On the theory of collective motion in nuclei. I. Classical theory

From the assumption that the collective Hamiltonian be invariant under the orthogonal group O(A ? 1, $\text{R}$) it is concluded that classical collective dynamics can be formulated on a symplectic manifold. This manifold is shown to be a coset space of the symplectic group $\text{L}$h(6, $\text{R}$) of dimension 12, 16 or 18. The first case corresponds to the dequantization of closed-shell collective dynamics and is described in terms of six complex s- and d-quasiparticles. In the limit A ? 1 it is shown that a transformation leads to interacting s- and d-bosons with the symmetry group $\text{u}$ (6) in the collective phase space.     

On the theory of collective motion in nuclei. III. From Hamiltonian dynamics to vortex-free fluidity

It is assumed that the Hamiltonian for collective motion in nuclei is invariant under the orthogonal group O(n, $\text{R}$). For degenerate orbits in phase space it is shown that the classical Hamiltonian equations reduce to the equations of a vortex-free fluid with a velocity field determined by independent equations of motion.     

On the linear response theory of infinite quantum systems

The perturbation of a C¹-dynamics α by a time-dependent unbounded 1-derivation of the form μ?(t)δ′ is considered. The existence of the perturbed dynamics and that of the linear responce oftthe C¹-dynamical system ($\text{A}$, $\text{R}$, α) is shown. The existence of the corresponding generalized susceptibility and some convergence problems are discussed.     

Symmetric space scalar field theory

Some quantum field theories, such as the chiral SU(2) ? SU(2) theory, can have a dynamics invariant under a group $\text{G}$ that is realized on a vacuum which is invariant only under a subgroup $\text{H}$ of $\text{G}$. These theories may be defined by scalar fields which are coordinates for the coset manifold $\text{G}$/$\text{H}$. They are thus non-polynomial theories on a symmetric space, with the group motions in this space described by a set of Killing vectors. We show how the Lagrange function may be constructed entirely from the Killing vectors. In particular, all physical quantities may be expressed in terms of the currents formed out of the Killing vectors. The current correlation functions do not exhibit the spurious wave function renormalizations which are encountered if ordinary Green's functions are computed. We illustrate the general method by calculating one-loop counter terms in a completely invariant fashion. An Appendix describes in simple terms the general theory of symmetric spaces, which should prove useful in other contexts.     

NMR investigation of the structure of NiZrF6·.6H2O at low temperatures

Previous studies have shown that NiZrF₆.6H₂O can be described by an axial spin Hamiltonian with parameters g = 2.33 and $\text{D}\text{k}\text{= ?3.14}\text{K}$, and exhibits ferromagnetic ordering at T_c = 164mK. Our room temperature X-ray measurements confirm that NiZrF₆·.6H₂O is isomorphous with NiSnCl₆·.6H₂O and give lattice parameters of $\text{a = 6.55}\text{A}\text{?}$ and α = 96°09′. Proton NMR measurements show that there exist important molecular motions at room temperature and that the space group remains $\text{R}\text{3}$ down to liquid helium temperatures. The positions of all twelve protons have been determined and are found to lie on a sphere of radius 2.9 Å centered on the nickel ion.     

The pair distributions and the osmotic coefficient in anomalous electrolytes up to concentrations c≈ 1 mol/1

The osmotic coefficient of anomalous electrolytes up to concentrations c ≈ 1 mol/l is explained by the pair distributions $\text{n(r) =}\text{exp}\text{[-β(}\text{V}{}_{\mathrm{c}}\text{(r) + V(}{}^{\mathrm{hs}}\text{)(r) + V}{}_1\text{(r))]}$. Here $\text{V}$_c(r) is a screened Coulomb potential, V(^hs)(r) a hard sphere potential and V₁(r) = ?A/r⁶ a short range attractive potential. For the contact distances R₊₊, R_?? and R_+? of the hard sphere potentials between ions with the same sign of their charges (++,??) and ions of opposite charges (+?) the relations R₊₊ = R_?? = R and R_+? = q₁R with 0 < q₁ < 1 are assumed. In contrast to a previous paper the parameter q₁ takes a fixed value q₁ ≈ 0,8. The constant A is determined by the fraction q₂ defined by A/R⁶ = q₂(Z²e²/DR) where the positive integer Z is the charge number of the ions and D the dielectric constant of the solvent. The numerical calculation of the osmotic coefficient of 1-1-valent hydrous electrolytes in the range of temperature 273 K ? T ? 293 K shows that the anomalous electrolytes are described by fractions q₂ in the range 0,25 ? q₂ ? 0,5 if the contact distances R are in the range $\text{3}\text{A}\text{?}\text{? R ? 7}\text{A}\text{?}$.     

