Spectral densities of molecular motion

Although the timescale and the principle of mobility filters are readily outlined in this way, the validity of the representation is limited, because relaxation times depend on the spectral densities of molecular motion at more than one frequency, which is neglected in Fig. 7.1.3. In addition, the spectral densities relevant to NMR of condensed matter... 

Hausser used a semi-classical approach to determine the dependence of a and p on the spectral densities of molecular motions for different relaxation mechanisms (S for scalar, D for dipolar) [49] ... 

Molecular reorientation in anisotropic media such as liquid crystals, described by the spectral densities of the motion of a molecule, are... 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Both Ti and T2 relaxations of water protons are mainly due to fluctuating dipole-dipole interactions between intra- and inter-molecular protons [62]. The fluctuating magnetic noise from all the magnetic moments in the sample (these moments are collectively tamed the lattice) includes a specific range of frequency components which depends on the rate of molecular motion. The molecular motion is usually represented by the correlation time, xc, i.e., the average lifetime staying in a certain state. A reciprocal of the correlation time corresponds to the relative frequency (or rate) of the molecular motion. The distribution of the motional frequencies is known as the spectral density function. 

The spectral density is a measure of the amplitude of the M-quantum component of the nuclear spin interaction oscillating at frequency Mlj0 as a result of molecular motion. 

If the reduced spectral densities do indeed mirror the pure singlemolecule contribution, then at least for symmetric-top liquids, which should have only one basic type of librational mode, it does not seem that the two observed bands can represent two distinct types of molecular motions. Similarly, the reduced spectral densities of liquids composed of less symmetric molecules also can often be fit to the same two types of bands, despite the existence of multiple possible librational modes. 

There is no reason why ultrasound should not be used to reduce 7j s in liquids if sound can be used to induce additional relative molecular motions and hence alter the spectral density of the fluemating magnetic field in the sample. 

Fig. 7.1.3 [Blii2] NMR-timescale of molecular motion and filter transfer functions of pulse sequences which can be utilized for selecting magnetization according to the timescale of molecular motion. The concept of transfer functions provides an approximative description of the filters. A more detailed description needs to take into account magnetic-field dependences and spectral densities of motion. The transfer functions shown for the saturation recovery and the stimulated-echo filter apply in the fast motion regime.

The spectral density is a measure of the amplitude of the M-quantum component of the nuclear spin interaction oscillating at frequency Mo as a result of molecular motion. Of course, we should also recognize that since H(t) varies randomly in time, otherwise identical spin systems will have different H(t) at any given time t. Thus, we need to perform an average over the ensembles of spin systems making up the total sample. We denote this ensemble average by a bar, and thus we replace CM in Eq. (11) with... 

This is a measure of the amount of molecular motion which is at the correct frequency, co j, to cause the transitions. Recall that molecular motion is the effect which makes the random fields vary with time. However, as we saw with RF pulses, the field will only have an effect on the spins if it is oscillating at the correct frequency. The spectral density is a measure of how much of the motion is present at the correct frequency. 

Soda and Chihara (1974) have pointed out that in the weak collision limit (x <spectral densities have fixed ratios independent of the type of isotropic molecular motion for each relaxation mechanism. This is a nice rule to know because only one J needs to be calculated from scratch for any given mode of motion and the other J s can be calculated from the first simply by choosing the proper argument, i.e., 0, u), or 2w. These authors also give a useful relation between the reduction factor of the second moment arising from a particular mode of molecular motion to the T minimum due to that same motion. 

NMR relaxation and its field dependence are a very important source of experimental information on dynamics of molecular motions. This information is conveyed through spectral density functions, which in turn are related to time-correlation functions (TCFs), fundamental quantities in the theory of liquid state. In most cases, characterizing the molecular dynamics through NMR relaxation studies requires the identification of the relaxation mechanism (for example the dipole-dipole interaction between a pair of spins) and models for the spectral densities/correlation functions." During the period covered by this review, such model development was concerned with both small molecules and large molecules of biological interest, mainly proteins. 

