Surface critical exponents using the renormalization group ϵ-expansion

A Ginzburg-Landau-Wilson (GLW) model extended to include surface energy terms is used to discuss the surface critical behaviour of a system of interacting spins. A straightforward extension of the bulk renormalization group transformation is worked out to first order in ε = 4 ? d (where d is the number of spatial dimensions). In contrast to previous work, the coefficients of the quadratic (r) and quartic (u) terms in the GLW free energy are allowed to depend on the distance from the surface. To lowest order in ε, we show that the fixed point $\text{u}{}^1\text{(z)}$ is given by the usual bulk value. The RG recursion formula we obtain has a line of possible fixed points $\text{r}{}^1\text{(z)}$ associated with both the “special” and “ordinary” transitions. The fact that the GLW model can only describe systems with short-range interactions allows one to select the correct fixed point, in which $\text{r}{}^1\text{(z)}$ is given by its bulk value up to atomic distances from the surface. We discuss the ordinary and special transitions in a unified way. The correlation functions involving surface spins are worked out and the critical exponents η_| and η_⊥ are evaluated to O(ε). Our results agree with those of Lubensky and Rubin (ordinary transition) and Bray and Moore (special transition), who used less transparent methods of carrying out the RG transformation.     

Quantum theory of pure liquid metals as two-component systems

With the ultimate aims of clarifying the interpretation and the utility of effective ion-ion interactions in liquid metals, and of understanding the unusual isotopic mass dependence of the shear viscosity of liquid metal Li, a fully quantum statistical mechanical theory is developed from the many-body Hamiltonian of the conduction electron-positive ion assembly.We have set up quantum equations of motion which are analogs of classical continuity and conservation equations by expanding the equation for the Wigner distribution function about its diagonal. The most important of these equations for our present purposes relates the time derivative of the current density j(r, t) to the flux of current and to density-density correlation functions for electrons, electron-ions, and ions.This theory is then applied to neutron scattering by liquid metals. While the theory is sufficiently general in principle to treat electron-ion interaction of arbitrary strength, it is shown that when the interacion is weak, the usual results are recovered, along with the effective ion-ion interaction. In this latter connection, it is also demonstrated how the effective Ornstein-Zernike function C(q) in a liquid metal is related to bare ion and bare electron direct correlation functions and to the bare electron partial structure factor. Combining C(q) with one of the classical equations of liquid structure such as Born-Green or Percus-Yevick then relates the effective ion-ion interaction to the partial correlation functions of the bare ions and electrons.It is further shown how gradient expansions of the correlation functions lead to equations of motion for the density, current, and energy density which are simply the hydrodynamic equations of the present quantum theory of two-component systems. It is pointed out that the analog of the Navier-Stokes equation for the two-component system may be used to identify the quantity $\text{4}\text{3}\text{η + ζ}$ for the liquid metal, η and ζ being respectively the shear and bulk viscosities. Finally, it is demonstrated that $\text{4}\text{3}\text{η + ζ}$ depends explicitly on functional derivatives of the nonequilibrium pair distribution functions of ion-ion, electron-ion, and electron-electron correlations.     

Microwave spectrum of 2-propene-1-imine,CH2CHCHNH

The reactive species, 2-propene-1-imine, has been identified by its microwave spectrum as a pyrolysis product of diallylamine vapor (100 mTorr, 400°C). Two entirely planar forms are observed, both with an “s-trans” CCCN configuration. The lower energy rotamer has an “anti” CCNH configuration, with $\text{r}\text{CC = 1.453(3)}\text{A}\text{?}$, $\text{r}\text{CC = 1.336(4)}\text{A}\text{?}$, and ∠ CCC = 122.9(3)° and a dipole moment of 2.01(2) D with 1.13(1) D and 1.66(2) D “a” and “b” components. The higher energy rotamer has a “syn” CCNH configuration and a dipole moment of 2.51(2) D with 2.39(2) and 0.77(3) D the “a” and “b” components. From relative intensity measurements, the ground state energy difference is determined to be 0.9 ± 0.1 kcal mole^?1.     

