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Surface acoustic wave (SAW) sensors are very promising for structural health monitoring (SHM) applications as they have the advantages of being robust, passive (batteryless), remotely interrogated (wireless) and can even be packageless. This paper describes ultralow-profile SAW resonators that can be directly fabricated and integrated on metallic parts in industrial facilities. They are based on piezoelectric thin films (ZnO) which are directly sputtered on polished industrial titanium (Ti) and stainless steel. With this approach, no sensor glue-bonding to the target is needed, and measurement errors related to this step are avoided. Demonstrator devices have been studied numerically and experimentally. The structural properties of the ZnO thin films were characterized through x-ray diffraction and atomic force microscopy. A preferred orientation (002) was achieved with a roughness of 50 nm on the top surface. Resonators were microfabricated and their functional parameters (i.e. resonance frequency, quality factor and electromechanical coupling) were extracted through impedance measurements and fitted with a Butterworth-van Dyke model. By increasing applied temperatures (up to 450 °C) and the strain (up to 1800 μϵ), a linear decrease of the resonance frequency has been shown. A temperature coefficient of frequency of −46.4 ppm °C−1 and a good strain sensitivity (1.49 ppmμϵ−1) were obtained, thus making the structure promising as a high temperature and strain sensing element in industrial SHM applications.

We study the nature of transition from stationary lasing to pulse-periodic oscillations when the phase of the delayed feedback and the diode laser pump current change. The appearance of oscillations can take place under the scenarios of soft or hard excitation of oscillations. We propose a semi-analytical approach to identify the nature of the transition and to determine the dynamic characteristics and stability of the arising spike regime with a change in the external parameters. Direct integration of the Lang - Kobayashi equations showed an acceptable accuracy of this approach. (control of laser radiation parameters)

The aim of this paper is to generate exact travelling wave solutions of the Harry—Dym equation through the methods of Adomian decomposition, He's variational iteration, direct integration, and power series. We show that the two later methods are more successful than the two former to obtain more solutions of the equation.

Using the truncated Painleve expansion, symbolic computation, and direct integration technique, we study analytic solutions of (2+1)-dimensional Boussinesq equation. An auto-Backlund transformation and a number of exact solutions of this equation have been found. The set of solutions include solitary wave solutions, solitoff solutions, and periodic solutions in terms of elliptic Jacobi functions and Weierstrass function. Some of them are novel.

Electrorheological (ER) fluids are a class of smart materials in which their rheological properties can be changed reversibly under the influence of an applied electric field. In recent years, many industrial applications of these smart fluids have been introduced by researchers, especially in the damping control of systems. In the present work, the applicability of squeeze-mode ER dampers in suppressing the vibration of a cantilever beam is investigated. The dynamic response of the beam for an impulse exciting force is obtained using a direct integration method based on a finite-element model for the structure. The nonlinear displacement and velocity-dependent characteristics of the squeeze-mode ER damper are considered in each of the iterations. Although the proposed ER damper has been found to have a significant influence on the dynamic response of the structure, adding a closed-loop control system could improve the damping behavior of the structure considerably. While the strength of the electric field depends on the gap between the electrodes, the control system uses displacement feedback for producing a controlling voltage to prevent the electric field exceeding its allowable bounds.

Recently, there have been many efforts to use the research techniques developed in the field of physics education research to improve the teaching and learning of quantum mechanics. In particular, part of this research is focusing on misconceptions held by students. For instance, a set of misconceptions is associated with the concept of stationary states. In this paper, we argue that a possible way to remove these is to solve the Schrodinger equation using the evolution operator method (EOM), and stress the fact that to find stationary states is only the first step in solving that equation. The EOM consists in solving the Schrodinger equation by direct integration, i.e. Ψ(x, t) = U(t)Ψ(x, 0), where is the time evolution operator, and Ψ(x, 0) is the initial state. We apply the evolution operator method in the case of the harmonic oscillator.

