Projected mean square displacement

The strongest contribution to the projected mean square displacement (ku)) and therefore to the absorption probability S(E) originates from C-Fe-C and N-Fe-C bending modes (8Ai, 22E, 23E, and 24E in Table 9.2). However, the energy range of these modes (8-15 meV) strongly overlaps with that of the acoustic modes (with composition factor = 0.17, = 337, wtpe = 57) and therefore... 

Single-chain m (k,t) and collective k,t) dynamical structure factors for projected dynamics in Eq. 113 reflect the fact that local density fluctuations around the tagged chain are relaxed by the projected motions both of the tagged chain and the matrix chains. The projected nature of the dynamics is hidden in Eqs. 114 and 115 in the projected mean squared displacement, which qualitatively describes typical distances over which elementary density fluctuations become dispersed around the tagged chain during the time t. 

To find the proper value of the projection parameter one can follow a self-consistent scheme based on the mean square displacement of the primitive path points (/)(s,s t) [22]. (j)(s,s-, t) is defined as... 

The relation between the means-square displacement (MSD) to the diffusion constant is valid in the physical limit of the observation time being larger compared with the mean-collision time (Haile 1991). We used Equation 10.6 to compute diffusivities through MD simulations. We computed diffusivity, D, along each direction using the projected Einstein equations ... 

A fully extended chain, without distortion of bond angles or deformation of bonds, is represented in Figure 9(b). In this conformation, the value of the end-to-end distance is nZp, where Zp is the length of the bond vector projected on the chain axis. If the equivalent random chain is now required to have the same end-to-end distance in full extension and also the same mean-square displacement length as the real chain, then equations (33) and (34) result. The maximum extension ratio of unperturbed macromolecules may be defined as the ratio of the fully extended length nZp to the root-... 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

The chain of subunit symmetry axis vectors (bond vectors) is projected onto a plane containing the first vector. In this plane, the mean squared angular displacement of the (m + 1 )th vector with respect to the first is(109) <<5, 2> = m + l kBT/k = RJP. Thus, we set... 

K, the static disorder is certainly maintained. The results are presented as plots of formula in Fig. 7. The deviations from linearity of the plots is small enough to support such method of analysis. The slopes of the curves give the 5a values tabulated in Table 4. It follows that in the (1 x l)Co/Cu(lll) case the anisotropy of surface vibrations clearly appears in the measured values of 8a and 5aT There are two reasons for such anisotropy the first is a surface effect due to the reduced coordination in the perpendicular direction. cF is a mean-square relative displacement projected along the direction of the bond Enhanced perpendicular vibrational amplitude causes enhanced mean-square relative displacement along the S—B direction. The second effect is due to the chemical difference of the substrate (Fig. 8). S—B bonds are Co—Cu bonds and the bulk Co mean-square relative displacement, cr (Co), is smaller than the bulk value for Cu, aJ(Cu). Thus for individual cobalt-copper bonds, the following ordering is expected ... 

Mean translational energy of dispersed particles ( ] ) is given by the equation = U2m.u = 3/2tg.T, where m = particle mass, = mean square speed of movement, = Boltzmann constant and T = absolute temperature. The movement ofparticles in acertain direction (e.g. along the x-axis) is characterised by the mean shift (A) A=[EA7/n], where A = projections of individual particle displacements on the given axis and n = number of projections. 

The essence of the first and second renormalization ansatzes to be described in the following refers to heuristic replacements of the mean squared segment displacement for projected dynamics, (R (())q. In the (once) renormalized Rouse model this unknown function is replaced by the result of the ordinary Rouse model, given in Eq. 58 [98]. In the... 

The reason for the numerous modifications originates in the structure, which is a three-dimensional array of comer-sharing metal-oxygen octahedra (Fig. 4.3). The idealized structure (cubic symmetry ReOs-type) is drawn in Fig. 4.4, showing the octahedra as squares in projection. The real stmcture is distorted. This means that the relatively small tungsten atoms tend to be displaced from the octahedral center and diese displacements are temperature-dependent. Moreover, the symmetry can be influenced by small amounts of impurities. 

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