Quantizing a classically ergodic system: Sinai's billiard and the KKR method

Sinai's “billiards on a torus,” i.e., free motion of a particle in a plane amongst reflecting discs of radius R centred on points of the unit square lattice, is a classically ergodic system with two freedoms, parametrized by R. Quantal energy levels E_n are given by the vanishing of the Korringa-Kohn-Rostoker (KKR) determinant of solid state theory. This gives a rapid computational scheme for computing E_n as functions of R. Except for the integrable case R = 0, no degeneracies are found, illustrating the theorem that two parameters, not one, are required to make levels cross in a generic system. The same theorem leads to the prediction that the probability distribution of the spacings S of neighbouring levels is $\text{O}$(S) as S → 0, in good agreement with computation. The KKR determinant is transformed analytically to give the level density d(E) semiclassically (i.e., as ? → 0) as the sum of a steady contribution $\text{d}\text{?}\text{(E)}$ and an oscillatory contribution d_osc(E). $\text{d}\text{?}$ is $\text{O}$(?^?2) and is given by the Weyl “area” formula plus “edge,” “corner” and “curvature” corrections, in excellent agreement with computation. d_osc is given by a sum over classical closed orbits (all unstable). Nonisolated closed orbits (not hitting discs) contribute terms with $\text{O}$$\text{(?}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext><mtext>)</mtext>}}$ to d_osc, while isolated closed orbits (bouncing between discs) contribute terms with $\text{O}$(?^?1) to d_osc. The isolated orbits are vastly more numerous than the nonisolated orbits and their contributions cannot be neglected. As a means of calculating the individual E_n (rather than the smoothed spectrum), the KKR method is much more efficient than the classical path sum.     

On the bifurcations occurring in the parameter space of the sixteen vertex model: II. Bifurcations arising from degeneracies in the N-matrix

The analysis of the bifurcation structure in the 16-dimensional parameter space of homogeneous (sixteen-) vertex models, started in part I, is completed in this paper. As before equivalence classes of models having the same partition function are identified by means of a 4 × 4 diagonal matrix N and a pair of characteristic (2 × 2)-matrices A and B. The bifurcation classes of models studied in this part include also classes for which the N-matrix shows degeneracies.The four primary bifurcation classes, uncovered in part I, i.e. the general- and complementary eight-vertex models and the one sided- and doubly cyclic models, each give rise to a hierarchy of two kinds of subbifurcations: on the one hand the models corresponding to the different patterns of degeneracies of the N-matrix, and on the other hand the models for which the “rotation angles” α and β characterizing the matrices A and B have values $\text{α =}\text{2π}\text{n}{}_{\mathrm{a}}$ and $\text{β =}\text{2π}\text{n}{}_{\mathrm{b}}$ respectively, with n_a and n_b integers.     

Time evolution of a quantum mechanical maser model

The time evolution of the Dicke maser model describing N spins ($\text{s =}\text{1}\text{2}$) interacting by a dipole coupling with one mode of an electromagnetic field, is studied for finite N. The mean photon number and its mean square deviation can be calculated as functions of time for various initial states. For not too large times, these quantities show a periodic behavior given by elliptic functions.     

Proposal of a new model for the modulated structure of α-uranium

The modulated structure of α-uranium is analysed in the framework of the theory of superspace groups. Arguments are given which lead to the (3 + 2)D superspace group Pmcm($\text{1}\text{2}$βγ) as the most probable one. The consequences for the structure are derived. The most important one is, that there are two independent atoms in the unit-cell, one with a displacement along b and c and the other with a displacement only along a. This model is quite different from that previously found, which has only one independent atom, and a modulation in all three directions.     

Pseudoscalar mesons in the c-c and b-b systems

It is shown that the pseudoscalar η_c and “η_c” (η_b and “η_b”) with M ≈ 2.80 and 3.51 GeV (9.17 and 9.88 GeV) can be predicted by using the radial mixing model, where the “η_c” and “η_b” are the excited 0^- states in the $\text{c-}\text{c}$ and $\text{b-}\text{b}$ systems, respectively, and the former would correspond to the observed 0^- meson with M ≈ 3.45 GeV in charmonium.     

Planar limit of the singlet spectrum for SU(N)-invariant quantum hamiltonians by the quantum collective field method

We calculate the quantum fluctuations around the classical vacuum of the quantum collective field in the case of SU(N)-invariant quantum mechanical systems. We obtain in this way the singlet spectrum up to corrections of order $\text{1}\text{N}$. Difficulties in the calculation of higher order terms in $\text{1}\text{N}$ are pointed out.     

