States on the current algebra

We introduce the field algebra Σ$\text{D}$(M;ⁿ?ⁿ$\text{g}$) associated with the current algebra $\text{D}$_r(M;$\text{g}$) for the Lie algebra $\text{g}$ over physical space M. The Heisenberg magnet model is generalized to this continuum. It is shown that the Hamiltonian can be given meaning as implementing a derivation of the field algebra in certain representations.We introduce new representations of the current algebra. For example, if G = SU(2), a representation in L²(R³)?³ is [σ(?)F]^j = ε^jkl?_kψ_l for (?_k) = ? in $\text{D}$_r(M;$\text{g}$)(ψ_l = F. This has cyclic subrepresentations with prime parts.     

General correction theorems on decomposition formulae of exponential operators and extrapolation methods for quantum Monte Carlo simulations

It is proved that the trace of the generalized Trotter formula $\text{Z(n) ≡ Tr[II exp(}\text{A}{}_{\mathrm{j}}\text{n]}{}^{\mathrm{n}}$ is an even function of n, when all A_j are symmetric, namely A^t_j = A_j, togethern with some generalizations. This yields a new extrapolatio n method of the form $\text{Z(n) ? Z(∞) + a/(n}{}^2\text{+ b)}$ for large n in quanntum Monte Carlo simulations.     

Nonorthogonality corrections in the method of correlated basis functions

A set of normalized linearly independent basis functions φ₁, φ₂, …, φ_j, … generates matrix representatives $\text{H}$ and $\text{N}$ of the Hamiltonian operator and the identity. An orthonormal basis $\text{φ}{}_1$, $\text{φ}{}_2$, …, $\text{φ}{}_{\mathrm{j}}$, … generated by a Löwdin transformation is characterized by the distance in Hilbert space between $\text{φ}{}_{\mathrm{j}}$ and φ_j. The choice of positive definite $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ minimizes these distances and maximizes the diagonal elements of $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Again for positive definite $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and a finite basis, 1 ? j ? p, the analysis yields a general theorem on Trace $\text{N}$${}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ (? p for all positive and negative integral values of n except n = ?1 and ? p for n = ?1).Sufficient conditions are determined which permit the application of the binomial theorem to the evaluation of the transform of $\text{H}$. Approximate formulas for the energy eigenvalues through third order in nondiagonal matrix elements are presented in a compact form containing characteristic nonorthogonality corrections depending on the exterior or interior location of the matrix element in the perturbation formulas.     

Spatial theory for algebras of unbounded operators

In this paper, the spatial theory of ¹-automorphisms is investigated in the context of algebras of unbounded operators. In particular, it is shown that ¹-automorphisms, satisfying some order relation, of the Op¹-algebra generated by the position and momentum operators q_j, p_j(j=1,...., n) on the Schwartz space $\text{I}\text{(}\text{R}{}^{\mathrm{n}}\text{)}$ are unitarily implemented.     

Exact approach to equilibrium for the glauber model of a one-dimensional Ising system with random ±J bonds

The (discrete-time) Glauber model is considered for a one-dimensional system of spins s_j = ±1 with nearest-neighbor Ising interaction $\text{H}\text{= -Σ}{}_{\mathrm{j}}\text{J}{}_{\mathrm{j}}\text{s}{}_{\mathrm{j}}\text{s}{}_{\mathrm{j+1}}$. The J_j = ±J are treated as random variables with an arbitrary joint probability p(J). The exact time-dependent average 〈s_j〉_t is determined, and from it the “quenched” average $\text{〈〈s}{}_{\mathrm{j}}\text{〉}{}_{\mathrm{t}}\text{〉}{}_{\mathrm{<mtext>av</mtext>}}\text{=Σ}{}_{\mathrm{<mtext>J</mtext>}}\text{p}\text{(}\text{J}\text{)〈s}{}_{\mathrm{j}}\text{〉}{}_{\mathrm{t}}$ is explicitly found.     

Invariance properties of the Dirac monopole

The quantum mechanical motion of a spinless electron in the external field of a magnetic monopole of magnetic charge μ is investigated. It is shown that Dirac's quantum condition $\text{2μe(}\text{h}\text{?}\text{c)}{}^{\mathrm{?1}}\text{= n}$ for the string being unobservable ensures rotation invariance and correct space reflection properties for any integer value of n. The rotation and space reflection operators are found and their group theoretical properties are discussed. When n is odd, half-integer j representations of the SU(2) group emerge without the introduction of spin. A method for constructing conserved quantities in the case when the potential is not explicitly invariant under the symmetry operation is also presented and applied to the discussion of the angular momentum of the electron-monopole system.     

