Motion resonance frequency

According to Eq. (4-62), when woTo < 1, T, is proportional to 1/Tc, whereas when woTc 1, Ti is proportional to Tc. When Tc = Wo, Tj has its minimum value. Figure 4-7 is a schematic representation of the relationship between T and Tc. The physical meaning of this relationship is that coupling between the spin system and the lattice is most efficient when the resonance frequency and the frequency of molecular motion are equal. Tc can be measured by studying the dependence of Ti on wq (by varying the field strength). For small molecules in solution Tc is commonly 10 to 10 s. 

Usually, nuclear relaxation data for the study of reorientational motions of molecules and molecular segments are obtained for non-viscous liquids in the extreme narrowing region where the product of the resonance frequency and the reorientational correlation time is much less than unity [1, 3, 5]. The dipolar spin-lattice relaxation rate of nucleus i is then directly proportional to the reorientational correlation time p... 

Here, the J terms are the spectral densities with the resonance frequencies co of the and nuclei, respectively. It is now necessary to find an appropriate spectral density to describe the reorientational motions properly (cf [6, 7]). The simplest spectral density commonly used for interpretation of NMR relaxation data is the one introduced by Bloembergen, Purcell, and Pound [8]. 

The Coriolis meter (Figure 6.28) contains a sensor consisting of one or more tubes which are vibrated at their resonant frequency by electromagnetic drivers, and their harmonic vibrations impart an angular motion to the fluid as it passes through the tubes which,... 

Most methods of testing bond type involve the motion of nuclei. The chemical method, such as substitution at positions adjacent to a hydroxyl group in testing for double-bond character, as used in the Mills-Nixon studies, is one of these. This method gives only the resultant bond type over the period required for the reaction to take place. Since this period is much longer than that of ordinary electronic resonance, the chemical method cannot be used in general to test for the constituent structures of a resonating molecule. Only in case that the resonance frequency is very small (less than the frequencies of nuclear vibration) can the usual methods be applied to test for the constituent structures and in this case the boundary between resonance and tautomerism is approached or passed. 

D11/Dj, from 1 to 10. Symbols correspond to synthetic experimental data generated assuming overall tumbling with rc = 5 ns and various degrees of anisotropy as indicated. Model-free parameters typical of restricted local backbone dynamics in protein core, S2=0.87, T oc =20 ps, were used to describe the effect of local motions. The H resonance frequency was set to 600 MHz. The solid lines correspond to the right-hand-side expression in Eq. (10). 

Fig. 4.49. Motion of positive ions in a uniform magnetic field B. (a) The radius is a function of ion velocity, but the frequencyof circulation is not. (b) Excitation of the ions by an RF electric field oscillating at their cyclotron resonance frequency. Adapted from Ref. [196] by permission. John Wiley Sons, 1986.

Irradiation of long narrowband RF-pulses with frequency offsets between 1 and 20 kHz relative to the resonance frequency of the free protons selectively influences transitions which correspond to the slopes of the broad resonance lines from the bound pool of spins. Therefore, only the spins of protons with restricted motion are saturated, whereas the free protons remain unaffected (e.g., see Refs. 13 and 43). 

The reason why one chose to follow the main liquid-crystalline to gel phase transition in DPPC by monitoring the linewidth of the various or natural abundance resonance is evident when we consider the expressions for the spin-lattice relaxation time (Ti) and the spin-spin relaxation time T2). The first one is given by 1/Ti oc [/i(ft>o) + 72(2ft>o)] where Ji coq) is the Fourier transform of the correlation function at the resonance frequency o>o and is a constant related to internuclear separation. The relaxation rate l/Ti thus reflects motions at coq and 2coq. In contrast, the expression for T2 shows that 1/T2 monitors slow motions IjTi oc. B[/o(0) -I- /i(ft>o) + /2(2u>o)], where /o(0) is the Fourier component of the correlation function at zero frequency. Since the linewidth vi/2 (full-width at half-maximum intensity) is proportional to 1 / T2, the changes of linewidth will reflect changes in the mobility of various carbon atoms in the DPPC bilayer. 

A more detailed analysis S, 62) shows Equation (12) to be reasonably rigorous and further shows that for random motion in the limit of t short compared to the reciprocal of the resonance frequency the line shape is Loren tzian. 

An interesting application of the motional narrowing concept arises in the double NMR technique BS). In this technique the contribution to the NMR line width of nuclei (A) in a solid by the dipolar fields of dissimilar nuclei (B) may be removed by application of a sufficiently strong rf field at the resonance frequency of the B nuclei. With Hib A/Ib, A/Ia where AH is the line width, flipping of B nuclei by the Hib field will cause fluctuations in the dipolar fields of B nuclei at the A nuclei which are rapid compared to T2a and hence cause narrowing of the NMR line of the A nuclei. This effect has been observed in several different solids of the AB type 5S,6A). 

Atomic polarization contributes to the relative motion of atoms in the molecule affected by perturbation by the applied field of the vibrations of atoms and ions having a characteristic resonance frequency in the IR region. The atomic polarization is large in inorganic materials which contain low-energy conductive bonds and approaches zero for nonconductive polymers. The atomic polarization is rapid, and this, as well as the electronic polarization, constitutes the instantaneous polarization components. 

Here, ks is the Boltzmann constant (1.38 x 10-23 J/K), T is the absolute temperature (300 K at room temperature), B is the bandwidth of measurement [typically about 1000 Hz for direct current (dc) measurement], /o is the resonant frequency of the cantilever, and Q is the quality factor of the resonance, which is related to damping. It is clear from Eq. (12.8) that lower spring constant, K, produces higher thermal noise. This thermal motion can be used as an excitation technique for resonance frequency mode of operation. 

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