Convergence properties of the Brillouin-Wigner type of perturbation expansion

We study the Brillouin-Wigner perturbation expansion of the model-space effective Hamiltonian corresponding to the full Hamiltonian H(x) = H₀ + xH₁, H₀ and H₁ being respectively the unperturbed and the interaction Hamiltonian and x being a strength parameter. The radius of convergence for the perturbation expansion is related to the poles of the energy-dependent effective interaction, and the location of these poles in the complex x-plane is discussed. The situation with poles lying off the real x-axis is examined. In terms of the spectrum of the unperturbed Hamiltonian H₀, some necessary conditions for convergence are derived, and the effects of intruder states are discussed. It is shown that the BW expansion of the ground-state energy can always be made convergent by a shift of the unperturbed energy spectrum.     

Coherent-state theory for the proton-neutron quasispin group

An extended coherent-state theory ^{1, 2}) is used to give a simple construction for the matrix elements of the proton-neutron sp(4) quasispin algebra. A very simple analytical expression is given for the matrix elements of the sp(4) generators valid for all cases where initial and final states are multiplicity-free with respect to the U(1) × SU(2) n, T subalgebra. In the more general case, involving multiplicity, the most natural orthonormal basis constructed by the coherent-state method leads to many-nucleon states in the seniority scheme which are approximately labelled by T_p, the isospin of the p nucleon pairs coupled to J = 0, T = 1.     

Magnetopolaritons in ZnTe

From an invariant expansion, we construct the exciton Hamiltonian for the Γ₆×Γ₈ excitons in theT _d -type material ZnTe represented by an 8×8 matrix including the influences of a finite wave vector and an external magnetic field. We diagonalize the Hamiltonian matrix to obtain the exciton states. Then the excitons are coupled to the electromagnetic radiation field thus giving the polariton states. The theoretical dispersion curves are fitted to the results of two-photon Raman scattering and reflection experiments in magnetic fields up to 22 T. From this fit we deduce precise values for the eigenergies, exciton masses,g-factors, and diamagnetic shifts.     

The torsional-wagging tunneling problem and the torsional-wagging-rotational problem in methylamine

A theoretical formalism is presented for fitting rotational energy levels in isolated (unperturbed) vibrational states of methylamine. This formalism is obtained by recasting and extending theoretical studies in the earlier literature, which were undertaken to help analyze the methylamine microwave spectrum. The present formalism is applicable when both the NH₂ umbrella (wagging) motion and the CH₃ internal rotation (torsion) motion take place near the high-barrier limit and leads to the usual Fourier sine and cosine series expansions for molecular energy levels. The derivation is separated into two parts, one treating the large-amplitude vibrational problem (the torsional-wagging problem) by itself, the other treating the torsional-wagging-rotational problem. In both treatments, permutation-inversion group and extended group ideas are used to determine the allowed terms in an effective rotational-tunneling Hamiltonian operator and to block diagonalize the matrix representation of this operator for a near-prolate symmetric top. The resulting energy levels and selection rules are discussed, but application of the method in detail to the methylamine spectrum is planned for a later paper.     

Superconductors containing impurities with crystal-field split energy levels

We investigate theoretically the problem of a superconducting matrix containing paramagnetic rare earth impurities with crystal-field split energy levels. There are two competing mechanisms which change the superconducting transition temperatureT _c. One is inelastic charge scattering of conduction electrons from the aspherical part of the 4f charge distribution, which leads to an increase inT _c similar to that of optical phonons. The other and often predominant mechanism comes from the exchange interaction, which depressesT _c and can be very effective even among non-magnetic levels via off-diagonal matrix elements. Crystalline fields serve to alter the effectiveness of the two kinds of scattering depending upon the symmetry character of the low-lying levels, and in favorable cases one may study separately the effects of the two types of scattering by adding different impurities to a given host. We find that crystal-field levels at energies quite high compared tok _B·T_c can still have an important effect onT _c. It is shown that the crystalline-field splitting should be directly observable as structure in the tunneling characteristics.     

Scalars coupled to fermions in 1+1 dimensions

Using a method introduced in an earlier paper, we study a Bose field coupled to a Fermi field in 1+1 space-time dimensions. We employ the standard Hamiltonian formalism in which one computes the eigenvalues and eigenvectors of the Hamiltonian matrix. The matrix elements are computed using states defined on a lattice in momentum space. The results are compared with known strong and weak coupling limits. Bound states and renormalization effects are studied. We find that the choice of bare masses which give specified physical masses can be non-unique once a critical couplingλ _μ has been exceeded.     

Spatially inhomogeneous states of charge carriers in graphene

The interaction of two-dimensional quasiparticles characterized by a linear dispersion E = ±u|p| (graphene) with impurity potentials is studied. It is shown that discrete levels corresponding to localized states are present in a one-dimensional potential well (quantum wire), whereas such states are absent in a two-dimensional well (quantum dot). The cross section for the scattering of electrons (holes) of graphene by an axially symmetric potential well is determined. It is shown that the cross section tends to a constant value in the limit of infinite particle energy. The effective Hamiltonian is derived for a curved quantum wire of graphene.     

