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DOE OSTI2026

Precise Modeling of a Complex Solenoidal Magnetic Field Using a Combination of Analytic Functions and a PINN

We demonstrate an iterative approach to modeling a sparsely measured magnetic field in a large-bore solenoid. This approach uses a hybrid of traditional and machine learning techniques. The traditional technique is a linear least-squares fit using a series solution to Laplace's equation, while the machine learning technique involves the training of a physics-informed neural network (PINN) on the least-squares fit residuals. We use a newly defined activation function "DELTAsnake," a modification to the snake activation function proposed by Ziyin et al. that allows for stronger curvature and non-monotonicity. The combined model approximately obeys Maxwell's equations to a level sufficient for producing high quality physics simulations and analysis. Our approach is applied to a highly realistic calculation of the expected magnetic field in the Mu2e experiment's Detector Solenoid which includes a simple model for the expected statistical measurement uncertainties. Using ten toy measurement simulations, we demonstrate the capabilities of our model in comparison to the least-squares method alone; the least-squares method alone results in a reduced chi-squared statistic of ${2.15 \pm 0.01}$, while our approach improves the reduced chi-square to ${1.034 \pm 0.005}$. Furthermore, for an average toy simulation, we show that the range of the RMS of the three field component residuals reduces from ${0.07-0.37}$ Gauss to ${0.05-0.07}$ Gauss. We find that this novel method is robust against a realistic systematic uncertainty deriving from Hall probe calibration bias and can be used to significantly reduce the number of measurements required to achieve an accurate model.

Kampa, Cole [Caltech] (ORCID:0000000192972920)↗

DOE OSTI2020

How to design a focusing guide: The large moderator case1

As continuously shaped super-mirrors are becoming available, the conceptual design of focusing guides should explore a wider range of possibilities to accomplish an efficient neutron beam extraction. Starting from a desired phase-space volume at the sample position and using an upstream ray-tracing approach, the acceptance diagram of any focusing guide can be calculated at the moderator position. To ensure high brilliance transfer and homogeneous coverage, the acceptance diagram should be fully included in the neutron source emission phase-space volume. Following this idea, the guide system can be scaled into dimensionless geometric figures that convey performance limits for a desired cross-section reduction. Moreover, if we impose a monotonic increase of the reflection angle with divergence angle at the sample position, the shape of the mirror is analytically determined. This approach was applied in the design of a focusing guide for SNAP instrument at SNS, at ORNL, USA. The results of McStas simulations are presented with different options included. Our approach facilitates finding an optimal solution for connecting multiple guide pieces to avoid excessive losses and ensure a homogeneous phase space coverage.

Frost, Matthew↗

DOE OSTI2026

Partially Ionized Plasma Physics and Technological Applications

Partially ionized plasma physics has attracted increased attention recently due to numerous technological applications made possible by the increased sophistication of computer modelling, the depth of the theoretical analysis, and the technological applications to a vast field of manufacturing for computer components. Partially ionized plasma is characterized by a significant presence of neutral particles in contrast to the fully ionized plasma. The theoretical analysis is based upon solutions of the kinetic Boltzmann equation, yielding the non-Maxwellian electron energy distribution function (EEDF), thereby emphasizing the difference with a fully ionized plasma. The impact of the effect on discharges in inert and molecular gases is described in detail, yielding the complex nonlinear phenomena resulting in plasma selforganization. A few examples of such phenomena are given, including the non-monotonic EEDFs in the discharge afterglow in a mixture of argon with the molecular gas NF3; the explosive generation of cold electron populations in capacitive discharges, hysteresis of EEDF in inductively coupled plasmas. Recently, highly advanced computer codes were developed in order to address the outstanding challenges in plasma technology. These developments are briefly described in general terms.

non-Maxwellian electron energy distribution functi↗

DOE OSTI2023

Diffusion of Hydrophilic to Hydrophobic Forms of Nile Red in Aqueous C 12 EO 10 Gels by Variable Area Fluorescence Correlation Spectroscopy

