Adiabatic Research Articles

Effect of Heat Leakage on Relativistic Quantum Lenoir Engine Performance with a Massless Boson as Working Substance in the Infinite Potential Box

A study on the effect of heat leakage on power output, thermal efficiency, and reversibility rate in a relativistic quantum Lenoir engine has been conducted. Initially, we analogize the quantum working substance of the engine, a massless boson trapped in an infinite potential box with a movable right wall, as an ideal gas confined in a pistoned cylinder. Then, the total work, heat input, and heat output of each engine cycle which consists of isochoric, adiabatic expansion, and isobaric compression are extracted by applying the concept of quantum thermodynamics. Finally, power output, thermal efficiency, and reversibility rate of the engine are calculated for different variations of the heat leakage constant. The results are the relationship between several parameters which are expressed in the graph of thermal efficiency vs. compression ratio, graph of efficiency/normal efficiency vs. compression ratio, power output vs. efficiency, and reversibility rate vs. compression ratio. The conclusion is that an increase in heat leakage has an effect on reducing the efficiency and reversibility rate of the engine but does not affect its power output. This work will provide a new chapter for further research related to the use of the boson particle as a working substance in the quantum heat engine, especially the study of the heat leakage effect on engine performance.

Mechanism of hot‐spot formation of emulsion explosives sensitized by hydrogen‐storage glass microballoons

AbstractIn order to investigate the primary factors influencing hot‐spot formation in emulsion explosives sensitized by hydrogen‐storage glass microballoons (GMBs), we conducted impact calculations on hydrogen‐storage GMBs. The calculations focused on tracking two main mechanisms: the brittle collapse of GMBs and the adiabatic compression of internal gas. Various parameters were considered, including loading pressures, initial porosities, gas types, and initial gas pressures. Our findings indicate that the contribution of brittle collapse to hot‐spot formation is negligible, while adiabatic compression emerges as the predominant intrinsic mechanism for hot‐spot ignition in GMB‐sensitized emulsion explosives. Moreover, we observed that the ignition time remains similar for low‐pressure nitrogen and high‐pressure hydrogen. The addition of hydrogen does not result in an increased number of hot‐spots; however, it elevates the energy of each individual hot‐spot, thereby enhancing power delivery. Optimal selection of GMB size is crucial for hot‐spot formation and hydrogen storage. GMBs that are excessively large are prone to shell breakage, while overly small GMBs have limited hydrogen storage capacity. GMBs within the size range of 20 μm to 100 μm are deemed more suitable for emulsion explosives.

Toward Cosmological Simulations of the Magnetized Intracluster Medium with Resolved Coulomb Collision Scale

We present the first results of one extremely high-resolution, nonradiative magnetohydrodynamical cosmological zoom-in simulation of a massive cluster with a virial mass of M vir = 2.0 × 1015 solar masses. We adopt a mass resolution of 4 × 105 M ⊙ with a maximum spatial resolution of around 250 pc in the central regions of the cluster. We follow the detailed amplification process in a resolved small-scale turbulent dynamo in the intracluster medium (ICM) with strong exponential growth until redshift 4, after which the field grows weakly in the adiabatic compression limit until redshift 2. The energy in the field is slightly reduced as the system approaches redshift zero in agreement with adiabatic decompression. The field structure is highly turbulent in the center and shows field reversals on a length scale of a few tens of kiloparsecs and an anticorrelation between the radial and angular field components in the central region that is ordered by small-scale turbulent dynamo action. The large-scale field on megaparsec scales is almost isotropic, indicating that the structure formation process in massive galaxy cluster formation suppresses any memory of both the initial field configuration and the amplified morphology via the turbulent dynamo. We demonstrate that extremely high-resolution simulations of the magnetized ICM are within reach that can simultaneously resolve the small-scale magnetic field structure, which is of major importance for the injection of and transport of cosmic rays in the ICM. This work is a major cornerstone for follow-up studies with an on-the-fly treatment of cosmic rays to model in detail electron-synchrotron and gamma-ray emissions.

