Hydrostatic Pressure Influence On Phase Transition And Physical Properties Of

On the basis of proposed earlier model the eeects of applied hydro-static pressure on the physical properties of DKDP type crystals are studied. Within the cluster approximation for proton model the dielec-tric, thermal and elastic characteristics of considered crystals are calculated. Detailed numerical analysis of obtained results is performed. It is shown that under the proper choice of the theory parameters, a satisfactory description of experimental results for the pressure dependencies of spontaneous polarization, static dielectric permittivities and transition temperature is obtained. To explain the experimental determined behaviour of elastic constants the anharmonic eeects must be taken into account.


Introduction
The ferroelectric properties of a potassium dihydrogen phosphate (KDP) were discovered about sixty years ago, but despite decades of intensive studies the mechanism of a phase transition in it remains a problem.This compound is known as a typical substance of hydrogen bonded crystals, which undergo ferroelectric structural phase transitions.The main feature of phase transitions in crystals of this family is a remarkable isotope e ect at transition temperature.It is strongly a ected by deuteration, increasing from 123K in KDP to 220K in its deuterated isomorph KD 2 PO 4 (DKDP).To account for this large isotopic e ect the tunnelling proton model was introduced and has become the established basis for models of phase transitions of this kind (see 1,2] and references therein).
In accordance with this model, the phase transition is triggered by proton ordering; the latter makes heavy atoms shift along the polar axis from their paraelectric positions, hence a spontaneous polarization arises.In high temperature phase H-atoms are disordered over two possible sites on a hydrogen bond and order onto one of those sites below T c .It was assumed that protons, moving in the double well potential, tunnel between its two minima.The isotope e ect is explained in terms of di erences in tunnelling matrix elements, arising from the mass di erence between H and D atoms: tunnelling prevents ordering; thus the larger tunnelling frequency (the lighter a particle), the lower transition temperature.In completaly deuterated DKDP crystal tunnelling e ects are practically negligible, so the model reduces to the regular order-disorder one.
As far as the static dielectric and thermal properties are concerned, the tunnelling model provides a satisfactory quantitative t to experimental data.It should be mentioned, that short range correlations between hydrogen atoms play an essential if not crucial role in the phase transition.Therefore, any result obtained in a way when these correlations are not taken into account adequately (e.g. in the mean eld approximation) is subject to doubt.The four particle cluster approximation, used, for instance, in 3{5] seems to be appropriate here.It has been shown that one can nd a set of tting parameters, which yields a good quantitative description of the isotope e ect on T c , Curie constant, as well as on the temperature behaviour of entropy and spontaneous polarization 5].Tunneling was found to be e ectively supressed by short-range correlations.Based on this statement calculations were performed in the 6,7], where the order-disorder model was used in the whole range of deuteron concentrations.The sets of free parameters, allowing one to describe the temperature and concentration dependencies of all static dielectric (including transverse and longitudinal permittivities) and thermal properties, were found.The number of theory parameters was much less than that of characteristics described.While the static properties were considered, heavy atoms were assumed to follow hydrogen motions instantaneously, and the polarization was directly determined by the level of H-ordering.Yet when the dynamics of the system is considered, this assumption is not appropriate anymore.Then a proton-phonon model with the motion of protons in the H-bonds coupled to the motion of heavy ions, especially to the optic mode along the polar axis, was proposed.Two modes appear in the system if the interaction of protons with only one optic mode is assumed; the instability of the crystal against the lower of them (the soft mode !? ) leads to a phase transition.
Both modes were detected in experiments on Raman scattering in KDP in x(yx)y geometry.It was found out that, rst, the soft mode is heavily overdamped in the whole temperature region, but can be made underdamped by application of high pressure 9].Second, in this geometry the corresponding mode in DKDP crystal is absent 10].Both facts are quantitatively accounted for by the soft mode model, giving a strong experimental evidence for it.
However, there do exist some experimental facts that do not accord with tunnelling (proton-phonon) picture (an excellent review of them is given in 11]).
