Correction of a misunderstanding about Olbers' paradox

It is sometimes said that, if the universe were not expanding, we all would burn up, (Olbers' paradox). In order to check on this, I have used a solution of Friedmann's equation that seems reasonable, and that predicts a contraction of the universe after it has reached a maximum size. We may then compare the radiation energy density U _n now at age t _n of the universe, with this density (=U ^* ) at a time t ^* during contraction when R(t) takes a value R ^* equal to its present value R _n .We simplify the calculation by choosing 2R _n as maximum of R(t). Then, t _n=(1/2 π?1)R _n/C=6.98 × 10⁹ year for R _n = 1.16× 10²⁸ cm.(The present density would then be ρ_n = 2.4× 10^?29 gram/cm³).We assume the number of galaxies per unit volume = N(t) = η/R(t) ³ with constant η,and we assume a constant average radiative power L per galaxy. Now at t _n we choose N _n L ≈ 10^?31 erg/cm³sec,but our conclusions would be the same for much larger values of this. We split U into three parts: U ₁ is the primordial energy density left over from t ≈ 0.We put U ₁(t _n ) ≈ 6× 10^?13 erg/cm³ corresponding to T ≈ 3°K.U ₂ is the density of the energy flux near the earth emitted by all stars in our own Milky Way. Most of it is the density U _S =L _⊙/4ηr ² c = 4.5 × 10^?5 erg/cm³ in the flux from the sun. Finally, U ₃ is the energy in the radiation emitted by all other galaxies since t ≈ 0, and was going to burn us according to Olbers. Since U ₁ is a function of R(t), we put U ₁(t^*) =U ₁ (t _n ).We shall also assume U ₂ to be unchanged, as it was not the effect of a change of U _s that we were investigating. In calculating U ₃ we shall overestimate it by neglecting the absorption by closer galaxies of some of the light emitted by farther ones.     

Hopping conductivity of carbynes modified under high pressures and temperatures: Galvanomagnetic and thermoelectric properties

The conductivity, thermopower, and magnetoresistance of carbynes structurally modified by heating under a high pressure are investigated in the temperature range 1.8–300 K in a magnetic field up to 70 kOe. It is shown that an increase in the synthesis temperature under pressure leads to a transition from 1D hopping conductivity to 2D and then to 3D hopping conductivity. An analysis of transport data at T ≤ 40 K makes it possible to determine the localization radius a ～ (56?140) Å of the wave function and to estimate the density of localized states _g(E _F) for various dimensions d of space: _g(E _F) ≈ 5.8 × 10⁷ eV^?1 cm^?1 (d=1), _g(E _F) ≈5×10¹⁴ eV^?1 cm ^?2 (d=2), and _g(E _F)≈1.1×10²¹ eV^?1 cm_?3 (d=3). A model for hopping conductivity and structure of carbynes is proposed on the basis of clusterization of sp ² bonds in the carbyne matrix on the nanometer scale.     

Mass spectrometry study of the kinetics of CdTe molecular-beam epitaxy: Part I. Cd, Te2, and CdTe

Surface processes in CdTe molecular-beam epitaxy were studied using in situ mass spectrometry. Modulated molecular Cd and Te₂ beams were used for measuring kinetic parameters. The experiments were performed at crystal temperatures of 600–730 K. The results were processed within a model in which condensation and evaporation occur through adsorption and desorption stages. The desorption rate was 2–10 s^?1 for Te₂ and more than 30 s^?1 for Cd. The CdTe evaporation activation energy and desorption energies were determined as E _ev = 1.1 eV, E _d (Cd) = 1.0 eV, and E _d (Te) = 0.6 eV. The adsorbate coverage was estimated as n(Cd) < 0.01 and n(Te) = 0.1–1 Te.     

