On the dynamic conductivity for an electron-impurity system

The correlation function formula for the dynamic conductivity of a system of non-interacting electrons in the field of impurities is analyzed in terms of proper connected diagrams. By selecting those diagrams appropriate in the region of weak coupling and low impurity concentration, a set of coupled equations for the energy broadening γ (ω, ε, n_s) and the energy shift Δ(ω, ε, n_s) is derived, where both γ and Δ depend on the frequency ω of a probing field, the energy ε of the electron, and the concentration, n_g, of impurities. With the assumption of a finite range potential, these equations are solved. It is found that γ (ω, n_s) is smaller than that extrapolated value which the conventional expression γ₀ for the low-concentration collision frequency would predict, in the entire region studied, that the difference γ₀-γ becomes appreciable when the ratio of the average time between scatterings, τ_c, to the average duration of a scattering, τ_d, is 100 or less, that γ (ω, n_s) decreases monotonically from its static value γ (0, n_s), and becomes vanishingly small in the region ω≈1/τ_d, and that in the static limit (ω=0), γ=γ₀[1?(2/π) (γ₀τ_d)+…], that the energy shift Δ is positive, and increases from 0 and reach a peak of magnitude γ₀ as ω is raised from 0. By using the γ and Δ obtained, the dynamic conductivity σ(ω, n_s) for degenerated electrons is calculated. The deviation, σ-σ₀, from the conventional expression σ₀=(?i) (n_ee²/M) [ω-iΓ₀]^?1, (n_e]=number density of electrons), for 0°K, is appreciable when the ratio τ_c/τ_d is 100 or less. The field-term correction, which arises from the modification of the scattering due to the probing field, is found to be negligible in the entire region studied.     

Analyticity properties and a convergent expansion for the inverse correlation length of the high-temperatured-dimensional Ising model

We show that the inverse correlation lengthm(β) (= mass of the fundamental particle of the associated lattice quantum field theory) of the spin-spin correlation function 〈s _x s _y 〉,x, y εZ ^d , of thed-dimensional Ising model admits the representation $$m(\beta ) = - ln\beta + r(\beta )$$ for small inverse temperaturesβ > 0.r(β) is ad-dependent function, analytic atβ = 0.c _n , the nth β = 0 Taylor series coefficient of r(β) can be computed explicitly from the Z^d limit of a finite number of finite lattice A spin-spin correlation functions 〈s₀s_x〉t>Afor a finite number ofx = (x ₁,x₂, ..., x_d), ¦x¦ = ∑ _i ^d 1¦x_i¦< R(n), where R(n) increases withn. Furthermore, there exists aβ' > 0, such that for eachβ ε (0,β')m(β) is analytic. Similar results are also obtained for the dispersion curve ω(p), ω(p)=ω(0)=m, pε(-π, π]^d?1, of the fundamental particle of the associated lattice quantum field theory.     

Pressure dependence of elastic constants of orthorhombic Li2Ge7O15 above and below the ferroelectric transition

The pressure derivatives of the elastic constants c_ij of orthorhombic Li₂Ge₇O₁₅ have been determined at 293 K by the method of pressure-induced shifts of resonance frequencies of thick plates at ca. 15 MHz in the range between 0 and 1500 bar. Approaching the transition at ca. 630 bar, all P_ij = dc_ij/dp (i, j = 1, 2, 3; p pressure) develop strongly negative values. At higher pressures a similar behaviour with reversed sign is observed. The pressure derivatives of the pure “shear resistances” c₄₄, c₅₅, and c₆₆ depend only slightly on pressure even in the vicinity of the transition. The main interactions driving the transition are of the totally symmetric type. The values dK^?1/dp (K volume compressibility) deviate strongly from the quasi-invariant value of ca. 5 observed in almost all stable crystals (dK^?1/dp = ? 1750 at 620 bar and 1380 at 700 bar). The anomalous piezoelastic behaviour reflects the anomalous thermoelastic behaviour: negative P_ij in the low pressure (high temperature) phase correspond to positive T_ij = d log c_ij/dT (T temperature) and vice versa.     

