KDP Model Research Articles

Modified KDP Model on the Kagome Lattice

Modified KDP Model on the Kagome Lattice

Thermodynamics of an elastic ferroelectric model

The modified KDP model of Wu on a compressible harmonic lattice is considered. An electric field and normal forces act on the system. The specific heats, mean lattice spacing, polarization and susceptibility are computed. Elasticity does not change the order of the transition, which remains of second order (without latent heat). The specific heat at constant volume is finite through the critical temperature, while the specific heat at constant force diverges at T + c. The dependence of the critical temperature on electric and force fields as well as the isotherms are discussed.

Modified KDP Model on the Kagomé Lattice

Modified KDP Model on the Kagomé Lattice

Quantum field theoretical treatment of a dynamical model of the ferroelectric KDP. II

For pt. I see ibid., vol. 4, 2097 (1971). The results of the previous paper with this title by the authors are applied to the electrical susceptibility and the specific heat of KDP and its deuterated isotope, in their paraelectric phase. It is found that the isotope effect and critical behaviour of these and other properties are explained in a selfconsistent way by our simple model if we allow for the expansion of the short hydrogen bonds on deuteration, an effect which has been discussed in connection with another model of KDP. The theory predicts the logarithmic behaviour of the specific heat above the critical point which has been observed recently by Reese. To explain the ferroelectric transition in full detail appears to require a more sophisticated model which would include short range forces.

Quantum field-theoretical treatment of a dynamical model of the ferroelectric KDP

The diagrammatic technique of Spencer (1968) is applied to the model of KDP and KDDP in which the long range electrostatic forces between protons and the long range coupling of protons with an optical phonon mode are considered. The results of the calculation are identified with the molecular field approximation and the random phase approximations.

Models for the Order—Disorder Transition in NaH3(SeO3)2

The entropy of the Makita—Miki model for NaH3(SeO3)2 is shown to be exactly solvable by transformation into a Slater KDP model problem. However, arguments are presented to show that the model is unsatisfactory. Two additional models are introduced. The first model is reduced to a dimer problem and is solved exactly. The solution is interesting in the context of critical phenomena, but it indicates that the model is not a good one for NaH3(SeO3)2. The second model is readily solved in the limit of high temperature using results for the two-dimensional Ising model. Although this model is too difficult to solve exactly at finite temperature except for one limiting case, it is encouraging that analogous models exhibit multiple phase transitions.

Solution of an elastic ferroelectric model

A solvable, elastic generalization of the Sutherland-Yang-Yang model is proposed. The specific heats of the elastic modified KDP model are finite at the critical temperature and their exponents are renormalized.

Zeros of the Partition Function for the Heisenberg, Ferroelectric, and General Ising Models

The Lee-Yang theorem for the zeros of the partition function of a ferromagnetic Ising model with real pair spin interactions is extended to general Ising models with complex many-spin interactions (satisfying appropriate ``ferromagnetic'' and spin inversion symmetry conditions). When many-spin interactions are present, all zeros lie on the imaginary Hz-axis for sufficiently low (but fixed) T, but, in general, some leave the imaginary axis as T → ∞. The extended Ising theorem is used to prove the same result for a Heisenberg system of arbitrary spin with the real anisotropic pair interaction Hamiltonian Hij=−(JijzSizSjz+JijxSixSjx+JijySiySjy)in an arbitrary transverse field (Hx, Hy) under the ``ferromagnetic'' condition Jijz≥|Jijx| and Jijz≥|Jijy|. The analyticity of the limiting free energy of such a Heisenberg ferromagnet and the absence of a phase transition are thereby established for all (real) nonzero magnetic fields Hz. The Ising theorem is also applied to hydrogen-bonded ferroelectric models to prove, in particular, that the zeros for the KDP model lie on the imaginary electric field axis for all T below the transition temperature Tc.

Distribution of Zeros of the Partition Function for the Slater Model of Ferroelectricity

The partition functions of the Slater's models of ferroelectricity (KDP model), antiferroelectricity (F model) and Wu's modified KDP model are calculated for lattices of finite size (up to 6×6). The distributions of the zeros of the partition function of the KDP model in the complex fugacity ( z ≡exp V / k T ) plane are on the unit circle at low temperatures and are two-dimensionally distributed at high temperatures. In case of the F model the distribution of zeros are two-dimensional in all temperature region. The zeros of the KDP model at low temperatures in the complex z 2 N plane are on the unit circle, and those at high temperatures lie on the negative real axis. The zeros of the F model in the complex z 2 N plane lie on the negative real axis at high and low temperatures. The distribution of zeros of the KDP model, the F model and the modified KDP model in the complex temperature plane are also obtained.

