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Influence of a proton subsystem on the structural phase transition in crystals such as KH2PO4
Journal of Molecular Structure 649 (2003) 1–5 www.elsevier.com/locate/molstruc
Influence of a proton subsystem on the structural phase transition in crystals such as KH2PO4 E.A. Shadchin, S.P. Sirenko* Institute of Physics, National Academy of Science of Ukraine, 46, Nauky Prosp., Kyiv 03028, Ukraine Received 26 March 2002; accepted 17 May 2002
Abstract A new mechanism of the phase transition (PT) in crystals such as KH2PO4 is proposed and theoretically studied. According to the model the proton subsystem plays a role of starting mechanism of the PT in the heavy ion system. It is shown for observing the isotopic replacement of the critical temperature it is enough to take into account the geometrical isotopic effect without special reference to the process of tunneling. q 2002 Published by Elsevier Science B.V. Keywords: Phase transition; Effective potential; Morse potential; H-bond
1. Introduction It is known that ferroelectric or antiferroelectric phase transition (PT) which temperature essentially depends on a degree of replacement of a proton (H) on a deuteron (D) is characteristic of crystals with hydrogen bonds such as KH2PO4 (KDP). The theory of the transition having been named a pseudo-spin model of Ising in a cross field received significant development in the literature [1]. According to this theory the ordering of protons (deuterons) is a result of a competition of two mechanisms: ordering dipole –dipole interaction and tunneling. Major argument for the benefit of this theory is significant anharmonics (double-minimum potential) for H or D on the H-bond. Besides, the increase of * Corresponding author. Address: NPP Operational Support Institute, 35-37 Radgospna Street, 252142 Kyiv, Ukraine. Fax: þ 380-44-265-4504. E-mail address: [email protected] (S.P. Sirenko).
transition temperature (Tc) at the replacement H on D (isotopic displacement of Tc) observed in experiments follows from this model. According to the theory the displacement of Tc is due to the reduction of tunneling frequency at replacement H on D. However, as was soon found out, a feature of the crystals is the occurrence of a ferroelectrical axis C almost perpendicular to H-bond grids. In the ferroelectrical phase the size of spontaneous polarization Ps is well explained by the observable displacement of ions Kþ1, Pþ5, O22 along the axis C relative to the symmetric position [2]. Thus, the hydrogen bond does not give the appreciable contribution to the saturated polarization Ps. Besides, Ichikawa and co-authors [3,4] found the correlation between the length of H-bond RH and DR ¼ ðRD 2 RH Þ; where RD is the length of O –D – O – bond. The correlation was named as the geometrical isotopic effect in the literature.
0022-2860/03/$ - see front matter q 2002 Published by Elsevier Science B.V. PII: S 0 0 2 2 - 2 8 6 0 ( 0 2 ) 0 0 3 1 8 - 6
2
E.A. Shadchin, S.P. Sirenko / Journal of Molecular Structure 649 (2003) 1–5
The linear dependence between the transition temperature Tc and the length of H(D)-bond was found. The dependence keeps in crystals contained hydrogen as well as deuterium. Later in Ref. [5] for KDP and DKDP was shown that only a part of increase of Tc (about 40 K) is directly due to the effect of tunneling. All these results subject significant doubt the Ising model with tunneling H(D). It is obvious that for describing the structural PT it is necessary to understand and to take into account a role of a heavy subsystem (an ion subsystem). It is actually reduced to a correct choice of a model of interaction H(D) (an light subsystem) with the heavy subsystem. As a rule [6, 7], this task was tried to decide on a phenomenological level by describing observed distortion of PO4-ions by pseudo-spin s. The inefficiency of such approach was especially shown in Ref. [6]. In the paper it was found the PT keeps after removing one of the subsystems. The bilinear connection between tunneling protons and the distortion of the electron environment of PO4-groups was examined in Ref. [8]. It is necessary also to note work in Ref. [7] where the PT mechanism based on inclusion of a phenomenological ratio between the length of H-bond and the distortion of PO4-ion without the reference to tunneling H(D) was proposed. In the present work we offer a new model of the PT based on an adequate choice of the potential of interaction of H(D) with a pair of oxygen on the H-bond. Thus, the proton subsystem plays an essential role in processes starting the PT mechanism in the heavy subsystem.
