arXiv subjects

A. Singer

Publications and source records attributed to A. Singer.

31 records · Page 2Linked to original sources

arXiv2012

Dynamics of colloidal crystals studied by pump-probe experiments at FLASH

We present a time-resolved infrared (IR) pump and extreme-ultraviolet (XUV) probe diffraction experiment to investigate ultrafast structural dynamics in colloidal crystals with picosecond resolution. The experiment was performed at the FLASH facility at DESY with a fundamental wavelength of 8 nm. In our experiment, the temporal changes of Bragg peaks were analyzed and their frequency components were calculated using Fourier analysis. Periodic modulations in the colloidal crystal were localized at a frequency of about 4-5 GHz. Based on the Lamb theory, theoretical calculations of vibrations of the isotropic elastic polystyrene spheres of 400 nm in size reveal a 5.07 GHz eigenfrequency of the ground (breathing) mode.

cond-mat.mtrl-sci↗

arXiv2012

Spatial and temporal coherence properties of single free-electron laser pulses

The experimental characterization of the spatial and temporal coherence properties of the free-electron laser in Hamburg (FLASH) at a wavelength of 8.0 nm is presented. Double pinhole diffraction patterns of single femtosecond pulses focused to a size of about 10 microns by 10 microns were measured. A transverse coherence length of 6.2 microns in the horizontal and 8.7 microns in the vertical direction was determined from the most coherent pulses. Using a split and delay unit the coherence time of the pulses produced in the same operation conditions of FLASH was measured to be 1.75 fs. From our experiment we estimated the degeneracy parameter of the FLASH beam to be on the order of $10^{10}$ to $10^{11}$, which exceeds the values of this parameter at any other source in the same energy range by many orders of magnitude.

physics.optics↗

arXiv2011

Three-dimensional structure of a single colloidal crystal grain studied by coherent x-ray diffraction

A coherent x-ray diffraction experiment was performed on an isolated colloidal crystal grain at the coherence beamline P10 at PETRA III. Using azimuthal rotation scans the three-dimensional (3D) scattered intensity in reciprocal space from the sample was measured. It includes several Bragg peaks as well as the coherent interference around these peaks. The analysis of the scattered intensity reveals the presence of a plane defect in a single grain of the colloidal sample. We confirm these findings by model simulations. In these simulations we also analyze the experimental conditions to phase 3D diffraction pattern from a single colloidal grain. This approach has the potential to produce a high resolution image of the sample revealing its inner structure, with possible structural defects.

cond-mat.mtrl-sci↗

arXiv2011

Coherence Properties of Individual Femtosecond Pulses of an X-ray Free-Electron Laser

Measurements of the spatial and temporal coherence of single, femtosecond x-ray pulses generated by the first hard x-ray free-electron laser (FEL), the Linac Coherent Light Source (LCLS), are presented. Single shot measurements were performed at 780 eV x-ray photon energy using apertures containing double pinholes in "diffract and destroy" mode. We determined a coherence length of 17 micrometers in the vertical direction, which is approximately the size of the focused LCLS beam in the same direction. The analysis of the diffraction patterns produced by the pinholes with the largest separation yields an estimate of the temporal coherence time of 0.6 fs. We find that the total degree of transverse coherence is 56% and that the x-ray pulses are adequately described by two transverse coherent modes in each direction. This leads us to the conclusion that 78% of the total power is contained in the dominant mode.

physics.optics↗

arXiv2009

Analysis of Coherence Properties of 3-rd Generation Synchrotron Sources and Free-Electron Lasers

A general theoretical approach based on the results of statistical optics is used for the analysis of the transverse coherence properties of 3-rd generation synchrotron sources and x-ray free-electron lasers (XFEL). Correlation properties of the wavefields are calculated at different distances from an equivalent Gaussian Schell-model source. This model is used to describe coherence properties of the five meter undulator source at the synchrotron storage ring PETRA III. In the case of XFEL sources the decomposition of the statistical fields into a sum of independently propagating transverse modes is used for the analysis of the coherence properties of these new sources. A detailed calculation is performed for the parameters of the SASE1 undulator at the European XFEL. It is demonstrated that only a few modes contribute significantly to the total radiation field of that source.

