Intertwining operators and S matrix

The class of one-dimensional many-body systems related to semisimple Lie groups G is studied. The Hamiltonians of these systems are expressed in terms of Casimir operators (or, equivalently, Laplace-Beltrami operators on symmetric spaces) of underlying symmetry groups G. It turns out that the S matrix for all these problems is related to the intertwining operators for groups G. This connection provides immediately the functional form of the S matrix. Moreover, this connection allows one to prove that multiparticle S-matrix elements can be expressed in terms of two-particle 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.     

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.     

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.     

On the influence of shell structure and matrix element fluctuations on the pairing correlation in nuclei

The uniform model for the nuclear pairing-force problem is extended to take into account the effect of fluctuations in nucleon orbital level densityρ and in pairing matrix elementsG _vv′. Simple formulas for the average dependence of level density on energy are presented, and alternative ways of estimating effects of shell bunching on the pairing correlation are derived. It is argued also that the pairing constantG might be systematically overestimated in the usual BCS numerical calculations; a semiconstant pairing force with diagonal elements larger than off-diagonal is thus suggested.     

G-matrix effective interaction with the paris potential

An effective interaction is derived by fitting the oscillator matrix elements of the sum of the OBEP functions to the G-matrix elements derived from the Paris nucleon-nucleon interaction. The functional form, the mass dependence and the ambiguities of the effective interaction are discussed. For the application of the present effective interaction, we study the mass dependence of the G-matrices and estimate the Landau-Migdal parameter g′. We have obtained reasonable results for these cases.     

Local forces and the 16O reaction matrix

It is demonstrated that the G-matrix elements obtained from a solution of the Bethe-Goldstone equation for finite nuclei and a given NN interaction can be very well approximated by an effective local interaction. The local approximation is determined from the reaction matrix in nuclear matter using the same realistic NN interaction. The comparison is performed on the level of RPA calculations for the excited states in ¹⁶O. Very good agreement is found between the results for both interactions except for scalar-isoscalar states. It is shown that this is due to the energy dependence of the reaction matrix and can be cured rather easily. A comparison with experiment for isovector states is very satisfactory.     

Quantum Mechanical Virial Theorem in Systems with Translational and Rotational Symmetry

Generalized virial theorem for quantum mechanical nonrelativistic and relativistic systems with translational and rotational symmetry is derived in the form of the commutator between the generator of dilations G and the Hamiltonian H. If the conditions of translational and rotational symmetry together with the additional conditions of the theorem are satisfied, the matrix elements of the commutator [G,H] are equal to zero on the subspace of the Hilbert space. Normalized simultaneous eigenvectors of the particular set of commuting operators which contains H, J ², J _z and additional operators form an orthonormal basis in this subspace. It is expected that the theorem is relevant for a large number of quantum mechanical N-particle systems with translational and rotational symmetry.     

Electronic theory for impurity segregation at lattice defects in metals

A tight-binding type electronic theory is used to study the impurity segregation at lattice defects in metals. It is shown that the heat of segregation E_segr results from a simple physical origin: It is roughly proportional to the difference in the diagonal matrix elements (V_i-V₀) of impurity potentials, where V_i (V₀) is determined for atomic sites in the vicinity of lattice defects (for a perfect lattice site). Applications to impurity segregation at cleaved surfaces, screw dislocations and tilt grain boundaries are discussed.     

Crossing resonance of wave fields in a medium with an inhomogeneous coupling parameter

The dynamic susceptibilities (Green’s functions) of the system of two coupled wave fields of different physical natures in a medium with an arbitrary relation between the mean value ? and rms fluctuation Δ? of the coupling parameter have been examined. The self-consistent approximation involving all diagrams with noncrossing correlation lines has been developed for the case where the initial Green’s function of the homogeneous medium describes the system of coupled wave fields. The analysis has been performed for spin and elastic waves. Expressions have been obtained for the diagonal elements G _mm and G _uu of the matrix Green’s function, which describe spin and elastic waves in the case of magnetic and elastic excitations, and for the off-diagonal elements G _mu and G _um , which describe these waves in the case of cross excitation. Change in the forms of these elements has been numerically studied for the case of one-dimensional inhomogeneities with an increase in Δ? and with a decrease in ? under the condition that the sum of the squares of these quantities is conserved: two peaks in the frequency dependences of imaginary parts of G _mm and G _uu are broadened and then joined into one broad peak; a fine structure appears in the form of narrow resonance at the vertex of the Green’s function of one wave field and narrow antiresonance at the vertex of the Green’s function of the other field; peaks of the fine structure are broadened and then disappear with an increase in the correlation wavenumber of the inhomogeneities of the coupling parameter; and the amplitudes of the off-diagonal elements vanish in the limit ? → 0.     