Fig. 1. W—oxygen atom, X—proton (deuteron).
the bond of oxygen atom in a heavy ion (it is PO4 for KDP, it is JO3 for NH4JO3 and so on). As usually [10], we take the potential as VðrÞ ¼ D0 {expð22aðr 2 r0 ÞÞ 2 2 expð2aðr 2 r0 ÞÞ} where D0, a, r0 are given parameters. From Fig. 1, the double Morse potential for H(D) is U ¼ Vðr 2 R1 Þ þ VðR2 2 rÞ: Replacing two variables R1, R2 with another R ¼ R1 2 R2 ; C ¼ ðR1 þ R2 Þ=2; after simple calculations we receive UðR; j 2 xÞ ¼ 0:5D0 {B22 ðRÞchð2aðj 2 xÞÞ 2 4B21 ðRÞchðaðj 2 xÞÞ};
ð1Þ
where R being the H-bond length, C being the mass center of two oxygen atoms, BðRÞ¼ exp{0:5ðR2Rc Þ}; Rc ¼2ðr0 þln2=aÞ; x being the coordinate of H(D), j being the coordinate of the mass center of two oxygen atoms referenced to the ‘absolute’ center of the H-bond chain in cells.
2. Model 3. Effective potential In modeling the PT in crystals contained hydrogen the correct choice of interaction between a proton (deuteron) and atoms of oxygen [7] is essential. For example, a potential Fij d describing moving PO4-groups was added to the double Morse potential in Ref. [9]. However, the mechanism of the interaction does not become clear. Let us consider a crystal as a set of onedimensional chains. In Fig. 1 the wavy line designates
The potential (1) describes the interaction of a proton (deuteron) with a system of two oxygen atoms bound to a heavy ion. It is important to note the proton subsystem is fast and heavy one (the pair of oxygen atoms) is slow. The appropriate proton frequency v is about 3 £ 103 cm21 (v q V; where V is the characteristic frequency of fluctuation of the mass center of O – O-system). This allows to average on a fast subsystem in Eq. (1) and to obtain the effective
E.A. Shadchin, S.P. Sirenko / Journal of Molecular Structure 649 (2003) 1–5
From the condition dUeff ðR; jÞ=dj ¼ 0; at
potential Ueff ðR; jÞ ¼ N 21
ð1
UðR; j 2 xÞexp{ 2 UðR;xÞ=kT}dx
where k being the Boltzman constant. Hereafter we go from dimensional variables j, x to dimensionless those aj ! j; ax ! x ð1
Rc # R # Rc þ 2 lnðch2wðRÞ=chwðRÞÞ=a;
ð9Þ
21
ð2Þ
N¼
3
exp{ 2 UðR; xÞ=kT}dx
ð3Þ
21
Ueff ðR; jÞ is a one-minima anharmonic potential. Taking into account Eqs. (5) –(7) it is easy to show condition (9) includes a wide class of crystals such as ˚ # R # 2.6 A ˚. KDP: 2.43A Thus, for the majority of crystals undergoing structural PT, the proton (deuteron) on H-bond due to the interaction with O – O-system provides an effective anhormonic symmetric potential for the heavy subsystem.
