A neutron scattering study of the effect of hydrostatic pressure on the incommensurate phase transformation in K2SeO4 was performed in order to obtain information concerning the nature of the competing forces that give rise to a structural instability at a general wave vector. It was found that an increase in pressure lowers the transition temperature (dTi/dPi = - 6.5 K/kbar) and results in a smaller value of the initial incommensurate wave vector. The pressure dependence of the incommensurate to ferroelectric phase transition was also investigated (dTc/dPc = - 9 K/kbar). The latter transition displays a distinct pressure hysteresis

Axe, J. D.

Hardy, John R.

Majkrzak, C. F.

Massa, N. E.

Press, W.

Ullman, F. G.

English

UNL | Libraries

University of Nebraska - Lincoln 
DigitalCommons@University of Nebraska - Lincoln 
John R. Hardy Papers Research Papers in Physics and Astronomy 
7-1-1980 
Effect of Hydrostatic Pressure on the Incommensurate Phase of 
K2SeO4 
W. Press 
Brookhaven National Laboratory, Upton, New York 
C. F. Majkrzak 
Brookhaven National Laboratory, Upton, New York 
J. D. Axe 
Brookhaven National Laboratory, Upton, New York 
John R. Hardy 
University of Nebraska - Lincoln 
N. E. Massa 
University of Nebraska - Lincoln 
See next page for additional authors 
Follow this and additional works at: https://digitalcommons.unl.edu/physicshardy 
 Part of the Physics Commons 
Press, W.; Majkrzak, C. F.; Axe, J. D.; Hardy, John R.; Massa, N. E.; and Ullman, F. G., "Effect of Hydrostatic 
Pressure on the Incommensurate Phase of K2SeO4" (1980). John R. Hardy Papers. 26. 
https://digitalcommons.unl.edu/physicshardy/26 
This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at 
DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in John R. Hardy Papers by an 
authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 
Authors 
W. Press, C. F. Majkrzak, J. D. Axe, John R. Hardy, N. E. Massa, and F. G. Ullman 
This article is available at DigitalCommons@University of Nebraska - Lincoln: https://digitalcommons.unl.edu/
physicshardy/26 
PHYSICAL REVIEW B VOLUME 22, NUMBER 1 1 JULY 1980 
Effect of hydrostatic pressure on the incommensurate phase of K2Se04 
W. Press 
Brookhavetr Nariorral Laboratot:~. Uptotr. New York 119 73 
arrd I/rstitut,fUr FeslkKpe~:fbrschi~rr~ der. Kerr(fbrsc~h~irrgsa,rla~e. 
Jiilich. Wrsr Grr/tlat(v 
C. F. Majkrzak and J.  D. Axe 
Brookhavetr Natiotral Laboratot:v, Uprotr. New York 11 973 
J.  R. Hardy, N .  E. Massa, and F. G.  Ullman 
U~ri~'ersi!v of Nebraska-Lirrcolt~ Litrcolrr. Nebraska 68588 
(Received 1 2  February 1980) 
A neutron scattering study of the effect of hydrostatic pressure on the incommensurate phase 
transformation in K2Se04 was performed in order to obtain information concerning the nature 
of the competing forces that  give rise to a structural instability at  a general wave vector. It was 
found that an increase in pressure lowers the transition temperature (dT,/rIP, = - 6 . 5  Klkbar) 
and results in a smaller value of the initial incommensurate wave vector. The pressure depen- 
dence of the incommensurate to ferroelectric phase transition was also investigated ( d T c / d P c  
= - 9 Klkbar). The latter transition displays a distinct pressure hysteresis. 
