Combinations Of Matrix Elements Research Articles

Topological indices are not necessarily invariant to graph labeling.

Each element of the Universal matrix U (vertex-degree vertex-distance weighted matrix) represents the mutual contribution of two vertices weighted for the vertex degrees and the distance between them. Regarding different labeling ways of graph vertices, particular matrix elements are not invariant to molecular labeling. Regarding the structural features they are invariants since there are only particular vertex combinations representing particular structural features. Some of the matrix elements are the best descriptors of a physicochemical property in question. Some combinations of matrix elements are very good descriptors of physicochemical properties of octanes regardless how we enumerate their vertices.

Information on photonuclear transition matrix elements from measuring the emitted nucleon polarization in the reaction plane.

A compact general expression for the differential cross section of the photonuclear reaction (..gamma.., N), in which the incident photon is circularly polarized and any polarization of the emitted nucleon observed, is presented. Angular distribution coefficients relevant to the nucleon polarization in the scattering plane are expressed as bilinear combinations of matrix elements for a special case of the /sup 16/O(..gamma..,N/sub 0/) reaction. The polarization P/sub x/, calculated within the direct-semidirect model at E/sub ..gamma../ = 60 MeV, is shown as an example of the theoretical result.

A program calculating the formulae for polarization effects in nuclear reactions

In the R-matrix theory of nuclear reactions, properly normalized observables can be expanded in terms of rotation matrices or Legendre functions. The corresponding expansion coefficients are then bilinear functions of the reaction matrix elements in the (1,s,J) coupling scheme. The numerical factor for each bilinear combination of matrix elements in a given coefficient is a function involving Clebsch-- Gordan, Racah, and X coefficients. By obtaining numerical values for the angular momentum functions and taking into account all relevant symmetries, the program FATSO computes the explicit formulas for any observable quantity for a given set of (1,s,J)-matrix elements. A set of 40 matrix elements can be accommodated, with up to 20 expansion coefficients. Approximate running time for cross sections involving vector- and tensorpolarized deuteron beams for a set of 20 matrix elements is 2 minutes. The program, in FORTRAN IV for the UNIVAC 1108 and the operating system EXEC 8, requires 44030 36- bit words of computer memory. (RWR)

The interplay of cluster structures in light nuclei: The model and the application to 7Li

Abstract The structure of light nuclei is treated as a superposition of different cluster structures. The wave function of a nucleus contains the components describing two-cluster structures constructed out of 1p shell and/or 1s shell clusters. The model is used to study the low-lying energy levels of the nucleus 7Li, whose structure is treated as a superposition of the cluster structures ( 4 He + 3 H ) and ( 6 Li + n ) . The space of the wave functions is spanned by antisymmetrized products of shell-model wave functions describing separated clusters. The separation S of the two shell-model potentials is taken as the generator coordinate. The projection of the generator coordinate wave function onto the eigenspace of angular momentum and parity operators is performed by an algebraic method based on an algorithm which expresses the effect of the rotation of the wave function in terms of the rotation of the generator coordinate S. This method enables one to express the matrix elements in the projected basis as a finite combination of matrix elements in the unprojected basis. The application of the model to the nucleus 7Li has shown the feasibility of the calculation with 1p shell clusters. The Volkov potentials were used in the calculations. The minimal energy of 7Li in the state described by the model function with fixed separation of the clusters 4He and 3He is lowered by 1.64 MeV by solving the Hill-Wheeler equation for the cluster structure ( 4 He + 3 H ) . The coupling of both cluster structures lowers the energy in addition by 0.32 MeV. The influence of the coupling on the shape of the nucleus is a follows: In the single cluster structure ( 4 He + 3 H ) the optimal separation of the clusters in the ground state is 3.5 fm and it is 0.5 fm larger than that in the first excited state 5 2 − . However, when both cluster structures are considered simultaneously, the optimal separation between the clusters 4He and 3He and the separation between the clusters 6Li and n in the ground state are equal to the corresponding separations in the excited state.

Electrostatic Interaction between Configurations Comprising Electrons and Holes

New identities are derived which enable the construction of closed formulas for the matrix elements of the electrostatic interaction between (two-particle) configurations comprising an electron and a hole. Relations between these matrix elements and those corresponding to two-electron configurations are obtained. Formulas are given in which the matrix elements of the electrostatic interaction between three-particle configurations are expressed as linear combinations of matrix elements between two-particle configurations.

Nuclear matrix elements in the first forbidden beta decay of 198Au

Abstract Nuclear matrix elements for the 2 − (0.959 MeV β − )2 + beta transition in 198 Au were extracted and comparison was made with CVC theory predictions for the vector matrix element ratio Λ cvc = ∫α/∫ir/p. It was found that the ratio Λ cvc was consistent with the CVC prediction based on the assumption that the Coulomb Hamiltonian is diagonal. However, contributions from off-diagonal matrix elements of the Coulomb Hamiltonian could not be ruled out. Fairly good limits were placed on the nuclear matrix elements in spite of the fact that the Coulombenhanced combinations of matrix elements dominate the transition.

The polarization of radioactive gold nuclei in iron

The y-radiations from the nuclei of 198 Au, 199 Au and 195 Au polarized as diamagnetic impurities in ferromagnetic iron have been studied down to 0.012 °K, reached by thermal contact with an adiabatically demagnetized chrome potassium alum ‘heat sink’. Specimen temperatures were deduced from measurement of the anisotropic γ-ray emission from 60 Co nuclei polarized in the same iron alloy, and the importance of this new technique in extending the temperature range and consequently the accuracy of interpretation of the measurements is discussed. From observation of the 412 keV γ-ray in 198 Au and of the 158 keV γ-ray in 199 Au, two independent and self-consistent values of |H eff |, the hyperfine coupling field present at gold nuclei in iron, are obtained. The mean value is |H eff | = 1.35 ± 0.05 x 10 9 Oe. With this value the 195 Au measurements yield a nuclear magnetic dipole moment of ± (0*13 ± 0*04) n.m. for this isotope. The agreement of the value of |He ff | found in this work with that from Mössbauer studies of 197 Au in iron justifies the assumed application of the £ approximation to the first forbidden β-transitions in the decays. Values for the ratio A of the linear combinations of matrix elements, V∘ and Y 1 corresponding to j β = 0 and 1 respectively in the 959 keV β-decay of 198 Au and the 250 keV /?-decay of 199 Au are found to be ±(0.82 ± 0.18) and ± ( 1.86 +0.82 -0.34 ) respectively. These results are compared with calculations based on the nuclear single particle model.