Expectation value field-dependent

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

Note that we have made use of the fact that for a homogeneous flow with an isotropic filter (u U ) = 0. More generally, the conditional expected value of the residual velocity field will depend on the filter choice. 

The expected value on the left-hand side is taken with respect to the entire ensemble of random fields. However, as shown for the velocity derivative starting from (2.82) on p. 45, only two-point information is required to estimate a derivative.14 The first equality then follows from the fact that the expected value and derivative operators commute. In the two integrals after the second equality, only /u,[Pg.264]

Figure Y shows the H NMRD profiles of water solutions of Fe(H20)g in 1 M perchloric acid at 298 K and in a glyceroTwater mixture (36). Only one dispersion is observed at about Y MHz. It corresponds to a correlation time Tc 3 X 10 s. The small increase of relaxivity above 20 MHz indicates that a field dependent is influential in the determination of at high field (see also Section II. C). From the fit to the SBM theory, is estimated to be around 5 x 10 s at room temperature, a value commonly found for small complexes in water solution and of the order expected for the mean lifetime between collisions with solvent molecules. The fit also provides a value for A = 0.095 cm , so that t o is calculated to be 9 x 10 s at room temperature. By increasing the viscosity through glycerol water mixture, it is shown that the relative influence of tb on with respect to becomes lower and lower with the increase in relaxivity in the high-field region being more and more evident. The fit of the profile acquired in the glycerol solution, performed by assuming that r, Ag, and A are not affected by the presence...

The NMRD profile of the hexaaqua nickel(II) complex is independent of the magnetic field up to about 100 MHz, as shown in Fig. 15. An increase in the proton relaxivity is then observed for higher fields, ascribed to a field dependence of the electron relaxation time, caused by fluctuations of the zero-field splitting. A value of Tig around 3 x 10 s can be estimated from the SBM theory, or, more appropriately, around 10 s if ZFS, expected to be around 3 cm 52), is taken into account in the fit. Data have been analyzed using the slow-motion theory (see Section IV of Chapter 2), which does not provide any value for the electron relaxation time, and indicates... 

Fig. 14. R.f. field dependence of the C-13 T, times. The T, values havfe been normalized by Tch (at 1 kHz spinning). The broken line estimates the field variation expected if the observed rotating frame relaxation were exclusively determined by spin-spin coupling. The dashed line represents the same field dependence and has been drawn through the 32 kHz data as an even more restrictive estimate there is no evidence whether or not the low field data are determined exclusively by spin-spin effects. As the relaxation times at 43 and 66 kHz are shorter than those predicted for purely spin-spin effects, the high field results (and perhaps even at 32 kHz) indicate molecular motion 62>.

Equation (13) shows that the complete temperature and field dependence of the strains can be calculated from static correlation functions (J Jj )7-,h (y, y — 1.2,3 label the cartesian components of the angular momentum J) where O7- h denote thermal expectation values (Callen and Callen 1965). As already mentioned above, a mean field theory may be used to evaluate (13) and calculate the magnetostriction. 

The general concepts of inverse control are simple to grasp, Suppose we specify, a priori, the time dependence of the expectation value of some observable, (O) = y(r). The Schrodinger equation for the system with applied control field is just... 

The expectation value of the property A at the space-time point (r, t) depends in general on the perturbing force F at all earlier times t — t and at all other points r in the system. This dependence springs from the fact that it takes the system a certain time to respond to the perturbation that is, there can be a time lag between the imposition of the perturbation and the response of the system. The spatial dependence arises from the fact that if a force is applied at one point of the system it will induce certain properties at this point which will perturb other parts of the system. For example, when a molecule is excited by a weak field its dipole moment may change, thereby changing the electrical polarization at other points in the system. Another simple example of these nonlocal changes is that of a neutron which when introduced into a system produces a density fluctuation. This density fluctuation propagates to other points in the medium in the form of sound waves. 

The H NMRD profiles of Mn(OH2)g+ in water solution show two dispersions (Fig. 5.43). The first (at ca. 0.05 MHz, at 298 K) is attributed to the contact relaxation and the second (at ca. 7 MHz, at 298 K) to the dipolar relaxation. From the best fit procedure, the electron relaxation time, given by rso = 3.5 x 10 9 and r = 5.3 x 10 12 s, is consistent with the position of the first dispersion, the rotational correlation time xr = 3.2 x 10 11 s is consistent with the position of the second dispersion and is in accordance with the value expected for hexaaquametal(II) complexes, the water proton-metal center distance is 2.7 A and the constant of contact interaction is 0.65 MHz (see Table 5.6). The impressive increase of / 2 at high fields is due to the field dependence of the electron relaxation time and to the presence of a non-dispersive zs term in the equation for contact relaxation (see Section 3.7.2). If it were not for the finite residence time, xm, of the water molecules in the coordination sphere, the increase in Ri could continue as long as the electron relaxation time increases. 

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