Amplitudes of motion

Point Imaginary frequency/cm l Relative amplitude of motion in reaction coordinate vibration (kyilkD)a Primary kHlkD... 

Fig. 7. The nature of information concerning the mean orientation and dynamics of an internuclear vector r, which can be obtained from RDC analysis. Upon diagonali-zation of the Cartesian dipolar interaction tensor R, described in the text, the mean vector orientation, r, will be described by the Euler angles a and /3. The eigenvalues will correspond to the axial and rhombic order parameters which describe the amplitude of motion. If the motion is asymmetric, as reflected in a nonzero rhombic order parameter, then the principal direction of asymmetry is described by the Euler angle y.

In the case of the higher frequency <3XH vibrations the motion of the hydrogen atom is already considerably restricted by the force constant controlling the R—X—H angle, i.e. it is a fairly normal low amplitude vibration. The extra effect of the H-bond is to restrict further the amplitude of motion, although at the same time it becomes possible for this vibration to interact- with low frequency motions of the type 6(RXH YR ). This latter type of interaction may be the cause of some breadth of dO H vibrations of the alcohols where the... 

In order to apply equation (44) to a specific vibration in a given molecule it is necessary to know the relative amplitudes of motion of the atoms, i. e., it is necessary to know the normal coordinate. Since in most cases this is not known, it would seem difficult to make use of this isotope rule. In many cases, however, the normal coordinate is quite closely approximated by a suitably chosen symmetry coordinate. Since the latter is usually determined quite readily, and since the frequencies are stationary with respect to small changes in the normal coordinates [Bernstein 14)], it can be expected that equation (44) will apply to a good approximation. This of course depends on the chosen symmetry coordinate being reasonably close to the actual normal coordinate. 

For the para-substituted phenyl rings (i) rapid oscillations about the para-axis of the ring between angles a and - ay corresponding to a 2a amplitude of motion, and (ii) fast yr-flips of the phenyl rings... 

A number of methods have been developed for assessing nitroxide dynamics based on the cw-EPR spectrum (see review by Sowa and Qin, 2008). In the semiquantitative approach, parameters measured directly from the EPR spectrum, such as the central fine-width (AHpp, Fig. 15.9A), the splitting of the resolved hyperfine extrema (2AeS, Fig. 15.9A), and the second moment (H2, characterizing how broad the spectrum is), are used to characterize nitroxide dynamics (Columbus and Hubbell, 2002, 2004 Mchaourab et al., 1996). These parameters report on the nitroxide mobility, which describes a combined effect of the rate and the amplitude of motion. For example, a broad center fine gives a small (AHpp) 1 value and indicates low mobility, which can result from low frequency but large amplitude motions, or small amplitude motions with fast rates. The line-shape parameters can be easily measured on a properly processed EPR spectrum, and... 

The device must be dimensioned so that at resonance a half-wavelength is accommodated in the distance L. The amplitude of motion at the outer metal surface will be related to that of the ceramic through the ratio q of their acoustic... 

Equation (1) has several consequences. The intensity depends directly on the scattering cross section. The scattering cross sections are element- and isotope-dependent, as shown in Table I. The amplitude of motion is larger for light atoms, and so, because hydrogen has the largest cross section and the smallest mass of any element, it dominates the scattering. 

To model quantitatively an IINS spectrum, it is only necessary to obtain the amplitudes of motion of the atoms in the vibrational modes. These can be calculated by a variety of methods, such as the balls-and-springs approach of the Wilson GF matrix method, ab-initio calculations, and molecular dynamics this point expresses what is undoubtedly the greatest strength of IINS spectroscopy. Examples are presented below. 

In the absence of a mathematical model that can take care of the large amplitudes of motion encountered in the higher torsional levels (note that the power expansion on which the MMFF method is based presupposes infinitely small amplitudes) the calculation of the exact geometry and energy of e.g. 

The evolution of the experimental anisotropy as a function of the temperature is shown in Fig. 8. As expected, the decay rate increases as the temperature increases. For the highest temperature (t > 50 °C), it can be noticed that the anisotropy decays from a value close to the fundamental anisotropy of DMA to almost zero in the time window of the experiment (about 60 ns). This means that the initial orientation of a backbone segment is almost completely lost within this time. This possibiUty to directly check the amplitude of motions associated with the involved relaxation is a very useful advantage of FAD. In particular, it indicates that in the temperature range 50 °C 80 °C, we sample continuously and almost completely the elementary brownian motion in polymer melts. Processes too fast to be observed by this technique involve only very small angles of rotation and cannot be associated with backbone rearrangements. On the other hand, the processes too slow to be sampled concern only a very low residual orientational correlation, i.e. they are important only on a scale much larger than the size of conformational jumps. 

Ubiquitin is a small (76 amino acids) extremely stable protein containing a broad collection of secondary structure elements including parallel and antiparallel beta strands assembled into a mixed beta sheet, alpha and 3io helices and a variety of turns (Vijay-Kumar et al., 1987 Di Stefano Wand, 1987). In previous work, we have examined the fast main chain dynamics of ubiquitin by use of 15n NMR relaxation methods (Schneider et al., 1992). These data were analyzed in terms of the so-called model free treatment of Lipari and Szabo (1982a,b). The amplitudes of motion of the backbone amide N-H vectors of the packed regions of the protein are generally highly restricted and show no apparent correlation with secondary stmcture context but do show a strong... 

Dynamic-shear measurements are of the complex viscosity rj ) as a function of the dynamic oscillation rate (o), at constant temperature. These tests are defined as isothermal dynamic frequency sweeps. Since the dynamic frequency sweeps are conducted at a given amplitude of motion, or strain, it is necessary to ensure that the sweeps are conducted in the region where the response is strain-independent, which is defined as the linear viscoelastic region. This region of strain independence is determined by an isothermal strain sweep, which measures the complex viscosity as a function of applied strain at a given frequency. This ensures that a strain at which the dynamic frequency sweep may be conducted in the linear viscoelastic region is selected. 

Stationary. For the optical modes at the gamma point, both atoms move against each other while keeping the centre of mass stationary the lighter atom has the larger amplitude of motion, Fig. 4.12. As the zone boundary is approached only the lighter atom moves while the heavier atom is stationary. 

Since they are much lighter, a correspondingly larger amplitude of motion is required and this results in significant INS intensity. 

The amplitudes of motion associated with one or two quantum excita-... 

It is also seen that the magnitude of the resonance depends on the constants of integration 0 and tj0, the amplitudes of motion of Xi and x2 varying between the limits /2 o and 0 in case that Vo = So, and retaining the constant value o/ /2 (no resonance ) in case that vo = 0. 

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