Eigenfunction expansions of operators admitting separation of an infinite number of variables

Let A_k be a self-adjoint operator in a Hilbert space H_k (k = 1, 2, …) and let $\text{L}$ be an operator of the form $\text{L}\text{= A}{}_{\mathrm{r}}\text{? 1 ? 1 ? … + 1 ? A}{}_2\text{? 1 ? 1 ? … + …}$ acting in the infinite tensor product $\text{?}\text{k=1}\text{∞}\text{H}{}_{\mathrm{k}}$. We construct the spectral theory of these operators. In particular, the expansion is generalized eigenvectors of this operator is constructed using the eigenvectors of the operators A_k.     

Density fluctuations in liquid argon

The formation of a collective mode and its dispersion law, ω(q), have been investigated using a generalised hydrodynamic approach, for values of the wave number $\text{q ? 0.2}\text{A}\text{?}{}^{\mathrm{?1}}$. They found to be sensitively dependent on the liquid structure.     

A mathematical interpretation of Dirac's formalism part b: generalized eigenfunctions in trajectory spaces

Starting with a Hilbert space $\text{L}{}_2\text{(}\text{R}\text{,μ)}$ we introduce the dense subspace $\text{R}\text{(}\text{L}{}_2\text{(}\text{R}\text{,μ))}$ where $\text{R}$ is a positive self-adjoint Hilbert–Schmidt operator on $\text{L}$₂(R,μ). For the space $\text{R}\text{(}\text{L}{}_2\text{(}\text{R}\text{,μ))}$ a measure-theoretical Sobolev lemma is proved. The results for the spaces of type $\text{R}\text{(}\text{L}{}_2\text{(}\text{R}\text{,μ))}$ are applied to nuclear analyticity spaces $\text{S}{}_{\mathrm{<mtext>X,</mtext><mtext>A</mtext>}}\text{=}\text{?}\text{t>0}\text{e}{}^{\mathrm{<mtext>-t</mtext><mtext>A</mtext>}}\text{(X)}$, where e^?t$\text{A}$ is a Hilbert–Schmidt operator on the Hilbert space X for each t>0. We solve the so-called generalized eigenvalue problem for a general self-adjoint operator $\text{P}$ in X.     

An effective method of investigation of positive maps on the set of positive definite operators

Let $\text{A}$₁ be the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$₁, and let $\text{A}$₂ be the algebra of linear operators of the m-dimensional Hilbert space $\text{H}$₂. Let $\text{L}$($\text{A}$₁, $\text{A}$₂) denote the complex space of linear maps from $\text{A}$₁ to $\text{A}$₂. By a positive map we mean an element of the space $\text{L}$($\text{A}$₁, $\text{A}$₂) (superoperator with respect to $\text{H}$₁) which maps positive definite operators in $\text{A}$₁ into positive definite operators in $\text{A}$₂. The aim of this paper is to present an effective method which allows to verify whether a given superoperator Λ∈$\text{L}$($\text{A}$₁, $\text{A}$₂) is a positive map. Besides that necessary and sufficient conditions for the positive definiteness of even-degree forms in many variables are given.     

On invariant states and the commutant of a group of quasi-free automorphisms of the car algebra

We study primary states of the CAR algebra which are left invariant under quasi-free automorphisms α_U corresponding to unitaries U of a von Neumann algebra $\text{M}$ on the one-particle Hilbert space, and show that they are quasi-free states ?_A corresponding to self-adjoint operators A in $\text{M}$′ with 0 ? A ? 1, under the assumption that $\text{M}$ does not contain any finite type Ifactor direct summands. Next we study automorphisms of the CAR algebra which commute with α_U for U in a von Neumann algebra $\text{M}$ and show that they are quasi-free automorphisms α_U with U in $\text{M}$′ under the same assumption on $\text{M}$ as above. Finally by using the latter result we obtain a generalization of a theorem of Hugenholtz and Kadison [3].     

Exact solutions in λϕn theory

Families of exact solutions of the λ?ⁿ theory in d + 1-dimensional space—time are obtained. Certain finite-energy static solutions are found in the case when $\text{n =}\text{2d}\text{(d ? 2)}$.     

