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Geometric isotope effect and Curie temperature in hydrogen-containing ferroelectric crystals
Journal of
MOLECULAR STRUCTURE ELSEVIER
Journal of MolecularStructure325(1994)59-64
Geometric isotope effect and Curie temperature in hydrogencontaining ferroelectric crystals* E.A. Shadchin*, A.I. Barabash Institute of Physics, Ukranian Academy of Sciences, 252650 Kiev-26,46 Prospect Nauki Str., Ukraine, Russian Federation
Received 30 September 1993
Abstract
The mutual effect of protons (deutrons) and heavy ions is examined within the limits of adiabatic approximation. The dependence of the length of the hydrogen bond on the proton state function is obtained. It is shown that the mutal effect of the proton and ion subsystems, taken into account by adiabatic corrections, should increase the phase transition temperature.
1. Introduction
The problem of phase transitions in the hydrogen-containing ferroelectrics can be reduced to Izing’s pseudo spin model in a transverse field [l]. At the same time the phase transitions are caused by a collective dipole-dipole interaction, which leads to the ordering of protons within the hydrogen bonds. However, experiments show [2] that, in particular, for a KH2P04 crystal at temperatures lower than T,, the behavior is spontaneous pdlarization of the crystal and can be described completely satisfactorily by distortion of a PO4 tetrahedron. Therefore, it is natural to assume that distortion of the behavior ions at T, is induced by the short-range order of proton configurations. However, since movement of the protons is determined by the state of heavy ions, * Presented at the second National Conference on Molecular Spectroscopy with International Participation held in WrocIaw, Poland, 27-30 September 1993. * Corresponding author.
it is necessary to take into account the mutual effect of the protons and heavy atoms or ions in the ordering mechanism in a phase transition. Within the limits of the pseudo spin model of Izing’s phase transition, the concept of separating movement of protons and heavy ions is realized in a model of the bounded proton-proton mode [3,4] which is expressed in renormalizing the dipoledipole interaction [5]. Tokunaga [6] proposed a different approach describing the phase transition according to which two orientations of PO4 groups in a KH2P04 crystal were regarded as synonymous with a certain pseudo spin. The problem of determining T, in this work is reduced to an attempt to explain the phase transition of the order-disorder type using a new Hamiltonian including the dipole-dipole interaction between the proton and the PO4 group. Such an attempt is assumed to be incorrect because, for heavy ions, there is no dynamic variable to which the pseudo spin operator would correspond. Over the last few years a series of works has appeared devoted to the geometric isotope effect
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E.A. Shadchin. A.I. BarabashlJ. Mol. Struck 325 (1994) 59-64
[7,8]. In these works the dependence of Curie temperature on the length of the hydrogen bond is considered. Both approaches to describing the phase transition of the order-disorder type are phenomenological in nature and are not sufficiently accurate. It is more strict to use an approach based on the adiabatic approximation. The use of this approximation appears to be sufficiently justified because the mass of protons is considerably smaller than the mass of heavy ions and this makes it possible to take into account selfconsistently the mutual effect on each other of the “light” (proton) and “heavy” (ion) subsystems. It is assumed that this approach includes the previously described model’s phase transitions and may lead, in particular, to a change in the equation for the parameter of the transition order. The adequacy of the proposed model is examined by either the existing experimental dependence of the Curie temperature on the hydrogen bond length or the square of the distances between the equilibrium proton sets on the hydrogen bond [7,8]. The aim of this work is to use the adiabatic approximation for describing the order-disorder phase transition and to obtain the dependence of the Curie temperature on the geometric parameters of hydrogen bonds in hydrogen-containing ferroelectrics.
