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A microscopic model of first order phase transition in squaric acid
Solid State Communications, Vol. 71, No. 1, pp. 45-48, 1989. Printed in Great Britain.
0038-1098/89 $3.00 + .00 Maxwell Pergamon Macmillan plc
A MICROSCOPIC MODEL OF FIRST O R D E R PHASE TRANSITION IN SQUARIC ACID C.L. Wang and Z.K. Qin Department of Physics, Shandong University, Jinan, P.R. China and D.L. Lin Department of Physics and Astronomy, State University of New York at Buffalo, Amherst, NY 14260, USA
(Received 2 December 1988 by B. Miihlschlegel) The contribution of four-body interaction is introduced into the pseudospin model in order to treat the first order phase transition in squaric acid. Using the Green's function method, we study the phase transition properties, the conclusion is that the first order phase transition is induced by the four-body interaction. RECENTLY special attention has been paid to the structural phase transition which appears in squaric acid crystal (H2C404) that first synthesised in 1959 [1]. It was shown by Semmingsen and Feder [2] and by Sammuelsen and Semmingsen [3] that the squaric acid undergoes an antiferroelectric phase transition around Tc = 373 K. The careful analysis of birefringence measurements [4] and ~3C-NMR [5, 6] has provided convincing evidence for a discontinuous transition. As shown in Fig. 1, layered structure is a distinctive characteristic of the squaric acid crystal. In the a-c plane of the low temperature structure the planar C404 units are linked at their four corners by asymmetric hydrogen bonds. The crystal is built up by strongly hydrogen bonded planar molecules with weak presumably Van der Waals interactions between the layers. Since the polarization vectors of adjacent layers are antiparallel to each other, no macroscopic dielectric dipole moment can be observed. Squaric acid is an order-disorder ferroelectrics which can be described by pseudo-spin model [7]. In the case of two-body interaction are included, only second order phase transition is predicted [8, 9], that is contradictionary to the experimental result. U. Deininghaus and M. Mehring [10] reexamined the structure of squaric acid, and indicated that there might be in principle a four-body interaction of the four hydrogen bonds on the corners of the squaric acid moleculars. In the calculation they omitted this contribution, and didn't get a first order phase transition. By considering the coupling between the proton and lattice distortion, Ishibashi [11] discussed the first order phase transition through the calculation
of the free energy in squaric crystal, and haven't got the result to fit the experimental data. In this communication we will introduce the fourbody interaction to the pseudo-spin model, then using Green's function method to study the first order phase transition in squaric acid. The Hamiltonian is 1
- 2pE Z $7
1
(1)
i
where ~ is the tunneling frequency, S/~ is the tunneling operator at ith site, and $7 is the dipole operator which measure the difference of the occupation number at the double-well potential. J~s is two-body interaction. J~k~is the four-body interaction. The last term is the contribution of the external field,/~ is the effective dipole moment. The first two terms is the pseudo-spin model proposed by Blinc [7]. As for the squaric acid crystal, two-body interaction consists of: intra-layer interaction J~ and J2 inter-layer interaction J3. The four-body interaction J4 comes from the four hydrogen bond at the corners of the same C404 unit, as shown in Fig. 2. In order to solve Hamiltonian (1), we construct the double time Green's function ((S~(t)[ST(t'))), where m = z, + , - and n -- z, x. Insert Hamiltonian (1) into the motion equation of the Green's function, a~((S"IS"))
=
1
2 ~ ( [ S " . S " ] ) + (([S", H ] I S " ) )
(2) 45
46
FIRST ORDER PHASE TRANSITION IN SQUARIC ACID
Vol. 71, No. 1
scheme [13] ((ABDFIC))
•
=
(B) (D)(F)((AIC)).
(3b)
Then the motion equation of Green's function reduce to
C
to
fl 2
f2
to-Q
-f2
0
0
0
0 0
f~ 2
0
0
0
0
0
0
0
0
0
0
tO
f2 2
0
0
fl
tO-Q
0
0
-f~
0 tO+Q
f2 2 0
0
tO+Q
azz
0
- ( S x)
~r+z J
G zx
Fig. 1. Schematic representation of two layers of the structure of squaric acid. Big circles - O atoms; mediate circles - atoms; small circles - H atoms.