where UðR; xÞ ¼ 0:5D0 {B22 ðRÞchð2xÞ 2 4B21 ðRÞchðxÞ} ð4Þ is the double-minima potential for H(D) at the fixed position of the mass center of O –O-system j ¼ 0. Taking into account that x p 1 (the characteristic distance between minima of the potential is ˚ in dimensional values), we can expand 2x0 , 0.3 A Eq. (4) up to fourth order and obtain UðR; xÞ ¼ 0:5D0 { 2 cðRÞx2 =2 þ bðRÞx4 =4} where cðRÞ ¼ 4{B21 ðRÞ 2 B22 ðRÞ};
4. Free energy of the model crystal Let us consider a crystal as a set of independent pairs of oxygen atoms being in the effective anhormonic potential (8). The distance between two oxygen atoms is proposed to be fixed. Thus we have the Einstein model for anhormonic oscillators with a Hamiltonian H ¼ K þ U where K being the kinetic energy of pair of oxygen atoms as whole. Taking into account the dipole –dipole interaction due to possible occurrence of average displacement kjk l we obtain in molecular field approximation
ð5Þ H¼
22
21
bðRÞ ¼ 2{4B ðRÞ 2 B ðRÞ}=3
ð6Þ
as D0 , 1.75 £ 104 cm21 [10] and for KDP cðRÞ , 0:32;
bðRÞ , 1:6
ð7Þ
at the temperature being characteristic for structural PT for the given class of crystals we have U=kT q 1: This inequality allows to integrate Eqs. (2) and (3) by the saddle-point technique. As a result we have
ðp2k =2M þUeff ðRk ; jk ÞÞ2
k
X
Jkk0 ðjk kjk0 l2kjk l2 =2Þ
k;k0
where M ¼2mox ; mox is the oxygen atom mass. Further, for finding the free energy of the model crystal in approach of the perturbation theory we choose as a ‘trial’ Hamiltonian H0 ¼K þU0 the Hamiltonian of harmonious oscillator H0 ¼K þU0 with a vibration frequency of V and a displaced vibration center of kjl: Then, according to Ref. [11] we obtain the free crystal energy F per one cell of the H-bond chain F ¼F0 þ0:5D0 kGðR; jÞl2kU0 l:
Ueff ðR; jÞ ¼ 0:5D0 {B2 ðRÞch2j 2 4B1 ðRÞchj} ; 0:5D0 GðR; jÞ;
X
ð8Þ
where B2 ðRÞ ¼ B22 ch2wðRÞ; B1 ðRÞ ¼ B21 chwðRÞ; wðRÞ ¼ ðcðRÞ=bðRÞÞ1=2 :
ð10Þ
Here averaging is carried out on states of harmonious oscillator Hamiltonian k···l¼sp{…expð2H0 =kTÞ}=sp{expð2H0 =kTÞ}:
ð11Þ
4
E.A. Shadchin, S.P. Sirenko / Journal of Molecular Structure 649 (2003) 1–5
Taking average in Eq. (10) according to Eq. (11), we obtain F ¼kT lnð2shð0:5"V=kTÞÞ þ0:5D0 {B2 ðRÞexpð2lj Þchð2kjlÞ 24B1 ðRÞexpðlj =2ÞchðkjlÞ} 20:5M V2 lj =a2 20:5J0 kjl2 =a2 :
ð12Þ
Here V, kjl are variational parameters of the free energy, lj ;kðj 2kjlÞ2 l¼0:5"a2 cthð0:5"V=kTÞ=M V is the dimensionless mean-square of fluctuation amplitude. The conditions of minimization of the free energy (1) result in a set of the equations for variational parameters
V2 2 ðmv20 =2MÞ›kGðR; jÞl0 =›lj ¼ 0
ð13Þ
22J0 kjl þ D0 a2 ›kGðR; jÞl0 =›j ¼ 0:
ð14Þ
Here v20 ¼ 2a2 D0 =m is the limiting frequency of the proton on the H-bond at the R ! 1. Taking into account Eq. (12) and excepting one paraphase solve from the sets (13) and (14) at kjl ¼ 0 we obtain chðkjlÞ ¼ {0:5g expð22lj Þ þ B1 ðRÞexpð23lj =2Þ}=B2 ðRÞ
V2 ¼ mv20 {B2 ðRÞ}chð2kjlÞexpð2lj Þ 2 B1 ðRÞexpðlj =2ÞchðkjlÞ}=M;
ð15Þ
ð16Þ
where g ¼ J0 a22 =D0 is a dimensionless parameter of the theory describing the dipole – dipole interaction. The numerical solution of the sets (15) and (16) for parameters of crystals KH2PO4 and KD2PO4 is given in Fig. 2. The agreement with experiments was observed at g ¼ 0:51: For KH2 PO 4 B1 ¼ 1, B2 ¼ 1.18. It corresponds to a choice of H-bond ˚ , RC,H ¼ 2.43 A ˚ [3]. It is necesslength RH ¼ 2.49 A ary to emphasize that for KD2PO4 not only the bond RC;D < length was varied but RD 2 RH < 0:01 A; RC;H þ 0:02 A were also varied [4]. Thus, for DKDP B1 ¼ 0.9, B2 ¼ 1.12. The numerical calculation shows that there is an essential dependence of the value of the average displacement of the mass center of oxygen atom pair below the critical temperature on particle grades in
Fig. 2. The dependence of the order parameter kjl on temperature (broken line is H-bond, solid one is D-bond).
