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DOE OSTI2025

Systematic study of the validity of the eikonal model including uncertainties

Nuclear reactions at intermediate beam energies are often interpreted using the eikonal model. In the analysis of complex reaction probes, where few-body reaction methods are needed, the eikonal method may be used as an efficient way for describing the fragment-target reaction process. In this work, we perform a systematic study to test the validity of the eikonal approximation for nucleon-nucleus reactions. We also quantify uncertainties due to the nucleon optical potential on reaction observables. We inspect the validity of the eikonal model and its semiclassical correction by comparing it to exact solutions (obtained from solving the optical-model equation with a finite-differences method) for a wide range of reactions. We also study the effect of relativistic corrections, both kinematic and dynamic, by effectively incorporating the relativistic effects at intermediate energies. The uncertainties from a Bayesian global optical potential (KDUQ) are propagated to the observables of interest. Our study includes neutron and proton reactions on 27 Al , 40 Ca , 90 Zr , and 208 Pb , for a wide range of energies 𝐸 lab = 0–400 MeV. We calculate neutron-total cross sections (elastic and reactions) as well as proton-absorption cross sections as a function of beam energy, using the eikonal model, the eikonal model with a semiclassical correction, and the exact solution. Here, we also compute angular distributions for the methods above. Our results show that for the proton-absorption cross section, the eikonal model can be used down to around 60 MeV and the semiclassical correction extends its use to 30 MeV. However, the validity of the eikonal model for the neutron-total cross section only goes down to ≈120 MeV, a range extended to ≈ 50 MeV when using the semiclassical correction. We find the semiclassical correction to the eikonal model to be less effective in describing the angular distributions. The 1⁢𝜎 uncertainty intervals on the observables we studied is less than 5% for most of the energies considered, but increases rapidly for higher energies, namely energies outside the range of KDUQ (𝐸 lab > 200MeV).

Cluster models↗

DOE OSTI2023

ExactPack: A python library of exact analytic solutions

Verification of multi-physics simulation software against problems with known analytic or semi-analytic solutions is an important aspect of research into a wide variety of fields involving the motion of fluids, shock physics and other dynamic material properties. Previous work comparing simulation results against analytic solutions has been ad-hoc, with developers frequently writing their own analytic solvers. This has resulted in a large amount of duplicated effort. The python library ExactPack has been developed as a collection of analytic and semi-analytic solvers to a variety of multi-physics problems, providing a consistent API to a set of well-tested solver implementations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Verification of MOOSE/Bison's Heat Conduction Solver Using Combined Spatiotemporal Convergence Analysis

Bison is a computational physics code that uses the finite element method to model the thermo-mechanical response of nuclear fuel. Since Bison is used to inform high-consequence decisions, it is important that its computational results are reliable and predictive. One important step in assessing the reliability and predictive capabilities of a simulation tool is the verification process, which quantifies numerical errors in a discrete solution relative to the exact solution of the mathematical model. One step in the verification process—called code verification—ensures that the implemented numerical algorithm is a faithful representation of the underlying mathematical model, including partial differential or integral equations, initial and boundary conditions, and auxiliary relationships. In this paper, the code verification process is applied to spatiotemporal heat conduction problems in Bison. Simultaneous refinement of the discretization in space and time is employed to reveal any potential mistakes in the numerical algorithms for the interactions between the spatial and temporal components of the solution. For each verification problem, the correct spatial and temporal order of accuracy is demonstrated for both first- and second-order accurate finite elements and a variety of time-integration schemes. Furthermore, these results provide strong evidence that the Bison numerical algorithm for solving spatiotemporal problems reliably represents the underlying mathematical model in MOOSE. The selected test problems can also be used in other simulation tools that numerically solve for conduction or diffusion.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2022

Error Study of FlecSim Using the Method of Manufactured Solutions [Slides]

Error was calculated using exact solution chosen with the method of manufactured solutions. Hydrodynamics match expected order of accuracy for spatial error within 1.5% for all variables tested. Radiation transport matches expected order of temporal accuracy within 0.5% for all variables tested.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2023

Gauge constrained algorithm of variational discrete action theory at N = 3 for the multiorbital Hubbard model

