The democratic nuclear interaction in relative S states

A simple relative l=0 interaction is obtained by setting all relative matrix elements of a given spin equal to each other. The nuclear matrix elements are nearly equal for this interaction and the standard l=0 interactions such as the Kallio-Koltveit interaction, although the relative matrix elements are quite different.     

Nuclear matrix elements in the 7/2−–9/2+ beta decay of175Yb

The angular correlation of the \({7 \mathord{\left/ {\vphantom {7 2}} \right. \kern-0em} 2}^ - \xrightarrow[{353keV}]{\beta }{9 \mathord{\left/ {\vphantom {9 2}} \right. \kern-0em} 2}^ + \xrightarrow[{114keV}]{\gamma }{7 \mathord{\left/ {\vphantom {7 2}} \right. \kern-0em} 2}^ + \) is measured atE _β =300 keV and found to beA ₂(β,γ)=0.108 ± 0.028. The result is combined with the data on longitudinal polarisation nuclear orientation to obtain the nuclear matrix elements which would give a simultaneous fit to all experimental data. The matrix elements thus obtained have predicted theβ spectrum shape correction factor. The sizes of the matrix elements indicate a cancellation effect in vector matrix elements which explains deviation from ξ-approximation. From the ratio of higher order matrix elements,λ, the deviation from theCVC ratio due to Fujita is found and thus the importance of off-diagonal elements inH _c is noted. The experimental matrix elements are compared with the model-predicted ones.     

The nuclear reaction matrix

Different difinitions of the reaction matrix G appropriate to the calculation of nuclear structure are reviewed and discussed. Qualitative physical arguments are presented in support of a two-step calculation of the G-matrix for finite nuclei. In the first step the high-energy excitations are included using orthogonalized plane-wave intermediate states, and in the second step the low-energy excitations are added in, using harmonic oscillator intermediate states. Accurate calculations of G-matrix elements for nuclear structure calculations in the A ≈ 18 region are performed following this procedure and treating the Pauli exclusion operator Q_2ν by the method of Tsai and Kuo. The treatment of Q_2ν, the effect of the intermediate-state spectrum and the energy dependence of the reaction matrix are investigated in detail. The present matrix elements are compared with various matrix elements given in the literature. In particular, close agreement is obtained with the matrix elements calculated by Kuo and Brown using approximate methods.     

An ensemble of random particle-hole matrices with collective eigenstates

An ensemble of random particle-hole matrices is studied as a function of the signs of the matrix elements. When all matrix elements have the same sign, a theorem due to Perron ensures the existence of a coherent state: the collective particle-hole state. The example of Gamow-Teller transitions between²⁰⁸Bi(1+) and²⁰⁸Pb(0+) is considered and the strength of the Gamow-Teller giant resonance as well as the fluctuations in the ensemble are studied. When a fractionp of matrix elements are of opposite sign, we find that a giant resonance may still exist providedp is small enough. Its magnitude and position in the spectrum are studied. The method proposed is valid for any particle hole excitation. The two body matrix elements of several interactions are analysed and are shown to fit into a unique valuep=0.2 for the 1 + states.     

The residual interaction of bound nucleons-two-nucleon matrix elements deduced from transfer experiments

Matrix elements for the effective two-nucleon interaction have been deduced from the population of multiplets near closed shells as observed in direct transfer reactions. In the evaluation, the limited purity of such multiples was taken into consideration, typically by weighting the observed fractions of the two-nucleon configurations by their spectroscopic strenghts and by using the resulting energy centroids. In a few cases, off-diagonal matrix elements are available from empirical wave funcitons. The systematic errors for particle-particle matrix elements extracted directly and those obtained from Pandya transformations were found to go in opposite directions. In some cases, this feautre of the empirical mehtod could be used to suggest upper and lower “bounds” for the extracted matrix elements. Diagonal matrix elements for the empirical residual interaction show a number of features suggestive of an underlying simplicity in the interaction of bound nucleons. Within experimental uncertainties (of about 10% for T=0 matrix elements) the monopole parts of the matrix elements are fit well with a simple A?0.75 dependence, and the data available to date do not reveal any significant monopole dependence on the quantum numbers of the interacting nucleons. The usefulness of scaling is suggested. Generally, diagonal matrix elements E_J(j₁, j₂) normalized by the extracted A-dependent monopole strength agree within expected experimental uncertainties whether derived from particle-particle or particle-hole multiples and whether extracted from the beginning or the end of a major shell. For values J≠0, the diagonal E_J(j²) matrix elements seem to follow two universal functions which depend on the semi-classical coupling angles θ₁₂, but are otherwise independent on j. For j₁≠j₂ several “typical” functions ?(θ₁₂) can be constructed which fit subsets of the data and differ in a predictable way. The general features of the bound-nucleon interaction appear consistent with those of theoretical matrix elements based on a number of short-range model forces or on calculations using the G matrix approach to deal with realistic free nucleon forces. For the latter, the available theoretical numbers for j₁=j₂ agree well with the T=1 set, but they differ quantitatively from the observed matrix elements for T=0, sometimes by many (experimental) standard deviations.     