proton and ion (for example, PO,& ensures that the adiabatic approximation can be used. In this case, Schriidinger’s equation for heavy and light particles has the form [TR+V(R)+U(R,x)+t,+J] x @(R)Q(R,x)= E+(R)!l'(R,x)
(2)
Here J is the long-range ordering proton interaction; in accordance with the method of adiabatic approximation, the wave function is selected in the form of a product of a wave function of the relative movement of the heavy ions Q(R) and the wave function of the proton Q(R,x) depending, in a parametric manner, on the distance between the oxygen atoms. It should be mentioned that in Eq. (2), all quantities are dimensionless: aR+R
cYyx+x
(3)
Kinetic energies of the relative motion of the heavy ions and of proton motion are represented in the units of energy of the hydrogen bond: TR = -2+(m/M)(dZ/dR')
t,=-qqayaxy
(4 = hZaZ/4mD) The function V(R) in Ref. [l] was chosen in Buckingham potential form which approximately may be represented as V(R)=(y/2DaZ)(R- RJZ E(P/~)(R-R~)~
2. Formulation of the phase temperature
(4)
According to Refs. [9,10], the proton energy on the hydrogen bond, which undergoes ordering during a phase transition, can be described by a two-well Morse potential
R, is the equilibrium distance Ro...o in the absence of the proton on the hydrogen bond. In the adiabatic approximation, Schrlidinger’s equation for a proton in the two-well potential (1) at fixed R has the form
U( R,x)= (D/2) [e-“(R-Rc)ch2ax-4e-“(R-%)‘2chax]
[tX+ U(R,x)]'dO)(R,x) = e(")(R)!I'(o)(R,x)
(1) Here R = &....is the distance between the oxygen atoms of a hydrogen bond, _REis the critical distance at which the two-well potential (1) transforms into a one-well potential, D, a are the parameters of the potential, and x is the coordinate of the proton counted from the center of the hydrogen bond. As shown previously, the inequality m < M, where m and M are respectively the mass of the
where, as follows from Eqs. (1) and (3) U(R,x) = (ch2x/chzxo)-4(chx/chxo)
(5)
x0 = arch-exp [(R- &)/2]
(6)
is the equilibrium position of the proton positioned at the minima of the potential energy (5). The wave function d')(R,x) of the Schrddinger equation for the proton in a two-well potential is
61
E.A. Shadchin, A.I. BarabashlJ. Mol. Struct. 325 (1994) 59-64
Also conside~ng that the final result must be the equation for T, for which the equality (x) rn 0, the solution of the Schrtidinger equation on the basis of the perturbation theory (0 > J, (x) M 0) can be written in the form
selected in the form of a linear combination: *f)(R,x)
= (6Jj/j)[e-%o)2/”
f e-‘(~+q12/‘]
where Si: = [(7r/d)‘j2 . (1 rfr e-‘X~)]P’~” where, according to Ref. ]9], the variable theoretical parameter d is defined from the condition that E 5 (H)has a minimal value. In this case &R)
= E(R) f (1/2)0(R)
(7)
are energies of protons in the even (+) and odd (-) states, E(R) 3 &j/2*
(1/4){exp(l/d)[2-
(I/chzxO)]
(0)(R) f (~=x~/~) E*(R) = E* Q*(R,x)
= @‘(R,x)
f (Xxo/Q)@(R,x)
(9) (10)
According to the adiabatic approximation, Eq. (9) can be examined as an additional potential in describing the energy of heavy ions. In this case, the Schrodinger equation for heavy ions has the form
- 4exp (1/4d)} a(R) = exp (--dx~)[2azx~ti, - exp (1/~)(2c~‘x~)-’ + 2 exp ( 1/4d)ch-‘x0]
(8)
d-t is the width of the wave function of the proton. here and in the rest of the article we assume that the square of the overlapping integral of the wave function is small, i.e. we can ignore the value exp(-2&) < 1. Since we are interested mainly in the mutual effect of the protons and heavy ions, the ordering interaction is selected in the simplest form of the mean self-consistent field, i.e. J rn -&(x)x
= -XX
As reported in Ref. [l 11, this approximation becomes accurate with an increase in the number of ordering particles situated in the interaction sphere. Therefore, when taking into account the long-range interaction, this approximation is completely sulhcient. For example, for the phase transition temperatures, determined in the cluster approximation $7:’ and in the molecular field approximation Tcmf,we can write
