arXiv subjects

G. F. Bertsch

Publications and source records attributed to G. F. Bertsch.

At least 19 recordsLinked to original sources

arXiv2024

Role of momentum in the generator-coordinate method applied to barrier penetration

Nuclear fission at barrier-top energies is conventionally modeled by a one-dimensional Schrödinger equation applied to internal fission channels, but that treatment is hard to justify in the configuration-interaction approach to nuclear Hamiltonians. Here we show that inclusion of states of finite momentum by the Generator Coordinate Method (GCM) considerably extends the range of energies at which GCM-based Hamiltonians could reproduce the Schrödinger treatment. The transmission probabilities for crossing the barrier are calculated by a discrete version of Kohn's variational method, which may also be useful for other systems of interacting fermions.

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arXiv2024

Microscopic description of induced fission in a configuration interaction approach

Even though more than 80 years have passed since the discovery of fission, its microscopic understanding has still been unclear. To clarify the underlying mechanics of induced fission, we analyze the distribution of a fission width using a miscropic framework based on a configuration-interaction approach. The distribution is known to follow a chi-squared distribution, which is characterized by the effective number of decay channels, $ν$. We introduce an effective Hamitonian for the space of compound nucleus states and estimate $ν$ from the rank of the imaginary part of the effective Hamiltonian. Applying the model to $^{235}$U(n,f), we succesfully reproduce the empirical value of $ν=2.3\pm1.1$. We also find that $ν$ is insensitve to the number of fission channels, which is consistent with an experimental finding.

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arXiv2024

Microscopic derivation of transition-state theory for complex quantum systems

The decay of quantum complex systems through a potential barrier is often described with transition-state theory, also known as RRKM theory in chemistry. Here we derive the basic formula for transition-state theory based on a generic Hamiltonian as might be constructed in a configuration-interaction basis. Two reservoirs of random Hamiltonians from Gaussian orthogonal ensembles are coupled to intermediate states representing the transition states at a barrier. Under the condition that the decay of the reservoirs to open channels is large, an analytic formula for reaction rates is derived. The transition states act as independent Breit-Wigner resonances which contribute additively to the total transition probability, as is well known for electronic conductance through resonant tunneling states. It is also found that the transition probability is independent of the decay properties of the states in the second reservoir over a wide range of decay widths.

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arXiv2024

Barrier penetration in a discrete-basis formalism

The dynamics of a many-particle system are often modeled by mapping the Hamiltonian onto a Schrödinger equation. An alternative approach is to solve the Hamiltonian equations directly in a model space of many-body configurations. In a previous paper the numerical convergence of the two approaches was compared with a simplified treatment of the Hamiltonian representation. Here we extend the comparison to the nonorthogonal model spaces that would be obtained by the generator-coordinate method. With a suitable choice of the collective-variable grid, a configuration-interaction Hamiltonian can reproduce the Schrödinger dynamics very well. However, the method as implemented here requires that the barrier height is not much larger than the zero-point energy in the collective coordinates of the configurations.

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arXiv2023

Modeling barrier-top fission dynamics in a discrete-basis formalism

A configuration-interaction model is presented for the barrier region of induced fission. The configuration space is composed of seniority-zero configurations constructed from self-consistent mean-field wave functions. The Hamiltonian matrix elements between configurations include diabatic and pairing interactions between particles. Other aspects of the Hamiltonian are treated statistically, guided by phenomenological input of compound-nucleus transmission coefficients. In this exploratory study the configuration space is restricted to neutron excitations only. A key observable calculated in the model is the fission-to-capture branching ratio. We find that both pairing and diabatic interactions are important for achieving large branching to the fission channels. In accordance with the transition-state theory of fission, the calculated branching ratio is found to be quite insensitive to the fission decay widths of the pre-scission configurations. However, the barrier-top dynamics appear to be quite different from transition-state theory in that the transport is distributed over many excited configurations at the barrier top.

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arXiv2022

Diabatic Hamiltonian matrix elements made simple

With a view to applying the Generator Coordinate Method to large configuration spaces, we propose a simple approximate formula to compute diabatic many-body matrix elements without having to evaluate two-body interaction matrix elements. The method is illustrated with two analytically solvable Hamiltonians based on the harmonic oscillator.

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arXiv2022

Generator coordinate method for transition-state dynamics in nuclear fission

Since its beginnings, fission theory has asumed that low-energy induced fission takes place through transition-state channels at the barrier tops. Neverthess, up to now there is no microscopic theory applicable to those conditions. We suggest that modern reaction theory is suitable for this purpose, and propose a methodology based on a configuration-interaction framework using the Generator Coordinate Method (GCM). Simple reaction-theoretic models are constructed with the Gaussian Overlap Approximation (GOA) to parameterize both the dynamics within the channels and their incoherent couplings to states outside the barrier. The physical characteristics of the channels examined here are their effective bandwidths and the quality of the coupling to compound-nucleus states as measured by the transmission factor $T$. We also investigate the spacing of GCM states with respect to their degree of overlap. We find that a rather coarse mesh provides an acceptable accuracy for estimating the bandwidths and transmission factors. The common numerical stability problem in using the GCM is avoided due to the choice of meshes and the finite bandwidths of the channels. The bandwidths of the channels are largely controlled by the zero-point energy with respect to the collective coordinate in the GCM configurations.