Investigation of the Strain–Stress Field in Nanoscale Multilayer Systems by the Phase Plane Method

This paper presents the results of the study of stress relaxation fields, deformation, and temperature of the system of nanostructured multilayer coatings. In the work, a nonlinear relationship between strain and stress was used to take into account nonlinear effects in the mechanism of nanostructure formation. The paper assumes that a friction surface is provided by the self-organization of shear components: both stress and strain on the one hand, and temperature on the other. The studied objects are described in the adiabatic approximation, taking into account the fact of the evolution of stresses and strains. With the help of phase portraits of the system, the dependence of the deformation processes on the stresses arising in the system without coating and with coating is shown. It is shown that the rate of change of deformation depends on the characteristics of the mechanical impact on the coating and on the amount of stress and deformation. A conclusion is drawn regarding the transition process in the presence of two regions (Hooke and plastic deformation) in the corresponding phase portrait of the strain–stress field of the system. The results of the work can be used to determine the effective parameters of a coating in the analysis of experimental time dependences of stresses.

A novel cycle engine for low-grade heat utilization: Principle, conceptual design and thermodynamic analysis

Efficient engine technologies to convert low-grade heat to electricity are urgently desired. In this work, a conceptual structure of engine for a novel cycle or one-way oscillating flow cycle (OOFC), which consists of two isochoric and two adiabatic processes, is described for low-grade heat utilization. Characteristics of OOFC allows for the working fluid temperature glide to be matched to the decrease in temperature of low-grade heat. Then, thermodynamic model is developed for evaluating the performance. Theoretical simulation results show that maximum specific output works are in the range of 12.2 kJ kg−1 – 79.7 kJ kg−1. Compared to Stirling cycle system, maximum specific output work in OOFC system could be improved by 16.2 %–24.8 %. Compared to ideal Carnot cycle engine system, maximum specific output works in OOFC system is nearly the same and 1.8 %–2.6 % lower. As Carnot cycle engine is ideal while thermodynamic cycle loss and heat transfer loss in cold heat exchangers are considered in OOFC engine, the ratios of maximum specific output work demonstrate that OOFC system could be very promising for low-grade heat utilization as a result of well-matched temperature profile in hot heat exchanger.

Analysis of spin-squeezing generation in cavity-coupled atomic ensembles with continuous measurements

We analyze the generation of spin-squeezed states via coupling of three-level atoms to an optical cavity and continuous quantum measurement of the transmitted cavity field in order to monitor the evolution of the atomic ensemble. Using analytical treatment and microscopic simulations of the dynamics, we show that one can achieve significant spin squeezing, favorably scaling with the number of atoms N. However, contrary to some previous literature, we clarify that it is not possible to obtain Heisenberg scaling without the continuous feedback that is proposed in optimal approaches. In fact, in the adiabatic cavity removal approximation and large N limit, we find the scaling behavior N−2/3 for spin squeezing and N−1/3 for the corresponding protocol duration. These results can be obtained only by considering the curvature of the Bloch sphere, since linearizing the collective spin operators tangentially to its equator yields inaccurate predictions. With full simulations, we characterize how spin-squeezing generation depends on the system parameters and departs from the bad cavity regime, by gradually mixing with cavity-filling dynamics until metrological advantage is lost. Finally, we discuss the relevance of this spin-squeezing protocol to state-of-the-art optical clocks.

Density functional theory beyond the Born–Oppenheimer approximation: exact mapping onto an electronically non-interacting Kohn–Sham molecule

This work presents an alternative, general, and in-principle exact extension of electronic Kohn–Sham density functional theory (KS-DFT) to the fully quantum-mechanical molecular problem. Unlike in existing multi-component or exact-factorization-based DFTs of electrons and nuclei, both nuclear and electronic densities are mapped onto a fictitious electronically non-interacting molecule (referred to as KS molecule), where the electrons still interact with the nuclei. Moreover, in the present molecular KS-DFT, no assumption is made about the mathematical form (exactly factorized or not) of the molecular wavefunction. By expanding the KS molecular wavefunction à la Born–Huang, we obtain a self-consistent set of ‘KS beyond Born–Oppenheimer’ electronic equations coupled to nuclear equations that describe nuclei interacting among themselves and with non-interacting electrons. An exact adiabatic connection formula is derived for the Hartree-exchange-correlation energy of the electrons within the molecule and, on that basis, a practical adiabatic density-functional approximation is proposed and discussed.

Non-adiabatic excited-state time-dependent GW molecular dynamics (TDGW) satisfying extended Koopmans' theorem: An accurate description of methane photolysis.

