Reciprocity Theorem Research Articles

Measurement of the cross section for the C13(α,n)16O reaction and determination of the cross section for the O16(n,α)13C reaction

The angular dependence of the differential cross sections for the $^{13}\mathrm{C}{(\ensuremath{\alpha},{n}_{0})}^{16}\mathrm{O}$ reaction was measured in the energy range 2.0--6.2 MeV using the time-of-flight method for separating neutrons corresponding to the ground state of the residual nucleus. The integrated total cross sections were derived from the measured data and the cross sections for the $^{16}\mathrm{O}{(n,{\ensuremath{\alpha}}_{0})}^{13}\mathrm{C}$ reaction were determined using the reciprocity theorem. The cross sections obtained for the reaction $^{16}\mathrm{O}{(n,{\ensuremath{\alpha}}_{0})}^{13}\mathrm{C}$ support the evaluation given in the ENDF/B-VIII.0 library.

An Approximate Solution for the Contact Problem of Profiles Slightly Deviating from Axial Symmetry

An approximate solution for a contact problem of profiles which are not axially symmetrical but deviate only slightly from the axial symmetry is found in a closed explicit analytical form. The solution is based on Betti’s reciprocity theorem, first applied to contact problems by R.T. Shield in 1967, in connection with the extremal principle for the contact force found by J.R. Barber in 1974 and Fabrikant’s approximation (1986) for the pressure distribution under a flat punch with arbitrary cross-section. The general solution is validated by comparison with the Hertzian solution for the contact of ellipsoids with small eccentricity and with numerical solutions for conical shapes with polygonal cross-sections. The solution provides the dependencies of the force on the indentation, the size and the shape of the contact area as well as the pressure distribution in the contact area. The approach is illustrated by linear (conical) and quadratic profiles with arbitrary cross-sections as well as for “separable” shapes, which can be represented as a product of a power-law function of the radius with an arbitrary exponent and an arbitrary function of the polar angle. A generalization of the Method of Dimensionality Reduction to non-axisymmetric profiles is formulated.

Pressure-driven flow of the viscoelastic Oldroyd-B fluid in narrow non-uniform geometries: analytical results and comparison with simulations

We analyse the pressure-driven flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, narrow channels and present a theoretical framework for calculating the relationship between the flow rate $q$ and pressure drop $\Delta p$ . We first identify the characteristic scales and dimensionless parameters governing the flow in the lubrication limit. Employing a perturbation expansion in powers of the Deborah number ( $De$ ), we provide analytical expressions for the velocity, stress and the $q$ – $\Delta p$ relation in the weakly viscoelastic limit up to $O(De^2)$ . Furthermore, we exploit the reciprocal theorem derived by Boyko $\&$ Stone (Phys. Rev. Fluids, vol. 6, 2021, L081301) to obtain the $q$ – $\Delta p$ relation at the next order, $O(De^3)$ , using only the velocity and stress fields at the previous orders. We validate our analytical results with two-dimensional numerical simulations in the case of a hyperbolic, symmetric contracting channel and find excellent agreement. While the velocity remains approximately Newtonian in the weakly viscoelastic limit (i.e. the theorem of Tanner and Pipkin), we reveal that the pressure drop strongly depends on the viscoelastic effects and decreases with $De$ . We elucidate the relative importance of different terms in the momentum equation contributing to the pressure drop along the symmetry line and identify that a pressure drop reduction for narrow contracting geometries is primarily due to gradients in the viscoelastic shear stresses. We further show that, although for narrow geometries the viscoelastic axial stresses are negligible along the symmetry line, they are comparable or larger than shear stresses in the rest of the domain.

Minimum-dose phase-contrast tomography by successive numerical optical sectioning employing the aberration-corrected STEM and a pixelated detector

Aberration correction combined with a pixelated detector enable atomic-resolution phase-contrast imaging in the scanning transmission electron microscope (STEM) using all elastically scattered electrons within the illumination cone. The review describes this possibility in detail revisiting the image formation in the STEM on a fundamental quantum-mechanical treatment of electron scattering within the object and the effect of the lenses on the electron wave. Describing electron scattering by means of scattering amplitudes enables a straightforward derivation of a) the reciprocity theorem, b) the optical theorem of electron scattering, and c) the precise formulation of the image intensity distribution in the STEM for different modes of operation. The second part of the review describes in detail a novel method for obtaining pure phase-contrast images in the STEM using the integrated differential phase-contrast (IDPC) procedure. The incorporation of a chromatic (Cc) and spherically (Cs) corrected objective lens and a pixelated detector in the STEM combined with numerical through-focusing enables optical sectioning with atomic 3D resolution of thick objects with about the same dose as that for a 2D object, at least in principle. Numerical simulations of the IDPC transfer function and the point spread function for the focal plane and several reconstructed defocused planes demonstrate the feasibility of the method.