Variational calculations of asymmetric nuclear matter

We report on variational calculations of the energy E(ρ, β) of asymmetric nuclear matter having ? = ?_n + ?_p = 0.05 to 0.35 fm^?3, and β = (?_n ? ?_p/g9 = 0 to 1. The nuclear h used in this work consists of a realistic two-nucleon interaction, called v₁₄, that fits the available nucleon-nucleon scattering data up to 425 MeV, and a phenomenological three nucleon interaction adjusted to reproduce the empirical properties of symmetric nuclear matter. The variational many-body theory of symmetric nuclear matter is extended to treat matter with neutron excess. Numerical and analytic studies of the β-dependence of various contributions to the nuclear matter energy show that at ? < 0.35 fm^?3 the β⁴ terms are very small, and that the interaction energy EI(ρ, β) defined as E(ρ, β) ? T_F(ρ, β), where T_F is the Fermi-gas energy, is well approximated by EI₀(?) + β²EI₂(ρ). The calculated symmetry energy at equilibrium density is 30 MeV and it increases from 15 to 38 MeV as ? increases from 0.05 to 0.35 fm^?3.     

Simple tests of quark-nucleon reciprocity

The inclusive cross-section for the production of a single hadron in deep inelastic electroproduction is studied in a dual resonance model. The Bjorken scaling behaviour in the virtual photon fragmentation region for finite x (≡ 2p_L^c.m./√s) is (1/σ^{T,L))d3σT,L/E?}d³p ～ (1/q²) F(x,p_⊥²/q²) and thus the transverse momentum grows like q², whereas in the parton model (1/σ^T,L)d³σ^T,L/E^?1d³p ～ F(x,p_⊥²). A related effect is the absence of two-jet structure in e⁺e^? annihilation. We believe that dual model results may give a more reliable indication of the deep inelastic behaviour for composite hadrons than the parton model.     

New bounds for dilepton production from heavy neutral leptons

Bounds on 〈E_?ⁿ〉/〈E₊ⁿ〈, 〉E₊E_?〈/〉E₂²〈 and 〈E₊E_?〉/〈E₊〉〈E_?〉 are direved for the processes ν_μN → μ^?μ⁺(e⁺) + X and μN → μ^?μ⁺ + X if dileptons are mediated by a $\text{spin}\text{-}\text{1}\text{2}$ heavy neutral lepton L₀. The bounds are shown to be independent of the production mechanism and mass of L₀. Useful conditional bounds are obtained relating the bounded quantities, which give information about the structure of the weak current responsible for L₀ decay.     

Reinvestigation of the decay scheme of the short-lived isomeric states in112In

The two short-lived isomeric states in¹¹²In have been reinvestigated by means of the¹¹²Cd(d, 2n) and¹⁰⁹Ag (α, n) reactions. Measurements of delayed X- andγ-ray, lifetime and TDPAD spectra have been performed. A 6.32(3) keVγ-ray has been identified in the isomeric decay and anM1 multipolarity has been assigned to it on the basis of the measured internal conversion coefficients:α _L =196(39) andα _tot=231(35). TheE1+M2(δ =+0.24(4)) andE2 multipolarities have been determined for the 263 and 188 keV transitions deexciting the 8^?(T _1/2=2.82(2) μs) and 7⁺ (T _1/2=0.93(2) μs) isomeric states, respectively. A revised isomeric decay scheme is presented.     

Virasoro type Lie algebras and deformations

The first and second cohomologies of Cartan Type Lie algebras with coefficients in irreducible tensor modules are calculated. The spaceH ¹(L, U) is interpreted as a space of deformations of (L, U)-modules.H ²(L, L)≠0 ifL=S ₂,S ₂ ⁺ orL=H _n ,H _n ⁺ . Lie algebra of divergenceless vector fieldsS ₂ ⁺ has only one nontrivial local deformation. The two-sided simple hamiltonian algebraH _n has 2n ²+n new local deformations in addition to Moyal cocycle. The Lie algebrasL=W _n (n>3),S _n?1(n>2),H _n (n>1),K _n+1(n>1) have 3, 1, 1, 3 nonisomorphic tensor modules with irreducible bases and nonzero 1-cohomologies; respectively, the corresponding numbers for 2-cohomologies are 9, 6, 7 and 9.     