1
g
(SX>
= ~
(4)
0
.~+~
-
J
using Tyablikov's decoupling scheme ((ABIC))
=
(B) ((A IC))
(3a)
where amn
to the ( ( A B I C ) ) - t y p e Green's function resulted from the two-body interaction, and for the ( ( A B D F I C ) ) type Green's function resulted from the four-body interaction, we introduced the following decoupling
J = J' (si>
_/ / f Z
((smIs")),
= =
J,j =
Q = j(s:)
2(J, + J 2 -
+ J'(SZ) 3
J3),
j
(5)
E Z~k, = 2]4
jkt
(s;)
=
(sz>.
From equation (4), we can get the Green's functions, then substitute into the spectral theorem [14] yielding
~
- f ~ ( S ~ > + (Sx> ( J ( S z) + J'(SZ> 2 + 2~E)
N
I I I
da I I I
.....
<~
Ja- -~"
/
-
-~.
x
/
\
\
\~ s~
/ \\
j
,' //
S3
/\
0
(6a)
S,
/
=
\al
(SX) 2[(SX> 2 -Jr- (SZ> 21
~ cth too too 2kr
(6b)
where too = [~2 q_ ( j ( S z) + j , ( S Z ) Z + 2pE)2],/2
(6c)
when E = 0, we have the equations that describe the phase transition process, [ - f ~ + ( S x) (J + J ' ( S ~ ) 2 ) ] ( S ") (SX) = 2[(SX) 2 + (SZ) 2]
=
--f~ cth tOo 090 2kT
0
(7a) (7b)
where Fig. 2. Pseudo-spin model of squaric acid.
too = [~2 + (J + a'(S~>2) ~ (SZ>q '/~
(8)
Vol. 71, No. 1
There is a trivial solution in equation (7), corresponding to the paraelectric phase. Equation (7) has another solution ( S z) :~ 0 which correspond to the antiferroelectric phase in squaric acid. From equation (7) we have f~ j ,
( S ~) = j +
L
11~-
J + J'(SZ)2 -
2~0
o
tanh
(1 l)
(10).
J(SZ)
+ J ' ( S : ) 3 d- 2#E
= ½tanh
2kT
(12)
Then we have the external field - - J ( S ~) - J ' ( S " ) : + k T l n
1 + 2 ] -
2
(13) and the free energy (S=)
F-
Fo =
f
=
Tc =
I
330
340
350 T (K)
360
370
380
1481.82KandJ'
=
2436.95K.
+ k T ( S z ) In
1.
4 ( S ~ ) 2)
1 + 2
-
2(SZ)
2. (14)
where Fo is the free energy of paraelectric phase. At the Curie temperature T = Tc the free energy of paraand ferroelectric phase are equal, i.e. F = Fo, hence ½J(S~)~ -- ¼ J ' ( S ~ ) ,4 + ½kT T~ In (1 -- 4(SZ)~) 1 +
2(S")c
1
2(S~),
-
372.2Kand(SZ),
=
0.258
REFERENCES
1j2 -- 1 j , ( S Z > 4
+ ½kTln (] -
3. 4. 5. 6.
--
0
(15)
7.
and T~, ( S : ) c also satisfy equation (12) with E = 0 (SZ)c =
5
which is in agreement with the experimental data [6]. The temperature dependence of ( S " ) is plotted in Fig. 3, also the curve of without considering the fourbody interaction is drawn as comparison. When the contribution of the four-body interaction in included, the pseudo-spin model can describe the first order phase transition satisfactorily. Without the four-body interaction, only a second order phase transition was obtained. As the conclusion the first order phase transition in squaric acid is induced by the four-body interaction and can be treated by the pseudo-spin model.
2#E.d
+ kTc
+ +++++++
The Curie temperature T~ and the discontinuity ( S z)c can be obtained from equations (15) and (16). For squaric acid we chose
0 --
+ +
Then from equations (15) and (16) we obtain
The determination of the Curie temperature will appeal to the free energy. In squaric acid crystal, the tunneling frequency is very small, and can be omitted. Setting ~ = 0 in equation (6) we have
2#E =
+ +
Fig. 3. Fitting of the temperature dependence of order parameters q = 2 ( S z) with the experimental data [6]. (a) J = 1481.82K and J ' = 2436.95K (b) J = 1489K and J ' = 0.
