Lattice systems

Iiifomiation about the behaviour of the 3D Ising ferromagnet near the critical point was first obtained from high- and low-temperatnre expansions. The expansion parameter in the high-temperatnre series is tanli K, and the corresponding parameter in the low-temperatnre expansion is exp(-2A ). A 2D square lattice is self-dual in the sense that the bisectors of the line joining the lattice points also fomi a square lattice and the coefficients of the two expansions, for the 2D square lattice system, are identical to within a factor of two. The singularity occurs when... 

Chesnut D A and Salsburg Z W 1963 Monte Carlo procedure for statistical mechanical calculation in a grand canonical ensemble of lattice systems J. Chem. Phys. 38 2861-75... 

Very recently, the scientific interests of several leading theoretical laboratories have turned to studies of quenched-annealed fluids. To the best of our knowledge, there has not been a comprehensive review of the theoretical studies of quenched-annealed fluid systems. Our intention in this chapter is to fill, at least partially, an existing vacuum. Evidently, it is impossible to discuss the state of the art in this rapidly developing area in every detail in a single paper with restricted dimensions. We will omit, for example, the discussion of the fundamentals of the replica method for lattice systems, referring the reader to a monograph [1]. 

Pseudomorphism received methodical study from about 1905. A micro-section taken across the interface between a substrate and an electrodeposit shows the grain boundaries of the former continue across the interface into the deposit (Fig. 12.5). As grain boundaries are internal faces of metal crystals, when they continue into the deposit the latter is displaying the form of the substrate. Hothersall s 1935 paper contains numerous excellent illustrations with substrates and deposits chosen from six different metals, crystallising in different lattice systems and with different equilibrium spacing. Grain boundary continuation and hence pseudomorphism is evident despite the differences. 

The density of states is the central function in statistical thermodynamics, and provides the key link between the microscopic states of a system and its macroscopic, observable properties. In systems with continuous degrees of freedom, the correct treatment of this function is not as straightforward as in lattice systems - we, therefore, present a brief discussion of its subtleties later. The section closes with a short description of the microcanonical MC simulation method, which demonstrates the properties of continuum density of states functions. 

Escobedo, F. A. de Pablo, 1. J., Gibbs-Duhem integrations in lattice systems, Europhys. Lett. 1997, 40, 111-116... 

In an effort to understand the mechanisms involved in formation of complex orientational structures of adsorbed molecules and to describe orientational, vibrational, and electronic excitations in systems of this kind, a new approach to solid surface theory has been developed which treats the properties of two-dimensional dipole systems.61,109,121 In adsorbed layers, dipole forces are the main contributors to lateral interactions both of dynamic dipole moments of vibrational or electronic molecular excitations and of static dipole moments (for polar molecules). In the previous chapter, we demonstrated that all the information on lateral interactions within a system is carried by the Fourier components of the dipole-dipole interaction tensors. In this chapter, we consider basic spectral parameters for two-dimensional lattice systems in which the unit cells contain several inequivalent molecules. As seen from Sec. 2.1, such structures are intrinsic in many systems of adsorbed molecules. For the Fourier components in question, the lattice-sublattice relations will be derived which enable, in particular, various parameters of orientational structures on a complex lattice to be expressed in terms of known characteristics of its Bravais sublattices. In the framework of such a treatment, the ground state of the system concerned as well as the infrared-active spectral frequencies of valence dipole vibrations will be elucidated. 

Now we consider the two principal challenges presented by the systems under study. The first involves determination of the ground state for a lattice system of static dipole moments and implies Ho minimization over all possible orientations of the vectors eR,. Molecules are assumed to have uniform dipole moments, = n with arbitrary orientations, eR> in the absence of dipole-dipole interactions. On... 

Treating vibrational excitations in lattice systems of adsorbed molecules in terms of bound harmonic oscillators (as presented in Chapter III and also in Appendix 1) provides only a general notion of basic spectroscopic characteristics of an adsorbate, viz. spectral line frequencies and integral intensities. This approach, however, fails to account for line shapes and manipulates spectral lines as shapeless infinitely narrow and infinitely high images described by the Dirac -functions. In simplest cases, the shape of symmetric spectral lines can be characterized by their maximum positions and full width at half maximum (FWHM). These parameters are very sensitive to various perturbations and changes in temperature and can therefore provide additional evidence on the state of an adsorbate and its binding to a surface. 