I. INTRODUCTION 
It is widely believed that the incomensurate struc- 
tural transformations that occur in quasi-one- and 
two-dimensional metals are due to electronic charge- 
density-wave (CDW) instabilities occurring at the 
Fermi surfaces.'t2 It is even clearer that recently stud- 
ied examples of incommensurate structural transfor- 
mations in insulators2 must proceed by other 
mechanism. A necessary (but not sufficient) condi- 
tion for a structural instability at a general wave vec- 
tor is that there exists strong competition between 
forces of various ranges. This competition is presum- 
ably supplied by the oscillation of conduction-electron 
screening [Ruderman-Kittel-Kasuya-Yosida (RKKY) 
interaction] in metals. It has been suggested3-' that 
competition between Coulomb dipolar and short- 
range repulsive interactions is crucial in the case of 
insulators. The extent to which observations con- 
vincingly support the above ideas for either insulators 
or metals is not overwhelming and additional experi- 
mental evidence seems necessary if they are to be 
tested further. It is in this spirit that the present in- 
vestigation of the effect of hydrostatic pressure on 
the phase transformation in K2Se04 was performed. 
K2Se04 is perhaps the best studied6 and best 
behaved member of a structural class of insulators 
[including Rb2ZnBr4, ' (NH4)2BeF4, and Na2C03 
(Ref. 911 which are susceptible to incommensurate 
instabilities. (See Ref. 6 for a description of the 
structure.) The incommensurate transformation is as- 
sociated with a soft phonon mode with wave vector, 
- I qo= T ( l  - 6 ) a * ,  along the orthorhombic a  axis, and 
6  = 0.07 at the transformation temperature ( TI 
= 130 K at atmospheric pressure) which decreases 
with decreasing temperature jumping discontinuously 
to 6  = 0 at Tc = 93 K. In this low-temperature 
"locked-in" phase the unit cell triples along a relative 
to the high-temperature phase and a spontaneous fer- 
roelectric moment appears. Reference 6 presents a 
Landau theory of these transformations which is rea- 
sonably complete and satisfactory on a phenomeno- 
logical level. According to Ref. 5, the detailed 
description of the transfo~mation in K2Se04 depends 
upon a delicate balance of Coulomb and short-ranged 
repulsive forces. 
11. EXPERIMENTAL PROCEDURE 
A single crystal of potassium selenate was grown 
from the saturated aqueous solution by slow evapora- 
tion. The crystal was then cut to the shape of a cube 
with a volume of approximately 0.25 cm3 and placed 
in an aluminum hydrostatic pressure cell. The pres- 
sure cell was subsequently mounted in a Cryogenics 
Associates CT-14 flow cryostat. The pressure on the 
sample could be varied continuously throughout the 
range from 0.5 to 5.0 kbar with an accuracy of 
+ 2.0 x kbar. Sample temperature was controlled 
to within + 0.05 K. 
The neutron scattering measurements were per- 
formed on a triple-axis spectrometer in the elastic 
scattering mode at the Brookhaven High-Flux Beam 
22 - 332 01980 The American Physical Society 
22 - EFFECT OF HYDROSTATIC PRESSURE ON THE . . . 333 
Reactor. Pyrolytic graphite monochromator and 
analyzer crystals were used in conjunction with a py- 
rolytic graphite filter to suppress the higher-order 
neutrons. The incident neutron wave vector was 
2.549 A-I. 
The scattering measurements were made in the 
[0101 zone. In referring to the incommensurate and 
ferroelectric phases, it is expedient to introduce the 
reciprocal-lattice constant a:= f a*,  so that, e.g., - q,= ( 1  - 8)~:. We use this new basis in what fol- 
lows. 
111. EFFECTS OF PRESSURE 
Figure 1 shows the intensity of the ( 4  - S , 0 , 2 )  sa- 
tellite reflection and the value of 6 as a function of 
temperature at a pressure of 2.1 kbar, Included as 
the dashed line is the temperature dependence of S at  
atmospheric pressure as determind by Iizumi et a/. 
The increase in pressure results in a larger value of S 
at  the onset of incommensurability (and, therefore, a 
smaller value of the initial incommensurate wave 
vector i&,) and a lowering of the paraelectric to in- 
commensurate phase-transition temperature from a 
value of = 127 K at atmospheric pressure6 to 
= 116 K at  2.1 kbar. 