J
(S-')
. ;20
(10)
-- 0
and equation
+
L°°
2kT"
Analysing the temperature dependence of ( S : ) in equation (10), we find that this equation describes a first order phase transition when J ' / J > 4/3 [15]. The limit temperature of paraelectric phase is To = J/4k, and the limit temperature of ferroelectric phase can be determined through 8T
+
(9)
substitute ( S x) into equation (7b) 1
47
FIRST O R D E R PHASE T R A N S I T I O N IN SQUARIC ACID
½tanh J ( S : ) , + J'(S-')~ 2kT,
(16)
8.
S. Cohen, J.R. Laher & J.D. Park, J. Amer. Chem. Soc. 81 (1959) 3480. D. Semmingsen & J. Feder, Solid State Commun. 15 (1974) 1396. E.J. Samuelsen & D. Semmingsen, (a) Solid State Commun. 17, 217 (1975); (b) J. Phys. Chem. Solids 38, 1275 (1977). W. Kuhn, H.D. Maier & J. Petersson, Solid State Commun. 32, 249 (1979). M. Mehring & J.D. Becker, Phys. Rev. Lett. 47, 366 (1981). G. Fisher, J. Petersson & D. Michel, Z. Phys. B - Condensed Matter 67, 387 (1987). R. Blinc & B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics, Chpt. 5, North-Holland (1974). V.I. Zinenko, Phys. Status Solidi (b) 78, 721 (1976).
48 9.
10. 11. 12.
FIRST ORDER PHASE TRANSITION IN SQUARIC ACID E. Matsushita, K. Yoshinitsu & T. Matsubaras, Prog. Theor. Phys. 64, 1176 (1980). U. Deininghaus & M. Mehring, Solid State Commun. 39 (1981) 1257. Y. Ishbashi, J. Phys. Soc. Jpn. 52, 200 (1983). N.N. Bogolyubov & S.V. Tyablikov, Doklady Akad. Nauk. S.S.S.R. 126, 53 (1959).
13. 14. 15.
Vol. 71, No. 1
C.L. Wang, Z.K. Qin & J.B. Zhang, Ferroelectrics 77, 21 (1987). J.B. Zhang & Z.K. Qin, Phys. Rev. A36, 915 (1987). C.L. Wang, Z.K. Qin & D.L. Lin (to be published in Phys. Rev).
0038-1098/89 $3.00 + .00 Maxwell Pergamon Macmillan plc
A MICROSCOPIC MODEL OF FIRST O R D E R PHASE TRANSITION IN SQUARIC ACID C.L. Wang and Z.K. Qin Department of Physics, Shandong University, Jinan, P.R. China and D.L. Lin Department of Physics and Astronomy, State University of New York at Buffalo, Amherst, NY 14260, USA
(Received 2 December 1988 by B. Miihlschlegel) The contribution of four-body interaction is introduced into the pseudospin model in order to treat the first order phase transition in squaric acid. Using the Green's function method, we study the phase transition properties, the conclusion is that the first order phase transition is induced by the four-body interaction. RECENTLY special attention has been paid to the structural phase transition which appears in squaric acid crystal (H2C404) that first synthesised in 1959 [1]. It was shown by Semmingsen and Feder [2] and by Sammuelsen and Semmingsen [3] that the squaric acid undergoes an antiferroelectric phase transition around Tc = 373 K. The careful analysis of birefringence measurements [4] and ~3C-NMR [5, 6] has provided convincing evidence for a discontinuous transition. As shown in Fig. 1, layered structure is a distinctive characteristic of the squaric acid crystal. In the a-c plane of the low temperature structure the planar C404 units are linked at their four corners by asymmetric hydrogen bonds. The crystal is built up by strongly hydrogen bonded planar molecules with weak presumably Van der Waals interactions between the layers. Since the polarization vectors of adjacent layers are antiparallel to each other, no macroscopic dielectric dipole moment can be observed. Squaric acid is an order-disorder ferroelectrics which can be described by pseudo-spin model [7]. In the case of two-body interaction are included, only second order phase transition is predicted [8, 9], that is contradictionary to the experimental result. U. Deininghaus and M. Mehring [10] reexamined the structure of squaric acid, and indicated that there might be in principle a four-body interaction of the four hydrogen bonds on the corners of the squaric acid moleculars. In the calculation they omitted this contribution, and didn't get a first order phase transition. By considering the coupling between the proton and lattice distortion, Ishibashi [11] discussed the first order phase transition through the calculation