The book covers a variety of questions related to orientational mobility of polar and nonpolar molecules in condensed phases, including orientational states and phase transitions in low-dimensional lattice systems and the theory of molecular vibrations interacting both with each other and with a solid-state heat bath. Special attention is given to simple models which permit analytical solutions and provide a qualitative insight into physical phenomena. 

Double pump experiments on an organic charge transfer complex TTF-CA by Iwai and coworkers demonstrated a new class of coherent control on a strongly correlated electron-lattice system [44]. While the amplitude of the coherent oscillation increased linearly with pump fluence for single pump experiments, the amplitude in the double pump experiments with a fixed pulse interval At = T exhibited a strongly super-linear fluence dependence (Fig. 3.16). The striking difference between the single- and double-pulse results indicated a cooperative nature of the photo-induced neutral-ionic transition. 

X-ray diffraction of pigment powder lends itself to the determination of pigment crystallinity. It is thus possible not only to determine the chemical configuration of a crystalline compound, but also the lattice system of the crystal through the diffraction pattern, in other words, the crystal quality size of crystallites, structural defects (Fig. 18). 

To take the above-mentioned ion-lattice coupling into account, the full ion-plus-lattice system must be considered, so that the static Hamiltonian given by Equation (5.2) must be replaced by... 

Rare earth consumption could, however, be affected by a change in host lattice system, with Eu3+ still being retained as activator. Either cost or luminescence efficiency could drive such a change. Costs could be decreased either by eliminating rare earth host lattice cations or by a decrease in the required concentration of Eu. However, in spite of considerable research effort, new host systems that accomplish these objectives have not been found. 

A recent extension of the fast-forward protocol to coherent control of site-to-site transfers in a lattice system [18] reveals that there is an inherent limit to which the rate of site-to-site transfer can be increased in the lattice system, while there is no such limit in a continuous system (in the nonrelativistic regime). The limit to the rate of site-to-site transfer is a consequence of the fact that the maximum phase difference allowed between the sites is limited to 2a . 

Population transfer between sites in a lattice is a ubiquitous phenomenon in condensed matter physics. Because a lattice space has a different symmetry than does continuous space, protocols to alter the evolution of population in a lattice system with an external field must differ slightly from the protocols used for the same purpose in continuous space. With this observation in mind, Masuda and Rice extended the fast-forward protocol to apply to lattice systems and applied then-formalism to enhancing site-to-site particle transfer in a BEC [ 18]. We now briefly review that formalism. 

As before, the time dependence of R(t) is chosen to satisfy the boundary conditions R(0) = Ri,R Tp) = Rp anddR 0)/dt = dR(Tp)/dt = 0, so that the additional phase vanishes at the initial and the final times. A comparison of Eqs. (3.20) and (3.39) reveals a remarkable difference between the acceleration possible in a lattice system and that in a continuous system, in that there is a lower limit to Tp in a lattice system, whereas there is no such limits for a continuous system. Thus, for a lattice system, there is also a higher limit to dR/dt that depends on cp , because the maximum phase difference allowed between the sites is limited to 2jt. 

The Fast-Forward Protocol in a Many-Body Lattice System... 

An interesting application of the fast-forward protocol is site-to-site population transfer of particles in a BEC confined on an optical lattice. As shown below, the features of that protocol differ somewhat between continuous and lattice systems. We consider a BEC subjected to an external potential that is a superposition of a lattice potential and a harmonic potential, with the representation... 

Figure 3.36 Schematic diagram of the lattice system and the harmonic potential. The solid and dashed curves show the harmonic potential at = 0 and t = respectively.

S. Masuda and S. A. Rice. Rapid coherent control of population transfer in lattice systems. Phys. Rev. A, 89(3) 033621-033627(2014). 

Equation (5.104) expresses the detailed balance in the closed lattice system. If the crossing from one plane to the next is thermally activated, so that v= v0-e i p)/kT, Eqn. (5.104) takes on the following form... 

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