In Fig. 2, we summarize the pressure dependence 
of the lattice constants as well as 6 at  T - T,, a and c 
are seen to decrease (almost linearly) with increasing 
pressure, as would be expected. On the other hand, 
the value of 6(Pl,Ti) increases as the pressure be- 
comes greater. 
The pressure-temperature phase diagram is 
represented by Fig. 3 .  Both the paraelectric to in- 
commensurate and incommensurate to ferroelectric 
phase-transition temperatures are lowered by an in- 
crease in pressure. The latter transition displays a 
\y/ PRESSURE=I bar 10.08 
TEMPERATURE I K I  
PRESSURE I k b o r l  
FIG. 2.  The values of a,  c, and S at the critical point 
(PI, T , )  as a function of pressure. 
distinct pressure hysteresis and the lower phase line 
indicated in Fig. 3 is for decreasing pressure. The 
cross-hatched area is indicative of the extent of the 
two-phase region at temperatures around 60 K as 
measured at  constant temperature. Presumably the 
two-phase region extends to both higher and lower 
temperatures. It is apparent that pressure destabilizes 
the commensurate phase more than the incommens- 
urate one so that it is unlikely that a direct transfor- 
mation from the normal to the commensurate phase 
would occur at  even higher pressures. The slope of 
the phase lines are dTildP, = - 6.5 Klkbar and 
dTc/dPc= - 9 Klkbar for the upper and lower transi- 
tions, respectively. 
The paraelectric to incommensurate phase transi- 
tion is second order6 and therefore dTlldPi is given 
by the Ehrenfest relation 
where Ap, is the difference in the coefficient of ther- 
mal expansion and AC, the difference in the specific 
heat at constant pressure just above and below Ti. 
T (K) 
FIG. 1 .  The intensity of the (4 - S,0, 2 )  satellite reflec- 
tions and the value of 6 as a function of temperature at a FIG. 3. Pressure-temperature phase diagram for K2Se04. 
pressure of 2.1 kbars. The temperature dependence of S at The vertical bar at 60 K is representative of the width of the 
atmospheric pressure is after lizumi er a/. (Ref. 6 ) .  coexistence region. 
334 PRESS, MAJKRZAK, AXE, HARDY, MASSA, A N D  ULLMAN - 22 
Taking values for AP, from the work of Shiozaki 
et al. lo and AC, from that of Aiki et a / . ,  I '  we esti- 
mate the value of dT,/dP, to be 2 - 21 k 10 Klkbar 
using Eq. (1). Because of the large uncertainties in- 
volved, more accurate measurements of the thermal 
expansion and specific heat are required before any 
quantitative comparison can be made. However, the 
negative slope is consistent with our findings. 
Samara et a/. l 2  have observed a striking regularity 
in the behavior of dTc/dPc for many structural phase 
transformations. Namely, they found that dTc/dPc 
was negative for soft zone-center phonon instabilities 
and positive for zone-boundary phonon instabilities, 
with no known exceptions (at the time of publica- 
tion). They tentatively interpreted this effect as due 
to a reversal of roles of short-range and long-range 
Coulomb forces in the two cases. Of course K2Se04 
strictly falls into neither category, but it does seem 
clear that the critical wave vector involved is far too 
large to be enhanced by singular q = 0 Coulomb 
terms. Thus conceptually, K2Se04 must be con- 
sidered in the 9 f 0 category, and therefore seems to 
constitute an exception to the relation noted by 
Samara et a/. (there appears to be one other known 
ferroelectr~c exceptionI3 ). 
