Root mean squared displacement

The situation becomes quite different in heterogeneous systems, such as a fluid filling a porous medium. Restrictions by pore walls and the pore space microstructure become relevant if the root mean squared displacement approaches the pore dimension. The fact that spatial restrictions affect the echo attenuation curves permits one to derive structural information about the pore space [18]. This was demonstrated in the form of diffraction-like patterns in samples with micrometer pores [19]. Moreover, subdiffusive mean squared displacement laws [20], (r2) oc tY with y < 1, can be expected in random percolation clusters in the so-called scaling window,... 

NMR Self-Diffusion of Desmopressin. The NMR-diffusion technique (3,10) offers a convenient way to measure the translational self-diffusion coefficient of molecules in solution and in isotropic liquid crystalline phases. The technique is nonperturbing, in that it does not require the addition of foreign probe molecules or the creation of a concentration-gradient in the sample it is direct in that it does not involve any model dependent assumptions. Obstruction by objects much smaller than the molecular root-mean-square displacement during A (approx 1 pm), lead to a reduced apparent diffusion coefficient in equation (1) (10). Thus, the NMR-diffusion technique offers a fruitful way to study molecular interactions in liquids (11) and the phase structure of liquid crystalline phases (11,12). 

For proteins the X-ray structures usually are not determined at high enough resolution to use anisotropic temperature factors. Average values for B in protein structures range from as low as a few A2 for well-ordered structures to 30 A2 for structures involving flexible surface loops. Using equation 3.6, one can calculate the root mean square displacement fu2 for a well-ordered protein structure at approximately 0.25 A (for B = 5 A2) and for a not-so-well-ordered structure at... 

It is of considerable practical importance to have some idea of how far an atom or ion will diffuse into a solid during a diffusion experiment. An approximate estimate of the depth to which diffusion is significant is given by the penetration depth, xp, which is the depth where an appreciable change in the concentration of the tracer can be said to have occurred after a diffusion time t. A reasonable estimate can be given with respect to the root mean square displacement of the diffusing... 

The statistics of the normal distribution can now be applied to give more information about the statistics of random-walk diffusion. It is then found that the mean of the distribution is zero and the variance (the square of the standard deviation) is na2), equal to the mean-square displacement, . The standard deviation of the distribution is then the square root of the mean-square displacement, the root-mean-square displacement, + f . The area under the normal distribution curve represents a probability. In the present case, the probability that any particular atom will be found in the region between the starting point of the diffusion and a distance of J (the root-mean-square displacement) on either side of it, is approximately 68% (Fig. 5.6b). The probability that any particular atom has diffused further than this distance is given by the total area under the curve minus the shaded area, which is approximately 32%. The probability that the atoms have diffused further than 2f is equal to the total area under the curve minus the area under the curve up to 2f. This is found to be equal to about 5%. Some atoms will have gone further than this distance, but the probability that any one particular atom will have done so is very small. 

Describe an experiment to determine Avogadro s number from the average root mean square displacement of a particle due to random walk. 

Table 65(a) Comparison of the Nitrogen Atom Thermal Motion for MI2MII[M(N02)IiJ Systems, Root-mean-square Displacements (A)540... 

The calculated root-mean-square displacement for a general sequence of jumps has two terms in Eq. 7.31. The first term, NT(r2), corresponds to an ideal random walk (see Eq. 7.47) and the second term arises from possible correlation effects when successive jumps do not occur completely at random. 

Consider sensory hairs 2 pm in diameter that are 20 pm apart. Inserting these values into equation (21.22) and assuming that D is 2.5 x 10 6 m2/s (the diffusion coefficient for bombykol, the main component of the commercial silkmoth sex pheromone Adam and Delbriick, 1968) results in the prediction that these hairs are likely to interfere with each other s odorant interception when the air speed between the hairs is below 0.0125 m/s. This is not a discontinuous function - the sensory hairs will interfere with each other more at slower speeds and less at faster speeds. Another way of appreciating what this means quantitatively is to recognize that the root mean square displacement of a molecule (considering movement in one dimension) is... 

Here, aL is the root-mean-square displacement, ACp is the heat capacity jump... 

Here (uj is the root-mean-square displacement of atoms along a direction of weak coupling, that is = (f4c f°r layered crystals (if... 

