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Laboratory coordinate frame

Coefficients Rlx form an orthogonal matrix iC transforming the (/-orbitals under rotation of the laboratory coordinate frame (LCF) to the local coordinate frame related to the ligand / and constructed such that its Oz axis is going through the metal atom and the ligand atom (DCF - diatomic coordinate frame). The perturbation caused by the ligand has a matrix representation elxx, in the DCF with A = a,7r(x),n(y), 5 xy), S(x2 - y2). These quantities are considered parameters of the AOM. [Pg.150]

Because in NMR the orientation of the field defines the z-axis of the laboratory coordinate frame of reference, Bq = (0,0, Bo) and = M cos0 is the projection of the magnetization vector on to the direction of the magnetic field. [Pg.25]

The difference is that B1 is applied in the transverse plane, for instance along the x-axis of the laboratory coordinate frame. Consequently, only the x-component of the spin angular momentum operator I defines the interaction energy together with the magnitude and time dependence of the x-component of B (cf. Section 2.2.1). [Pg.70]

Fig. 5.4.1 Relationship between the Cartesian laboratory coordinate frame with axes x and y and the rotated Cartesian frame with axes r and s. The field gradient is applied parallel to r at an angle

Fig. 5.4.1 Relationship between the Cartesian laboratory coordinate frame with axes x and y and the rotated Cartesian frame with axes r and s. The field gradient is applied parallel to r at an angle <p defining the variables of the projection P(r, <p) of the object.
This equation is written, of course, in the laboratory coordinate frame, and the orientational average requires knowledge of the absorption and emission transition dipole directions in the molecular coordinate system. The average may be evaluated through the use of Euler matrix terms in the standard manner (Riehl and Richardson, 1976a). The reader is referred to previous publications in which various excitation emission geometries and molecular transition moments have been evaluated (Riehl and Richardson, 1976a, 1986). [Pg.297]

The solute is a polyatomic system characterized by the intramolecular potential V(Rp. The solute Hamiltonian in the laboratory coordinate frame can be cast into a matrix form as follows ... [Pg.446]

The differential equations that are derived according to the axial symmetry of the laboratory coordinate frame may be simplified by switching to a coordinate frame that rotates (about the z-axis). The rotating frame of reference (x, y, z) is associated with field H of frequency v, and rotates about field Ho with angular... [Pg.34]

Large, positive values of 2 correspond to positions far ahead of the wavefront or to early times (the temporal initial state) large, negative values of z indicate the long-time conditions (the temporal final state) after the wave has passed a given station in the laboratory coordinate frame. [Pg.490]

In order to transform the interaction vector in the laboratory coordinate frame defined by the stationary magnetic field to the final coordinate frame F, defined by the jump axis, which is diagonal with the DD vector or aligned with the principal axis system of the CSA tensor, it may be useful to express the overall transformation as a series of transformations through intermediate coordinate frames N. This will enable the amplitude and frequency of each motion to be explicitly included in the correlation function. In the case of overall isotropic motion with diffusion constant D, the dipolar correlation function can be expressed as (Keepers and James, 1982)... [Pg.364]

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 ... [Pg.1290]

Theorists calculate cross sections in the CM frame while experimentalists usually measure cross sections in the laboratory frame of reference. The laboratory (Lab) system is the coordinate frame in which the target particle B is at rest before the collision i.e. Vg = 0. The centre of mass (CM) system (or barycentric system) is the coordinate frame in which the CM is at rest, i.e. v = 0. Since each scattering of projectile A into (v[i, (ji) is accompanied by a recoil of target B into (it - i[/, ([) + n) in the CM frame, the cross sections for scattering of A and B are related by... [Pg.2007]

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... [Pg.2063]

