q-DEPENDENT CORRELATION FUNCTIONS AND DIELECTRIC PERMITTIVITIES OF DKDP AND DADP TYPE CRYSTALS. INFLUENCE OF EXTERNAL PRESSURE

Within the four-particle cluster approximation, we obtain an equation for the q-dependent deuteron pair correlation functions of DKDP and DADP crystals to which external hydrostatic and uniaxial 3 pressure is applied. Their wavevector, temerature and pressure dependences are studied. Expressions for the dielectric permittivities of the strained crystals are derived. It is shown that under the proper choice of theory parameters, the obtained results agree well with the available experimental data.


Introduction
Lately, a great attention has been paid to investigations of external pressure e ects in KDP-type crystals.In our recent papers 1{6] we developed an approach (based on the model 7,8]) which allowed one to describe the pressure dependences of the transition temperature, dielectric and thermal properties of deuterated ferroelectrics and antiferroelectrics of DKDP and DADP type.The calculations were carried out within the four particle cluster approximation.It allowed us to take into account the strong shortrange correlations between deuterons adequately.A good description of the available experimental data was obtained, and some predictions concerning the e ects of uniaxial pressure 3 on the considered responses of the crystals were made.Further experimental studies of hydrostatic and, especially, uniaxial pressure in uence on these crystals are required to verify our predictions and determine the theory parameters more precisely.
In understanding the nature of phase transitions in KDP type crystals and studying their internal structure, the thermal neutron scattering technique is extremely useful.Since the calculation of neutron cross-sections may be reduced to calculation of appropriate correlation functions 9], to study the latters becomes exceedingly important.Besides that, analytical expressions for correlation functions enable one to calculate the corresponding dielectric susceptibility tensor.
Deuteron q-dependent correlation functions of unstrained DKDP and DADP crystals have been calculated in a few papers.Thus, in 10] the equation for the q-dependent correlation functions of DKDP and DADP crystals was suggested.No consistent derivation of it was given.The calculations were carried out for the case of q = 0 only.The obtained results correspond to the cluster approximation only at T > T C .The method of calculation of the q-dependent correlation functions of paraelectric DKDP for a model with short-range interactions alone was proposed in 11].In the four-particle cluster approximation, the dielectric susceptibility as a function of wavevector of some directions was calculated, and some numerical results were presented.
The consistent procedure for calculation of q-dependent correlation functions of the model of a DKDP crystal with both short-range and long-range interactions (the latter being taken into account in the mean eld approximation) in the cluster approximation was proposed in 12].An equation of the Ornstein-Zernike type for pair correlation functions was derived both for T > T C and T < T C .Components of the dielectric susceptibilty tensor were calculated for some directions of wavevector.
In this work, following the method of 12], we derive an equation for pair q-dependent correlation functions of DKDP and DADP crystals to which external hydrostatic and uniaxial 3 pressure is applied.In Section 2, we suggest the uni ed model of strained DKDP and DADP crystals.In Section 3, the equations for pair correlation functions are derived.In Section 4, the expressions for dielectric permittivities are presented.Results of numerical calculations are given in Section 5.