KL0-KS0 transmission regeneration on deuterons and neutrons in a momentum range of 10–50 GeV/c

The energy dependence of the K_L⁰-K_S⁰ transmission regeneration amplitudes on deuterons and neutrons in the momentum region 10–50 GeV/c is determined. The moduli of the modified transmission amplitudes are momentum dependent. These dependences are fitted by the expression A_jp^?nj, where A_j and n_j (j = d, n) are constants:

$\text{A}{}_{\mathrm{<mtext>d</mtext>}}\text{=2.88 ±0.04}\text{mb}\text{, n}{}_{\mathrm{<mtext>d</mtext>}}\text{=0.546±0.030,}\text{for deuterons}\text{,}$

$\text{A}{}_{\mathrm{<mtext>n</mtext>}}\text{=1.97 ±0.14}\text{mb}\text{, n}{}_{\mathrm{<mtext>n</mtext>}}\text{=0.530±0.019,}\text{for neutrons}\text{,}$

The amplitude phases do not depend on the kaon momentum and are equal to $\text{?}{}_{\mathrm{<mtext>d</mtext>}}\text{= (?130.9 ± 2.7)°}$?_n = (?132.3 ± 1.7)°. The mean value of the ratio of the total cross-section differences for K⁰ and K⁰ interactions with neutrons and protons is determined. The residues of the partial ω and ? amplitudes, which contribute to the kaon-nucleon interaction amplitudes, are also obtained.     

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.     

Inverse solutions to the inelastic scattering problem

Exact inverse solutions to the integral equation $\text{φ(r}{}_{\mathrm{s}}\text{|r}{}_0\text{, k) = ?}{}_{\mathrm{D}}{}^3\text{f (r, ω)g(r|r}{}_0\text{, k)g(r|r, k)d}{}^3\text{r}$, where g(r|r_j, k); j = 0 or s is the free space Green function, are derived in plane and cylindrical coordinates for fixed ω. These solutions allow an inelastic scattering potential f(r, ω) which is of compact support r ? $\text{D}$³ to be recovered from scattering data collected over the surfaces of a plane and cylinder respectively.     

Study of the reactions 118Sn(d,t) and98Mo(d,t): (I). Spectroscopy of 117Sn and 97Mo

The vector analyzing power and differential cross section have been measured at a deuteron energy of 12.0 MeV for ${}^{118}\text{Sn}\text{(}\text{d}\text{,}\text{t}\text{)}$ transitions to six states of ¹¹⁷Sn (E_x = 0.0, 0.16, 0.31, 0.71, 1.02 and 1.18 MeV), for ${}^{98}\text{Mo}\text{(}\text{d}\text{,}\text{t}\text{)}$ transitions to eight states of ⁹⁷Mo (E_x = 0.0, 0.68, 0.72, 0.89, 1.12, 1.28, 2.39 and 2.52 MeV), and for ${}^{118}\text{Sn}\text{(}\text{d}\text{,}\text{d}\text{)}\text{and}{}^{98}\text{Mo}\text{(}\text{d}\text{,}\text{d}\text{)}$. Deuteron optical model potentials were obtained from analysis of the elastic scattering measurements, and were used in a DWBA analysis of the $\text{(}\text{d}\text{,}\text{t}\text{)}$ results. Comparison of the measurements and DWBA predictions for σ(θ) and for iT₁₁(θ) allows unambiguous determination of tl_n and j_n for all ¹¹⁸Sn(d, t) and most ⁹⁸Mo(d, t) transitions. Differences in the triton energy relative to the Coulomb barrier cause marked qualitative differences in the measured cross sections and analyzing powers between ${}^{118}\text{Sn}\text{(}\text{d}\text{,}\text{t}\text{)}\text{and}{}^{98}\text{Mo}\text{(}\text{d}\text{,}\text{t}\text{)}$ transitions of the same l_n and j_n.     