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arXiv2021

Porter-Thomas fluctuations in complex quantum systems

The Gaussian Orthogonal Ensemble (GOE) of random matrices has been widely employed to describe diverse phenomena in strongly coupled quantum systems. An important prediction is that the decay rates of the GOE eigenstates fluctuate according to the distribution for one degree of freedom, as derived by Brink and by Porter and Thomas. However, we find that the coupling to the decay channels can change the effective number of degrees of freedom from $ν= 1$ to $ν= 2$. Our conclusions are based on a configuration-interaction Hamiltonian originally constructed to test the validity of transition-state theory, also known as Rice-Ramsperger-Kassel-Marcus (RRKM) theory in chemistry. The internal Hamiltonian consists of two sets of GOE reservoirs connected by an internal channel. We find that the effective number of degrees of freedom $ν$ can vary from one to two depending on the control parameter $ρΓ$, where $ρ$ is the level density in the first reservoir and $Γ$ is the level decay width. The $ν= 2$ distribution is a well-known property of the Gaussian Unitary ensemble (GUE); our model demonstrates that the GUE fluctuations can be present under much milder conditions. Our treatment of the model permits an analytic derivation for $ρΓ\gtrsim 1$.

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arXiv2021

Transition-state dynamics in complex quantum systems

A model is proposed for studying the reaction dynamics in complex quantum systems in which the complete mixing of states is hindered by an internal barrier. Such systems are often treated by the transition-state theory, also known in chemistry as RRKM theory, but the validity of the theory is questionable when there is no identifiable coordinate associated with the barrier. The model consists of two Gaussian Orthogonal Ensembles (GOE) of internal levels coupled to each other and to the wave functions in the entrance and decay channels. We find that the transition-state formula can be derived from the model under some easily justifiable approximations. In particular, the assumption in transition-state theory that the reaction rates are insensitive to the decay widths of the internal states on the far side of the barrier is fulfilled for broad range of Hamiltonian parameters. More doubtful is the common assumption that the transmission factor $T$ across the barrier is unity or can be modeled by a one-dimensional Hamiltonian giving $T$ close to unity above the barrier. This is not the case in the model; we find that the transmission factor only approaches one under special conditions that are not likely to be fulfilled without a strong collective component in the Hamiltonian.

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arXiv2021

Configuration-interaction approach to nuclear fission

We propose a configuration-interaction (CI) representation to calculate induced nuclear fission with explicit inclusion of nucleon-nucleon interactions in the Hamiltonian. The framework is designed for easy modeling of schematic interactions but still permits a straightforward extension to realistic ones. As a first application, the model is applied to branching ratios between fission and capture in the decay modes of excited fissile nuclei. The ratios are compared with the Bohr-Wheeler transition-state theory to explore its domain of validity. The Bohr-Wheeler theory assumes that the rates are insensitive to the final-state scission dynamics; the insensitivity is rather easily achieved in the CI parameterizations. The CI modeling is also capable of reproducing the branching ratios of the transition-state hypothesis which is one of the key ingredients in the present-day theory of induced fission.

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arXiv2020

Least action and the maximum-coupling approximations in the theory of spontaneous fission

We investigate the dynamics of spontaneous fission in a configuration-interaction (CI) approach. In that formalism the decay rate is governed by an effective interaction coupling the ground-state configuration and a fission doorway configuration, with the interaction strength determined by inverting a high-dimensioned CI Hamiltonian matrix that may have a block-tridiagonal structure. It is shown that the decay rate decreases exponentially with the number of blocks at a rate determined by the largest eigenvalue of a matrix in the block space for Hamiltonians with identical off-diagonal blocks. The theory is greatly simplified by approximations similar in spirit to the adiabatic and the least-action approximations in continuum representations. Here each block is replaced by a single matrix element. While the adiabatic reduction underestimates the coupling, a reduction based on a maximum-coupling approximation works well in a schematic CI model.

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arXiv2020

Force and pressure in many-particle quantum dynamics

The Newtonian concept of force may be useful in some aspects of the dynamics of many-particle quantum systems such as fissioning nuclei. Following Ehrenfest's method, we show that the quantum kinetic force between parts of an extended quantum system can be described by an operator acting on the boundary between the two subsystems. The contribution to the force due to a short-ranged particle interaction can also be treated in the same way. This includes interaction effects treated in density functional theory by local functionals. The force operators are applied to several simple models to demonstrate the method.