There is a longstanding difficulty that time-dependent density functional theory relying on adiabatic local density approximation is not applicable to the electron dynamics, for example, for an initially excited state, such as in photochemical reactions. To overcome this, we develop non-adiabatic excited-state time-dependent GW molecular dynamics (TDGW) on the basis of the extended quasiparticle theory. Replacing Kohn-Sham orbitals/energies with correlated, interacting quasiparticle orbitals/energies allows the full correspondence to the excited-state surfaces and corresponding total energies, with satisfying extended Koopmans' theorem. We demonstrate the power of TDGW using methane photolysis, CH4→CH3•+H, an important initiation reaction for combustion/pyrolysis and hydrogen production of methane. We successfully explore several possible pathways and show how this reaction dynamics is captured accurately through simultaneously time-tracing all quasiparticle levels. TDGW scales as O(NB3-4), where NB is the number of basis functions, which is distinctly advantageous to performing dynamics using configuration interaction and coupled cluster methods.

A semiclassical model for charge transfer along ion chains in silicates

It has been observed in fossil tracks and experiments in the layered silicate mica muscovite the transport of charge through the cation layers sandwiched between the layers of tetrahedra-octahedra-tetrahedra. A classical model for the propagation of anharmonic vibrations along the cation chains has been proposed based on first principles and empirical functions. In that model, several propagating entities have been found as kinks or crowdions and breathers, both with or without wings, the latter for specific velocities and energies. Crowdions are equivalent to moving interstitials and transport electric charge if the moving particle is an ion, but they also imply the movement of mass, which was not observed in the experiments. Breathers, being just vibrational entities, do not transport charge. In this work, we present a semiclassical model obtained by adding a quantum particle, electron or hole to the previous model. We present the construction of the model based on the physics of the system. In particular, the strongly nonlinear vibronic interaction between the nuclei and the extra electron or hole is essential to explain the localized charge transport, which is not compatible with the adiabatic approximation. The formation of vibrational localized charge carriers breaks the lattice symmetry group in a similar fashion to the Jahn-Teller Effect, providing a new stable dynamical state. We study the properties and the coherence of the model through numerical simulations from initial conditions obtained by tail analysis and other means. We observe that although the charge spreads from an initial localization in a lattice at equilibrium, it can be confined basically to a single particle when coupled to a chaotic quasiperiodic breather. This is coherent with the observation that experiments imply that a population of charge is formed due to the decay of potassium unstable isotopes.

Secular evolution of resonant planets in the coplanar case

Aims. We study the secular evolution of two planets in mutual deep mean-motion resonance (MMR) in the planar elliptic three-body problem framework for different mass ratios. We do not consider any restriction in the eccentricity of the inner planet e1 or in the eccentricity of the outer planet e2. Methods. The method we used is based on a semi-analytical model that consists of calculating the averaged resonant disturbing function numerically. It is assumed for this that all the orbital elements (except for the mean longitudes) of both planets are constant on the resonant timescale. In order to obtain the secular evolution inside the MMR, we make use of the adiabatic invariance principle, assuming a zero-amplitude resonant libration. We constructed two phase portraits, called the ℋ1 and ℋ2 surfaces, in the three-dimensional spaces (e1, Δϖ, σ) and (e2, Δϖ, σ), where Δϖ is the difference between the planetary longitude of perihelia and σ is the critical angle. These surfaces, which are related through the angular moment conservation, allow us to find the apsidal corotation resonances (ACRs) and to predict the secular evolution of e1, e2, Δϖ, and σ (libration center). Results. While studying the 1:1, 2:1, 3:1, and 3:2 MMR, we found that large excursions in eccentricity can exist in some particular cases. We compared the secular variations of e1, e2, Δϖ, and σ predicted by the model with a numerical integration of the exact equations of motion for different mass ratios. We obtained good matches. Finally, the model was applied to study the secular evolution of the resonant exoplanet systems HD 73526 and HD 31527. They both have a pair of planets and are very close to the deep MMR condition. In the first system, we found that the pair of planets that constitutes the system evolves in a symmetrical ACR, whereas in the second system, we found that planets c and d, which are in an unusual 16:3 MMR, are close to an ACR, but outside its dynamical region, where Δϖ circulates.