It should be noted that the JmL ( ) re quantities that are obtained from experiment without reference to any molecular dynamics model. When measured as a function of temperature and frequency, these spectral densities of motion provide the best test of motional models for liquid crystals. 

For an isolated spin-1 system, it is convenient to define sum and difference magnetizations [Eqs. (2.84)-(2.85)] in the J-B experiment. The decay of the difference (quadrupolar order) proceeds exponentially at a rate T q, while the sum (Zeeman order) recovers exponentially towards equilibrium at a different rate. The J-B experiment allows simulataneous determination of these rates from which Ji uJo) and J2 2ujo) can be separated. Table 5.1 briefly summarizes thermotropic liquid crystals in which spectral density measurements were reported. Figure 5.4 illustrates the temperature and frequency dependences of spectral densities of motion (in s by including the interaction strength Kq factor) for 5CB-di5. The result is fairly typical for rod-like thermotropic liquid crystals. The spectral densities increase with decreasing temperature in the nematic phase of 5CB. The frequency dependence of Ji uJo) and J2(2a o) indicate that molecular reorientation is likely not in the fast motion regime. However, the observed temperature dependence of the relaxation rates is opposite to what is expected for simple liquids. This must be due to the anisotropic properties (e.g., viscosity) of liquid crystals. 

The small step rotational diffusion model has been employed to extract rotational diffusion constants Dy and D from the measured deuterium spectral densities in liquid crystals [7.25, 7.27, 7.46, 7.49 - 7.53]. Both the single exponential correlation functions [Eq. (7.54)] and the multiexponential correlation functions [Eq. (7.60)] have been used to interpret spectral densities of motion. However, most deuterons in liquid crystal molecules are located in positions where they are rather insensitive to motion about the short molecular axis. Thus, there is a large uncertainty in determining D or Tq (t o) because of Dy > D and the rather small geometric factor [doo( )] for most deuterons in liquid crystal molecules. For 5CB, it is necessary to fix [7.52] the value of D using the known activation... 

The most difficult problem in any relaxation theory is the calculation of correlation functions or spectral densities of motion. It is often possible to determine the mean square spin interaction H t)) where H t) is a component of the spin Hamiltonian which fluctuates randomly in time owing to molecular motions. The time dependence of the correlation function - r)) can often be approximated... 

Fig. 18. Schematic representation of the spectral density (or intensity function) for spin couplings in the frame of the three-component analysis of molecular motions in polymers. The dipolar broadening region represented as the hatched section at low fluctuation rates is predominantly responsible for the transverse relaxation rate (compare Eq. 40 and Ref. [2]). Variation of the temperature shifts the components across the fluctuation rate defined by the motional-averaging condition, so that the influence of the individual components changes one by one. Variation of the molecular weight or the polymer concentration likewise shifts the molecular weight or concentration dependent components across the motional averaging fluctuation rate...

Levine et al. (1973), Tropp (1980), King and Jardetzky (1978), King et al. (1978), Wittebort and Szabo (1978), Keepers and James (1982), Allison et al. (1982), Hart et al. (1981), and Lipari and Szabo (1982) have published specific spectral density formulations relevant to specific models of molecular motion. 

It can be seen that Eq. (38) is identical to Eq. (37) when Xj = 1/r, (i.e., when the motions are independent and all occur at much different rates). In general, however, this is not the case, and Xj will be determined as a composite of different motions. Spectral densities of the form of Eq. (38) can be employed with Xj and Cj values as unknown parameters to be determined from the measured NMR relaxation data. Although the Xj and Cj values thus determined have no physical meaning, such a procedure should enable the minimum number of molecular motions contributing to relaxation to be determined. However, whenever a specific motional model is utilized, it is readily apparent when an additional motion is required to fit the data. 

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