Mean squared relative displacement

Note Gold neighbor, coordination number N), interatomic distance [i (A)], and mean square relative displacement... 

The quasi-ideality of the (1 x l)Co/Cu(lll) and (1 x l)Co/Cu(110) monolayer interfaces allows a temperature dependent study of the polarisation dependent Debye Waller damping of the EXAFS oscillations i.e the analysis of the amplitude of the mean square relative displacements of the Co atoms parallel to the adsorbate layer, or perpendicular to it. The results are based on the analysis of data collected with the sample temperature T = 77 K and T = 300 K. The S—S and S—B (see above)... 

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 ... 

The obtained Ao gi = 5.7 x 10 is even larger than the value of Acr (Cu) X (= 4.7 X 10 A ), and of the hypothetical Co—Cu crystal with intermediate elastic properties than bulk cobalt and copper (4.1 x lO" A ). The derived effect of the effect of the lower coordination of the surface atoms on the mean-square relative displacement (perpendicular vs. parallel motions) is 1.4 times larger amplitude of the perpendicular vs. parallel motions, in agreement with lattice dynamics calculations. This SEXAFS study has produced a measure of the surface effect on the atomic vibrations. This has been possible due to the absence of surface or adsorbate reconstruction (i.e. no changes in bond orientations with respect to the bulk) and of intermixing. 

The Debye Waller analysis of the S—B bonds gives A05 g2(HO) = 2.9 x 10 A. This value is lower than the pure Co value (3.6 x 10 A ). Due to the low density of the (110) face, one mi t have expected a large mean-square relative displacement. The measured small value reveals a stiffening of the force constant of the Co—Cu bond. This is consistent with the large eontraction of the Co—Cu interlayer distance (sell % see above). The stiffening in strongly relaxed surfaces has been observed before and overcompensate the effect of the reduced surface coordination in the perpendicular direction. Reversed surfaee anisotropy of the mean square relative atomic displacements has also been found on an other low-density surfaee C2 x 2 Cl/Cu(l 10) i.e. one half density of Cl vs. Cu(l 10) in plane density where the Cl atoms moves with amplitudes parallel to the surface eomparable with those of the Cu subtrate, but with a much reduced amplitude in the perpendicular direction... 

Results of best fits to SEXAFS modulations, recorded at normal incidence, from Al203(0001)lxl-Cu and Al203(0001)(V31xV31)R 9°-Cu at 0.5 ML equivalent coverage of Cu [110]. N is the effective coordination number of a shell of scatterers, R is its distance, and Aa2 the mean square relative displacement. 

Very recently, experimental results that indicate a direct connection between the local Cu-O bond fluctuation and occurrence of superconductivity have been obtained (Fig. 12) [16]. The mean squared relative displacement (MSRD) of Cu-O bond lengths in the CUO2 plane, cr o > shows an anomalous increase below T and a sudden decrease around T. Ibe 5% substitution of Cu by magnetic atoms, Co or Ni, suppresses superconductivity completely, and so does the anomalous behavior... 

Photoionization (and therefore EXAFS) takes place on a time scale that is very short relative to atomic motions, so the experiment samples an average configuration of the neighbors around the absorber. Thus, one needs to consider the effects of thermal vibration and static disorder, both of which will have the effect of reducing the EXAFS amplitude. These effects are considered in the so-called Debye-Waller factor which represents the mean-square relative displacement along the absorber-backscatterer direction and is given by... 

In Eq. (1), k is the photoelectron wave vector relative to Eq (k = 0) N is the the number of neighboring atoms of the same kind at a distance r., of is the mean-square relative displacement (MSRD) of the absorber-scatterer atom pair from their equilibrium inter-atomic distance or in molecular spectroscopy terminology, the mean-square amplitude of vibration other terms have their usual meaning Using standard Fourier transform and curve fitting procedures, we can derive the coordination number, bond length and local dynamics (MSRD) from EXAFS. 

The chemical implications of the EXAFS bond length and the mean-square relative displacement of the absorber-scatterer pair are best illustrated with a system of metal ions in aqueous solution and its correlation with the rate of water substitution reaction... 

RMSRD = Root-Mean-Square-Relative-Displacement... 

Table 2. EXAFS Bond Length", Root-Mean-Square Relative Displacement, Reorganization and Symmetric Vibrational Amplitude of Fe(HjO)j and...

As both particles are undergoing Brownian motion, suppose that in a time interval they experience displacements dv and dvi- Then their mean square relative displacement is... 

As an example of the capabilities of EXAFS spectroscopy, the mean Cd-S distances as a function of the diameters of a series of CdS nanocrystals are depicted in Figure 3.15 [163]. These data are gained from a thorough temperature-depen-dent study of the size dependence of various structural and dynamic properties of CdS nanoparticles ranging in size from 1.3 to 12.0 nm. The properties studied include the static and the dynamic mean-square relative displacement, the asymmetry of the interatomic Cd-S pair potential, with conclusions drawn as to the crystal structure of the nanoparticles, the Debye temperatures, and the Cd-S bond lengths. As seen from Figure 3.15, the thiol-stabilized particles (samples 1-7) show an expansion of the mean Cd-S distance, whereas the phosphate-stabilized particles (samples 8-10) are slightly contracted with respect to the CdS bulk values. 