Hydrodynamic density functional theory of simple dissipative fluids

In this paper, a statistical physical derivation of thermodynamically consistent fluid mechanical equations is presented for non-isothermal viscous molecular fluids. The coarse-graining process is based on (i) the adiabatic expansion of the one-particle probability density function around local thermodynamic equilibrium, (ii) the assumption of decoupled particle positions and momenta, and (iii) the variational principle. It is shown that there exists a class of free energy functionals for which the conventional thermodynamic formalism can be naturally adopted for non-equilibrium scenarios, and describes entropy monotonic fluid flow in isolated systems. Furthermore, the analysis of the general continuum equations revealed the possibility of a non-local transport mode of energy in highly compressible dense fluids.

Quantum simulations of time-dependent Hamiltonians beyond the quasistatic approximation

Existing approaches to analog quantum simulations of time-dependent quantum systems rely on perturbative corrections to quantum simulations of time-independent quantum systems. We overcome this restriction to perturbative treatments with an approach based on flow equations and a multimode Fourier expansion. The potential of the quantum simulations that can be achieved with our approach is demonstrated with the pedagogical example of a Λ system and the quench in finite time through a quantum phase transition of a Chern insulator in a driven noninteracting Hubbard system. The example of the Λ system demonstrates the ability of our approach to describe situations beyond the validity of adiabatic approximations. Published by the American Physical Society 2024

High-pressure liquid refrigerant injection for reciprocating compressors

Isentropic (adiabatic) compression of superheated vapor in vapor compression refrigeration cycles usually results in some discharge superheat, i.e. the discharge temperature exceeding the condensing temperature. Large discharge superheat reduces both operating range and compression efficiency. Sub-isentropic compression with heat rejection to the heat sink is therefore superior to adiabatic compression, as it results in lower discharge superheat and improved efficiency. However, significant temperature reduction by heat transfer through the working chamber walls is difficult to facilitate in common refrigeration compressors, particularly reciprocating compressors. Refrigerant and oil injection is a well-established method for reducing temperatures and increasing efficiency in screw and scroll compressors, as well as multi-stage systems. In comparison, effective methods for temperature reduction in single-stage reciprocating compressors are fairly limited. Therefore, a concept for high-pressure liquid injection is proposed. By means of a pump and injection valve, high-pressure liquid refrigerant is injected into the working chamber and atomized. The compressed gas is cooled through the evaporation of the injected liquid. The injection valve allows for control over the injection timing and quantity, enabling adjustment of the temperature profile, which is a major distinction from existing refrigerant injection methods. The coupled compression-injection process is evaluated for various refrigerants using an energetic chamber model. The injection is found to be particularly suitable for refrigerants with a large evaporation heat and a shallow dew line slope. For the example of ammonia, a theoretical reduction in compression work of 8.9% at an evaporation temperature of −10 °C and a condensing temperature of 45 °C is found.

Systematic investigation of nucleon optical model potentials in (p, d) transfer reactions* *Supported by the National Natural Science Foundation of China (U2067205, 12205098), and the National Key Research and Development Program of China (2022YFA1602403)

The consistent three-body model reaction methodology (TBMRM) proposed by J. Lee et al. [ Phys. Rev. C 69, 064313 (2004); Phys. Rev. C 73, 044608 (2006); Phys. Rev. C 75, 064320 (2007)], which includes adopting the simple zero-range adiabatic wave approximation, constraining the single-particle potentials using modern Hartree–Fock calculations, and using global nucleon optical model potential (OMP) geometries, are widely applied in systematic studies of transfer reactions. In this study, we investigate the influence of different nucleon OMPs in extracting spectroscopic factors (SFs) from ( ) reactions. Our study covers 32 sets of angular distribution data of ( ) reactions on four targets and a large range of incident energies (20-200 MeV/nucleon). This study uses two semi-microscopic nucleon OMPs, i.e., Jeukenne, Lejeune, and Mahaux (JLM) [Phys. Rev. C 16, 80 (1977); Phys. Rev. C 58, 1118 (1998)] and CTOM [Phys. Rev. C 94, 034606 (2016)], and a pure microscopic nucleon potential, i.e., WLH [Phys. Rev. Lett. 127, 182502 (2021)]. The results are compared with those using the phenomenological global optical potential KD02 [Nucl. Phys. A 713, 231 (2003)]. We find that the incident energy dependence of spectroscopic factors extracted from ( ) reactions is evidently suppressed when microscopic OMPs are employed for 12C, 28Si, and 40Ca. In addition, spectroscopic factors extracted using the systematic microscopic optical potential CTOM based on the Dirac-Brueckner-Hartree-Fock theory are more in line with the results obtained from ( ) measurements, except for 16O and 40Ca at high energies (> 100 MeV), necessitating an exact treatment of double-magic nuclei. The results obtained by using the pure microscopic optical potential, WLH, based on the EFT theory show the same trend as those of CTOM but are generally higher. The JLM potential, which relies on simplified nuclear matter calculations with old-fashioned bare interactions, produces results that are very similar to those of the phenomenological potential KD02. Our results indicate that modern microscopic OMPs are reliable tools for probing the nuclear structure using transfer reactions across a wide energy range.