Time-nonlocal six-phase-lag generalized theory of thermoelastic diffusion

Lagging behavior is also expected in the process of mass diffusion similar to the process of heat transport during the ultra fast transient (D. Y. Tzou, Macro to microscale heat transfer: The lagging behavior. John Wiley and Sons, Ltd., 2015). We therefore propose a novel multi-field coupled mathematical model for a linear, isotropic and homogeneous thermoelastic diffusive continuum by taking into account the modified Fourier’s law of heat conduction, the modified Fick’s law of mass diffusion and extending the concept of phase lag from heat conduction to mass diffusion. The modified Fourier’s law includes temperature gradient and thermal displacement gradient among the constitutive variables with three different phase lags for the heat flux vector, the temperature gradient and the thermal displacement gradient. The modified Fick’s law includes chemical potential gradient and the chemical potential displacement gradient among the constitutive variables with three different phase lags for the mass flux vector, the chemical potential gradient and the chemical potential displacement gradient. The Fourier’s law of heat conduction is replaced by a fractional order approximation of the modified Fourier’s law whereas the Fick’s law of mass diffusion is replaced by a fractional order approximation of the modified Fick’s law. The proposed model includes some of the existing models in the field of thermoelastic diffusion as special cases. A variational principle is obtained and uniqueness of solution of the governing field equations under suitable conditions is proved. Finally, a reciprocity theorem is established for the proposed generalized thermoelastic diffusion model.

Identification of Vibration Modes and Wave Propagation of Operational Rails by Multipoint Hammering and Reciprocity Theorem.

Vertical bending vibration modes and rail wave propagation, including the damping characteristics, are the factors that cause rail corrugation. However, the ability to identify actual railways has been limited because of the huge number of sensors required for field tests. In this study, a novel and field-applicable method for identifying rail vibration modes and wave propagation characteristics is developed by multipoint hammering and the reciprocity theorem instead of multipoint measuring. Additionally, the proposed method is applied to an actual rail with a direct fastening track system on a bridge that has corrugation with a wavelength of approximately 0.04 m. As a result, the wavelength (wavenumber)-, group velocity-, and distance damping (attenuation) frequency relationship of the wave propagation is clarified in addition to the rail frequencies and mode shapes up to approximately 1500 Hz, including the pinned-pinned mode. Finally, the identified wavelength-frequency relationships and the measured rail irregularity can empirically demonstrate that the generated corrugation on the rail is produced by wave interference on the two axles in the bogie.

An inverse method to reconstruct crack-tip cohesive zone laws for fatigue by numerical field projection

Cohesive zone failure models are widely used to simulate fatigue crack propagation under cyclic loadings, but the functional form of such phenomenological models does not have a strong micromechanical basis. While numerical techniques have been developed to extract the crack-tip cohesive zone law for monotonic fracture from experimental strain information, elucidating the unloading-reloading hysteresis representing accumulated and irreversible damage under fatigue cycling is still challenging. Here, we introduce a novel field projection method (FPM) to reconstruct the cohesive zone laws for steady-state fatigue crack growth from elastic strain field information at various loading and unloading stages within a single steady-state fatigue cycle. The method is based on the Maxwell-Betti’s reciprocal theorem, which considers the material response to be linear elastic during loading and unloading, albeit with a reciprocity gap to account for the inhomogeneous residual elastic strain accumulated at the end of each fatigue cycle. Through numerical experiments, we demonstrate that this FPM is capable of accurately extracting the crack-tip cohesive tractions and separations, along with the elusive unloading-reloading hysteresis, which constitute the cohesive zone law for fatigue crack growth. We discuss the errors and uncertainties associated with this inverse approach.

