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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

X-ray scattered halo around IGR J17544–2619

X-ray photons coming from an X-ray point source not only arrive at the detector directly, but also can be strongly forward-scattered by the interstellar dust along the line of sight (LOS), leading to a detectable diffuse halo around the X-ray point source. The geometry of small-angle X-ray scattering is straightforward, namely, the scattered photons travel longer paths and thus arrive later than the unscattered ones; thus, the delay time of X-ray scattered halo photons can reveal information of the distances of the interstellar dust and the point source. Here we present a study of the X-ray scattered halo around IGR J17544–2619, which is one of the so-called supergiant fast X-ray transients. IGR J17544–2619 underwent a striking outburst when observed with Chandra on 2004 July 3, providing a near δ-function light curve. We find that the X-ray scattered halo around IGR J17544–2619 is produced by two interstellar dust clouds along the LOS. The one that is closer to the observer gives the X-ray scattered halo at larger observational angles, whereas the farther one, which is in the vicinity of the point source, explains the halo with a smaller angular size. By comparing the observational angle of the scattered halo photons with that predicted by different dust grain models, we are able to determine the normalized dust distance. With the delay times of the scattered halo photons, we can determine the point source distance, given a dust grain model. Alternatively, we can discriminate between the dust grain models, if the point source distance is known independently.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2010

Multiple Scattering of Laser Pulses in Snow Over Ice: Modeling the Potential Bias in ICESat Altimetry

The primary goal of NASA's current ICESat and future ICESat2 missions is to map the altitude of the Earth's land ice with high accuracy using laser altimetry technology, and to measure sea ice freeboard. Ice however is a highly transparent optical medium with variable scattering and absorption properties. Moreover, it is often covered by a layer of snow with varying depth and optical properties largely dependent on its age. We describe a modeling framework for estimating the potential altimetry bias caused by multiple scattering in the layered medium. We use both a Monte Carlo technique and an analytical diffusion model valid for optically thick media. Our preliminary numerical results are consistent with estimates of the multiple scattering delay from laboratory measurements using snow harvested in Greenland, namely, a few cm. Planned refinements of the models are described.

Davis, A. B.↗
DOE OSTI2014

The iterative thermal emission method: A more implicit modification of IMC

For over 40 years, the Implicit Monte Carlo (IMC) method has been used to solve challenging problems in thermal radiative transfer. These problems typically contain regions that are optically thick and diffusive, as a consequence of the high degree of “pseudo-scattering” introduced to model the absorption and reemission of photons from a tightly-coupled, radiating material. IMC has several well-known features that could be improved: a) it can be prohibitively computationally expensive, b) it introduces statistical noise into the material and radiation temperatures, which may be problematic in multiphysics simulations, and c) under certain conditions, solutions can be nonphysical, in that they violate a maximum principle, where IMC-calculated temperatures can be greater than the maximum temperature used to drive the problem. We have developed a variant of IMC called iterative thermal emission IMC, which is designed to have a reduced parameter space in which the maximum principle is violated. ITE IMC is a more implicit version of IMC in that it uses the information obtained from a series of IMC photon histories to improve the estimate for the end of time step material temperature during a time step. A better estimate of the end of time step material temperature allows for a more implicit estimate of other temperature-dependent quantities: opacity, heat capacity, Fleck factor (probability that a photon absorbed during a time step is not reemitted) and the Planckian emission source. We have verified the ITE IMC method against 0-D and 1-D analytic solutions and problems from the literature. These results are compared with traditional IMC. We perform an infinite medium stability analysis of ITE IMC and show that it is slightly more numerically stable than traditional IMC. We find that significantly larger time steps can be used with ITE IMC without violating the maximum principle, especially in problems with non-linear material properties. The ITE IMC method does however yield solutions with larger variance because each sub-step uses a different Fleck factor (even at equilibrium)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1974

Coherent propagation of charged particle bunches in random magnetic fields

When the distribution function for energetic particles moving in interplanetary space is expressed in terms of eigenfunctions of an operator which describes pitch angle scattering by random magnetic fields, the familiar phenomenon of diffusion is the dominant solution of the transport equations provided that the spectrum of magnetic fluctuations is not too steep. The present paper describes a novel mode of particle transport when the latter condition is not satisfied (i.e., when the exponent for the spectrum of magnetic fluctuations is greater than 2). In this mode, the density inhomogeneities propagate coherently, in wavelike fashion, and with very little dispersion along the guiding field at a characteristic velocity equal to half the particle speed. The most striking manifestation of this phenomenon is the observed scatter-free propagation of solar flare electrons below 1 MeV.-