In particular, the experiments on light scattering in the low frequency region, carried out in z(yx) z geometry have revealed 12] that, in contrast to the x(yx)y spectrum, the peak corresponding the soft mode in DKDP does exist.Furthermore, its frequency is larger than that in the KDP crystal, whereas the reverse was expected.In Raman spectra of internal modes region 13] there have been found some peaks ( 3 and 4 ) which are forbidden, if PO 4 groups are of S 4 symmetry above T c , but must appear in the case of C 2 symmetry.Besides, it is stated that no evidence for PO 4 groups of C 1 symmetry ("lateral" con gurations) had been found.The very recent experiments on Raman scattering under high pressure asserted 14,15], that the soft mode frequency !? in forward z(yx) z + scattering was larger than that in 90deg x(yx)y scattering.Hence, !?was found to decrease when the scattering vector increases, although it was supposed to increase, as a frequency of a usual ferroelectric soft mode at ? point.This result was claimed as that which suggests a breakdown of Hydrostatic pressure in uence 131 the coupled proton-phonon model, since it predicts an unreasonable !?q relation of the soft mode.
Another model was proposed in order to explain the facts which seem to be inconsistent with tunnelling picture 16,17].Within this model, PO 4 groups are assumed to be of C 2 symmetry both below and above T c ; the phase transition originates from ordering of distorted PO 4 tetrahedrons.
The low-lying overdamped peak, which was thought to correspond to the soft mode, is assigned to a libration mode of PO 4 groups now.Yet it should be mentioned that the order-disorder model of PO 4 dipoles meets some di culties as well.The rst, and the most important one is that it doesn't account for the isotope e ects.Usually, the increase in transition temperature is explained in terms of the so-called geometric isotope e ect (see below), but the speculations here are rather qualitative, and no quantitative explanation has been developed yet.Besides, there arise some complications, concerning the ice rule constraint (two hydrogens near each PO 4 group and one hydrogen on a bond).
So, now we are in a situation when after the years of investigations, there were developed two di erent approaches to the problem, and the experimental evidence for and against both of them hase been found.The question of how and why the ferroelectric phase transition in KDP crystal occurs in fact, still remains the issue of the day.
Perhaps, an external hydrostatic pressure may become a probe which is able to shed some light on this long-pending problem.External pressure essentially in uences physical properties of the crystal 18{24].In modifying the geometrical parameters of internal crystal structure, we change molecular potentials of the system and thereby signi cantly change its characteristics.Among the most striking pressure e ects is the decrease of the transition temperature and ultimate vanishing of the ordered phase.At p of about 17kbar the transition temperature of KDP crystal falls to zero with an inde nite slope in accordance with the third law of thermodynamics.
Pressure reduces the magnitude of the static dielectric constant and shifts the whole " c (T) curve to the lower temperatures.In the paraelectric phase " c obeys the Curie-Weiss law in a wide temperature range, showing a marked discontinuous change at the transition point, so " c = " 1 + C T ?T 0 ; at T > T c .Here T 0 is the Curie-Weiss temperature, which is, for KDP, the same as T c to within 0.1K .At ambient pressure the Curie constant C decreases with pressure linearly with d ln C=dp = ?6:60:2%GPa ?1 in KDP and ?14:3 1:0%GPa ?1 in 82% DKDP 21].
Hydrostatic pressure can induce a transition from the ferroelectric to the paraelectric phase.At a constant temperature " c (p) obeys "the Curie-Weiss law" with respect to p: " c = C p ? p 0 ; (1.1) p 0 is strongly temperature dependent, whereas C is not so.The spontaneous polarization P s decreases with the increase of pressure linearly with @ ln P s =@p = ?23:51:5GPa ?1 in KDP at 76 K and ?8:7 1:0%GPa ?1 at 190 K in 94%DKDP 21].The pressure in uence on P s arises from the intrinsic changes in P s due to the changes in the density of dipoles and in dipole momentum as well as from the shift of the whole P s (T) curve to lower temperatures due to the lowering of T c .It is clear that the shift in T c does not a ect P s at temperatures far from the transition point.
The pressure dependencies of all the studied characteristics are found to be strongly dependent on the deuteration level.It is interesting that in KDP crystal @ ln P s =@p is much larger than @ ln C=@p, and in DKDP the reverse is true.( 18,21]).
Pressure dependence of the elastic constants C ij of paraelectric KDPtype crystals have been reported in 25].It was found that in all studied compounds (KDP, DKDP, ADP, DADP) C 44 and C 66 constants exhibited an anomalous non-linear (appr.parabolic) behaviour, whereas the others (C 11 and C 33 ) increased linearly with increasing pressure.One should expect that at points where the constants C 44 and C 66 extrapolate to zero the pressure-induced phase transitions take place, although the corresponding pressures (appr.100kbar) are well beyond the limits of the apparatus used.
