On the kinetics of a two-level system with a diffusion-modulated interaction

For a two-level system corresponding to a particle of spin ½ in a random field in the Z direction, the relaxation function

has been estimated, the magnitude Ω(t) being the sum of the isotropic interactions of the particle in question with particles j executing diffusional motion. Specifically Ω(t)=ω₀ + Σ ω(Mj, rj), where ω₀ = constant, M_j is a random time-independent parameter, ω(M, r) decreases with r faster than r ^-3 and r _j = r _j (t) is a diffusional, random function. From the expression for <σ⁺(t)>, we establish general features of the relaxation phenomenon for diffusional processes, and calculate the relaxation rate 1/T ₂ and relaxation shift Δω to be 1/T ₂-iΔω = 4π CAv_M λ _M , where C is the concentration of particles and λ _M is the scattering length for an equation of the Schrödinger type with an imaginary potential -iω(M, r) instead of U/?, and diffusion coefficient instead of ?/2m. We also found that for the case of ‘external’ relaxation, the Redfield approach proved valid only under the simultaneous restrictions of low concentration and weak interaction.     

On the Taylor expansion of the expression for the transition probability for normal absorption processes

Molecular reorientation of 2-chloropyrimidine dissolved in CS₂ (0·1 M) has been investigated by means of ¹³C and proton relaxation. Although weakly coupled, the proton system subjected to non-selective 180-τ-90 pulse sequences allows the determination of one autocorrelation and one cross-correlation dipolar spectral density. The proton and carbon-13 relaxation data allow the complete determination of the rotational diffusional tensor:

and

It is shown that scalar relaxation due to nitrogen-14, has no effect on proton longitudinal relaxation time, because of a cross term due to the symmetry of the molecule, although this mechanism could, a priori, have been thought to be important. Finally, the nitrogen relaxation time recalculated with D_xx , D_yy , D_zz and the quadrupolar coupling tensor is in agreement with the observed linewidth.     

Molecular dynamics calculation of the dielectric constant without periodic boundary conditions. I

We outline the difficulties in obtaining a reliable value of the dielectric constant of a fluid using molecular dynamics calculations with periodic boundary conditions, and give some explanation of the observed asymptotic behaviour of h_D (r) and hΔ(r) in Monte Carlo simulations of dipolar hard spheres. An alternative method consisting in simulating a dielectric in vacuum is described. This is applied to two dimensional systems. The pertinent theoretical relations for a dielectric disc in vacuum are therefore derived. It is concluded that relations involving MC or MD computation of <m ²> must be carefully handled.     

A closed-form solution of the Schroedinger equation describing the harmonic oscillator with time dependent frequency and acted on by a time dependent perturbative force

A new form of the semiclassical quantum conditions in non-separable systems is proposed. In two dimensions (2D) it has the form (? = 1)

where C_Σ is the path of a classical trajectory closed in phase space, N_x and N_y are the number of circuits in the x and y ‘senses’ on the invariant toroid and J_x and J_y are the ‘good’ action variables on the toroid; these action variables, J_x and J_y , must have the values 2π(n ₁ + ½) and 2π(n ₂ + ½) respectively where n ₁ and n ₂ are the integer quantum numbers. Closed classical trajectories occur only for the exceptional toroids with rational frequency ratios. In the general case we imply that the trajectory has closed on itself to some arbitrary accuracy. Results for the 2D potentials studied are in agreement with previously published work. It is shown how the method may be extended to 3D systems.     

On Integration of the Nonlinear d’Alembert-Eikonal System and Conditional Symmetry of Nonlinear Wave Equations

Abstract

We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d’Alembert equation □u = F (u) and nonlinear eikonal equation u _xµ u _x µ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility conditions and construct a general solution of the d’Alembert-eikonal system for all cases when it is compatible. The results obtained can be applied, in particular, to construct principally new (non-Lie, non-similarity) solutions of the non-linear d’Alembert, Dirac, and Yang-Mills equations. Solutions found in this way are shown to correspond to conditional symmetry of the equations enumerated above. Using the said approach, we study in detail conditional symmetry of the nonlinear wave equation □w = F ₀(w) in the four-dimensional Minkowski space. A number of new (non-Lie) reductions of the above equation are obtained giving rise to its new exact solutions which contain arbitrary functions.     

Weak and Partial Symmetries of Nonlinear PDE in Two Independent Variables

Abstract

Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and partial symmetries introduced by the author are analyzed. For a family of nonlinear heat equations of the form u _t = (k(u) u _x)_x + q(u), pairs of functions (k(u), q(u)) are pointed out such that the corresponding equations admit nontrivial two-dimensional modules of partial symmetries. These modules yield explicit solutions that look like u(t, x) = F (θ(t) x + φ(t)) or u(t, x) = G(f(x) + g(t)).     