It is hoped that a comparison of the present results 
with further model calculations of the type discussed 
in Ref. 5 will lead eventually to an improved micro- 
scopic picture. In the meantime, there is one simple 
manipulation of the results which is possible. It was 
noted in Ref. 6 that the dispersion of the soft phonon 
branch as measured by neutron scattering could be 
Fourier transformed to yield effective "force con- 
stants" between entire planes of atoms lying perpen- 
dicular to f. It was found in this way that third- 
neighbor interlayer forces were dominant over first- 
and second-neighbor forces whereas fourth- and 
higher-neighbor forces were weaker still. Figure 4 
compares, for example, the measured dispersion of 
the soft branch at T = 130 K with that calculated by 
adjusting the three lowest-order force constants and 
FIG. 4. The measured dispersion of the soft branch at 
T = 130 K [after lizumi cJ/ 01. (Ref. 6)l and that calculated by 
adjusting the three lowest-order force constants as described 
in the text. 5 is the magnitude of the wave vector along Z'* 
in an extended zone scheme. 
neglecting longer-ranged contributions; i.e., we ap- 
proximate 
where 5 is the magnitude of the wave vector along f * 
in an extended zone ~ c h e m e . ~  Assuming that the 
transformation remains continuous at  higher pres- 
sure, the wave vector of the soft mode, to( Ti,Pi) is 
then given by Eq. (2) with the additional constraints. 
By solving Eqs. (2) and (3) simultaneously using the 
measured values of P,, T,) we can determine the 
two ratios (F1/F3) and (F2/F3) at various points 
along the phase line. The values so obtained are 
shown in Table I. Notice that the magnitudes of 
( FI/F3)  and ( F2/F3) both increase along the phase 
line in a way which would be roughly consistent with 
F1 and F2 remaining relatively constant and F3 de- 
TABLE 1. Values of the ratios F1/F3 and Fd/F3 of the effective force-constants coupling layers 
1 
in the crystal separated by a distance i t l a  at a number of critical points ( T , , P , ) .  
Ti ( K )  Pi (kbar) y o  (units of aL*) 5 F l l F 3  F2/F3 
22 - EFFECT OF HYDROSTATIC PRESSURE ON THE . . . 335 
creasing. While the agreement between the mea- 
sured and calculated curves in Fig. 4 makes it clear 
that the three-force-constant approximation has limit- 
ed validity, it does make specific predictions about 
the pressure dependence of the soft phonon branch. 
The present sample was too small to investigate the 
soft-mode dynamics, but we hope to make such mea- 
surements in the near future. 
ACKNOWLEDGMENTS 
Research for the work at Brookhaven was support- 
ed by the Division of Basic Energy Sciences, Depart- 
ment of Energy, under Contract No. DE-AC02- 
76CH00016. Work at the University of Nebraska 
was supported by the United States Army Research 
Office. 
'Fo r  a review o f  one-dimensional CDW's ,  see  R. C o m e s  
and  G .  Shirane, in'Hlghlv C ~ I ~ ~ L I ~ . / I I I ~  O t ~ r - D i t ~ e ~ ~ s i o t l a /  
Sollrls, edited by J .  T .  Devreese  (Plenum,  New York, 
19791, Chap.  2. For  two-dimensional CDW's ,  see  J. A. 
Wilson, F .  J .  Di Salvo, and S.  Mahajan, Adv. Phys. 24, 
117 (1975). 
2For  a general review of neutron scattering measurements  
o n  incommensurate  systems, see J. D. Axe ,  I~~c~ las t i c  New- 
 rot^ Scat/erillg ( IAEA, Vienna, 1978). 
3Y. Yamada and  T. Yamada,  J .  Phys. Soc. Jpn. a, 2167 
(1966). 
4 ~ .  M. Fortuin ,  Physica (Utrecht) m, 224 (1977). 
5M. S. Haque,  J. R .  Hardy, Q. Kim,  and  F. G .  Ullman, Solid 
State C o m m u n .  27, 813 (1978); M .  S. Haque and  J .  R. 
Hardy, Phys. Rev.  B (in press).  
6M. Iizumi, J. D.  Axe,  G .  Shirane, and K .  Shimaoka,  Phys. 
Rev. B 15, 4392 (1977). 
'C. J .  d e  Paler and C.  van Dijk, Phys. Rev.  B 18, 1281 
(1978). 
8M.  l ~ z u m i  and K. Gesi ,  Solid State C o m m u n .  22, 37 
(1977). 