physics.optics↗

arXiv2008

Narrow escape and leakage of Brownian particles

Questions of flux regulation in biological cells raise renewed interest in the narrow escape problem. The often inadequate expansions of the narrow escape time are due to a not so well known fact that the boundary singularity of Green's function for Poisson's equation with Neumann and mixed Dirichlet-Neumann boundary conditions in three-dimensions contains a logarithmic singularity. Using this fact, we find the second term in the expansion of the narrow escape time and in the expansion of the principal eigenvalue of the Laplace equation with mixed Dirichlet-Neumann boundary conditions, with small Dirichlet and large Neumann parts. We also find the leakage flux of Brownian particles that diffuse from a source to an absorbing target on a reflecting boundary of a domain, if a small perforation is made in the reflecting boundary.

math-ph↗

arXiv2007

Partially Reflected Diffusion

The radiation (reaction, Robin) boundary condition for the continuum diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary, however, the relation between the reaction (radiation) constant in the Robin boundary condition and the reflection probability is still unclear. In this paper we clarify the issue by finding the relation between the reaction (radiation) constant and the absorption probability of the diffusing trajectories at the boundary. We analyze the Euler scheme for the underlying Itô dynamics, which is assumed to have variable drift and diffusion tensor, with partial reflection at the boundary. Trajectories that cross the boundary are terminated with a given probability and otherwise are reflected in a normal or oblique direction. We use boundary layer analysis of the corresponding Wiener path integral to resolve the non-uniform convergence of the probability density function of the numerical scheme to the solution of the Fokker-Planck equation with the Robin boundary condition, as the time step is decreased. We show that the Robin boundary condition is recovered in the limit iff trajectories are reflected in the co-normal direction. We find the relation of the reactive constant to the termination probability. We show the effect of using the new relation in numerical simulations.

math-ph↗

arXiv2005

Attenuation of the Electric Potential and Field in Disordered Systems

We study the electric potential and field produced by disordered distributions of charge to see why clumps of charge do not produce large potentials or fields. The question is answered by evaluating the probability distribution of the electric potential and field in a totally disordered system that is overall electroneutral. An infinite system of point charges is called totally disordered if the locations of the points and the values of the charges are random. It is called electroneutral if the mean charge is zero. In one dimension, we show that the electric field is always small, of the order of the field of a single charge, and the spatial variations in potential are what can be produced by a single charge. In two and three dimensions, the electric field in similarly disordered electroneutral systems is usually small, with small variations. Interestingly, in two and three dimensional systems, the electric potential is usually very large, even though the electric field is not: large amounts of energy are needed to put together a typical disordered configuration of charges in two and three dimensions, but not in one dimension. If the system is locally electroneutral--as well as globally electroneutral--the potential is usually small in all dimensions. The properties considered here arise from the superposition of electric fields of quasi-static distributions of charge, as in nonmetallic solids or ionic solutions. These properties are found in distributions of charge far from equilibrium.

cond-mat.stat-mech↗

arXiv2005

Brownian Simulations and Uni-Directional Flux in Diffusion

Brownian dynamics simulations require the connection of a small discrete simulation volume to large baths that are maintained at fixed concentrations and voltages. The continuum baths are connected to the simulation through interfaces, located in the baths sufficiently far from the channel. Average boundary concentrations have to be maintained at their values in the baths by injecting and removing particles at the interfaces. The particles injected into the simulation volume represent a unidirectional diffusion flux, while the outgoing particles represent the unidirectional flux in the opposite direction. The classical diffusion equation defines net diffusion flux, but not unidirectional fluxes. The stochastic formulation of classical diffusion in terms of the Wiener process leads to a Wiener path integral, which can split the net flux into unidirectional fluxes. These unidirectional fluxes are infinite, though the net flux is finite and agrees with classical theory. We find that the infinite unidirectional flux is an artifact caused by replacing the Langevin dynamics with its Smoluchowski approximation, which is classical diffusion. The Smoluchowski approximation fails on time scales shorter than the relaxation time $1/γ$ of the Langevin equation. We find the unidirectional flux (source strength) needed to maintain average boundary concentrations in a manner consistent with the physics of Brownian particles. This unidirectional flux is proportional to the concentration and inversely proportional to $\sqrt{Δt}$ to leading order. We develop a BD simulation that maintains fixed average boundary concentrations in a manner consistent with the actual physics of the interface and without creating spurious boundary layers.

