Numerical evaluation of the 1S12-state radiative level shift

The one-photon self-energy radiative level shift for an electron in a Coulomb potential is evaluated numerically for the $\text{1S}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ state. The evaluation is done for values of the nuclear charge Z = 10, 20,…, 110. The errors in the values obtained are estimated to be less than 0.1 %. The results are compared with the results of previous calculations. The evaluation is based on the expressions given in the preceding paper.     

A comment on the potential in heavy quarkonium quantum-mechanical systems

I discuss the characteristics of the non-relativistic potential responsible for the ψ and ? systems. In particular a comparison is made between the recently advocated flavour independent power behaved potential and the QCD inspired Coulomb + linear potential. The ratios $\text{ΔM}{}_{\mathrm{?}}\text{ΔM}{}_{\mathrm{\psi }}$ of radical excitation energy level differences are used as sensitive tests to the r dependence of V(r). Deviations of these ratios from a constant value may indicate a gradual increase in the effective power of the potentials as r increases (in agreement with the Coulomb + linear potential). Decisive tests have to come from the not yet discovered toponium family where the ratios $\text{ΔM}{}_{\mathrm{T}}\text{ΔM?}$ should start to become greater than 1 if a singularity of the potential exists at r → 0.     

A model for phase transitions in Tetrathiafulvalene-Tetracyanoquinodimethane (TTF-TCNQ)

The observed phase transitions in Tetrathiafulvalene-Tetracyanoquinodimethane (TTF-TCNQ) are discussed using a simple model for the interchain coupling of charge density waves. Estimates based on Coulomb energies show that for 38 K < T < 49 K the components $\text{q}{}_{\mathrm{x}}\text{=}\text{π}\text{a}\text{+ q′}{}_{\mathrm{x}}$ and q_z of the wave vector associated with the charge density wave satisfy $\text{q}{}_{\mathrm{z}}\text{c}\text{q′}{}_{\mathrm{x}}\text{a?0.1}$, with $\text{q′}{}_{\mathrm{x}}\text{a～(T}{}_2\text{? T)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and T₂ = 49 K. A possible mechanism for the first order transition at 38 K is proposed. The results are compared with neutron and X-ray scattering and with isotope shifts of the transition temperatures.     

PI-Muonium: A determination of the pion charge radius

The $\text{2}\text{S}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? 2}\text{P}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ energy level shift of the π-gm atom is studied. It is shown that an experimental determination of the $\text{2}\text{S}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? 2}\text{P}\text{f}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ energy level shift accurate to one part per thousand can provide an independent determination of the pion charge radius.     

Spectroscopic information on ground-state Ar2, Kr2, and Xe2 from interatomic potentials

Calculations of vibrational and rotational level spacings of homonuclear inert gas diatomic molecules by numerical integration of the radial Schrödinger equation are presented. The potentials which were used for the ground states of Ar₂, Kr₂, and Xe₂ were obtained from accurate fits to the molecular beam scattering data. From the calculated $\text{Δ}\text{G}{}_{\mathrm{<mtext>v+</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{'}\text{s}$ and B_v's, the following spectroscopic constants (in cm^?1) were fitted: for Ar₂ω_e = 31.92, ω_ex_e = 3.31, ω_ey_e = 0.11, B_e = 0.060, α_e = 0.004; for $\text{Kr}{}_2\text{ω}{}_{\mathrm{e}}\text{? 23.99}$, $\text{ω}{}_{\mathrm{e}}\text{x}{}_{\mathrm{e}}\text{? 1.30}$, $\text{ω}{}_{\mathrm{e}}\text{y}{}_{\mathrm{e}}\text{? 0.021}$, $\text{B}{}_{\mathrm{e}}\text{? 0.024}$, $\text{α}{}_{\mathrm{e}}\text{? 0.001}$; for $\text{Xe}{}_2\text{ω}{}_{\mathrm{e}}\text{? 21.26}$, $\text{ω}{}_{\mathrm{e}}\text{x}{}_{\mathrm{e}}\text{? 0.75}$, $\text{ω}{}_{\mathrm{e}}\text{y}{}_{\mathrm{e}}\text{? 0.008}$, $\text{B}{}_{\mathrm{e}}\text{? 0.013}$, $\text{α}{}_{\mathrm{e}}\text{? 0.0004}$.     