9W. Van Aalst, J .  den  Hollander,  W .  J. A .  M .  Peterse, and  
P. M.  d e  Wolff, Acta Crystallogr. Sect. B 32 ,  47 (1976). 
I0S. Shiozaki, A .  Sdwada, Y. Ishibashi, and  Y. Takagi,  J.  
Phys. Soc. Jpn. 42, 353 (1977). 
"K. A i k ~ ,  K. Hukuda,  H .  Koga, and  T .  Kobayashi,  J .  Phys. 
Soc. Jpn. 28, 389 (1970). 
I2G. A ,  Samara ,  T. Sakudo,  and  K.  Yoshimitsu, Phys. Rev.  
Lett .  35 ,  1767 (1975). 
I3F. Smutny,  J. Fousek,  and M .  Kotrbova,  Ferroelectrics Z, 
143 (1974). 


Effect of Hydrostatic Pressure on the Incommensurate Phase of K\u3csub\u3e2\u3c/sub\u3eSeO\u3csub\u3e4\u3c/sub\u3e

DigitalCommons@University of Nebraska

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln John R. Hardy Papers Research Papers in Physics and Astronomy 7-1-1980 Effect of Hydrostatic Pressure on the Incommensurate Phase of K2SeO4 W. Press Brookhaven National Laboratory, Upton, New York C. F. Majkrzak Brookhaven National Laboratory, Upton, New York J. D. Axe Brookhaven National Laboratory, Upton, New York John R. Hardy University of Nebraska - Lincoln N. E. Massa University of Nebraska - Lincoln See next page for additional authors Follow this and additional works at: https://digitalcommons.unl.edu/physicshardy  Part of the Physics Commons Press, W.; Majkrzak, C. F.; Axe, J. D.; Hardy, John R.; Massa, N. E.; and Ullman, F. G., "Effect of Hydrostatic Pressure on the Incommensurate Phase of K2SeO4" (1980). John R. Hardy Papers. 26. https://digitalcommons.unl.edu/physicshardy/26 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in John R. Hardy Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Authors W. Press, C. F. Majkrzak, J. D. Axe, John R. Hardy, N. E. Massa, and F. G. Ullman This article is available at DigitalCommons@University of Nebraska - Lincoln: https://digitalcommons.unl.edu/physicshardy/26 PHYSICAL REVIEW B VOLUME 22, NUMBER 1 1 JULY 1980 Effect of hydrostatic pressure on the incommensurate phase of K2Se04 W. Press Brookhavetr Nariorral Laboratot:~. Uptotr. New York 119 73 arrd I/rstitut,fUr FeslkKpe~:fbrschi~rr~ der. Kerr(fbrsc~h~irrgsa,rla~e. Jiilich. Wrsr Grr/tlat(v C. F. Majkrzak and J.  D. Axe Brookhavetr Natiotral Laboratot:v, Uprotr. New York 11 973 J.  R. Hardy, N .  E. Massa, and F. G.  Ullman U~ri~'ersi!v of Nebraska-Lirrcolt~ Litrcolrr. Nebraska 68588 (Received 1 2  February 1980) A neutron scattering study of the effect of hydrostatic pressure on the incommensurate phase transformation in K2Se04 was performed in order to obtain information concerning the nature of the competing forces that  give rise to a structural instability at  a general wave vector. It was found that an increase in pressure lowers the transition temperature (dT,/rIP, = - 6 . 5  Klkbar) and results in a smaller value of the initial incommensurate wave vector. The pressure depen- dence of the incommensurate to ferroelectric phase transition was also investigated ( d T c / d P c  = - 9 Klkbar). The latter transition displays a distinct pressure hysteresis. I. INTRODUCTION It is widely believed that the incomensurate struc- tural transformations that occur in quasi-one- and two-dimensional metals are due to electronic charge- density-wave (CDW) instabilities occurring at the Fermi surfaces.'t2 It is even clearer that recently stud- ied examples of incommensurate structural transfor- mations in insulators2 must proceed by other mechanism. A necessary (but not sufficient) condi- tion for a structural instability at a general wave vec- tor is that there exists strong competition between forces of various ranges. This competition is presum- ably supplied by the oscillation of conduction-electron screening [Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction] in metals. It has been suggested3-' that