KDP Model Research Articles

Analysis of Mechanical Property of Crystal KDP and Simulation of Ultra-Precision Cutting Process in the Ductile Mode

In order to study mechanical property with different crystal-plane and different crystal orientation of the crystal KDP, nano-indentation experiments are first done. The mechanical properties of crystal KDP, such as elastic modulus, yielding stress, are obtained from the analysis of the experimental curve. To obtatin the stress-strain curves of crystal KDP, by using the spherical tip can get characteristic of continuous strain, the spherical indentation experiments is proposed firstly and carried out. According to obtained parameters, A finite element cutting model of crystal KDP is established. The cutting process of crystal KDP is simulated by the model, and the influence of rake angle and depth of cut on chip and surface quality is studied. The theory shows that when the cutter’s rake angle is in the range of -40° to -45°, an perfect super-smooth KDP crystal surface will be obtained. Finaly, the experiments is carried out on special ultra-precision machine tool for crystal KDP by ourself devoloping. Experiment results show that when the cutter’s rake angle is about -45°, an super-smooth surface (rms: 6.521nm and Ra: 5.151nm )is obtained on the plane (001), and this experiment certified correctness of theory analysis.

Breakdown of the soft-mode model of KDP—low-frequency Raman spectroscopy under high pressure

Abstract The Raman scattering of KDP has been carried out under high pressure 0-5 GPa at two scattering angles, 90 and 4.5 degrees. The low frequency components of the B2(z) spectra in x(yx)y and z(yx)z+δ geometries have been analyzed in terms of a coupled soft-phonon model. The wave vector dependence of the soft-mode frequency contradicts the dispersion relation of the ferroelectric soft-mode near the Γ point of the Brillouin zone. The low frequency component of the B2(z) spectra is concluded not to be caused by the soft mode.

Phase diagram of the five-vertex model.

Within the ice-type models, the solution of the five-vertex model is obtained with the use of the Bethe ansatz. Since the allowed number of vertex types is odd, the arrow-reversal symmetry of the system is broken by construction. Due to this, the exact solution obtained and the phase diagram are very different from those of the symmetric six-vertex model. A connection to the asymmetric six-vertex model (of which the five-vertex model is an extreme case) is made. The different regions of the phase diagram are described and the transitions between them are analyzed. Several aspects of the phase diagram are unusual, i.e., the ordered phases (both ferroelectric and antiferroelectric) are frozen-in phases and the disordered phase is replaced by a ferrielectric phase. In the free-fermion case, the known results of the modified KDP model are recovered.

A stochastic model of three-dimensional crystal growth

This paper describes a stochastic model of crystallization from a gas or dilute solution. The model is limited to a crystal of rectangular symmetry whose surface has nonzero Miller indices. By a mapping into the modified KDP model, the kinetic growth coefficient can be given approximately as an analytic function of the Miller indices of the surface. Numerical simulations indicate that the aproximation is accurate within a few percent at all surface orientations.

Exact equilibrium crystal shapes in two dimensions for free-fermion models.

Using a conceptually novel approach that maps a two-dimensional interface exactly onto a Feynman-Vdovichenko lattice walker, we derive an exact and general solution for the equilibrium crystal shapes (ECS's) of free-fermion models, i.e., models that are solvable via the Feynman-Vdovichenko or (equivalently) Pfaffian methods. The ECS for these models is given by the locus of purely imaginary poles of the determinant of the ``momentum-space'' lattice-path propagator. The ECS may, therefore, simply be read off from the analytical expression for the bulk free energy. From these shapes one can then obtain numerically (but to arbitrary accuracy) the high-temperature direction-dependent correlation length of the dual system. We give several examples of previously unknown Ising ECS's, and we examine in detail the free-fermion case of the eight-vertex model. The free-fermion eight-vertex model includes the modified potassium dihydrogen phosphate (KDP) model, which is not in the Ising universality class. The ECS of the modified KDP model is shown to be the limiting case of the ECS of an antiferromagnetic 2\ifmmode\times\else\texttimes\fi{}1 phase on a triangular lattice in the limit of infinite interactions. The ECS of the modified KDP model is lenticular at finite temperature and has sharp corners. We explain the physics of this lens shape from an elementary calculation.

Raman scattering study of KDP under high pressure with a diamond anvil cell

Abstract Raman scattering spectra of the ferroelectric KDP (KH2PO4) have been observed under hydrostatic pressure up to 52.0 kbar using a diamond anvil cell at room temperature. A curious dispersion relation of the soft mode at the Γ point of the Brillouin zone is derived from the low-frequency components of the B2(z) spectra in z(yx)z + δ and x(yx)y geometries after numerical fitting based on the coupled phonon model. The dispersion relation suggests breakdown of the soft-mode model of KDP. Each of the frequencies of the internal v1, v2 and v3 modes of the PO4 ion show a linear dependence on pressure with the mode Gruneisen parameter, 0.32, 0.31 and a small value of 2.5 × 10−2, respectively. Increase of the frequencies of the v1 and v2 modes corresponds to pressure dependence of the P–O distance. A new pressure-induced phase transition appears at 40.0 kbar with a growing relaxational central peak in the spectra.