In this work, solute diffusion within lyotropic liquid crystal gels prepared from a series of water and decaethylene glycol monododecyl ether (C 12 EO 10 ) mixtures was explored by variable area fluorescence correlation spectroscopy. Aqueous C 12 EO 10 gels were prepared in concentrations ranging from 55:45 to 70:30 wt% of surfactant and water. Small angle X-ray scattering revealed that these gels comprise hexagonal mesophases of cylindrical micelles. Micelle spacing was found to decrease with increasing C 12 EO 10 concentration. Three different Nile red (NR) dyes were employed as model solutes and were separately doped into the gels at nanomolar levels. These include a hydrophilic form of NR incorporating an anionic sulfonate group (NRSO 3 - ), a hydrophobic form incorporating a fourteen-carbon alkane tail (NRC 14 ), and commercial NR as an intermediate case. FCS data acquired from the gels revealed that NRSO 3 - diffused primarily in 3D. Its diffusion coefficient exhibited a monotonic decrease with increasing gel concentration and micelle packing density, consistent with confinement of its motions as a result of its exclusion from the micelle cores. NRC 14 exhibited the smallest diffusion coefficient, most likely due to its larger size and enhanced interactions with the micelle cores. NR yielded an intermediate diffusion coefficient and the most anomalous behavior of the three dyes, attributable to its facile partitioning between core and corona regions, and greater participation by 1D diffusion. The results of these studies afford an improved understanding of molecular mass transport through soft-matter nanomaterials like those being developed for use in drug delivery and membrane based chemical separations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Hydrogen Bond Network Disruption by Hydration Layers in Water Solutions with Salt and Hydrogen-Bonding Polymers (PEO)

A mean field theory model describing the interaction of ion hydration layers with the network of hydrogen bonds of both water and the nonionic polymer poly(ethylene oxide) (PEO) is presented. The predictions of the model for types and statistics of hydrogen bonds, the number of water molecules bound to PEO, or their dependence on temperature are successfully verified from all-atom simulations at different NaCl and PEO concentrations. Furthermore, our simulations show that the binding of cations to PEO increases monotonically with salt concentration, in agreement with recent experimental results, through a mechanism in which the sum of the number of bound water and cations is independent of salt concentration. Here, the model introduced is general and can describe any salt or hydrogen-bond-forming polymer.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Effect of isopropanol cosolvent on the rheology and spinnability of aqueous polyacrylic acid solutions

Abstract We investigate the effect of alcohol fraction (isopropanol, IPA) in a binary water‐alcohol solvent mixture on the shear and extensional rheological properties, as well as the role of viscoelasticity on fiber formation of poly(acrylic acid) (PAA) in electrospinning. Comparison of the scaling of both specific viscosities η sp and extensional relaxation times λ E of PAA in water–IPA mixtures, showed stronger scaling compared to salt‐free aqueous polyelectrolyte solutions, except for the η sp in the unentangled regime displaying a polyelectrolyte‐like scaling η sp ~ c 0.5 for all IPA%. Such deviation suggested IPA induces association/aggregation of PAA. However, the trends between η sp and λ E magnitudes as a function of IPA% differ for concentrations compared in the entangled regime. The η sp as well as their elastic moduli exhibit a maximum, whereas λ E increases monotonically with IPA%, suggesting a complex interplay of various interactions are dictating their structure in water‐IPA mixtures, affecting their shear and extensional response differently. Electrospinning experiments showed increasing IPA% reduces the onset of both beaded and uniform fibers. Analysis using dimensionless numbers indicated the enhancement of their elasticity by IPA, and the consequent stabilizing effect on their jets/filaments against break‐up during electrospinning, plays a role in the improvement of their fiber formation.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Vanishing adsorption limit of Riemann problem solutions for the polymer model

Here we examine the vanishing adsorption limit of solutions of Riemann problems for the Glimm–Isaacson model of chemical flooding of a petroleum reservoir. A contact discontinuity is deemed admissible if it is the limit of traveling waves or rarefaction waves for an augmented system that accounts for weak chemical adsorption onto the rock. We prove that this criterion justifies the admissibility criteria adopted previously by Keyfitz–Kranzer, Isaacson–Temple and de Souza–Marchesin, provided that the fractional flow function depends monotonically on chemical concentration. We also demonstrate that the adsorption criterion selects the undercompressive contact discontinuities required to solve the general Riemann problem in an example model with non-monotone dependence.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Specular-reflection contributions to static and dynamic radiation forces on circular cylinders