competition between Coulomb dipolar and short- range repulsive interactions is crucial in the case of insulators. The extent to which observations con- vincingly support the above ideas for either insulators or metals is not overwhelming and additional experi- mental evidence seems necessary if they are to be tested further. It is in this spirit that the present in- vestigation of the effect of hydrostatic pressure on the phase transformation in K2Se04 was performed. K2Se04 is perhaps the best studied6 and best behaved member of a structural class of insulators [including Rb2ZnBr4, ' (NH4)2BeF4, and Na2C03 (Ref. 911 which are susceptible to incommensurate instabilities. (See Ref. 6 for a description of the structure.) The incommensurate transformation is as- sociated with a soft phonon mode with wave vector, - I qo= T ( l  - 6 ) a * ,  along the orthorhombic a  axis, and 6  = 0.07 at the transformation temperature ( TI = 130 K at atmospheric pressure) which decreases with decreasing temperature jumping discontinuously to 6  = 0 at Tc = 93 K. In this low-temperature "locked-in" phase the unit cell triples along a relative to the high-temperature phase and a spontaneous fer- roelectric moment appears. Reference 6 presents a Landau theory of these transformations which is rea- sonably complete and satisfactory on a phenomeno- logical level. According to Ref. 5, the detailed description of the transfo~mation in K2Se04 depends upon a delicate balance of Coulomb and short-ranged repulsive forces. 11. EXPERIMENTAL PROCEDURE A single crystal of potassium selenate was grown from the saturated aqueous solution by slow evapora- tion. The crystal was then cut to the shape of a cube with a volume of approximately 0.25 cm3 and placed in an aluminum hydrostatic pressure cell. The pres- sure cell was subsequently mounted in a Cryogenics Associates CT-14 flow cryostat. The pressure on the sample could be varied continuously throughout the range from 0.5 to 5.0 kbar with an accuracy of + 2.0 x kbar. Sample temperature was controlled to within + 0.05 K. The neutron scattering measurements were per- formed on a triple-axis spectrometer in the elastic scattering mode at the Brookhaven High-Flux Beam 22 - 332 01980 The American Physical Society 22 - EFFECT OF HYDROSTATIC PRESSURE ON THE . . . 333 Reactor. Pyrolytic graphite monochromator and analyzer crystals were used in conjunction with a py- rolytic graphite filter to suppress the higher-order neutrons. The incident neutron wave vector was 2.549 A-I. The scattering measurements were made in the [0101 zone. In referring to the incommensurate and ferroelectric phases, it is expedient to introduce the reciprocal-lattice constant a:= f a*,  so that, e.g., - q,= ( 1  - 8)~:. We use this new basis in what fol- lows. 111. EFFECTS OF PRESSURE Figure 1 shows the intensity of the ( 4  - S , 0 , 2 )  sa- tellite reflection and the value of 6 as a function of temperature at a pressure of 2.1 kbar, Included as the dashed line is the temperature dependence of S at  atmospheric pressure as determind by Iizumi et a/. The increase in pressure results in a larger value of S at  the onset of incommensurability (and, therefore, a smaller value of the initial incommensurate wave vector i&,) and a lowering of the paraelectric to in- commensurate phase-transition temperature from a value of = 127 K at atmospheric pressure6 to = 116 K at  2.1 kbar. In Fig. 2, we summarize the pressure dependence of the lattice constants as well as 6 at  T - T,, a and c are seen to decrease (almost linearly) with increasing pressure, as would be expected. On the other hand, the value of 6(Pl,Ti) increases as the pressure be- comes greater. The pressure-temperature phase diagram is represented by Fig. 3 .  