Position vectors

In general, p (r) is a function of the vector position of the point of observation r. However, if one is concerned mainly with the inhomogeneity of the confined fluid in the normal (z) direction, the average over the interfacial area is adequate. Averaging yields... 

Point charge 0a is located at vector position fa and point charge Qb is at vector position fb. The vector joining to Qb is also shown fb - Fa points in the direction from Qa to Qb. 

The physical significance of 1/ab is that it represents the work done in bringin up (2b from infinity to the point whose position vector is Tb, under the influenc of (2a which is fixed at Ta- Because of the symmetry of the expression, it als represents the work done in bringing up point charge (2a from infinity to th point with vector position Ta, under the influence of point charge Qb which fixed in space at position Tb-... 

Here the symbol q, indicates the 3-dimensional vector position of the f particle. To apply Hilbert space concepts to this theory we now postulate a one-to-one correspondence between the rays of and the points of the Schrodinger 3N hyperspace. Thus there exists a continuum of normalized eigenvectors in represented by the symbol 4i.4a.". > ... 

Following is a pictorial vector representation of a doublet. It shows the evolution of the components of a signal. Draw vector positions in the fourth and fifth frames, along with their directions of rotation. 

M and are magnetization components of the X vector of a heteronuclear AX spin system, shown here at a certain delay after the application of a 90° pulse. Draw the vector positions and their direction of rotation after each of the following radiofrequency pulses ... 

The disturbance carries momentum and energy, but by a mechanism that seems to be inconsistent with the accepted ideas of classical dynamics. According to Newton s second law the force exerted on a particle of mass m at vector position r, is F = mr. Seen from another frame of reference, moving at a constant velocity v, relative to the first, the position of the particle is perceived to be... 

The wave function, V /(r), is a function of the vector position variable r. To determine it at every point in space it is convenient to take advantage of the fact that the potential V(r) depends only on the scalar interatomic distance r. In spherical coordinates (Figure 1.2), the Laplacian operator V2 has the form... 

Fig. 4 Effect of nanoclay loading on neat SEBS a Lorentz -corrected SAXS profiles (vertically shifted for better clarity) showing effect of nanoclay arrows indicate peak positions, b Lengths corresponding to first- and second- order scattering vector positions along with the 2D SAXS patterns for each sample of clay-loaded nanocomposites...

Point charge QA is located at vector position rA and point charge (2b is at vector position rB. The vector joining QA to (2b is also shown rB — rA points in the direction from QA to (2b-... 

The flexible helix modeled here is best described by the entire array of conformations it can assume. A comprehensive picture of this array is provided by the three-dimensional spatial probability density function Wn(r) of all possible end-to-end vectors (25, 35). This function is equal to the probability per unit volume in space that the flexible chain terminates at vector position relative to the chain origin 0,as reference. An approximate picture of this distribution function is provided by the three flexible single-stranded B-DNA chains of 128 residues in Figure 5(a). The conformations of these molecules are chosen at random by Monte Carlo methods (35, 36) from the conformations accessible to the duplex model. The three molecules are drawn in a common coordinate system defined by the initial virtual bond of each strand. For clarity, the sugar and base moieties are omitted and the segments are represented by the virtual bonds connecting successive phosphorus atoms. 

Color cluster rotation Described in Section 6.6. The cloud of points is aligned with the gray vector, positioned at the center of the RGB cube, and rescaled to fill most of the cube using the inverse of the square root of the largest eigenvalue as a scaling factor. 

Hudson et al. [26] describes a method called the Most Descriptive Compound (MDC) method for selecting representative subsets and a sphere exclusion method for selecting sets of compounds that cover the available property space. The MDC method aims to select subsets that most effectively represent the compounds in the original collection. It operates by calculating a vector / of N elements where there are N compounds. For each compound, the other compounds are ranked in order of distance to it. The reciprocal of the rank of each compound n is then stored in vector position / . The process... 

Figure 8.12. Coordinate system defining the vector positions of the electrons (1) and nuclei (A or a), relative to the centre-of-mass.

Crystal lattice vectors Momentum Wave vector Position vector Velocity... 

The Cauchy and Finger tensors may be expressed in terms of the mapping fimction / and its derivatives [26,60], using the vector positions Xj. and Xx and... 

Expressing the vibration-dejjendent potential as a pair sum, where r, and are the vector positions of particles a and P,... 

A comparison between the values deduced from charge densities and those obtained with the vector position operator, using ah initio SCF wave functions of unsaturated heterocycles 107 3>, shows that the dipole... 

Notice that if we had taken the convention for the Fourier series Equation (44) that puts the global vector position of the atom, R/y = R/ + Xy, in the exponential term the Fourier coefficients T y could not be identified with magnetic moments because of the phase factor containing the atom positions. 

Let us apply this general philo.sophy to the grand potential for the system depicted schematically in Fig. 5.12. Clearly, Eq. (1.66) represents the (exact differential of the) grand potential for the situation shown in Fig. 5.12. Because the fluid substrate potential in Eq. (5.83) depends on the (vector) position r of a fluid molecule rather than a subset of Cartesian coordinates, it is immediately clear- that we are dealing with a fluid that is inhomogeneous in all three spatial directions. According to our discussion in Section 1.6.1,... 

Figure 1.5 Optically induced potential energy landscapes for four identical interacting particles as a function of the vector positions of one of them, when the other three are at (0, 0, 0), (0, 0, 3), and (0, 0, 6). Scale in lla k lAits c units. Polarization and wavevector as in Figure 1.3. Black circular shapes represent local divergences in energy shift in the proximity of the fixed particles. Copyright (2008) by the American Physical Society.

The unit cell group description of the normal modes of vibration within a unit cell, many of which are degenerate, given above is adequate for the interpretation of IR or Raman spectra. The complete interpretation of vibronic spectra or neutron inelastic scattering data requires a more generalized type of analysis that can handle 30N (N=number of unit cells) normal modes of the crystal. The vibrations, resulting from interactions between different unit cells, correspond to running lattice waves, in which the motions of the elementary unit cells may not be in phase, if ky O. Vibrational wavefunctions of the crystal at vector position (r+t ) are described by Bloch wavefunctions of the form [102]... 

The yth atom has a scattering factor fj and is located at the lateral vector position r, = x a -I- y b and at the vertical position Zj (in angstroms). Becanse the sample consists of a two-dimensional powder, the observed intensity 4 for a given (or 2Qf,p) and q position contains contribntions from both the (h,k) and (h,k) reflections, with possibly several h, k integer valnes. 

This chain, of fundamental importance, was studied in the preceding chapter. Its properties can be summarized as follows. The various points of the chain have, vector positions r with j = 0,.. ., N, and the vectors uy... 

We consider a set of N Gaussian chains. Each chain is labelled with an index j, with j = 1,..., IV. It will be assumed that all chains have the same number N of links and that the vector position of a point belonging to a chain j is denoted by the function ij(n) with 0 < n < N,... 

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