Relativistic correction to the deuteron magnetic moment and angular condition

The relativistic correction (RC) to the deuteron magnetic moment is calculated using the light-cone dynamics. The restrictions imposed by the angular condition on the electromagnetic current operator of the deuteron are discussed in detail. It is shown that the additive model for the current operator of interacting constituents is consistent with the angular condition only for the two first terms of the expansion of the “good” current component $\text{j}{}_{\mathrm{+}}\text{=}\text{1}\text{2}\text{(j}{}_0\text{+ j}{}_{\mathrm{<mtext>z</mtext>}}\text{)}$ in powers of the momentum transfer q. The RC to μ_d is expressed through the matrix element of the “good” component j₊ and is found to be equal to $\text{(0.6–0.8) × 10}{}^{\mathrm{?2}}\text{e}\text{h}\text{?}\text{/2m}{}_{\mathrm{<mtext>p</mtext>}}\text{c}$ for realistic NN potentials. Taking account of RC decreases essentially the discrepancy between the theoretical and experimental values of μ_d. Possible solutions of the angular condition for squared q-terms of the j₊ current component are also discussed.     

Rigorous spin tests from angular distribution of single two-body decays

We extend to the several-amplitude decays j→j₁+j₂, (j?j₁+j₂,j₁+j₂?2) the rigorous spin tests given by Doncel, Michel and Minnaert for the usual one-amplitude strong decays, in quadrupole and quadrupole-hexadecapole spaces. We also derive spin tests in dipole-quadrupole space for the parity violating decays $\text{j →}\text{1}\text{2}\text{0}\text{and}\text{j →}\text{1}\text{2}\text{1}\text{2}$.     

On observability of N-level quantum systems

The purpose of this paper is to discuss necessary and sufficient conditions for observability of N-level quantum systems. We assume that the information about a physical system is given by the mean values Tr(?(t_j)A_i) = m_Ai(t_j), of n self-adjointoperators A₁,…,A_n on $\text{H}$ at some instants t₁ < t₂ <…<t_s. The question of theminimal number n of operators A₁,…,A_n (physical quantities $\text{A}{}_1\text{, …,}\text{A}{}_{\mathrm{n}}$) for which the quantum system $\text{S}$ is (A₁,…,A_n)-observableis discussed.     

Dispersion of empty surface states on Ni(1 1 0)

We have used k-resolved bremsstrahlung isochromat spectroscopy at $\text{h}\text{?}\text{ω=9.5eV}$ to investigate Ni(1 1 0) along the ΓKL bulk mirror plane. Empty surface states of both the crystal-induced and image-potential-induced type have been detected besides two bulk direct transitions. We studied their different behaviour against oxygen contamination and mapped their energy dispersion E(k_∥) along the ΓY direction of the surface Brillouin zone.     

Toroidal hydromagnetics

The complete set of hydromagnetic equations is transformed into Poisson equations and equations of motion for flux densities and their associated variables. The toroidal components of the vector potential A and of the momentum density $\text{a}\text{≠}\text{πv}$ are represented by the po loidal flux densities Ψ and Ψ, respectively, for which the equations of motion are derived. The poloidal components A_⊥ and a_⊥ are represen ed by the potentials atΦ, U and φ, u, for which we obtain Poisson equations in the poloidal plane. Thus one has to solve two Dirichlet and two von Neumann problems at every time step. The source terms of the four Poisson equations define the remaining four variables, namely, $\text{Λ = ▽ ·}\text{A}$,$\text{Ω=(▽×}\text{A}\text{)ζ/R}$, $\text{λ=?·}\text{a}$, and $\text{ω=(?×}\text{a}\text{)ζ/R}$, for which equations of motion are also derived. In the limit of small toroidicity ? we look fo r a selfconsistent scaling of the equations with v_⊥～ε. But the curl of v_⊥×B in Faraday's law creates a toroidal plasma component of B which is one order of magnitude larger than in the case of a low β equilibrium; therefore, the motion becomes fully three-dimensional. Finally, an artificial pressure law is needed to balance the lowest order of the Lorentz force. The conclusion is then that the scaling laws previously used are not applicable for toroidal geometry, and that the effort to obtain numerical solutions is not dramatically higher than without using any scaling law.     

Relativistic spin selection rules obtained by SL(3, R) symmetry

All irreducible (unitary or not) ray representations of SL(3, R) obeying the Δj = 2 selection rule imposed by Regge trajectories are constructed. They provide irreducible ray representations of $\text{SL(4,}\text{R}\text{) · T}{}^4$ which restricted to the Poincaré subgroup yield unitary representation of real mass and of spin spectrum which statisfies the Δj = 2 selection rule.     