In this paper, a complete nonlinear finite element model for coupled-domain MEMS devices with electrostatic actuation and squeeze film effect is developed. For this purpose, a corotational finite element formulation for the dynamic analysis of planer Euler beams is employed. In this method, the internal nodal forces due to deformation and intrinsic residual stresses, the inertial nodal forces, and the damping effect of squeezed air film are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory using d'Alamber and virtual work principles. An incremental-iterative method based on the Newmark direct integration procedure and the Newton-Raphson algorithm is used to solve the nonlinear dynamic equilibrium equations. Numerical examples are presented and compared with experimental findings which indicate properly good agreement.

It is well known that moment equations for a multivariate linear system with multiplicative red noise are not closed: equations for low-order moments involve higher-order moments. Further, the probability density for this system is complicated enough that analytic expressions for the moments are difficult to obtain by direct integration. We introduce a closure approximation so that the vector mean of such systems may be estimated with high accuracy. The approximation is accurate for red noises with a wide range of timescales, and approaches the correct limits for both very small and very large correlation times. We include an application relevant to climate modeling and comment on the implications for numerical-model investigations of global warming.

Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigated systematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcation parameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence of breather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutions are given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials is utilized to give simpler forms for the pulse-like solitary wave solutions. In view of the references, our results are the most complete and the theoretical methods are the simplest hitherto.

We analyse the dispersive properties of one-dimensional photonic band gap structures, and discuss applications to phase matching conditions in the case of second harmonic generation. Band edge effects such as increased density of modes, large field enhancement, and low group velocity induce highly efficient parametric amplification, provided the proper phase matching conditions can be established. Direct integration of Maxwell's equations in the time domain confirms these conclusions, and shows that parametric amplification in one-dimensional photonic band gap structures provide much larger conversion efficiencies compared with ordinary quasi-phase-matching devices.

A novel micromachined composite beam structure has been designed and fabricated using the maskless anisotropic etching technology for 100 steps. The composite beam consists of two beam sections, a horizontal beam and a vertical beam, linked in series along the 100 direction. The composite beam is compliant in two orthogonal directions: the direction normal to the wafer due to the horizontal beam and the direction parallel with the wafer surface and perpendicular to the beam direction due to the vertical beam. The geometries of the two beam sections can be independently optimized through the process control. By attaching a mass to the end of the beam and integrating piezoresistive bridges on both beam sections, a composite beam-mass structure sensitive to two acceleration components can be formed. This structure can work as a rate gyroscope if the mass is electrostatically driven into vibration in the vertical direction and the Coliois force in the lateral direction is detected by a piezoresistive bridge. The principle, mask design and the process of the 100 masked-maskless etching for the composite beam structure are described and the experimental results on acceleration sensitivities and angular rate are presented.

A reduced variational technique is developed to describe the interaction of opposite-handed, circularly-polarized, d-dimensional solitons propagating in an isotropic material with both cubic and quintic nonlinearities. Dynamical equations for the simultaneous collapse and interaction of the two solitons are developed and are shown to be analogous to a particle in a (d+2)-dimensional potential well. This potential simultaneously describes the dynamics of the interaction as well as the self- and cross-focusing of the two solitons. Stability and escape velocity formulas are derived and compared to analytic and numerical results. The possibility of a "slingshot" interaction is revealed, even for d = 1 dimensional solitons, and its dynamics are illustrated by direct integration of the variational state equations.

The magnetic field generated by an electron bound in a spherically symmetric potential is calculated for eigenstates of the orbital and total angular momentum. General expressions are presented for the current density in such states and the magnetic field is calculated through the vector potential, which is obtained from the current density by direct integration. The method is applied to the hydrogen atom, for which we reproduce and extend known results., This article is a long version of our article which will appear in Eur. J.phys.20. It contains 22 pages 3 figures

An improved method for solving time-dependent problems in quantum mechanics, in the customary cases of constant or harmonic perturbation, is applied to the calculation of the self-energy of electrons interacting with phonons in solids. The mixing of unperturbed Bloch states, resulting from the actual coupling, is self-consistently taken into account, and the related quantum probability amplitudes are determined through direct integration over the quasiparticle spectrum. Laplace transform and elementary mathematics are used, thereby enhancing the physical transparency, and bringing out approximations in every stage. Explicit illustrative results are worked out in the simple case of slowly varying self-energy parameters. The method is critically compared with the standard Green function approach, and further encourages more detailed applications.