How inertial lift affects the dynamics of a microswimmer in Poiseuille flow

The transport of motile microorganisms is strongly influenced by fluid flows that are ubiquitous in biological environments. Here we demonstrate the impact of fluid inertia. We analyze the dynamics of a microswimmer in pressure-driven Poiseuille flow, where fluid inertia is small but non-negligible. Using perturbation theory and the reciprocal theorem, we show that in addition to the classical inertial lift of passive particles, the active nature generates a ‘swimming lift’, which we evaluate for neutral and pusher/puller-type swimmers. Accounting for fluid inertia engenders a rich spectrum of complex dynamics including bistable states, where tumbling coexists with stable centerline swimming or swinging. The dynamics is sensitive to the swimmer’s hydrodynamic signature and goes well beyond the findings at vanishing fluid inertia. Our work will have non-trivial implications on the transport and dispersion of active suspensions in microchannels.

Huygens’ principle derived by using momentum reciprocity theorem of electromagnetic field

The classical reciprocity theorem of electromagnetic field proposed by Lorentz H.A. in 1896 is one of the important theories of electromagnetics. The Lorentz reciprocity theorem is widely used in many fields such as communication, antenna signal transmission, and electromagnetic imaging. The Lorentz reciprocity theorem is an “energy-based” reciprocity theorem. Over the past hundred years, some new reciprocity theorems of electromagnetic field have been discovered, including reciprocity theorems in frequency domain and time domain. In 2020, Lindell et al. extended the concept of the 'Rumsey reaction' in a differential form to include both the Lorentz force density reaction term and the power density reaction term. In the same year, Liu et al. derived the momentum reciprocity theorem from the Maxwell's equations. The momentum reciprocity theorem, like the Lorentz reciprocity theorem, can be used for both theoretical analysis and practical applications. Huygens’ principle is usually derived by the Lorentz reciprocity theorem. In this paper, the momentum reciprocity theorem is employed to derive the Huygens’ principle.

Computation of nonreciprocal dynamics in nonlinear materials

<p style='text-indent:20px;'>The reciprocity theorem in elastic materials states that the response of a linear, time-invariant system to an external load remains invariant with respect to interchanging the locations of the input and output. In the presence of nonlinear forces within a material, circumventing the reciprocity invariance requires breaking the mirror symmetry of the medium, thus allowing different wave propagation characteristics in opposite directions along the same transmission path. This work highlights the application of numerical continuation methods for exploring the steady-state nonreciprocal dynamics of nonlinear periodic materials in response to external harmonic drive. Using the archetypal example of coupled oscillators, we apply continuation methods to analyze the influence of nonlinearity and symmetry on the reciprocity invariance. We present symmetry-breaking bifurcations for systems with and without mirror symmetry, and discuss their influence on the nonreciprocal dynamics. Direct computation of the reciprocity bias allows the identification of response regimes in which nonreciprocity manifests itself as a phase shift in the output displacements. Various operating regimes, bifurcations and manifestations of nonreciprocity are identified and discussed throughout the work.</p>

A General Form of the Electromagnetic Reciprocity Theorem

A General Form of the Electromagnetic Reciprocity Theorem

On the Far Field Pattern for Acoustic Scattering by a Piecewise Homogeneous Obstacle in Two Dimensions

A time-harmonic plane acoustic wave is scattered by a piecewise homogeneous obstacle with a penetrable or impenetrable core. We construct in the close form an integral representation for the far field pattern in which we have incorporated the physical and geometrical characteristics of the scatterer. Through this representation, we obtain the far field pattern for this scatterer. We prove scattering relations between the far field patterns of two scattering problems due to two distinct incident waves on the same scatterer. In particular, we prove reciprocity and general scattering theorems. The optical theorem, connecting the total power that the scatterer extracts from the incident plane wave either by radiation or by absorption with the corresponding far field pattern of an incident plane wave, is recovered as a corollary of the general scattering theorem. Moreover, if we consider incident waves to be both a plane and a spherical, we derive a mixed reciprocity theorem. We define the corresponding far field operators and using these relations, we prove some properties that can be used for solving inverse scattering problems.