The model
We consider a system of deuterons moving on O-D: : :O bonds in a crystal of KD 2 PO 4 (DKDP) or ND 4 D 2 PO 4 (DADP) type.The primitive cell 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 four nearest structural elements surrounding it.An external hydrostatic ( h ) or uniaxial stress ( 3 ), which does not lower the system symmetry h = (?p; ?p; ?p); 3 = (0; 0; ?p) (2.1) and electric eld E i (i = 1; 2; 3) directed along the crystallographic axes a, b, c are applied.
The Hamiltonian of the system has the following form 7] Hamiltonian (2.3) describes short-range con gurational interactions between deuterons near tetrahedra of "A" and "B" type; r f is a relative position vector of a hydrogen bond in a cell.Two eigenvalues of Ising spin qf = 1 are assigned to two equilibrium positions of a deuteron on the f-th bond in the q-th unit cell.c (0) ij are the "seed" elastic constants; " i are the components of the strain tensor; v = v=k B ; v is the unit cell volume; k B is the Boltzmann constant.
A.P.Moina The pressure dependence of can be taken into account in the following way.According to 13,14], in KDP and DKDP crystals is a linear function of hydrostatic pressure.Assuming that the character of this dependence on hydrostatic and uniaxial 3 stress in DADP crystal and uniaxial stress in DKDP is the same and bearing in mind (2.1) we can write = 0 + 1 p: Expanding Jff 0 in terms of pressure and considering the fact that it is proportional to 2 , we get The parameters ff 0 (qq 0 ) for a given crystal are the same for all pressures which do not lower its symmetry, whereas the ratio 1 = 0 is di erent for hydrostatic and uniaxial stresses.
The following calculations will be carried out in the cluster approximation.Considering the structure of the crystals, it is natural to divide the quasispin lattice into the four-particle clusters, vertices of which we choose in the centers of the O-D: : :O bonds.As f2 q1f1 we denote an e ective eld acting on the spin q1f1 from the side of its nearest neighbour f 2 .Let us make an identical transformation (the long-range interactions have been taken into account in the mean eld approximation) where H 0 is the Hamiltonian of the so-called reference system, and H (1)  qf = ?zqf qf 2 ; The summation in P R is carried out over clusters instead of lattice sites.
Hereafter, it is implied that the clusters R, R 0 belong to the cell q, clusters R 1 , R 0 1 belong to the cell q 1 etc; f denotes the set of the nearest neighbours of the site f.zqf = qf ?X q-dependent correlation functions : : : 97 Let us de ne the generating function F of the reference system as F = ln Sp e ?H 0 = ln Sp 8 < : exp(?X qf H (1)   qf ) exp (?
Then the free energy of the crystal is equal to Our goal is to calculate the correlation functions (cumulant averages of spin operators products, in particular, single-particle and pair ones) which are related to the free energy in the following way: h q1f1 : : : qnfn i H c = 2 n h q1f1 : : : (? F) h qnfn : (2.10) respectively, the correlation functions of the reference system are h q1f1 : : : qnfn i c = 2 n q1f1 : : : F qnfn : (2.11) From (2.8)-(2.11) it follows that the single-particle correlation functions of the reference and general systems coincide h qf i H = h qf i, whereas for the pair correlation functions the following relations hold (2.12) Let us introduce the notations h(:::)i 0 Spf(:::) exp(?P qf H (1)   qf )g Sp exp(?P qf H (1)   qf ) ; F (1)   qf ln Sp exp(?H (1)   qf ) = ln 2 cosh zqf 2 : Then F = X qf F (1)   qf + ln We restrict our consideration to the rst order of the cluster expansion, that is ln Then, the generating function F becomes F = X R ln Sp exp ?H (4)  R ] ?
X qf F (1)  qf ; (2.15) 98 A.P.Moina where The elds z Rf are z Rf = Rf ?X (2.17) the sum contains e ective elds created by the neighbouring quasispins outside the cluster; for the lattice considered, Rf is an e ective eld created by the whole neighbouring cluster R 0 , which also contains the site Rf.Let the eld R 0 f is created by the spins of the cluster R.
From the condition of a minimum of the generating function F with respect to Rf @F @ Rf = 0 (2. 19) it follows that @F (4)   R @ Rf = @F (4) R @z Rf = @F (1)   Rf @ Rf = @F (1) R @z Rf ; @F (4) R 0 f @ Rf ; (2.20) and h Rf i = 2 @F (4) R @ Rf = 2 @F (1) R @ Rf ; (2.21) (provided that the site f belongs to the clusters R and R 0 ), that is, the mean values of spins, calculated with the single-particle Hamiltonian H (1)   qf and the four-particle one H (4)   R , must coincide.The relations (2.21) comprise the system of equations for the unknown elds Rf .

