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Elimination the CentreofMass Motion in the Nuclear ShellModel with Isospin 1

A. Deveikis, A. Fedurtsya

Published Jan 1, 2011• 0 citations

Abstract

oretical classification of manyparticle antisymmetri� cal states allows one to significantly simplify the anti� symmetrization procedure for Aparticle states. The efficiency of the proposed procedure for calculation of coefficients of fractional parentage uprise from the possibility to perform the calculation of translationally invariant CFP's in the ordinary oneparticle basis. The introduced CESO's may be used most effectively for ab initio No Core Nuclear Structure calculations with oscillator intrinsic density matrices within the framework of Reduced Hamiltonian method (2). Abstract—An efficient procedure for largescale calculations of the twoparticle translational invariant coef� ficients of fractional parentage (CESO's) for several jsh ells with isospin is presented. The approach is based on a simple enumeration scheme for antisymmetric manyparticle states and efficient algorithms for calcu� lation of the coefficients of fractional parentage for a single jshell and several jshells with isospin. The CESO's may be obtained by diagonalizing the centreofmass Hamiltonian in the basis set of antisymmetric Aparticle oscillator functions with singled out dependenc e on intrinsic coordinates of two last particles and choosing the subspace of its eigenvectors corresponding to the minimal eigenvalue equal to 3/2. An arbitrary number of oscillator quanta can be involved. The characteristics of the introduced CESO's basis are investi�

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A new procedure for large-scale calculations of the coecients of fractional parentage (CFP) for manyparticle systems is presented. The approach is based on a simple enumeration scheme for antisymmetric N particle states, and we suggest an ecient method for constructing the eigenvectors of two-particle transposition operator PN1;N in a subspace where N1 and N2 = N N1 nucleons basis states are already antisymmetrized. The main result of this paper is that according to permutation operators PN1;N eigenvalues we can distinguish totally asymmetrical N particle states from the other states with lower degree of asymmetry.

Abstract A new procedure for large-scale calculations of the coefficients of fractional parentage (CFP) for many-particle systems is presented. The approach is based on a simple enumeration scheme for antisymmetric N particle states, and we suggest an efficient method for constructing the eigenvectors of two-particle transposition operator

P_{N_1 ,N}

in a subspace where N 1 and N 2 = N − N 1 nucleons basis states are already antisymmetrized. The main result of this paper is that according to permutation operators

P_{N_1 ,N}

eigenvalues we can distinguish totally asymmetrical N particle states from the other states with lower degree of asymmetry.

A new method for calculation of shell model intrinsic density matrices, defined as two-particle density matrices integrated over the centre-of-mass position vector of two last particles and complemented with isospin variables, has been developed. The intrinsic density matrices obtained are completely antisymmetric, translation-invariant, and do not employ a group-theoretical classification of antisymmetric states. They are used for exact realistic density matrix expansion within the framework of the reduced Hamiltonian method. The procedures based on precise arithmetic for calculation of the intrinsic density matrices that involve no numerical diagonalization or orthogonalization have been developed and implemented in the computer code.

Abstract In the independent particle model, the total wave function is expressed as a linear combination of products of single particle wave functions. Fractional parentage coefficients give the part of the expression depending on the symmetry properties of the wave function. An operator is introduced which picks out the antisymmetric part of any n-particle wave function. The operator is given explicit form, and an explicit formula for the fractional parentage coefficients is deduced. The total fractional parentage coefficients can be split into a part depending on the orbital vectors, and a part depending on the spin (and isotopic spin) vectors. The separate parts can have symmetries other than total antisymmetry. Operators are introduced which pick out states of definite symmetry from any state. The operators are given explicit form, and explicit formulae for orbital, or spin, fractional coefficients are deduced. Detailed results are given for the three, four and five particles cases.

The one-body and two-body density matrices in coordinate space and their Fourier transforms in momentum space are studied for a nucleus (a nonrelativistic, self-bound finite system). Unlike the usual procedure, suitable for infinite or externally bound systems, they are determined as expectation values of appropriate intrinsic operators, dependent on the relative coordinates and momenta (Jacobi variables) and acting on intrinsic wavefunctions of nuclear states. Thus, translational invariance (TI) is respected. When handling such intrinsic quantities, we use an algebraic technique based upon the Cartesian representation, in which the coordinate and momentum operators are linear combinations of the creation and annihilation operators a^+ and a for oscillator quanta. Each of the relevant multiplicative operators can then be reduced to the form: one exponential of the set {a^+} times other exponential of the set {a}. In the course of such a normal-ordering procedure we offer a fresh look at the appearance of "Tassie-Barker" factors, and point out other model-independent results. The intrinsic wavefunction of the nucleus in its ground state is constructed from a nontranslationally-invariant (nTI) one via existing projection techniques. As an illustration, the one-body and two-body momentum distributions (MDs) for the 4He nucleus are calculated with the Slater determinant of the harmonic-oscillator model as the trial, nTI wavefunction. We find that the TI introduces important effects in the MDs.