The recently developed variational discrete action theory (VDAT) provides a systematic variational approach to the ground state of the quantum many-body problem, where the quality of the solution is controlled by an integer N, and increasing N monotonically approaches the exact solution. VDAT can be exactly evaluated in the d = ∞ multiorbital Hubbard model using the self-consistent canonical discrete action theory (SCDA), which requires a self-consistency condition for the integer time Green's functions. Previous work demonstrates that N = 3 accurately captures multiorbital Mott/Hund physics at a cost similar to the Gutzwiller approximation. Here we employ a gauge constraint to automatically satisfy the self-consistency condition of the SCDA at N = 3, yielding an even more efficient algorithm with enhanced numerical stability. We derive closed form expressions of the gauge constrained algorithm for the multiorbital Hubbard model with general density-density interactions, allowing VDAT at N = 3 to be straightforwardly applied to the seven-orbital Hubbard model. We present results and a performance analysis using N = 2 and N = 3 for the SU⁡(2⁢N orb ) Hubbard model in d = ∞ with N orb = 2–8, and compare to numerically exact dynamical mean-field theory solutions where available. Finally, the developments in this work will greatly facilitate the application of VDAT at N = 3 to strongly correlated electron materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Dynamics of inspiraling dark energy

Abstract We investigate the dynamics of a multifield dark energy model, which arises from certain rapid-turning and inspiraling trajectories in field space. We find the speed of sound $$c_s$$ c s of the dark energy perturbations around the background and show that $$c_s$$ c s is monotonically decreasing with time. Furthermore, it has a positive-definite lower bound that implies a certain clustering scale. We also extend the previously known background solution for dark energy to an exact solution that includes matter. This allows us to address the implications of our model for two cosmological tensions. More precisely, we argue that the $$\sigma _8$$ σ 8 tension can be alleviated generically, while reducing the Hubble tension requires certain constraints on the parameter space of the model. Notably, a necessary condition for alleviating the Hubble tension is that the transition from matter domination to the dark energy epoch begins earlier than in $$\Lambda $$ Λ CDM.

Anguelova, Lilia↗

DOE OSTI2021

Kinklike structures in an arcsin real scalar dynamics

Highlights: • Kinklike analyses in a real scalar theory with an arcsin dynamic. • First-order formalism implementation and deformation procedure in an arcsin scalar dynamics. In this paper, we analyze kink-like analytical solutions in a real scalar theory with an arcsin dynamics inspired by the arcsin electrodynamics presented in Kruglov (2015). This analysis is done by means of the first-order formalism. This formalism provides a framework where the equations of motion can be simplified by preserving the linear stability of the theory. In this work, the deformation procedure is implemented with the aim of finding exact solutions in systems with generalized dynamics. Along the paper, we explore how the first-order formalism is implemented in the arcsin kinetics and how such a term influences the kink-like solutions. As a part of the result of our paper, we show that the kink-like solutions are similar to the ones obtained in the standard scalar kinetic theory. We also show that the extra parameter, that controls the non-linearities of the model, plays an essential role in the energy densities and stability potentials. These quantities vary according to this parameter. The goals here are to show how the first-order framework is implemented in this arcsin scenario and to present the analytical kink-like solutions that can be found by means of the first-order framework and deformation method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Electric and magnetic black holes in a new nonlinear electrodynamics model

Highlights: • We introduce a new NED model which is comparable with the BI model in the weak-field limit. • We find an electric black hole solution in the context of Einstein’s gravity minimally coupled with the new NED model. • We study the thermal stability of the electric black hole solution. • Modified Smarr’s formula consistent with the first law of black hole thermodynamics is obtained. A new nonlinear electrodynamics (NED) model is introduced in the form of a nonpolynomial Lagrangian which admits static spherical and also plane wave solutions in a flat space. Upon coupling with gravity the electric field is finite and comparable with the Born–Infeld counterpart. The electric and magnetic black hole solutions in the Einstein’s gravity coupled with this NED model are presented. The solutions give both asymptotically and in the weak field limit Reissner–Nordström (RN) black hole and unlike the other known models our electric solution is expressed in terms of elementary functions in a closed form. We study the first law and derive the modified Smarr’s formula for the electric type extension of our model. Considerable rich structure, especially thermodynamic ones, ranging from first to the second order phase transitions are added to the RN black hole of linear electrodynamics with this NED model. Having the exact solution for the metric function at our disposal we investigate the stability of the electric black hole from both the thermodynamical and causal points of view.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Toward perturbation theory methods on a quantum computer

Perturbation theory, used in a wide range of fields, is a powerful tool for approximate solutions to complex problems, starting from the exact solution of a related, simpler problem. Advances in quantum computing, especially over the past several years, provide opportunities for alternatives to classical methods. Here, we present a general quantum circuit estimating both the energy and eigenstates corrections that is far superior to the classical version when estimating second-order energy corrections. We demonstrate our approach as applied to the two-site extended Hubbard model. In addition to numerical simulations based on qiskit, results on IBM’s quantum hardware are also presented. Our work offers a general approach to studying complex systems with quantum devices, with no training or optimization process needed to obtain the perturbative terms, which can be generalized to other Hamiltonian systems both in chemistry and physics.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Energy density and spectrum of single-cycle and sub-cycle electromagnetic pulses