Relativistic comparison theorems for the Klein-Gordon equation with scalar and vector potentials in d-dimensions

Relativistic comparison theorems are established for discrete eigenvalues of Klein-Gordon equation with vector and scalar potentials in d-dimensions. Theorem 1: If V(λ) and S(λ) depend on a parameter λ, ∂S/∂λ?0, S?0, ∂V/∂λ?0, V?0, E>0, then it follows that ∂En/∂λ?0. Theorem 2: If S₂?S₁?0, 0?V₂?V₁, E>0, then the corresponding eigenvalues are ordered as En(2)?En(1). Theorem 3: If 0?V₂?V₁, S₂?|S₁|, En(1)>0, En(2)>0, then En(2)?En(1). Some illustrative examples are given.     

On a Semiclassical Formula for Non-Diagonal Matrix Elements

Let H(?)=?? ²d²/dx ²+V(x) be a Schrödinger operator on the real line, W(x) be a bounded observable depending only on the coordinate and k be a fixed integer. Suppose that an energy level E intersects the potential V(x) in exactly two turning points and lies below V _∞=lim?inf?_|x|→∞ V(x). We consider the semiclassical limit n→∞, ?=? _n →0 and E _n =E where E _n is the nth eigenenergy of H(?). An asymptotic formula for 〈n|W(x)|n+k〉, the non-diagonal matrix elements of W(x) in the eigenbasis of H(?), has been known in the theoretical physics for a long time. Here it is proved in a mathematically rigorous manner.     

The long-time behaviour of the electric-field autocorrelation function in a finite photon gas

We investigate an autocorrelation function of a soluble three-dimensional system, namely the temporal coherence functionC _E(t)∝<E(0)E(t)> of the thermal radiation field in a cube-shaped cavity for the stochastic electrical fieldE. In the thermodynamic limit,C _E(t) relaxes exponentially at intermediate times, but a “long-tail” behaviourC ₀(t)=At^?4 withA<0 is predominant for long times. In the case of a finite, but not too small, cavity lengthL obeyingΛ=hc/k _BT?L and at timest withct?L, C _E(t) is described by an asymptotic expansion in powers ofL ^?1 using generalized Riemann zeta functions. Surface-and shape-effects enhance the long-tail. In the case of very small cavities withL«Λ, we calculate an expansion ofC _E(t) in terms of exp(?L ^?1) and cosines. An oscillatory, but not strictly periodic, long-time behaviour is observed in this case.     

Messung der Winkelverteilung der Neutronenpolarisation bei Streuung von 15,85 MeV-Neutronen am Kohlenstoff

AtE _n=15.85 MeV the angular distributions of neutron polarizationP(θ) for¹²C(n, n)¹²C and of scattering asymmetry A(θ) for¹² C(n,n′)¹² C ^*(Q=?4.43 MeV) have been measured. In a neutron time-of-flight method with a plastic scintillator as scatterer carbon recoil nuclei were used for detection. Polarized neutrons were produced in thed-t reaction atE _d=1.90 MeV at a reaction angle of 70° (lab.). WithP _n=?0.135 scattering polarizations P(θ) are forθ _lab=30, 40, 50, 60, 70, and 80° respectively ?27.0±2.1, ?48.4±2.7, ?68.7±3.6, ?20.7±6.2, +5.3±3.9, and +2.1±4.5%.     

Thermochemistry of azide salts and the azide ion using direct minimisation equations to obtain the lattice energy of the univalent azide salts

The opportunity to test a new equation for the computation of the lattice energy and at the same time examine a disparity in the literature data for the enthalpy of formation of the azide ion, $\text{ΔH}{}^{\mathrm{\theta }}{}_{\mathrm{?}}\text{(}\text{N}{}_3{}^{\mathrm{?}}\text{) (}\text{g}\text{)}$ was the motivation for this study. The results confirm our earlier calculation and show the new equation to be reliable. Thermodynamic data produced in the study take values: $\text{ΔH}{}^{\mathrm{\theta }}{}_{\mathrm{?}}\text{(}\text{N}{}_3{}^{\mathrm{?}}\text{)(}\text{g}\text{) = 144}\text{kJ mor}{}^{\mathrm{?1}}$ΔH^θ_hyd(N₃^?) = ?315 KJ mol^?1 or ΔH^θ_hyd(N₃^?) = ?295 KJ mol^?1U_POT(NaN₃) = 732 kJ mol^?1U_POT(KN₃) = 659 kJ mol^?1U_POT(RbN₃) = 637 kJ mol^?1U_POT(CsN₃) = 612 kJ mol^?1U_POT(TIN₃) = 689 kJ mol^?1. The lattice energies of azides whose enthalpies of formation are documented have been calculated as well as the enthalpy of formation of the azide radical.     

