The Hartree–Fock Approximation Calculation of Electric Polarizability of Atoms with an Open Shell

Within the framework of the restricted Hartree–Fock method, equations of stationary coupled perturbation theory have been obtained for atomic-molecular systems with an open shell in orbital representation. Corrections to the Hartree–Fock orbitals are sought in the form of expansions in unperturbed orbitals which are assumed calculated in the LCAO approximation. The resulting inhomogeneous algebraic system for the expansion coefficients admits an exact solution. The static polarizability of atoms with an open shell from Li to Sc has been calculated with the use of an optimized basis set of Slater-type atomic orbitals.     

Levinson's Theorem for the Schrödinger Equation in One Dimension

Levinson's theorem for the one-dimensional Schrödinger equation with asymmetric potential which decays at infinity faster thanx ^–2 is established by theSturm-Liouville theorem. The critical case where the Schrödinger equation hasa finite zero-energy solution is also analyzed. It is demonstrated that the numberof bound states with even (odd) parityn ₊(n _–) is related to the phase shift ₊ (0)[_– (0)] of the scattering states with the same parity at zero momentum as ₊ (0)+ /2 =n ₊ and _– (0) =n _– for the noncritical case, and ₊ (0) =n ₊ and_– (0) – /2 =n _– for the critical case.     

Spectral fluctuations and zeta functions

We study theoretically and numerically the role of the fluctuations of eigenvalue spectra {it_n in a particular analytical continuation process applied to the (generalized) zeta functionZ(s)= _{n n} ^–s fors large and positive. A particularly interesting example is the spectrum of the Laplacian on a triangular domain which tessellates a compact surface of constant negative curvature (of genus two). We indeed find that the fluctuations restrict the abscissa of convergence, and also affect the rate of convergence. This then initiates a new approach to the exploration of spectral fluctuations through the convergence of analytical continuation processes.work done while at the Service de Physique Théorique, Saclay, and Institut de Physique Nucléaire, Orsay.     

Classical solutions ofℂP n−1 models and their generalizations

We discuss finite action solutions ofP ^n–1 models, of their Grassmannian generalizations and of a classical fermion — boson model based on the supersymmetric extension of theP ^n–1 model.Invited talk presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.I would like to thank Dr. A. M. Din for his collaboration with me on all the topics discussed here. Also I would like to thank E. F. Corrigan, D. B. Fairlie, M. Günyadin, J. Lukierski, I. Singer, R. Stora and many others for discussions and constructive criticisms and Professor J. Niederle for his invitation to present this talk at the symposium and for hospitality in Czechoslovakia.     

Relativistic spherical polytropes: An analytical approach

Using the technique of Padé (2, 2) approximant we present, in this paper, an approximate analytical solution to the field equations of general relativity for time-independent, spherically symmetric systems in which the pressureP and density are related by a polytropic equation of state:P = K ^1+1/n. The boundary values of coordinate radius ₁, for polytropic indicesn = 0, 1.0 (0.5) 3.0, are given in Table I. Table II contains the values of other physical parameters, (₁) (mass), (the density concentration), and 2GM/c²R (the ratio of gravitational radius to the coordinate radius) forn = 0 and 1.Work done at Azerbaijan State University, Baku, USSR (1977–79).     

On the variation of hopping transport in amorphous silicon with preparation conditions

Thedc conductivity of amorphous silicon prepared by two successive ion bombardments at different temperatures has been measured as a function of temperature. The results may be expressed in terms of a generalized hopping formula =₀ exp [–(T ₀/T) ⁿ where the parameter set {n,T ₀, ₀} varies with the irradiation conditions. In particular, the hopping exponent has been found to assume the limiting values ofn1/4 at irradiation temperatures ofT _i100 K and ofn1/2 atT _i500 K, whereas intermediate values ofn have been observed for temperatures inbetween. It is concluded that thermally activated redistribution processes of radiation defects control the final state of disorder in the irradiated samples, which in turn determines the particular hopping characteristics. Within the framework of existing theories the two limiting cases can be explained to be due to a disordered solid of homogeneous and granular structure, respectively.     

Negative muon precession in nuclei with spin

Polarized negative muons were stopped in various materials containing nuclei with nonzero spin. The TF-SR precession signal of theF ⁺ hyperfine state (frequencyv ₊ > 0) was pronounced for ^–Li and ^–Be, faint for ^–Cl (in NiCl₂), and undetectable for ^–F (in CaF₂ or NiF₂) and ^–P. TheF ^– signal (frequencyv _– < 0) was observed clearly for ^–Be, ^–Al and ^–Na, marginally for ^–K, ^–V and ^–Nb, and not at all for ^–Ga (at 4 kOe) or ^–Co (ferrromagnetic, zero field). In the heavier elements theF ^– signal is fed by transitions from theF ⁺ to theF ^– state at a rateR, as long asR(v ₊ –v _–). [See separate paper on ^–Al in these Proceedings, p. 879.]I am grateful to Alex Schenck and Bruce Patterson for the loan of equipment and samples, to Fred Gygax for helping set up the apparatus, to Jun Imazato for help with the experiment, and to Toshi Yamazaki and Tak Suzuki for valuable comments. I am also indebted to SIN for several weeks of free ^– beam time and to SIN, BOOM, and the University of Tokyo for free time on their VAX computers, which were kept busy for several months in the analysis.     