of the free energy in squaric crystal, and haven't got the result to fit the experimental data. In this communication we will introduce the fourbody interaction to the pseudo-spin model, then using Green's function method to study the first order phase transition in squaric acid. The Hamiltonian is 1
- 2pE Z $7
1
(1)
i
where ~ is the tunneling frequency, S/~ is the tunneling operator at ith site, and $7 is the dipole operator which measure the difference of the occupation number at the double-well potential. J~s is two-body interaction. J~k~is the four-body interaction. The last term is the contribution of the external field,/~ is the effective dipole moment. The first two terms is the pseudo-spin model proposed by Blinc [7]. As for the squaric acid crystal, two-body interaction consists of: intra-layer interaction J~ and J2 inter-layer interaction J3. The four-body interaction J4 comes from the four hydrogen bond at the corners of the same C404 unit, as shown in Fig. 2. In order to solve Hamiltonian (1), we construct the double time Green's function ((S~(t)[ST(t'))), where m = z, + , - and n -- z, x. Insert Hamiltonian (1) into the motion equation of the Green's function, a~((S"IS"))
=
1
2 ~ ( [ S " . S " ] ) + (([S", H ] I S " ) )
(2) 45
46
FIRST ORDER PHASE TRANSITION IN SQUARIC ACID
Vol. 71, No. 1
scheme [13] ((ABDFIC))
•
=
(B) (D)(F)((AIC)).
(3b)
Then the motion equation of Green's function reduce to
C
to
fl 2
f2
to-Q
-f2
0
0
0
0 0
f~ 2
0
0
0
0
0
0
0
0
0
0
tO
f2 2
0
0
fl
tO-Q
0
0
-f~
0 tO+Q
f2 2 0
0
tO+Q
azz
0
- ( S x)
~r+z J
G zx
Fig. 1. Schematic representation of two layers of the structure of squaric acid. Big circles - O atoms; mediate circles - atoms; small circles - H atoms.
1
g
(SX>
= ~
(4)
0
.~+~
-
J
using Tyablikov's decoupling scheme ((ABIC))
=
(B) ((A IC))
(3a)
where amn
to the ( ( A B I C ) ) - t y p e Green's function resulted from the two-body interaction, and for the ( ( A B D F I C ) ) type Green's function resulted from the four-body interaction, we introduced the following decoupling
J = J' (si>
_/ / f Z
((smIs")),
= =
J,j =
Q = j(s:)
2(J, + J 2 -
+ J'(SZ) 3
J3),
j
(5)
E Z~k, = 2]4
jkt
(s;)
=
(sz>.
From equation (4), we can get the Green's functions, then substitute into the spectral theorem [14] yielding
~
- f ~ ( S ~ > + (Sx> ( J ( S z) + J'(SZ> 2 + 2~E)
N
I I I
da I I I
.....
<~
Ja- -~"
/
-
-~.
x
/
\
\
\~ s~
/ \\
j
,' //
S3
/\
0
(6a)
S,
/
=
\al
(SX) 2[(SX> 2 -Jr- (SZ> 21
~ cth too too 2kr
(6b)
where too = [~2 q_ ( j ( S z) + j , ( S Z ) Z + 2pE)2],/2
(6c)
when E = 0, we have the equations that describe the phase transition process, [ - f ~ + ( S x) (J + J ' ( S ~ ) 2 ) ] ( S ") (SX) = 2[(SX) 2 + (SZ) 2]
=
--f~ cth tOo 090 2kT
0
(7a) (7b)
where Fig. 2. Pseudo-spin model of squaric acid.
too = [~2 + (J + a'(S~>2) ~ (SZ>q '/~
(8)
Vol. 71, No. 1
There is a trivial solution in equation (7), corresponding to the paraelectric phase. Equation (7) has another solution ( S z) :~ 0 which correspond to the antiferroelectric phase in squaric acid. From equation (7) we have f~ j ,
( S ~) = j +
L
11~-
J + J'(SZ)2 -
2~0
o
tanh
(1 l)
(10).
J(SZ)
+ J ' ( S : ) 3 d- 2#E
= ½tanh
2kT
(12)
Then we have the external field - - J ( S ~) - J ' ( S " ) : + k T l n
1 + 2
2
(13) and the free energy (S=)
F-
Fo =
f
=
Tc =
I
330
340
350 T (K)
360
370
380
1481.82KandJ'
=
2436.95K.