The rotation of a sedimenting spheroidal particle in a linearly stratified fluid

We derive analytically the angular velocity of a spheroid, of an arbitrary aspect ratio$\kappa$, sedimenting in a linearly stratified fluid. The analysis demarcates regions in parameter space corresponding to broadside-on and edgewise settling in the limit$Re, Ri_v \ll 1$, where$Re = \rho _0UL/\mu$and$Ri_v =\gamma L^3\,g/\mu U$, the Reynolds and viscous Richardson numbers, respectively, are dimensionless measures of the importance of inertial and buoyancy forces relative to viscous ones. Here,$L$is the spheroid semi-major axis,$U$an appropriate settling velocity scale,$\mu$the fluid viscosity and$\gamma \ (>0)$the (constant) density gradient characterizing the stably stratified ambient, with the fluid density$\rho_0$taken to be a constant within the Boussinesq framework. A reciprocal theorem formulation identifies three contributions to the angular velocity: (1) an$O(Re)$inertial contribution that already exists in a homogeneous ambient, and orients the spheroid broadside-on; (2) an$O(Ri_v)$hydrostatic contribution due to the ambient stratification that also orients the spheroid broadside-on; and (3) a hydrodynamic contribution arising from the perturbation of the ambient stratification whose nature depends on$Pe$;$Pe = UL/D$being the Péclet number with$D$the diffusivity of the stratifying agent. For$Pe \ll 1$, this contribution is$O(Ri_v)$and orients prolate spheroids edgewise for all$\kappa \ (>1)$. For oblate spheroids, it changes sign across a critical aspect ratio$\kappa _c \approx 0.41$, orienting oblate spheroids with$\kappa _c < \kappa < 1$edgewise and those with$\kappa < \kappa _c$broadside-on. For$Pe \ll 1$, the hydrodynamic component is always smaller in magnitude than the hydrostatic one, so a sedimenting spheroid in this limit always orients broadside-on. For$Pe \gg 1$, the hydrodynamic contribution is dominant, being$O(Ri_v^{{2}/{3}}$) in the Stokes stratification regime characterized by$Re \ll Ri_v^{{1}/{3}}$, and orients the spheroid edgewise regardless of$\kappa$. Consideration of the inertial and large-$Pe$stratification-induced angular velocities leads to two critical curves which separate the broadside-on and edgewise settling regimes in the$Ri_v/Re^{{3}/{2}}$–$\kappa$plane, with the region between the curves corresponding to stable intermediate equilibrium orientations. The predictions for large$Pe$are broadly consistent with observations.

Soft-lubrication interactions between a rigid sphere and an elastic wall

The motion of an object within a viscous fluid and in the vicinity of a soft surface induces a hydrodynamic stress field that deforms the latter, thus modifying the boundary conditions of the flow. This results in elastohydrodynamic interactions experienced by the particle. Here, we derive a soft-lubrication model, in order to compute all the forces and torque applied on a rigid sphere that is free to translate and rotate near an elastic wall. We focus on the limit of small deformations of the surface with respect to the fluid-gap thickness, and perform a perturbation analysis in dimensionless compliance. The response is computed in the framework of linear elasticity, for planar elastic substrates in the limiting cases of thick and thin layers. The EHD forces are also obtained analytically using the Lorentz reciprocal theorem.

Optical forces and directionality in one-dimensional PT -symmetric photonics

We discuss the optical forces exerted on parity-time $(\mathcal{PT})$ symmetric heterostructures under normal incidence of a single and two counterpropagating plane waves. The underlying strategy is through generalized parametric space, stemming from consideration of the $\mathcal{PT}$-symmetry condition and Lorentz reciprocity theorem. In such a generalized parametric space, we are able to not only exhaustively indicate various $\mathcal{PT}$ phases and extraordinary wave phenomena but also deduce the directionality and magnitudes of optical forces. We find that when the system is illuminated by a normally incident wave, it can exhibit the symmetric pushing effect in the exact symmetry phase, unidirectional null, and bidirectional null forces (BNF) at the exceptional point, and pulling-pushing flipped forces in the broken symmetry phase, with BNF found at the pushing-pulling turning point. In two counterpropagating plane wave interferences, the magnitudes as well as the directionality of the resultant optical force can be tuned by a relative phase of incident waves. More interestingly, we observe that a null force independent of the relative phase occurred in a specific region of the broken symmetry phase and exceptional point. In addition, we offer several $\mathcal{PT}$-symmetric heterostructures to support our findings. Our results may benefit applications in $\mathcal{PT}$ optomechanics and force rectifiers.

