Rms displacement

The RMS displacement as a function of time for the shifted potential simulations are shown in Figure 3. As expected for this functional form, the longer cutoff distances result in a smaller RMS deviation from X-ray. The results for the 100 picosecond analysis section of all of the simulations are summarized in Table IV. For Table IV, the term "rdie" indicates that a distant dependant dielectric was used, cdie indicates that a constant dielectric was used, and eps2 indicates that the electrostatic forces have been scaled by 0.5. 

Figure 3. The RMS displacement for all atoms from the X-ray structure as a function of time for the simulations using the "shift" potential.

In order to verify the conditions of this averaging process, one has to relate the displacements during the encoding time - the interval A between two gradient pulses, set to typically 250 ms in these experiments - with the characteristic sizes of the system. Even in the bulk state with a diffusion coefficient D0, the root mean square (rms) displacement of n-heptane or, indeed, any liquid does not exceed several 10 5 m (given that = 2D0 A). This is much smaller than the smallest pellet diameter of 1.5 mm, so that intraparticle diffusion determines the measured diffusion coefficient (see Chapter 3.1). This intrapartide diffusion is hindered by the obstades of the pore structure and is thus reduced relative to D0 the ratio between the measured and the bulk diffusion coeffident is called the tortuosity x. More predsely, the tortuosity r is defined as the ratio of the mean-squared displacements in the bulk and inside the pore space over identical times ... 

We are often interested in the rms thermal displacements in A. They correspond to the contravariant components UJk along covariant axes of unit length, rather than along the non-unit length a, b, c axes. The rms displacements are obtained from... 

This shows that the mean displacement is simply not a useful parameter to characterize the trajectory of the particle and suggests we should seek an alternate quantity. Instead of averaging the displacements directly, suppose we first square them to eliminate the differences in sign, then average them and take the square root. This quantity, called the root mean square (rms) displacement, will give a better measure of the meanderings of the marker since the sign differences have been eliminated. 

What is the difference between the mean of the displacement and rms displacement Can they be equal under any circumstance ... 

What is the relation between the diffusion coefficient and the rms displacement ... 

Brownian fluctuations, inertia, nonhydrodynamic interactions, etc.) to lead to exponential divergence of particle trajectories, and (2) a lack of predictability after a dimensionless time increment (called the predictability horizon by Lighthill) that is of the order of the natural logarithm of the ratio of the characteristic displacement of the deterministic mean flow relative to the RMS displacement associated with the disturbance to the system. This weak, logarithmic dependence of the predictability horizon on the magnitude of the disturbance effects means that extremely small disturbances will lead to irreversibility after a very modest period of time. 

The value of the NMR spin-lattice relaxation time in each of the pixels of an image may be converted to a pore size by the adoption of a relaxation model. For a liquid imbibed in a pore space the relaxation rate is enhanced. This is believed to be due to interactions between a thin layer of liquid and the solid matrix at the solid/liquid interface increasing the relaxation rate. There is then also difllisional exchange between this surface-affected layer and the rest (bulk) of the liquid in the rest of the pore. In the case here where the pore sizes are several orders of magnitude smaller than the rms displacement of the probe water molecules employed, the "two-fraction fast-exchange" model of Brownstein and Tarr [7] will be used, where the overall measured value of T, is given by ... 

In addition, the polarity of the surface modes probably increases at a defect (either charge localisation in the midgap state or oxide bond formation would greatly increase the polarity of the modes (ctp) associated with the surface defect). The increase in polarity naturally leads to a larger modulation of the electron energy within the band and mixing of states for the same rms displacement. 

Figure 21. Normal-mode distribution rms average displacement of atoms within a residue for a 1-A rms displacement along selected modes.

Figure 37. Dihedral-angle variations for hinge mode (a) versus residue number (b) versus residue number (c) sidechain angles x1 versus residue number. Each bar corresponds to the change in the dihedral angle in going from the 1- A rms displaced closed structure to the 1-A rms displaced open structure.

The rms displacement of surface atoms in several cubic metals has been determined in this manner. In most cases the mean displacement component perpendicular to the surface plane, l x ), was obtained from low-energy electron-diffraction... 

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