For a shock wave in a solid, the analogous picture is shown schematically in Fig. 2.6(a). Consider a compression wave on which there are two small compressional disturbances, one ahead of the other. The first wavelet moves with respect to its surroundings at the local sound speed of Aj, which depends on the pressure at that point. Since the medium through which it is propagating is moving with respect to stationary coordinates at a particle velocity Uj, the actual speed of the disturbance in the laboratory reference frame is Aj - -Ui- Similarly, the second disturbance advances at fl2 + 2- Thus the second wavelet overtakes the first, since both sound speed and particle velocity increase with pressure. Just as a shallow water wave steepens, so does the shock. Unlike the surf, a shock wave is not subject to gravitational instabilities, so there is no way for it to overturn. [Pg.18]

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. [Pg.526]

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... [Pg.379]

We present here some very general exact results, which hold for arbitrary reorientation mechanisms of any molecule in an equilibrium isotropic fluid (but not a liquid crystal). A coordinate frame (R) is rigidly attached to the molecule of interest. Its orientation in the laboratory frame (L) is defined by the Euler rotation = (affy) that carries a coordinate frame from coincidence with the laboratory frame L to coincidence with the molecular frame R/ The conditional probability per unit Euler volume [( (0r at time t must depend only on the Euler rotation A = 1 (i.e., rotate first by < 0 then... [Pg.145]

The orientation of the coordinate frame R fixed in a particular rod with respect to the laboratory frame L is specified by the Euler rotation, = (afiy), as before. Under the preceding assumptions, a diffusion equation for the probability density [/(if, /)] is derived,<29) namely,... [Pg.151]

A sufficiently rarefied gas, or a mixture of gases, consists of a number of neutral molecules of species 1 and 2 (which may or may not be the same). We may assume a distribution of velocities (measured in the laboratory frame), fi ( ) d3u, that may be modeled by a Maxwellian distribution function, with i = 1 or 2, as long as the duration of the average collision is short compared to the time between collisions. For binary collisions, one usually transforms from laboratory coordinates, Vj, to relative ( >12) and center-of-mass (1>cm) velocities,... [Pg.29]

Whereas the group jr and its representations are relevant and sufficient for problems which are completely defined by relative nuclear configurations (RNCs) of a SRM, primitive period isometric transformations have to be considered as nontrivial symmetry operations in all those applications where the orientation of the NC w.r.t. the frame and laboratory coordinate system is relevant, e.g. the rotation-internal motion energy eigenvalue problem of a SRM. Inclusion of such primitive period operations leads to the internal isometric group ( ) represented faithfully by... [Pg.15]

Using primed coordinates in the laboratory-centered frame to help comparisons with Pack-Hirschfelder s work [8,15], the molecular hamiltonian... [Pg.28]

The product state distributions and the populations of the various chemical and electronic channels belong to the category of so-called scalar properties which are defined without reference to a particular coordinate frame. They have a magnitude but no direction. However, since the electromagnetic field vector E of the photolysis laser defines a specific direction in the laboratory frame, all vectors inherent to a photodissociation process can be measured relative to E. The vectors of interest are ... [Pg.15]


See other pages where Laboratory coordinate frame is mentioned: [Pg.220]    [Pg.6]    [Pg.324]    [Pg.483]    [Pg.55]    [Pg.319]    [Pg.75]    [Pg.231]    [Pg.356]    [Pg.560]    [Pg.324]    [Pg.69]    [Pg.591]    [Pg.220]    [Pg.6]    [Pg.324]    [Pg.483]    [Pg.55]    [Pg.319]    [Pg.75]    [Pg.231]    [Pg.356]    [Pg.560]    [Pg.324]    [Pg.69]    [Pg.591]    [Pg.24]    [Pg.41]    [Pg.381]    [Pg.52]    [Pg.259]    [Pg.260]    [Pg.263]    [Pg.64]    [Pg.88]    [Pg.89]    [Pg.346]    [Pg.5]    [Pg.320]    [Pg.346]   
See also in sourсe #XX -- [ Pg.297 ]




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