A comprehensive analysis of pressure in uence on crystal structure of paraelectric KDP-type crystals has been performed by R.J.Nelmes (see 26{ 32]).
The crystal structure of KDP-type crystals is known to be tetragonal It is known that the deuteration signi cantly changes geometrical parameters of a hydrogen bond; an increase of T c is accompanied by a corresponding increase in H-bond length 2R and the H-site separation .This fact invites the question, whether the isotope e ect on T c may be completely or partially attributable to the changes in the H-bond geometry (so called "geometrical isotopic e ect" 33]) without explicit invoking tunnelling motions.
In order to answer this question, an experiment with an application of external pressure can be extremely useful, since hydrostatic pressure inuences the H-bond structure in the same way as going from the deuterated to undeuterated crystals does.
It has been found 26,28] that the bond parameters (H-site separation) and 2R vary linearly with temperature at a constant pressure and linearly with pressure at a constant temperature, so (p; T) = (p 0 ; T 0 ) + c 1 (T ?T 0 ) + c 2 (p ?p 0 ) + c 3 (p ?p 0 )(T ?T 0 ); and the same is true for 2R.It enables one to extrapolate to the critical values c and 2R c , appearing at the pressure when the ordered phase vanishes.
It turns out that T c tends to 0K at the same non-zero c of approximately 0.2 Ain all the studied compounds, independently of their crystal structure and dimensionality of H-bond framework (1D for PbH 2 PO 4 , 2D for the squaric acid H 2 C 4 O 4 and 3D for KDP or DKDP).Moreover, at the same value of (within 0.01 A) both KDP and DKDP have the same transition temperature!It means that deuteration e ects are almost completely suppressed when is kept constant 30{32].
A comprehensive microscopic theory of high pressure e ects in the crystals considered above is not yet developed.Usually, the pressure dependence of ferroelectricity in KDP family crystal is qualitatively accounted for in the framework of tunnelling or coupled proton-phonon models, assuming that the proton tunnelling frequency increases and the dipolar proton-proton interaction J (enhanced by the proton-lattice interaction in the case of the coupled proton-phonon model) decreases with pressure 18,21].Within the mean-eld approximation the transition temperature is given by the equation 4 =J = tanh( =kT c ): The equation has solutions (a nite T c does exist) only if 4 < J.If 4 becomes larger than J at some su ciently high pressure, the ordered phase must vanish.In the same way the pressure dependencies of the Curie constant and spontaneous polarization are described.
A more appropriate four-particle cluster approximation (FPCA) for the tunnelling model was used in 34].E ects of both deuteration and external pressure are explained in a consistent way with a quite conservative number of tting parameters.The free energy calculated numerically from the theory is analysed in terms of Landau expansions in the polarization.It is assumed that energies of non-ionized and singly ionized groups " and w are in the lowest order proportional to the squared distance between Hsites, these parameters being dependent on pressure and deuteration only through their dependence on .The pressure dependence of the parameter , introduced to describe the long-range dipolar interactions, is governed by and by the constant Q L h (this constant speci es the contribution to the volume electrostriction due to explicit dependence of the free energy on the lattice constants).Proper choice of and Q L h gives acceptable agreement with experiment for the logarithmic pressure derivatives of Curie-Weiss temperature, saturation polarization and the Curie-Weiss constant.
In 35] the model of coupled anharmonic oscillators was used to investigate the e ects of pressure on the ferroelectric phase transition.It is assumed that the model parameters depend on distance between ions of the lattice according to some power laws.Expressions are obtained for the pressure dependence of the transition temperature within MFA, the theoretical results being in good agreement with the observed experimental ndings both for KDP and DKDP.
A rather simple model of a one-dimensional hydrogen bond with a double Morse potential, analogous to that developed in 36] was studied by E. Matsushita and T.Matsubara in 37].An empirical relation between bond length 2R and H-site distance was explained.Within this theory, the geometric isotope e ect is entirely caused by the tunnelling motion of hydrogen and cannot be a reason, apart from tunnelling, for the shift of T c with deuteration.
In 38] the dependence of the tunnelling frequency and exchange integral on the H-bond length and pressure with the same potential was examined.It appears that this dependence does not coincide with that assumed in the phenomenological theories.The pressure dependence of the transition temperature calculated within MFA is found to be in good qualitative agreement with experiment.