Mass spectrometry study of the kinetics of CdTe molecular-beam epitaxy: Part II. Cd, Te2, Te, and CdTe

The results of an in situ mass-spectrometric study of surface processes occurring during CdTe molecular beam epitaxy are presented. The measurements of kinetic parameters are performed with modulated Cd and Te₂ molecular beams with an intensity of 0.1–5.0 ML/s at a crystal temperature of 550–730 K. The experimental results are treated using a model in which condensation and evaporation proceed through adsorption and desorption steps. The desorption rates are 2–15 and 150 s^?1 for Te₂ and Cd, respectively. The activation energy of CdTe “evaporation” is found to be 1.2 eV; the desorption energies are E _d(Cd) = 1.0 eV and E _d(Te₂) = 0.3 eV. The adsobate coverage with cadmium atoms and tellurium molecules is estimated to be n(Cd) < 0.01, n(Te₂) = 0.02–0.20, and n(Te) = 0.2–1.0.     

Stability of bipolarons in the presence of a magnetic field

The stability of large Fröhlich bipolarons in the presence of a static magnetic field is investigated with the path integral formalism. We find that the application of a magnetic field (characterized by the cyclotron frequence ω _c) favors bipolaron formation: (i) the critical electronphonon coupling parameter α _c (above which the bipolaron is stable) decreases with increasing ω _c and (ii) the critical Coulomb repulsion strength U _c (below which the bipolaron is stable) increases with increasing ω _c. The binding energy and the corresponding variational parameters are calculated as a function of α, U and ω _c. Analytical results are obtained in various limiting cases. In the limit of strong electron-phonon coupling (α ? 1) we obtain for ω _c ? 1 that E _estim ? E _estim(ω _c = 0) + c(u)ω _c/α ⁴ with c(u) an explicitly calculated constant, dependent on the ratio u = U/α where U is the strength of the Coulomb repulsion. This relation applies both in 2D and in 3D, but with a different expression for c(u). For ω _c ? α ²? 1 we find in 3D E _estim ? ω _c - α ² A(u) ln²(ω _c/α ²), (also with an explicit analytical expression for A(u)) whereas in 2D E _estim ^2D ? ω _c - α√ω _cπ(u-2-√2)/2. The validity region of the Feynman-Jensen inequality for the present problem, bipolarons in a magnetic field, remains to be examined.     

Surface-induced finite-size effects for first-order phase transitions

We consider classical lattice models describing first-order phase transitions, and study the finite-size scaling of the magnetization and susceptibility. In order to model the effects of an actual surface in systems such as small magnetic clusters, we consider models with free boundary conditions. For a field-driven transition with two coexisting phases at the infinite-volume transition pointh=h _t , we prove that the low-temperature, finite-volume magnetizationm _free(L, h) per site in a cubic volume of sizeL ^d behaves like $$m_{free} (L,h) = \frac{{m_ + + m_ - }}{2} + \frac{{m_ + - m_ - }}{2}tanh\left[ {\frac{{m_ + - m_ - }}{2}L^d (h - h_\chi (L))} \right] + O\left( {\frac{1}{L}} \right)$$ whereh _x (L) is the position of the maximum of the (finite-volume) susceptibility andm _± are the infinite-volume magnetizations ath=h _t +0 andh=h _t ?0, respectively. We show thath _x (L) is shifted by an amount proportional to 1/L with respect to the infinite-volume transition pointh _t provided the surface free energies of the two phases at the transition point are different. This should be compared with the shift for periodic boundary conditions, which for an asymmetric transition with two coexisting phases is proportional only to 1/L ^2d . One can consider also other definitions of finite-volume transition points, for example, the positionh _U (L) of the maximum of the so-called Binder cumulantU _free(L,h). Whileh _U (L) is again shifted by an amount proportional to 1/L with respect to the infinite-volume transition pointh _t , its shift with respect toh _χ (L) is of the much smaller order 1/L ^2d . We give explicit formulas for the proportionality factors, and show that, in the leading 1/L ^2d term, the relative shift is the same as that for periodic boundary conditions.     