The finite-element method of analysis is unrestricted by size considerations and is well suited for the study of very small structures such as MEMS devices embedded in structures. This paper presents the numerical approach and results for a silicon-based micro-flow sensor for pulsed-flow sensing. The new approach presented here is the treatment of transient problems. A finite-element formulation is presented for modeling the dynamic response of piezoelectric ceramic sensors embedded in a micro-cantilever subjected to mechanical loading resulting from fluid flow. An unconditionally stable method (called the -method) is used for the direct integration of the equations of motion and implicit - explicit procedures are used for the transient analysis of the linear system of equations. For verification of the code, the device is tested for step, rectangular pulse and sinusoidal loading. For the case of the micro-fluid flow sensor, numerical results are in good agreement with the available experimental data based on a piezoresistive sensor. The numerical approach presented here may be used in CAE models for microsensors under more realistic, transient excitation.

The theoretical description of the propagation of an ultrashort pulse in a dispersive and dissipative medium, e.g. a plasma, in the presence of a magnetic field is performed in the space-time domain using a Green's function approach. For the purpose a model equation is constructed in which the source term is represented by a δ-function in space and time. The model equation, which is of the Klein-Gordon type can be given an explicit solution directly in the form of a δ-function part and a wake-field part, for which the argument of a Bessel function contains all the parameters of the problem. The results thus reveal the influence of the Green's function of the plasma frequency, the gyro-magnetic frequency and the damping rate. The resulting Green's function can then be used to derive the plasma response of an ultrashort pulse of arbitrary shape and structure by direct integration of the product of the Green's function and the initial pulse over the spatial source domain. The results should be of interest for diagnostics of plasmas and other media with dispersive and dissipative properties.

Calculations of the ionization yield obtained for the strong circularly polarized light ionization of m=0, +or-1 2p initial states from the so-called single Floquet state approximation are compared with the exact results reported by Gajda, Piraux and Rzazewski (1994) obtained via direct integration of the time-dependent Schrodinger equation. For m=-1, 0 excellent agreement is found despite the fact that the pulse length is only 20 optical cycles. The differences between the two approaches for m=1 support the different mechanism of stabilization in that case.

The equations describing tokamak plasmas constitute a coupled set of nonlinear partial parabolic equations whose time evolution and stationary solutions can only be found numerically. The 'transport codes' written for this purpose are usually too complicated to run for a nonspecialist, and provide little insight into some basic features of the problem. Here we concentrate on the problem of obtaining the stationary solutions, through direct integration of the time independent equations, which should require orders of magnitude less CPU time. This relates with the question of obtaining physically acceptable solutions, a problem present in other areas of science as well. We provide results at three levels of accuracy. A simple 'shooting method', a 'genetic code' method, and the exact but time consuming asymptotic solution obtained after typically a few thousand iterations of the time dependent equations. This last method is mainly interesting here as a check on the other two.

The pressing of hard metal components is analysed with numerical methods. The analysed components are selected from produced components for which the density distribution in the material after pressing has been measured. The expected results from the analyses are the density distribution and the springback after unloading and ejection of the components. The highly non-linear quasistatic problem is analysed with the use of explicit integration of the equations of motion. A contact constraint method based on direct integration of the equations of the contact interface is used in the analyses. The contact and friction algorithms have been developed in earlier work and are further verified by analyses of a test problem that has an analytical solution. The behaviour of the powder is described by a special cap plasticity material model developed for powder applications. In one example the results obtained using the cap model are compared with results obtained with a multisurface plasticity model. The parameters of the constitutive models are fitted to triaxial experimental data through optimization methods. The presented methods are evaluated by comparing the results with experimental data from density measurements where a technique based on gamma ray absorption is used. The density distributions are qualitatively in good agreement with experimental results. The springback obtained in the simulation of unloading and ejection is in good agreement with measured values.

