minimise the free energy for any value of Q the condition is 
@ A F / ~ Q ) ~ ~  = 0. 
The elastic constants would be renormalised to become explicitly temperature depen- 
dent, driving the transition. If a sublattice distortion of the polyhedral framework is the 
driving mechanism then Q ,  the order parameter itself, is used for the description of the 
transition with the temperature dependence of the potential arising from the prefactor 
U .  This arises from the condition that the crystal is free of stress at any temperature 
( d A F / d ~ , ) l , ~  = 0. 
In either case symmetry allows coupling between Q and E which may itself give rise to 
0022-3719/88/020277 + 09 $02.50 @ 1988 IOP Publishing Ltd 277 
278 
the transformation via an explicit temperature dependence of A. Bismayer et a1 (1986) 
show that it is possible to distinguish between the third mechanism and the first two by 
considering the temperature dependence of the elastic constants. Furthermore their 
Raman observations reveal that softening of optic modes occurs without full recovery 
in the high-temperature phase. This may indicate that the structural phase transition 
shows aspects of order-disorder behaviour in addition to the pure displacive model 
previously considered. 
The temperature evolution of the optical birefringence of As205 has been explained 
within the scope of Landau theory which was shown to apply over a large temperature 
interval between room temperature and T, - 5 K, where T, = 578 K (Salje et a1 1987). 
There is no evidence of crossover behaviour between order-disorder and displacive 
behaviour. Landau theory is appropriate, therefore, for the description of this ferro- 
elastic transition and it is suggested that this is due to long correlation lengths in 
framework structures which limit the temperature interval over which deviations from 
mean-field behaviour can occur. Although the dynamical properties and the morphic 
birefringence have been studied in much detail, nothing is known about the basic 
ferroelastic parameter-the spontaneous strain. This paper reports the results of inves- 
tigations employing x-ray diffraction to study both the temperature evolution of the 
spontaneous strain and the variation in correlation length through the transition, and 
examines the role of acoustic softening as a driving force for the transition. 
S A T Redfern and E Salje 
2. Experimental details 
Extremely hygroscopic As205 powder was loaded into 0.3 mm diameter silica glass 
capillaries, and sealed under argon. Diffraction experiments were carried out using 
a Huber 632 heating Guinier powder camera configured to record transmission powder 
patterns. Cu K a l  was selected from a bent quartz monochromator. A Huber 633 
Figure 1. The continuous movement heating Guinier diffraction pattern of As205. Reflec- 
tions in the tetragonal phase have been indexed. 
The ferroelectric phase transition in As205 279 
temperature controller allowed relative temperature stability better than 1 K,  as 
measured by a chromal-alumel thermocouple. Exposure were made under isothermal 
conditions, the controller allowing automatic isothermal conditions, the controller 
allowing automatic film movement and temperature variation between exposures. The 
absolute temperature scale was calculated and correction facilitated by measuring 
known transitions in the temperature range 370 to 700K. Care was taken to avoid 
errors arising from fluctuations in thermal gradient through the furnace. 
Figure 1 shows a photograph obtained from a continuous heating experiment, 
which is effectively a plot of line position with temperature. The large spontaneous 
strain is evident from the obvious peak splitting of pairs of reflections in the ortho- 
rhombic phase. Lines showing no peak splitting correspond to those with equal 
multiplicities in the orthorhombic and tetragonal phases. 
Cell parameters were calculated by least-squares refinement of about 20 measured 
reflections at each temperature, giving one standard deivation error of around 0.0014 A 
in a and b and 0.0007 A in c.  The temperature evolution of these lattice parameters 
is shown in figure 2. Line profiles were obtained from scans of exposed films by a 
Joyce-Loebl microdensitometer. Five densitometer scans were made for the exposure 
at each temperature and averaged to reduce statistical effects arising from film grain. 
The densitometer traces shown in figure 3 demonstrate the splitting of the tetragonal 
411 reflection into the orthorhombic 411 and 141 reflections through the transition, 
and show the behaviour of the 222 reflection, which is unaffected by the transition. 
Temperature IK) 
Figure 2. Lattice parameters of As20, from room temperature to 770 K. The size of each 
data point indicates the average standard error. 
280 S A T Redfern and E Salje 
Figure 3. Densitometer traces of As20,  powder x-ray patterns sho 
the 411 reflection. 
wing peak splittii ng of 
Figure 4. The spontaneous strain of As205 from room temperature to T,. Error bars 
indicate two standard deviations. The full curve shows the strain predicted by Landau 
theory. 
