Reference frame laboratory-fixed

An important issue in reaction stereodynamics must be eonsidered. Experiment and vector preparations are usually performed in the laboratory reference frame (space fixed) whereas the important preparation for the reaction is the molecular reference frame (body fixed) which rotates during the collisions at a non-constant angular velocity. This leads to numerous difficulties, which have motivated an important literature reviewed in Ref [18], As far as stereodynamics considerations are... 

When applied to a volume-fixed frame of reference (i.e., laboratory coordinates) with ordinary concentration units (e.g., g/cm3), these equations are applicable only to nonswelling systems. The diffusion coefficient obtained for the swelling system is the polymer-solvent mutual diffusion coefficient in a volume-fixed reference frame, Dv. Also, the single diffusion coefficient extracted from this analysis will be some average of concentration-dependent values if the diffusion coefficient is not constant. 

Here, ak is the isotropic chemical shift referenced in ppm from the carrier frequency co0, SkSA is the anisotropy and tfk SA the asymmetry of the chemical-shielding tensor, here also expressed in ppm. Note that for heteronuclear cases different reference frequencies co0 are chosen for different nuclei (doubly rotating frame of reference). The two Euler angles ak and pk describe the orientation of the chemical-shielding tensor with respect to the laboratory-fixed frame of reference. The anisotropy dkSA defines the width and the asymmetry t]kSA the shape of the powder line shape (see Fig. 11.1a). 

Figure 3-26 Quartz crystal growth and diffusion profile in (a) a laboratory-fixed reference frame and (b) an interface-fixed reference frame. At a given time, a given kind of curve is used to outline the crystal shape and plot the concentration profile.

The new reference frame is known as the interface-fixed reference frame, and the old reference frame is called the laboratory-fixed reference frame. The melt consumption rate u depends on whether the growth is controlled by interface reaction, or by diffusion, or by externally imposed conditions such as cooling. 

Figure 1 In the laboratory frame (A) the molecule is tumbling with the static magnetic field fixed along the z-axis. The reference frame of a rigid molecule, instead, moves with the compound and leads to an effective variation of the magnetic field vector (B).

As an example, consider a chain of atoms— 1,/ ,/ + 1... with a fixed valence angle so that the angle between bonds / — 1, / and j>. [I + 1 is fixed at 6. Let dj. i = 1,2,3, be the unit base vectors of the fixed laboratory reference frame. In addition, we introduce a moving local Cartesian system with base vectors dj = 1,2,3, defined in Fig. 11. Then any vector r can be written in terms of components with respect to either frame as r = r, c, = fj dj, where we are adopting throughout the convention that superimposed bars denote components with respect to the local frame a,. The vector components with respect to the two frames are related in the usual way as... 

The particles are numbered from 1 to N with Mi the mass of particle i, R, = [A, Yt Zi) a column vector of Cartesian coordinates for particle i in the external, laboratory fixed, frame, Vr the Laplacian in the coordinates of R, and Ri — Rj the distance between particles i and j. The total Hamiltonian, eqn.(l), is, of course, separable into an operator describing the translational motion of the center of mass and an operator describing the internal energy. This separation is realized by a transformation to center-of-mass and internal (relative) coordinates. Let R be the vector of particle coordinates in the laboratory fixed reference frame. 

We show here only the theoretical framework developed in the previous paper. Our analysis is restricted in two dimensions of the vertical plane (x,y) of the inclined plane, and for simplicity we assume that the region of the crystal is semi-infinite. The x axis is parallel to the inclined plane and the y axis is normal to it. We choose a reference frame moving in the y direction with a mean growth rate V with respect to a fixed laboratory frame of reference. The origin of y axis is the solid-liquid interface. An experimental observation shows that the decrease in the temperature and the increase in the wind speed in the air lead to the increase in V. The typical values of V measured are on the order of 10 (m/s). Here we assume that V takes a constant value and there is no wind in the air. 