(n is the number of the nearest neighbors). Taking these considerations into account, the Scbrodinger equation has the form I& + WR, x) - W**(R,
x) = RTt(R)@*(R,x)
In the harmonic approximation, taking into account only zero vibrations, we have s* = c,fiw_+/2
(11)
where c* = W*(R*)
(12)
R* are the equilibrium positions of the oxygen atoms included in the structure of heavy ions corresponding to the proton state. Taking into account the fact that at the transition temperature (x) = 0, the last term in Eq. (10) can be ignored as a low value of a higher order. Therefore W*(R) = V(R)+ E(R)f (1/2)fl(R)
(13)
and R* will be determined from the condition CT W,/aR = 0.From Eqs. (4) and (13) we obtain P(R- - Re) + [(WaR)
+ (1/2)(LWaR)],=,_
P(R+ - Ro) + [(a&IdR) -
(1/2)@fl/dR)],=,+
= 0
= 0
The solution of these equations with respect to the variables R, is difficult and we shall therefore try to find approximate values R, on the basis of the condition that the protons change only relatively slightly the equilibrium coordinates of the oxygen atoms of the heavy ions. Consequently, we obtain R+ = Ro + (1/2~)(~/~R)~~=~
(14)
62
E.A. Shadchin. A.I. Baraba.vh/J. Mol. Struct. 325 (1994) 59-64
where & =
R2 -
(15)
WP@W%=R~
Eqs. (6)-(8) show that LITJaR < 0. Taking this into account, analysis of Eqs. (14) and (15) shows that the presence of a proton in the hydrogen bond reduces the distance between the oxygen atoms of the heavy ions. This result is in agreement with the results obtained in structural examination [12]. In addition, it should be mentioned that the protons, present in the hydrogen bond in the symmetric position, “pull” the oxygen atoms (or reduce the distance between the heavy ions) to the distance which is greater than in the case of the nonsymmetric protons, by the value 2A = (l/P)]afl/aR]~=R, We can easily obtain the quantity c& determined by Eq. (12), taking into account the small size of value A: c* = c(-Jf Cl
- A[(a@Ro)
wherei1,2, . . . It is necessary to notice that the dipole moment of the proton is really non-proportional to x0. The value of the proton moment d is proportional to the distance x,(R) between two minima of the wave function for a proton in a hydrogen bond and d, determined by the mass m of the ordering particle d M x,(R)
= x0(R) - AZ/(2mUo) 112
where U. is parameter of the Morse potential. Taking into account Eqs. (6) and (17), we obtain the specified Curie temperature T,__as a function of the distances between oxygen atoms of two heavy ions T,(R)
+ @IV&)]
z a,(R
= 1
Subsequently, we determine the mean values
- RJ + b,(R
- RJ1’2
(18)
= /@:(R)Q:(R,x)xBRax
=
l=
f‘W&WWd
T)4/(4l(x)=o =1 T=Tc
Jox”ou4l) fvo) x
th
(1/2)R(Ro)-A(Ro)(a&/a% + waw b T,
Taking into account (15), it can easily be seen that
avIa& G a&/aR,
It is well known that the equation for T, has the form SP[P((X),
0, = k)jT,
Consequently,
Taking into account that the mass M of the heavy ion is sufficiently large, it can approximately be assumed that the amplitude of zero vibrations of the oxygen atoms of the heavy ions is small in comparison with the value A. Consequently, ignoring broadening of the wave functions a*(R), we can easily obtain (*I+)
(17)
+ v(R,) - (1/2)A(afi/aRo)
Cl = (1/2)n(Ro)
(k[x+)
T&o) = Jo&R)
which will correlate with experiment below. A more strict examination taking into account the tunneling proton process leads to
where co = 0s)
of oscillator wells in the vicinity of R s R* and ignoring the smallest members which are proportional to (fl/Jo$)’ in Eq. (16) we obtain
(16)
Examining the simplest case in which Eq. (11) w+ = w_, which corresponds to the same form
Subsequently, since according to Ref. [9] dZ 3 (1 /24)thzxo >> 1
we obtain the relation R M 2dZ&$ exp (-dxi)
63
E.A. Shadchin, A.I. Barubmh/J. Mol. Struct. 325 (1994) 59-64
Taking into account that bx(@RIJ = (l/2) cthx, we obtain
Jod l= wo)
~(~)~
(19)
--m-
,
I.“”
Specifically, if the tunneling frequency is considerably lower than the transition t~~rature from Eq. (19) we have
I+ (~ohwa~a~iaRo- waR2) ‘b Tc= (Joxz,P) 1+ Jo~/~~~
(20)
2q40
,
I‘ I
2.50
2.45
(I.