+ k T ( S z ) In
1.
4 ( S ~ ) 2)
1 + 2
-
2(SZ)
2. (14)
where Fo is the free energy of paraelectric phase. At the Curie temperature T = Tc the free energy of paraand ferroelectric phase are equal, i.e. F = Fo, hence ½J(S~)~ -- ¼ J ' ( S ~ ) ,4 + ½kT T~ In (1 -- 4(SZ)~) 1 +
2(S")c
1
2(S~),
-
372.2Kand(SZ),
=
0.258
REFERENCES
1j
+ ½kTln (] -
3. 4. 5. 6.
--
0
(15)
7.
and T~, ( S : ) c also satisfy equation (12) with E = 0 (SZ)c =
5
which is in agreement with the experimental data [6]. The temperature dependence of ( S " ) is plotted in Fig. 3, also the curve of without considering the fourbody interaction is drawn as comparison. When the contribution of the four-body interaction in included, the pseudo-spin model can describe the first order phase transition satisfactorily. Without the four-body interaction, only a second order phase transition was obtained. As the conclusion the first order phase transition in squaric acid is induced by the four-body interaction and can be treated by the pseudo-spin model.
2#E.d
+ kTc
+ +++++++
The Curie temperature T~ and the discontinuity ( S z)c can be obtained from equations (15) and (16). For squaric acid we chose
0 --
+ +
Then from equations (15) and (16) we obtain
The determination of the Curie temperature will appeal to the free energy. In squaric acid crystal, the tunneling frequency is very small, and can be omitted. Setting ~ = 0 in equation (6) we have
2#E =
+ +
Fig. 3. Fitting of the temperature dependence of order parameters q = 2 ( S z) with the experimental data [6]. (a) J = 1481.82K and J ' = 2436.95K (b) J = 1489K and J ' = 0.
J
(S-')
. ;20
(10)
-- 0
and equation
+
L°°
2kT"
Analysing the temperature dependence of ( S : ) in equation (10), we find that this equation describes a first order phase transition when J ' / J > 4/3 [15]. The limit temperature of paraelectric phase is To = J/4k, and the limit temperature of ferroelectric phase can be determined through 8T
+
(9)
substitute ( S x) into equation (7b) 1
47
FIRST O R D E R PHASE T R A N S I T I O N IN SQUARIC ACID
½tanh J ( S : ) , + J'(S-')~ 2kT,
(16)
8.
S. Cohen, J.R. Laher & J.D. Park, J. Amer. Chem. Soc. 81 (1959) 3480. D. Semmingsen & J. Feder, Solid State Commun. 15 (1974) 1396. E.J. Samuelsen & D. Semmingsen, (a) Solid State Commun. 17, 217 (1975); (b) J. Phys. Chem. Solids 38, 1275 (1977). W. Kuhn, H.D. Maier & J. Petersson, Solid State Commun. 32, 249 (1979). M. Mehring & J.D. Becker, Phys. Rev. Lett. 47, 366 (1981). G. Fisher, J. Petersson & D. Michel, Z. Phys. B - Condensed Matter 67, 387 (1987). R. Blinc & B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics, Chpt. 5, North-Holland (1974). V.I. Zinenko, Phys. Status Solidi (b) 78, 721 (1976).
48 9.
10. 11. 12.
FIRST ORDER PHASE TRANSITION IN SQUARIC ACID E. Matsushita, K. Yoshinitsu & T. Matsubaras, Prog. Theor. Phys. 64, 1176 (1980). U. Deininghaus & M. Mehring, Solid State Commun. 39 (1981) 1257. Y. Ishbashi, J. Phys. Soc. Jpn. 52, 200 (1983). N.N. Bogolyubov & S.V. Tyablikov, Doklady Akad. Nauk. S.S.S.R. 126, 53 (1959).
13. 14. 15.
Vol. 71, No. 1
C.L. Wang, Z.K. Qin & J.B. Zhang, Ferroelectrics 77, 21 (1987). J.B. Zhang & Z.K. Qin, Phys. Rev. A36, 915 (1987). C.L. Wang, Z.K. Qin & D.L. Lin (to be published in Phys. Rev).