Lattice Green's function for the simple cubic and tetragonal lattices at arbitrary points

The lattice Green's function for the simple cubic lattice (γ = 1) and tetragonal lattice at an arbitrary point (l, m, n)

$\text{I(a;l,m,n;γ) =}\text{1}\text{π}{}^3\text{∫∫}\text{∫}\text{o}\text{π}\text{cos}\text{lx}\text{cos}\text{my}\text{cos}\text{nx}\text{a?i??γ}\text{cos}\text{x?}\text{cos}\text{y?}\text{cos}\text{x}\text{dxdydz}$

is evaluated, assuming a ? 0, γ ? 0 without loss of generality. The integral I(a; l, m, n; γ) which has singularities at a = ± γ ± 1 ± 1, is expressed in all regions of (a, γ), i.e., for (i) a > 2 + γ, (ii) 2 ? γ > a > γ(γ < 2), (iii) a <γ ? 2 (γ > 2), (iv) a < 2 ? γ (1 < γ < 2) and γ > a (0 < γ < 1), (v) |a ? γ| < 2 and a + γ > 2, in terms of Kampé de Fériet function by the method of the analytic continuation using the Mellin-Barnes type integral. The numerical values are shown in figures. The high temperature susceptibilities of the Heisenberg model of the ferro- and antiferromagnets are calculated using the results of I(a; l, m, n; γ), showing a shift from three to two dimensions and that from three to one dimensions. The correlation function of the isotropic ferromagnet is calculated and the critical index ν is observed to be 1.     

Scaling function for two-point correlations with long-range interactions to order 1/η

The two-point correlation function ? (q, ξ) is calculated in the critical region of momentum space q in terms of a suitable correlation lenght ξ, by means of perturbation expansion to order 1/n, for an n-vector system with long-range interactions decaying as $\text{|}\text{R}\text{/a|}{}^{\mathrm{?(d\; +\; \sigma )}}$, for $\text{|}\text{R}\text{/a|}\text{a}\text{?}\text{1}$, where a is the spacing on a d-dimensional lattice, σ < d < 2σ and 0 < σ ? 2 ? η_SR. The calculations are done in zero field for T ? T_c. Explicit expansions for long-range propagators are developed for $\text{σ « 1}$ and for the neighborhood of σ ? 2 ? η_SR, in terms of which a universal, cut-off independent scaling function is obtained over the whole range of $\text{x = |}\text{q}\text{| ξ}$, and it is shown that the amplitude of the correlation-length dependence of the susceptibility becomes a universal parameter. Both the exponents and the coefficients of the expansion for fixed q as $\text{(T ? T}{}_{\mathrm{<mtext>c</mtext>}}\text{)}\text{T}{}_{\mathrm{<mtext>c</mtext>}}\text{→0}$ are calculated explicitly. The former are shown to require the validity of the operator-product expansion and explicit logarithmic correction terms are obtained for $\text{d = d}{}^1\text{= 3σ/2}$. For these and other dimensionalities, the coefficients are shown to be finite functions of d and σ. The correction to the Ornstein-Zernike form is given explicitly, with non-integer powers of x that have finite coefficients.     

Spin-dependence and glueball mixing with θ(1640) in ordinary meson spectroscopy

It is shown from a fairly general point of view that meson spectroscopy implies that the spin-spin and tensor forces are due to very short-ranged interactions. The (Q₁, Q₂) ? (Q_A, Q_B) mixing of axial vector $\text{I =}\text{1}\text{2}$ mesons implies the presence of a substantial repulsive Thomas term as well as an attractive short-ranged spin-orbit force of comparable magnitude. This analysis makes no reference to detailed potentials or wave functions. Inverted multiplets are predicted as a consequence of the large repulsive Thomas term. The spin-dependent potentials can be interpreted as effective exchanges dominated by short-ranged vector exchange and a confining potential transforming as a Lorentz scalar, although small amounts of other exchanges are also possible. A model-dependent analysis of the gluon annihilation contribution to the mass matrix and two-body decays of the I = 0 2⁺⁺ mesons indicates significant gluon mixing in these states. The presence of a $\text{non-q}\text{q}$ state (glueball?) which mixes with f'(1514) and another I = 0 2⁺⁺ state is required by the mixing model. The possibility that this additional state is θ(1640) is considered. The mass of such a state satisfies f'(1514)<M(G₀)<θ(1640). The model predicts $\text{0.01 < Λ(θ → ηη)/Λ(θ →}\text{K}\text{K}\text{)<0.18}$, with the actual widths sensitive to the details of singlet-octet mixing in the η wave function.     