Earl, J. A.↗
DOE OSTI2022

Study of the water dynamics near hydrophilic, nanostructured CuO surfaces by quasielastic and inelastic neutron scattering

We have used quasielastic and inelastic neutron scattering to investigate the structure, dynamics, and phase transitions of water interacting with superhydrophilic CuO surfaces that not only possess a strong affinity for water but also a “grass-like” topography that is rough on both micro- and nanoscales. Here, we report quasielastic neutron scattering (QENS) measurements on two samples differing in water content at five temperatures below 280 K. The QENS spectra show water undergoing two different types of diffusive motion near the CuO surfaces: a “slow” translational diffusion occurring on a nanosecond time scale and a faster rotational motion. Further from the surfaces, there is “fast” translational diffusion comparable in rate to that of bulk supercooled water and the rotational motion occurring in the interfacial water. Analysis of the QENS spectra supports wetting of water to the CuO blades as seen in electron microscopy images. In addition, we observe an anomalous temperature dependence of the QENS spectra on cooling from 270 to 230 K with features consistent with a liquid–liquid phase transition. We suggest that the solvent-like properties of the coexisting bulk-like water in our CuO samples are a significant factor in determining the temperature dependence of the interfacial water’s dynamics. Our results are compared with those obtained from two well-studied substrate classes: (1) silicas that contain ordered cylindrical nanopores but have weaker hydrophilicity and (2) nanoparticles of other transition-metal oxides, such as TiO 2 , which share the strong hydrophilicity of our samples but lack their porosity.

36 MATERIALS SCIENCE↗
DOE OSTI2012

Effect of carbon dioxide and nitrogen on the diffusivity of methane confined in nano-porous carbon aerogel

The microscopic diffusivity of methane (CH{sub 4}) confined in nano-porous carbon aerogel was investigated as a function of added carbon dioxide (CO{sub 2}) and nitrogen (N{sub 2}) pressure using quasi-elastic neutron scattering (QENS). In the range of the external pressure of 1-2.5 MPa, the self-diffusivity of methane was found to increase with CO{sub 2} pressure and remain practically unchanged in the N{sub 2} environment. Increasing mobility of methane with CO{sub 2} pressure suggests that the adsorbed CH4 molecules become gradually replaced by CO{sub 2} on the surface of carbon aerogel pores, whereas the presence of N{sub 2} does not induce the replacement. The molecular mobility of the methane, with or without added carbon dioxide and nitrogen, is described by the unrestricted diffusion model, which is characteristic of methane compressed in small pores. On the other hand, both nitrogen and carbon dioxide molecules in carbon aerogel, when studied alone, with no methane present, follow a jump diffusion process, characteristic of the molecular mobility in the densified adsorbed layers on the surface of the aerogel pores.

03 NATURAL GAS↗
DOE OSTI2026

Structuring, stochastic behavior, and charge storage capacity of redox-active microemulsions formulated with mixtures of toluene and ionic liquid as oil phase

Oil/water microemulsions (µEs) are promising electrolytes for redox flow batteries (RFBs) as they simultaneously improve charge capacity and ionic conductivity. Here, we report the successful formulation of bicontinuous µEs where the oil phase is a solution of trihexyltetradecylphosphonium bis(trifluoromethylsulfonyl)amide ionic liquid in toluene with redox-active ferrocene. We examined the effect of supporting electrolyte anion (NO 3 – , Cl – , ClO 4 – ) on the structure, reactivity, transport, and electrolytic performance of redox-active µEs. Neutron scattering and nuclear magnetic resonance showed that the domain size increased as Cl – > NO 3 – > ClO 4 – while the ferrocene diffusion coefficient increased as NO 3 – > Cl – > ClO 4 – . Scanning electrochemical microscopy indicated anion-dependent current fluctuations during electrolysis, with ClO 4 – exhibiting the least high-frequency oscillations, which correlate to the highest charge and discharge capacity and reversibility. All ionic liquid containing systems improved the performance of toluene-based µEs, highlighting new design principles for these electrolytes.