and for the H-bond. So, for KD2PO4 kjl ¼ 0:012 A KD2PO4 kjl ¼ 0:04 A:
5. Conclusion In the paper a new mechanism of structural PT in crystals contained hydrogen based on the adequate choice of the potential of the interaction H(D) with two oxygen atoms laying on H-bond is offered. In the framework of the model of the crystal consisting of one-dimensional chains of Hbond it is shown that the light subsystem (proton (deuteron)) plays a role of starting mechanism for processes resulting to PT in the system of heavy
Fig. 3. The dependence of oscillation frequencies on temperature (broken line is H-bond, solid one is D-bond).
E.A. Shadchin, S.P. Sirenko / Journal of Molecular Structure 649 (2003) 1–5
ions at temperatures below critical. It is also shown that the observed isotopic displacement of the transition temperature for KD2PO4 can be obtained due to geometrical isotopic effect without the reference to the process of tunneling. The model is general enough and covers a wide class of crystals notwithstanding the numerical result check being performed for KH2PO4 (KD2PO4). The relative simplicity of the theory is that there is only one free parameter determining the dipole – dipole interaction. Notwithstanding there is the quite good agreement with experiment, it is necessary to note that the numerical calculation of PT temperature is rather sensitive to parameters of the Morse potential, values of which are known approximately. It is also necessary to note that while the H-bond length RH, RC,H are not bad known for KH2PO4, RC,D is calculated for KD2PO4. It results in an error in calculating Tc(D). Besides, the possible processes of tunneling are not taken into account in the model.
5
The results given in Fig. 3 show that there is a dependence of vibration frequency of the mass center on kind of a particle on the H-bond.
References [1] R. Blinc, D. Svetina, Phys. Rev. 147 (1966) 430. [2] G.E. Bacon, R.S. Pease, Prog. R. Soc. A230 (1956) 359. [3] M. Ichikawa, K. Motida, N. Yamada, Phys. Rev. B36 (1) (1987) 874. [4] M. Ichikawa, Acta Crystallogr. B34 (1978) 2074. [5] R.J. Nelmes, J. Phys. C 21 (1988) L881. [6] M. Tokunaga, T. Matsubara, Prog. Theor. Phys. 35 (4) (1966) 581. [7] S. Tanaka, Phys. Rev. 50 (22) (1994) 16247. [8] A. Bussman-Holder, K.H. Michel, Phys. Rev. Lett. 80 (10) (1998) 2173. [9] H. Sugimoto, S. Ikeda, Phys. Rev. Lett. 67 (1991) 1306. [10] T. Matsubara, E. Matsushita, Progr. Theor. Phys. 67 (1982) 1. [11] R.P. Feyuman, Statistical Mechanics—CA, Institute of Technology, W.A. Benjamin Inc, 1972, p. 404.
Influence of a proton subsystem on the structural phase transition in crystals such as KH2PO4 E.A. Shadchin, S.P. Sirenko* Institute of Physics, National Academy of Science of Ukraine, 46, Nauky Prosp., Kyiv 03028, Ukraine Received 26 March 2002; accepted 17 May 2002
Abstract A new mechanism of the phase transition (PT) in crystals such as KH2PO4 is proposed and theoretically studied. According to the model the proton subsystem plays a role of starting mechanism of the PT in the heavy ion system. It is shown for observing the isotopic replacement of the critical temperature it is enough to take into account the geometrical isotopic effect without special reference to the process of tunneling. q 2002 Published by Elsevier Science B.V. Keywords: Phase transition; Effective potential; Morse potential; H-bond
1. Introduction It is known that ferroelectric or antiferroelectric phase transition (PT) which temperature essentially depends on a degree of replacement of a proton (H) on a deuteron (D) is characteristic of crystals with hydrogen bonds such as KH2PO4 (KDP). The theory of the transition having been named a pseudo-spin model of Ising in a cross field received significant development in the literature [1]. According to this theory the ordering of protons (deuterons) is a result of a competition of two mechanisms: ordering dipole –dipole interaction and tunneling. Major argument for the benefit of this theory is significant anharmonics (double-minimum potential) for H or D on the H-bond. Besides, the increase of * Corresponding author. Address: NPP Operational Support Institute, 35-37 Radgospna Street, 252142 Kyiv, Ukraine. Fax: þ 380-44-265-4504. E-mail address: [email protected] (S.P. Sirenko).