The purpose of this paper is to reexamine the usefulness of an proton model for a DKDP crystal in describing the behaviour of a crystal under high hydrostatic pressure.It opens up the series of publications where we study the in uence of external hydrostatic and uniaxial of di erent geometries stresses on the phase transition in crystals of KH 2 PO 4 family.Our calculations are based on the model of the strained crystals of this type, proposed in 39,40].
The structure of this paper is the following.In Section 2 the Hamiltonian of the system under consideration is formulated and the equation for the transition temperature as a function of external pressure is derived.In Section 3 and 4 the in uence of pressure on elastic constants and piezoelectric modules and on thermal properties of a crystal is studied.Expressions for the longitudinal and transverse components of static dielectric susceptibility tensor are derived in Section 5. Section 6 contains a discussion on the obtained results.

Cluster approach
We consider a system of deuterons moving on O-D: : :O bonds in a crystal of KD 2 PO 4 type.The unit cell of Bravais lattice of such a crystal is composed of two neighbouring PO 4 tetrahedra together with four hydrogen bonds attached to one of them ("A"type tetrahedra).Hydrogen bonds going to another ("B" type) tetrahedron belong to the four nearest structural elements surrounding it 1 The Hamiltonian of the system at presence of hydrostatic pressure ?p = 1 = 2 = 3 and an external electric eld E i (i = 1; 2; 3) along the crystallographic axes a, b, c has the following form 39]: + fji E i fj (R qj ) 2 : (2.1) The Hamiltonian (2.1) describes the short-range con gurational interactions between deuterons near tetrahedra of "A" and "B" type; r fj is a relative position vector of a deuteron on a hydrogen bond.Two eigenvalues of Ising spin fi (R qi ) = 1 are assigned to two equilibrium positions of a deuteron on the f i -th bond in the q i -th unit cell.c (0) ij are the "seed" elastic constants; " i are the components of the deformations tensor; v = v=k B , v is the unit cell volume; k B is the Boltzmann constant.F i fj are internal elds, created by, rst, e ective long-range forces, including as well an indirect interaction between deuterons through lattice vibrations, and, second, by a hydrostatic pressure p.They read: 39,40] where (1)   fi = h fi (R qj )i; J fifj is a long-range dipole-dipole interaction be- tween deuterons; 1i , 2i , 3i are the so-called deformation potentials; = e is the dipole moment of a hydrogen bond; is a distance between two H-sites on a hydrogen bond.According to 26,28], is a linear function both of pressure and temperature, so: = (p 0 ; T 0 ) + c 1 (T ?T 0 ) + c 2 (p ?p 0 ) + c 3 (p ?p 0 )(T ?T 0 ) = 0 + 1 p; with (p 0 ; T 0 ) = 0:387 A, c 1 = ?5 10 ?5 A/K, c 2 = ?3:8 10 ?4 A/kbar, c 3 = 4 10 ?6 A/K/kbar, p 0 = 16:5kbar, T=295K 26]; 0 > 0 and 1 < 0.
Taking into account the fact that J fifj = 2 J fifj we can write: =-1.77 10 ?2 kbar ?1 at 250K and slightly decreases with lowering temperature.
The static and dynamic properties of deuterated orthophosphates will be considered in the four-particle cluster approximation.In terms of density matrices 41] it reads: 0 = e ?Hi Spe ?Hi = e ?H iA (2.4) where H iA 4 , H iB 4 are the four-particle cluster Hamiltonians, which describe behaviour of deuterons near "A" and "B" tetrahedra.Since the equilibrium distribution functions of deuterons around "A" and "B" tetrahedra are equal within the cluster approximation, then we can consider the static and dynamic properties of the KD 2 PO 4 crystal on the basis of H iA 4 solely.
Within the proposed model the energies of deuteron con gurations ", w, and w 1 are assumed to be linear functions of deformations " i : T < T c T > T c " = " 0 + ?
11 " 1 + c 0? 12 " 2 + c 0? 13 " 3 ? 2 equal to zero at zero pressure, one must assume the parameters 2i to be temperature dependent.For the sake of simplicity and in order to keep 2i constant we put deformations " i to be equal to zero only at the transition point.Thus the parameters 2i can be determined from the equation: a is taken at T = T c .At all other temperatures small residual deformations ( 10 ?6 ) exist at pressure of 1 bar.
