Beam coordinate system

Beam coordinate system, XB, YB, and ZB a Cartesian coordinate system with the origin on the central ray of the incident flux at the sample surface, the XB axis in the PLIN and the ZB axis normal to the surface as shown in Fig. Al. Note The angle of incidence, scatter angle, and incident and scatter azimuth angles are defined with respect to the beam coordinate system ... 

Incident direction the central ray of the incident flux specified by and in the beam coordinate system ... 

Incident azimuth angle (< j) the fixed 180° angle from the XB axis to the projection of the incident direction onto the XB-YB plane. Note It is convenient to use a beam coordinate system (see Fig. A2) in which = 180° because this makes the correct angle to use directly in the familiar form of the grating equation. Conversion to a sample coordinate system is straight forward, provided the sample location and rotation are known ... 

The Z and ZB axes are always the local normal to the sample face. Locations on the sample face are measured in the sample coordinate system. The incident and scatter directions are measured in the beam coordinate system. If the sample fiducial mark is not an X axis mark, the intended value must be indicated on the sample (Fig. Al). 

Fio. Al. Relationship between sample and beam coordinate systems (1). 

A-2. Angle Conventions for the Incident and Scattered Light in THE Beam Coordinate System... 

Fig. 1.6. The particle coordinate system Oxyz and the beam coordinate system OxhVhZh have the same spatial orientation...

Fig. 1.7. General orientations of the particle and beam coordinate systems where...

Fig. 1.9. Reference frames (a) global coordinate system and (b) beam coordinate system...

The orthotropic stress and strain relationships of Equations 8.42 and 8.43 were defined in principal material directions, for which there is no coupling between extension and shear behavior. However, the coordinates natural to the solution of the problem generally will not coincide with the principal directions of orthotropy. For example, consider a simply supported beam manufactured from an angle-ply laminate. The principal material coordinates of each ply of the laminate make angles 0 relative to the axis of the beam. In the beam problem stresses and strains are usually defined in the beam coordinate system (jc,y), which is off-axis relative to the lamina principal axes (L, T). 

In a crossed-beam experiment the angular and velocity distributions are measured in the laboratory coordinate system, while scattering events are most conveniently described in a reference frame moving with the velocity of the centre-of-mass of the system. It is thus necessary to transfonn the measured velocity flux contour maps into the center-of-mass coordmate (CM) system [13]. Figure B2.3.2 illustrates the reagent and product velocities in the laboratory and CM coordinate systems. The CM coordinate system is travelling at the velocity c of the centre of mass... 

Figure B2.3.2. Velocity vector diagram for a crossed-beam experiment, with a beam intersection angle of 90°. The laboratory velocities of the two reagent beams are and while the corresponding velocities in the centre-of-mass coordinate system are and U2, respectively. The laboratory and CM velocities for one of the products (assumed here to be in the plane of the reagent velocities) are denoted if and u, respectively.

Maximum information is obtained by making Raman measurements on oriented, transparent single crystals. The essentials of the experiment are sketched in Figure 3. The crystal is aligned with the crystallographic axes parallel to a laboratory coordinate system defined by the directions of the laser beam and the scattered beam. A useful shorthand for describing the orientational relations (the Porto notation) is illustrated in Figure 3 as z(xz) y. The first symbol is the direction of the laser beam the second symbol is the polarization direction of the laser beam the third symbol is the polarization direction of the scattered beam and the fourth symbol is the direction of the scattered beam, all with respect to the laboratory coordinate system. 

A simple Michelson interferometer. If we place two mirrors at the end of two orthogonal arms of length L oriented along the x and y directions, a beamsplitter plate at the origin of our coordinate system and send photons in both arms trough the beamsplitter. Photons that were sent simultaneously will return on the beamsplitter with a time delay which will depend on which arm they propagated in. The round trip time difference, measured at the beamsplitter location, between photons that went in the a -arm (a -beam) and photons that went in the y arm (y-beam) is... 

Because the orientation of the reciprocal space coordinate system is rigidly coupled to the orientation of the real-space coordinate system of the sample, the reciprocal space can be explored8 by tilting and rotating the sample in the X-ray beam (cf. Chap. 9). 

The zone axis coordinate system can be used for specifying the diffraction geometry the incident beam direction and crystal orientation. In this coordinate, an incident beam of wavevector K is specified by its tangential component on x-y plan = k x + k y, and its diffracted beam at Kt+gt, for small angle scatterings. For each point inside the CBED disk of g, the intensity is given by... 

Considerable difficulties arise in finding an exact solution, but at the initial stage of the process when the interface differs little from a plane, analytical expressions can be obtained. If the incident light beams are flat, the relief of the surface is described by the relation (Tyagai et al, 1978) in an appropriate coordinate system ... 

Figure 13. Cartesian [center-of-mass (CM)] contour diagrams for NH+ produced from reaction of N+ with H2. Numbers indicate relative product intensity corresponding to each contour. Direction of N+ reactant beam is 0° in center-of-mass system. For clarity, beam profiles have been displaced from their true positions (located by dots and 0°). Tip of velocity vector of center of mass with respect to laboratory system is located at origin of coordinate system (+). Scale for production velocities in center-of-mass system is shown at bottom left of each diagram (a) reactant N+ ions formed by impact of 160-eV electrons on N2 two components can be discerned, one approximately symmetric about the center of mass and the other ascribed to N+(IZ3), forward scattered with its maximum intensity near spectator stripping velocity (b) ground-state N+(3/>) reactant ions formed in a microwave discharge in N2. Only one feature is apparent—contours are nearly symmetric about center-of-mass velocity.12 ...

Fig. 2.1.1 Idealized molecular-beam experiment for the reaction A(i, va) + B(y, vb) ( (l, ( ) + D(m. I m, ). The coordinate system is fixed in the laboratory. The reactants move with the relative speed v = a — wb. ...

The relation between the deflection angles in the two coordinate systems is derived below, in the special case where the target atom is at rest before the collision. This case represents of course not the typical situation in a crossed molecular-beam experiment. However, it greatly simplifies the relation and the derivation displays the essence of the problem. The general case is considered in Appendix C. 

Fig. 6.6. Schematic of realization of alignment-orientation Stark conversion, (a) Choice of coordinate systems, (b) Possible realization scheme for AB molecules seeded in a free jet of X atoms, (c) Symbolic polar plot of J distribution (dashed line refers to initial cylindrical symmetry over beam axis z ).

Usually the coordinate system is chosen that way, defining the x, v-planc by the sample surface with the x-axis oriented in the direction of the X-ray beam [13,25], As a consequence, the z-axis is perpendicular to the sample surface (see Fig. 1). In this case... 

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