‘5.a I@ *
*,
2.55
Fig. 1.
Eq. (7) shows that
at/ah= exp[-(R
- &)I
Since R2 > 8.4 A (a G 2.9 A-‘), and in accordance with (14) & = 7.04A, we can assume that
(a&/a~-a~/aR~) M 0.7
3. Comparison with experiment
Also taking into account that In the first instance, it can be seen that the obtained dependence (18) agrees well qua~tatively with the experimental data represented in Fig. 1. The idea of plotting dependence of the Curie temperature on &.... distance was first suggested in [13] and [14]. It should be mentions that Fig. 1 is derived from experimental data taken from [ 151 for different hydrogen-containing crystals: I-KH2 P04, 2-NH4H2P04, 3-RbH2P04, 4-NH4H2As O,, S-RbHSO‘,, 6-NH.,HSO+ To compare the received results with the experimental data, we determined the ratio of the temperature of the phase transition with the transition temperature of a fully deuterated crystal. Taking into account that x0” x fi - x!, and also $ M 0.3 + 0.6A, we obtain
/(l + 2v)(l + /J) where
(21)
p4 = (&2*=/4mD)(+zD) - (M/4m)(Awo/D)Z M 10-t where we is the frequency of the ions containing the oxygen atoms, we can easily con&m the validity of the following inequality (1 + V)(l + @)/(l
+ 2v)(l +/A) < 1
(22)
Thus, according to Eqs. (21) and (22), the dependence of the ratio of the transition temperature of the proton-containing crystal on the transition temperature of the fully deuterated crystal as the function (~/~)’ should deviate from the linear dependence
(23) and the deviation should be toward the side of lower values of TcH/TcD and, as indicated by Eqs. (20) and (21), it should decrease with increasing x0. Figure 2 shows the dependency of TcH/T, on (~/~)’ for a number of ferroelectric crystals and their deuterated analogs: 1, l’-KH2P04, 2, 2’-RbHzPOd, 3-NaH2(Se03)2, 4-KH2(Se03)2,
64
E.A. Studchin, A.I. Bara&ash/J. Mol. Struct. 325 (1994) 59-64
Irp), (#/x~)‘], should correspond to a downward deviation of dependence (Eq. (23)) which is not now linear.
Acknowledgments The authors express their thanks to Prof. Dr. N.D. Sokolov for many useful discussions and making si~ificant remarks to this work.
_ , / I
-,
0.0
f”‘,““““‘,‘,,‘19 0.5
0.0
cx*vx
DlZ 1.0
Fig. 2.
5-~~)~H~IO~, 6- CsH2P04, 7-CsH#eO& (experimental points were taken from [16]). It can be seen that, as predicted by theoretical examination, the points are distributed below those discussed in [17] dependence (Eq. (23)).
4. Conclusions In concluding, it may be said that by taking into account the adiabatic correlations, the distance between the oxygen atoms of the adjacent heavy ions decreases, i.e. the hydrogen bonds shorten. It is shown that the length of the hydrogen bonds is determined by the state of the protons, a proton situated in the symmetric state in the hydrogen bond is characterized by a shorter hydrogen bond. In addition, the results show that by taking into account the mutual effect of the heavy ions and protons within the limits of the adiabatic approximation we increase the phase transition temperature which, in the coordinates [(TcH/
References [I] E.R. Lippinkot and R. Schroeder, J. Chem. Phys., 23 (1955) 1099. [2] G.E. Bacon and R.S. Pease, Proc. R. Sot. London, A, 230 (1955) 581. [3] R. Brout, K.A. Muller and H. Thomas, Solid State Commun., 4 (1966) 507. [4] L. Novakovic, J. Phys. Chem. Solids, 27 (1966) 1469. [5] K. Kobayashi, J. Phys. Sot. Jpn., 24 (1968) 497. [6] M. Tokunaga, Prog. Theor. Phys. Suppl., 80 (1984) 156. [7l S. Tonaka, Phys. Rev. B., 42 (1990) 10489. [8] K. Fujii, J. Phys. Sot. Jpn., 61 (1992) 342. [9] E. Matsushita and T. Matsubara, Prog. Theor. Phys. Suppl., 67 (1982) 1. [IO] T. Matsubara and E. Matsushita, Prog. Theor. Phys. Suppl., 71 (1984) 209. [I I] V.G. Vaks, Intr~u~tion into Fe~oel~t~~ty Theory, Nauka, Moscow, 1973, p. 327 (in Russian). [12] N.D. Sokolov in N.D. Sokolov (Ed.) Proceedings Hydrogen Bond, Mir, Moscow, 1981, p. 63 (m Russian). [I9 M. Ichikava, K. Motida and N. Yamada, Phys. Rev. B, 36 (1987) 874. [14] M. Ichikava and K. Motida, J. Phys. Sot. Jpn., 56 (1987) 3750. [IS] M. Ichikava, Acta Crystallogr., Sect. B, 34 (1978) 2074. [ 161 N.D. Sokolov, M.V. Vener and V.A. Saveliev, J. Mol. struct., 177 (1988) 93. [17] E.A. Shadchin and AI. Barabash, Khim. Fiz., 9 (1990) 852 (in Russian).