On the charged-impurity-limited mobility for a very pure semiconductor

The low temperature mobility μ limited by charged impurities in very pure semiconductors (Ge) is interpreted in terms of the Coulomb collision in medium. The theory yields a peculiar dependence in temperature T and charged impurity concentration $\text{n}{}_{\mathrm{s}}\text{: μ ∞n}{}_{\mathrm{s}}{}^{\mathrm{<mtext>-</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$.     

Coupling constants for several light nuclei from a dispersion analysis of nucleon and deuteron scattering amplitudes

A forward dispersion relation cannot be applied to charged particle scattering amplitudes unless the influence of the Coulomb interaction is explicitly considered. Earlier studies have shown how Coulomb effects can be taken into account when direct (s-channel or bound-state) poles are investigated. In this paper we extend the Coulomb modification to include I = 0 exchange (u-channel) processes as well. We then apply a forward dispersion relation to empirical d + α, p + d and n + d elastic scattering amplitudes which contain both direct and exchange poles with and without Coulomb effects. We obtain detailed and model-independent information on the following vertices: ⁶Li-α-d (S- and D-state) ⁴He-d-d, ³He-d-p, ³H-d-n and d-p-n. From the coupling constants we calculate the asymptotic normalization (spectroscopic factors) C²₁ of the corresponding cluster wave functions, which become: $\text{C}{}^2{}_0\text{(}{}^6\text{Li}\text{, α}\text{d}\text{) = 4.62 ± 0.23}$, $\text{C}{}^2{}_2\text{(}{}^6\text{Li}\text{, α}\text{d}\text{) = (1 ± 6) × 10}{}^{\mathrm{?4}}$, C²₀(α, dd) < 2, $\text{C}{}^2{}_0\text{(}{}^3\text{He, dp}\text{) = 3.5 ± 0.4}$, $\text{C}{}^2{}_0\text{(}{}^3\text{H, dn}\text{) = 2.6 ± 0.3}\text{and}\text{C}{}^2{}_0\text{(}\text{d, np}\text{) = 1.66 ± 0.1}$.     

The pair correlations down to r = 0 in a fully ionized dilute and homogeneous hydrogen plasma in equilibrium

The pair correlations in a fully ionized dilute and homogeneous hydrogen plasma in equilibrium are calculated down to r = 0 on the basis of the equilibrium hierarchy of statistical mechanics and the linear superposition approximation. Thereby the Coulomb potential between electron and proton is modified by a damped potential with finite value at r = 0 which is due to quantum corrections given by Hagenow and Koppe. The solution of the hierarchy equations is quite simple: For $\text{r > γ}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{l}{}_{\mathrm{D}}\text{(γ = 5 × plasma parameter, l}{}_{\mathrm{D}}\text{= Debye-length)}$ one obtains the well-known result by Debye and Hückel, for $\text{r < γ}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{l}{}_{\mathrm{D}}$ the pair distribution functions are simply given by the corresponding Boltzmann factors. The calculation of the energy of the system leads to the result that the nonideal part is coming from the Debye-Hückel part of the pair correlations only.     