Interest in the response of highly reflecting objects in water to modulated acoustical radiation forces makes it appropriate to consider contributions to such forces from perfectly reflecting objects to provide insight into radiation forces. The acoustic illumination can have wavelengths much smaller than the object's size, and objects of interest may have complicated shapes. Here, the specular contribution to the oscillating radiation force on an infinite circular cylinder at normal incidence is considered for double-sideband-suppressed carrier-modulated acoustic illumination. The oscillatory magnitude of the specular force decreases monotonically with increasing modulation frequency, and the phase of the oscillating force depends on the relative phase of the sidebands. The phase dependence on the modulation frequency can be reduced with the appropriate selection of a sideband relative-phase parameter. That is a consequence of the significance of rays that are incident on the cylinder having small impact parameters that are nearly backscattered. For one choice of a relative sideband phase, a prior partial wave series (PWS) solution is available, which supports the specular analysis when the PWS is evaluated for a rigid cylinder. The importance of specular contributions for aluminum cylinders in water is noted. A specular analysis for an analogous spherical reflector is also summarized.

Marston, Philip L. (ORCID:0000000306778537)↗

DOE OSTI2025

Minimal implicit-solvent coarse-grained simulation of Pluronic block copolymers with ionic liquids

Pluronic block copolymers, composed of poly(ethylene oxide) (PEO) and poly(propylene oxide) (PPO) in a triblock structure (PEO–PPO–PEO), are well known for their amphiphilic character and ability to self‐assemble into micelles in aqueous solution. The addition of ionic liquids (ILs) can further modulate the core–shell structures of these copolymers, influencing their stability, critical micellization temperature, and size. However, fully atomistic simulations often become prohibitively expensive due to the size and complexity of these systems. In this work, coarse‐grained simulations using a minimal implicit‐solvent model were performed to examine how two classes of ILs, namely, 1‐alkyl‐3‐methylimidazolium ([C n C 1 im]) and 1‐alkyl‐3‐methylpyrrolidinium ([C n C 1 pyrr]), change the micellization of Pluronic block copolymers in aqueous solution. The effects of IL concentration and alkyl group length were investigated, and the model greatly improved the efficiency of simulating large‐scale micelle systems. Furthermore, the numerical simulations are qualitatively compared with experimental investigations. Our results show that adding ILs expands the micelle core by embedding IL tails among the PPO blocks, thereby increasing overall micelle size. Less polar ILs generally induce more pronounced micellar growth. However, the effect of IL tail length on conformation and micellar packing is non‐monotonic. Up to moderate chain lengths (around C8–C10), the IL tails can extend sufficiently to increase local separation within the micelle; at longer tail lengths, enhanced hydrophobic clustering and steric hindrance cause the tails to bend or fold, capping further expansion. In addition, although block copolymer chains tend to pack more closely in the presence of longer‐tailed ILs, the random coil size of an individual polymer chain does not necessarily shrink. Meanwhile, these insights provide a deeper understanding of how Pluronic/IL systems interact, informing applications in drug delivery, cosmetics, food, and environmental engineering. Finally, our minimal implicit‐solvent model can be applied to larger systems and longer timescales, substantially reducing computational cost while reproducing key structural trends observed experimentally.

Atomistic simulations↗

DOE OSTI2023

Electron kinetics in a positive column of AC discharges in a dynamic regime

Abstract We have performed hybrid kinetic-fluid simulations of a positive column in alternating current (AC) argon discharges over a range of driving frequencies f and gas pressure p for the conditions when the spatial nonlocality of the electron energy distribution function (EEDF) is substantial. Our simulations confirmed that the most efficient conditions of plasma maintenance are observed in the dynamic regime when time modulations of mean electron energy (temperature) are substantial. The minimal values of the root mean square electric field and the electron temperature have been observed at f/p values of about 3 kHz Torr −1 in a tube of radius R = 1 cm. The ionization rate and plasma density reached maximal values under these conditions. The numerical solution of a kinetic equation allowed accounting for the kinetic effects associated with spatial and temporal nonlocality of the EEDF. Using the kinetic energy of electrons as an independent variable, we solved an anisotropic tensor diffusion equation in phase space. We clarified the role of different flux components during electron diffusion in phase space over surfaces of constant total energy. We have shown that the kinetic theory uncovers a more exciting and rich physics than the classical ambipolar diffusion (Schottky) model. Non-monotonic radial distributions of excitation rates, metastable densities, and plasma density have been observed in our simulations at pR > 6 Torr cm. The predicted off-axis plasma density peak in the dynamic regime has never been observed in experiments so far. We hope our results stimulate further experimental studies of the AC positive column. The kinetic analysis could help uncover new physics even for such a well-known plasma object as a positive column in noble gases.

