Mean square amplitude

The intensity of light scattering, 7, for an isolated atom or molecule is proportional to the mean squared amplitude... 

This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /iLorentzian distribution of the system s normal modes. In the absence of friction (2.27) describes thermally activated as well as tunneling processes when < 1, or fhcoo > 1, respectively. At first glance it may seem surprising... 

A similar treatment applies for the unstable regime of the phase diagram (v / < v /sp), where the mixture decays via spinodal decomposition.For the linearized theory of spinodal decomposition to hold, we must require that the mean square amplitude of the growing concentration waves is small in comparison with the distance from the spinodal curve. 

The isotropic equivalent thermal parameters are on the whole larger than in the PbTX-1 dimethyl acetal structure or the structure of the natural product. The B values for atoms on the fused ring skeleton range from 4.7 to 12.6 A (mean square amplitudes of 0.059 and 0.16 A ). Curiously, the largest values are associated with C17-C20 of the 9-membered E ring—the ring that adopts two conformations in crystalline PbTX-1. The acyclic atoms do not have appreciably higher thermal parameters, with the exception of hydroxyl 013, which has a B of 22.4 A 2. 

Other information that can be obtained from a wide-angle diffraction pattern includes the detection of stacking faults and microstrains, the lattice parameters, and the mean-square amplitude of vibratioiu... 

Reduction by hydrogen completely alters the chemical reactivity and its variation with size. At the same time, the Pt particle size is reduced. The mean-square amplitude of vibration follows this reactivity. 

An important early paper on fluctuation processes is that of Harry Nyquist (1928), who suggested an equation linking the mean-square amplitude of thermal noise in an electrical circuit to the resistance R of the noise EMF (or current) generator ... 

In their first paper on noise, Tyagai and Luk yanchikova (1967) considered a very simple redox reaction of the type of O + ze = R . According to these authors, when the reaction rate is limited by a slow-discharge step, the mean-square amplitude of the noise is given by... 

When it is limited by slow diffusion of the reactants, the mean-square amplitude is... 

According to these equations, in kinetically controlled reactions the mean-square amplitude is about 10 V, while in reactions occurring under diffusion control it is almost an order of magnitude smaller. Thus, the size of electrochemical (thermal) equilibrium fluctuations is extremely small. 

The mean-square amplitude of nonequilibrium noise, quite in contrast to that of equilibrium thermal noise, may reach rather high values for instance, hundreds of millivolts during anodic polarization of semiconductors (Parkhutik and Timashev, 2000). 

With h 6) - 1/sin 0)5(0 — Oq), one obtains the same result as given by (4.58), which implies that the anisotropy of the/factor cannot be derived from the intensity ratio of the two hyperfine components in the case of a single crystal. It can, however, be evaluated from the absolute/value of each hyperfine component. However, for a poly-crystalline absorber (0(0) = 1), (4.66) leads to an asymmetry in the quadrupole split Mossbauer spectrum. The ratio of l-Jh, as a function of the difference of the mean square amplitudes of the atomic vibration parallel and perpendicular to the y-ray propagation, is given in Fig. 4.19. 

Fig. 12.1 (continued) (b) Mean square amplitudes of hypothetical nondissociating HCI (squares) and DCI (circles) as a function of temperature... 

Fig. 12.1 (continued) (c) Isotope effects on mean square amplitudes (upper curve) and root mean square amplitudes (lower curve) as a function of temperature for hypothetical nondissociating molecules. At low temperatures the molecules are in the ground state and the amplitude is nearly independent of temperature. At higher temperature the vibrational amplitudes increase due to excitation into upper levels (Fig. 12.1) but the ratios drop smoothly to the classical value of unity at very high temperature (Fig. 12.1)... 

Vibrational amplitude effects for diatomic molecules can be straightforwardly calculated from spectroscopic data. Mean amplitudes and mean square amplitudes for diatomics are expressed in Equations 12.8a and 12.8b (see, e.g. Levine 1975)... 

Cyvin S. J., Molecular Vibrations and Mean Square Amplitudes, Elsevier, Amsterdan, 1968... 

While the Rouse model predicts a linear time evolution of the mean-square centre of mass coordinate (Eq. 3.14), within the time window of the simulation t<9 ns) a sublinear diffusion in form of a stretched exponential with a stretching exponent of (3=0,83 is found. A detailed inspection of the time-dependent mean-squared amplitudes reveals that the sublinear diffusion mainly originates from motions at short times t[Pg.39]

Mean square amplitudes of vibration were calculated by Baran (14) based on the frequencies and estimated geometry reported by Smith et al. (270). The UV absorption spectrum of FCIO2 was studied by Sicre and Schumacher (264) and Pilipovich et al. (228). From a mass spectro-... 

Thus, in the high-temperature limit, the mean-square displacement of the harmonic oscillator, and therefore the temperature factor B, is proportional to the temperature, and inversely proportional to the frequency of the oscillator, in agreement with Eq. (2.43). At very low temperatures, the second term in Eq. (2.51a) becomes negligible. The mean-square amplitude of vibrations is then a constant, as required by quantum-mechanical theory, and evident in Fig. 2.5. 

The frequency vD at the edge of the Brillouin zone is thus equal to vsqD/2n. The Debye temperature 0D is defined as hvD/(kB). As shown below, 0D is an inverse measure for the vibrational mean-square amplitudes of the atoms in a crystal at a given temperature. 

FIG. 2.6 Temperature dependence of the rigid-body translational mean-square amplitudes of quinolinic acid. The line represents the results from Eq. (2.51) with v = 44 cm. Source Takusagawa and Koetzle (1979). 

A better alternative is to use the difference structure factor AF in the summations. The electrostatic properties of the procrystal are rapidly convergent and can therefore be easily evaluated in direct space. Stewart (1991) describes a series of model calculations on the diatomic molecules N2, CO, and SiO, placed in cubic crystal lattices and assigned realistic mean-square amplitudes of vibration. He reports that for an error tolerance level of 1%, (sin 0/2)max = 1-1.1 A-1 is adequate for the deformation electrostatic potential, 1.5 A-1 for the electric field, and 2.0 A 1 for the deformation density and the deformation electric field gradient (which both have Fourier coefficients proportional to H°). 

---