The temperature of the rst order phase transition T c is determined from the criterion that g z ( (1)z ; T c ; p) = g z (0; T c ; p); (2.17) the order parameter and deformations obey the system (2.15).Polarization of the crystal, when the electric eld along the z-axis is applied, is equal to an e ective dipole momentum 3 and the unit cell volume v being linear functions of pressure: (2.19) Respectively, P 1 = 1 v (1)x 1 ? (1)x 3 ]; P 2 = 2 v (1)y 2 ? (1)y 4 ]: (2.20)

Elastic and piezoelectric properties
Let us study now how the hydrostatic pressure a ects the elastic properties of crystals under consideration.Matrix of elastic constants in ferroelectric phase has the following symmetry: c p? ij = c 0? ij ?Here we used the following notations: = cosh 2z + b cosh z] ? (1)sinh 2z + 2b sinh z]; t i = ?2b? 2i inh z + (1) M ?i ; In the present approach we cannot determine how the constants c p? 44 , c p?  In direct piezoelectric e ect, coe cients of piezoelectric strain e 3i determine a polarization caused by mechanical deformations, and in reverse piezoe ect, a pressure, which, at a given electric eld, must be applied to a crystal to maintain it undeformated.They read: e 3i = @P 3 @" i E3 = 3 v 2 T 2 ci (1) + t i D ?2' ; e h = e 31 + e 32 + e 33 : where ' = 1 1 ? (1)] 2 + ; At direct piezoe ect, constants of piezoelectric strain give an increase of an electric eld, necessary to maintain polarization constant, if this increase is caused by mechanical deformations.At reverse piezoe ect, they give an increase in pressure due to a polarization, which maintains a crystal undeformated.
h 3i = ?@E 3 @" i P3 = 2 3 D '(2 ci (1) + t i ); h h = h 31 + h 32 + h 33 : At direct piezoe ect, coe cients of piezoelectric deformations g 3i characterize an electric voltage in open circuit at a given pressure, and, at the where c E? ij = c P? ij ?h 3i e 3j ; i; j = 1; 2; 3: Piezoelectric and elastic characteristics are connected through the following relations: X i s p? ij h 3j = g 3j ; X i c E? ij d 3j = e 3j ; Let us note that all piezoelectric coe cients, calculated above (i.e. e 3i , h 3i , g 3i , d 3i ) are equal to zero in paraelectric phase.

Thermal properties
Let us consider now the thermal properties of a deuteron subsystem of a KD 2 PO 4 crystal.

Discussion
Before going into the discussion of the proposed in previous sections theory, let us note that, strictly speaking, this theory can be valid for a completely deuterated KD 2 PO 4 crystal only, whereas the vast majority of experimental data concern the crystals, deuterated partially.Nevertheless, as the relaxational character of dielectric dispersion in K(H 1?x D x ) 2 PO 4 crystals implies, tunnelling e ects in them are most likely suppressed by the short-range correlations between hydrogens.So at least in the case of high deuteration, we are allowed to neglect tunnelling and apply the theory to K(H 1?x D x ) 2 PO 4 crystals as well.In 6,7] it has been shown that under the proper choice of free parameters one can get a satisfactory description of a number of thermodynamic characteristics of considered crystals at ambient pressure.
For a numerical estimate of the pressure and temperature dependencies of dielectric, elastic, piezoelectric and thermal characteristics of KD 2 PO 4 crystals obtained in previous sections it is necessary to set the values of the cluster parameters ", w, w 1 , long range interaction parameters c (0), dipole moments per unit cell c and a , deformation potentials ij , cj and "seed" elastic constants c 0 ij .As the theory parameters for undeformated K(H 1?x D x ) 2 PO 4 we took those obtained in 6, 7].The deformation potentials were chosen such that the best agreement with experimental data was attained for the variation of transition temperature T c , spontaneous polarization P s and longitudinal static dielectric permittivity " 3 (0; T; p) with pressure (see Table 1).
Further experimental study of transverse dielectric, piezoelectric, elastic and thermal characteristics as functions of hydrostatic pressure is required to de ne the values of deformation potentials more precisely.
It should be mentioned that amount of deuteration is di cult to determine exactly: the discrepancies in T c (x) dependence reported by di erent authors reach 10%.In this work we used the data by E.N.Volkova (see 6] and references therein).