Long-wavelength excitations in a Bose gas at zero temperature

The long-wavelength excitations in a simple model of a dilute Bose gas at zero temperature are investigated from a purely microscopic viewpoint. The role of the interaction and the effects of the condensate are emphasized in a dielectric formulation, in which the response functions are expressed in terms of regular functions that do not involve an isolated single-interaction line nor an isolated single-particle line. Local number conservation is incorporated into the formulation by the generalized Ward identities, which are used to express the regular functions involving the density in terms of regular functions involving the longitudinal current. A perturbation expansion is then developed for the regular functions, producing to a given order in the perturbation expansion an elementary excitation spectrum without a gap and simultaneously response functions that obey local number conservation and related sum rules.Explicit results to the first order beyond the Bogoliubov approximation in a simple one-parameter model are obtained for the elementary excitation spectrum ω_k, the dynamic structure function $\text{S}$(k, ω), the associated structure function $\text{S}$_m(k), and the one-particle spectral function $\text{A}$(k, ω), as functions of the wavevector k and frequency ω. These results display the sharing of the gapless spectrum ω_k by the various response functions and are used to confirm that the sum rules of interest are satisfied. It is shown that ω_k and some of the $\text{S}$_m(k) are not analytic functions of k in the long wavelength limit. The dynamic structure function $\text{S}$(k, ω) can be conveniently separated into three parts: a one-phonon term which exhausts the f sum rule, a backflow term, and a background term. The backflow contribution to the static structure function $\text{S}$₀(k) leads to the breakdown of the one-phonon Feynman relation at order k³. Both $\text{S}$(k, ω) and $\text{A}$(k, ω) display broad backgrounds because of two-phonon excitations. Simple arguments are given to indicate that some of the qualitative features found for various physical quantities in the first-order model calculation might also be found in superfluid helium.     

A probabilistic analysis of the difficulties of unifying quantum mechanics with the theory of relativity

A procedure is given for the transformation of quantum mechanical operator equations into stochastic equations. The stochastic equations reveal a simple correlation between quantum mechanics and classical mechanics: Quantum mechanics operates with “optimal estimations,” classical mechanics is the limit of “complete information.” In this connection, Schrödinger's substitution relationsp _x → -i? ?/?x, etc, reveal themselves as exact mathematical transformation formulas. The stochastic version of quantum mechanical equations provides an explanation for the difficulties in correlating quantum mechanics and the theory of relativity: In physics “time” is always thought of as a numerical parameter; but in the present formalism of physics “time” is described by two formally totally different quantities. One of these two “times” is a numerical parameter and the other a random variable. This last concept of time shows all the properties required by the theory of relativity and is therefore to be considered as the relativistic time.     

Comments on the Kolmogorov-Sinai-Sąsiada entropy and the quantum information theory

Commenting the recent generalization by S$\text{a}\text{?}$siada of the Kolmogorov-Sinai entropy to the quantum case (KSSentropy), it is remarked that this entropy refers to the process of evolution as a whole and to the initial state (t = 0), not to the state at any time (t ? 0). Therefore, the KSS entropy has no direct relation to the von Neumann entropy or A-entropy at time t. Secondly, the proof of the no-increase theorem of S$\text{a}\text{?}$siada (referring to the initial time) is valid only for the Markov type of time evolution, while the KSS entropy can be generalized to time evolution with arbitrary time correlations. Some important consequences of the new concept for the formulation of the quantum information theory are also presented.     

Can the optical excitations of surface plasmons be used to study (liquid) metal surfaces?

The surface plasmon dispersion relation is obtained for a metal with a free electron density given by N(z) = N_b + (N_s ? N_b) exp $\text{(}\text{?z}\text{a}\text{)}$ for z ? 0 and N = 0 for z < 0. We have used a local theory which includes the effects of retarded fields and found the dispersion relation to be sensitive to the parameters $\text{(N}{}_{\mathrm{s}}\text{? N}{}_{\mathrm{b}}\text{)}\text{N}{}_{\mathrm{b}}$ and a, which characterize our density profile.     

On the bifurcations occuring in the parameter space of the sixteen vertex model: I. Discrete bifurcations

The 16-dimensional parameter space of homogeneous sixteen vertex models is scanned for bifurcation points, i.e. points corresponding to models which possess extra symmetries not existing in nearby points. Equivalence classes of models having the same partition function are identified by means of a characteristic “normal” model, represented by a (4 x 4)-diagonal matrix N, and a pair of (2 x 2)-matrices A and B. In this paper the matrix N is assumed to be non-degenerate and the only bifurcations found are those associated with special types of matrices A and B, i.e. matrices whose decomposition in terms of Pauli-matrices corresponds to a vector a ≡ (a₁, a₂, a₃) or b ≡ (b₁, b₂, b₃) that is invariant with respect to one or more elements of the cubic symmetry group.The various bifurcation classes of models found include: a) the families of general- and “complementary” eight vertex models, b) discrete sets of doubly- and one-sided cyclic models and c) a number of secondary bifurcation classes within the eight-vertex families, among which is Baxters symmetric eight-vertex model.     

Two-dimensional decorated Ising model with classical vector spins and Ising spins of magnitude s

A two-dimensional decorated Ising model with ν-dimensional vector spins and Ising spins of magnitude s is considered. The partition function, magnetization and correlation functions are expressed in terms of the average of functions of the spins of the Ising model with effective exchange constants. These results, although derived for a two-dimensional lattice, are valid for a lattice of arbitrary dimensionality. The phase diagram is obtained exactly in the zero external field and two-dimensional lattice for arbitrary values of s and ν, and, as expected, three transition temperatures are obtained for some values of the parameters. It is also shown that for $\text{|}\text{S}\text{|=1, s>}\text{1}\text{2}$ there is an additional ordered phase (up-down/up-down), and for $\text{|}\text{S}\text{|=ν}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ this additional phase can be either up-down/up-down or up-up/down-down depending on the values of ν and s.     