The ferroelectric phase transition in As205 281 
3. Spontaneous strain 
The transition from the high-temperature space group P41212 to the low-symmetry 
subgroup P212121 involves no loss of translational symmetry and is due solely to the 
critical behaviour of phonons associated with the centre (r point) of the Brillouin 
zone. Bismayer et a1 (1986) give the correlation table between these two space groups 
showing that the active representation of the order parameter corresponds to the 
representation B in the supergroup P41212. The spontaneous strain must also trans- 
form according to this active representation, in other words the symmetry-adapted 
strain is proportional to xz - y z .  Hence the form of the spontaneous strain (in Voigt 
notation) is E, a ( E ~  - E ~ )  or, in the nomenclature of Aizu (1970), E, = 2-’/’1~, - E ~ / .  
There are two orientational states of the ferroelastic P212121 structure with respect to 
the high-symmetry para-elastic phase. These are denoted S and S’ by Aizu, and 
correspond to the two possible twin domains observed in AsZO5 with equal and 
opposite resultant strains. 
Applying the approach of Salje er a1 (1985a) it is now possible to determine the 
twin law for ferroelastic domains in AsZO5. The spontaneous strains for the two 
adjacent twin domains are given by 
Ex,  0 0 
s = [; ;yy  gl = ( E 1  E 2  0 0 0 0 )  
S’ = [ ; - ; y y  ;] = ( - E l  - E 2  0 0 0 0 ) .  - E x x  0 0 
At the domain boundary all strain components must be equal. Adopting the 
arguments of Sapriel (1975) the domain wall is the plane defined by 
6, - S&x, = 0 
which leads to the condition that x = y or x = -y at the domain boundary. The twin 
planes are therefore of the form (110). According to David (1983) this implies that 
the soft acoustic modes associated with the transition propagate in the (110) directions. 
The spontaneous strain may be calculated from the lattice parameters measured 
by x-ray powder diffraction (figure 2). The strains and E~ are given by 
E1 =./ao - 1 
E 2  = b/bo - 1 
where the subscript 0 refers to those parameters expected by extrapolation from the 
high-symmetry para-phase to the ferroelastic phase. For the transition considered 
here a. = bo = (a + b) /2;  hence eS is given by 
E ,  = I ( u  - b) / (a  + b)l. 
The variation of the spontaneous strain, E $ ,  with temperature is shown in figure 4. 
The full curve shows the strain predicted by a Landau potential with critical exponent 
p = 4. We see that the spontaneous strain follows Landau behaviour within the limit 
or error. 
282 S A T Redfern and E Salje 
4. Discussion 
The temperature dependence of the spontaneous strain in As205 is in agreement with 
that expected from Landau theory of the structural phase transition. The good 
agreement with theory may be related to the restricted directions of propagation of 
the critical acoustic phonons. The softening is limited to waves in the (110) directions 
only. Cowley (1976) points out that in such cases (which he denotes as type I) a 
continuous phase transition may be expected to occur. Deviations from Landau 
behaviour result from fluctuations in the order parameter near T,. As fluctuations 
become more pronounced and comparable with the value of Q itself and if they extend 
over distances corresponding to the correlation range E then the Ginzburg criterion is 
no longer fulfilled and Landau theory breaks down. In the case of type I structural 
instabilities, however, fluctuations of the acoustic phonons are restricted to cones 
around the two perpendicular directions of wave propagation in k-space, and thus 
Landau theory will be obeyed even at temperatures close to T,. 
In addition to the onset of spontaneous strain, the phase transition is also reflected 
by a singularity in the width of those powder diffraction lines that split in the 
orthorhombic phase. Figure 5 shows the temperature evolution of the linewidth of 
such reflections, namely the orthorhombic 141 and 411 reflections, which merge to 
411 in the tetragonal phase. The observed increase in r does not occur for reflections 
whose Bragg angle does not depend on the order parameter, for example 222, as can 
be seen in figure 3. Above T, we observe significant but decreasing line broadening 
of the tetragonal 411 peak, corresponding to the increasing correlation length of 
fluctuations beyond that seen by x-rays. The linewidth of the 411 peak falls to that 
of the original low-temperature orthorhombic peaks. The possible order-disorder 
behaviour inferred from the non-recovery of the softened Raman mode observed by 
0.6 
0.5 
2 0.L 
'c) 
E 
w 
i 
0.3 
0.2 
450 MO 550 600 650 70( 
Temperature I K )  
Figure 5. The absolute linewidth, r, of the 411 reflection through the transition. r is in 
units of 28 degrees. 
