Angular Momentum Expansion Research Articles

Diffraction of non-relativistic electron waves by a cylindrical capacitor

The diffraction of non-relativistic electron waves by a cylindrical capacitor is considered for an electric voltage at the capacitor small compared with the energy of the incident electrons. On the basis of the exact general solution of the Schrodinger equation for an electron in an attractive logarithmic potential, asymptotic solutions inside the capacitor which are similar to JWKB-type solutions, but with a significant modification, are derived. Application of appropriate boundary conditions to the asymptotic solutions yields an angular momentum expansion of the scattered wave which is further evaluated by means of the Sommerfeld-Watson transformation. The change of various well known diffraction phenomena with increasing electric voltage at the capacitor is calculated explicitly and discussed in detail; in particular, the convergence of electron interference fringes towards the optical axis is one of the main results of the investigation.

An angular momentum expansion of energy and structure of high spin states

A new angular momentum expansion of level energies of ground-state band of even-even nuclei has been obtained which is found to converge rapidly even for the most back-bending nuclei. Attempts have been made to interpret the parameters and calculate them microscopically. It is found that nuclear structure in the forward bending region is quite different compared to that in the back-bending region.

The atomic string approximation in high energy electron diffraction

It is often useful to study dispersion surfaces in order to understand the complicated diffraction effects which are observed in transmission electron micrographs. To obtain these dispersion surfaces accurately from the many-beam equations in cross-grating high energy electron diffraction it is necessary to diagonalise matrices at least of the order 60×60 even for the simplest structures. Such a method cannot easily be used to relate the form of the dispersion surface to the projected potential U(R) seen by the fast electrons. It is possible, however, to solve an equivalent two-dimensional band structure problem for the fast electrons in the crystal. The KKR method, which exploits the near cylindrical symmetry of the projected potential within each atomic string in order to expand the Bloch waves in terms of angular momentum basis functions eiℓθ, enables such an approach to be made and requires the solution of small secular determinants.Defining the transverse energy S to be zero at the maxima of the atomic string potentials, the atomic string approximation(ASA) reduces the KKR equations for one string per unit cell of the projected potential to the formwhere the eigenvector components C are simply related to the coefficients in the angular momentum expansion of the Bloch waves and the structure constants Mℓℓ'(K) only depend on the arrangement of the atomic strings and the orientation K.

Effect of alloying on the Fermi surface

We obtain a theoretical expression for the change in Fermi-surface areas when dilute impurities are introduced in a nearly-free-electron-like host metal or in a $d$-band host metal having no resonances in the immediate vicinity of the Fermi energy. Following Stern, we obtain the Fermi surface from the real part of the self-energy. The self-energy is derived within a pseudopotential $T$-matrix formalism which is developed as an extension of usual scattering theory. We approximate the crystal lattice Green's function by a free-electron propagator in the $T$-matrix equation. The final expression for Fermi-surface changes involves impurity scattering phase shifts and pseudo-wave-function angular momentum expansion coefficients. These are the same parameters which enter the expression for the electron lifetime. The theoretical results are in agreement with recent de Haas-van Alphen measurements in dilute lead and magnesium based alloys.

Adiabatic dilations and essential Regge spectrum

Adiabatic perturbation theory works for non-conserved dilation operators. At each effective coupling, all renormalization group functions are dilation eigenvalues of a “tangent” field theory, which preserves scale invariance instead of translation invariance. The adiabatic expansion reconstructs the true field theory from this tangent bundle. For complex angular momentum, the short and long distance expansions mix indecomposably at certain turning points. By connecting the UV and IR limits across them, the adiabatic method determines Regge intercepts from integrals over the renormalization group. Fixed Regge cuts or accumulations of poles are insensitive to the IR region.

Comparison of the angular velocity and the angular momentum expansions for yrast band level energies in even-even nuclei

The results from the three- and the four-parameter expressions of the angular velocity expansion (AVE) and the angular momentum expansion (AME) are compared with the available experimental information on transition energies for yrast band levels in even-even nuclei. It is found that from the point of view of range of applicability, degree of fit to experimental transition energies, and high spin behavior, AME gives distinctly better results than AVE. The critical spin for backbending calculated from three-parameter AME is found to be in a fair agreement with the experimental data.NUCLEAR STRUCTURE Even-even rare earth nuclei, calculated yrast band transition energies, three- and four-parameter expansions in angular velocity and angular momentum, compared with experimental data.