Using the slave boson method for the 1-band Hubbard model, we derive an effective Hamiltonian in terms of charge and spin variables. For large values of the Hubbard U we recover the standard t-J model. For smaller values of U the range of interaction between the spin and charge variables is increased. We also consider the effect of an onsite electron-phonon interaction and show that the effect of integrating the double occupancy and the integration of phonons is in general not additive.

An analytical model for the electrothermomechanical analysis of out-of-plane deformation in resistively heated single-stepped beams is presented. The model takes into account the conductive heat transfer from the beam to the substrate in which it is anchored. It also considers the temperature dependence of the beam material properties and accounts for the locally enhanced resistive heating effect around the release holes in the beam to predict temperature distribution along the beam. Energy method and Euler–Bernoulli beam theory are used for the prediction of out-of-plane deformation and stress distribution of the beam, as well as the out-of-plane rotation at the middle of the beam. The model considers the nonuniformity of the air gap between the beam and the substrate and captures the resultant asymmetric temperature distribution along the beam. The out-of-plane rotations in the middle of the single-stepped beam predicted by the analytical model and measured experimentally agree within 10%. The analytical model is then used to predict the maximum actuation current, which results in high temperature plastic deformation and agrees with the experiments within 5%. The proposed analytical model provides a good approach for systematic design and analysis of out-of-plane electrothermal microactuators based on single-stepped beam design.

A novel Fourier method for rigorous diffraction computation at periodic structures is presented. The procedure is based on a differential equation for the S-matrix, which allows direct integration of the S-matrix blocks. This results in a new method in Fourier space, which can be considered as a numerically stable and well-parallelizable alternative to the conventional differential method based on T-matrix integration and subsequent conversions from the T-matrices to S-matrix blocks. Integration of the novel differential equation in implicit manner is expounded. The applicability of the new method is shown on the basis of 1D periodic structures. It is clear however, that the new technique can also be applied to arbitrary 2D periodic or periodized structures. The complexity of the new method is similar to the conventional differential method with N being the number of diffraction orders.

An analytical model is proposed for time-averaged ion energy distributions (IEDs) in collisional rf sheaths (?i?sm, where ?i is the ion mean free path and sm is the sheath thickness), in which charge transfer is the dominant ion-neutral collision mechanism. Our model is different from the model in Israel et al 2006 J. Appl. Phys. 99 093303 in two aspects. Firstly, to calculate the ion flux, we consider ions created in both the space charge region and the quasi-neutral region instead of only the quasi-neutral region. Secondly, to calculate the ion energy, we use an rf-modulated transit time instead of only the average transit time. Consequently, our model is valid over a wider pressure range (from a weakly collisional sheath to a strongly collisional sheath) compared with the model in Israel et al 2006 J. Appl. Phys. 99 093303. Our model shows that, in a collisional rf sheath (argon, 5?Pa, 27.12?MHz and 100?W), 65% of the ion flux in the IED comes from secondary ions created in the space charge region. Results of our model are compared with those obtained by measurement, direct integration method and particle-in-cell/Monte Carlo collision simulation.

We obtain the energies, widths and wave functions of hydrogen atom under a constant electric field by direct integration of the Hamiltonian in parabolic coordinates.