Functional derivative δTc/δα2 (Ω)F(Ω) for weak coupling superconductors

We present approximate analytic calculation of the functional derivative $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{(Ω)F(Ω)}$, where T_c is the superconducting critical temperature and $\text{α}{}^2\text{(Ω)F(Ω)}$ is the electron-phonon spectral function, within the “square-well model” for the phonon mediated electron-electron interaction and weak coupling limit $\text{ω}{}_{\mathrm{D}}\text{(2πT}{}_{\mathrm{c}}\text{)? 1 (ω}{}_{\mathrm{D}}$ is the Debye energy). It is found that $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{(Ω)F(Ω) = (1 + λ)}{}^{-1}\text{G(Ω)}$ where λ is the familiar electron-phonon coupling parameter and $\text{G(Ω)}$ is a universal function of the reduced frequency $\text{Ω}\text{=}\text{Ω}\text{T}{}_{\mathrm{c}}$. We compare this formula with accurate numerical results for several weak coupling superconductors. The overall agreement is good     

On some recent results concerning the superconducting transition temperature

The Eliashberg gap equations relate the transition temperature T_c of an isotropic superconductor to its electron-phonon spectral function α²F(ω) and Coulomb pseudopotential parameter $\text{μ}{}^1$. Recently the Eliashberg theory has been used to derive some supposedly rigorous results bearing on the problem of attaining higher superconducting transition temperatures: Bergmann and Rainer derived an expression for the functional derivative $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{F(ω)}$; Allen and Dynes showed that in the asymptotic limit of very large $\text{λ(λ?10)k}{}_{\mathrm{B}}\text{T}{}_{\mathrm{c}}\text{=f(μ}{}^1\text{)(λ〈ω}{}^2\text{〉)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and Leavens proved that for any isotropic superconductor k_BT_c ?0.2309A, where A is the area under its electron-phonon spectral function. In this letter we show that the result of Allen and Dynes is not compatible with the other results and is, in fact, incorrect.     

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.     

Information on the dynamics of the reaction KN → K1N from KLN interactions

If $\text{K}{}_{\mathrm{S}}{}^1\text{,}{}_{\mathrm{L}}\text{is a K}{}^1$ resonance decaying into K_S,L (the short- and long-lived kaon) and a neutral system S^o of pions, one can isolate the C-even and C-odd, crossed-channel contributions to $\text{KN → K}{}^1\text{N}$ by using the reactions $\text{K}{}_{\mathrm{L}}\text{N → K}{}_{\mathrm{S}}{}^1\text{,}{}_{\mathrm{L}}\text{N}$ whether S^o is a C-eigenstate, or a mixture of C-even and C-odd states. Applications to the production of K¹(890) and the Q-meson are discussed, and simple numerical predictions made for Q_S,L production. Q-production data indicate approximate t-channel helicity conservation for the ω and P' exchanges at vertices involving a spin change, in similarity to the belief for the pomeron. Q_S,L production data can give information also on Q-decays.     

The muonium energy levels for n = 2

The initial results of a calculation of the muonium energy levels for n = 2 is reported. The results, which contain all recoil corrections through order α⁵ are combined with previously calculated higher-order binding corrections to yield 1047.03 MHz for the muonium Lamb shift and 10 921.50 for the $\text{2}\text{P}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{?2P}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ fine structure interval. A lower- order, quadratic recoil term, which is insignificant for the hydrogen Lamb shift, contributes ?0.17 MHz to the muonium Lamb shift.     

Valence band parameters and free exciton reduced mass of zinc telluride derived from free exciton magnetoreflectance

Reflectance spectra were measured on ZnTe in magnetic fields up to 18 T for B ? [100] and B ? [110]. The experiments yield renormalized valence band parameters $\text{γ}{}^1{}_2\text{= 0.83 ± 0.08}$ and $\text{γ}{}^1{}_3\text{= 1.30 ± 0.12}$, corresponding to bare parameters γ₂ = 0.95 ± 0.09 and γ₃ = 1.48 ± 0.14. From the free exciton Rydberg energy $\text{R}{}^1{}_0\text{= 12.8}\text{meV}$ we derive a reduced exciton polaron mass m₀ 0.080 ± 0.005 and a bare reduced mass m₀ 0.074 ± 0.005, corresponding to $\text{γ}{}^1{}_1\text{= 3.9 ± 0.7}$ and γ₁ = 4.4 ± 0.7 for an electron effective polaron mass $\text{m}{}^1{}_{\mathrm{e}}\text{= 0.116 m}{}_0$. We further calculate the exciton diamagnetic shift rate according to existing low-field theories modified by a variational calculation taking into account polaron effects and valid up to γ ? 1. The difference between experiment and theory is 10% and the agreement is considered satisfactory.     