MOLECULAR STRUCTURE ELSEVIER
Journal of MolecularStructure325(1994)59-64
Geometric isotope effect and Curie temperature in hydrogencontaining ferroelectric crystals* E.A. Shadchin*, A.I. Barabash Institute of Physics, Ukranian Academy of Sciences, 252650 Kiev-26,46 Prospect Nauki Str., Ukraine, Russian Federation
Received 30 September 1993
Abstract
The mutual effect of protons (deutrons) and heavy ions is examined within the limits of adiabatic approximation. The dependence of the length of the hydrogen bond on the proton state function is obtained. It is shown that the mutal effect of the proton and ion subsystems, taken into account by adiabatic corrections, should increase the phase transition temperature.
1. Introduction
The problem of phase transitions in the hydrogen-containing ferroelectrics can be reduced to Izing’s pseudo spin model in a transverse field [l]. At the same time the phase transitions are caused by a collective dipole-dipole interaction, which leads to the ordering of protons within the hydrogen bonds. However, experiments show [2] that, in particular, for a KH2P04 crystal at temperatures lower than T,, the behavior is spontaneous pdlarization of the crystal and can be described completely satisfactorily by distortion of a PO4 tetrahedron. Therefore, it is natural to assume that distortion of the behavior ions at T, is induced by the short-range order of proton configurations. However, since movement of the protons is determined by the state of heavy ions, * Presented at the second National Conference on Molecular Spectroscopy with International Participation held in WrocIaw, Poland, 27-30 September 1993. * Corresponding author.
it is necessary to take into account the mutual effect of the protons and heavy atoms or ions in the ordering mechanism in a phase transition. Within the limits of the pseudo spin model of Izing’s phase transition, the concept of separating movement of protons and heavy ions is realized in a model of the bounded proton-proton mode [3,4] which is expressed in renormalizing the dipoledipole interaction [5]. Tokunaga [6] proposed a different approach describing the phase transition according to which two orientations of PO4 groups in a KH2P04 crystal were regarded as synonymous with a certain pseudo spin. The problem of determining T, in this work is reduced to an attempt to explain the phase transition of the order-disorder type using a new Hamiltonian including the dipole-dipole interaction between the proton and the PO4 group. Such an attempt is assumed to be incorrect because, for heavy ions, there is no dynamic variable to which the pseudo spin operator would correspond. Over the last few years a series of works has appeared devoted to the geometric isotope effect
0022-2860/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDZ 0022-2860(94)08253-E
60
E.A. Shadchin. A.I. BarabashlJ. Mol. Struck 325 (1994) 59-64
[7,8]. In these works the dependence of Curie temperature on the length of the hydrogen bond is considered. Both approaches to describing the phase transition of the order-disorder type are phenomenological in nature and are not sufficiently accurate. It is more strict to use an approach based on the adiabatic approximation. The use of this approximation appears to be sufficiently justified because the mass of protons is considerably smaller than the mass of heavy ions and this makes it possible to take into account selfconsistently the mutual effect on each other of the “light” (proton) and “heavy” (ion) subsystems. It is assumed that this approach includes the previously described model’s phase transitions and may lead, in particular, to a change in the equation for the parameter of the transition order. The adequacy of the proposed model is examined by either the existing experimental dependence of the Curie temperature on the hydrogen bond length or the square of the distances between the equilibrium proton sets on the hydrogen bond [7,8]. The aim of this work is to use the adiabatic approximation for describing the order-disorder phase transition and to obtain the dependence of the Curie temperature on the geometric parameters of hydrogen bonds in hydrogen-containing ferroelectrics.