Uniform, compact, and thick InGaN layers are grown on Si(111) substrates by plasma-assisted molecular beam epitaxy without any buffer layers at low temperatures of around 320 °C. By adjusting the Ga/In flux ratio, InGaN layers with In compositions between 10 and 33% are obtained, providing emission covering the whole visible spectral range. The In composition varies less than 2% over large areas, and the single-crystalline hexagonal InGaN layers have a well-defined epitaxial relationship with the Si substrate. Photoluminescence is observed up to room temperature, opening the prospect for the direct integration of InGaN light-emitting devices with Si technology.

X-ray beam deflection control method using a single flexible glass capillary is proposed to illuminate a fixed sample at a target position with different incidence angle. A 700 mm-long capillary with a bore diameter of 50 micron and an outer diameter of 2 mm, which gives suitable flexibility for the critical curvature, was employed for the test experiment in SPring-8. The X-ray beam with a wavelength of 0.1 nm was introduced into the capillary, whose axis at the input side was finely adjusted to be parallel to the X-ray beam axis with swivel-, rotation- and translation stages. The divergence angle of output beam was measured and is 1–2 mrad. By moving the output-side capillary-support transversely to the beam axis, the beam deflection angle was changed over the range of about 80 mrad. The maximum throughput was larger than 60 % in efficiency, and 8 × 1010 photons/s in flux. Mapping with the beam deflection system has also been demonstrated for X-ray absorption measurement of a test sample composed of copper and nickel films. The materials were identified by changing the X-ray photon energy around their absorption edges.

In this article we investigate the beam deflection by a one-dimensional binary diffraction grating. We propose a simple theoretical expression that can be used to predict the beam deflection angle due to both periodic and aperiodic binary grating profiles. We show that the theoretically calculated beam deflection angles agree well with the experimentally obtained deflection angles for various grating patterns. Thus the expression can be used to precisely position the deflected beam at a predetermined location. Further, we show that the theoretical expression can be used to construct a map between the spatial frequency of the grating and the deflection angle which can be employed to deflect the beam at equal intervals by sequentially changing the spatial frequency in accordance with the map. We also demonstrate the superior beam repeatability of a binary grating based beam scanner.

This paper is concerned with the forced nonlinear vibration of multi-layered graphene sheets modelled at the atomic level by the lattice structure approach. In this, the covalent bond between two carbon atoms is assumed as a structural member with prescribed physical and material properties. An atom is treated as a nodal point with its own mass and six degrees of freedom. The highly nonlinear van der Waals interaction between adjacent graphene layers is fully incorporated in the model by placing it in the force vector. This adjustment significantly reduces the computational hardships due to nonlinearity and increases the efficiency of the method. Newmark's direct integration method is modified to address the nonlinearity in the load vector and used for the solution of the matrix equation governing the motion of the multi-layered graphene sheet. Double-layered square graphene with simply supported and clamped boundary conditions is analysed to examine the out-of-plane and in-plane vibrational characteristics. Also, in order to illustrate the applicability of the numerical method, analyses are carried out with the first- and second-order Taylor series approximations of the van der Waals interactions, influence of which is found to be quite significant in the bending modes of vibration, but it essentially does not have a role in the in-plane modes. The numerical method developed herein is quite appropriate with reference to the structural formation at the atomic scale and also more efficient than previous computational approaches by others.

This paper describes an analytical method for the wave field induced by a moving load on a periodically supported beam. The Green's function for an Euler beam without support is evaluated by using the direct integration. Afterwards, it introduces the supports into the model established by using the superposition principle which states that the response from all the sleeper points and from the external point force add up linearly to give a total response. The periodicity of the supports is described by Bloch's theorem. The homogeneous system thus obtained represents a linear differential equation which governs rail response. It is initially solved in the homogeneous case, and it admits a no null solution if its determinant is null, this permits the establishment the dispersion equation to Bloch waves and wave bands. The Bloch waves and dispersion curves contain all the physics of the dynamic problem and the wave field induced by a dynamic load applied to the system is finally obtained by decomposition into Bloch waves, similarly to the usual decomposition into dynamic modes on a finite structure. The method is applied to obtain the field induced by a load moving at constant velocity on a thin beam supported by periodic elastic supports.