An experimental determination of the critical exponents at the metal-insulator transition

This work deals with a determination of the magnitude of the critical exponents and their ratio from an analysis of the temperature dependence of the conductivity and magnetoresistivity of doped Ge 〈 As 〉 with an impurity concentration making it an insulator near the metal-insulator transition. The values found are $\text{?? = 0.60 ± 0.04}$, ξ = 1.38 ± 0.07, $\text{ξ}\text{??}\text{= 2.3 ± 0.2}$; neither the critical exponents nor their ratio depend on compensation within 0 < k < 60%.     

Hydrodynamics of an n-component phonon system

The dynamic properties of an n-component phonon system in d dimensions, which serves as a model for a structural phase transition of second order, are investigated. The symmetry group of the hamiltonian is the group of orthogonal transformations $\text{O}$(n). For n ≥ 2 a continuous symmetry is broken for T<T_c, where T_c is the transition temperature. We derive the hydrodynamic equations for the generators of this group, the $\text{1}\text{2}\text{n (n ? 1)}$ angular-momentum variables. Besides the usual hydrodynamics of a phonon system, there are $\text{1}\text{2}\text{n (n ? 1)}$ additional independent diffusive modes for T > T_c. In the ordered phase we find 2 (n ? 1) propagating modes with linear dispersion and quadratic damping. Formally the hydrodynamics is similar in the isotropic Heisenberg ferromagnet (n = 2) or the isotropic antiferromagnet (n ≥ 3). The relaxing modes for T < T_c require special care. We study the critical dynamics by means of the dynamical scaling hypothesis and by a mode-coupling calculation, both of which give the critical dynamical exponent $\text{z =}\text{1}\text{2}\text{d}$. The results are compared with the 1/n expansion. It is shown that for large n there is a non-asymptotic region characterized by an effective exponent $\text{z}\text{?}\text{= φ/2ν}$, where φ is the crossover exponent for a uniaxial perturbation, and ν the critical exponent of the correlation length.     

Systems of ordinary differential equations with nonlinear superposition principles

This work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations.

$\text{x}\text{?}\text{= η(x,t).}\text{(x, η ?}\text{R}{}^{\mathrm{n}}\text{, t ?}\text{R}\text{)}$

. in terms of a finite number of particular solutions. The point of departure is Lie's criterion according to which such a rule exists if and only if the vector fields η(x,t). ? generate a finite dimensional Lie algebra. We provide three different constructive methods for deriving superposition rules and apply them to systems of coupled Riccati equations of the projective and conformal types based, respectively, on the Lie algebra sl(n + 1, R) and o(p + 1, n ? p + 1).     

Scaling function for order-parameter correlations in expansion to order 1/n: I. Short-range interactions

The order-parameter correlation function $\text{G}\text{?}\text{(}\text{q}\text{, ξ}{}_1\text{)}$ is calculated in the critical region of momentum space $\text{q}$ in terms of a second-moment correlation length ξ₁ by means of perturbation expansion to order 1/n, for an n-vector system with short-range interactions, in zero field above T_c, for 2 < d < 4. The scaling function of the $\text{q}$ dependence is obtained in closed form with a precisely identified cutoff-dependent factor which is the amplitude of the correlation-length dependence of the susceptibility. Both the exponents and the coefficients of the expansion for fixed q as t = (T?T_c)/T_c → 0 are given explicitly and the former are shown to be in accordance with the operator product expansion. The coefficients of order 1/n in the terms associated with a t^k(1?α) dependence of the energy density, for integer k ≥ 1, are expected to be explicitly cutoff-dependent and this is verified by the detailed calculations for k = 1. The behaviour for fixed t and q → 0 is shown to be markedly different from the Ornstein-Zernike approximation. Detailed comparison is provided with the scaling function of the t dependence of the correlations appearing in parallel work.     

Determination of the one-pion exchange potential from pn and pp charge exchange scattering

The πNN vertex function is determined from $\text{d}\text{σ}\text{d}\text{t}$ for pn → np and $\text{p}\text{p →}\text{n}\text{n}$ at 8 GeV/c in the interval 0 < ? t < 0.1 GeV². A “regularor mass” of 3.5m_π=488 MeV is found, corresponding to an “extension” of 0.40 fm of the πNN vertex. The resulting OPE potential is discussed.     