Low temperature structural distortion in the high Tc Chevrel-phase superconductors PbMo6S8 and SnMo6S8

High-resolution powder neutron diffraction data reveal that, at temperatures below 100 K, oxygen-free, high-T_c samples of PbMo₆S₈ and SnMo₆S₈ exhibit a small structural distortion from the R$\text{3}$ rhombohedral space group. The data cannot be refined in a simple P$\text{1}$ triclinic space group as previously reported for EuMo₆S₈ and BaMo₆S₈. Thus, a supercell ordering is suggested.     

KMS condition in a Schrödinger picture of the dynamics

Given the C¹-algebra $\text{A}$ of observables, the KMS condition is formulated in terms of the time evolution α¹_t of a set $\text{S}$₀ ? $\text{S}$($\text{A}$) of “physical” states subject to certain natural conditions. α¹_t need not be defined by an automorphism group of $\text{A}$. It is shown that, for a KMS state ω, α¹_t induces a 1-automorphism α^ω_t of the von Neumann algebra π_ω($\text{A}$)″ generated by the representation π_ω belonging to ω.     

A convergent expansion about mean field theory: I. The expansion

We give a convergent expansion for nearly Gaussian quantum field theory in the multiphase region. The expansion combines (1) an expansion in phase boundaries, (2) a cluster expansion, and (3) a perturbation expansion to isolate dominant behavior. We study in detail the ground state of the $\text{P}$(φ)₂ = (λφ⁴ ? φ² ? μφ)₂ model, with ∥ μ ∥ ? λ² ? 1. The ground state is close to the classical free field, obtained by replacing $\text{P}$(φ) by the quadratic mean field polynomial $\text{P}$_c(φ), tangent to $\text{P}$ at a global minimum. Selecting one minimum gives a pure phase (ergodic ground state) satisfying the Wightman-Osterwalder-Schrader axioms with a positive mass. We also establish analyticity in λ for μ = 0 in the sector ∥ Im λ ∥ < ? Re λ ? 1, for ? ? 1.     

Cation ordering in the tetrahedral sites of the thiospinel FeIn2S4 (indite)

Recent X-ray and electron diffraction studies have substantiated earlier indications, based on dielectric behavior, e.s.r. studies, elastic constants, magnetic measurements, specific heat and spectral data, that in many cases the cubic spinel structure exhibits a small deviation from Fd3m symmetry. This discrepancy has been associated with a small displacement (less than 0.10Å) of the octahedrally coordinated metal atom along [111] in space group F4?3m but has not hitherto been tested using precise single crystal X-ray diffraction data. Although more than thirtyhk0 reflections of the type h + k = 4n + 2, structurally forbidden for the space group Fd3m, are clearly visible in diffraction patterns from the thiospinel indite ($\text{FeIn}{}_2\text{S}{}_4\text{; a = 10.618(3)}\text{A}\text{?}$), a refinement in space group Fd3m using 412 unique X-ray data (u = 0.25907(7); inversion parameter = 0.947(7); R = 0.048) indicates that the structure may be satisfactorily described in terms of the conventional symmetry. A refinement of the data in space group F4?3m shows that the displacement, if present, is not detectable within experimental error but allows a rationalization of the observed diffraction symmetry in terms of an unequal distribution of In atoms over the two nonequivalent tetrahedral sites in F4?3m (100 and 83 ± 1% In, respectively) and the presence of a nonspherical distribution of residual electron density about the octahedral site.     

Proprietes magnetiques et etude par effet Mössbauer du thiosilicate Fe2SiS4

Fe₂SiS₄$\text{(a = 12,407}\text{A}\text{?}\text{, b = 7,198}\text{A}\text{?}\text{, c = 5,812}\text{A}\text{?}$, space group Pnma) has the olivine structure; the ferrous ions are located in two kinds of sites: one half in planes of mirror symmetry m, the other half in centers of symmetry. 1?.The magnetic study of this compound by means of magnetization measurements and Mössbauer effect, indicates from 127 K an antiferromagnetic arrangement along oy for the m sites, and an induced partial order of the same kind for the 1? sites. At 33 K, the Fe²⁺ spins in 1? sites are completely ordered and at the same time takes place a rearrangement of the magnetic structure. The observed complex model is analogous to that of Fe₂GeS₄, i.e. antiferromagnetic along 0x and ferrimagnetic along Oz.