Comparison of calculated photoionization cross-sections of diamond and silicon in the approximations of plane wave and orthogonalized plane wave

In the approximation of the orthogonalized plane wave the shape of X-ray photoelectron spectra of the valence band in diamond and silicon has been calculated. It is shown that consideration of orthogonality terms influences greatly the value of the ratio between photoionization cross-sections of s- and p-electrons. X-ray emission and photoelectron spectroscopy allow us to define the density of states for silicon valence electrons.X-ray photoelectron spectroscopy is at present widely used for the study of the electronic structure of solids. The photoelectron spectra of crystals clearly reveal all the variations in the density of states. However the curves for the photoelectron energy distribution may be different from the calculated density of states for valence electrons. This indicates the importance of the transition probability for the formation of a photoelectron spectrum. In refs. 1 and 2 the X-ray photoelectron spectra obtained with high resolution for diamond and silicon crystals are compared with the density of states and the conclusion made that the s-states lie higher than the p-states. Densities of states and X-ray photoelectron spectra for diamond and silicon have been calculated³. The wave functions of the valence electrons were found by the tight-binding method⁴ while plane waves were used as wave functions for the excited electrons. From a theoretical point of view it is more reasonable to use orthogonalized plane waves to describe the electron states in the conduction band. Both the core and valence states are to be orthogonalized.The present work reports calculation of the shape expected for X-ray photo-electron spectra of diamond and silicon valence electrons in the approximation of orthogonalized plane wave. Investigation was also made of the influence caused by the orthogonality terms on the ratio of photoionization cross-sections of s- and p-electrons. The electronic structure of diamond and silicon was calculated also by the tight-binding method with the use of the parameters from ref. 3. X-ray photo-electron spectra of valence electrons were calculated over 5230 points in 1/48 part of the Brillouin zone. For simplicity it was assumed that the polarization vector of the electromagnetic wave A is directed along the crystal Z-axis. As was done in ref. 3, the X-ray photoemission intensity was averaged over angular variables. As an example we shall give the formula used for calculation of the X-ray photoelectron spectrum for diamond I(ω, E) ～ ∝E σ_n,k [($\text{1}\text{3}$ET_2s² + $\text{1}\text{3}$T_2p²U_2p?2s² - $\text{2}\text{3}$ ∝ET_2sT_2pU_2p?2s)|C_sⁿ(K|² + $\text{1}\text{5}$ET_2p² (|C_xⁿ(K|² + $\text{3}\text{5}$ET_2p² + $\text{1}\text{3}$T_1sU_1s-2p + T_2sU_2s-2p)² + $\text{2}\text{3}$ √ET_2p(T_1sU_1s-2p + T_2sU_2s-2p) √C_zⁿ(k)|²] where T_nl(E) = fx_O^∞ r²R_nl(tr)_jl(∝Erdr and U_ij = (E_i-E_j fx8r³R_iR_j(r)R_j(r)dr R_i(r) is the radial part of the atomic wave function, C_iⁿ(k), is found from the Schrödinger equation by the tight-binding method. A similar formula is valid for silicon but the number of integrals T_nl and U_ij will be larger owing to the fact that there are more electron states in the silicon atomic core. In eqn. (1) the terms | C_xⁿ|²,|C_yⁿ|² and |C_zⁿ|² have different factors because the A vector is directed along the crystal Z-axis. These factors will be the same when the A vector is directed along (111). Therefore the contribution of p-electrons to the photoelectron spectrum will be proportional to the partial density of p-states.The formula (1) is simplified in the approximation of a plane wave I(ω, E ～ E^{$\text{3}\text{2}$} Σ_n,k[T_s²$\text{1}\text{3}$|C_sⁿ(k)|² + T_p² ($\text{1}\text{5}$|C_xⁿ(k)|² + C_yⁿ(k)|² + C_zⁿ(k)|²)] (2) Figures 1 and 2 show the results obtained from eqns. (1), (2). Here both the spectra calculated in the plane wave approximation and those found experimentally are given. The ratio between maximum III (p-states prevail) and maximum I (s-states dominate) is given in Table 1.As can be seen from Table 1 and Figs. 1 and 2, the spectra calculated in the

4. Relationship between maximum III and maximum I and between photoionization cross-sections of s- and p- electrons in diamond and silicon

     

18.