On stochastic spatial patterns and neuronal polarity

To investigate the statistical behavior in the sizes of finite clusters for percolation, cluster size distribution n _s (p) for site and bond percolations at different lattices and dimensions was simulated using a modified algorithm. An equation to approximate the finite cluster size distribution n _s (p) was obtained and expressed as: log?(n _s (p)) = as ? b log?s + c. Based on the analysis of simulation data, we found that the equation is valid for p from 0 to 1 on site and for the bond percolation of two-dimensional (2D) and three-dimensional (3D) lattices. Furthermore, the relationship between the coefficients of the equation and the occupied ratio p was studied using the finite-size scaling method. When \(x = D(p - p_c )L^{y_t }\) , p < p _c , and D was a nonuniversal metric factor. a was found to be related only to p, and the a-x curves of different lattices were nearly overlapped; b was related to the dimensions and p, and the scaled data of the b of all lattices with the same dimension tended to fall on the same curves. Unlike a and b, c apparently had a quadratic relation with x in 2D lattices and linear relation with x in 3D lattices. The results of this paper could significantly reduce the amount of tasks required to obtain numerical data of on the cluster size distribution for p from 0 to p _c .     

Observation of πo mesons with large transverse momentum in high-energy proton-proton collisions

Invariant cross-sections are presented for the inclusive reaction p + p → π^o + anything, Measurements of large transverse momentum π^o's (2.5 GeV/c<p_⊥<9 GeV/c) were made near 90° at the CERN ISR at five centre-of-mass energies (√s = 23.5, 30.6, 44.8, 52.7 and 62.4 GeV. At large p_⊥, the invariant cross-sections are seem to vary with s and p_⊥, in good agreement with a fit of the form Ap_⊥^?nF(p_⊥/√s), with n≈8 and F(p_⊥/√s)≈exp(?26p_⊥/√s).     

Pressure effect on magnetization for (CoTm)90Zr10 (Tm = Cr,Mo) amorphous alloys

Temperature dependences of the forced volume magnetostriction dω/dH and the saturation magnetization σ_s for (CoTm)₉₀Zr₁₀ (Tm = Cr, Mo) amorphous alloys have recently been measured by the 3-terminal capacitance method and a vibrating sample magnetometer and magnetic balance at temperatures from 77 K to the Curie temperature T_c or the crystallization temperature. The pressure coefficient of σ_s0 at 0 K, d ln σ_s0/dp, is estimated from (dω/dH)₀ extrapolated to 0 K using the thermodynamical relation. The values of d ln σ_s0/dp increase in negative value with increasing Tm concentration. The relation between d ln σ_s0/dp and the pressure coefficient of T_c, d ln T_c/dp, estimated indirectly from dω/dH is discussed.     

Height-height correlations for surface growth on percolation networks

The height-height correlations of the surface growth for equilibrium and nonequilibrium restricted solid-on-solid (RSOS) model were investigated on randomly diluted lattices, i.e., on infinite percolation networks. It was found that the correlation function calculated over the chemical distances reflected the dynamics better than that calculated over the geometrical distances. For the equilibrium growth on a critical percolation network, the correlation function for the evolution time t?1 yielded a power-law behavior with the power ζ^′, associated with the roughness exponent ζ via the relation ζ=ζ^′d_f/d_l, with d_f and d_l being, respectively, the fractal dimension and the chemical dimension of the substrate. For the nonequilibrium growth, on the other hand, the correlation functions did not yield power-law behaviors for the concentration of diluted sites x less than or equal to the critical concentration x_c.     

Measurements of relative transition probabilities of ns-4p,nd-4p,and nf-3d transitions in neutral potassium

Relative transition probability (A-value) measurements are reported for the ns-4p (n = 6–15), nd-4p (n = 5–13), and nf-3d (n = 7–14) series transitions in neutral potassium (KI). The results are based on intensity measurements of optically-thin KI spectral lines from a steady-state potassium emission source. The source employed was a radially-symmetric potassium-seeded argon plasmajet containing ? 5 mole per cent K-atoms. Local emission coefficients were obtained by means of the Abel transform. For the prevailing free-stream conditions (P ～ 11 torr, T ～ 3200 K, n_e ～ 10¹⁴ cm^-3), the potassium excited state populations are described by a Boltzmann distribution down to and including the ground state.The reported A-values are normalized to that of the 580.2 nm line (7s-4p_{$\text{3}\text{2}$} transition). The relative accuracy is estimated to be 5–20% for the ns-4p series and 15–25% for the nf-3d transitions. With the exception of the data for the 6s-4p and all nf-3d transitions, the measurements agree to within estimated experimental uncertainties with the values given in the NBS tabulation [Wiese et al. (1969)], although systematic differences are observed. For the exceptions noted, the measurements lie 25–50% below the relative NBS values. Comparison is also made to recent calculations of Lindgard and Nielson, and Ormonde.     