The nature of the vibronic coupling in a metalloporphyrin and its Raman scattering

It has been shown that a Raman line of a nontotally symmetric vibration (b_1g, b_2g, or a_2g) of a metalloporphyrin (D_4h) can be caused, not only by a vibronic coupling between the first ($\text{A}\text{?}$) and second ($\text{B}\text{?}$) excited electronic states but also by a vibronic coupling within one of the electronic states (let us call the former the QS mechanism and the latter the QQ and/or SS mechanism). A simple formulation has been made for each of those different mechanisms, so that a numerical calculation can be made of the excitation profile of a Raman scattering for a given set of coupling constants and damping factors. Next, the result of an examination is given of the excitation profile of the Raman scatterings of some of the b_1g, b_2g, and a_2g vibrations of nickel octaethylporphyrin and nickel octaethyltetrachloroporphryin. Most of the Raman lines of these nontotally symmetric vibrations show a resonance effect only in the Q-band ($\text{A}\text{?}\text{←}\text{X}\text{?}$) region but not in the Soret-band ($\text{B}\text{?}\text{←}\text{X}\text{?}$) region. On the basis of this fact, it has been suggested that those Raman lines are caused mainly by the QQ mechanism rather than the QS mechanism. The observed vibrational structure of an excitation profile also seems to support this suggestion.     

On dielectric and magnetic relaxation phenomena and vectorial internal degrees of freedom in thermodynamics

It has been shown by the author in a previous paper that thermodynamic vectorial internal degrees of freedom which influence the polarization or the magnetization of a medium may give rise to dielectric or magnetic relaxation phenomena. Snoek's equation for magnetic relaxation phenomena was derived and it was shown that Debye's theory for dielectric after-effects in polar liquids is a special case of the developed theory. In this paper it is shown that if Z is some vectorial internal degree of freedom which influences the polarization a new internal degree of freedom bip^(int) may be defined which is a function of biZ, which may replace biZ as vectorial internal degree of freedom and which is a part of the total specific polarization. Furthermore, p^(int) may be introduced in such a way that the remaining part of the polarization, p^(el) (defined by $\text{p}{}^{\mathrm{<mtext>(</mtext><mtext>el</mtext><mtext>)</mtext>}}\text{=}\text{p}\text{?}\text{p}{}^{\mathrm{<mtext>int</mtext><mtext>)</mtext>}}$, where p is the total polarization per unit of mass), has the property that it vanishes for all values of p^(int) if the medium is in a state where the electric field E and the mechanical elastic stresses vanish and the temperature of the medium equals some reference temperature. If the equations of state are linearized the latter result implies for an isotropic medium $\text{E}\text{=ρa}{}^{\mathrm{(0,0)}}{}_{\mathrm{(bdp)}}\text{p}{}^{\mathrm{<mtext>(</mtext><mtext>el</mtext><mtext>)</mtext>}}$, where ρ is the mass density and a^(0,0)_(P) a constant. On the other hand p^(int) satifies a relaxation equation. It is seen that the use of p^(int) as an internal degree of freedom is of great advantage. This is connected with the fact that p^(int) is a measurable quantity in contradistinction to an arbitrary “hidden” vectorial internal degree of freedom. Analogous results may be obtained for magnetic after-effects.     

Photoelectron spectra of pyromellitic dianhydride,dithioanhydride and diimide and of trimellitic anhydride

The photoelectron (He(I)) spectra of the tricyclic tetracarbonyl compounds pyromellitic dianhydride, dithioanhydride and diimide and of the tetracyclic hexacarbonyl compound trimellitic anhydride have been investigated. To aid the interpretation of the main features of the spectra, i.e. the ordering and splitting of the ⁿCO ionisations and the behaviour of the ‘benzenic’ and heteroatom π ionisations, MO calculations based on a ZDO pragmatic model and semiempirical SCF-PP calculations have been carried out. The evolution of the ⁿCO and π_X ionisations upon progressive fusion of anhydride moieties with a benzene nucleus is analysed in detail. The proposed orbital sequences for the n orbitals are: a_g(S⁺) $\text{\$}\text{?}$b_1u(AS⁺) $\text{\$}\text{?}$b_2u(S^?) $\text{\$}\text{?}$b_3g(AS^?) for the tetracarbonyls and a′₁(S⁺) $\text{\$}\text{?}$e′(AS⁺) $\text{\$}\text{?}$a′₂ (AS^?) ≈ e′(S^?) for the hexacarbonyl.