Correlation functions of the transverse Ising chain at the critical field for large temporal and spatial separations

We study the behavior of 〈σ₀^x(t)σ_n^x(0)〉 and 〈σ₀^y(t)σ_n^y(0)〉 for the transverse Ising chain at the critical magnetic field at T = 0. Explicit results are obtained for the three distinct regions where t → ∞ and n → ∞with $\text{0 ?}\text{n}\text{t}\text{<1}$, $\text{1 <}\text{n}\text{t}$, or $\text{t = n + n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{(}\text{z}\text{2}\text{)}$ where z is fixed of order one. In this latter region the general Painlevé V solution is shown to reduce to a Painlevé II function. We use our results to discuss the general problem of long-time behavior of Toda equations with slowly decaying initial values.     

Change of photon statistics due to multi-photon absorption

Simple approximative equations governing the temporal behaviour of both the mean photon number, $\text{n}$, and its mean square deviation, Δn², in the process of k-photon absorption (k = 1, 2, …) are derived and solved for initial photon distributions characterized by $\text{n}$ ? 1 and Δn² ? $\text{n}$^-2. It is readily shown that such an initial distribution, in the course of attenuation, tends to a distribution for which $\text{Δn}{}^2\text{= k (2 k - 1)}{}^{-1}\text{n}$. Hence, for k > 1, the distribution is narrower than a Poisson distribution which means that photon antibunching occurs. The feasibility of a Hanbury Brown and Twiss type experiment allowing to detect this effect utilizing two-photon absorption is discussed, and an estimation of the required order of magnitude of the two-photon absorption cross-section is presented.     

Space-time symmetries and the gravitational-neutrino field equations in general relativity

We consider a neutrino field with geodesic rays in interaction with a gravitational field admitting a Killing vector field n^μ. It is found that for solutions of the Einstein-Weyl field equations the neutrino field ξ^A and the neutrino flux vector l^μ are restricted by the equations: $\text{L}{}_{\mathrm{n}}\text{ξ}{}^{\mathrm{<mtext>A\; =\; ?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{is}\text{ξ}{}^{\mathrm{A}}$ and $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= 0}$, whereas s is a real constant. In the case of pure radiation neutrino fields these equations become: $\text{L}\text{ξ}{}^{\mathrm{A}}\text{=}\text{case1}\text{2}\text{(p ? is)ξ}{}^{\mathrm{A}}$, $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= pl}{}^{\mathrm{\mu }}$, where p and s are in general real functions of the coordinates.     

Radial matrix elements of rn for twofold and threefold isotropic simple harmonic oscillators

A generating function for associated Laguerre polynomials and Louck's recursion relations have been employed independently to formulate radial matrix elements of rⁿ for g-fold (g = 2, 3) isotropic simple harmonic oscillators in terms of quantum numbers v and l. This paper gives all relevant matrix elements (vl|rⁿ|v′l′) of rⁿ for n = 1,2, …, 7 and the diagonal element of r⁸. Introduction of the quantities, $\text{V =}\text{1}\text{2}\text{(v + v′ + g)}$ and $\text{L =}\text{1}\text{2}\text{(l + l′ + g ? 2)}$, simplifies the expressions and reveals symmetries and patterns occurring as n increases. Results are compared with those for the one-fold oscillator and sample computer routines have been prepared for numerical evaluation of the matrix elements for g = 1, 2, 3.     

Effects of cation order on the energy bands of GaAs-AlAs heterostructures

Pseudopotential band structure calculations of (GaAs)_n ? (A$\text{l}$As)_n with n = 1, 2, 3 and of Ga_0.5A$\text{l}$_0.5As show that the effects of cation order are small due to the weakness of the Ga-A$\text{l}$ potential difference. The energy bands of the n = 1 heterostructure can be obtained by folding those of the alloy except for small energy shifts and splittings which can be handled in perturbation theory. The superlattice energy gap decreases with increasing n, the shrinkage being 4 meV for n = 2 and 10 meV for n = 3.     

Spins and mixing ratios in 138Ba

The spin assignments to the 1899 (4⁺), 2308 (3⁺ or 4⁺) and 2446 (3⁺) keV levels in ¹³⁸Ba have been confirmed by γ-γ directional correlation measurements. In addition, the multipolarity and $\text{E2}\text{M1}$ mixing parameters for a number of transitions have been established as follows: 409 keV (M1+E2, ?0.75 < δ < ?0.45 or ?0.85 < δ < ?0.05 depending on the choice of J^π = 3⁺ or 4⁺ for the initial state), 463 keV (E2, 0 < δ < 0.15 for $\text{M3}\text{E2}$ admixture), 547 keV (M1+E2, ?0.06 < δ < ?0.015), 872 keV (M1+E2, δ undefined) and 1010 keV (M1+E2, ?0.015 < δ < +0.020).     