The experimental values of the elastic constants are available for the K(H 1?x D x ) 2 PO 4 crystals only with x = 0 x = 0:89 above T c 43].Nevertheless, we can estimate the values of c + ij at x = 0:98, 0.87, 0.84, assuming their linear dependence on deuteration.Those data were taken as the "seed" elastic constants c 0+ ij .Since the values of elastic constants of ferroelectric DKDP were not available at all, we carried out calculations of T c , " i and other characteristics at di erent pressures and di erent trial values of c 0? ij close to c 0+ ij .The set of c 0? ij , providing the best t to experimental data is given in Table 2.We met some di culties, trying to solve the system of equations (2.15) by Newton-Raphson method directly.Instead, we had to minimize the thermodynamical potential g( (1)z ; T; p) with respect to the order parameter (1)z and determine the deformations " i from the three latter equations of (2.15).The transition point was found from the (2.17) criterion.
In uence of hydrostatic pressure on the spontaneous polarization P s vs temperature curves of K(H 1?x D x ) 2 PO 4 was studied by G.A.Samara in 21], where the data for P s (T) for x = 0:94 and T c = 220 K at p =0, 2.07 and 4.14 kbar are presented.According to table I of this paper the value of P s (T) at 190 K is 6:15 0:10 C/cm 2 .However, in 42] (also by G.A.Samara) it is asserted that at nearly the same transition temperature of 219:8 0:1K deuteration is x = 0:98 with a saturation polarization of 6:21 0:07 C/cm 2 .Furthermore, at all temperatures P s (T) at x = 0:94 21] is greater than P s (T) at x = 0:98 42], whereas the reverse is expected.Since the dependence of T c on deuteration x obtained in 42] is in agreement with the independent data of E.N.Volkova (see 6]), we assumed that in 21] x had been merely understated { its more precise value should be about 1.0.
The calculated dependencies of spontaneous polarization P s on temperature at di erent pressures and on pressure at di erent temperatures along with experimental points of 21] are plotted in gs.4 and 5.A good description of experimental data is obtained at ? c = ?c ? k ?p, with k = 0:051 10 ?18 esu cm/kbar, (or, d ln ?c =dp = ?1:07%/kbar(compare with d ln ?c =dp = ?1:2%/kbar of Samara 21]).As one can see, an increase in pressure leads to a decrease in P s saturation value, while the jump of P s at the transition point hardly depends on pressure.The strength of the rst-order transition increases with pressure.
In gs.6-9 the temperature dependencies of piezoelectric coe cients e 3i , d 3i h 3i and g 3i of K(H 0:02 D 0:98 ) 2 PO 4 crystal at di erent pressures are depicted.In the paraelectric phase all piezocoe cients are equal to zero.
When the temperature tends to T c the magnitudes of e 3i and d 3i sharply increase, and abrupt jumps of them are found at the transition point.Pressure reduces the size of the jumps.The temperature range T where e 3i and d 3i di er from zero, decreases with pressure.
Coe cients h 3i and g 3i increase slightly with temperature and fall to zero with a discontinuity at T = T c ; the size of the discontinuity increases with pressure.The values of e 31 , e 32 ; : : : d 31 , d 32 are negative, and those of e 33 , e h ; : : : d 33 , d h are positive.This fact is consistent with the statement that the sign of resultant piezoelectric coe cients e h ; : : : d h determines the direction of the transition temperature shift with pressure.
by external pressure.In the paraelectric phase the symmetry of elastic constants matrix some- gure 2 the calculated pressure dependence of the transition temperature on of K(H 1?x D x ) 2 PO 4 crystal for x = 0:98; 0.87, and 0.84 is shown along with experimental data 21, 24].Up to 25kbar, T c decreases with pressure linearly as T c (p) = T c ? k T p, k T (0:98) = 2:02 K/kbar, k T (0:87) = 2:67 K/kbar, k T (0:84) = 3:02 K/kbar.The further raising of pressure leads to the deviation of T c (p) dependence from linearity, and the slope j@T c =@pj signi cantly increases.Calculations show that at p 65 kbar the transition temperature falls to zero, so the ordered phase disappears.T c , K p, kbar Figure 2. The transition temperature of K(H 1?x D x ) 2 PO 4 crystals as a function of pressure at di erent deuteration levels x: 1, 24] { 0.98; 2 { 0.87; 3, 2 21] { 0.84

Table 1 .
Parameters of the theory (all in K) at di erent deuterations.