Bimolecular annihilation reactions: Immobile reactants and multipolar interactions

We study the A + B 0 annihilation reaction via multipolar interactions wr^–s (r distance) in one and two dimensions. For equal numbers of immobile A and B particles we present computer simulations and numerical calculations of the decay. We find at large times that ind dimensions the particle concentration follows nt^–d/(2s–d)This work is dedicated to Prof. George H. Weiss.     

Asymptotic behavior of the density for two-particle annihilating exclusion

We consider a stochastic process which presents an evolution of particles of two types,A andB, onZ ^d with annihilations between particles of opposite types. Initially, at each site ofZ ^d, independently of the other sites, we put a particle with probability 2<1 and assign to it one of two types with equal chances. Each particle evolves onZ ^d in the following manner: independently from the others, it waits an exponential time with mean 1, chooses one of its neighboring sites on the latticeZ ^d with equal probabilities, and jumps to the site chosen. If the site to which a particle attempts to move is occupied by another particle of the same type, the jump is suppressed; if it is occupied by a particle of the opposite type, then both are annihilated and disappear from the system. The considered process may serve as a model for the chemical reactionA+Binert. Let (t) denote the density of particles in this process at timet. We prove that there exist absolute finite constantsc(d) andC(d) such that for all sufficiently larget,c(d)t ^–d/4 (t)C(d)t ^–d/4 in the dimensionsd4 andc(d)t ^–1 (t)C(d)t ^–1 in all higher dimensions. This completes and makes more precise the results obtained by us earlier and shows that asymptotically the density behaves like that in a similar process called two-particle annihilating random walks which was studied by Bramson and Lebowitz. Our proofs are based on the approach developed in their and our works. We use the basic properties of random walk and various tools which have been designed to study simple symmetric exclusion processes.     

A rigorous bound on the critical exponent for the number of lattice trees,animals, and polygons

The number ofn-site lattice trees (up to translation) is believed to behave asymptotically asCn ^{–0 n} , where is a critical exponent dependent only on the dimensiond of the lattice. We present a rigorous proof that (d–1)/d for anyd2. The method also applies to lattice animals, site animals, and two-dimensional self-avoiding polygons. We also prove that v whend=2, wherev is the exponent for the radius of gyration.     

Electrooptical Kerr constant and third-order susceptibility measurements in nematic liquid crystals and their mixtures

We report for the first time the determination of the real part of the third order nonlinear susceptibilities ⁽³⁾ (–, , 0, 0) in the isotropic phase of two nematic liquid crystalsp-(ethoxybenzylidine)-p-butylaniline (EBBA) andp-(methoxybenzylidene)-p-butylaniline (MBBA) from electrooptic Kerr effect experiments. The highest value of ⁽³⁾ observed near nematic-to-isotropic transition temperature, at 632.8 nm is found to be 1.067×10^–20 m²V^–2 for EBBA and 6.602×10^–20 m²V^–2 for MBBA.     

Anomalous fluctuations in random walk dynamics

The anomalous dispersion of noninteracting particles randomly walking in a network is considered. It is shown that the existence of large dangling branches attached to a backbone induces a l/f-like behavior in the current autocorrelation function at low frequencies. The waiting times associated with dangling loops scale liket ^–3/2. The size of the dangling branches provides a lower cutoff to the power law behavior. When the side branches are infinite, self-similar structures, the power law behavior persists up to a zero frequency. The currents we consider are created either by a bias on the random walk or by a current source. We consider both the total current, which is often referred to in the literature, and the current measured at endpoints of a specimen attached to a (model) battery. The differences and similarities between the two corresponding correlations are analyzed. In particular, we find that in the second case l/f noise exists only for large bias. When a statistical distribution of dangling branches is considered, we find that the largest power of frequency in the spectrum is 1.13. Much of our results are true when the dangling branches are replaced by traps having waiting time distributions that equal those of the branches. The waiting time associated with a power law distribution of dangling loops (m ^–x:m is the length of the loop) scales liket ^–1 –(x/2). However, it is shown that geometry alone can be responsible for the appearance of power laws in the spectra. Random geometry can be regarded as a model (or source) of random hopping times.     

Relativistic Schrödinger Theory and the Hartree–Fock Approach

Within the framework of Relativistic Schrödinger Theory (RST), the scalar two-particle systems with electromagnetic interactions are treated on the basis of a non-Abelian gauge group U(2) which is broken down to the Abelian subgroup U(1)×U(1). In order that the RST dynamics be consistent with the (non-Abelian) Maxwell equations, there arises a compatibility condition which yields cross relationships for the links between the field strengths and currents of both particles such that self-interactions are eliminated. In the non-relativistic limit, the RST dynamics becomes identical to the well-known Hartree–Fock equations (for spinless particles). Consequently the original RST field equations may be considered as the relativistic generalization of the Hartree–Fock equations, and the exchange interactions of the conventional theory (induced by the anti-symmetrization postulate) do reappear here as ordinary gauge interactions due to a broken symmetry.     