Both the paraelectric to in- commensurate and incommensurate to ferroelectric phase-transition temperatures are lowered by an in- crease in pressure. The latter transition displays a \y/ PRESSURE=I bar 10.08 TEMPERATURE I K I  PRESSURE I k b o r l  FIG. 2.  The values of a,  c, and S at the critical point (PI, T , )  as a function of pressure. distinct pressure hysteresis and the lower phase line indicated in Fig. 3 is for decreasing pressure. The cross-hatched area is indicative of the extent of the two-phase region at temperatures around 60 K as measured at  constant temperature. Presumably the two-phase region extends to both higher and lower temperatures. It is apparent that pressure destabilizes the commensurate phase more than the incommens- urate one so that it is unlikely that a direct transfor- mation from the normal to the commensurate phase would occur at  even higher pressures. The slope of the phase lines are dTildP, = - 6.5 Klkbar and dTc/dPc= - 9 Klkbar for the upper and lower transi- tions, respectively. The paraelectric to incommensurate phase transi- tion is second order6 and therefore dTlldPi is given by the Ehrenfest relation where Ap, is the difference in the coefficient of ther- mal expansion and AC, the difference in the specific heat at constant pressure just above and below Ti. T (K) FIG. 1 .  The intensity of the (4 - S,0, 2 )  satellite reflec- tions and the value of 6 as a function of temperature at a FIG. 3. Pressure-temperature phase diagram for K2Se04. pressure of 2.1 kbars. The temperature dependence of S at The vertical bar at 60 K is representative of the width of the atmospheric pressure is after lizumi er a/. (Ref. 6 ) .  coexistence region. 334 PRESS, MAJKRZAK, AXE, HARDY, MASSA, A N D  ULLMAN - 22 Taking values for AP, from the work of Shiozaki et al. lo and AC, from that of Aiki et a / . ,  I '  we esti- mate the value of dT,/dP, to be 2 - 21 k 10 Klkbar using Eq. (1). Because of the large uncertainties in- volved, more accurate measurements of the thermal expansion and specific heat are required before any quantitative comparison can be made. However, the negative slope is consistent with our findings. Samara et a/. l 2  have observed a striking regularity in the behavior of dTc/dPc for many structural phase transformations. Namely, they found that dTc/dPc was negative for soft zone-center phonon instabilities and positive for zone-boundary phonon instabilities, with no known exceptions (at the time of publica- tion). They tentatively interpreted this effect as due to a reversal of roles of short-range and long-range Coulomb forces in the two cases. Of course K2Se04 strictly falls into neither category, but it does seem clear that the critical wave vector involved is far too large to be enhanced by singular q = 0 Coulomb terms. Thus conceptually, K2Se04 must be con- sidered in the 9 f 0 category, and therefore seems to constitute an exception to the relation noted by Samara et a/. (there appears to be one other known ferroelectr~c exceptionI3 ). It is hoped that a comparison of the present results with further model calculations of the type discussed in Ref. 5 will lead eventually to an improved micro- scopic picture. In the meantime, there is one simple manipulation of the results which is possible. It was noted in Ref. 6 that the dispersion of the soft phonon branch as measured by neutron scattering could be Fourier transformed to yield effective "force con- stants" between entire planes of atoms lying perpen- dicular to f. It was found in this way that third- neighbor interlayer forces were dominant over first- and second-neighbor forces whereas fourth- and higher-neighbor forces were weaker still. Figure 4 compares, for example, the measured dispersion of the soft branch at T = 130 K with that