Equation for q-dependent correlation functions
In this Section, following the procedure suggested in 12] for DKDP-type crystal at ambient pressure, we derive an equation of Ornstein-Zernike type for the q-dependent pair correlation functions of strained DKDP and DADP crystals.
The values of " 0 , w 0 , 0 c (0) for DKDP and a (k Z ) for DADP have been found in 6,15,16].They provide a satisfactory description of the temperature dependences of static and dynamic dielectric permittivities, spontaneous polarization, speci c heat at ambient pressure as well as values of transition temperatures of the crystals.
The values of the deformation potentials 1i of DKDP and DADP, ci (0) of DKDP and ai (k Z ) of DADP have been chosen in 1{5].They yield a good t to the available experimental data for variation of transition temperature, spontaneous polarization, longitudinal dielectric permittivity of DKDP and transition temperature of DADP with hydrostatic pressure.The values of 2i parameters were determined from the criteria (2.34) and (2.35).The dependence of the D-site distance on hydrostatic pressure in DKDP has been reported in 13,14].The value of 1 = 0 in the case of DADP was found from the available data for the T N ( ) and T N (p) dependences 17,18].In the case of the uniaxial pressure 3 , the 1 = 0 parameter was chosen such that a good t to the reported in 4] T C ( 3 ) dependence was obtained.Let us mention that the transition temperature T C (T N ) decreases with the uniaxial pressure 3 several times more rapidly that with hydrostatic h : q-dependent correlation functions : : : 109 As one can see, b 11 (q) > b 12 (q) > b 13 (q).Overall, the correlation functions have maxima in the center of the Brillouin zone, except for the b 11 (q; 0; 0) which has a minimum at q = 0.The amplitude of spin correlations increases when temperature approaches T C .Therefore, an increase in temperature and increase in pressure in the paraelectric phase and decrease in temperature and decrease in pressure in the ferroelectric phase in uence the correlations in a similar way, because all of them move the system from the transition point.A strong anisotropy of correlation functions is observed along the (q; 0; 0) and (0; 0; q) directions; that agrees with the result of 11].
In gure 2 we plot the wavevector dependences of the pair correlation functions of a DADP crystal at di erent values of hydrostatic pressure and temperature at T > T N .Here, also b 11 (q) > b 12 (q) > b 13 (q).However, the maxima of b 12 (q; 0; 0) and b 13 (q; 0; 0) shift from the center to the boundary of Brillouin zone.Furthermore, b 12 (q) and b 13 (q) increase with pressure and temperature.The temperature dependence of the inverse longitudinal dielectric permittivity " ?1 3 (0; T; p) of a DKDP crystal at di erent hydrostatic and uniaxial ?p = 3 pressures is plotted in gures 3 and 4, respectively.In gure 4 we also depicted the " ?1 3 (0; T; p) curve at hydrostatic pressure of 1 kbar.
As can be seen, in the paraelectric phase the Curie-Weiss law is obeyed in a wide temperature range at di erent pressures, the Curie constant decreasing with hydrostatic pressure and being independent of uniaxial pressure.
The main e ect of the uniaxial pressure is the shift of the transition point and, thereby, of the whole " 3 (T) curve to lower temperatures.Therefore, at constant T, the " 3 (0; T; p) almost does not depend on pressure in the paraelectric phase and only slightly varies in the ferroelectric phase.In gures 5 and 6 we plot the calculated temperature dependences of the static transverse " 1 (0; T; p) and longitudinal " 3 (0; T; p) permittivities of a DADP crystal at di erent pressures along with the experimental points for the ambient pressure.We are not aware of any experimental measurement of pressure e ects on dielectric characteristics of deuterated DADP.Theoretical values of "free" permittivities are calculated using relations (4.16) and experimental data for appropriate piezoelectric constants and elastic shear compliances 20].Since in DADP crystal d 36 d 14 , the difference between clamped and free values of the longitudinal permittivity is several times larger than that of transverse permittivity.A good description of experimental data is obtained for " 1 (0; T; p), while for " 3 (0; T; p) a small discrepancy between theory and experiment is observed.Probably, q-dependent correlation functions : : : 111 we could remove the discrepancy by explicit taking into account of shear strain " 6 .
The main pressure e ect is in the shift of the " i (0; T; p) curves to lower temperatures.Besides, " 1 (0; T; p) and " 3 (0; T; p) of DADP decrease with pressure in the paraelectric phase and increase in the antiferroelectric phase, with the pressure e ect being much stronger at T > T N .

Conclusions
In this paper, on the basis of previously proposed model 1{6], we study the in uence of external hydrostatic and uniaxial 3 pressure on the qdependent pair correlation functions and dielectric properties of DKDP and DADP crystals.In the four-particle cluster approximation, we derive an A.P.Moina equation for these functions and nd expressions for static dielectric permittivities of these crystals as functions of pressure and temperature.Theoretical results are compared with available experimental data.It is shown that under the proper choice of tting parameters, the theory provides a satisfactory description of variation of studied characteristics with hydrostatic pressure.
We also state the possible changes in the dielectric properties of DKDP and DADP crystals with the uniaxial pressure, assuming some plausible changes in the crystal structure with pressure.The main feature of the predicted e ects is that even low uniaxial stress can induce a signi cant shift of the transition temperature, whereas the intrinsic changes in the responses of DKDP and DADP crystals (i.e.beyond the shift) are rather small.
It is necessary to carry out comprehensive experimental studies of the uniaxial pressure e ects on these crystals, especially on their structure.We hope that these measurements will allow us to de ne the theory parameters more precisely and verify our predictions.

2 is
the four-particle cluster Hamiltonian.Here we use the fact that since each site of the given lattice belongs to two neighbouring clusters then

1 )
Di erentiating the expression for the single particle correlation function(2.21) with g = f 1 occurs only once.