Bounds on theK-Π matrix elements

Using Machet's method, we derive bounds on theK-Π matrix elements. Our bounds on matrix elements with left-left operators show that quark model calculations always overestimate these matrix elements. In all cases the vacuum insertion method gives an upper bound. No conclusive indication of an enhancement of the penguin matrix element is observed. Its contribution to the ΔI=1/2 amplitude could possibly be destructive and would rule out the penguins as an explanation to the ΔI=1/2 rule.     

Nuclear reaction matrix calculations with a shell-modelQ

The Barrett-Hewitt-McCarthy (BHM) method for calculating the nuclear reaction matrixG is used to compute shell-model matrix elements forA=18 nuclei. The energy denominators in intermediate states containing one unoccupied single-particle (s.p.) state and one valence s.p. state are treated correctly, in contrast to previous calculations. These corrections are not important for valence-shell matrix elements but are found to lead to relatively large changes in cross-shell matrix elements involved in core-polarization diagrams.     

Many-electron exchange Hamiltonian in the theory of ferromagnetism

A Hamiltonian describing effects of many-electron interatomic exchange in magnetic crystals is derived by application of an appropriate unitary transformation to the many-electron, many-nucleus Hamiltonian. It is expressed in terms of atomic annihilation and creation operators and j-fold exchange matrix elements for 1 ? j ? N, where N is the number of electrons localized on each crystal site. Explicit formulas are derived for the case of not more than half-filled shells with Hund's rule factorization. An appropriate spin algebra is introduced and the coefficient of the bilinear (Heisenberg) and biquadratic interactions evaluated in terms of single and double-exchange matrix elements. The relative magnitudes of such matrix elements are roughly estimated using determinantal wave functions.     

Experimental determination of some matrix elements of the effective residual interaction among particle-hole configurations in 208Pb

Matrix elements of the residual interaction in ²⁰⁸Pb are derived from the wavefunctions and the energies of states and configurations in ²⁰⁸Pb. From the experimentally determined wavefunctions of 20 low-lying states in ²⁰⁸Pb with spins I^π = 2^?, 3^?, 4^?, 5^?, 6^?, 7^? one obtains matrix elements of the effective residual interaction among particlehole configurations defined with respect to the physical ground state of ²⁰⁸Pb. The average value of these matrix elements is about 100 keV. No significant difference in the average value of the off-diagonal matrix elements is found for the different spin values. Also many nondiagonal matrix elements have values approaching the diagonal matrix elements.     

Renormalized shell model matrix elements derived from nucleon-nucleon phase shifts

We apply theG matrix elements calculated by Singh et al. (using the DWBA approach) to some spectra calculations forA=18, 19, 20 after first renormalizing the matrix elements by including second-order processes inG. The corrections introduced by the Distorted-Wave-Born-Correction Term are attractive for both the H.J. and Reid potentials. The magnitude and uncertainties in these corrections are discussed. Differences between these potentials are attributed chiefly to their different off-the-energy-shell behaviour. The results are compared to calculations with the Kuo matrix elements and with experiment.     

Computation of electric dipole matrix elements for hydrogen fluoride

The electric dipole matrix elements of hydrogen fluoride have been calculated by numerical integration for transitions involving large quantum numbers υ, J. Overtones have been included through Δυ = 5. Molecular wave functions obtained by numerical integration of the Schrödinger equation were used. The influence of the mechanical motion on the matrix elements has been determined for Morse and Rydberg-Klein-Rees (RKR) potential functions. The influence of the electric dipole-moment function approximations has been investigated by a comparison of matrix elements obtained with approximations having the form of a truncated polynomial and a wave-function expansion. The inaccuracies in the matrix elements caused by uncertainties in the dipole-moment coefficients have been investigated.     