Dielectric properties of a mixture of dipolar hard spheres

A theory for the dielectric constant, ε, of a fluid mixture of dipolar hard spheres is formulated by generalizing the methods developed by Ramshaw and Wertheim for the pure fluid case. The resulting expression for ε depends on the pair distribution functions, g _αβ(r ₁, θ₁, r ₂, θ₂) for a dipolar mixture. Due to the unavailability of exact representations for these dipolar pair distribution functions, the results of the mean spherical approximation are employed in the formalism developed. Numerical results are given for ε as calculated from the pair distribution functions for a spherical volume of macroscopic dimensions. The compositional dependence of the ε obtained in this way for a specific mixture is compared with the corresponding properties of the well established theories of Clausius-Mossotti-Debye and Onsager. In addition, the relative importance of the dipole moment and size of the hard sphere parameters in determining ε for a dipolar mixture (the correlative behaviour of which is described by the mean spherical approximation) is evaluated. It is found that the differences in hard core diameters can be largely ignored, in that ε for an ‘effective’ single component fluid can be given to within 2–5 per cent relative error (at worst) of the mean spherical approximation's result. Such an ‘effective pure fluid’ is described as having the same polarization content as the actual mixture being considered. Thereby, the properties of the effective fluid are determined by the quantity y = 4πβ(m ₁ ² ρ₁ + m ₂ ² ρ₂)/9 where m_i and ρ _i are the dipole moment and number density of component i in the binary mixture, with β = (kT)^-1.     

Molecular orientation correlations and reorientational motions in liquids carbon monoxide,nitrogen and oxygen at 77 K; A Raman and Rayleigh light-scattering study

Reorientational autocorrelation functions have been determined from measurements of depolarized vibrational Raman scattering for liquid carbon monoxide, nitrogen and oxygen at 77 K and atmospheric pressure. The autocorrelation functions, which for these liquids are not significantly affected by vibration-rotation interaction, reveal that free rotation is an important feature of the molecular motion in liquid nitrogen but is less important for carbon monoxide and oxygen. The differences in behaviour are discussed in terms of intermolecular forces.

New values for the depolarized Rayleigh scattering cross section have been determined from intensity measurements made relative to the 992 cm^-1 Raman line of benzene. These values are compared to those reported previously by the authors using a different intensity standard (Chem. Phys. Lett., 31, 355 (1975)). The scattering cross sections yield the following values , where ?_ij is the angle between the major axes of molecules i and j (i≠j) and P ₂ indicates the second Legendre polynomial: -0·15 ± 0·2 for CO, +0·30 ± 0·2 for N₂ and +0·40 ± 0·2 for O₂. The large errors result from uncertainties in the local field correction factor. The negative value for CO can be explained as a result of strong quadrupole interactions which tend to align neighbouring molecules perpendicular to one another. The forms of the reorientational cross-correlation functions determined from the current Raman data and previous Rayleigh data are briefly discussed.     

Nuclear magnetic spin-lattice relaxation and molecular motion in solid white phosphorus and in liquid phosphorus

The ³¹P spin-lattice relaxation rates have been measured in solid white phosphorus and in liquid phosphorus over the temperature range 110 K to 400 K and at Larmor frequencies of 10 MHz and 30 MHz. The contributions to the measured relaxation rate from the different interactions have been separated. In the low-temperature, crystalline phase there are important contributions to the relaxation rate from the anisotropic chemical shielding and the intramolecular dipole-dipole interactions which are modulated by the reorientational motion of the molecule. Interference effects between these two interactions, which are important in liquids, are demonstrated to be quenched by the strong dipolar interactions in the solid. The reorientational correlation time is given by

and the chemical shielding anisotropy by

In the high-temperature, plastic-crystalline phase the reorientational correlation time is

as obtained from the anisotropic chemical shielding relaxation rate which is separated from the other contributions by its quadratic dependence on the Larmor frequency. Using this τ _R the intramolecular dipole-dipole relaxation rate is calculated. The contribution from the translational diffusion modulated intermolecular dipole-dipole interaction is calculated from the self-diffusion coefficient. When these contributions are subtracted from the observed relaxation rate, there remains a frequency-independent relaxation rate, proportional to 1/δ _R , which is attributed to the spin-rotational interaction. The latter is shown to be quantitatively consistent with large-angle reorientational jumps of the P₄ molecules by 120° about their C _3v axes. The relaxation in the liquid phase is dominated by the spin-rotational interaction and the expression representing the spin-rotational relaxation rate is the same as the one derived in the plastic-crystalline phase. The mechanism of molecular reorientation in the liquid is therefore the same as in the plastic-crystalline phase.