The T matrix and the K matrix in cluster scattering

The authors investigate the scattering properties of small clusters of atoms using the multiple scattering formalism developed by Lloyd (1967). The transition and reaction matrices for different ensembles of scatterers are evaluated and used to calculate the density of electron eigenstates and the angular scattering cross sections. The dependence of the convergence of the angular momentum expansions for these quantities on the orientation, geometry and size of the cluster is examined. The calculations have been made for clusters of carbon atoms arranged in the geometry of the diamond lattice. It is found that it is more expedient to determine the density of states from an expression derived by Lloyd than to employ the cluster scattering matrices. An initial insight is obtained into the scattering properties of the tetrahedrally bonded elemental semiconductors and correlation with the known existence of an absolute energy gap in the corresponding crystalline and amorphous solids is discussed. Definite evidence is given for the 'channelling' of electrons along the bonds at energies well above the muffin tin zero.

Problems with Perturbation Treatments of Anharmonic Nuclear Oscillations

Microscopic expressions are obtained for the coefficients in the angular momentum expansion ${E}_{I}={a}_{2}(\frac{1}{2}I)+{a}_{4}{(\frac{1}{2}I)}^{2}+\ensuremath{\cdots}$ of the energies of quasirotational members of quadrupole vibrational bands. Some limitations on the applicability of perturbation treatments of anharmonic corrections to the random-phase approximation are noted.

Singularities in Forward Multiparticle Scattering Amplitudes and the S-Matrix Interpretation of Higher Virial Coefficients

The role of singular forward multiparticle scattering amplitudes in S-matrix formulas for the higher virial coefficients is studied in detail. It is shown that by means of a simple limiting process one can give a precise, unambiguous meaning to the traces over on-shell scattering amplitudes which appear in our previous formulas for the virial coefficients. The general arguments, in which we maintain a reasonable level of rigor, are supported by explicit calculations for the third and fourth virial coefficients. Also, the angular momentum expansion used in earlier work is shown to converge.

Self-Consistent Perturbation of Hartree-Fock-Bogoliubov Equations and Nuclear Rotational Spectra. II

The coefficient $\mathcal{B}$, occurring in the angular-momentum expansion ${E}_{I}=\mathcal{A}I(I+1)+\mathcal{B}{I}^{2}{(I+1)}^{2}+\ensuremath{\cdots}$ of the energy levels of the ground-state rotational band of even-even deformed nuclei, is numerically calculated on the basis of a microscopic model derived in a previous paper. Numerical estimates of the $z$ factors, which measure the effect of band-mixing on $\ensuremath{\gamma}$-ray branching ratios from vibrational to ground-state bands, are also presented.

Crossing-Symmetric Watson-Sommerfeld Transformation

The derivation of a crossing-symmetric WatsonSommerfeld transformation that simultaneously displays the contributions of Regge poles in all three channels is presented. An analysis that attempts to free Regge poles from their strict attachment to angular momentum and Legendre expansions is used. (C.E.S.)

Angular Momentum Expansions in Relativistic Field Theory

As a step toward more general applications of angular momentum expansions in quantum field theory the expansion of the scattering matrix is examined in the case of scattering of spin 0 by spin 1/2 particles. The matrix is represented in terms of its eigenvalues and eigenvectors, the latter being eigenstates of total angular momentum. Using eigenstates of helicity to simplify the discussion, the eigenvectors raay sometimes be obtained from the conservation laws alone. The eigenvalues are computed only to second order in a Yukawa interaction, but the results are more useful than the usual second-order matrix elements. Since angular momentuni expansions lead effectively to solutions of operator equations, the expressions derived facilitate the relativistic application of Heitler-s unitary approximation (with an exact solution of the equation relating the transition operator T and the Hermitian reaction operator K) or the determinantal method of Schwinger and Baker. (auth)

An analysis of the spin and parity of the τ-meson

The generality of the double angular momentum expansion for the final state in T-meson decay (1) is pointed out. The significance of the relativistic corrections is emphasized. A relativistic analysis is made of the energy and angular distribution of 481 collected events. One or more terms of the expansion are considered. In addition, a classical analysis is made of partially integrated distributions. On the basis of the analysis it is concluded that the τ-meson may have [spin, parity] of [0-], [1 + ] or [3 + ] on the basis of a weak set of assumptions concerning the interaction. If stronger assumptions are utilized one can eliminate the [1 + ] case or even both the [1 + ] and [3 + ] cases.