Mori Formalism Research Articles

Magnetic relaxation in highly anisotropic systems

Using the Mori formalism a general theory is developed for the calculation of paramagnetic relaxation effects in a general crystal field doublet. For systems with extreme magnetic anisotropy, the relaxation rates show a strong angular dependence, being severely peaked in a direction perpendicular to the magnetic easy axis. Application to the RRh 4B 4 (R = rare-earth) compounds is discussed.

Nuclear response function in the Mori formalism.

The Mori formalism is used to study the nuclear response function at finite temperatures. This formalism rather naturally leads to an extension of the zero-temperature random-phase approximation for finite temperatures, and it provides a prescription for the calculaton of the damping widths of vibrational states at finite temperatures.

Dynamics of mixed valent Europium and Samarium systems

For mixed valent Europium and Samarium systems dynamical magnetic and charge susceptibilities are calculated at temperatures above their fluctuation temperature using Mori's formalism. The low-lying spin-orbit excited level of the 4f6-configuration leads to additional relaxation modes and to conspicuous temperature effects.

Shubnikov-de haas effect in kinetic approximation

The transverse and longitudinal magneto-conductivity of an electron-impurity system is calculated using a kinetic approximation based on the Mori formalism. The numerical results agree qualitatively with measurements on narrow gap semiconductors.

Generalized Langevin equation versus the equation-of-motion method for an interacting many-electron system.

Mori's formalism [Prog. Theor. Phys. 33, 413 (1965)] as applied to an interacting many-electron system is shown to lead to the random-phase-approximation (RPA) form of the dynamic susceptibility in the limit of small momentum transfer where RPA is a valid theory.

Binary collisions in fluids: II: velocity auto correlation function

A calculation based on the Zwanzig–Mori formalism and the binary collision expansion is presented for the velocity auto correlation function (VCF) of simple classical fluids. The binary collision expansion is truncated after the second term, thus incorporating exactly all the effects of a single collision between two particles, and this expansion is applied to the memory function associated with the VCF. The only inputs required are the pair distribution function g(r) and a central potential U(r). The potential chosen is the Lennard–Jones potential. It is seen that the approximation of the memory function by the binary force auto correlation function yields results for VCF that are in very good agreement with molecular dynamics results over a wide range of densities from dilute gases to liquids.

Influence of nondiagonal dynamic susceptibility on the EPR signal of Heisenberg magnets

In EPR with linearly polarised microwaves if just the diagonal contributions chi ++( omega ) and chi --( omega ) of the dynamic susceptibility are considered then a discrepancy occurs between the experimentally observed signal and the predictions of theory. By means of the Mori formalism the authors have calculated the non-diagonal contributions chi +-( omega ) and chi -+( omega ) for insulators with dominating isotropic exchange. For rather small linewidth these terms result in a new kind of dynamic shift of the resonance field, whereas for broad lines they give rise to a drastic anomaly of lineshape. All these effects could be verified experimentally in both one- and three-dimensional spin lattices of axial symmetry in the following ways: the dynamic shift expected for narrow lines could be observed in TMMC; lineshape anomalies were studied in the very broad line of CsNiF3; and in MnF2 the critical increase of linewidth near TN allowed variation between the two extreme cases.

Towards a molecular theory of the solid—liquid phase boundary. Elementary kinetic processes

AbstractStarting from the molecular, structural theory of the f.c.c. crystal—liquid phase boundary for onecomponent Lennard‐Jones systems near the triple point given by the author, the elementary kinetic processes of particles undergoing solid—liquid transitions are studied, and the absolute transition time is determined which is about 10−10 s. Starting with calculations of the mean square of the fluctuating force for extremely inhomogeneous systems such as the interface, and calculations of a space‐dependent friction coefficient via generalizations of the Mori formalism, the investigations are based on a description of the stochastic single‐particle motion by means of the Smoluchowski equation. It is shown that the stochastic single‐particle motion is closely related to the cell dynamics of stochastic lattice models; the corresponding absolute transition probabilities are calculated.

