Nuclear structure in a hierarchy of ideal spaces

1981 ◽  
Vol 59 (11) ◽  
pp. 1670-1673
Author(s):  
M. Banville ◽  
P.-A. Simard

A new method permitting nuclear structure calculations for a system with an arbitrary number of fermions in an arbitrary number of subshells is developed through a generalization of the ideal space concept used in the boson methods. The nuclear shell problem is transcribed into a hierarchy of ideal spaces; the one-to-one correspondence between the states in each ideal space permits the generation of complete bases of antisymmetric states. The Hamiltonian matrix elements for the system are given. A generalization of the fractional parentage coefficients for such systems is obtained. The symmetries of those coefficients lead to a very important reduction in the complexity of the matrix elements.

Formulae are derived for the matrix elements of a non-central force in a configuration of N particles, involving fractional parentage coefficients for a reduction by two particles. The case when this configuration contains a number of closed shells and one unfilled shell is considered, although the methods used are applicable to less simple configurations. Specializing to the single 2 p -shell configurations (1 s ) 4 (2 p ) m for which the necessary coefficients have been derived, the matrices of central, tensor and two-body spin-orbit forces are given in terms of their two-particle elements. The determination of these two-particle elements is discussed and values are given for the 2 p -shell. Finally, the matrices derived are used in an attempt to derive the spins and moment data of certain 2 p -shell nuclei from a two-body charge symmetric interaction containing central, tensor and spin-orbit forces.


A method is derived for calculating matrix elements of a two-body interaction in wave functions which were classified in part I interms of the group U 2- . For simplicity, a Cartesian basis of intrinsic functions is introduced in which the one-dimensional oscillators in x, y and z are separately diagonal. An application to 24 Mg in L-S coupling shows very little mixing of the quantum number K but an appreciable (10 to 20 %) mixing of U 3 representations (λμ). Overall agreement with experiment is quantitatively only tolerable but the main pattern of the spectrum is undoubtedly given by the lowest representation (84). On this basis, suggestions are made concerning the type of spectra to be expected for even and odd parity levels of the even-even nuclei in the mass region 16 < A < 40.


2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
D. Maître ◽  
H. Truong

Abstract In this article we present a neural network based model to emulate matrix elements. This model improves on existing methods by taking advantage of the known factorisation properties of matrix elements. In doing so we can control the behaviour of simulated matrix elements when extrapolating into more singular regions than the ones used for training the neural network. We apply our model to the case of leading-order jet production in e+e− collisions with up to five jets. Our results show that this model can reproduce the matrix elements with errors below the one-percent level on the phase-space covered during fitting and testing, and a robust extrapolation to the parts of the phase-space where the matrix elements are more singular than seen at the fitting stage.


1984 ◽  
Vol 62 (12) ◽  
pp. 1369-1373 ◽  
Author(s):  
Josef Pliva ◽  
J. W. C. Johns

The absorption spectrum of cyclopropane, C3H6, was measured in the region between 790 and 950 cm−1 on a high-resolution Fourier transform spectrometer. The section containing the Q-branches of the perpendicular band of the ν11 vibration of species E′ was deconvolved to an effective line width of 0.0020–0.0025 cm−1. The structure of the ν11 band is strongly affected by l-type resonance. A total of 88 sub-bands with KΔK = −42 to 45 have been assigned in this band. The K = 4–3 and 2–3 sub-bands both exhibit K doubling of the lines with high J resulting from a combined effect of the off-diagonal matrix elements [Formula: see text], [Formula: see text], and [Formula: see text] with the l doubling in the K = 1, l = 1 state. Otherwise, the ν11 band is found to be free of perturbations by other vibrational states, in spite of the fact that a Jx,y Coriolis interaction is allowed by selection rules with the ν7 band (species [Formula: see text]) whose band origin is only 14.38 cm−1 below that of ν11. This shows that the value of [Formula: see text] is essentially zero. Also, the allowed Jz Coriolis interaction with the ν10 state, which lies 160.01 cm−1 above ν11, does not noticeably affect the two bands. A Hamiltonian matrix, including the matrix elements responsible for the K doubling and l-type resonance, was used for the treatment of the ν11 band. A set of accurate ground state constants and spectroscopic constants for the upper state ν11 is reported that reproduces 3240 observed lines of this band with a standard deviation of 0.0009 cm−1. Lines of the parallel band ν7 are just barely seen between the ν11 lines, which are perhaps 30–50 times stronger. Spectroscopic constants for the ν7 band have been obtained from 135 individual lines assigned to the Q- and R-branches of sub-bands with K = 6–21.