Determination of hole diffusion length in n-GaN

The paper presents the derivation of a model for minority carriers collection based on the reciprocity theorem and its application for determination of hole diffusion length in n-GaN by means of photoluminescence. The estimated hole diffusion lengths at room temperature are 110 nm and 194 nm in the case of low and high excitation, respectively, which could be explained by saturation of non-radiative recombination centers in bulk GaN and at the surface with photogenerated carriers.

On Channel Reciprocity in Reconfigurable Intelligent Surface Assisted Wireless Networks

Channel reciprocity greatly facilitates downlink precoding in time-division duplexing (TDD) multiple-input multiple-output (MIMO) communications without the need for channel state information (CSI) feedback. Recently, reconfigurable intelligent surfaces (RISs) emerge as a promising technology to enhance the performance of future wireless networks. However, since the artificial electromagnetic characteristics of RISs do not strictly follow the normal laws of nature, it brings up a question: does the channel reciprocity hold in RIS-assisted TDD wireless networks? After briefly reviewing the reciprocity theorem, in this article, we show that there still exists channel reciprocity for RIS-assisted wireless networks satisfying certain conditions. We also experimentally demonstrate the reciprocity at the sub-6 GHz and the millimeter-wave frequency bands by using two fabricated RISs. Furthermore, we introduce several RIS-assisted approaches to realizing nonreciprocal channels. Finally, potential opportunities brought by reciprocal/nonreciprocal RISs and future research directions are outlined.

Variational principles of laminated plates considering interlayer slip

A model for analyzing laminated plates is proposed, considering the interlayer slip. The proposed model is suitable for civil engineering structures, such as undertrack structures and layered pavements. The interlayer slip at the interface was introduced in the model to reflect the effect of imperfect bonding between adjacent layers. The deformation of each layer follows the Kirchhoff–Love assumption of the classical plate theory. The principle of virtual work and the reciprocal theorem of laminated plates were established to account for interlayer slip, based on which the principles of minimum potential and complementary energy were derived and proven. Two- and three-field generalized variational principles were then developed by introducing the Lagrange multiplier method. Moreover, the variational principles for the free vibration and buckling of laminated plates were obtained. Finally, analytical solutions of the bending, free vibration and buckling responses of a simply supported laminated plate were obtained, and they were consistent with the numerical results.

Pulsed Electromagnetic Scattering by Metasurfaces—A Numerical Solution Based on the Cagniard–DeHoop Method of Moments

A novel time-domain (TD) integral-equation (IE) technique for analyzing the pulsed electromagnetic (EM) scattering by a 2-D metasurface, represented by a thin highly contrasting screen with combined dielectric and magnetic properties, is presented. The problem under consideration is formulated with the aid of the EM reciprocity theorem of the time-convolution time. Subsequently, upon discretizing the space–time solution domain, the reciprocity relation is cast into two independent matrix forms whose elements are derived analytically using the classic Cagniard–DeHoop (CdH) technique. The thus obtained convolution-type systems of equations are solved for induced electric and magnetic currents through a stable step-by-step updating procedure. The presented numerical method is finally validated using alternative TD solutions.

Coupled Faxen relations for non-uniform slip Janus spheres

A non-uniform surface slip can cause a symmetry breaking in the geometry of an otherwise homogeneous spherical particle to give rise to an anisotropic hydrodynamic resistance to the particle. Here, we develop a more general theoretical framework capable of decoding the surface-pattern-dependent hydrodynamic features for single heterogeneous spheres having arbitrary non-uniform slip length distributions in small variations, especially for those of weakly stick–slip or slip–slip Janus spheres in either the two-faced or striped type. Utilizing the Lorentz reciprocal theorem in conjunction with surface spherical harmonic expansion, we derive a new coupled set of Faxen formulas for the hydrodynamic force and torque on a non-uniform slip sphere by expressing impacts of slip anisotropy in terms of surface dipole and quadrupole without solving detailed flow fields. Our results reveal not only how various additional forces/torques arise from surface dipole and quadrupole, but also that it is the anti-symmetric dipole responsible for distinctive force-rotation/torque-translation coupling. These features are very distinct from those of no-slip or uniform-slip particles, possibly spurring new means to characterize or sort Janus particles in microfluidic experiments. In addition, the coupled Faxen relations with surface moment contributions reported here may infer potential changes in the collective nature of hydrodynamic interactions between non-uniform slip spheres. Furthermore, the present framework can also be readily applied to heterogeneous self-propelled squirmers whose swimming velocities are sensitive to slip anisotropy.