Phase Transition of a Molecular Zipper

Two systems of biomolecules in solution are known to be represented by simple models which can be solved to give phase transitions in one dimension. The systems are double-stranded DNA and the helical polypeptides. We treat the very simple problem of the single-ended zipper. There is an interesting simialrity with the work of Nagle on the one-dimensional analog of the Slater model of KDP.

Proof of the first order phase transition in the Slater KDP model

The Slater KDP model defined on d-dimensional tetrahedral lattices is proved to have a phase transition for which the entropy and energy are discontinuous functions at a transition temperaturekT c =ɛ/ln2, independent of dimensionality.

The One-Dimensional KDP Model in Statistical Mechanics

It is observed that the one-dimensional analog of the Slater KDP model provides a simple introduction to the study of strongly interacting systems and phase transitions as well as to recent research on two-dimensional KDP models. Although the first-order phase transition of the one-dimensional Slater model technically disappears when more realistic interaction strengths are taken into account, an anomalously large specific heat remains in the modified one-dimensional model.

Remarks on the Modified Potassium Dihydrogen Phosphate Model of a Ferroelectric

The modified potassium dihydrogen phosphate (KDP) model of a two-dimensional ferroelectric with the inclusion of an arbitrary electric field $\mathcal{E}$ is considered. The nature of the phase transition is unaffected by the inclusion of an external field, whereas the transition temperature depends on $\mathcal{E}$ with ${T}_{c}\ensuremath{\rightarrow}\ensuremath{\infty}$ as $\mathcal{E}\ensuremath{\rightarrow}\ensuremath{\infty}$, and ${T}_{c}\ensuremath{\rightarrow}0$ as $\mathcal{E}\ensuremath{\rightarrow}\mathrm{some}\mathrm{special}\mathrm{values}$. We also find that ${T}_{c}$ is determined by different relations in different regions in the $\mathcal{E}$ plane, a property not shared by the Slater KDP model. Among the other thermodynamic properties discussed, the critical behavior of the polarizability is shown to be $\ensuremath{\chi}\ensuremath{\sim}{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\frac{1}{2}}$. It is shown that this model is identical to the problem of close-packed dimers on a hexagonal lattice.

Note on Singularity of Specific Heat in the Second Order Phase Transition

Singularity of specific heat near the transition point is investigated in terms of the dis­ tribution of. zeros of the partition function in the complex temperature plane. In the case of a centrally symmetric two-dimensional distribution of zeros, the singularity of the radial dis­ tribution function of zeros reflects directly the anomaly of specific heat. Consequently, it is shown that in general the specific heat does not necessarily take the same critical indices above and below the transition point. The modified Slater KDP model solved exactly by Wu gives a nice example to our theory. The radial distribution function of zeros in the complex z(=eC!lcT) plane for this model is given by the equation g(r) =1/(rc2rv4-=i2). Then, the sin­ gularity of the specific heat for this model takes the form C+'(T-Tc)- 1 1 2 for T>Tc and

Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric

The Slater KDP model is solved for all temperatures and with an electric field. Above Tc the specific heat behaves lik (T−T c )−1/2 and the polarizability like (T−T c )−1. There is first-order phase transition at T c (latent heat). Below T c the free energy is simply -∣ℰ∣d (ℰ = electric field, d = dipole moment).

Exact Solution of a Two-Dimensional Model for Hydrogen-Bonded Crystals

A model of two-dimensional hydrogen-bonded crystals satisfying the ice rule is presented and solved exactly in the presence of an external electric field. The transfer-matrix method is used. The model includes, for special values of the parameters, the ice problem, the Rys $F$ model of an antiferroelectric, and the Slater KDP model of a ferroelectric.

Lattice Statistics of Hydrogen Bonded Crystals. II. The Slater KDP Model and the Rys F-Model

The Slater KDP model, a simple hydrogen bonded ferroelectric model, and the Rys F-model, a simple hydrogen bonded antiferroelectric model, have been treated using both high- and low-temperature series for the partition function. The high-temperature series is a modification of the residual entropy of ice series discussed in I. For each model a temperature is found at which the high-temperature series and the low-temperature series are identically equal. For the KDP model this equality gives a transition temperature and a latent heat is easily calculated, both of which are exact. It so happens that these exact results agree with previous analyses which used only mean field approximations. For the F-model the formal equality of the series gives the first evidence for a phase transition. Although the latent heat calculation throws some doubt on the existence of a transition, after further discussion of the series it is concluded that there most probably is a phase transition.