Physics↗

DOE OSTI2024

Low-Temperature Characterization of a Nonaqueous Liquid Electrolyte for Lithium Batteries

Rechargeable batteries exhibit poor performance at low temperatures due to sluggish ion transport through the electrolytic phase. Ion transport is governed by three transport parameters—conductivity, diffusion coefficient, and the cation transference number with respect to the solvent velocity—and the thermodynamic factor. Understanding how these parameters change with temperature is necessary for designing improved electrolytes. In this work, we combine electrochemical techniques with electrophoretic NMR to determine the temperature dependence of these parameters for a liquid electrolyte, LiTFSI salt dissolved in tetraglyme between −20 and 45 °C. At colder temperatures, all species in the electrolyte tend to move more slowly due to increasing viscosity, which translates to a monotonic decrease in conductivity and diffusion coefficient with decreasing temperature. Surprisingly, we find that the field-induced velocity of solvent molecules at a particular salt concentration is a nonmonotonic function of temperature. The cation transference number with respect to the solvent velocity thus exhibits a complex dependence on temperature and salt concentration. The measured thermodynamic and transport properties are used to predict concentration gradients that will form in a lithium-lithium symmetric cell under a constant applied potential as a function of temperature using concentrated solution theory. The calculated steady current at −20 °C is lower than that at 45 °C by roughly two orders of magnitude.

25 ENERGY STORAGE↗

DOE OSTI2022

Insights into Sorption and Molecular Transport of Aqueous Glucose into Zeolite Nanopores

Liquid-phase heterogeneous catalysis using zeolites is important for biomass conversion to fuels and chemicals. There is a substantial body of work on gas-phase sorption in zeolites with different topologies; however, studies investigating the diffusion of complex molecules in liquid medium into zeolitic nanopores are scarce. Here, we present a molecular dynamics study to understand the sorption and diffusion of aqueous β-d-glucose into β-zeolite silicate at T = 395 K and P = 1 bar. Through 2-μs-long molecular dynamics trajectories, we reveal the role of the solvent, the kinetics of the pore filling, and the effect of the water model on these properties. We find that the glucose and water loading is a function of the initial glucose concentration. Although the glucose concentration increases monotonically with the initial glucose concentration, the water loading exhibits a nonmonotonic behavior. At the highest initial concentration (~20 wt %), we find that the equilibrium loading of glucose is approximately five molecules per unit cell and displays a weak dependence on the water model. Glucose molecules follow a single-file diffusion in the nanopores due to confinement. Here, the dynamics of glucose and water molecules slows significantly at the interface. The average residence time for glucose molecules is an order of magnitude larger than that in the bulk solution, while it is about twice as large for the water molecules. Our simulations reveal critical molecular details of the glucose molecule’s local environment inside the zeolite pore relevant to catalytic conversion of biomass to valuable chemicals.

09 BIOMASS FUELS↗

DOE OSTI2024

A chemical kinetics theory for interpreting the non-monotonic temperature dependence of enzymatic reactions

Abstract. One notable observation of enzymatic chemical reactions is that, for a given abundance of enzymes and substrates, temperature increases cause reaction rates to first increase consistent with the Arrhenius relationship, then plateau, and finally fall off quickly to zero at high temperatures. While many mathematical functions have been used to describe this pattern, we here propose a chemical kinetics theory which successfully replicates this observation and provides insights into the processes responsible for these dynamics. The chemical kinetics theory combines the law of mass action, von Smoluchowski's diffusion-limited chemical reaction theory, and Eyring's transition state theory. This new theory reveals that the thermally reversible enzyme denaturation ensured by the ceaseless thermal motion of molecules and ions in an enzyme solution explains the plateau and subsequent decrease in chemical reaction rates with increasing temperature. The temperature-dependent affinity parameter (K) that relates enzymes and substrates through their binding also affects the shape of the emergent temperature response. We demonstrate that with an increase in substrate availability, K shifts the optimal temperature, where reaction rates plateau, towards higher values. Further, we show that the chemical kinetics theory accurately represents 12 sets of published enzyme assay data and includes the popular mechanistic model by Ratkowsky et al. (2005) as a special case. Given its good performance and solid theoretical underpinning, we believe this new theory will facilitate the construction of more mechanistic-based environmental biogeochemical models.