Calculation of kinetic energy terms for the vibrational Hamiltonian: Application to large-amplitude vibrations using one-, two-, and three-dimensional models

One-, two-, and three-dimensional models involving large-amplitude vibrations have been used to calculate kinetic energy terms. Principle G matrix elements as well as cross terms in the kinetic energy were determined. Calculations were done on models involving the ring-puckering and PH inversion vibrations for 3-phospholene and the ring-puckering, ring deformation, and SiH₂ in-phase rocking vibrations for 1,3-disilacyclobutane. Kinetic energy expansions for g₄₄ and g₄₅ type terms were determined. Calculations show a coordinate dependence of the principle G matrix elements as well as of the g₄₅ terms. The vectorial models used in these calculations make it possible to treat vibrations in a one-, two-, or three-dimensional model separate from the other vibrations without carrying out a coordinate transformation, which would be necessary for the Wilson GF high- or low-frequency separation.     

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.     

The unitary representations of G̶L(4, R)

In the paper, nine series of unitary representations of the group $\text{G}\text{?}$L(4, R) are found. Moreover, there are obtained matrix elements of the generators of $\text{G}\text{?}$L(4, R) in an orthonormal basis. The elements of the basis are classified over the subgroup SU(2)?SU(2).     

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.     

Some unexpected proportionalities between components of the spin—other-orbit interaction in the f shell

The reduced matrix elements of the component z _i of the inter-electronic spin-other-orbit interaction have been calculated for all the states of the atomic f shell. Three of the z _i , each of which belongs to the irreducible representation (30) of Racah's group G₂, are found to exhibit matrix elements that are often related to one another in ways that go beyond what the Wigner-Eckart theorem, generalized to G₂, would predict. Examples are presented for matrix elements whose bras belong to the IR (20) of G₂ and whose kets belong to (31). The challenges to current theory are discussed.     

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.     

The structure of Heesch groups and its relation to material property tensors

The magnetic crystal point groups (Heesch groups) are classified according to their structure with respect to the three inversion operations: space, time, and total inversion. Accordingly the tensors are classified by the irreducible representations of the full inversion group. The groups and tensors are considered under the action of the elements A_i of the group of automorphisms of the full inversion group. The following correspondence theorem is proved: The matrix form of the tensor representation T of the group G coincides with the matrix form of the representation A_iT of the group A_iG. The theorem gives a clear explanation of the so-called “magic numbers” and provides a suitable short cut for the calculation and tabulation of material property tensors.     

On the physical meaning of Sachs form factors and on the violation of the dipole dependence of G E and G M on Q 2

The questions of how a dipole character of the dependence of the form factors G _E and G _M on the square of the momentum transfer to a proton, Q ², arise and why a violation of this dependence occurs, which was first observed in a JLab polarization experiment, are investigated. The answers to these questions could be obtained owing to the use of the simplest QCD concepts of the proton structure and the results obtained by calculating the matrix elements of the proton current in the case of non-spin-flip and spin-flip transitions for protons in the diagonal spin basis (DSB), where the Little Lorentz group common to the initial and final proton states is realized. In DSB, the form factors G _E and G _M are determined by the matrix elements J _p ^??,?? and J _p ^???,?? of the proton current in the cases of non-spin-flip and spin-flip transitions for protons. In an arbitrary reference frame, the relations between these matrix elements and the form factors are J _p ^??,?? ?? G _E and J _p ^???,?? $\sqrt \tau G_M$ , where ?? = Q ²/4m ², with m being the proton mass. In considering the problem in question at the quark level, use is made of the model where the proton consists of three pointlike quarks having identical masses and where the respective matrix element of the proton current is the product of three quark-current amplitudes having the form J _p ^??,?? ?? 1 and J _p ^???,?? ?? $\sqrt \tau$ . It is shown that the aforementioned dipole dependence arises if the proton spin flip is due to spin flip for only one of the three quarks. As to violations of this dependence, they are caused by the contributions to J _p ^??,?? from spin-flip transitions for two quarks or by the contribution to J _p ^???,?? from spin-flip transitions for all three quarks constituting the proton.     

The average effective interaction in linked cluster theory

We derive a theoretical method for calculating the averages (over J and T) of matrix elements of the nuclear effective interaction, in any order of the linked cluster expansion. We apply this to calculate all terms through fourth order in the G-matrix, for the averages in mass-6 and mass-18 nuclei. We find that the averages in fourth order are as large as in second and third order. This behavior is associated with a small number of terms, which suggests a partial summation method of improving the series. The relation to other work is discussed.