Sine-Gordon form factors in finite volume

We compare form factors in sine-Gordon theory, obtained via the bootstrap, to finite volume matrix elements computed using the truncated conformal space approach. For breather form factors, this is essentially a straightforward application of a previously developed formalism that describes the volume dependence of operator matrix elements up to corrections exponentially decaying with the volume. In the case of solitons, it is necessary to generalize the formalism to include effects of non-diagonal scattering. In some cases it is also necessary to take into account some of the exponential corrections (so-called μ-terms) to get agreement with the numerical data. For almost all matrix elements the comparison is a success, with the puzzling exception of some breather matrix elements that contain disconnected pieces. We also give a short discussion of the implications of the observed behavior of μ-terms on the determination of operator matrix elements from finite volume data, as occurs e.g. in the context of lattice field theory.     

Effects of mode-mixing and non-Condon interaction in the vibronic spectra of multimode systems

A method of calculation of optical spectra, which allows one to take into account the mixing of an arbitrary number of modes in the transition, is developed. Thus the dependence of electronic transition matrix elements on nuclear coordinates is also taken into consideration. In this method, the calculation of the Fourier transform of the spectrum is reduced to the calculation of the determinant of the 2N×2N matrix and its reciprocal matrix; the elements of the matrix are given by time-dependent pair-correlation functions.     

Calculation of high-order rotational centrifugal-distortion matrix elements for a Hund's case (a) basis set

The derivation of explicit expressions for the Hund's case (a) matrix elements of R^2k is discussed, where R is the mechanical rotational angular momentum operator of the molecule. A recursion relation is developed that permits matrix elements of R^2k to be expressed in terms of those of R^2(k?1), thus affording a straightforward means of calculating the case (a) matrix elements of rotational centrifugal-distortion constants D_v, H_v, L_v, M_v, etc., to an arbitrarily high order. The explicit matrix elements of L_v are listed.     

Flavouring the Gribov process

The exchange of flavour carrying trajectories is studied in the non-covariant parton interpretation of reggeon field theory. While pomeron exchange is described by wee partondensities, i.e. diagonal elements of the density matrix of a fast hadron, meson exchange is described by density matrix elements which are diagonal in parton number but off-diagonal in flavour. The reggeon field theory “hamiltonian” describes a markoffian evolution of this block-diagonal density matrix during a boost. This interpretation is possible both if there are two distinctf and ? trajectories and in case of ? identity. The meson trajectories are superpositions of odd and even signature trajectories.     

The dependence of the photodisintegration cross section on the off-shell T-matrix

Nucleon-nucleon scattering phase shifts determine the diagonal element of the transition matrix. The off-diagonal elements are not completely arbitrary but have conditions imposed on them by the range and the tail of the potential. Electromagnetic interaction can also be used to place restrictions on the off-diagonal elements. We find that the cross section of the deuteron photodisintegration is sensitive to the off-shell transition matrix. The integrated cross section can be varied by as much as 30 % or more, and the matrix element for the El transition by a factor of 2. While the matrix element for the photodisintegration depends on the off-shell elements of the T-matrix, it cannot be used to discriminate between alternative off-shell T-matrices. We have constructed classes of different off-shell T-matrices, which produce identical photo-disintegration cross sections and other two-body scattering and bound-state properties.     

Representation of the Coulomb Matrix Elements by Means of Appell Hypergeometric Function <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>

Exact analytical representation for the Coulomb matrix elements by means of Appell’s double series F₂ is derived. The finite sum obtained for the Appell function F₂ allows us to evaluate explicitly the matrix elements of the two-body Coulomb interaction in the lowest Landau level. An application requiring the matrix elements of Coulomb potential in quantum Hall effect regime is presented.     

Nuclear structure effects on the neutrinoless double beta decay

The nuclear matrix elements of the 0ν ββ decay of⁷⁶Ge,⁸²Se,¹⁰⁰Mo,^128,130Te,¹³⁶Xe and¹⁵⁰Nd are calculated in the proton-neutron quasiparticle RPA with theG-matrix of the Paris potential. It is shown that the matrix elements are not sensitive to details of nuclear structure, in contrast to the 2ν ββ decay. We investigate effects of ground-state correlations and those of short-range correlations on the suppression of the nuclear matrix elements. We also discuss effective values of the neutrino mass which are deduced from experimental 0ν ββ decay half-lives.     

Isospin mixing between T=0 and 1 states in the 1p-shell

The effect of using different proton and neutron wave functions to evaluate matrix elements of a charge independent nucleon-nucleon interaction is examined. It is shown that this is predominantly an isovector effect and in the 1p-shell can give rise to large off diagonal matrix elements (? 100 keV). The magnitude of these matrix elements are extremely sensitive to the detailed structure of the single particle wave functions. Until this effect is satisfactorily taken into account it will be difficult to demonstrate the need for a change symmetry breaking nucleon-nucleon interaction from a measurement of isospin mixing between T=0 and 1 states.