proton and ion (for example, PO,& ensures that the adiabatic approximation can be used. In this case, Schriidinger’s equation for heavy and light particles has the form [TR+V(R)+U(R,x)+t,+J] x @(R)Q(R,x)= E+(R)!l'(R,x)
(2)
Here J is the long-range ordering proton interaction; in accordance with the method of adiabatic approximation, the wave function is selected in the form of a product of a wave function of the relative movement of the heavy ions Q(R) and the wave function of the proton Q(R,x) depending, in a parametric manner, on the distance between the oxygen atoms. It should be mentioned that in Eq. (2), all quantities are dimensionless: aR+R
cYyx+x
(3)
Kinetic energies of the relative motion of the heavy ions and of proton motion are represented in the units of energy of the hydrogen bond: TR = -2+(m/M)(dZ/dR')
t,=-qqayaxy
(4 = hZaZ/4mD) The function V(R) in Ref. [l] was chosen in Buckingham potential form which approximately may be represented as V(R)=(y/2DaZ)(R- RJZ E(P/~)(R-R~)~
2. Formulation of the phase temperature
(4)
According to Refs. [9,10], the proton energy on the hydrogen bond, which undergoes ordering during a phase transition, can be described by a two-well Morse potential
R, is the equilibrium distance Ro...o in the absence of the proton on the hydrogen bond. In the adiabatic approximation, Schrlidinger’s equation for a proton in the two-well potential (1) at fixed R has the form
U( R,x)= (D/2) [e-“(R-Rc)ch2ax-4e-“(R-%)‘2chax]
[tX+ U(R,x)]'dO)(R,x) = e(")(R)!I'(o)(R,x)
(1) Here R = &....is the distance between the oxygen atoms of a hydrogen bond, _REis the critical distance at which the two-well potential (1) transforms into a one-well potential, D, a are the parameters of the potential, and x is the coordinate of the proton counted from the center of the hydrogen bond. As shown previously, the inequality m < M, where m and M are respectively the mass of the
where, as follows from Eqs. (1) and (3) U(R,x) = (ch2x/chzxo)-4(chx/chxo)
(5)
x0 = arch-exp [(R- &)/2]
(6)
is the equilibrium position of the proton positioned at the minima of the potential energy (5). The wave function d')(R,x) of the Schrddinger equation for the proton in a two-well potential is
61
E.A. Shadchin, A.I. BarabashlJ. Mol. Struct. 325 (1994) 59-64
Also conside~ng that the final result must be the equation for T, for which the equality (x) rn 0, the solution of the Schrtidinger equation on the basis of the perturbation theory (0 > J, (x) M 0) can be written in the form
selected in the form of a linear combination: *f)(R,x)
= (6Jj/j)[e-%o)2/”
f e-‘(~+q12/‘]
where Si: = [(7r/d)‘j2 . (1 rfr e-‘X~)]P’~” where, according to Ref. ]9], the variable theoretical parameter d is defined from the condition that E 5 (H)has a minimal value. In this case &R)
= E(R) f (1/2)0(R)
(7)
are energies of protons in the even (+) and odd (-) states, E(R) 3 &j/2*
(1/4){exp(l/d)[2-
(I/chzxO)]
(0)(R) f (~=x~/~) E*(R) = E* Q*(R,x)
= @‘(R,x)
f (Xxo/Q)@(R,x)
(9) (10)
According to the adiabatic approximation, Eq. (9) can be examined as an additional potential in describing the energy of heavy ions. In this case, the Schrodinger equation for heavy ions has the form
- 4exp (1/4d)} a(R) = exp (--dx~)[2azx~ti, - exp (1/~)(2c~‘x~)-’ + 2 exp ( 1/4d)ch-‘x0]
(8)
d-t is the width of the wave function of the proton. here and in the rest of the article we assume that the square of the overlapping integral of the wave function is small, i.e. we can ignore the value exp(-2&) < 1. Since we are interested mainly in the mutual effect of the protons and heavy ions, the ordering interaction is selected in the simplest form of the mean self-consistent field, i.e. J rn -&(x)x
= -XX
As reported in Ref. [l 11, this approximation becomes accurate with an increase in the number of ordering particles situated in the interaction sphere. Therefore, when taking into account the long-range interaction, this approximation is completely sulhcient. For example, for the phase transition temperatures, determined in the cluster approximation $7:’ and in the molecular field approximation Tcmf,we can write