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 ... 

Over the years several semi-empirical melting or freezing rules have emerged in an attempt to correlate SFE with other features of the solid or fluid phases [153]. A distinguishing characteristic of these rules is that they are all formulated in terms of properties of just one of the phases. The best known of these is the Lindemann rule [154], which states that the melting point of a solid correlates with the mean-squared atomic displacement. When this quantity exceeds a particular value, the solid is observed to melt. This rule is actually a loose statement about the mechanical stability of the solid, rather than a statement of its thermodynamic stability relative to the fluid. 

The absolute values of the squares of the structure amplitudes (F were determined for AIN in the temperature range 85-670 K using monochromatic Cu radiation. These values were used to calculate the mean-square dynamic displacements and the atomic scattering factors of the A1 and the N ions. The values of were used also to find the shortest relative distance (uq /c ) between the A1 and the N ions along the c axis. This distance was 0.386 0.001, which is different from 0.375 for a perfect structure (c/a = 1.633) and from 0.380 for the case of equal values of all the shortest atomic spacings (c/a = 1.600). The temperature dependences indicated that the mean-square dynamic displacements (u ) in AIN were anisotropic. Thus, at room temperature, these displacements were (0.30 0.02) 10" A, u = (0.65 0.03) 10 A ... 

The values of below the critical temperature, derived from the areas of the spectra, are associated with the local thermal motions of the individual iron atoms relative to their neighbours. The. total mean square displacements of the iron atoms above 7, derived from the areas of the narrow subspectra, were resolved into three components associated with different modes of thermal motions ., s and fc corresponding to local, slow collective and fast collective modes, respectively. This is shown in Figure 6.17 for metmyoglobin and ferritin. The value of ,o,. is obtained by a linear extrapolation from the values of below the critical temperature. The difference between and ioc gives the mean-square collective displacement associated with large-scale motions of parts of the surrounding protein and is resolved into the two parts and fc. The values of correspond to slow... 

Because of the relatively short displacement time or length scales typically probed by NMR diffusometry, it is particularly well suited to detect anomalies in the segment displacement behavior expected on a time scale shorter than the terminal relaxation time, that is for root mean squared displacements shorter than the random-coil dimension. All models discussed above unanimously predict such anomalies (see Tables 1-3). Therefore, considering exponents of anomalous mean squared displacement laws alone does not provide decisive answers. In order to obtain a consistent and objective picture, it is rather crucial to make sure that (i) the absolute values of the mean squared segment displacement or the time-dependent diffusion coefficient are compatible with the theory, (ii) the dependence on other experimental parameters such as the molecular weight are correctly rendered, and (iii) the values of the limiting time constants are not at variance with those derived from other techniques. 

Fig. 21 Mean-square displacement vs. evolution time for 16-mers with an occupation density of 0.9375 in a 32-sized cubic lattice. The triangles are for four middle chain units, the circles are for the mass center, and the crosses are for the chain units relative to the center of mass. The lines with slopes of 1.0 and 0.5 indicate the scaling expected according to the Rouse model of polymer chains [56]... 

K for myoglobin (Parak et al., 1981). Thus, measurements of (x ) at temperatures below this value should show a much less steep temperature dependence than measurements above, if nonharmonic or collective motions (whose mean-square displacement is denoted (x )c) are a significant component of the total (x ). Figure 21 illustrates the expected behavior of (x )v, x, and their sum for a simple model system in which a small number of substates are separated by relatively large barriers. In practice, the relative contributions of simple harmonic vibrations and coUective modes will vary from residue to residue within a given protein. 

The damping factors take into account 1) the mean free path k(k) of the photoelectron the exponential factor selects the contributions due to those photoelectron waves which make the round trip from the central atom to the scatterer and back without energy losses 2) the mean square value of the relative displacements of the central atom and of the scatterer. This is called Debye-Waller like term since it is not referred to the laboratory frame, but it is a relative value, and it is temperature dependent, of course It is important to remember the peculiar way of probing the matter that EXAFS does the source of the probe is the excited atom which sends off a photoelectron spherical wave, the detector of the distribution of the scattering centres in the environment is again the same central atom that receives the back-diffused photoelectron amplitude. This is a unique feature since all other crystallographic probes are totally (source and detector) or partially (source or detector) external probes , i.e. the measured quantities are referred to the laboratory reference system. 

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

Atoms taking part in diffusive transport perform more or less random thermal motions superposed on a drift resulting from field forces (V//,-, Vrj VT, etc.). Since these forces are small on the atomic length scale, kinetic parameters established under equilibrium conditions (i.e., vanishing forces) can be used to describe the atomic drift and transport, The movements of atomic particles under equilibrium conditions are Brownian motions. We can measure them by mean square displacements of tagged atoms (often radioactive isotopes) which are chemically identical but different in mass. If this difference is relatively small, the kinetic behavior is... 

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