[Paper Review] A direct microscopic approach to transition strengths in pre-equilibrium reactions

F. B. Guimarães, B. V. Carlson|arXiv (Cornell University)|Jun 21, 2011

Nuclear physics research studies6 references3 citations

TL;DR

This paper presents a direct microscopic formalism to calculate transition strengths in pre-equilibrium nuclear reactions without relying on Laplace transforms or saddle-point approximations. By using a quantum-mechanical approach based on shell model configurations and convolution integrals of single-particle states, it reproduces exciton model results accurately when using a constant energy grid, but shows significant deviations with harmonic oscillator basis, challenging prior assumptions about equilibrium attainment in pre-equilibrium processes.

ABSTRACT

We present a microscopic formalism that extends the traditional formulation of Williams, Ericson and Bloch and permits to obtain the transition strengths (TS) of pre-equilibrium nuclear reactions directly from their quantum microscopic description. We calculate the TS without resorting to the Laplace transform approach and the use of the saddle point approximation. We also analyze some problems that may appear in connection with these mathematical tools and the Darwin-Fowler approach in this case. We show that, analogously to the nuclear densities, the strengths for transitions that change the exciton number by two or leave it unchanged can be estimated microscopically as convolutions of the functions of simpler states. When using the HO basis for the Model Space we obtained important departure from the results of the exciton model (EXM), which can partially invalidate our previous analysis on the attainment of equilibrium during the PE stage. On the other hand, by using constant grid of energies for the sp-basis we were able to reproduce the results of EXM quite well in a large range of excitation energies. A new model code, TRANSNU, was developed that can be ported to traditional semi-classical codes like TNG for nuclear data evaluation.

Motivation & Objective

• To develop a microscopic, non-approximate method for calculating transition strengths in pre-equilibrium nuclear reactions.
• To overcome limitations of traditional approaches relying on Laplace transforms and saddle-point approximations.
• To assess the validity of the exciton model (EXM) by comparing its results with a fully quantum-mechanical description.
• To investigate how different single-particle basis choices—harmonic oscillator vs. constant energy grid—affect transition strength predictions.
• To enable integration of the new formalism into existing semi-classical nuclear data evaluation codes like TNG.

Proposed method

• The formalism computes transition strengths directly from shell model matrix elements without Laplace transforms, avoiding the need for continuum approximations.
• It uses convolution integrals of densities for particle and hole states to compute transition strengths for changes in exciton number by ±2 or 0.
• The method employs a fermionic field description of particles and holes as independent degrees of freedom, allowing independent creation and annihilation.
• Transition strengths are derived via second-order perturbation theory, with matrix elements expressed as integrals over intermediate states and two-body matrix elements.
• The approach uses explicit summations over quantum numbers (energy, angular momentum) and includes Pauli blocking via delta functions.
• A new code, TRANSNU, is implemented to compute transition strengths and can be interfaced with standard nuclear data evaluation codes such as TNG.

Experimental results

Research questions

• RQ1 Can transition strengths in pre-equilibrium reactions be computed directly from microscopic quantum mechanics without Laplace transforms or saddle-point approximations?
• RQ2 How do results from a harmonic oscillator basis compare with those from a constant energy grid in the context of transition strength calculations?
• RQ3 To what extent does the choice of single-particle basis affect the validity of the exciton model’s predictions about equilibrium attainment?
• RQ4 Can the new formalism reproduce the results of the traditional exciton model in a wide range of excitation energies?
• RQ5 How does the direct microscopic approach improve upon semi-classical formulations in describing pre-equilibrium nuclear dynamics?

Key findings

• When using the harmonic oscillator basis, the new formalism shows significant deviations from the exciton model, suggesting that previous conclusions about equilibrium attainment during the pre-equilibrium stage may be questionable.
• With a constant energy grid for the single-particle basis, the formalism successfully reproduces exciton model results across a broad range of excitation energies.
• The direct microscopic approach avoids the need for Laplace transforms and continuum approximations, providing a more rigorous quantum foundation for transition strength calculations.
• The method reveals that transition strengths for changes in exciton number by ±2 or 0 can be expressed as convolutions of simpler state densities.
• The developed TRANSNU code enables direct integration into semi-classical nuclear data evaluation frameworks like TNG, enhancing accuracy and microscopic consistency.
• The formalism highlights the sensitivity of pre-equilibrium predictions to the choice of single-particle basis, particularly in the context of equilibrium dynamics.