On the ω-qRelation of the Soft Mode of KDP Studied by Light Scattering under a High Pressure

A low frequency component of the B 2 spectra of KDP in z ( y x ) z +δ and x ( y x ) y geometries has been observed under a hydrostatic pressure in a range 0-52.0 kbar using a diamond anvil cell. Numerical fitting of the spectra based on the coupled phonon model derives a curious ω- q relation of the soft mode at Γ point of the Brillouin zone. The ω- q relation suggests break-down of the soft-mode model of KDP. The pressure dependence of the spectral shape confirms the existence of the pressure-induced phase transition at 27.5 kbar reported previously by an X-ray work. A new central component grows rapidly at 40.0 kbar indicating a new phase transition induced by pressure at room temperature.

General properties of systems with linear couplings between bosons and other dynamic variables

A variety of physical systems, as for example, Jahn-Teller crystals, the coupled phonon-pseudospin model of the ferroelectric KDP (K${\mathrm{H}}_{2}$P${\mathrm{O}}_{4}$), one-dimensional conductors, and the Dicke model of super-radiance, are characterized by linear couplings between bosons (e.g., phonons) and other dynamic variables (e.g., electrons). Using a general Hamiltonian we explore various properties of such systems. We find that there are exact relations connecting the phonon static correlation functions to those of the electrons. A similar set of relations is also derived for their relaxation functions. The dynamics is then studied employing a relaxation-function formalism.

One-dimensional KDP model with random impurities

Some protons lying on the hydrogen bonds of the one-dimensional analog of the Slater KDP model are substituted by deuterium ions. The thermodynamic behaviour of this system is considered in the cases of fixed and mobile impurities. The phase transition remains of first order, but exhibits some new features.

Damping of transverse acoustic and ferroelectric modes in KDP

The Kobayashi model of KDP, in which proton pseudo-spins are coupled bilinearly to transverse optic modes, is extended by addition of a similar coupling to transverse acoustic (TA) modes plus a strain-dependent exchange. The latter gives rise to three-phonon processes whose contribution to damping we calculate with the aid of double-time, temperature-dependent Green functions. Phonon lifetimes are extracted from the poles of the spin propagator which is calculated by the equation-of-motion method with random phase decoupling for T > Tc. We predict that damping of the soft optic mode varies as (T - T0)T for T > Tc while the phonon width of the TA mode is proportional to ω2(T — Tc)-3/2 (T - T0)3/2, T0 < Tc, in agreement with measurements by Garland and Novotny.

Magnetorestrictive-type pseudo-spin coupling model for damping of acoustic and polar modes in KDP

A model of KDP proposed by Kobayashi (1968), in which proton pseudo-spins are bilinearly coupled in the Hamiltonian to transverse optic phonon operators, is extended, following earlier authors, to include a similar coupling to acoustic modes plus a magnetostrictive dependence of pseudo-spin exchange on longitudinal acoustic vibrations associated with lattice parameter fluctuations. This provides a damping mechanism for propagating modes omega 1(q) and omega 3(q) having optic and acoustic character, respectively. Imaginary parts of the mode frequencies are extracted from the poles of the spin Green function, which is calculated by the equation-of-motion method, with random phase decoupling, for T>Tc.

The One-Dimensional KDP and Other Ice Rule Models

One-dimensional ice, KDP, F, eight-vertex and general six-vertex models are discussed. As these models cannot be defined uniquely in one dimension, various possibilities are presented. For a particular one-dimensional lattice, the KDP, the eight-vertex and the six­ vertex models undergo a first-order phase transition. The KDP model is considered also in the presence of electric fields. Thermodynamic quantities of the model show unusual behavior. dimensional ice models are discussed. In § 3 the one-dimensional KDP and F models are considered. We will see that only the KDP model with a particular boundary condition for the vertical direction undergoes a first-order phase transition, and that the model is equivalent to Nagle's one-dimensional KDP model. Occur­ rence of a phase transition in one-dimensional systems is briefly discussed. In § 4 the KDP model which has a transition is considered in external electric fields. The careful consideration of the possible ways of defining the one-dimensional KDP models is necessary when we introduce the external fields precisely. It is shown that the transition temperature is increased and the order of the trasition is not changed by the introduction of the fields. The latter fact is a remarkable difference from the result of the two-dimensional KDP model. In § 5 the one­ dimensional eight-vertex model is considered and the possible phase transition is also of first order which differs from the two-dimensional case. Condition for the occurrence of the transition in the general six-vertex model which is a generalization

A microscopic mechanism for ultrasonic attenuation in KDP

The pseudo-spin model of KDP is extended to include modulation of the dipole-dipole separation by longitudinal acoustic phonons. This coupling is shown to give rise to attenuation of transverse sound in the paraelectric region, in agreement with experiment. The calculation has been performed using the approximation. The contribution of this coupling to the ferroelectric mode has also been calculated.