diffusion↗
NASA NTRS1999

Dynamics of Single Chains of Suspended Ferrofluid Particles

We present an experimental study of the dynamics of isolated chains made of super-paramagnetic particles under the influence of a magnetic field. The motivation of this work is to understand if the chain fluctuations exist and, if it does, how does the fluctuation affect chain aggregation. We find that single chains strongly fluctuate and that the characteristic frequency of their fluctuations is inversely proportional to the magnetic field strength. The higher the field the lower the characteristic frequency of the chain fluctuations. In the high magnetic field limit, chains behave like rigid rods without any internal motions. In this work, we used ferrofluid particles suspended in water. These particles do not have any intrinsic magnetization. Once a magnetic field is applied, a dipole moment is induced in each particle, proportional to the magnetic field. A dipolar magnetic interaction then occurs between particles. If dipole-dipole magnetic energy is higher than the thermal energy, the result is a structure change inside the dipolar fluid. The ratio of these two energies is expressed by a coupling constant lambda as: lambda = (pi(a(exp 3))(chi(exp 2))(mu(sub 0))(H(sub 0))(exp 2))/18kT Where a is the particle radius, mu(sub 0) is the vacuum magnetic permeability, H(sub 0) the applied magnetic field, k the Boltzmann constant and T the absolute temperature. If lambda > 1, magnetic particles form chains along the field direction. The lateral coalescence of several chains may form bigger aggregates especially if the particle volume fraction is high. While many studies and applications deal with the rheological properties and the structural changes of these dipolar fluids, this work focuses on the understanding of the chain dynamics. In order to probe the chain dynamics, we used dynamic light scattering (DLS) in self-beating mode as our experimental technique. The experimental geometry is such that the scattering plane is perpendicular to the magnetic field. Therefore, only motions in this plane are probed. A very dilute sample of a ferrofluid emulsion with a particle volume fraction of 10(exp -5) is used in this experiment. We chose such a low volume fraction to avoid multiple light scattering as well as lateral chain-chain aggregation. DLS measures the dynamic structure factor S(q,t) of the sample (q is the scattering wave vector, t is the time). In the absence of the magnetic field, identical particles of ferrofluid droplets are randomly distributed and S(q,t) reduces to exp(-q(exp 2)2D(sub 0)t). D(sub 0)=(kT/(6(pi)(eta)(a)) is the diffusion coefficient of Brownian particles (where Xi = (6(pi)(eta)(a)) is the Stokes frictional coefficient of a spherical particle in a fluid of viscosity eta). If interactions or polydispersity can not be ignored, an effective diffusion coefficient is introduced. Formally, D(sub eff) is defined as: D(sub eff) = - q(exp -2) partial derivative of (ln(S(q,t)) with respect to time, as t goes to 0. D(sub eff) reduces to D(sub 0) if no interactions and only a few particles size are present. Therefore, we can use DLS to measure particle size. The particle radius was found to be a=0.23 mu m with 7% of polydispersity. In this case, if we vary the scattering angle theta (and so q) we do not have any change in the measured diffusion coefficient: it is q-independent. When a magnetic field is applied, particles aggregate into chains if lambda > 1. We first studied the kinetics of the chain formation when lambda = 406. At a fixed scattering angle, we measured diffusion coefficient D(sub eff) as a function of time. Experimentally, we find that D(sub eff) decreases monotonously with time. Physically, this means that chains are becoming longer and longer. Since we are only sensitive to motions in the scattering plane and since chains have their main axis perpendicular to this plane, the measured diffusion coefficient is the trans-verse diffusion coefficient. We can relate D(sub eff) to the mean number of particles per chain N(t) at a given time and to the diffusion coefficient of an isolated particle D(sub 0) as D(sub eff)=f(N(t))D(sub 0). Since f(N) is known from other recent work, N can be expressed as a function of the time. We found a square root dependency: N(t) proportional to the square root of t. As expected for very low volume fraction, this behavior is characteristic of a diffusion-limited aggregation as suggested by several authors and by our previous work. In this study, we focus on the dependence of the effective diffusion coefficient on the scattering angle and the magnetic field strength. After the magnetic field is applied (lambda = 406) for a long time, typically 6 hours, kinetics of chain formation becomes very slow. Chain size does not vary much over the next hour period. Thus, we can perform different interesting experiments. First, at a fixed magnetic field, we measure the effective diffusion coefficient as a function of the scattering angle (from 5 to 130 deg). Our results show that the measured diffusion coefficient increases linearly with the scattering angle: D(sub eff) proportional to q. If we do the same experiment for different lambda values, D(sub eff) depends on lambda as D(sub eff) proportional to lambda(exp -1/2). We also find for different lambda values that the same asymptotic D(sub eff) value is obtained when q approaches zero. The angle dependency of D(sub eff) suggests that an additional motion exists besides chain drifting. Chain size is constant during experiment, which was verified by measuring the same diffusion coefficient at the beginning and at the end of the angle switching. If chains are rigid, D(sub eff) is independent of q. Therefore, we found that D(sub eff) not only measures the motion of the entire chain but also its internal fluctuations. These internal motions are the fluctuations of the particles in the chain. To understand the q dependency of D(sub eff), let us look at the probing length used. In our study, the characteristic length scale probed is l=2pi/q which is in the range of 0.9<l/a<20. When l is much larger than the particle radius (l/a going to infinity) we are mainly sensitive to the center of mass diffusion of the chain. D(sub eff) is then the diffusion of the entire chain and depends only on N but not on q and lambda. The value D(sub eff) allows us to obtain the number of particles per chain N. In the opposite limit (i.e. l/a < 1), we are sensitive to motions on the size of individual particles. The main contributions to the measured diffusion coefficient come from internal motions of the chains (i.e. particles' fluctuations inside the chain). We investigated also the effect of the chain size on D(sub eff). For the same value of the magnetic interaction lambda and different chain sizes, we found that the slope of D(sub eff) versus q decreases when N increases as a power law D(sub eff) proportional to N(exp -0.7). This is an important behavior because this means that the longer the chain the slower its internal fluctuations besides the whole chain motion.

