Incidence, plane

In this section, two illustrative numerical results, obtained by means of the described reconstruction algorithm, are presented. Input data are calculated in the frequency range of 26 to 38 GHz using matrix formulas [8], describing the reflection of a normally incident plane wave from the multilayered half-space. 

The first paper about NiFe layers evaporated under an angle was pubHshed in the early 1960s (40). The films prepared this way are often called obHque-incidence or angle-of-incidence films. It was found that these kinds of films show an anisotropy whose strength depends on the angle of incidence of the atoms (a/ during deposition (Fig. 15). If is between 0 and 65 the anisotropy Hes parallel to the film plane and perpendicular to the incidence plane. 

Figure 4 Interference pettern created when regularly spaced atoms scatter an incident plane wave. A spherical wave emanates from each atom diffracted beams form at the directions of constructive interference between these waves. The mirror reflection—the (00) beam—and the first- and second-order diffracted beams are shown.

Information about the properties of the sample are contained in the complex ratio, p, of the Fresnel coefficients of reflection of the parallel (rp) and perpendicular (rg) incident plane polarized electrical field vectors. 

Four polarized ATR spectra can be recorded to characterize the three-dimensional (3D) orientation of a sample, p- and s-polarized spectra are recorded with the sample clamped with its Z- and X-axes sequentially aligned perpendicular to the incidence plane (that is, parallel to the s-polarized electric field). The absorbance measured in these different configurations is related to the anisotropic absorption indices of the sample, kj, as... 

The theory of this aberration was worked out in the 1920s by Schwarzchild. For simphcity we shall discuss the case of a beam conditioner comprising a single crystal and an aperture as in the classic double-crystal arrangement. If the Bragg planes are tilted about an axis contained in the incidence plane and the... 

Bragg planes, then rays which are not contained in the incidence plane will not see equal angles with respect to the specimen and the reference. If we set the crystals so that the median ray (in the incidence plane) makes equal angles, then an inclined ray may make the Bragg angle for the reference crystal but will not be diffracted from the specimen (Figure 2.21). The result is that only a band of rays satisfies the Bragg conditions for both crystals. The band moves up (or down) as the crystals are rotated. The consequences are ... 

Tilting about axis in incidence plane and Bragg plane... 

For symmetric reflections the peak search may now begin. For asymmetric reflections, the specimen must be rotated about its normal until the desired diffraction vector lies in the incidence plane of the beam conditioner. This is normally the diffractometer surface. An accurate knowledge of the orientation of the specimen in two axes is required to set asymmetric reflections this is usually taken from the position of the orientation flat or groove. 

The second issue concerns the spatial resolution achievable with this setting. To a good approximation, the spatial resolution normal to the incidence plane is related to the specimen to source distance D, the specimen to plate distance L and projected height of the source H by the simple geometrical relation ... 

Figure 8.4 Schematic diagram showing the geometrical resolution limit set by the projected source height normal to the incidence plane...

These equations give the optimum conditions for a rapid high resolution topograph when only the resolution in the incidence plane is considered. However we must also consider the resolution normal to the incidence plane (Figure 8.4) and this is given geometrically by =VL/D (8.12)... 

Under optimum conditions the resolntion normal to the incidence plane of a Lang topograph will be... 

In the laboratory. X-ray topography is nsnally performed with characteristic radiation and it is straightforward to show that the exposnre time for a section topograph scales with and for a traverse topograph as where P is the X-ray tnbe power and S is the source dimension in the incidence plane. The continnons... 

The high degree of X-ray polarisation in the electron orbit plane provides means of controlling both the signal/noise ratio and the penetration of the X-rays into the specimen. Depending on whether the incidence plane is chosen vertically or horizontally, sigma or pi polarisation may be selected. The strain sensitivity and the extinction distance can thus be varied while the normal photoelectric absorption conditions remain identical. 

In this section we derive an approximate expression for the absorption cross section of a large weakly absorbing sphere. We assume that the incident plane wave can be subdivided into a large number of rays the behavior of which at interfaces is governed by the Fresnel equations and Snell s law (Section 2.7). A representative ray incident on the sphere at an angle 0, is shown in Fig. 7.1. At point 1 on the surface of the sphere the incident ray is divided into externally reflected and internally transmitted rays these lie in the plane of incidence, which is determined by the normal to the sphere and the direction of the incident ray. If the polar coordinates of point 1 are (a, 0f, ), the plane of incidence is the plane = constant. At point 2 the transmitted ray encounters another boundary and therefore is partially reflected and partially transmitted. In a like manner we can follow the path of the rays within the sphere, a path that does not deviate outside the plane of incidence. At each point where a ray encounters a boundary it is partially reflected internally and partially transmitted into the surrounding medium. On physical grounds we know that the absorption cross section cannot depend on the polarization of the incident... 

Were it not for the particle, of course, A would just be a unit vector parallel to the direction of propagation of the incident plane wave, and the field lines would be parallel lines. At sufficiently large distances from the particle the field lines are nearly parallel, but close to it they are distorted. It is the nature of this distortion in the neighborhood of a small sphere and its relation to the optical properties of the sphere that we now wish to investigate. 

The presence of a crack or other discontinuity presents a serious difficulty for the standard V z) theory, because the reflectance function is defined for infinite plane waves that are reflected into infinite plane waves, and this requires a reflecting surface that is uniform. If a surface contains a crack then this requirement is violated, and an incident plane wave may be scattered into a whole family of waves (Tew et al. 1988). This scattering can be described in k-space by a scattering function S(kx, k x), where the prime refers to incident waves and the unprime to scattered waves. The x-direction is taken as tangential to the surface, and at this stage the theory is confined to two dimensions in the plane normal to both the surface and the crack. The response of the microscope can then be written in terms of the scattering function by integrating over the incident and reflected waves separately... 

The incident plane wave has only field components perpendicular to the direction of propagation. In contrast, the evanescent field has components along all directions X, y, and z of a Cartesian coordinate system attached to the IRE, as shown in Fig. 2. The direction of the incident field vector can be selected by use of a polarizer. The symbols II and denote electric field vectors parallel and perpendicular to the... 

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