Double helical coils are becoming more and more attractive for accelerator magnets and other applications. Conceptually, a sinusoidal modulation of the longitudinal position of the turns allows virtually any multipolar field to be produced and maximizes the effectiveness of the supplied ampere turns. Being intrinsically three-dimensional, the modeling of such structures is very complicated, and several approaches, with different degrees of complexity, can be used. In this paper we present various possibilities for solving the magnetostatic problem of a double helical coil, through both finite element analyses and direct integration of the Biot–Savart law, showing the limits and advantages of each solution and the corresponding information which can be derived.

Crustal faults and sharp material transitions in the crust are usually represented as triangulated surfaces in structural geological models. The complex range of volumes separating such surfaces is typically three-dimensionally meshed in order to solve equations that describe crustal deformation with the finite-difference (FD) or finite-element (FEM) methods. We show here how the Boundary Element Method, combined with the Multipole approach, can revolutionise the calculation of stress and strain, solving the problem of computational scalability from reservoir to basin scales. The Fast Multipole Boundary Element Method (Fast BEM) tackles the difficulty of handling the intricate volume meshes and high resolution of crustal data that has put classical Finite 3D approaches in a performance crisis. The two main performance enhancements of this method: the reduction of required mesh elements from cubic to quadratic with linear size and linear-logarithmic runtime; achieve a reduction of memory and runtime requirements allowing the treatment of a new scale of geodynamic models. This approach was recently tested and applied in a series of papers by [1, 2, 3] for regional and global geodynamics, using KD trees for fast identification of near and far-field interacting elements, and MPI parallelised code on distributed memory architectures, and is now in active development for crustal dynamics. As the method is based on a free-surface, it allows easy data transfer to geological visualisation tools where only changes in boundaries and material properties are required as input parameters. In addition, easy volume mesh sampling of physical quantities enables direct integration with existing FD/FEM code.

In this paper the nonlinear vibration of an aircraft panel structure modelled as an isotropic cracked plate and subjected to transverse harmonic excitation is considered for studying the dynamic response, both analytically and experimentally. A crack is arbitrarily located at the centre of the plate, consisting of a continuous line. This mathematical model is in the form of Duffing equation with a cubic nonlinear term. The perturbation method of multiple scales is used to solve the algebraic equation, and then investigated with the results of the direct integration within Mathematica™ and finite element analysis in ABAQUS for the first mode only. In addition, experimental measurements are also carried out to verify the dependence of the cracked plate's fundamental mode shape and resonance frequency on the vibration displacement amplitude. An extermely close agreement between these results is observed.

This paper derives the exact solution of the model #3 proposed by Zhang et al for the rectangular cantilever beam. The model is solved by a reduced differential equation and the designated exact solution is formally presented in the integration form in terms of the Airy functions, Ai(X) and Bi(X). Both the deflection and curvature are presented as containing the constituents contributed by Ai(X) and Bi(X) and an additional portion attributed to the interaction between the Airy functions. The results show that the curvature at the fixed end is primarily determined by the contribution of Bi(X), while the dominant constituent for the deflection at the free end is dependent on the magnitude of the loading scenario. The relations for the directions of the constituent deflections and curvatures are deduced from the derived exact solution and by the properties of the Airy functions, which reveal that the contribution of Ai(X) results in forward deflection, and the curvature and the deflection are of the same sign. However, the curvature contributed by Bi(X) induces retrograde deflection, which means that the curvature and the deflection are of opposite sign. The results also suggest that the hardening effect of the present model can be represented by the contribution of Bi(X), with which the beam resists being deflected as if it were stiffened by Bi(X) under all loading scenarios. Nonetheless, there is no simple relation for the curvature and deflection attributed to the interaction of the Airy functions, which should be evaluated by direct integration and differentiation of the derived exact solution.