Calculation of relaxation functions: A new development within the Mori formalism

Calculation of relaxation functions: A new development within the Mori formalism

Dynamic response of a two-dimensional dense electron gas: Equivalence of random-phase-approximation and approximate-memory-function approach

It is shown that the expressions for the polarizability and intrinsic conductivity of a two-dimensional (2D) dense electron gas, obtained recently by Lee and Hong with the use of Mori formalism, can be derived more simply within the same scheme and that they are equivalent to the corresponding expressions in the random-phase approximation in the limit of small wave number, i.e., $k\ensuremath{\rightarrow}0$ with $\frac{\ensuremath{\omega}}{k}$ finite. Similar expressions have been derived for the three-dimensional and one-dimensional electron gas. In the former the behavior of the intrinsic conductivity is similar to that in the 2D case, whereas in the latter it is very different.

ON THE BROWNIAN MOVEMENT OF A ROTATOR AND ITS APPLICATION TO NONLINEAR DIELECTRIC RELAXATION

The Debye theory of dielectric relaxation as corrected for inertial effects has as yet been only considered in the linear approximation. There, the rise and decay transients are identical. Here a method recently developed for the treatment of a rotator in a periodic potential is applied to calculate the transient behaviour when the linear approximation is discarded. The Kramers equation for the problem is expanded in a set of orthogonal functions which lead to a set of linear differential difference equations giving the relaxation behaviour. It is shown that the Mori formalism for the problem leads to the same set of differential difference equations as the Kramers equation.

Sum rules and dynamical properties of two-dimensional classical electron liquid

Expressions for the fourth frequency moment sum rule for the longitudinal and transverse current correlation functions are obtained for a classical system of electrons trapped on the surface of liquid helium. The expressions involving static triplet correlation functions are considerably simplified so that the numerical results for the moments can be easily obtained for all momentum transfers. The results for the moments as a function of wavenumber k are presented for 2.23<or= Gamma <or=90.0. The authors make use of these results to analyse the recent molecular dynamics (MD) data on the dynamical structure function S(k, omega ) and the transverse current correlation function Ct(k, omega ) using the Mori formalism. It is found that a relaxation kernel essentially with single decay constant provides an overall good description of MD data over the entire range of Gamma .

Nuclear dissipation with residual interactions studied by means of the Mori formalism

We apply correlation function techniques to the calculation of nuclear friction within the framework of a linear response theory. We make use of the fluctuation dissipation theorem to relate the response function to the correlation function which is evaluated by exploiting projection operator techniques. We go beyond the one-body dissipation approximation in the sense that we have taken into account the decay of particle-hole excitations into more complicated configurations. A rather simple formula for the frequency and temperature dependence of the friction coefficient is derived which we have applied to the high energy fission of238U. The friction coefficients for deformations around the first and second minima of the fission barrier have been calculated using this approach.

Dielektrisdie Verluste starrer polarer Molekule in verdiinnter Losung bei Mikrowellen und im Fernen Infrarot

Abstract With regard to their Debye and Poley absorption the dielectric loss of twelve species of rigid polar molecules has been measured in very dilute solutions (n-heptane cyclohexane and decalin) at frequencies from 300 MHz to 5 THz. Within the limits of experimental error for t-butylchloride no Poley absorption is observed. Its ε" (ω) plot may be fitted using the three-variable Mori formalism because e" finally decreases with the fifth power of frequency. The absorption data of all other dipole molecules cannot be fitted by a single Mori curve. Therefore we propose to separate the Debye absorption approximating the low frequency wing and the predominant loss region as done with t-butylchloride. In this way the dispersion step of the Poley absorption itself is obtained, and it is discussed according to the libration model. The observed step is too large in benzophenone only perhaps due to the fast orientation of a mesomeric moment by a ring twist. The Poley ε" (ω) data are formally matched by one or two three-variable Mori curves, and the frequency behaviour of the dipole ensemble is discussed in terms of librations damped by soft collisions, using the absorption coefficient α (ω)

Dynamics of proton-lattice interaction in hydrogen-bonded ferroelectrics

Abstract The dynamics of proton-lattice interaction in hydrogen-bonded ferroelectrics are studied using Mori's formalism. Detailed expressions for the various spin and photon relaxation functions are obtained. Using approximate static correlation functions, the behaviour of the phonon relaxation function is discussed.

