A projected Hartree-Fock calculation for 12C

doi:10.1016/0375-9474(67)90441-1

Nuclear Physics A

Volume 94, Issue 3, 20 March 1967, Pages 687-697

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Abstract

An approximate projected Hartree-Fock calculation for ¹²C is carried out. An increase of 11 MeV in the binding energy can be obtained by admixing 0d and 0f oscillators states to the orbitals. These admixtures enhance the E2 transition from the first excited state to the ground state and bring it in agreement with the experimental value.

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Configuration mixing calculations have been performed in ²⁰Ne using basis states which are energetically the lowest-lying solutions of the constrained Hartree-Fock equations with an angular momentum constraint of the form $〈 J ̂^{2} 〉 = J(J + 1)$ . For J = 6, very good agreement with the lower-lying 6⁺ states in an exact eigenvalue spectrum has been obtained with relatively few PAV-K mixed CHF basis states.
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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 $52^{−}$ . 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.
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Translationally invariant nuclear wave functions are constructed from deformed harmonic oscillator shell-model wave functions, with an exact projection of angular momentum quantum numbers. It is shown that the computation of matrix elements with the translationally invariant wave functions is as simple as the standard calculation, and formulae are obtained for (i) the potential energy, (ii) the kinetic energy and rms radius, and (iii) the charge form factor.

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Nuclear Physics A

Abstract

References (15)

Phys. Lett.

Nuclear Physics

Nuclear Physics

Nuclear Physics

Phys. Lett.

Phys. Rev.

Cited by (22)

Configuration mixing calculations with basis states obtained from constrained variational methods

Intermediate coupling calculations for light nuclei in a projected hartree-fock basis

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

Some calculations for <sup>12</sup>C in the α-particle model

The structure of <sup>12</sup>C in a deformed translationally invariant model:. Variational determination of the energy and static properties

On the construction of translationally invariant deformed wave functions