The orbital and charge-spin fractional parentage coefficients for the nuclear d 3 and d 4 con­figurations are derived using group theory. The orbital coefficients are given in a form appropriate to the new subclassification of the states according to irreducible representations of R 5 discussed in part I (Jahn 1950). Using these coefficients the complete energy matrices for the d 3 and d 4 configurations are derived, for a general charge-symmetric central two-body interaction, from the known energy matrix for the d 2 configuration.


1970 ◽  
Vol 48 (7) ◽  
pp. 819-826 ◽  
Author(s):  
M. Banville ◽  
P. A. Simard

The transcription of the quasi-particle in the ideal space has been studied in such a way that all terms in the Hamiltonian in the QTD approximation for the system of four quasi-particles can be found. It is found that an infinity of solutions exists which verify the commutator {α, α+} while yielding correct Hamiltonian matrix elements. Finally, particular solutions which do not verify this commutator are found. Their particular invariance properties under canonical transformations make them relatively easy to obtain. Nonphysical states are eliminated by these transcriptions and all the physical states appear properly antisymmetrized.


1973 ◽  
Vol 51 (1) ◽  
pp. 40-49 ◽  
Author(s):  
M. Banville ◽  
P. A. Simard

The concept of an ideal space of bosons and ideal particles is applied directly to systems of ordinary particles with a transcription of the particle into that ideal space that preserves both the Pauli principle and the transformation properties of the one- and two-body operators. We treat systems up to four particles.


The formulae of Redmond are used to construct expressions for the fractional parentage coefficients relating the configurations l 3 and l 2 . The explicit occurrence of godparent states is avoided for the quartet states of f 3 and also for a sequence of doublet states. The latter are defined by the set of quantum numbers f 3 WUSLJJ 2 , where W and U are irreducible representations of the groups R 7 and G 2 . Matrix elements of the type ( f 3 WUSL || U k || f 3 W'U'SL' ), where U k is the sum of the three irreducible tensor operators u k corresponding to the three f electrons, are tabulated for k = 2, 4 and 6 and for all values of W, U, S and L .


2010 ◽  
Vol 19 (04) ◽  
pp. 541-547
Author(s):  
EDWARD KWAŚNIEWICZ

The generalized Wick theorem for coupled operators is applied for a description of a system of N interacting bosons. The matrix elements of the N-bosons Hamiltonian containing one- and two-body terms are expressed in terms of overlaps between N-boson states spanning a boson space. A possibility of transfering of the presented formalism for the description of 2N-fermion systems is also considered.


2020 ◽  
Vol 17 (2) ◽  
pp. 0502 ◽  
Author(s):  
Ahmed H Ali

Nuclear shell model is adopted to calculate the electric quadrupole moments for some Calcium isotopes 20Ca (N = 21, 23, 25, and 27) in the fp shell. The wave function is generated using a two body effective interaction fpd6 and fp space model. The one body density matrix elements (OBDM) are calculated for these isotopes using the NuShellX@MSU code. The effect of the core-polarizations was taken through the theory microscopic by taking the set of the effective charges. The results for the quadrupole moments by using Bohr-Mottelson (B-M) effective charges are the best. The behavior of the form factors of some Calcium isotopes was studied by using Bohr-Mottelson (B-M) effective charges.


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