Laboratory coordinate system

Fig. 6. (a) Different coordinate systems (laboratory L, director D, and magnetic m) nsed to define motion parameters for a nitroxide spin label, (b) Diffusion rotation angles used to define the magnetic axes relative to the diffusion axes. Note that the reference system for these angles is the diffusion frame, whereas the reference system is the magnetic (g) frame for the magnetic tilt angles (cf. Fig. 3). 

Here the ijk coordinate system represents the laboratory reference frame the primed coordinate system i j k corresponds to coordinates in the molecular system. The quantities Tj, are the matrices describing the coordinate transfomiation between the molecular and laboratory systems. In this relationship, we have neglected local-field effects and expressed the in a fomi equivalent to simnning the molecular response over all the molecules in a unit surface area (with surface density N. (For simplicity, we have omitted any contribution to not attributable to the dipolar response of the molecules. In many cases, however, it is important to measure and account for the background nonlinear response not arising from the dipolar contributions from the molecules of interest.) In equation B 1.5.44, we allow for a distribution of molecular orientations and have denoted by () the corresponding ensemble average ... 

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.

Equation (B2.3.10) shows that the scattered intensity observed in the laboratory is distorted from that hr the CM coordinate system. Those products which have a larger laboratory velocity or a smaller CM velocity will be observed in the laboratory with a greater intensity. 

The Hamiltonian in this problem contains only the kinetic energy of rotation no potential energy is present because the molecule is undergoing unhindered "free rotation". The angles 0 and (j) describe the orientation of the diatomic molecule s axis relative to a laboratory-fixed coordinate system, and p is the reduced mass of the diatomic molecule p=mim2/(mi+m2). 

The angles 0, (j), and x are the Euler angles needed to specify the orientation of the rigid molecule relative to a laboratory-fixed coordinate system. The corresponding square of the total angular momentum operator fl can be obtained as... 

We assume that in (4.38) and (4.39), all velocities are measured with respect to the same coordinate system (at rest in the laboratory) and the particle velocity is normal to the shock front. When a plane shock wave propagates from one material into another the pressure (stress) and particle velocity across the interface are continuous. Therefore, the pressure-particle velocity plane representation proves a convenient framework from which to describe the plane Impact of a gun- or explosive-accelerated flyer plate with a sample target. Also of importance (and discussed below) is the interaction of plane shock waves with a free surface or higher- or lower-impedance media. 

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. 

These simple relations motivate a more formal approximation in which we first re-expand the interaction potential in a space-fixed ("laboratory-frame") coordinate system as... 

Laboratory frame model A means of visualising the processes taking place in an NMR experiment by observing these processes at a distance, i.e., with a static coordinate system. See Rotating frame model. 

The subscripts refer to Cartesian coordinates in the laboratory coordinate system. The terms E(go) and B(go) are the electric and magnetic induction fields, respectively. In addition, the nonlinear induced magnetic moment (magnetization) is defined as ... 

Physics and chemistry are carried out in laboratory frames using coordinate systems to set up experimental devices. Before discussing quantum mechanical processes let us recall the form of the total Hamiltonian for a set of particles having charges qa and masses ma interacting with an electromagnetic field A. This Hamiltonian is given by ... 

Let ijs now apply this concept of the RRF to the case where an rf field Hi is present. We choose a Cartesian coordinate system with tlje z axis along the dc field Hq and the y axis along the rf field Hi. The total field is given in the laboratory reference frame by... 

In (3.1) not all tensors are necessarily coaxial or diagonal. If the principal axes system of the g tensor is chosen as the molecular coordinate system eM, g has diagonal form. The laboratory frame eL is then related to eM by the rotation matrix R according to... 

Equation (4.4), which connects the known variables, unbumed gas pressure, temperature, and density, is not an independent equation. In the coordinate system chosen, //, is (lie velocity fed into the wave and u2 is the velocity coming out of the wave. In the laboratory coordinate system, the velocity ahead of the wave is zero, the wave velocity is uh and (u — u2) is the velocity of the burned gases with respect to the tube. The unknowns in the system are U, u2, P2, T2, and p2. The chemical energy release is q, and the stagnation adiabatic combustion temperature is T, for n-> = 0. The symbols follow the normal convention. 

The equations of internal motion defined within the local (molecular) fixed coordinate system have to be transformed to the laboratory fixed coordinate system in which all experiments are performed. Thus, introducing Euler s angles, (b, 0, and X, the coordinates of the electrons (e) and nuclei (n) are transformed in the following... 

For a diatomic species, the vibration-rotation (V/R) kinetic energy operator can be expressed as follows in terms of the bond length R and the angles 0 and < ) that describe the orientation of the bond axis relative to a laboratory-fixed coordinate system ... 

Restriction (i) implies that history of the system is not a relevant thermodynamic property. Restriction (ii) implies that position or orientation of the system are not considered thermodynamic properties, because different observers must be free to select their own preferred laboratory coordinate systems. (Note that omission of position r as a relevant property strongly distinguishes thermodynamics from classical dynamics, where spatial location r of the center of mass is a prominent variable of the system.)... 

The state of stress in a flowing liquid is assumed to be describable in the same way as in a solid, viz. by means of a stress-ellipsoid. As is well-known, the axes of this ellipsoid coincide with directions perpendicular to special material planes on which no shear stresses act. From this characterization it follows that e.g. the direction perpendicular to the shearing planes cannot coincide with one of the axes of the stress-ellipsoid. A laboratory coordinate system is chosen, as shown in Fig. 1.1. The x- (or 1-) direction is chosen parallel with the stream lines, the y- (or 2-) direction perpendicular to the shearing planes. The third direction (z- or 3-direction) completes a right-handed Cartesian coordinate system. Only this third (or neutral) direction coincides with one of the principal axes of stress, as in a plane perpendicular to this axis no shear stress is applied. Although the other two principal axes do not coincide with the x- and y-directions, they must lie in the same plane which is sometimes called the plane of flow, or the 1—2 plane. As a consequence, the transformation of tensor components from the principal axes to the axes of the laboratory system becomes a simple two-dimensional one. When the first principal axis is... 

Fig. 1.1. Laboratory coordinate system x direction of flow (also 1-direction), y direction of velocity gradient (also 2-direction), I, II principal directions of stress, y orientation angle if stress-ellipsoid, vx velocity (in -direction), q velocity gradient...

In this equation is the internal friction factor of thej-th normal mode and Qjj1 is the inverse transformation matrix of Zimm. In other words, Cerf assumed that one can ascribe a separate internal friction factor to every normal mode. This assumption is critisized by Budtov and Gotlib (183) as, in this way, the elements of the internal friction matrix in the laboratory coordinate system x, y, z, viz. 

Solution. The laboratory coordinate system is used and there is no change in the overall specimen volume. The integral in Eq. 4.50 is proportional to the sum area 1 + area 2 in Fig. 4.7. Area 1 is positive and area 2 is negative. When x = 0 is set at the position of the original interface, area 2 is proportional to the amount of diffusant that has left... 

A series of relationships have been derived between the stationary coordinate system (the scientist in his or her laboratory) and a moving (intrinsic, invariant) coordinate system that can be compared to classical calculations of dynamic variables (Table 1.3). 

For additional insight, let us now consider the same reaction as described in the center-of-mass (cm) coordinate system. In the cm system the total momentum of the particles is zero, before and after the collisions. The reaction as viewed in the laboratory, and cm system is shown in Figure 10.2. 

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 ...

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