Inclusive single-particle distributions in π±p reactions at 8 and 16 GeV/c

Invariant single-particle cross sections for pion and proton production in π^±p interactions at 8 and 16 GeV/c are presented in terms of integrated distributions as functions of x, reduced rapidity ζ and p_⊥², and also in terms of double differential cross sections E d²σ/(dx dp_⊥²) and dζ dp_⊥²). A comparison of π^± and π^? induced reactions is made and the energy dependence is discussed. It is shown that the single-particle structure function cannot be factorized in its dependece on transverse and longitudinal momentum. For the beam-unlike pion, there is an indication for factorizability in terms of rapidity and transverse momentum in a small central region.     

Peripheral pn production and decay angular distributions in the reaction π−p → (pn)p at 12 GeV/c incident momentum

The reaction $\text{π}{}^{\mathrm{?}}\text{p}\text{→ (}\text{p}\text{n}\text{)}\text{p}{}_{\mathrm{s}}$, where p_s is a slow proton, was measured at 12 GeV/c incident momentum with the CERN-OMEGA spectrometer. Both antiproton and proton were identified uniquely by electronics information. We obtained 1844 events with four-momentum Transfer squared in the range 0.13 ? |t| ? 0.33 GeV² and with invariant masses $\text{M(}\text{p}\text{n}\text{)}$ up to 2.5 GeV. The corresponding cross section in this t range is determined to be σ = 4 ± 0.4 μb. Extrapolating the differential cross section over the whole t range assuming dσ/dt ≈ exp(5.3t) we estimate a cross section of σ = 9.3 ± 2.0 μb. Comparison with data on $\text{π}{}^{\mathrm{?}}\text{p}\text{→ (}\text{p}\text{p}\text{)}\text{n}{}_{\mathrm{s}}$ (where n_s is a slow neutron) in the same t range shows that the $\text{(}\text{p}\text{n}\text{)}\text{p}{}_{\mathrm{s}}\text{and}\text{(}\text{p}\text{p}\text{)}\text{n}{}_{\mathrm{s}}$ cross sections have approximately the same magnitude.     

Effect of hydrostatic pressure on the Curie temperature of the Heusler alloys Ni2MnZ(Z = Al,Ga, In,Sn and Sb)

The pressure derivative of the Curie temperature dT_c/dp of the Heusler alloys Ni₂MnZ(Z = Al, Ga, In, Sn and Sb) has been obtained from the results of temperature dependence of initial permeability under pressure up to about 6 kbar. For all alloys the Curie temperatures increase linearly with increasing pressure at the rate of dT_c/d_p: +0.7 K/kbar for Ni₂MnAl, +1.0 K/kbar for Ni₂MnGa, +0.9 K/kbar for Ni₂MnIn, +1.4 K/kbar for Ni₂MnSn and +4.1 K/kbar for Ni₂MnSb. On the basis of these results, the interatomic dependence of the exchange interaction for Heusler alloys is discussed. The magnetic susceptibilities of those alloys are also reported.     

On the quantum density of states and partitioning an integer

This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For N bosons in a one-dimensional harmonic oscillator potential, it is well known that the asymptotic (N→∞) density of states is identical to the Hardy-Ramanujan formula for the partitions p(n), of a number n into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for p^s(n), the latter being the number of partitions of n into a sum of sth powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain d^s(n) for distinct partitions. We find that the distinct square partitions d²(n) show pronounced oscillations as a function of n about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdos-Lehner formula for restricted partitions for the s=1 case, we use the modified technique to obtain a new formula for distinct restricted partitions.     

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.     