math-ph↗

arXiv2005

Langevin Trajectories between Fixed Concentrations

We consider the trajectories of particles diffusing between two infinite baths of fixed concentrations connected by a channel, e.g. a protein channel of a biological membrane. The steady state influx and efflux of Langevin trajectories at the boundaries of a finite volume containing the channel and parts of the two baths is replicated by termination of outgoing trajectories and injection according to a residual phase space density. We present a simulation scheme that maintains averaged fixed concentrations without creating spurious boundary layers, consistent with the assumed physics.

math-ph↗

arXiv2004

Narrow Escape, Part I

A Brownian particle with diffusion coefficient $D$ is confined to a bounded domain of volume $V$ in $\rR^3$ by a reflecting boundary, except for a small absorbing window. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. We construct an asymptotic approximation for the case of an elliptical window of large semi axis $a\ll V^{1/3}$ and show that the mean escape time is $Eτ\sim\ds{\frac{V}{2πDa}} K(e)$, where $e$ is the eccentricity of the ellipse; and $K(\cdot)$ is the complete elliptic integral of the first kind. In the special case of a circular hole the result reduces to Lord Rayleigh's formula $Eτ\sim\ds{\frac{V}{4aD}}$, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion $Eτ=\ds{\frac{V}{4aD}} [1+\frac{a}{R} \log \frac{R}{a} + O(\frac{a}{R}) ]$. This problem is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function.

math-ph↗

arXiv2004

Narrow Escape, Part II: The circular disk

We consider Brownian motion in a circular disk $Ω$, whose boundary $\pΩ$ is reflecting, except for a small arc, $\pΩ_a$, which is absorbing. As $ε=|\partial Ω_a|/|\partial Ω|$ decreases to zero the mean time to absorption in $\pΩ_a$, denoted $Eτ$, becomes infinite. The narrow escape problem is to find an asymptotic expansion of $Eτ$ for $ε\ll1$. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously derived expansion for a general smooth domain, $Eτ= \ds{\frac{|Ω|}{Dπ}}[\log\ds{\frac{1}ε}+O(1)],$ ($D$ is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is $E[τ| \x(0)=\mb{0}]=\ds{\frac{R^2}{D}}[\log\ds{\frac{1}ε} + \log 2 +\ds{1/4} + O(ε)]$. The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because $\log\ds{\frac{1}ε}$ is not necessarily large, even when $ε$ is small. We also find the singular behavior of the probability flux profile into $\pΩ_a$ at the endpoints of $\pΩ_a$, and find the value of the flux near the center of the window.

math-ph↗

arXiv2004

Narrow Escape, Part III: Riemann surfaces and non-smooth domains

We consider Brownian motion in a bounded domain $Ω$ on a two-dimensional Riemannian manifold $(Σ,g)$. We assume that the boundary $\pΩ$ is smooth and reflects the trajectories, except for a small absorbing arc $\pΩ_a\subset\pΩ$. As $\pΩ_a$ is shrunk to zero the expected time to absorption in $\pΩ_a$ becomes infinite. The narrow escape problem consists in constructing an asymptotic expansion of the expected lifetime, denoted $Eτ$, as $ε=|\partial Ω_a|_g/|\partial Ω|_g\to0$. We derive a leading order asymptotic approximation $Eτ= \ds{\frac{|Ω|_g}{Dπ}}[\log\ds{\frac{1}ε}+O(1)]$. The order 1 term can be evaluated for simply connected domains on a sphere by projecting stereographically on the complex plane and mapping conformally on a circular disk. It can also be evaluated for domains that can be mapped conformally onto an annulus. This term is needed in real life applications, such as trafficking of receptors on neuronal spines, because $\log\ds{\frac{1}ε}$ is not necessarily large, even when $ε$ is small. If the absorbing window is located at a corner of angle $α$, then $Eτ= \ds{\frac{|Ω|_g}{Dα}}[\log\ds{\frac{1}ε}+O(1)],$ if near a cusp, then $Eτ$ grows algebraically, rather than logarithmically. Thus, in the domain bounded between two tangent circles, the expected lifetime is $Eτ= \ds{\frac{|Ω|}{(d^{-1}-1)D}}(\frac{1}ε + O(1))$.

math-ph↗