Tang, Jinyun↗

DOE OSTI2026

Evaluating the Limits of QAOA Parameter Transfer at High-Rounds on Sparse Ising Models With Geometrically Local Cubic Terms

The emergent practical applicability of the Quantum Approximate Optimization Algorithm (QAOA) for approximate combinatorial optimization is a subject of considerable interest. One of the primary limitations of QAOA is the task of finding a set of good parameters, which is usually done using a variational optimization loop. Parameter transfer, or parameter concentration, is a phenomenon where QAOA angles trained on problem instances that are self-similar tend to perform well for other problem instances from that similar class. This suggests a potentially highly efficient and scalable non-variational learning method for QAOA angle finding. In this work, we systematically study QAOA parameter transferability from small problem sizes (16 and 27 decision variables) onto large problem instances (up to 156 qubits) for heavy-hex graph Ising models with geometrically local higher order terms using the Julia based QAOA simulation tool \texttt{JuliQAOA} to perform classical angle finding for up to $49$ QAOA layers ($p$). Parameter transfer of the fixed angles is validated using a combination of full statevector, Projected Entangled Pair States (PEPS), Matrix Product State (MPS), and LOWESA numerical simulations. We find that the QAOA parameter transfer from single instances applied to other (unseen) problem instances does not in general provide monotonically improving performance as a function of $p$ - there are many cases where the performance temporarily decreases as a function of $p$ - but despite this the transferred angles have a general trend of improved expectation value as the QAOA depth increases, in many cases converging close to the true ground-state energy of the $100+$ qubit instances. We also sample the hardware-compatible Ising models using the ensemble of transfer-learned QAOA parameters on several superconducting qubit IBM Quantum processors with 127, 133, and 156 qubits. We find continuous solution quality improvement of the hardware-compatible QAOA circuits run on the IBM NISQ processors up to $p=5$ on \texttt{ibm\_fez}, up to $p=9$ on \texttt{ibm\_torino}, and up to $p=10$ on \texttt{ibm\_pittsburgh}.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Metrics for Intercomparison of Remapping Algorithms (MIRA) protocol applied to Earth system models

Abstract. Strongly coupled nonlinear phenomena such as those described by Earth system models (ESMs) are composed of multiple component models with independent mesh topologies and scalable numerical solvers. A common operation in ESMs is to remap or interpolate component solution fields defined on their computational mesh to another mesh with a different combinatorial structure and decomposition, e.g., from the atmosphere to the ocean, during the temporal integration of the coupled system. Several remapping schemes are currently in use or available for ESMs. However, a unified approach to compare the properties of these different schemes has not been attempted previously. We present a rigorous methodology for the evaluation and intercomparison of remapping methods through an independently implemented suite of metrics that measure the ability of a method to adhere to constraints such as grid independence, monotonicity, global conservation, and local extrema or feature preservation. A comprehensive set of numerical evaluations is conducted based on a progression of scalar fields from idealized and smooth to more general climate data with strong discontinuities and strict bounds. We examine four remapping algorithms with distinct design approaches, namely ESMF Regrid (Hill et al., 2004), TempestRemap (Ullrich and Taylor, 2015), generalized moving least squares (GMLS) (Trask and Kuberry, 2020) with post-processing filters, and WLS-ENOR (Li et al., 2020). By repeated iterative application of the high-order remapping methods to the test fields, we verify the accuracy of each scheme in terms of their observed convergence order for smooth data and determine the bounded error propagation using challenging, realistic field data on both uniform and regionally refined mesh cases. In addition to retaining high-order accuracy under idealized conditions, the methods also demonstrate robust remapping performance when dealing with non-smooth data. There is a failure to maintain monotonicity in the traditional L2-minimization approaches used in ESMF and TempestRemap, in contrast to stable recovery through nonlinear filters used in both meshless GMLS and hybrid mesh-based WLS-ENOR schemes. Local feature preservation analysis indicates that high-order methods perform better than low-order dissipative schemes for all test cases. The behavior of these remappers remains consistent when applied on regionally refined meshes, indicating mesh-invariant implementations. The MIRA intercomparison protocol proposed in this paper and the detailed comparison of the four algorithms demonstrate that the new schemes, namely GMLS and WLS-ENOR, are competitive compared to standard conservative minimization methods requiring computation of mesh intersections. The work presented in this paper provides a foundation that can be extended to include complex field definitions, realistic mesh topologies, and spectral element discretizations, thereby allowing for a more complete analysis of production-ready remapping packages.