The diffusion fluxes defined earlier are seen to be special cases of the more general definitions presented above. Table 1.3 summarizes the most commonly encountered diffusion fluxes and Table 1.4 summarizes those fluxes that are measured with respect to a laboratory fixed coordinate reference frame. 

TABLE 1.4 Fluxes With Respect to a Laboratory Fixed Frame of Reference"... 

Thereafter, in the second step of the procedure the calculation is carried out using a model formulation considering a fixed laboratory reference frame... 

We should note that this description of the system as being two fluids separated by a flat interface already has inherent in it the spatial averaging to a scale of resolution that is much larger than the individual pore level of description. The volume-averaged velocity in each fluid is determined by Darcy s law. As noted earlier, we assume that the fluids (and the interface between them) move with a uniform velocity V in the positive z direction. It is therefore convenient to consider the problem with respect to a moving reference frame that is fixed at the unperturbed fluid interface, i.e., we introduce z, which is related to the original laboratory frame of reference as... 

The Cartesian reference frames p, a, b, and r are defined as follows p denotes a laboratory-fixed frame in which the laser polarization vectors are fixed a signifies a frame of reference that has been chosen to have its origin at center A similarly b has its origin at center B r represents a frame in whieh the R vectors are rotationally invariant. 

In the context of motion and flux, it is clear that the flux should be defined with respect to a reference frame (Chakraborty 1995). In crystalline silicates, because diffusion of oxygen and silicon can be much slower than that of other cations (with possible exception of Al), this can be achieved quite easily by using the fixed silicate lattice as a reference frame in which ions jump from site to site. This is the so-called lattice fixed frame, which commonly coincides with the laboratory frame. Note that the motion of a dilute isotope of oxygen (e.g. O) can still be treated in this frame. Pick s first law can readily be modified to take into account the variability of reference frames. 

This chapter describes mass transport to electrodes by diffusion and migration. It is assumed throughout that there is no convective motion of the solution, and transport is described with respect to a reference frame fixed to the laboratory [1, 2]. Thus, many of the equations derived in this chapter cannot be directly applied to systems with bulk motion. 

The velocity vectors of the fragments wiU be correlated with the laser polarization vector the polarization vector of the laser beam also defines a laboratory-fixed coordinate frame. But one also has to remember that, in the molecular reference frame, the angular recoil distribution depends on the symmetries of the electronic (initial and final) states involved in the absorption process, namely that the relative orientation of the transition dipole moment of a molecule fi and the direction of the laserpolarization vector (see Chapter 16 for details). 

As a mathematical preliminary, we note the following. To deal with molecules, we are required to transform freely between the laboratory reference frame and the body-fixed frame that rotates with the molecule. The rotation from the lab frame (x,y,z)... 

Spatial instabilities arise from disturbances localized in space and in the present problem these can be in the bulk (as formulated) or at the interfaces. The latter produces a slightly different formulation which is analyzable in exactly the same way as for bulk disturbances. The appropriate reference frame is a fixed laboratory frame since we are interested in instabilities that grow as they convect downstream. 

Figure 1 illustrates the geometry of a typical laser light scattering experiment within a laboratory-fixed frame of reference x,y,z. 

The torque /t x Bj will cause each magnetic moment /i, which is stationary in the rotating frame, to process around the direction of the field B]". This direction is fixed in the rotating fl-ame (let us call it the x -direction) and rotates around the z-axis with the Larmor frequency in the laboratory reference frame. If the tfequency of the RF field S2 is not equal to col, i e., in the off-resonance case, the precession of the magnetic moments in the rotating frame is around an axis defined by an effective magnetic field given by ... 

The orientation of the collision frame with respect to the laboratory-fixed Z-axis, is designated k=(0k,cpk). Note that the projection quantum numbers in Eq. 25 all refer to the same lal ratory system. In the case of excitation by circularly polarized light, only the mj=+l or mj=-l projection states are prepared, where the Z-axis refers now to the direction of propagation of the laser. Again, the appropriate expression for the scattering amplitude is given by Eq. 25 with now mj=+l or mj=-l. 

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