Spin displacement distribution

We define the spin-displacement density function, q(z, Z), so that the density of spins - the number of spins divided by the voxel volume - that have displacements between Z and Z + dZ in a voxel at z is q(z, Z)dZ. The density function q(z, Z) can be expressed in terms of local spin density p(z) and the normalized displacement distribution function P(z, Z) ... 

The spin-displacement density function, c (z, Z), and the normalized displacement distribution function, P(z, Z), can be converted readily into the joint spin-velocity density function, q(z, vn), and the normalized velocity distribution function, P(z, vn), respectively, with the net velocity vn defined as vn = Z/A. Once the velocity density function is determined for each of the volume elements, the superficial average velocity, v, is calculated by [23] ... 

In order to obtain spatial resolution of the molecular translations within each voxel, it is necessary to combine the velocity-encoding gradient sequence shown in Section 3.1.1 with a standard imaging sequence such as frequency or phase encoding. The displacement distribution function, P A (R), must now be generalized to a spatial-displacement joint density function Aso that it describes the total number of spins located at r with displacement R during time A 28... 

Figure 5 Sequences using stimulated echoes for flow velocity imaging (A) and tor the measurement of the displacement distribution function without imaging (B). The use of stimulated echoes often proves very convenient since the NMR signal can be observed for longer delays A, limited only by the T, value. The pair of gradient pulses, when exactly matched, produce no phase shift for nonmoving spins, and for moving spins they induce a phase shift A(j> equal to the product of the wavevector q = yGd, by the displacement (r(A) - r(0)). For a uniform velocity field v, (r(A) - r(0)) = Av everywhere in space and v can be found from the phase-shift measurement at a given value of q (A). In more complex flow situations, the full displacement distribution can be obtained from a Fourier transform analysis of data acquired with incremented G, and thus q, values (B).

One particular model of a biological system is a set of barriers in an otherwise homogeneous medium. If diffusion is observed over a short enough time, very little of the substance experiences the effect of the barriers, and the observed motion is characteristic of the medium alone. As the time of observation is extended, more of the substance is reflected at barriers and thus, its total displacement is less than would have been the case without the barrier bringing about less attenuation of the PG spin-echo [48, 49]. From an analysis of such data one can, in principle, obtain information on the geometry of the domains in which restricted diffusion occurs information on cell size, droplet size distribution [50] etc. 

Before turning to many-electron molecules, it is useful to ask Where does the energy of the chemical bond come from In VB theory it appears to be connected with exchange of electrons between different atoms but in MO theory it is associated with delocalization of the MOs. In fact, the Hellmann-Feynman theorem (see, for example, Ch.5 of Ref.[7]) shows that the forces which hold the nuclei together in a molecule (defined in terms of the derivatives of the total electronic energy with respect to nuclear displacement) can be calculated by classical electrostatics, provided the electron distribution is represented as an electron density P(r) (number of electrons per unit volume at point r) derived from the Schrodinger wavefunction k. This density is defined (using x to stand for both space and spin variables r, s, respectively) by... 

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