A connection between ν-dimensional Yang-Mills theory and (ν−1)-dimensional,non-linear σ-models

We study non-linear σ-models and Yang-Mills theory. Yang-Mills theory on the ν-dimensional lattice ? ^v can be obtained as an integral of a product over all values of one coordinate of non-linear σ-models on ? ^v?1 in random external gauge fields. This exhibits two possible mechanisms for confinement of static quarks one of which is that clustering of certain two-point functions of those σ-models implies confinement of static quarks in the corresponding Yang-Mills theory. Clustering is proven for all one-dimensional σ-models, for theU(n) ×U(n) σ-models,n=1, 2, 3, ..., in two dimensions, and for the SU(2) × SU(2) σ-models for a large range of couplingsg ² ? O(ν). Arguments pertinent to the construction of the continuum limit are discussed. A representation of the expectation of Wilson loops in terms of expectations of random surfaces bounded by the loops is derived when the gauge group is SU(2),U(n) or O(n),n=1, 2, 3, ..., and connections to the theory of dual strings are sketched.     

Image potential eigenstates and resonances on the (110) surfaces of noble metals: Energies and lifetimes

The energies and lifetimes of the image potential resonances at the ? point on the Cu(110), Ag(110), and Au(110) surfaces are studied, and the energies of the image potential states on the Pd(110) surface are analyzed. It is shown that every quantum number n corresponds to a pair of image potential states (resonances) n ⁺ and n ^? at the ? point. The average energy of a pair of eigenstates (resonances) at n ≥ 2 is well described by the formula ē _n = (E _n+ + E _n?) = E ₀ ? (0.85 eV)/(n + δ)², where δ is the quantum defect, E ₀ = (1/2m _e )(?π/a)², and a is the lattice parameter. The splitting energy of a pair of eigenstates (resonances) obeys the law ΔE _n = E _n+ — E _n? ∝ n ^?3. The lifetimes τ _n± of image potential resonances are proportional to n ³.     

Perturbation of the Laplacian supported by null sets,with applications to polymer measures and quantum fields

We discuss hamiltonians in L²(R^d, dx) of the form H = ?Δ + V, with V a potential supported by a zero measure set C. In particular if C is a path of a brownian motion b such that V(x) = ∫₀¹λ(x, ω)δ(x-b(s, ω)) ds, we show that H exists as a nontrivial, self-adjoint, lower bounded perturbation of ?Δ when d ?5. We must choose λ to be an infinitesimal, negative function for d = 4,5, but for d ? 3 any bounded real-valued function λ will do. The connection with Edward's model of polymers as well as with quantum fields of the ?_d⁴-type is also discussed. The proofs use methods of nonstandard analysis.     

4f ground state properties of cerium in AuCu3 intermetallic compounds studied by x-ray absorption spectroscopy

L_III absorption edges of cerium in ternary AuCu₃ compounds Ce(Pd_1?xM_x)₃ (M = Ag and Rh) and (Ce_1?xY_x)Pd₃ are presented. We find that Ce in these compounds is mixed valent with n_f ? 0.7. The valence V of Ce(Pd_1?xRh_x)₃ becomes concentration independent of V = 3.29 ± 0.02 for x ? 0.18. In Ce_1?xY_xPd₃ we observe the valence to increase up to V = 3.32 ± 0.02 at the highest concentration of Y. We show, that a critical volume, available for the Ce atom, is associated with the concentration independent valence around 3.3. For both systems, linewidths W₁, W₀ and Δ = E(4f¹) ? E(4f⁰), as derived from L_III double-bump structure, are given. Possible falsifications due to “final states effects” on the valence are discussed.