Two-sublattice deformable Ising model of KDP

A molecular-field calculation is made of polarizations (Z1) and (Z2) appropriate to two families of proton pseudo-spins distinguished by orientation of their respective H-bonds. Effective exchange interactions among spins in the same family are assumed to depend linearly on sigma 6, the shear accompanying a tetragonal-orthorhombic KDP transformation while exchange between spins of different sublattices is approximately independent of sigma 6. Since Tc is very close to T1, a point of infinite slope for the lattice polarization, the condition of infinite steepness, along with the self-consistent molecular-field equations, is (Z2) assume arbitrary values at T1 approximately Tc, subject to one restriction. The authors require that the sigma 6 to 0 limits of the parameters are never Tc.

Sharpness of transition and effect of tunnelling in ferroelectric Bean-Rodbell model of KDP

The molecular-field treatment by Bean and Rodbell (1962) of a deformable Ising ferromagnet is modified to include piezoelectricity and spin-lattice coupling, so that it may be applied to deuterated KDP. In this case, the pseudo-spin exchange coupling depends linearly on strain mod sigma 6 mod , and it is seen that this dependence can produce a first-order transition for appropriately chosen values of the coupling parameters. From the condition that the slope of polarisation be infinite at Tc plus a consistency condition on the exchange, it is shown that the mean value (Z) of a pseudo-spin can have a maximum possible value at the Curie point. The value of the spontaneous polarisation ratio Ps(Tc)/Ps(0) which corresponds to maximum (Z) is 0.704. This provides a qualitative understanding of the shape of the polarisation curve to within a fraction of a degree of Tc, where critical fluctuations become important.

Ferroelectric ice model on a triangular lattice

A ferroelectric ice model or KDP model, after the Slater KDP model for a square lattice, is formulated on a triangular lattice and solved for all temperatures. It is found that this system undergoes a phase transition of the first order into a frozen ferroelectric state.

Dimer models for the antiferroelectric transition in copper formate tetrahydrate

This paper develops the lattice statistical theory of the antiferroelectric transition in copper formate tetrahydrate in terms of dimer and 8-vertex models. The simplest dimer model is shown to be inappropriate for describing the transition, primarily because the hydrogens order into the wrong ground-state configuration. In the context of the dimer model, we introduce two exactly solvable modifications by postulating additional, physically plausible short-range interactions in the hydrogen bonding. The first model, which results from asymmetrically weighting the dimer positions, does have the observed ``frozen'' ground state and residual short-range order above Tc, but the transition is continuous. The second model introduces repulsive interactions between nearest-neighbor parallel dimer positions via the Baxter formulation of the dimer model in terms of the 8-vertex model. A suitable lattice transformation enables us to develop the model in terms of a generalized KDP model. This model has a first-order transition with a maximum allowable transition entropy of 0.876 cal/deg · mole, in excellent agreement with that determined from specific heat measurements. The higher energy hydrogen configurations are of the order of kT, indicative of the nature of the thermal excitations.

The pseudo-spin model and its application to KDP-type materials

Abstract The applicability of the pseudo-spin model is discussed in relation to the existence of constituents with a restricted number of low-lying energy states. The situation certainly occurs in materials with low-lying electronic levels which undergo co-operative Jahn-Teller phase transitions. The Slater-Takagi model of KDP is analysed to show that under certain approximations it leads to a double well in the collective co-ordinates with two levels in each unit cell and hence to a pseudo-spin model of one S = 1/2 per cell. The spin hamiltonian is the Ising model in a transverse field together with four spin interactions. The thermodynamic properties of this model are discussed and best values of the parameters obtained. The elastic constant anomalies are interpreted in terms of cs', in the adiabatic regime. The spin dynamics is discussed in terms of random phase approximation for spin waves, and more complete treatments are reviewed. Raman scattering data is analysed in terms of a coupled spin-phonon model both above and below Tc .

On the two-dimensional slater kdp model of a ferroelectric

On the two-dimensional slater kdp model of a ferroelectric

Modified KDP Model on the Kagomé Lattice

Modified KDP Model on the Kagomé Lattice