Shock Cooling and Breakout Emission for Optical Flares Associated with Gravitational-wave Events

The astrophysical origin of stellar-mass black hole (BH) mergers discovered through gravitational waves (GWs) is widely debated. Mergers in the disks of active galactic nuclei (AGNs) represent promising environments for at least a fraction of these events, with possible observational clues in the GW data. An additional clue to unveil AGN merger environments is provided by possible electromagnetic emission from postmerger accreting BHs. Associated with BH mergers in AGN disks, emission from shocks emerging around jets launched by accreting merger remnants is expected. Here we compute the properties of the emission produced during breakout and the subsequent adiabatic expansion phase of the shocks, and we then apply this model to optical flares suggested to be possibly associated with GW events. We find that the majority of the reported flares can be explained by breakout and shock cooling emission. If the optical flares are produced by shock cooling emission, they would display moderate color evolution, possibly color variations among different events, and a positive correlation between delay time and flare duration and would be preceded by breakout emission in X-rays. If the breakout emission dominates the observed lightcurve, we predict the color to be distributed in a narrow range in the optical band and the delay time from GW to electromagnetic emission to be longer than ∼2 days. Hence, further explorations of delay time distributions, flare color evolution, and associated X-ray emission will be useful to test the proposed emission model for the observed flares.

Design and evaluation of a kind of polymer-bonded explosives with improved mechanical sensitivity and thermal properties

The emergence of TKX-50, an energetic ionic salt with a high enthalpy of formation and low sensitivity, has opened a new path for the development of high-energetic, insensitive composite explosives. However, due to the poor interfacial binding properties of TKX-50 with conventional binders, there is a lack of effective guidance for the design of TKX-50 based composite explosives. To address the above issues, the interactions between carboxymethyl cellulose acetate butyrate (CMCAB) and other binders with explosives TKX-50/HMX were compared using the molecular dynamics method. Based on the simulations, TKX-50/HMX/CMCAB-based polymer-bonded explosives (PBXs) were prepared with CMCAB as binder, which displays a high binding energy (Ebind) with TKX-50 and high cohesive energy density (CED), and the effect of TKX-50 content on the performance of PBXs was investigated. The physical properties of PBXs, specifically the morphology, mechanical sensitivity, and thermal conductivity, were analyzed by SEM, sensitivity apparatus, and thermal conductivity meter, respectively. The specific heat capacity (Cp) and non-isothermal decomposition temperature of PBXs were tested by DSC, and then the corresponding thermal kinetic parameters were analyzed to evaluate their thermal safety. The adiabatic thermal decomposition processes of PBXs were tested using an ARC instrument. The decomposition mechanism and kinetics were also explored to further analyze their thermal stability and thermal safety under adiabatic conditions. The computer code EXPLO5 was used to predict the detonation parameters of PBXs. The results showed that CMCAB and TKX-50 displayed favorable interfacial bonding properties, and TKX-50 can be bonded with HMX to form a molding powder with a desirable morphology and safety profile. The TKX-50 in PBXs effectively improves the mechanical sensitivity and thermal safety of PBX and has a significant effect on the detonation performance of PBX. This research demonstrates a novel method suitable for screening and investigating high-energetic insensitive explosive systems compatible with TKX-50.