Transverse spin correlation function of the one-dimensional spin-$\text{1}\text{2}$ XY model     

Takashi Tonegawa 《Solid State Communications》1981,40(11):983-986

The transverse spin pair correlation function p^x_n=<S^x_mS^x_m+n>=<S^x_mS^x_m+n> is calculated exactly in the thermodynamic limit of the system described by the one-dimensional, isotropic, spin-$\text{1}\text{2}$, XY Hamiltonian

$\text{H=?2J}\text{∑}\text{l=1}\text{N}\text{(S}{}^{\mathrm{x}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{x}}{}_{\mathrm{l+1}}\text{+S}{}^{\mathrm{y}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{y}}{}_{\mathrm{l+1}}\text{)}$

. It is found that at absolute zero temperature (T = 0), the correlation function ρ^x_n for n ≥ 0 is given by

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p}}\text{=}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p}}\text{Π}\text{j=1}\text{p?1}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p?2j}}\text{if}\text{n=2p}$

,

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p+1}}\text{=±}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p+1}}\text{Π}\text{j=1}\text{p}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p+2j}}\text{if}\text{n=2p+1}$

, where the plus sign applies when J is positive and the minus sign applies when J is negative. From these the asymptotic behavior as n → ∞ of |?^x_n| at T = 0 is derived to be $\text{|ρ}{}^{\mathrm{x}}{}_{\mathrm{n}}\text{| ～}\text{a}\text{n}$ with a = 0.147088?. For finite temperatures, ρ^x_n is calculated numerically. By using the results for ?^x_n, the transverse inverse correlation length and the wavenumber dependent transverse spin pair correlation function are also calculated exactly.     

19.

Activation energy of field emission flicker noise power due to adsorbed potassium on tungsten     

J. Beben  Ch. Kleint  R. Meclewski 《Surface science》1980,93(1):33-46

The temperature dependence of the field emission flicker noise spectral density functions has been investigated for potassium adsorbed on tungsten (112) planes by a probe hole technique. By integration of the spectral density functions $\text{W(?) = B}{}_{\mathrm{i}}\text{?}{}^{\mathrm{?ge}}\text{i}$ the noise power $\text{(δn}{}^2{}_{\mathrm{\Delta ?}}$ for different frequency intervals Δ? is obtained. From the exponential temperature dependence of $\text{(δn}{}^2{}_{\mathrm{\Delta ?}}$ noise power “activation energies” q_Δ? are determined. Plots of these energies versus coverage show a similar “oscillating” behaviour as recently found for $\text{W(?}{}_{\mathrm{j}}\text{)}$ or which indicates phase transitions of the adsorbed potassium submonolayers. The noise activation energies are discussed in terms of existing models and a comparison is made between the experimental q values and surface diffusion energies E_d as determined by conventional methods.     

20.

Far-infrared reflection spectra,TO- and LO-phonon frequencies,coupled and decoupled plasmon-phonon modes,dielectric constants,and effective dynamical charges of manganese,iron, and platinum group pyrite type compounds     

H.D. Lutz  G. Schneider  G. Kliche 《Journal of Physics and Chemistry of Solids》1985,46(4):437-443

The far-infrared reflection spectra of hot-pressed samples of the pyrites MX₂ with M = Mn, Fe, Ru, and Os and X = S, Se, and Te and PtY₂ with Y = P, As, and Sb are presented in the range from 40 to 700 cm^?1. The spectra show five reststrahlen bands and more or less free carrier contributions due to deviation from stoichiometry. The oscillator parameters ω_j, ρ_j, γ_j, ω_p, and γ_o, the transverse optical phonon frequencies ω_TO, and the coupled plasmon-phonon frequencies Ω₊ and Ω_? were calculated. The uncoupled longitudinal optical phonon frequencies ω_LO were determined from ?Im $\text{(}\text{1}\text{}\text{?}\text{ge}\text{)}$ of the plasmon-free phonon spectra calculated from the oscillator parameters, neglecting the free carrier contributions. The obtained effective ionic charges (Szigeti charges) reveal an increasing covalency of the pyrites in the order pnictides > chalcides and Fe > Ru > Os > Mn compounds. The phonon frequencies reflect the increasing bond strengths on going from 3d to 4d and 5d metal compounds, discussed in a former work (Phys. Chem. Miner.9, 109 (1983)). The true intensities of the phonon modes for using them with respect to lattice dynamical calculations are discussed.     