Cutillas, S.↗
DOE OSTI2022

On the effect of scalar flux weighing of linearly anisotropic scattering matrices in few-group transport calculations

The majority of the codes available for homogenized group constant generation for deterministic transport calculations apply the approximation of scalar flux weighting during energy group condensation of higher-order anisotropic scattering matrices. In this paper, we discuss the effect of scalar flux weighting of the linearly anisotropic scattering matrices in the frame of S P{sub 3} and S{sub 12} calculations performed for a two-dimensional VVER-440 reactor benchmark. To compare group constants generated for two neutron energy groups, an infinite pin cell was homogenized with Serpent 2 and ERANOS ECCO. Serpent 2 applies scalar flux, while ERANOS ECCO performs current weighting of the linearly anisotropic scattering matrices during energy group condensation. For analyzing the effect of the various weighting options, three simple reactor models were built assuming different core sizes using standard rectangular assemblies with 15*15 fuel pins. Diffusion, SP{sub 3} and S{sub 12} calculations were performed for the 3 models using group constants generated with Serpent 2 and ERANOS ECCO. The effect of scalar flux weighting of linearly anisotropic scattering matrices is shown by comparing the decrease in reactivity due to the decreased reactor size, as well as the assembly power distribution to reference results obtained with Serpent 2 Monte Carlo calculations. Neglecting higher than linearly anisotropic scattering and indirect application of diffusion coefficients in higher-order transport calculations is advised if angular flux-moment spectra weighted higher-order scattering matrices cannot be generated. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2026

Structure and Dynamics of Water Confined in Transition Metal Carbide MXenes: Implications for Electrochemical Applications

Two-dimensional transition metal carbides and nitrides (MXenes) are an important family of electrochemically active 2D materials. MXenes combine high conductivity with hydrophilicity, making them attractive materials for many applications, including electrochemical energy storage, sensing, desalination, and others. In order to better understand the role of structure on MXene properties, here, we investigated the vibrational properties and diffusion of water in MXenes with differing layer thicknesses and transition metal compositions using inelastic, quasi-elastic, and small-angle neutron scattering. We found that all of the Mo-containing MXenes studied here exhibited comparable vibrational dynamics and diffusion coefficients to each other and to previously studied Ti 3 C 2 T x . However, Ti 2 CT x was distinguished by its faster diffusion and more hydroxyl groups compared to the other MXenes studied. These results can help guide the selection of appropriate MXenes for energy storage and electrochemical water purification applications.