We report the parallel fabrication of miniaturized chemical sensors by the direct integration of nanostructured transition metal oxide films onto micro-hotplate platforms based on micromachined suspended membranes. This has been achieved by local deposition on a 10 × 10 membrane wafer of a supersonic cluster beam through a microfabricated auto-aligning silicon shadow mask. The sensing properties of the obtained devices were tested with respect to various gaseous species. For reducing and oxidizing species such as ethanol and NO2, very good performance in terms of linearity and sensitivity was observed. These results demonstrate the feasibility of the coupling of a bottom-up nanofabrication technique such as supersonic cluster beam deposition to a top-down microfabricated platform for a direct and parallel integration methodology of nanomaterials in MEMS.

An experimental investigation on ion beam deflection by grid translation was performed. The ion beam deflection in ion optics is a desired technology for ion thrusters because thrust vector control utilizing this technique can eliminate the need for conventional gimbaling devices and thus reduce propulsion system mass. A grid translation mechanism consisting of a piezoelectric motor, a ceramic lever, and carbon-based grids with slit apertures was fabricated and high repeatability in beam deflection characteristics was obtained using this mechanism. Results showed that the beam deflection angle was proportional to the grid translation distance and independent of slit width and grid voltage. A numerical simulation successfully reproduced the beam deflection characteristics in a qualitative and quantitative sense. A maximum beam deflection angle of approximately plus or minus 6 degrees, which was comparable to that of the ordinary gimbaling devices used in space, was obtained without a severe drain current. Therefore, the beam deflection by grid translation is promising as a thrust vectoring method in ion thrusters.

Eigenvalues of the three-dimensional Schrodinger equation with a radial Gaussian potential are obtained using the Liouville-Green uniform asymptotic method. The results are compared with those obtained by direct integration of the Schrodinger equation.

Calculations of incoherent scattering factors for sodium, aluminium, argon and krypton have been carried out by direct integration over the impulse approximation Compton profiles (IACP). The authors compare their results with those obtained by the Waller-Hartree (WH) theory (1929). It is found that at low scattering angles, the IACP results differ significantly from the WH scattering factors.

The effects of beam focusing and the influence of total beam deflection on injection efficiency are estimated numerically for the neutral beam injectors of large tokamaks such as the JT-60. It is shown that beam focusing is important in improving injection efficiency in such a large device. As one method of focusing, beam focusing by aperture displacement was investigated in the two-stage acceleration system on the basis of a thin lens approximation. The simultaneous displacement of apertures in the plasma and gradient grids (or suppressor and exit grids) was found to be adequate for beam focusing, because the focal point hardly depends on the field intensity ratio which determines the two-stage ion beam optics.

The geometric factor of a particle detector having either circular or rectangular geometry is derived analytically. Both isotropic and 'cos2 theta ' variations of intensity are considered. An approach based on 'shadow areas' is applied to the case of particles passing through two unequal circular areas, and it is proved that the results are identical with those obtained previously using an approach based on the direct integration of 'infinitesimal areas'. This latter approach is then used for the case of two unequal rectangular areas, which has not been considered previously. The analytical results confirm those obtained previously using numerical techniques.

A method of evaluation of lattice Green functions using a Fourier series expansion of their imaginary parts is developed. It is shown, through an example, that numerical evaluation of the real and imaginary parts of the Green functions can be done by this method with much greater ease than by direct integration of the Green function integrals, with the added advantage that the real and imaginary parts of the Green functions can be represented by simple analytic expressions with good approximation.

Peoglos (1988) reports forces on portions of a current loop due to the remainder of the loop which are in excellent agreement with the direct integration of Ampere's original force law. Peoglos has not, however, confirmed the non-physical absurd Biot-Savart law: it predicts ambiguous, and thus non-testable, results for the force on Ampere's bridge. It predicts merely fortuitously the correct Ampere force on a straight portion, since only lateral forces are measured. The force is not generally given by 2Fx=I2dL/dx and could not, in any case, distinguish between the Ampere and Biot-Savart laws.