The ferroelectric phase transition in As205 283 
Bismayer et a1 (1986) should lead to broadening of lines in the tetragonal phase, which 
do we not observe. The broadening may be explained solely in terms of order parameter 
fluctuations near T,: the linewidth increases due to inhomogeneous variations of 
the spontaneous strain, and lines that are independent of the order parameter remain 
sharp. The correlation length of the fluctuations increases as T, is approached from 
below , and becomes comparable with and then greater than the correlation length of 
the x-rays. The linewidth is therefore directly proportional to the susceptibility x 
of the order parameter. Landau theory predicts that the susceptibility shows the 
temperature dependence 
x CK IT - T,(-Y 
and the slope of x-’ against T below T, is twice that above T,. The data have been 
plotted in figure 6 for r-’ close to T, and it can be seen that this relationship between 
the slopes above and below T, is followed. This corresponds to the critical exponent 
y = 1 as expected for a phase transition following Landau theory. Hence the line 
broadening is entirely consistent with Landau theory, and results from the expected 
fluctuations of the order parameter near T,. 
Softening of the (110) transverse acoustic phonons at the zone centre might be 
expected to lead to dynamical broadening of the powder lines, but this diffuse scattering 
would affect all Bragg peaks and we have not been able to observe such an effect. 
Nevertheless such a phonon instability, resulting in softening of the mode near q = 0 
as T, is approached, may be used to describe the driving order parameter. The 
softening of the acoustic mode may be considered in terms of the elastic constants in 
the tetragonal form; the transition arises from an instability in the quantity C, ,  - Cl2. 
In this respect the transition is similar to that observed in DCN (MacKenzie and Pawley 
1979), where the phonon dispersion curves have been measured by neutron scattering 
and are observed to soften. Similar acoustic softening has been observed in s-triazine 
Temperature IK) 
Figure 6. The inverse linewidth, r. (corrected for instrumental broadening) for the 411 
reflection versus temperature. r is in units of 28 degrees. 
284 S A T Redfern and E Salje 
(Dove er a1 1983), where the ferroelastic transition at 198 K is accompanied by 
significant changes in the dispersion curves above T,. 
The softening of the hard-mode Raman lines observed by Bismayer et a1 (1986) 
arises from direct coupling between the acoustically driven spontaneous strain and the 
optic modes. The spontaneous strain behaves as the order parameter, and is pro- 
portional to the shift in the hard-mode frequency (Am): 
Am2 a E :  Q2 a ( T ,  - T)2P p = 3. 
These dependences obviously violate the Gruneisen relation: 
Aw/w = y i  AV/V 
where V is the molar volume and U is the hard-mode frequency. In the orthorhombic 
phase the cell volume is given by a X b X c,  and the lattice parameters a and b can be 
expressed in terms of the extrapolated tetragonal parameters a. and bo (ao = bo) as 
U = - &(TC - T)1’2 
b = bo + &(T,  - T)1’2 
j V  = co[aobo - a2(Tc - T ) ]  
j A u  0~ A V K  ( T ,  - T )  a E :  
which conflicts with the observed result, Am a E , .  Hence the Gruneisen relation is not 
obeyed in the ferroelastic phase of As205,  where the dilation along b counteracts the 
contraction along a in the expression for volume. 
In conclusion, the x-ray scattering experiments described here imply a transition 
mechanism in which the spontaneous strain directly reflects the order parameter, and 
which can be understood as being driven by acoustic mode softening. The transition 
is correctly described by a Landau model for the free energy with the distortion of the 
polyhedral framework playing the major role in the transition. In this respect As205 
is a model ferroelastic. Landau theory has also been successfully applied to a number 
of framework silicates to describe the behaviour of the spontaneous strain (Salje er ai 
1985b, Redfern and Salje 1987). It seems that for displacive transitions in framework 
structures, Landau theory accurately describes the observed behaviour over most of 
the range of the reduced temperature scale. This is attributed to the long length scales 
over which correlated ordering may occur. 
Acknowledgments 
The authors wish to thank U Bismayer for providing powdered samples of As205. 
SATR acknowledges financial support from the Natural Environmental Research 
Council. This is Cambridge Earth Sciences contribution number 1024. 
References 
Aizu K 1970 J .  Phys.  Soc. Japan 28 70616 
Bismayer U ,  Salje E, Jansen M and Dreher S 1986 J. Phys. C: Solid State Phys.  19 4537-45 
Cowley R A 1976 Phys. Rev. B 13 4877-85 
David W I 1983 J .  Phys.  C: Solid State Phys.  16 2455-70 
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Dove M T, Heilmann I U ,  Kjems J K, Jurittu J and Pawley G S 1983 Phys. Status Solidi b 120 172-81 
Jansen M 1978 Z. Anorg. (Allg.) Chem 441 5-12 
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Mackenzie G A and Pawley G S 1979 J .  Phys. C: Solid State Phys.  12 2717-35 
Redfern S A T and Salje E 1987 Phys. Chem. Minerals 14 189-95 
Salje E, Bismayer U and Jansen M 1987 J .  Phys. C:  Solid State Phys. 20 3613-20 
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Spontaneous strain and the ferroelastic phase transition in As,O

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Spontaneous strain and the ferroelastic phase transition in As,O

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