Root-mean-square amplitude

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)... 

T12S04, Cs2W04, Cs2S04 and Cs2Mo04 all385 possess a structure which is presumably close to the D2(i structure (86). The extraordinary large root-mean-square amplitudes of vibration observed for these four compounds have been inter-... 

Fig.117 Stacked plots of 13C exchange NMR spectra taken with tm = 50 ms. a T= - 40 °C. b T = 60 °C. c Simulated diagonal powder spectrum corresponding to a. b Simulated spectrum of b obtained by considering rotation around the 033 direction with a root-mean-square amplitude of 12° (from [77])...

Figure 7.5 Operating principle of an LVDT. The dotted line represents the output of the left secondary, the dot-dashed line is the output of the right secondary. The solid line is their sum. The root mean squared amplitude of the solid line represents the core position.

Long, D. A., and E. A. Seibold Root-Mean-Square Amplitudes of Vibration in Some Group 4 Tetrahalides. Trans. Faraday Soc. 56, 1105-1109 (1960). 

Shot noise — originates from the movement of charge carriers when they cross n-p junctions or arrive at an electrode interface. It is much smaller than thermal noise and depends on the signal as follows C/shot = R(2IeA/) /2. [/shot is the -> root mean square amplitude... 

Flicker noise — is common to all solid-state devices and predominates in measurements at frequencies, / < 300 Hz. Although the physical origin of this noise is not well understood, it can be described by the following empirical equation [/fiicker = (KI2If)1/2, [/flicker is the - root mean square amplitude of this noise, K is a constant depending on factors such as resistor materials and geometry, and I is the DC current [i]. 

On a molecular scale liquid surfaces are not flat, but subject to Jluctuations. These irregularities have a stochastic nature, meaning that no external force is needed to create them, that they cannot be used to perform work and are devoid of order. Their properties can only be described by statistical means as explained in sec. 1.3.7. Surface fluctuations are also known as thermal ripples, or thermal waves, in distinction to mechanically created waves that will be discussed in detail in sec. 3.6. Except near the critical point, the amplitudes of these fluctuations are small, in the order of 1 nm, but they can, in principle, be measured by the scattering of optical light. X-ray and neutron beams. From the scattered intensity the root mean square amplitude can be derived and this quantity can, in turn, be related to the surface tension because this tension opposes the fluctuations ). 

The magnitudes of these displacements may interest the reader. If iliso, found from a Wilson plot (Figure 7.14, Chapter 7), is 4 A, then U = (w ) is about 0.05 and the root-mean-square amplitude is... 

In the mid 1950s Durwaxd W. J. Cruickshank " noted that atoms in, for example, a rotating molecule, are displaced towards the rotation axis. Rotational oscillations of molecules, such as found in crystalline benzene near its melting point, will cause an apparent displacement of atomic positions from their true positions because the best fit to the electron density should be curvilinear but, with the limitations of present-day techniques, is generally linear (Figure 13.15). If the root-mean-square amplitude of libration about an axis is o> (in radians), then the apparent (but not real) shortening of the bond, d, is ... 

Essentially the characteristic temperature is a measure of the temperature at which the atomic heat capacity is changing from zero to 6 cal deg for silver (0 = 215 K) this occurs around 100 K, but for diamond (0 = 1860 K) with a much more rigid structure, the atomic heat capacity does not reach 5 cal deg i until 900 K. Those elements that resist compression and that have high melting points have high characteristic temperatures. Equations have been derived relating y/ u ) to the characteristic temperature 0. At room temperature diamond, with a characteristic temperature of 1860 K, has a root-mean-square amplitude of vibration, / u ) of 0.02 A, while copper and lead, with characteristic temperatures of 320 and 88 K, respectively, have values of 0.14 and 0.28 A for (u ). - Similar types of values are obtained for crystals with mixed atom (or ion) types. For example, average values of / u ) for Na+ and Cl in sodium chloride (0 = 281 K) are 0.14 A at 86 K and 0.23 A at 290 K. ° ... 

Results on a large number of linear-chain complexes of platinum are summarised in Table 11. Harmonic wavenumbers and anharmonicity constants have been determined in all cases. The normal coordinate seems to be related to the halogen movements involved in the proposed hopping process for the conductivity of these linear-chain mixed-valence complexes (95). The chain halogen atoms would need to move, on average, 0.54,0.38 and 0.22 A for chlorides, bromides and iodides, respectively, in order to reach the point midway between the two platinum atoms, i.e. to the situation of a platinum (III) chain. These values only differ by a factor of about two from the root-mean-square amplitudes of vibration of Vi in the Vj = 16 states these are calculated (91) to be 0.22 A for X = Cl (wi = 319.5 cm-i) and 0.20 A for X = Br (cji = 179.6 cm ). These distance changes are related to the shift in the equilibrium... 

It is generally agreed that thermally induced vibrations of atoms in solids play a major role in melting [2.144]. The simple vibrational model of Linde-mann predicts a lattice instability when the root-mean-square amplitude of the thermal vibrations reaches a certain fraction / of the next neighbor distances. However, the Lindemann constant/varies considerably for different substances because lattice anharmonicity and soft modes are not considered, thus limiting the predictive power of such a law. Furthermore, Born proposed the collapse of the crystal lattice to occur when one of the effective elastic shear moduli vanishes [2.138], Experimentally, it is found instead that the shear modulus as a function of dilatation is not reduced to zero at Tm and would vanish at temperatures far above Tm for a wide range of different substances [2.145]... 

A valence density map of uracil is shown in Figure 3. Notice that contrast between O and N is clearly evident, but the density about the protons and C nuclei does not appear as a distinct maximum. For data that extend to v 1.0 A-1 in sin0/X and for root-mean-square amplitudes of vibration larger than 0.14 A, these valence density maps should be free from series termination error.15... 

Figure 6.4.7 shows the interpretation of two sets of relaxation data obtained from a specific site, Trp9, in the polypeptide backbone of gramicidin A [24]. From powder pattern studies it has been shown that the local motions occur about an axis consistent with the C, —C ,+i axis, and that the motion is a librational motion of about 20° [21], quite similar to the indole side-chain described earlier. The data in Fig. 6.4.7 has been interpreted in light of this experimentally defined motional model. However, the field-dependent relaxation data suggests that the amplitude is much less than 20°, in fact it is closer to a root mean square amplitude of 5°. However, this apparent... 

The root mean square amplitude of the signal noise should decrease according to the following equation ... 

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