Exact two-point correlation functions of turbulence without pressure in three dimensions

We investigate exact results of isotropic turbulence in three dimensions when the pressure gradient is negligible. We derive exact two-point correlation functions of the density in three dimensions and show that the density-density correlator behaves as , where $\text{α}{}_3\text{=2+}\text{1}\text{6}\text{√33}$. It is shown that, in three dimensions, the energy spectrum E(k) in the inertial range scales with exponent $\text{2−}\text{1}\text{12}\text{√33⋍1.5212}$. We also discuss the time scale for which our exact results are valid for strong 3D turbulence in the presence of pressure. We confirm our predictions by using the recent results of numerical calculations and experiment.     

The tensor J8 contribution to the rotational energy of a tetrahedral molecule

Two methods are outlined for calculating the tensor distortion energy of a tetrahedral molecule to the eighth degree in the rotational angular momentum. In the first, the entire distortion Hamiltonian including the eighth-rank distortion tensor $\text{Ω}{}_8$ is diagonalized. In the second, the basis which diagonalizes only the fourth rank tensor $\text{Ω}{}_4$ is used to generate tables which allow the direct determination of the octic energy terms. For J ≤ 20, numerical values are given of the diagonal matrix elements of $\text{Ω}{}_8$ and of the second-order perturbation correction due to offdiagonal matrix elements of the sixth-rank tensor $\text{Ω}{}_6$. Explicit expressions in terms of angular momentum operators are given for $\text{Ω}{}_8$ and a selection of the operator equivalents T_8,m.     

Surface plasmons in layered structures and their role in superconductivity

The contribution of surface plasmons to electron-electron interaction is examined by calculating the effective Coulomb repulsion μ in the case of periodically spaced conducting planes separated by an insulator of dielectric constant ?_i. Substantial reduction in μ is obtained when $\text{?}{}_{\mathrm{i}}\text{》 1}$.     

Neutron-proton elastic scattering at 325 MeV

Accurate measurements of the P and D_T parameters in np elastic scattering at 325 MeV resolve a major ambiguity in the phase shift analysis. The mixing parameters $\text{ε}{}_1\text{= 8.4 ± 0.6°}$ and $\text{ε}{}_3\text{= 7.2 ± 0.4°}$ are particularly well determined, and in good agreement with theoretical expectations.     

Energies and orbital sizes for some Rydberg and valence states of the nitrogen molecule

Accurate SCF computations are reported on the Rydberg states of N₂ of electron configurations $\text{---1π}{}_{\mathrm{u}}{}^3\text{2π}{}_{\mathrm{u}}$, $\text{---1π}{}_{\mathrm{u}}{}^3\text{3σ}{}_{\mathrm{u}}$, and ---3σ_g2π_g, also on the valence states of the configuration $\text{---1π}{}_{\mathrm{u}}{}^3\text{1π}{}_{\mathrm{g}}$. The Rydberg state calculations supplement those of Lefebvre-Brion and Moser. A comparison is made between the $\text{---1π}{}_{\mathrm{u}}{}^3\text{2π}{}_{\mathrm{u}}$ states and the parallel set of states of the $\text{1π}{}_{\mathrm{u}}{}^3\text{1π}{}_{\mathrm{g}}$ configuration. This comparison shows a sharp difference in the ¹Σ⁺ states of the two configurations, the ¹Σ⁺ state being very high in the latter but relatively low in the former configuration. Recknagel coefficients are given for the several states of the two configurations; as expected, these are much smaller for the $\text{1π}{}_{\mathrm{u}}{}^3\text{2π}{}_{\mathrm{u}}$ configuration. Also, the ¹Δ state is relatively lower for the latter configuration.