Temperature dependent structural Phase transition in Lead Mono Hydrogen Aresenate crystal

With the use of the modified pseudospin lattice coupled-mode (PLCM) model by adding third-order and fourth-order phonon anharmonic interactions terms and the statistical Green’s function technique a phenomenological explanation of phase transition of the lead monohydrogen arsenate (PbHAsO4) crystal has been given. Expressions for shift, width, dielectric constant, loss tangent and modified ferroelectric soft mode frequency have been derived theoretically. By fitting model values of various quantities obtained from literature in these derived expressions, their temperature dependence has been evaluated numerically. Present theoretically results agree well with the experimental results of others.


INTRODUCTION
Due to promising applications in the field of electronics and technology ferroelectric crystals are continuously being attracted to both scientists and physicists. These crystals are used for preparation of memory devices like capacitors of small size, piezoelectric acoustic  Corresponding Author Email: arvindsgfi@gmail.com transducers and pyroelectric infrared detectors 1,2 . Lead mono hydrogen arsenate (PbHAsO4) belongs to lead hydrogen phosphate (PbHPO4) type ferroelectric crystals which are called monetites. In these crystals the direction of the spontaneous polarization is almost parallel to the direction of H-bond O-H…O projecting on the (010) bonds in the form of one dimensional chain along c-axis. The PO4 chains in this salt are bound to one another by the O-H…O bonds. Thus the intra chain coupling (within a chain) is stronger than the interchain coupling between the chains. If one compares PbHAsO4 crystal with largely studied KH2PO4 crystal, one finds that there are three major differences 3 (i) one dimensional ordering of protons (ii) unusual large isotopes effect (iii) spontaneous polarization direction is not along c-axis. Therefore it can be said that on the basis of the simple pseudospin lattice coupled mode model, the nature of ferroelectric transition of PbHAsO4 crystal cannot be adequately explained. So here in order to explain phase transition in lead monohydrogen arsenate crystal, we should use two-sublattice coupled mode model. Experimental investigations in lead mono hydrogen arsenate crystal have been made by a large number of workers. Deguchi and Nakamura 4 have made crystal growth studies on lead monohydrogen arsenate crystal.
Kroupa et al. 5 have made experimental dielectric and far infrared studied in lead mono hydrogen arsenate crystal. Charykova et al. 6 have studied dielectric properties of lead mono hydrogen arsenate crystal. Lee et al. 7 have carried out experimental determination of stability constants of lead mono hydrogen crystal. Zachak et al. 8 have studied thermodynamic properties of lead mono hydrogen arsenate crystal. In order to explain dielectric properties and ferroelectric structural phase transition, in past, theoretical work have been done by many workers. Using pseudospin model without tunneling term De Carvalho and Salinas 9 have studied this crystal. By adding two-sublattice term in pseudospin model calculation were made by Blinc et al. 10 . Two sublattice PLCM model with fourth-order phonon anharmonic term was used by Chaudhuri et al. 11 in their calculation. However these authors have made decoupling of correlations in early stage, as a result of which some important interactions disappeared in their theoretical derived results. In this paper, with modified PLCM model, by adding third-and fourth-order phonon anharmonic interactions terms and double time temperature dependent Green's function method 12 are third-and fourth-order atomic force constants.
The last two terms in above equation are called third-and fourth order anharmonic interactions terms.

Green's functions method
We consider the Green's function where z i S 1 and z j S 1 are spin operators, on sites i and j, θ is unit step function: . The angular bracket  ...... denotes ensemble average over a grand canonical ensemble.
Differentiating Green's function (2) two times with respect to time t and t using model Hamiltonian (Eq.1) and multiplying both sides by i and after Fourier transforming we obtain (putting in the form of Dyson equation) and   , Eq. (3) gives value of Green's function (2) In Eqs (8) and (9) Now we obtain Green's function (7) finally as If we simplify Eq. (17), we obtain   This frequency with negative sign is called the modified ferroelectric soft mode frequency which becomes zero at transition temperature and gives rise to ferroelectric transition.
Applying condition of stability i.e.
The loss of power which is known as loss factor in ferroelectrics (or dielectrics) due to orientation of dipoles is expressed as loss tangent (tan  ): We obtain from Eq. (28)  tan as