Based on the exact solution of the Maxwell equations in the form of a collapsing spherical vector wave specified by an arbitrary function of time, we have calculated the maximum energy density that can be achieved when focusing extremely short pulses of various shapes. It is shown that our earlier formula expressing the maximum energy density in terms of the spectrum parameters for Gaussian quasi-monochromatic pulses is approximately (with an accuracy of ±40 %) valid for the main types of extremely short pulses. (extremely short laser pulses)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Benchmarking the multiconfigurational Hartree method by the exact wavefunction of two harmonically trapped bosons with contact interaction

Highlights: • Comparison of exact and numerical single-particle density matrices. • Benchmark for multiconfigurational Hartree. • Convergence for large number of orbitals. We consider two bosons in a one-dimensional harmonic trap, interacting by a contact potential, and compare the exact solution of this problem to a self-consistent numerical solution by using the multiconfigurational time-dependent Hartree (MCTDH) method. We thereby benchmark the predictions of the MCTDH method with a few-body problem that has an analytical solution for the most commonly experimentally realized interaction potential in ultracold quantum gases. It is found that exact ground state energy and first order correlations are accurately reproduced by MCTDH up to the intermediate dimensionless coupling strengths corresponding to typical background scattering lengths of magnetically trapped ultracold dilute Bose gases. For larger couplings, established for example by (a combination of) Feshbach resonances and optical trapping, the MCTDH approach overestimates the depth of the trap-induced correlation dip of first order correlations in position space, as well as overestimates the fragmentation, defined as the average relative occupation of orbitals other than the energetically lowest one. We anticipate that qualitatively similar features in the correlation function may arise for larger particle numbers, paving the way for a quantitative assessment of the accuracy of MCTDH by experiments with ultracold atoms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Accurate localization of Kosterlitz-Thouless-type quantum phase transitions for one-dimensional spinless fermions

We investigate the charge-density wave (CDW) transition for one-dimensional spinless fermions at half band filling with nearest-neighbor electron transfer amplitude t and interaction V. The model is equivalent to the anisotropic XXZ Heisenberg model for which the Bethe Ansatz provides an exact solution. For V>V c =2t, the CDW order parameter and the single-particle gap are finite but exponentially small, as is characteristic for a Kosterlitz-Thouless transition. It is notoriously difficult to locate such infinite-order phase transitions in the phase diagram using approximate analytical and numerical approaches. Second-order Hartree-Fock theory is qualitatively applicable for all interaction strengths, and predicts the CDW transition to occur at V$^{(2)}_{c,2}$≈1.5t. Second-order Hartree Fock theory is almost variational because the density of quasiparticle excitations is small. We apply the density-matrix renormalization group (DMRG) for periodic boundary conditions for system sizes up to 514 sites, which permits a reliable extrapolation of all physical quantities to the thermodynamic limit, apart from the critical region. We investigate the ground-state energy, the gap, the order parameter, the momentum distribution, the quasiparticle density, and the density-density correlation function to locate V c from the DMRG data. In conclusion, tracing the breakdown of the Luttinger liquid and the peak in the quasiparticle density at the band edge permits us to reproduce V c with an accuracy of one percent.

1-dimensional spin chains↗

DOE OSTI2024

Nonlinear Poisson–Boltzmann solutions for charged parallel plates: When opposite charges repel

I present an exact solution of the Poisson–Boltzmann equation for two parallel plates and discuss the solution properties. I discuss in more detail plates with opposite charges: In this case, there are two critical separations, L c,1 < L c,2 . For separations less than L c,1 , the force between plates is repulsive. It switches to attractive at L c,1 , but with the electric potential having the same sign on both plates. For L > L c,2 , the force remains attractive, and the potential at the plates has the same sign as the charge on each plate. I also describe charge regulation, determined by pK a , and provide formulas for both the critical distance where oppositely charged plates repel and their charging process. Finally, the implications of these results for the nanoparticle assembly, as driven by electrostatic interactions, are also discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Exact multi-instanton solutions to self-dual Yang–Mills equation on curved spaces

We find exact multi-instanton solutions to the self-dual Yang–Mills equation on a large class of curved spaces with SO(3) isometry, generalizing the results previously found on R 4 . Here, the solutions are featured with explicit multi-centered expressions and topological properties. As examples, we demonstrate the approach on several different curved spaces, including the Einstein static universe and R × dS$^{E}_{3}$, and show that the exact multi-instanton solutions exist on these curved backgrounds.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Sensitivity analysis of a layered piezoelectric system using ZFEM