The role of the configuration interaction in excitation of the Cd(II) principal series at electron-atom collisions

The excitation functions of Cd(II) spectral lines, namely, the lines of the 4d ¹⁰5s ² S _1/2?4d ¹⁰ np ² P ₃ ^2/0 (n≤10) principal series and the line corresponding to a transition from the Beitler level 4d ⁹(5s5p ¹ P ⁰)² P ₃ ^2/0 , are analyzed under the conditions of electron-cadmium atom collisions in the electron energy range from excitation thresholds to 400 eV. It is found that the excitation functions of the spectral lines attributed to transitions from the Beitler level are similar to those of the principal series lines corresponding to transitions from the 4d ¹⁰ np ² P ₃ ^2/0 (n=9, 10) levels nearest to the Beitler level. This similarity is explained by the significant admixing of the Beitler level to the initial levels of the principal series lines. It is shown that the admixing substantially affects the excitation functions of this series for the states more distant from the Beitler level (up to n=6).     

Semi-inclusive scaling for largepT events

We consider semi-inclusive reactions of the type p + p → (particle with large p_T) + n charged particles + neutrals, and propose the following scaling law

$\text{E}\text{d}{}^3\text{σ}{}_{\mathrm{n}}\text{d}{}^3\text{p}\text{=}\text{1}\text{(}\text{s}\text{)}{}^{\mathrm{k+1}}\text{H}{}^{\mathrm{2p}}\text{T}\text{s}\text{,}\text{n}\text{s}$

for the distribution function of the large-p_T particle produced in association with n charged particles. This scaling rule is shown to be consistent with present information on single-particle spectra and average associated multiplicities at large p_T. Also, we show that if the associated multiplicity were to continue to increase linearly with p_T, then moments of the multiplicity distribution would increase like powers of s.     

Isometric immersions of pseudo-Riemannian space forms

In this paper we study local isometric immersions f:M_sⁿ(K)→N_s+q²ⁿ⁻¹(c) of a time-like n-submanifold M_sⁿ(K) with constant sectional curvature K and index s into a pseudo-Riemannian space form N_s+q²ⁿ⁻¹(c) with constant sectional curvature c and index s+q, where q≥0, 1≤s≤n−1 and K≠c. We first prove the existence of Chebyshev coordinates of a time-like submanifold M_sⁿ(K) in certain conditions. Afterwards, we generalize the classical Bäcklund theorem for space-like (or time-like) submanifolds in N_n−1²ⁿ⁻¹(c) and N₁²ⁿ⁻¹(c). Finally as an application, in the Chebyshev coordinates, we use the Bäcklund theorem to give a Bäcklund transformation and a permutability formula between the generalized sine-Laplace equation and the generalized sinh-Laplace equation.     

Oscillator strengths for 3dn4s → 3dn4p transitions in the iron series

The observed oscillator strengths for the 3dⁿ4s → 3dⁿ4p transitions in the iron series show anomalous behaviour for Cr and Mn : the former is exceedingly small whereas the latter is unusually large.Theoretical Hartree-Fock gf-values are reported and a considerable discrepancy with experimental values is noted for Cr. The effect of the interaction of 3dⁿ4p with 3d^n?14s4p is studied using a fixed core, multi-configuration Hartree-Fock approximation. This interaction does not improve significantly the agreement with observed values.     

μ-capture and ortho-para transitions in the muonic molecule ppμ

The capture of a muon by protons at different hydrogen densities is considered. More precise values of γ-factors for ortho- and para-states of the muonic molecule ppμ are obtained: 2γ_o = 1.009±0.001 and 2γ_p = 1.143±0.001. Relativistic effects are taken into account in the muonic molecule, and the rate of ortho-para transitions λ_op = (7.1±1.2) × 10⁴s^?1, caused by these effects, is calculated. With this value of λ_op the μ-capture rate in liquid hydrogen is found to be Λ_c = (490±10)s^?1 that is in agreement with the value Λ_c = 460±20 s^? recently measured by the Saclay-CERN-Bologna collaboration.     

The crossover behaviour of the susceptibility of the Ising model on anisotropic lattices

The behaviour of the reduced susceptibility χ′ of a d-dimensional Ising system in the limit t_d = tanhβJ_d → 0 is investigated. The critical exponent γ_n of (∂ⁿχ′/∂tⁿ_d)_td = 0 is, for n = 2,3, found to satisfy γ_n ? (n + 1)γ₀, and hence, by inequalities found by Liu and Stanley, γ_n = (n + 1) γ₀. For d = 2, the latter identify is valid for all n.     

New classifications in the spectrum of helium-like fluorine (F VIII)

New transitions, belonging to the series 2p¹P-nd¹D, 2s³S-np³P and 2p³D, are reported in the spectrum of helium-like fluorine (F VIII).