MXenes↗
DOE OSTI2011

Preparation of Pt deposited nanotubular TiO{sub 2} as cathodes for enhanced photoelectrochemical hydrogen production using seawater electrolytes

The purpose of this study was to develop effective cathodes to increase the production of hydrogen and use the seawater, an abundant resource in the earth as the electrolyte in photoelectrochemical systems. In order to fabricate the Pt/TiO{sub 2} cathodes, various contents of the Pt precursor (0-0.4 wt%) deposited by the electrodeposition method were used. On the basis of the hydrogen evolution rate, 0.2 wt% Pt/TiO{sub 2} was observed to exhibit the best performance among the various Pt/TiO{sub 2} cathodes with the natural seawater and two concentrated seawater electrolytes obtained from single (nanofiltration) and combined membrane (nanofiltration and reverse osmosis) processes. The surface characterizations exhibited that crystal structures and morphological properties of Pt and TiO{sub 2} found the results of XRD pattern and SEM/TEM images, respectively. - Graphical abstract: On the basis of photoelectrochemical hydrogen production, 0.2 wt% Pt/TiO{sub 2} was observed to exhibit the best performance among the various Pt/TIO{sub 2} cathodes with natural seawater. In comparison of hydrogen evolution rate with various seawater electrolytes, 0.2 wt% Pt/TiO{sub 2} was found to show the better performance as cathode with the concentrated seawater electrolytes obtained from membrane. Highlights: > Pt deposited TiO{sub 2} electrodes are used as cathode in PEC H{sub 2} production. > Natural and concentrated seawater by membranes are used as electrolytes in PEC. > Pt/TiO{sub 2} shows a good performance as cathode with seawater electrolytes. > H{sub 2} evolution rate increases with more concentrated seawater electrolyte. > Highly saline seawater is useful resource for H{sub 2} production.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1995

Interactions in Undersaturated and Supersaturated Lysozyme Solutions: Static and Dynamic Light Scattering Results

We have performed multiangle static and dynamic light scattering studies of lysozyme solutions at pH=4.7. The Rayleigh ratio R(sub g) and the collective diffusion coefficient D(sub c) were determined as function of both protein concentration c(sub p) and salt concentration c(sub s) with two different salts. At low salt concentrations, the scattering ratio K(sub c)(sub p)/R(sub theta) and diffusivity increased with protein concentration above the values for a monomeric, ideal solution. With increasing salt concentration this trend was eventually reversed. The hydrodynamic interactions of lysozyme in solution, extracted from the combination of static and dynamic scattering data, decreased significantly with increasing salt concentration. These observations reflect changes in protein interactions, in response to increased salt screening, from net repulsion to net attraction. Both salts had the same qualitative effect, but the quantitative behavior did not scale with the ionic strength of the solution. This indicates the presence of salt specific effects. At low protein concentrations, the slopes of K(sub c)(sub p)/R(sub theta) and D(sub c) vs c(sub p) were obtained. The dependence of the slopes on ionic strength was modeled using a DLVO potential for colloidal interactions of two spheres, with the net protein charge Z(sub e) and Hamaker constant A(sub H) as fitting parameters. The model reproduces the observed variations with ionic strength quite well. Independent fits to the static and dynamic data, however, led to different values of the fitting parameters. These and other shortcomings suggest that colloidal interaction models alone are insufficient to explain protein interactions in solutions.