Eine Integralgleichungsmethode zur Behandlung der Streuung an gebundenen Teilchen

First it is pointed out that various methods known for the treatment of multi-particle scattering problems such as the methods ofLax, Watson, andFaddeev are based on the same type ofT-operator equations with eliminated interactions. They only differ by the identification of the interactions. — Then an integral equation treatment for the scattering of a particle by a system ofn bound particles is developed. If the scattering occurs via local two-body forces, the interaction matrix element splits into that of the two-particle case and a momentum-dependent factor. This fact is used to simplify the scattering equations which then get a mathematical structure similar to that of theT-operator equations discussed at the beginning (however, involving sums of bound states rather than sums of interactions) and which, therefore, can be handled in a similar way. When the interactions are eliminated by means of the two-particle scattering amplitudes, the off-shell energy parameter of these amplitudes may be chosen to be dependent on quantum numbers of the bound system. Such a choice shows indeed to be favorable if one likes to keep only the lowest order approximation of the integral equations. The resulting approximate formula leads, after some further approximations, for resonant scattering to a formula ofLamb, and for weakly energy-dependent amplitudes to a formula ofFermi (being related to the impulse approximation). — The resonant scattering formula is applied to a quantum-mechanical derivation of a method for the determination of nuclear lifetimes which had been proposed on semiclassical arguments byCiocchetti et al. — Finally the method developed for the scattering of a particle by a system of bound particles is extended to collisions between composite particles.     

Analyses of inelastic proton scattering from 20Ne

Differential cross sections and analyzing powers for inelastic proton scattering to the 2⁺₁, 4⁺₁ and 6⁺₁ states of ²⁰Ne are analyzed using an antisymmetrized distorted wave approximation in direct reaction theory. The spectroscopy of these states was obtained by angular momentum projection from an axially symmetric minimal-energy Hartree-Fock intrinsic state.     

6Li electromagnetic form factors and phenomenological cluster models

The longitudinal form factors of the ground and 2.18 MeV (3⁺, T = 0) states, and the transverse form factors of the 3.56 MeV (0⁺, T = 1) and 5.37 MeV (2⁺, T = 1) states of ⁶Li are compared with the predictions based on fully antisymmetrized α-d and t-τ cluster models. The longitudinal form factors are adequately described by the α-d model, but the transverse form factors seem to be more consistent with a t-τ model which is close to the shell-model limit. Estimates are made for the ground state t and α spectroscopic factors. The 3.56 MeV M1 transition current density is calculated for both models and compared with experiment.     

Extension of Yafet's theory of spin relaxation to ferromagnets

By making use of Kramer's degeneracy of the electronic states in a nonmagnetic material, Yafet has derived an expression for the spin relaxation time T₁ due to scattering of electrons at phonons which involves the properties of electronic and phononic states and the matrix elements for the scattering. It is shown that an analogous expression for T₁ can be derived for ferromagnets (where Kramer's degeneracy does not hold) when taking into account the conservation of the total number of electrons. This expression can be used as a starting point for the ab initio calculation of T₁, and this quantity is required for an interpretation of the ultrafast demagnetization of ferromagnets after excitation with a femtosecond laser pulse.     

Microscopic derivation of a complex heavy ion potential

The energy dependence of a complex heavy ion optical potential is derived in Brueckner Hartree Fock using a local density approximation and a selfconsistent single particle spectrum. The two ions are described by an antisymmetrized cluster model wavefunction. Both real and imaginary part are found to increase with energy. Results are given for the elastic scattering of ⁴⁰Ca+¹⁶O.     

The hyperfine structure of diatomic molecules: Hund's case (cα)

The hyperfine splitting in diatomic molecules is treated according to an unperturbed model that is denoted Hund's coupling case (c_α). The advantages of this model are: (i) Rapid convergence of the perturbation expansion (spin-orbit interaction is included in the unperturbed Hamiltonian), (ii) hyperfine interactions of the type ΔJ ≠ 0 are automatically included, (iii) simple rotational-independent matrix elements result for the hyperfine Hamiltonian, and (iv) it is rather easy to judge whether the various hyperfine parameters are correlated or not. Numberical diagonalization of a secular matrix of dimension (2S + 1)(2I + 1) (or less for singular levels) yields the hyperfine splittings. The interaction with remote electronic states is included through a Van Vleck transformation. The coupling case (c) matrix elements are readily specialized to coupling case (a), and the present method is applied to the a³Π₁ substate of InH (optical), the X³Σ^? state of ³³SO and ¹⁷O¹⁶O (microwave) and to the X²Π state of ⁷LiO (radio frequency).     

Algebraic solution of the Hubbard model on the infinite interval

We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite line at zero density. This enables us to diagonalize the Hamiltonian algebraically. The eigenstates can be classified as scattering states of particles, bound pairs of particles and bound states of pairs. We obtain the corresponding creation and annihilation operators and calculate the S-matrix. The Hamiltonian on the infinite line is invariant under the Yangian quantum group Y(su(2)). We show that the n-particle scattering states transform like n-fold tensor products of fundamental representations of Y(su(2) ) and that the bound states are Yangian singlet.     

