In particular, we rely heavily on the power and versatility of generating functions to derive many of the present results.
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The goal of this paper is to derive the separation-method formalism for Gaussian matrix elements in a cylindrical harmonic-oscillator basis, with particular emphasis placed on the details of the derivation because of its relevance to other types of interactions, and other applications involving the harmonic-oscillator basis. The separation method is especially well-suited to the HFB algorithm, because the coefficients needed to calculate the two-body matrix elements derived in this paper can be calculated quickly once and for all, and stored with relatively little computer memory. Thus microscopic fission calculations must rely on fast and accurate algorithms to evaluate the two-body matrix elements, such as the separation method. The resulting large-scale computations can become very time-consuming and are prone to errors in accuracy. Therefore many sets of matrix elements need to be calculated, each set corresponding to a harmonic-oscillator basis optimized for a particular nuclear shape, and each set requiring a large number of oscillator shells. Fission also implies the evolution of the nucleus through a variety of exotic shapes leading to scission. The proper identification of scission configurations and the calculation of their properties depend sensitively on accurate calculations of the matrix elements of the effective interaction. Scission configurations are then found by driving the nucleus to such exotic shapes that the delicate balance between its surface tension and the Coulomb repulsion between the nascent fission fragments is broken. In the microscopic description of fission, the matrix elements of the nucleon-nucleon interaction are typically used in a Hartree–Fock–Bogoliubov (HFB) procedure to construct a Slater-determinant wave function for the nucleus. (1) have had considerable success in recent years, ,, and are therefore of great interest. On the other hand, microscopic calculations of fission using the interaction in Eq. These results are crucial, for example, in microscopic calculations of nuclear fission using the Gogny force, where the nucleus elongates along a symmetry axis, until scission occurs.įission calculations in particular bring to the fore many of the technical difficulties involved in the computation of Gaussian matrix elements.
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In this paper, we derive the separation method for a wider class of systems that exhibit axial symmetry. In the separation method, two-body matrix elements are expressed as a more manageable finite sum of products of one-body matrix elements. In previous work, the separation method was introduced as a way of calculating the Gaussian matrix elements efficiently and accurately for systems with spherical symmetry.
![gaussian software harmonic osscialtor approximation gaussian software harmonic osscialtor approximation](https://slideplayer.com/slide/14932042/91/images/3/Motivation+Schrödinger+Equation+can+only+be+solved+exactly+for+simple+systems.+Rigid+Rotor%2C+Harmonic+Oscillator%2C+Particle+in+a+Box%2C+Hydrogen+Atom..jpg)
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The calculation of Gaussian matrix elements in a harmonic-oscillator basis, however, poses definite technical challenges in accuracy as well as execution time. Finally the Coulomb interaction V Coul ∼ 1 / | r → 1 − r → 2 | between protons is clearly not of Gaussian form, but the mathematical framework presented in this paper can be applied equally well to a Coulomb potential.įor the calculation of matrix elements in molecular, atomic, and nuclear physics, harmonic-oscillator functions provide a convenient and popular orthogonal basis. A spin-orbit term with strength W L S uses a Dirac-delta function, but extensions of the Gogny force have been proposed that introduce a Gaussian form for this term. Two Gaussian terms appear explicitly with range parameters μ 1 and μ 2. In nuclear physics, for example, the Gogny interaction V ( r → 1, r → 2 ) = ∑ i = 1 2 ( W i + B i P ˆ σ − H i P ˆ τ − M i P ˆ σ P ˆ τ ) e − ( r → 1 − r → 2 ) 2 / μ i 2 + i W L S ( ∇ ← 1 − ∇ ← 2 ) × δ ( r → 1 − r → 2 ) ( ∇ → 1 − ∇ → 2 ) ⋅ ( σ → 1 + σ → 2 ) + t 0 ( 1 + x 0 P ˆ σ ) δ ( r → 1 − r → 2 ) ρ γ ( r → 1 + r → 2 2 ) + V Coul, where P ˆ σ and P ˆ τ are spin- and isospin-exchange operators and ρ is the total nuclear density, gives the effective (in-medium) potential between nucleons. The Gaussian form represents a relatively simple two-body potential with a finite range, which is needed in many realistic descriptions of many-body systems. Gaussian interactions play an important role in the microscopic description of molecular and nuclear processes.