On the homotopical significance of the type of von Neumann algebra factors

The set of all projections and the set of all unitaries in a von Neumann algebra factorA are studied from the homotopical point of view relative to the operator norm topology.Two projectionsE andF can be deformed continuously to each other if and only ifEF and 1–E1–F where denotes the equivalence of projections inA in the sense of von Neumann. In other words, the relative dimension and co-dimension are a complete homotopical invariants of projections inA and label pathwise connected components of the set of projections.The first homotopy group ₁(U(A)) of unitaries inA is shown to be 0 forA of infinite type. For typeII ₁ and typeI _n factors, ₁(U(A)) are isomorphic to additive groups of realsR and integersZ, respectively, in which the first homotopy group ₁(F U(A)) of the center ofU(A) is imbedded asZ andnZ, respectively.     

Decomposition of many-body Schrödinger operators

The complex-dilated many-body Schrödinger operatorH(z) is decomposed on invariant subspaces associated with the cuts {+z ^–2 R ⁺}, where is any threshold, and isolated spectral points. The interactions are dilation-analytic multiplicative two-body potentials, decaying asr ^–1+ atr=0 and asr ^–1+ atr=.     

Scaling solutions of Smoluchowski's coagulation equation

We investigate the structure of scaling solutions of Smoluchowski's coagulation equation, of the formc _k (t)s(t)^– (k/s(t)), wherec _k (t) is the concentration of clusters of sizek at timet,s(t) is the average cluster size, and(x) is a scaling function. For the rate constantK(i, j) in Smoluchowski's equation, we make the very general assumption thatK(i, j) is a homogeneous function of the cluster sizesi andj:K(i,j)=a ^– K(ai,aj) for alla>0, but we restrict ourselves to kernels satisfyingK(i, j)/j0 asj. We show that gelation occurs if>1, and does not occur if1. For all gelling and nongelling models, we calculate the time dependence ofs(t), and we derive an equation for(x). We present a detailed analysis of the behavior of(x) at large and small values ofx. For all models, we find exponential large-x behavior: (x)A x ^– e ^–x asx and, for different kernelsK(i, j), algebraic or exponential small-x behavior: (x)Bx ^– or (x)=exp(–Cx ^–|| + ...) asx0.     

Poisson–Lie Diffeomorphism Groups

We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of ⁿ . Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra W _n of formal vector fields on ⁿ . We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson–Lie diffeomorphism groups gives rise to large classes of compatible Poisson structures on ⁿ , thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff( ^m ) × FDiff ⁿ ) gives rise to classes of compatible Poisson structures on the space J ( ^m , ⁿ ) of infinite jets of smooth maps ^{m n} , which makes it also a Poisson space for this action. Poisson modules of generalized densities are also constructed. Initial steps towards a classification of these structures are taken.     

Application of151Eu Mössbauer spectroscopy to studies of electric field gradients in high-temperature superconductors

The rare-earth site of samples having the compositions (Fe,Cu)Sr₂(Eu,Ce) _n Cu₂O_4+2n+z (n=2 and 3) were studied by¹⁵¹Eu Mössbauer spectroscopy. The results obtained are compared with measurements of EuBa₂Cu₃O_7– (123) and Bi₂Sr₂(Ca_1–yEu_y)Cu₂O_8+E (2212) samples. The full quadrupole Hamiltonian of the 21.5-keV -transition was applied in analyzing the measured spectra. Simultaneous fitting of spectra recorded from crystallites oriented in various directions was used to determine the hyperfine parameters of then= 2 and 3 samples. Two different electric field gradients, not discernible in simple random crystallite measurements, were obtained for then=3 series.     

Direction-Dependent Free Energy Singularity of the Antiferroelectric Asymmetric Six-Vertex Model

The transition from the ordered commensurate phase to the incommensurate Gaussian phase of the antiferroelectric asymmetric six-vertex model is investigated by keeping the temperature constant below the roughening point and varying the external fields (h, v). In the (h, v) plane, the phase boundary is approached along straight lines v = k h, where (h, v) measures the displacement from the phase boundary. It is found that the free energy singularity displays the exponent 3/2 typical of the Pokrovski–Talapov transition f const(h)^3/2 for any direction other than the tangential one. In the latter case f shows a discontinuity in the third derivative.     

Microwave dielectric properties of Ti4+/MgO and Ni2+/MgO

The dielectric properties of titanium-doped magnesium oxide (Ti/MgO) and nickel-doped magnesium oxide (Ni/MgO) single crystals have been measured in the range of temperature from 300 to 450 K at the microwave frequency of 9.31 GHz. For both crystals the dielectric properties are found similar. From the conductivity data, the activation energy in the measured temperature region has been estimated to be 0.15 eV. The values of the temperature dependence (–1)^–1(+2)^–1 (/T) _p have been calculated. The data confirms the Bosmann and Havinga postulate that, for materials in which the dielectric constant; is less than 20 the temperature dependence should be positive.