In multilayered lattices, even in such ones, for which the macroscopic characteristics are not distinguished by an appreciable anisotropy (as, for example, HTSC type 1-2-3), the interaction between separate atoms or atomic groups can be strongly anisotropic. The "damping" interaction propagation between layers inherent in substances of the specified class may result in appreciable manifestation of such local anisotropy both in the phonon spectrum [15] and in the behaviour of some vibrationary characteristics, in particular the root-mean-square displacement of atoms from separate layers along various crystal directions. 

Some multilayered HTSC, for example Bi2Sr2CaCu20x, show an anisotropy of the elastic moduli inherent for layered crystals, and negative thermal expansion in a direction within the layer [16], which can be described by formula (2). At the same time for multilayered structures such as HTSC 1-2-3, where the interlayer interaction between all layers is of the same order, the intralayer interaction essentially varies from one layer to another layer. Local anisotropy of chain type is characteristic for layers with weak intralayer interaction (a layer of the rare earth and a layer of chains Cu-O). In these layers the root-mean-square displacement of atoms in a direction within the plane is beyond the classical limit at lower temperatures, and is appreciably higher than the root-mean-square... 

Table 2. A comparison of the nitrogen atom thermal motion26 for M 2Mn[M(N02)6] systems, root-mean-square displacements(A), displacements along the M-N bonds are indicated by an asterisk ( )...

Teller systems the opposite will apply, Table 2 (c), and for the pseudo compressed Jahn-Teller systems, Table 2(d), which involve a static distortion along Cu-N(l), but a dynamic distortion along Cu-N(2) and Cu-N(3), smaller root-mean-square displacements are predicted to lie along the Cu-N bonds for N (1), and larger root-mean-square displacements for N(2) and N(3). These predictions are clearly born out for A2Pb-[Cu(N02)6], [A Cs, Rb], but only partially for the less accurate data for K2Pb[Cu(N02)6] (276 K). 

When a particle moves in brownian motion, the chance that it will ever return to its initial position is negligibly small. Thus, there will be a net displacement with time of any single particle, even though the average displacement for all particles is zero. For example, during a short time interval one particle may move a distance sls another a distance s2 and so on. Some of these displacements will be positive, others negative some up, others down but with equilibrium conditions the sum of the displacements will be zero. It is possible to estimate the displacement of any particle in terms of its root-mean-square displacement. 

Example 9.6 Find the ratio of tlt such that the root-mean-square displacement estimated considering particle inertia (Eq. 9.28) is 10 percent less than the estimate when inertia is not considered (Eq. 9.19). 

Estimate the root-mean-square displacement for a 2-p.m silica dust particle (p = 2.65 g/cm3) over a 10-min period. 

Equation 6.33 states that the root-mean-square displacement is proportional to the square root of the number of jumps. For very large values of n, the net displacement of any one atom is extremely small compared to the total distance it travels. It turns out, that the diffusion coefficient is related to this root-mean-square displacement. It was shown independently by Albert Einstein (1879-1955) and Marian von Smoluchowski (1872-1917) that, for Brownian motion of small particles suspended in a liquid, the root-mean-square displacement, is equal to V(2Dt), where t is the time... 

Note that in Eq. 6.34 the mean-square displacement is used, rather than the root-mean-square displacement. For a one-dimensional random walk, the mean-square displacement is given by 2Dt, and for a two-dimensional random walk, 4Dt. Since the jump distance (a vector) is A, if the jump frequency is now defined as F = n/t (the average number of jumps per unit time), then on combining Eq. 6.33 and 6.34 gives ... 

Alternatively, Einstein s equation [6] for the relation between the root mean square displacement X and the diffusion coefficient D,... 

Both p and m can be expressed in terms of the basic quantities p (the encounter diameter), o (the root mean square displacement for relative diffusion motion), and v (the frequency of relative diffusion displacements) as [3]... 

The root mean square displacement due to Brownian motion in the time interval t is [7] ... 

Thus the root mean square displacement in 1 s for a 1 pm particle settling in water, viscosity 0.001 Pa s, at an absolute temperature 300 K is 0.938 pm this is almost the same as the distance settled under gravity by a quartz particle (density 2650 kg m" ) in 1 s (0.90 pm). A comparison of Brownian movement displacement and gravitational settling displacement is given by Fuchs [8]. For a size determination to be meaningful the displacement of the particles due to Brownian diffusion must be much smaller than their displacement due to gravity, hence the condition ... 

The molecular root mean square displacement, r t)), of the diffusing molecules during the observation time, t, has to be much smaller than the crystal radius, R, in order to guarantee that the measured r.m.s. displacement reflects the undisturbed intracrystalline self-diffusion. Assuming... 

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