58 GEOSCIENCES↗

DOE OSTI2026

pyEF: A Python Framework for QM and QM/MM Atom-Wise Electric Field Analysis

We introduce pyEF, a software package for computing molecular electric fields, electrostatic interaction energies, and electrostatic potentials from quantum mechanical (QM) atom-centered multipole expansions with atom-wise decomposable contributions. We demonstrate the computational efficiency and accuracy of this QM-derived electric field evaluation tool through several tests. To assess the influence of the underlying QM method and charge partitioning scheme on these electrostatic quantities, we analyze over 250 configurations of an acetone solute molecule in five solvents of variable polarity. We find that electric field calculations are highly sensitive to the choice of charge partitioning method. Even among real-space charge schemes, acetone Stark tuning rates differ by up to a factor of 2. Benchmarking computed solvent dipole moments against experimental bulk values, we conclude that the CM5, ADCH, and Hirshfeld-I charge schemes most reliably capture solvent electrostatics and therefore provide a more faithful foundation for computing electric fields. When constructed from these real-space charges, electric fields are nearly insensitive to basis set size and monotonically increase in magnitude with higher Fock exchange. We also demonstrate efficient convergence of QM electrostatics when more distant molecules are represented solely by MM point charges, reducing computational overhead. Leveraging these findings, we demonstrate the use of pyEF to deduce environmental effects on a transition metal complex from a Ga 4 L 6 12– nanocage and quantify the dominant role of organic linkers in orchestrating electrostatic preorganization.

electric fields↗

DOE OSTI2023

Kinetics of Shear-Induced Structural Ordering in Dense Colloids

The macroscopic rheological response of a colloidal solution is highly correlated with the local microscopic structure, as revealed by an in situ Rheo-SAXS experiment with a high temporal resolution. Oscillatory shear can induce a strain-controlled ordering-to-disorder transition, resulting in a shear-thickening process that is different from the normal shear-thickening behavior that is driven by hydrodynamics and particle friction. Here, we reveal that there is a complex time-dependent kinetics toward structural ordering under different applied strains. When the strain amplitude reaches a critical value that starts to induce disordering in the system, the pathway toward the dynamic equilibrium can also become highly non-monotonic. Within the same oscillatory cycle, there is a strong correlation of ordering with different phases of the oscillation, with the system oscillating between two dynamic metastable states.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Focusing of laser-accelerated ions with spherically curved targets

A theoretical model for laser-accelerated ion focusing from spherically curved targets using a Gaussian self-similar solution is presented. This model describes the evolution of the focal location and focal radius with final ion energy and the initial radial boundary of the ions, the latter being a function of both target geometry and the ion acceleration radius for a given ion energy. The theory is supported by particle-in-cell simulations of a variety of target shapes with varying radii of curvature and target opening angles, as well as variations in the injected electron beam radius and energy spectrum. The theory and simulations suggest that the focal location varies linearly with the radius of curvature, with the square root of the ratio of ion energy to effective electron temperature, and monotonically increases with the energy-dependent ratio of the initial ion radial boundary to the radius of curvature. Considering ponderomotive acceleration, this suggests that the focal length should scale inversely with intensity, $(I_Lλ_L^2)$ −1/4 , suggesting a 10× increase in intensity will reduce the focal distance by ̃1.8× for a given ion energy.

Physics - Plasma physics↗