	Diamond	Silicon
IIII/II
Density of states of valence electrons	2.64	2.29
XPS (in PW-approximation)	0.27	1.65
XPS (in OPW-approximation)	0.41	1.12
XPS (experimental)	0.44	1.40
σs/σp
PW-approximation	25.0	1.7
OPW-approximation	13.6	2.4
Free atom	12.0	3.4

Transverse spin correlation function of the one-dimensional spin-12 XY model

The transverse spin pair correlation function p^x_n=<S^x_mS^x_m+n>=<S^x_mS^x_m+n> is calculated exactly in the thermodynamic limit of the system described by the one-dimensional, isotropic, spin-$\text{1}\text{2}$, XY Hamiltonian

$\text{H=?2J}\text{∑}\text{l=1}\text{N}\text{(S}{}^{\mathrm{x}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{x}}{}_{\mathrm{l+1}}\text{+S}{}^{\mathrm{y}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{y}}{}_{\mathrm{l+1}}\text{)}$

. It is found that at absolute zero temperature (T = 0), the correlation function ρ^x_n for n ≥ 0 is given by

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p}}\text{=}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p}}\text{Π}\text{j=1}\text{p?1}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p?2j}}\text{if}\text{n=2p}$

,

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p+1}}\text{=±}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p+1}}\text{Π}\text{j=1}\text{p}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p+2j}}\text{if}\text{n=2p+1}$

, where the plus sign applies when J is positive and the minus sign applies when J is negative. From these the asymptotic behavior as n → ∞ of |?^x_n| at T = 0 is derived to be $\text{|ρ}{}^{\mathrm{x}}{}_{\mathrm{n}}\text{| ～}\text{a}\text{n}$ with a = 0.147088?. For finite temperatures, ρ^x_n is calculated numerically. By using the results for ?^x_n, the transverse inverse correlation length and the wavenumber dependent transverse spin pair correlation function are also calculated exactly.     

Activation energy of field emission flicker noise power due to adsorbed potassium on tungsten

The temperature dependence of the field emission flicker noise spectral density functions has been investigated for potassium adsorbed on tungsten (112) planes by a probe hole technique. By integration of the spectral density functions $\text{W(?) = B}{}_{\mathrm{i}}\text{?}{}^{\mathrm{?ge}}\text{i}$ the noise power $\text{(δn}{}^2{}_{\mathrm{\Delta ?}}$ for different frequency intervals Δ? is obtained. From the exponential temperature dependence of $\text{(δn}{}^2{}_{\mathrm{\Delta ?}}$ noise power “activation energies” q_Δ? are determined. Plots of these energies versus coverage show a similar “oscillating” behaviour as recently found for $\text{W(?}{}_{\mathrm{j}}\text{)}$ or which indicates phase transitions of the adsorbed potassium submonolayers. The noise activation energies are discussed in terms of existing models and a comparison is made between the experimental q values and surface diffusion energies E_d as determined by conventional methods.     

Far-infrared reflection spectra,TO- and LO-phonon frequencies,coupled and decoupled plasmon-phonon modes,dielectric constants,and effective dynamical charges of manganese,iron, and platinum group pyrite type compounds

The far-infrared reflection spectra of hot-pressed samples of the pyrites MX₂ with M = Mn, Fe, Ru, and Os and X = S, Se, and Te and PtY₂ with Y = P, As, and Sb are presented in the range from 40 to 700 cm^?1. The spectra show five reststrahlen bands and more or less free carrier contributions due to deviation from stoichiometry. The oscillator parameters ω_j, ρ_j, γ_j, ω_p, and γ_o, the transverse optical phonon frequencies ω_TO, and the coupled plasmon-phonon frequencies Ω₊ and Ω_? were calculated. The uncoupled longitudinal optical phonon frequencies ω_LO were determined from ?Im $\text{(}\text{1}\text{}\text{?}\text{ge}\text{)}$ of the plasmon-free phonon spectra calculated from the oscillator parameters, neglecting the free carrier contributions. The obtained effective ionic charges (Szigeti charges) reveal an increasing covalency of the pyrites in the order pnictides > chalcides and Fe > Ru > Os > Mn compounds. The phonon frequencies reflect the increasing bond strengths on going from 3d to 4d and 5d metal compounds, discussed in a former work (Phys. Chem. Miner.9, 109 (1983)). The true intensities of the phonon modes for using them with respect to lattice dynamical calculations are discussed.