Muschol, Martin↗
DOE OSTI2016

Light scattering management of dye-sensitized solar cells based on double-layered photoanodes aided by uniform TiO{sub 2} aggregates

Highlights: • A new architecture of double-layered TiO{sub 2} electrodes is presented. • The electrode contains two alternate layers of TiO{sub 2} nanoparticles and aggregates. • The aggregates layers are deposited onto the nanocrystalline layer. • The new design showed improved efficiency compared to conventional cells. - Abstract: This study presents a new double-layered TiO{sub 2} film containing a nanocrystalline under-layer and a uniform, sponge-like light scattering over-layer for dye-sensitized solar cells (DSCs) application. The over-layer is composed of 2-μm-diameter uniform aggregates, containing small nanoparticles with the average grain size of 20 nm. X-ray diffraction reveals that the light scattering layer has a mixture of anatase and rutile phases, whereas the nanocrystalline layer has a pure anatase phase. Ultraviolet–visible (UV–vis) spectra show that the light scattering layer has lower band gap energy than the nanocrystalline under-layer, extending the absorption of TiO{sub 2} into visible region. Diffuse reflectance spectroscopy demonstrates that the double-layered electrode enjoyed better light scattering ability. The double-layered DSC shows the highest power conversion efficiency of 7.69% and incident photon-to-current efficiency of 88% as a result of higher light harvesting and less recombination which is demonstrated by electrochemical impedance spectroscopy.

36 MATERIALS SCIENCE↗
DOE OSTI2019

The role of mid-gap phonon modes in thermal transport of transition metal dichalcogenides

We present a comprehensive theoretical study on thermal transport in monolayer transition metal dichalcogenides MX 2 (M: Mo, W; X: S, Se) with various sample sizes. An unusually high anharmonic scattering strength is found in MoSe 2 compared to the other three family members, which arises from its unique phonon band dispersion, specifically the mid-frequency phonon branches associated with the vibrations of Se atoms of MoSe 2 . The mid-frequency modes almost completely span the gap that exists between the high-frequency phonon branches and the acoustic ones, allowing the former to readily decay into the latter. The resultant high anharmonic scattering gives rise to a short mean free path which makes the room temperature in-plane thermal conductivity in MoSe 2 even lower than WSe 2 when the sample length is larger than 51.5 nm. With varying sample sizes, the ordering of thermal conductivity among the four materials changes as phonon transport transits from the ballistic to diffusive regime, driven by the competition between the phonon frequency spectrum range and the scattering strength. In conclusion, our work provides a microscopic picture of phonon transport in TMDs and guidance to tailor their thermal conductivities for electronic and thermoelectric applications.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS1974

Scattering of light in a spherical multilayer atmosphere

The problem of radiation diffusion in the atmosphere of a planet illuminated by solar rays was examined. The basic equations of the problem were obtained and the solution was found for the case when the absorption coefficient in the atmosphere is constant. The assumption was introduced of exponential variation of absorptivity with altitude, and a first approximation solution was proposed.

Titarchuk, L. G.↗
NASA NTRS1993

Diffusion coefficients of energetic water group ions near Comet Giacobini-Zinner

Data from the ultralow-energy charge analyzer and energetic particle anisotropy spectrometer sensors, acquired when the ICE spacecraft flew past Comet Giacobini-Zinner on September 11, 1985, are combined, and a single, self-consistent analysis technique is applied to derive a single-particle spectrum from about 200 to 1600 km/s. This information, together with the deduced bulk flow speed of the ions, is used to calculate a parallel diffusion coefficient in the transition region downstream of the bow wave (2.3 +/- 0.5) x 10 exp 17 sq cm/s; the corresponding scattering mean free path is (6 +/- 1) x 10 exp 4 km. The parallel diffusion coefficient is found to depend on the collision frequency of water group ions with Alfven waves, which are assumed to be propagating parallel (antiparallel) to the magnetic field.

Tan, L. C.↗
NASA NTRS1977

The velocity correlation function in cosmic-ray diffusion theory

It is shown that Earl's (1973) eigenvalue sum for the cosmic-ray spatial diffusion coefficient parallel to the mean magnetic field is precisely equivalent to the time integral of the particle-velocity correlation function parallel to the mean field. A derivation due to Kubo (1957) is applied to cosmic-ray pitch-angle scattering, and it is proven that all nine components of the cosmic-ray diffusion tensor can be expressed as integrals over the velocity correlation function. A pitch-angle correlation function is derived, and the effect of long-wavelength turbulence on the velocity correlation function and spatial diffusion coefficients is examined. Application of the velocity-correlation method to a realistic case involving both pitch-angle scattering and appreciable fluctuation in the direction of the local field indicates that long-wavelength turbulence in the local field reduces the parallel diffusion coefficient and places an upper limit on the ratio of the perpendicular to parallel diffusion coefficients.

Forman, M. A.↗