Root mean squared width

As noted earlier, the diffraction of X-rays, unlike the diffraction of neutrons, is primarily sensitive to the distribution of 00 separations. Although many of the early studies 9> of amorphous solid water included electron or X-ray diffraction measurements, the nature of the samples prepared and the restricted angular range of the measurements reported combine to prevent extraction of detailed structural information. The most complete of the early X-ray studies is by Bon-dot 26>. Only scanty description is given of the conditions of deposition but it appears likely his sample of amorphous solid water had little or no contamination with crystalline ice. He found a liquid-like distribution of 00 separations at 83 K, with the first neighbor peak centered at 2.77 A. If the pair correlation function is decomposed into a superposition of Gaussian peaks, the area of the near neighbor peak is found to correspond to 4.23 molecules, and to have a root mean square width of 0.50 A. 

The four-atom molecules in Fig. 4.1 provide an excellent example of this direct connection between the moments of the eigenspectrum and the local topology. Firstly, as predicted by eqn (4.49), we see from Fig. 4.4 that the centre of gravity of all the eigenspectra is identically zero. Secondly, the root mean square width of the eigenspectrum is predicted by eqn (4.50) to be... 

The distribution of site energies (DOS, for density of states ) is generally assumed to be Gaussian, with an r.m.s. (root mean square) width several times larger than k T (the thermal energy). For qualitative purposes, some important consequences of such a DOS are the following. 

For a known, simple material at an interface, another strategy for generating a structural model is to generate p(z) as a sum of contributions, pj(z), from individual atoms at heights z. Each atom (or pseudo atom, say, CH2) in an aliphatic chain is modelled by, e.g, a Gaussian-shaped Pi(z) function (of root-mean-square width a) to smear out the 8 electrons as shown in Figure 7. 

Figure 7 The electron density profile p(z) (left) of a close-packed monolayer of arachidic acid (right). The electron density profile p(z) can be constructed either as the sum of densities of individual atoms (or pseudo atoms, e.g., CH2), represented by Gaussian-shaped Pi(z) functions of root-mean-square width a (full lines) or as the sum of two slabs of constant densities (dotted lines). In either case the subphase contributes to the total p(z). The electron density due to the slabs is smeared by a roughness a, giving the dashed lines that are barely distinguishable ii om the full line. Typically o is of order 3 A but for display purposes, the Figure was constructed using o = 1 A). Adapted from ref. [16],...

The theoretical prediction is supported by the experiments. Patterns that spontaneously form from the uniform state have multiple domains with different characteristic angles. The root-mean-square width A rms of the observed angular distribution function changes with the bifurcation parameter in qualitative accord with theory compare Figure 8b with Figure 8a [13]. A quantitative comparison of experiment and theory would require an evaluation of the coefficients in the Landau-Ginzburg equation from the chemical kinetics and diffusion coefficients of the reactants [47]. 

Fig. 8. Predicted and observed bifurcation diagrams for the transition from a uniform state to rhombic patterns, (a) The band of allowed values ofA6 = 0- 60° for linearly stable rhombs, as a function of the bifurcation parameter n. The bifurcations from the uniform state to rhombic patterns (or regular hexagons, 0 — 60°) and from rhombic patterns (or hexagons) to stripes are both slightly subcritical. (b) The root-mean-square width of the distribution, A rms, of angles measured for the observed Turing patterns, as a function of malonic acid concentration. Other control parameters were the same as in Figure 3b. (From [13])...

The r.m.s. (root mean square) width a contains infonnation about the power distribution within the pulse. This is useful when the pulse is no longer rectangular and cannot be described just by its overall width. To evaluate a we define to be the mth moment of the impulse response, whence we deduce from Eq. (4-55) that [6]... 

Although the power spectral density contains information about the surface roughness, it is often convenient to describe the surface roughness in terms of a single number or quantity. The most commonly used surface-finish parameter is the root-mean-squared (rms) roughness a. The rms roughness is given in terms of the instrument s band width and modulation transfer function, M(p, q) as... 

Fig. 16 Root-mean-square-error (RMSE) plots for the deconvolutions of Fig. 15. The response-function widths are (a)7, (b) 7.5, (c) 8, (d) 8.5, (e) 9, and (f) 9.5 points.

At constant temperature, the observed widths of the spectral functions decrease with increasing mass of the collisional pair. This fact is a simple consequence of the mean translational energy of a pair, jm v = kT, which is the same for all pairs. The interaction time is roughly proportional to the reciprocal root mean square speed, and thus to the square root of the reduced mass. 

A particularly interesting case is when the inhomogeneous width is comparable to the root-mean-square of the fluctuation amplitude (A2n < 2KkBT). In this case the interpretation of the simple two-pulse echo measurement is ambiguous [39, 40]. Again, in the classical limit, in this region,... 

To find the width of the distribution, we evaluate the measured average (mean) concentration and the root-mean-squared deviation from the average ... 

We used p instead of = in Equation 5.37 because the exact numerical value depends on the definition of the uncertainties—you will see different values in different books. If we define At in Figure 5.13 as the full width at half maximum or the root-mean-squared deviation from the mean, the numerical value in Equation 5.37 changes. It also changes a little if the distribution of frequencies is not Gaussian. Equation 5.37 represents the best possible case more generally we write... 

When one has obtained the local propagator for each pbcel, the details of the local motion can be easily calculated. For example, the width of P(Z, A) is determined by the root mean square Brownian motion (2DA), while the displacement of P(Z, A) along the z-axis is determined by the... 

If the strain field is not homogeneous on the length scale of the crystallite size or smaller, according to Equation (9), different parts of the material diffract at slightly different angles, thus producing a broadened profile. Profile width and shape will evidently depend on the strain distribution across the sample. Considering the root mean square strain (or microstrain), Equation... 

However, while the SFD method is the best of the approximate procedures considered, the level energies it yields are typically in error by more than a full level width, and the root mean square relative error associated with its widths is ca. 352. Thus, while the SFD method does qualitatively explain the bulk of the differences among the various values in Table 11, its inadequacies are unfortunately too large to allow its predictions to be the basis of attempts to refine the potential energy surface. A more detailed discussion of the nature and weaknesses of the various approximate methods may be found in Ref.(19). 

There are two types of noise relevant to chromatographic-determinations. Detector, or electrical, noise is the random fluctuation of the baseline signal in the presence of mobile-phase flow. Noise values can be reported as either peak-to-peak values, or as a root-mean-square (rms) value. The rms value can be estimated easily as A the peak-to-peak value. This estimation follows from statistical considerations noise is a randomly occurring phenomenon, and as such the values should follow Gaussian statistics. Ninety-nine percent of the values should then fall within the mean value 2.5 standard deviations. It has been recommended that the baseline region measured be sufficiently wide as to encompass at least 20 base widths of the analyte peak (23). The measurement of noise in chromatographic systems has been addressed in detail (23). 

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