(n is the number of the nearest neighbors). Taking these considerations into account, the Scbrodinger equation has the form I& + WR, x) - W**(R,
x) = RTt(R)@*(R,x)
In the harmonic approximation, taking into account only zero vibrations, we have s* = c,fiw_+/2
(11)
where c* = W*(R*)
(12)
R* are the equilibrium positions of the oxygen atoms included in the structure of heavy ions corresponding to the proton state. Taking into account the fact that at the transition temperature (x) = 0, the last term in Eq. (10) can be ignored as a low value of a higher order. Therefore W*(R) = V(R)+ E(R)f (1/2)fl(R)
(13)
and R* will be determined from the condition CT W,/aR = 0.From Eqs. (4) and (13) we obtain P(R- - Re) + [(WaR)
+ (1/2)(LWaR)],=,_
P(R+ - Ro) + [(a&IdR) -
(1/2)@fl/dR)],=,+
= 0
= 0
The solution of these equations with respect to the variables R, is difficult and we shall therefore try to find approximate values R, on the basis of the condition that the protons change only relatively slightly the equilibrium coordinates of the oxygen atoms of the heavy ions. Consequently, we obtain R+ = Ro + (1/2~)(~/~R)~~=~
(14)
62
E.A. Shadchin. A.I. Baraba.vh/J. Mol. Struct. 325 (1994) 59-64
where & =
R2 -
(15)
WP@W%=R~
Eqs. (6)-(8) show that LITJaR < 0. Taking this into account, analysis of Eqs. (14) and (15) shows that the presence of a proton in the hydrogen bond reduces the distance between the oxygen atoms of the heavy ions. This result is in agreement with the results obtained in structural examination [12]. In addition, it should be mentioned that the protons, present in the hydrogen bond in the symmetric position, “pull” the oxygen atoms (or reduce the distance between the heavy ions) to the distance which is greater than in the case of the nonsymmetric protons, by the value 2A = (l/P)]afl/aR]~=R, We can easily obtain the quantity c& determined by Eq. (12), taking into account the small size of value A: c* = c(-Jf Cl
- A[(a@Ro)
wherei1,2, . . . It is necessary to notice that the dipole moment of the proton is really non-proportional to x0. The value of the proton moment d is proportional to the distance x,(R) between two minima of the wave function for a proton in a hydrogen bond and d, determined by the mass m of the ordering particle d M x,(R)
= x0(R) - AZ/(2mUo) 112
where U. is parameter of the Morse potential. Taking into account Eqs. (6) and (17), we obtain the specified Curie temperature T,__as a function of the distances between oxygen atoms of two heavy ions T,(R)
+ @IV&)]
z a,(R
= 1
Subsequently, we determine the mean values
- RJ + b,(R
- RJ1’2
(18)
= /@:(R)Q:(R,x)xBRax
=
l=
f‘W&WWd
T)4/(4l(x)=o =1 T=Tc
Jox”ou4l) fvo) x
th
(1/2)R(Ro)-A(Ro)(a&/a% + waw b T,
Taking into account (15), it can easily be seen that
avIa& G a&/aR,
It is well known that the equation for T, has the form SP[P((X),
0, = k)jT,
Consequently,
Taking into account that the mass M of the heavy ion is sufficiently large, it can approximately be assumed that the amplitude of zero vibrations of the oxygen atoms of the heavy ions is small in comparison with the value A. Consequently, ignoring broadening of the wave functions a*(R), we can easily obtain (*I+)
(17)
+ v(R,) - (1/2)A(afi/aRo)
Cl = (1/2)n(Ro)
(k[x+)
T&o) = Jo&R)
which will correlate with experiment below. A more strict examination taking into account the tunneling proton process leads to
where co = 0s)
of oscillator wells in the vicinity of R s R* and ignoring the smallest members which are proportional to (fl/Jo$)’ in Eq. (16) we obtain
(16)
Examining the simplest case in which Eq. (11) w+ = w_, which corresponds to the same form
Subsequently, since according to Ref. [9] dZ 3 (1 /24)thzxo >> 1
we obtain the relation R M 2dZ&$ exp (-dxi)
63
E.A. Shadchin, A.I. Barubmh/J. Mol. Struct. 325 (1994) 59-64
Taking into account that bx(@RIJ = (l/2) cthx, we obtain
Jod l= wo)
~(~)~
(19)
--m-
,
I.“”
Specifically, if the tunneling frequency is considerably lower than the transition t~~rature from Eq. (19) we have
I+ (~ohwa~a~iaRo- waR2) ‘b Tc= (Joxz,P) 1+ Jo~/~~~
(20)
2q40
,
I‘ I
2.50
2.45
(I.
