Displacement distribution function

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] ... 

Applying the Gaussian approximation, i.e. assuming that the atomic displacement distribution function are Gaussian Eq. 2.6 transforms into ... 

Table 4.2 Two-dimensional displacement distribution function W on W 110 heating periods were 60 s each experimental values were averaged over equivalent directions. Theoretical values calculated from the experimental mean square displacments are listed in... 

The factor R may be called a diffusion anisotropy factor for the surface. For diffusion of a W on the W (110), Tsong Casanova find a diffusion anisotropy factor of 1.88 from a set of data taken at 299 K, and of 2.14 from a set of data taken at 309 K. The average is 2.01, which agrees with the theoretical value, 2, to within 0.5%. A more detailed general analysis has since then been reported,137 and diffusion anisotropy on the W (110) surface has also been observed in field emission experiments.138 It should be noted, however, that the same ratio can be expected if an adatom jumps instead in the [001] and [110] directions with an equal frequency. Thus a measurement of surface diffusion anisotropy factor alone does not establish uniquely the atomic jump directions. The atomic jump directions can, of course, be established from a measurement of the displacement distribution function in two directions as discussed in the last section. Such measurements can only be done with atomic resolution field ion microscopy. 

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... 

Another interesting and powerful approach for obtaining detailed information on the flow field inside porous materials (without imaging) is to study the displacement distribution function, which can be... 

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).

It should be shifted to shorter times as indicated by the dashed lines. The kernel once again moves to the right and at reasonably short times will begin to overlap the displaced distribution function this accounts for... 

We have solved the set of integral equations on the pair distribution functions by discretizing them. This is equivalent to allowing the atomic displacements to finite number of points. When they are discretized, to solve them is a straightforward application of the exsisting CVM. 

In conclusion, we have presented a new formulation of the CVM which allows continuous atomic displacement from lattice point and applied the scheme to the calculations of the phase diagrams of binary alloy systems. For treating 3D systems, the memory space can be reduced by storing only point distribution function f(r), but not the pair distribution function g(r,r ). Therefore, continuous CVM scheme can be applicable for the calculations of phase diagrams of 3D alloy systems [6,7], with the use of the standard type of computers. 

Fig. 4) for relatively long chains37. However, under conditions of molecular orientation, the distribution function is usually displaced towards higher fi and the analysis of the crystallization process should be carried out over a wide range of fi values. 

Figure 11 shows that the molecular weight distribution in the melt (presence of short chains) can account for the coexistence of two types of crystals in the absence of molecular orientation or at a slight stretching of the melt. However, there is a purely thermodynamic reason for the appearance of this main structural feature of samples crystallized under conditions of molecular orientation, even at high degrees of orientation, when virtually the whole distribution function is displaced into the region of /S > /3cr. 

Figure 14 shows the displacement of the distribution function towards high / , i.e. the uncoiling of molecules under the influence of stretching for polyethylene (A = 3 x 10-9 m, N = 100 and T = 420 K). This displacement will be characterized by the position of the maximum of the distribution curve, the most probable value of / , i.e. j3m, as a function of x (Fig. 15). Figure 15 also shows the values of stresses a that should be applied to the melt to attain the corresponding values of x (o = xkT/SL, where S is the transverse cross-section of the molecule). 

It will be assumed for the moment that the non-bonded atoms will pass each other at the distance Tg (equal to that found in a Westheimer-Mayer calculation) if the carbon-hydrogen oscillator happens to be in its average position and otherwise at the distance r = Vg + where is a mass-sensitive displacement governed by the probability distribution function (1). The potential-energy threshold felt is assumed to have the value E 0) when = 0 and otherwise to be a function E(Xja) which depends on the variation of the non-bonded potential V with... 

Leadbetter AJ, Norris EK (1979) Molec Phys 38 669. There are different contributions which give rise to a broadening a of the molecular centre of mass distribution function f(z). The most important are the long-wave layer displacement thermal fluctuations and the individual motions of molecules having a random diffusive nature. The layer displacement amplitude depends on the magnitude of the elastic constants of smectics ... 

Fig. 78.—Radial distribution function W r) of the chain displacement vectors for the same polymer chains as in Fig. 77. W(r) is expressed in A"h...

With the availability of lasers, Brillouin scattering can now be used more confidently to study electron-phonon interactions and to probe the energy, damping and relative weight of the various hydro-dynamic collective modes in anharmonic insulating crystals.The connection between the intensity and spectral distribution of scattered light and the nuclear displacement-displacement correlation function has been extensively discussed by Griffin 236). 

Figure 9.4 Radial distribution functions for liquid (left) and amorphous (right) InP. In each case, distribution functions for In In, P P, In P atom pairs and for all atoms are shown. The zeros for the P P, In P, and total distribution functions are displaced vertically by 1, 2, and 3 units, respectively. (Reprinted by permission from the source cited in Fig. 9.2.)...

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