calculated by adjusting the three lowest-order force constants and FIG. 4. The measured dispersion of the soft branch at T = 130 K [after lizumi cJ/ 01. (Ref. 6)l and that calculated by adjusting the three lowest-order force constants as described in the text. 5 is the magnitude of the wave vector along Z'* in an extended zone scheme. neglecting longer-ranged contributions; i.e., we ap- proximate where 5 is the magnitude of the wave vector along f * in an extended zone ~ c h e m e . ~  Assuming that the transformation remains continuous at  higher pres- sure, the wave vector of the soft mode, to( Ti,Pi) is then given by Eq. (2) with the additional constraints. By solving Eqs. (2) and (3) simultaneously using the measured values of P,, T,) we can determine the two ratios (F1/F3) and (F2/F3) at various points along the phase line. The values so obtained are shown in Table I. Notice that the magnitudes of ( FI/F3)  and ( F2/F3) both increase along the phase line in a way which would be roughly consistent with F1 and F2 remaining relatively constant and F3 de- TABLE 1. Values of the ratios F1/F3 and Fd/F3 of the effective force-constants coupling layers 1 in the crystal separated by a distance i t l a  at a number of critical points ( T , , P , ) .  Ti ( K )  Pi (kbar) y o  (units of aL*) 5 F l l F 3  F2/F3 22 - EFFECT OF HYDROSTATIC PRESSURE ON THE . . . 335 creasing. While the agreement between the mea- sured and calculated curves in Fig. 4 makes it clear that the three-force-constant approximation has limit- ed validity, it does make specific predictions about the pressure dependence of the soft phonon branch. The present sample was too small to investigate the soft-mode dynamics, but we hope to make such mea- surements in the near future. ACKNOWLEDGMENTS Research for the work at Brookhaven was support- ed by the Division of Basic Energy Sciences, Depart- ment of Energy, under Contract No. DE-AC02- 76CH00016. Work at the University of Nebraska was supported by the United States Army Research Office. 'Fo r  a review o f  one-dimensional CDW's ,  see  R. C o m e s  and  G .  Shirane, in'Hlghlv C ~ I ~ ~ L I ~ . / I I I ~  O t ~ r - D i t ~ e ~ ~ s i o t l a /  Sollrls, edited by J .  T .  Devreese  (Plenum,  New York, 19791, Chap.  2. For  two-dimensional CDW's ,  see  J. A. Wilson, F .  J .  Di Salvo, and S.  Mahajan, Adv. Phys. 24, 117 (1975). 2For  a general review of neutron scattering measurements  o n  incommensurate  systems, see J. D. Axe ,  I~~c~ las t i c  New-  rot^ Scat/erillg ( IAEA, Vienna, 1978). 3Y. Yamada and  T. Yamada,  J .  Phys. Soc. Jpn. a, 2167 (1966). 4 ~ .  M. Fortuin ,  Physica (Utrecht) m, 224 (1977). 5M. S. Haque,  J. R .  Hardy, Q. Kim,  and  F. G .  Ullman, Solid State C o m m u n .  27, 813 (1978); M .  S. Haque and  J .  R. Hardy, Phys. Rev.  B (in press).  6M. Iizumi, J. D.  Axe,  G .  Shirane, and K .  Shimaoka,  Phys. Rev. B 15, 4392 (1977). 'C. J .  d e  Paler and C.  van Dijk, Phys. Rev.  B 18, 1281 (1978). 8M.  l ~ z u m i  and K. Gesi ,  Solid State C o m m u n .  22, 37 (1977). 9W. Van Aalst, J .  den  Hollander,  W .  J. A .  M .  Peterse, and  P. M.  d e  Wolff, Acta Crystallogr. Sect. B 32 ,  47 (1976). I0S. Shiozaki, A .  Sdwada, Y. Ishibashi, and  Y. Takagi,  J.  Phys. Soc. Jpn. 42, 353 (1977). "K. A i k ~ ,  K. Hukuda,  H .  Koga, and  T .  Kobayashi,  J .  Phys. Soc. Jpn. 28, 389 (1970). I2G. A ,  Samara ,  T. Sakudo,  and  K.  Yoshimitsu, Phys. Rev.  Lett .  35 ,  1767 (1975). I3F. Smutny,  J. Fousek,  and M .  Kotrbova,  Ferroelectrics Z, 143 (1974). 

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Effect of Hydrostatic Pressure on the Incommensurate Phase of K\u3csub\u3e2\u3c/sub\u3eSeO\u3csub\u3e4\u3c/sub\u3e

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