‘5.a I@ *
*,
2.55
Fig. 1.
Eq. (7) shows that
at/ah= exp[-(R
- &)I
Since R2 > 8.4 A (a G 2.9 A-‘), and in accordance with (14) & = 7.04A, we can assume that
(a&/a~-a~/aR~) M 0.7
3. Comparison with experiment
Also taking into account that In the first instance, it can be seen that the obtained dependence (18) agrees well qua~tatively with the experimental data represented in Fig. 1. The idea of plotting dependence of the Curie temperature on &.... distance was first suggested in [13] and [14]. It should be mentions that Fig. 1 is derived from experimental data taken from [ 151 for different hydrogen-containing crystals: I-KH2 P04, 2-NH4H2P04, 3-RbH2P04, 4-NH4H2As O,, S-RbHSO‘,, 6-NH.,HSO+ To compare the received results with the experimental data, we determined the ratio of the temperature of the phase transition with the transition temperature of a fully deuterated crystal. Taking into account that x0” x fi - x!, and also $ M 0.3 + 0.6A, we obtain
/(l + 2v)(l + /J) where
(21)
p4 = (&2*=/4mD)(+zD) - (M/4m)(Awo/D)Z M 10-t where we is the frequency of the ions containing the oxygen atoms, we can easily con&m the validity of the following inequality (1 + V)(l + @)/(l
+ 2v)(l +/A) < 1
(22)
Thus, according to Eqs. (21) and (22), the dependence of the ratio of the transition temperature of the proton-containing crystal on the transition temperature of the fully deuterated crystal as the function (~/~)’ should deviate from the linear dependence
(23) and the deviation should be toward the side of lower values of TcH/TcD and, as indicated by Eqs. (20) and (21), it should decrease with increasing x0. Figure 2 shows the dependency of TcH/T, on (~/~)’ for a number of ferroelectric crystals and their deuterated analogs: 1, l’-KH2P04, 2, 2’-RbHzPOd, 3-NaH2(Se03)2, 4-KH2(Se03)2,
64
E.A. Studchin, A.I. Bara&ash/J. Mol. Struct. 325 (1994) 59-64
Irp), (#/x~)‘], should correspond to a downward deviation of dependence (Eq. (23)) which is not now linear.
Acknowledgments The authors express their thanks to Prof. Dr. N.D. Sokolov for many useful discussions and making si~ificant remarks to this work.
_ , / I
-,
0.0
f”‘,““““‘,‘,,‘19 0.5
0.0
cx*vx
DlZ 1.0
Fig. 2.
5-~~)~H~IO~, 6- CsH2P04, 7-CsH#eO& (experimental points were taken from [16]). It can be seen that, as predicted by theoretical examination, the points are distributed below those discussed in [17] dependence (Eq. (23)).
4. Conclusions In concluding, it may be said that by taking into account the adiabatic correlations, the distance between the oxygen atoms of the adjacent heavy ions decreases, i.e. the hydrogen bonds shorten. It is shown that the length of the hydrogen bonds is determined by the state of the protons, a proton situated in the symmetric state in the hydrogen bond is characterized by a shorter hydrogen bond. In addition, the results show that by taking into account the mutual effect of the heavy ions and protons within the limits of the adiabatic approximation we increase the phase transition temperature which, in the coordinates [(TcH/
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