Vibrational correlation-function modeling in the context of the Zwanzig–Mori formalism. Application to liquid ethane

The relationship between the memory-function approach to correlation-function modeling and the Mori continued-fraction scheme suggests that unsuspected dynamical couplings may require higher-order procedures to adequately model some correlation functions. The application of first-, second-, and third-order memory-function procedures and their significance is discussed. Some guidelines for the experimentalist are given, and the problem of the truncation of isotropic Raman bands is discussed. The spontaneous Raman-scattering spectrum of the ν3 band of C2H6 has been reexamined for sample temperatures in the range 93–168 K. Both the isotropic and anisotropic spectra have been obtained, and the vibrational-correlation function has been modeled within experimental error. The need to go to third order in the modeling procedure and the values of the second moments of the IVH spectra suggest a vibrational–rotational coupling. The depolarization ratio for the ν3 band was observed to be 0.21.

The mobility of a quantum particle in a three-dimensional random potential

Abstract The approximation scheme recently proposed for the dynamical conductivity and density fluctuation spectrum of a quantum particle moving in a random potential is rederived within Mori's formalism. The equations are solved by asymptotic expansions (i) for small coupling to study the corrections to the kinetic equation approach for the mobility, (ii) for strong coupling to analyse the insulator instability caused by increasing spontaneous fluctuations, and (iii) for the regime close to the Anderson phase transition from a state with metallic conductivity to an insulating one. The equations are solved numerically and an extensive discussion of the solutions is presented.

Diffusion in a deformable lattice: Theory and numerical results

We develop a transport theory of one-dimensional diffusion of a single particle in a three-dimensional deformable lattice with a Debye spectrum. Our treatment is based on the Mori formalism; the memory kernel is evaluated in the mode-mode coupling approximation. This allows us to evaluate the absolute magnitude of the diffusion constant, the frequency-dependent conductivity, and the incoherent neutron scattering cross section of the diffusing particle. The main results are: (a) The diffusion constant obeys an Arrhenius law at very low temperatures. At higher temperature (${k}_{B}T\ensuremath{\gtrsim}\frac{\ensuremath{\Delta}E}{4}$, $\ensuremath{\Delta}E$ is the activation energy) the apparent activation energy increases with temperature. This increase depends on the mass ${M}^{A}$ of the diffusing particle leading to an apparent activation energy that increases slightly with decreasing ${M}^{A}$. The prefactor (attempt frequency) shows only a very small dependence on ${M}^{A}$ for not too small masses. As a result, the diffusion coefficient is nearly independent of ${M}^{A}$. Only for small ${M}^{A}$ we recover the isotopic effect as predicted, for instance, by the theory of Vineyard. (b) The frequency-dependent conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ shows a two-peak structure and a temperature dependence which is similar to that obtained in the Fokker-Planck treatments. The frequency of the upper peak of $\ensuremath{\sigma}(\ensuremath{\omega})$ exhibits a square-root dependence on ${M}^{A}$ whereas $\ensuremath{\sigma}(\ensuremath{\omega})$ depends very little on ${M}^{A}$ at low frequencies. Detailed results for the dependence on the elastic constant of the crystal and the masses of the particles are given. (c) Analogous results and plots have been obtained for the incoherent cross section of the diffusing particle.

Dynamical susceptibilities of intermediate valence compounds

Relaxation rates of magnetic moment and charge are discussed for systems showing intermediate valence. The periodic Anderson model and a more realistic model with periodic 4f shells are investigated by the use of the Mori formalism. It is shown that the relaxation of the 4f moment and the charge above the “spin fluctuation” temperature is dominated by the second order process with respect to the hybridization interaction and is nearly independent of temperature. The moment relaxation rate is significantly larger than that in the integral valent case where the fourth order Korringa process is dominant. The theory can explain the anomalously large width and the momentum independent nature of the quasi-elastic component of the neutron scattering on intermediate valence compounds.

Central peak due to thermal diffusion mode in the tunnelling model

Coupling of the soft mode to a thermal diffusion mode is investigated near the structural phase transition on the basis of the Mori formalism (1965) applied to the tunnelling model. These modes couple linearly in the ordered phase through the order parameter, which produces a narrow central peak in the structure factor. It is also shown that the ratio of integrated intensities of the central peak to the soft mode in the ordered phase satisfies the generalised Landau-Placzek relation. These modes in the disordered phase do not couple linearly, so the mode-mode coupling theory is used in order to study the non-linear coupling which produces a logarithmic singularity in the response function to given an extremely narrow central peak. The numerical value of the coupling constant is estimated from experimental values and is shown to be sufficiently large, at least in the ordered phase, to observe the central peak due to the thermal diffusion mode. When the soft mode is one-dimensionally anisotropic, a pole-like singularity is found in place of the logarithmic one.