The complex variable finite element method (ZFEM) is a numerical technique which aims to find the partial derivatives of the independent variables with respect to variation in dependent parameters declared in the physics. This is done by combining the complex Taylor series expansion within the weak formulation of the governing equation in a coupled system of linear equations forming a complex valued block matrix given by the Cauchy–Riemann matrix representation. In this work, two-dimensional linear first-order elements have been implemented in ZFEM to predict the design derivatives of the mechanical displacement field and the voltage potential field for a layered piezoelectric system in a steady-state study with Dirichlet boundary condition applied at the top and bottom edges of the geometry. This approach allows the standard FEM solution to quantify the sensitivity of the mechanical displacement and voltage potential fields with respect to small variations in the material properties through the information obtained from the computation of the derivatives. The domain is formed by a layered body with PZT-4 and PZT-5 stacked together. For result verification, the numerical solution obtained with ZFEM was compared to results from a commercial FEM package and the solution from the imaginary part was compared to the exact solution of a well-known benchmark problem. In conclusion, comparison of the results showed good agreement for both the real and imaginary parts of the solution and the largest sensitivities were found in PZT-5 specifically in C 13 , C 33 , and ε 33 .

42 ENGINEERING↗

DOE OSTI2021

Mass enhancement in 3 d and s - p perovskites from symmetry breaking

In some d-electron oxides, the measured effective mass m* exptl has long been known to be significantly larger than the model effective mass m* model deduced from mean-field band theory, i.e., m* exptl =βm* model , where β>1 is the “mass-enhancement” or “mass-renormalization” factor. Previous applications of density functional theory (DFT), based on a symmetry-restricted structure with the smallest number of possible magnetic, orbital, and structural degrees of freedom, missed such mass enhancement. This fact has been taken as evidence of strong electronic correlation, often described via the symmetry-restricted dynamic mean-field approach of the many-body theory, being the exclusive enabling physics. This paper uses instead a static density functional approach that does not restrict positional or spin degrees of freedom (symmetry-broken structures). This approach analyzes the contributions of different symmetry-broken modalities to mass enhancement for a few nominally highly correlated d-electron perovskites as well as the nominally uncorrelated, closed-shell s-p bonding perovskites. It shows that the energy-lowering symmetry-broken spin effects (e.g., nonzero local moment in the paramagnetic phase) and structural effects (e.g., atomic displacement) as described in mean-field DFT already manifest mass enhancement for both electrons and holes in a range of d-electron perovskites SrVO 3 , SrTiO 3 , BaTiO 3 , and LaMnO 3 , as well as p-electron perovskites CsPbI 3 and SrBiO 3 , including both metals (SrVO 3 ) and insulators (the rest). This is revealed only when enlarged unit cells of the same parent global symmetry, which are large enough to allow for symmetry-breaking distortions and concomitant variations in spin order, are explored for their ability to lower the total energy. Positional symmetry breaking that leads to mass enhancement includes octahedral rotation in halide perovskites such as CsPbI 3 , Jahn-Teller-like Q${+}\atop{2}$ distortion in LaMnO 3 , and bond disproportionation in SrBiO 3 , while magnetic symmetry breaking resulting in mass enhancement includes the formation of a distribution of local moments in SrVO3 that averages to zero in the paramagnetic phase. Not all symmetry breaking leads to significant mass enhancement, e.g., the rather small octahedral rotations in the nearly perfectly cubic SrTiO 3 cause negligible mass enhancement, as do the paraelectric displacements in BaTiO 3 . In principle, by ergodicity, the two descriptions, i.e., the symmetry-restricted dynamic approach with a single time-fluctuating magnetic moment and the symmetry-broken mean-field approach with a static distribution of spatially fluctuated local moments, are not mutually exclusive but are a choice of representation and consequently, a choice of computational efficiency. In approximate implementations, the symmetry-broken mean-field approach appears to remove much of what was strong correlation in dynamically correlated symmetry-restricted solutions, leaving smaller (“weak”) residual correlation with respect to the exact solution.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Effects of finite element discretization and model simplification on calculations of ductile failure initiation

The finite element method is a scheme to discretize the infinite number of degrees of freedom in continuum-level problems down to a finite number of degrees of freedom. This discretization is done in conjunction with methods that also reduce the field differential equations to sets of algebraic ones that can be solved by arithmetical operations. Therefore, solutions attained by finite element models are approximations to the exact solutions of the field equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Reference solutions for linear radiation transport: the Hohlraum and Lattice Benchmarks

Radiation transport describes the propagation of energetic particles through space as they interact with a surrounding material medium. In a kinetic description, radiation transport is modeled by a radiation transport equation (RTE) that prescribes the density of the radiation in position-momentum phase space. The purpose of this dataset is to provide highly resolved solutions to two benchmark problems. These two benchmarks do not possess exact solutions; moreover, the construction of a manufactured solution may require a non-physical source that is not desirable, especially if it spoils the physical nature of the solution. Thus the goal of this computational study is to provide a highly resolved reference solution for testing newer, more cost efficient methods that are currently being developed in the research community.

97 MATHEMATICS AND COMPUTING↗