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arXiv2020

A microscopic model for spontaneous fission: validity of the adiabatic approximation

We investigate microscopically the tunneling dynamics in spontaneous fission of atomic nuclei. To this end, we employ a schematic solvable model with a pairing-plus-quadrupole interaction. The spontaneous decay of a system is simulated by introducing a small imaginary part to the energy of a fission doorway state. We show that the many-body Hamiltonian can be reduced to an effective 2$\times$2 Hamiltonian, from which one can derive a simple approximate formula for the decay width. We particularly investigate the applicability of the adiabatic approximation, which has often been used in the literature. With typical value of the parameters, we find that the adiabatic approximation works within a factor of around 5 for the decay width.

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arXiv2020

Derivation of K-matrix reaction theory in a discrete basis formalism

The usual derivations of the S and K matrices for two-particle reactions proceed through the Lippmann-Schwinger equation with formal definitions of the incoming and outgoing scattering states. Here we present an alternative derivation that is carried out completely in the Hamiltonian representation, using a discrete basis of configurations for the scattering channels as well as the quasi-bound configurations of the combined fragments. We use matrix algebra to derive an explicit expression for the K matrix in terms of the Hamiltonian of the internal states of the compound system and the coupling between the channels and the internal states. The formula for the K matrix includes explicitly a real dispersive shift matrix to the internal Hamiltonian that is easily computed in the formalism. That expression is applied to derive the usual form of the S matrix as a sum over poles in the complex energy plane. Some extensions and limitations of the discrete-basis Hamiltonian formalism are discussed in the concluding remarks and in the Appendix.

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arXiv2019

A schematic reaction-theory model for nuclear fission

The K-matrix formalism is applied to a schematic model for nuclear fission. The purpose is to explore the dependence of observables on the assumptions made about the configuration space and nucleon interaction in the Hamiltonian of the fissile nucleus. As expected, branching ratios in induced fission are found to depend sensitively on the character of the residual interaction, whether it is pairing in form or taken from a random ensemble. On the other hand, the branching ratio is not much affected by the presence of additional configurations that do not introduce new fission paths.

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arXiv2019

Decay widths at the scission point in nuclear fission

An outstanding problem in the theory of nuclear fission is understanding the Hamiltonian dynamics at the scission point. Here we apply the Generator Coordinate Method to calculate decay widths for pre-scission configurations into the two-fragment continuum. Transitions that are allowed under diabatic dynamics can have widths up to several MeV. For non-diabatic decays through the pairing interaction, typical widths to a specific final state channel are 2-3 orders of magnitude smaller. The nucleus U-236 is taken as a representative example in the calculations.

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arXiv2019

Monopole moments and nuclear compressibility

The nuclear compressibility has a role in nuclear physics in several ways. Its relationship to the giant monopole is well known and has been subject of much theoretical work. Less well known is its affect on in nuclear structure, namely monopole transitions between low-lying states in the spectrum. Here I revisit the relationship between monopole and quadrupole moments in deformed nuclei. It has often been assumed without any microscopic justification that the nucleus can be treated as an incompressible fluid. An analytic formula is derived here for the resulting relationship between monopole and quadrupole moments. The formula is shown to be well satisfied within self-consistent mean-field theory calculated with several energy functionals. The formula can be tested when both monopole and quadrupole matrix elements of band-to-band transitions are known. Data is available for the decay of the first excited $K^π= 0^+$ band to the ground-state band in $^{156}$Gd, and the extracted matrix elements are found to be consistent with the incompressible fluid picture.

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arXiv2019

Diabatic scission paths

An outstanding problem in the theory of nuclear fission is to understand the Hamiltonian dynamics at the scission point. In this work the fissioning nucleus is modeled in self-consistent mean-field theory as a set of Generator Coordinate (GCM) configurations passing through the scission point. In contrast to previous methods, the configurations are constructed in the Hartree-Fock approximation with axially symmetric mean fields and using the K-partition numbers as additional constraints. The goal of this work is to find paths through the scission point where the overlaps between neighboring configurations are large. A measure of distance along the path is proposed that is insensitive to the division of the path into short segments. For most of the tested K-partitions two shape degrees of freedom are adequate to define smooth paths. However, some of the configurations and candidate paths have sticking points where there are substantial changes in the many-body wave function, especially if quasiparticle excitations are present. The excitation energy deposited in fission fragments arising from thermal excitations in the pre-scission configurations is determined by tracking orbital occupation numbers along the scission paths. This allows us to assess the validity of the well-known scission-point statistical model, in which the scission process is assumed to be fully equilibrated up to the separated fission fragments. The nucleus 236U is taken as a representative example in the calculations.

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