transition temperature (Tc) at the replacement H on D (isotopic displacement of Tc) observed in experiments follows from this model. According to the theory the displacement of Tc is due to the reduction of tunneling frequency at replacement H on D. However, as was soon found out, a feature of the crystals is the occurrence of a ferroelectrical axis C almost perpendicular to H-bond grids. In the ferroelectrical phase the size of spontaneous polarization Ps is well explained by the observable displacement of ions Kþ1, Pþ5, O22 along the axis C relative to the symmetric position [2]. Thus, the hydrogen bond does not give the appreciable contribution to the saturated polarization Ps. Besides, Ichikawa and co-authors [3,4] found the correlation between the length of H-bond RH and DR ¼ ðRD 2 RH Þ; where RD is the length of O –D – O – bond. The correlation was named as the geometrical isotopic effect in the literature.
0022-2860/03/$ - see front matter q 2002 Published by Elsevier Science B.V. PII: S 0 0 2 2 - 2 8 6 0 ( 0 2 ) 0 0 3 1 8 - 6
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E.A. Shadchin, S.P. Sirenko / Journal of Molecular Structure 649 (2003) 1–5
The linear dependence between the transition temperature Tc and the length of H(D)-bond was found. The dependence keeps in crystals contained hydrogen as well as deuterium. Later in Ref. [5] for KDP and DKDP was shown that only a part of increase of Tc (about 40 K) is directly due to the effect of tunneling. All these results subject significant doubt the Ising model with tunneling H(D). It is obvious that for describing the structural PT it is necessary to understand and to take into account a role of a heavy subsystem (an ion subsystem). It is actually reduced to a correct choice of a model of interaction H(D) (an light subsystem) with the heavy subsystem. As a rule [6, 7], this task was tried to decide on a phenomenological level by describing observed distortion of PO4-ions by pseudo-spin s. The inefficiency of such approach was especially shown in Ref. [6]. In the paper it was found the PT keeps after removing one of the subsystems. The bilinear connection between tunneling protons and the distortion of the electron environment of PO4-groups was examined in Ref. [8]. It is necessary also to note work in Ref. [7] where the PT mechanism based on inclusion of a phenomenological ratio between the length of H-bond and the distortion of PO4-ion without the reference to tunneling H(D) was proposed. In the present work we offer a new model of the PT based on an adequate choice of the potential of interaction of H(D) with a pair of oxygen on the H-bond. Thus, the proton subsystem plays an essential role in processes starting the PT mechanism in the heavy subsystem.
Fig. 1. W—oxygen atom, X—proton (deuteron).
the bond of oxygen atom in a heavy ion (it is PO4 for KDP, it is JO3 for NH4JO3 and so on). As usually [10], we take the potential as VðrÞ ¼ D0 {expð22aðr 2 r0 ÞÞ 2 2 expð2aðr 2 r0 ÞÞ} where D0, a, r0 are given parameters. From Fig. 1, the double Morse potential for H(D) is U ¼ Vðr 2 R1 Þ þ VðR2 2 rÞ: Replacing two variables R1, R2 with another R ¼ R1 2 R2 ; C ¼ ðR1 þ R2 Þ=2; after simple calculations we receive UðR; j 2 xÞ ¼ 0:5D0 {B22 ðRÞchð2aðj 2 xÞÞ 2 4B21 ðRÞchðaðj 2 xÞÞ};
ð1Þ
where R being the H-bond length, C being the mass center of two oxygen atoms, BðRÞ¼ exp{0:5ðR2Rc Þ}; Rc ¼2ðr0 þln2=aÞ; x being the coordinate of H(D), j being the coordinate of the mass center of two oxygen atoms referenced to the ‘absolute’ center of the H-bond chain in cells.