The microwave spectrum of 14NH3 in the v = 000011 state

The frequencies and assignments of 50 lines in the pure inversion spectrum of ¹⁴NH₃ in the 0001¹ vibrational state are reported in the microwave frequency region 18–53 GHz and in selected regions up to 58 GHz.The J = 0 inversion frequency, K-type doubling constant K, l = 2, ?1 and molecular dipole moment in this state are 32 904.7 ± 2.0 MHz, 1.958 ± 0.040 MHz and 1.459 ± 0.002 D, respectively, where model inadequacies are included in the uncertainties of the first two parameters. The dipole moment measurements for this and the ground state are in excellent agreement with Stark laser measurements. An expression containing the effective l-type doubling constant is obtained from the combination of frequencies $\text{[ν(1, 1, 1) ? ν(1, 1, ?1) ? ν(2, 1, 1) + ν(2, 1, ?1)]}\text{8}\text{= 10 361.894 ± 0.004}\text{MHz}$. A preliminary value for the l-type doubling constant is 10 655 ± 20 MHz.     

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.     

Photon statistics in saturated m-photon amplification and attenuation of coherent light

The change of photon statistics of a nearly coherent single-mode light beam travelling through a saturable m-photon-amplifying or absorbing medium (m = 1, 2, …) is theoretically investigated. With the help of an approximation scheme valid for $\text{n}$ ? 1, Δn² ? $\text{n}$² ($\text{n}$ mean value and Δn² variance of the photon number) an analytical expression is derived for the quantity Δn²/$\text{n}$. The results holds for arbitrary values of the initial population inversion and of the degree of saturation. It turns out that for both amplification and attenuation processes after sufficiently strong interaction Δn²/$\text{n}$ tends to an asymptotic value.     

Intensities of rotational lines in first overtone bands for axially symmetric molecules of the group C3v

The interaction of vibration and rotation is considered in the computation of the intensities of rotational lines in the first overtone bands of axially symmetric molecules of the group C_3v. The calculation utilizes the contact transformation method through first order of approximation as outlines by Hanson and Nielsen. General formulas for the intensities of the lines in the first overtone bands 2ν_n and 2ν_m are obtained, where n and m denote normal modes of species A₁ and E, respectively. It is found that to this order of approximation the usual selection rules ΔJ = 0, ±1 and ΔK = 0 are observed for the parallel overtone band 2ν_n. For the overtone band 2ν_m, the selection rules are more complicated, being ΔJ = 0, ±1; Δl_m = 0 and ΔK = 0, Δl_m = ±2 and $\text{Δ}\text{K = ?1}$, or Δl_m = ±2 and ΔK = ±2.     

Spectroscopy of electron subbands on Ge-(111)

We report on the resonant excitation of electronic surface subbands at far-infrared energies. Distinct lines for transitions from the lowest-lying n = 0 subband to the n = 1 and 2 states are observed. The transition energies are compared to a subband calculation. Typical linewidths are of order 2 meV and indicate an intersubband relaxation rate which is ～ $\text{1}\text{10}$ of the scattering rate derived from the surface mobility. The 0→ 1 transition shows additional structure on the low energy side.     

A study of high transverse momentum π0's at ISR energies

We present a study of the transverse momentum spectrum of π⁰'s produced at c.m. angles θ = 90° and 53° in pp collisions at √s = 23.6, 30.8, 45.1, 53.2, and 62.9 GeV. The experiment was performed with a lead-glass detector. The data can be described with the usual parametrization $\text{p}{}_{\mathrm{<mtext>T</mtext>}}{}^{\mathrm{?n}}\text{?;(x}{}_{\mathrm{<mtext>T</mtext>}}\text{, θ)}$, with n = 7.2 ± 0.2. Comparison between the 90° and 53° data further indicates no appreciable dependence on θ, at least for x_T < 0.3. Two-particle inclusive cross sections for π⁰'s produced alongside are also presented. They are observed to have a dependence upon the transverse momentum of the dipion similar to that of single-particle cross sections and with the same value of n. Two-photon decays of η mesons are observed between 3 and 4 GeV/c transverse momentum with a production cross section half of that of π⁰.