Truncation of the nuclear shell-model space using spectral strength distribution

A procedure, based on spectral distribution methods, is introduced to truncate a large shell-model matrix and to renormalize the matrix for the effects of the truncation. The basis states are selected depending on their contribution in the ground-state region and the renormalization is achieved by requiring that the strength distributions of each configuration in the truncated space to have the same widths as those in the full space. The calculation starts with a given effective Hamiltonian in the original space without having to introduce any free parameters due to truncation. Tests against full shell-model calculations in the ds shell are performed for energy levels of ²⁸Si and for electromagnetic transitions in the (ds)⁶T = 0 space.     

Multiple scattering theory in condensed materials

Second-order elliptic differential equations (such as the time-independent single particle Schrödinger equation) may be solved in a finite closed disjoint region of space independently of the rest of space. The solution in all space may then be determined by solving the equations in the exterior region together with boundary conditions at the junction of the two regions. These boundary conditions are determined by the previously found interior solution. This means that such regions may be taken as ‘black boxes’ whose exact details do not matter. The simplest example of this is phase-shift scattering theory from a single scatterer where all the scattering properties are described by the phase shifts, and the exact details of the scattering potential are unimportant. In a macroscopic condensed system, however, there are many core regions and one is really concerned with the multiple scattering which takes place between these different scattering centres. Much of this article is devoted to investigating the formal properties of scattering theory when there are many non-overlapping spherical regions of radius R _M, each of which is described by its own scattering matrix, or, equivalently for a spherically symmetric potential, by its phase shifts. Non-spherically symmetric and spin-dependent potentials are permitted, but for simplicity we assume initially that the interstitial region between each disjoint scattering region has zero potential. The generalization of the multiple scattering formalism for non-zero interstitial potential is also given at a later stage.

It is shown that in such a system a generalized T-matrix may be defined which describes the radiation from one of the core regions when another one has been excited. It is then a many channel T-matrix in which the channels are the different disjoint scattering regions. It is shown that the formal properties of this T matrix are the same as for a normal T matrix. In § 2 we review the properties of ordinary scattering theory, and then in § 3 we show that analogous properties for the generalized T matrix hold. An exact expression for the density of particle eigenstates is derived in terms of the positions and scattering matrices of the individual scattering centres. This expression reduces to the standard KKR band structure equation for the infinite regular lattice. We also consider how to construct the density of eigenstates and the charge density for such a system. These latter quantities may be approached in two different ways: the usual way is to consider the scattering material to occupy all space, but from a multiple scattering viewpoint one must consider the total volume of condensed material to be small compared with all space, even if both limit to infinity. It is not obvious that the latter method leads to the same results as the former (formally the density of eigenvalues is identical to the free electron density of eigenvalues in the latter method) and it is shown how the differences in the two approaches are resolved. We also discuss the expansion of some of these results for a perfect lattice. While the usual expansions are pseudo-potential expansions, a manifestly ‘on-energy shell’ expansion is derived which does not contain the arbitrary parameters of the pseudo-potential expansions. Finally, in § 4, we review the most significant contributions of other authors to the theory of multiple scattering.     

Theoretical investigation of parity violation in pα scattering

Calculations are presented of the parity-non-conserving (PNC) analyzing power A_z for longitudinally polarized incoming protons in elastic $\text{p}\text{α}$ scattering below ～50 MeV. The results are given in terms of the PNC as well as the regular, parity-conserving (PC) meson-nucleon coupling constants for single π-, ρ- and ω-exchange. The reliability and parameter dependence of the calculations are discussed in detail. The PC on-shell scattering parameters play an important role in particular for the angular dependence of A_z, which must be known for the actual analysis of measurements and are taken from experiments directly (phase analysis) rather than model calculations. The PNC scattering amplitudes are calculated in DWBA with optical-model wave functions properly antisymmetrized. Short-range correlations between nucleon pairs are taken into account by a Jastrow factor whose appropriate choice is discussed in detail. While absorption from the elastic channel is taken into account by the optical-model wave function, intermediate breakup channels are not included explicitly in the matrix elements.     

Diagrammatic structure of the general coupled cluster equations

The general formula for the number of diagrammatic terms occurring in the T_n equation within a particular coupled cluster model is derived. Both the antisymmetrized and Goldstone diagrams are considered. In addition to the full coupled cluster equation approximate approaches are discussed, and for each the general formula for the number of terms is given. Analogous expressions are presented for the number of diagrammatic terms contributing to the elements of the transformed Hamiltonian [Hbar] = e^?T He^T .     

Scattering of 21.1 MeV polarized protons from 89Y

The elastic and inelastic scattering of 21.1 MeV protons from ⁸⁹Y have been investigated. The excited states with a collective nature can be described quite well with standard OM and DWBA techniques. The transitions to the low-lying single-particle states are described with antisymmetrized microscopic DWBA including tensor and spin-orbit forces. The agreement with the experimental data is encouraging.