2. Model 3. Effective potential In modeling the PT in crystals contained hydrogen the correct choice of interaction between a proton (deuteron) and atoms of oxygen [7] is essential. For example, a potential Fij d describing moving PO4-groups was added to the double Morse potential in Ref. [9]. However, the mechanism of the interaction does not become clear. Let us consider a crystal as a set of onedimensional chains. In Fig. 1 the wavy line designates
The potential (1) describes the interaction of a proton (deuteron) with a system of two oxygen atoms bound to a heavy ion. It is important to note the proton subsystem is fast and heavy one (the pair of oxygen atoms) is slow. The appropriate proton frequency v is about 3 £ 103 cm21 (v q V; where V is the characteristic frequency of fluctuation of the mass center of O – O-system). This allows to average on a fast subsystem in Eq. (1) and to obtain the effective
E.A. Shadchin, S.P. Sirenko / Journal of Molecular Structure 649 (2003) 1–5
From the condition dUeff ðR; jÞ=dj ¼ 0; at
potential Ueff ðR; jÞ ¼ N 21
ð1
UðR; j 2 xÞexp{ 2 UðR;xÞ=kT}dx
where k being the Boltzman constant. Hereafter we go from dimensional variables j, x to dimensionless those aj ! j; ax ! x ð1
Rc # R # Rc þ 2 lnðch2wðRÞ=chwðRÞÞ=a;
ð9Þ
21
ð2Þ
N¼
3
exp{ 2 UðR; xÞ=kT}dx
ð3Þ
21
Ueff ðR; jÞ is a one-minima anharmonic potential. Taking into account Eqs. (5) –(7) it is easy to show condition (9) includes a wide class of crystals such as ˚ # R # 2.6 A ˚. KDP: 2.43A Thus, for the majority of crystals undergoing structural PT, the proton (deuteron) on H-bond due to the interaction with O – O-system provides an effective anhormonic symmetric potential for the heavy subsystem.
where UðR; xÞ ¼ 0:5D0 {B22 ðRÞchð2xÞ 2 4B21 ðRÞchðxÞ} ð4Þ is the double-minima potential for H(D) at the fixed position of the mass center of O –O-system j ¼ 0. Taking into account that x p 1 (the characteristic distance between minima of the potential is ˚ in dimensional values), we can expand 2x0 , 0.3 A Eq. (4) up to fourth order and obtain UðR; xÞ ¼ 0:5D0 { 2 cðRÞx2 =2 þ bðRÞx4 =4} where cðRÞ ¼ 4{B21 ðRÞ 2 B22 ðRÞ};
4. Free energy of the model crystal Let us consider a crystal as a set of independent pairs of oxygen atoms being in the effective anhormonic potential (8). The distance between two oxygen atoms is proposed to be fixed. Thus we have the Einstein model for anhormonic oscillators with a Hamiltonian H ¼ K þ U where K being the kinetic energy of pair of oxygen atoms as whole. Taking into account the dipole –dipole interaction due to possible occurrence of average displacement kjk l we obtain in molecular field approximation
ð5Þ H¼
22
21
bðRÞ ¼ 2{4B ðRÞ 2 B ðRÞ}=3
ð6Þ
as D0 , 1.75 £ 104 cm21 [10] and for KDP cðRÞ , 0:32;
bðRÞ , 1:6
ð7Þ
at the temperature being characteristic for structural PT for the given class of crystals we have U=kT q 1: This inequality allows to integrate Eqs. (2) and (3) by the saddle-point technique. As a result we have
ðp2k =2M þUeff ðRk ; jk ÞÞ2
k
X
Jkk0 ðjk kjk0 l2kjk l2 =2Þ
k;k0
where M ¼2mox ; mox is the oxygen atom mass. Further, for finding the free energy of the model crystal in approach of the perturbation theory we choose as a ‘trial’ Hamiltonian H0 ¼K þU0 the Hamiltonian of harmonious oscillator H0 ¼K þU0 with a vibration frequency of V and a displaced vibration center of kjl: Then, according to Ref. [11] we obtain the free crystal energy F per one cell of the H-bond chain F ¼F0 þ0:5D0 kGðR; jÞl2kU0 l:
Ueff ðR; jÞ ¼ 0:5D0 {B2 ðRÞch2j 2 4B1 ðRÞchj} ; 0:5D0 GðR; jÞ;
X
ð8Þ
where B2 ðRÞ ¼ B22 ch2wðRÞ; B1 ðRÞ ¼ B21 chwðRÞ; wðRÞ ¼ ðcðRÞ=bðRÞÞ1=2 :
ð10Þ
Here averaging is carried out on states of harmonious oscillator Hamiltonian k···l¼sp{…expð2H0 =kTÞ}=sp{expð2H0 =kTÞ}:
ð11Þ
4
E.A. Shadchin, S.P. Sirenko / Journal of Molecular Structure 649 (2003) 1–5
Taking average in Eq. (10) according to Eq. (11), we obtain F ¼kT lnð2shð0:5"V=kTÞÞ þ0:5D0 {B2 ðRÞexpð2lj Þchð2kjlÞ 24B1 ðRÞexpðlj =2ÞchðkjlÞ} 20:5M V2 lj =a2 20:5J0 kjl2 =a2 :
ð12Þ
Here V, kjl are variational parameters of the free energy, lj ;kðj 2kjlÞ2 l¼0:5"a2 cthð0:5"V=kTÞ=M V is the dimensionless mean-square of fluctuation amplitude. The conditions of minimization of the free energy (1) result in a set of the equations for variational parameters
V2 2 ðmv20 =2MÞ›kGðR; jÞl0 =›lj ¼ 0
ð13Þ
22J0 kjl þ D0 a2 ›kGðR; jÞl0 =›j ¼ 0:
ð14Þ
Here v20 ¼ 2a2 D0 =m is the limiting frequency of the proton on the H-bond at the R ! 1. Taking into account Eq. (12) and excepting one paraphase solve from the sets (13) and (14) at kjl ¼ 0 we obtain chðkjlÞ ¼ {0:5g expð22lj Þ þ B1 ðRÞexpð23lj =2Þ}=B2 ðRÞ
V2 ¼ mv20 {B2 ðRÞ}chð2kjlÞexpð2lj Þ 2 B1 ðRÞexpðlj =2ÞchðkjlÞ}=M;
ð15Þ
ð16Þ
where g ¼ J0 a22 =D0 is a dimensionless parameter of the theory describing the dipole – dipole interaction. The numerical solution of the sets (15) and (16) for parameters of crystals KH2PO4 and KD2PO4 is given in Fig. 2. The agreement with experiments was observed at g ¼ 0:51: For KH2 PO 4 B1 ¼ 1, B2 ¼ 1.18. It corresponds to a choice of H-bond ˚ , RC,H ¼ 2.43 A ˚ [3]. It is necesslength RH ¼ 2.49 A ary to emphasize that for KD2PO4 not only the bond RC;D < length was varied but RD 2 RH < 0:01 A; RC;H þ 0:02 A were also varied [4]. Thus, for DKDP B1 ¼ 0.9, B2 ¼ 1.12. The numerical calculation shows that there is an essential dependence of the value of the average displacement of the mass center of oxygen atom pair below the critical temperature on particle grades in
Fig. 2. The dependence of the order parameter kjl on temperature (broken line is H-bond, solid one is D-bond).
and for the H-bond. So, for KD2PO4 kjl ¼ 0:012 A KD2PO4 kjl ¼ 0:04 A:
5. Conclusion In the paper a new mechanism of structural PT in crystals contained hydrogen based on the adequate choice of the potential of the interaction H(D) with two oxygen atoms laying on H-bond is offered. In the framework of the model of the crystal consisting of one-dimensional chains of Hbond it is shown that the light subsystem (proton (deuteron)) plays a role of starting mechanism for processes resulting to PT in the system of heavy
Fig. 3. The dependence of oscillation frequencies on temperature (broken line is H-bond, solid one is D-bond).
E.A. Shadchin, S.P. Sirenko / Journal of Molecular Structure 649 (2003) 1–5
ions at temperatures below critical. It is also shown that the observed isotopic displacement of the transition temperature for KD2PO4 can be obtained due to geometrical isotopic effect without the reference to the process of tunneling. The model is general enough and covers a wide class of crystals notwithstanding the numerical result check being performed for KH2PO4 (KD2PO4). The relative simplicity of the theory is that there is only one free parameter determining the dipole – dipole interaction. Notwithstanding there is the quite good agreement with experiment, it is necessary to note that the numerical calculation of PT temperature is rather sensitive to parameters of the Morse potential, values of which are known approximately. It is also necessary to note that while the H-bond length RH, RC,H are not bad known for KH2PO4, RC,D is calculated for KD2PO4. It results in an error in calculating Tc(D). Besides, the possible processes of tunneling are not taken into account in the model.
5
The results given in Fig. 3 show that there is a dependence of vibration frequency of the mass center on kind of a particle on the H-bond.
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