A SIMPLIFIED ANALYTICAL MODEL OF STIRLING ENGINE EFFICIENCY CALCULATION. A THEORETICAL ANALYSIS OF WORKING GAS

The indicated power and thermal efficiency are the most important features of the Stirling engine. In the most general way, the efficiency depends on a set of parameters, such as the ratio between the volumes and temperatures of the working gas in the hot and cold compartment, the adiabatic function and energy losses due to friction processes between engine components. Stirling engines can use mono, bi or polyatomic molecules such as helium, hydrogen or carbon dioxide as working gases. The first part of the contribution is dedicated verification and completion of our results obtained on the basis of the technical design data of the Genoa03 Stirling engine. These were determined based on an analytical model of the thermal efficiency of a simplified thermodynamic cycle described in the pressure-volume diagram by two adiabatic transformations and two isochoric transformations. The input data used are those used by our αSETS calculation code. The second part of the article is dedicated to the analytical elaboration of the adiabatic function of the working gas for the case of a mixture consisting of two or three types of working gases at different molar ratios. The adiabatic functions are used to estimate thermal efficiency. It has been shown that a mixture of two or three types of gas does not improve the thermal efficiency of the Stirling engine compared to the use of one-component gas. The results of these models related to the Beale relationship can be used successfully in the design process of Stirling engines

A mathematical analysis of the adiabatic Dyson equation from time-dependent density functional theory

In this article, we analyse the Dyson equation for the density–density response function (DDRF) that plays a central role in linear response time-dependent density functional theory (LR-TDDFT). First, we present a functional analytic setting that allows for a unified treatment of the Dyson equation with general adiabatic approximations for discrete (finite and infinite) and continuum systems. In this setting, we derive a representation formula for the solution of the Dyson equation in terms of an operator version of the Casida matrix. While the Casida matrix is well-known in the physics literature, its general formulation as an (unbounded) operator in the N-body wavefunction space appears to be new. Moreover, we derive several consequences of the solution formula obtained here; in particular, we discuss the stability of the solution and characterise the maximal meromorphic extension of its Fourier transform. We then show that for adiabatic approximations satisfying a suitable compactness condition, the maximal domains of meromorphic continuation of the initial DDRF and the solution of the Dyson equation are the same. The results derived here apply to widely used adiabatic approximations such as (but not limited to) the random phase approximation and the adiabatic local density approximation. In particular, these results show that neither of these approximations can shift the ionisation threshold of the Kohn–Sham system.

The chemistry and excitation of H2 and HD in the early Universe

ABSTRACT We have critically reviewed the literature pertaining to reactions that are significant for the chemistry of hydrogen-, deuterium-, and helium-bearing species in the homogeneous early Universe. For each reaction rate coefficient, we provide a fit in the modified-Arrhenius form, specifying the corresponding uncertainty and temperature range. This new network, limited to 21 reactions, should be the most reliable to date. Combined with accurate state-to-state rate coefficients for inelastic and reactive collisions involving H2 and HD, it allows us for the first time to follow the evolution of the abundances of atomic and molecular species, level populations of H2 and HD, and the ortho:para ratio (OPR) of H2, in a self-consistent fashion during the adiabatic expansion of the universe. The abundances of H2 and HD change only marginally compared to previous models, indicating that the uncertainties on the main reaction rate coefficients have essentially been removed. We also find that the adiabatic expansion has a dramatic effect on the OPR of H2, which freezes-out at redshifts z ≲ 50. In contrast, at higher redshifts, the populations of the rotational levels of H2 and HD are predicted to be fully thermalized at the temperature of the cosmic background radiation field, a result that conflicts with some recent, independent calculations. This new network allows the chemistry of primordial gas to be followed during the early phase of collapse towards Population III star progenitors.

Universality and hysteresis in slow sweeping of bifurcations.

Bifurcations in dynamical systems are often studied experimentally and numerically using a slow parameter sweep. Focusing on the cases of period-doubling and pitchfork bifurcations in maps, we show that the adiabatic approximation always breaks down sufficiently close to the bifurcation, so the upsweep and downsweep dynamics diverge from one another, disobeying standard bifurcation theory. Nevertheless, we demonstrate universal upsweep and downsweep trajectories for sufficiently slow sweep rates, revealing that the slow trajectories depend essentially on a structural asymmetry parameter, whose effect is negligible for the stationary dynamics. We obtain explicit asymptotic expressions for the universal trajectories and use them to calculate the area of the hysteresis loop enclosed between the upsweep and downsweep trajectories as a function of the asymmetry parameter and the sweep rate.