Dipole correlation time

The dipole correlation time of the system is a valuable quantity to calculate as it is related to the sample s absorption spectrum. Liquids usually absorb in the infrared region of the electromagnetic spectrum, a typical spectrum being shown in Figure 7.12. As can be seen, the spectrum is very broad with none of the sharp peaks characteristic of a well-resolved spectrum for a species in the gas phase. This is because the overall dipole of a liquid does not change at a constant rate but, rather, there is a distribution of frequencies. The intensity of absorption at any frequency depends upon the relative contribution of that frequency to the overall distribution. If, on average, the overall dipole changes very rapidly (i.e. the relaxation time is short) then the maximum in the absorption spectrum will occur at a... 

The hrst molecular dynamics computation of single molecule orientational correlation functions at liquid interfaces was reported by Benjamin. In bulk water, the water dipole correlation time (4 0.2 ps) and the water HH vector correlation time (1.5 0.1 ps, which can be approximately deduced from the NMR line shape) are in reasonable agreement with experiments. The reorientation was found to be faster at the water liquid/vapor interface. The reorientation dynamics of water molecules at the water/l,2-dichloroethane interface is, in contrast, slightly slower (to 6 0.3 and 2.3 0.2 ps for the dipole and the HH vectors, respectively).Similar results were found in a recent study by Chowdhary and Ladanyi of water reorientation near hydrocarbon liquids having different structure (different branching). The slower reorientation was limited to water molecules immediately next to the organic phase. Slower dynamics were observed when the reorientation was calculated in the intrinsic frame (thus eliminating the effect of capillary fluctuations). 

In addition to the qualitative explanation of the appearance and shape of different dispersion regions in nCB, temperature dependencies of dielectric relaxation times observed experimentally were compared with the estimated temperature dependencies of mean dipole correlation times for F (g)) and F[ coy For 7CB, very good qualitative agreement was observed (see Fig. 4.8(a)). However, when the temperature range of the liquid crystalline phase is sufficiently broad, as in the case of 5CB, the dielectric relaxation time of the low-frequency process observed parallel... 

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

Given the specific, internuclear dipole-dipole contribution terms, p,y, or the cross-relaxation terms, determined by the methods just described, internuclear distances, r , can be calculated according to Eq. 30, assuming isotropic motion in the extreme narrowing region. The values for T<.(y) can be readily estimated from carbon-13 or deuterium spin-lattice relaxation-times. For most organic molecules in solution, carbon-13 / , values conveniently provide the motional information necessary, and, hence, the type of relaxation model to be used, for a pertinent description of molecular reorientations. A prerequisite to this treatment is the assumption that interproton vectors and C- H vectors are characterized by the same rotational correlation-time. For rotational isotropic motion, internuclear distances can be compared according to... 

The rate at which dipole-dipole relaxation occurs depends on several factors (a) the nature of the nucleus, (b) the internuclear distance, r, and (c) the effective correlation time, t of the vector joining the nuclei (which is inversely proportional to the rate at which the relevant segment of the... 

In addition to the dipole-dipole relaxation processes, which depend on the strength and frequency of the fluctuating magnetic fields around the nuclei, there are other factors that affect nOe (a) the intrinsic nature of the nuclei I and S, (b) the internuclear distance (r,s) between them, and (c) the rate of tumbling of the relevant segment of the molecule in which the nuclei 1 and S are present (i.e., the effective molecular correlation time, Tf). 

While the rate of change of dipolar interaction depends on t its magnitude depends only on the internuclear distance and is independent of t,. Thus the dipole-dipole relaxation depends on the molecular correlation time T the internuclear distance r, and the gyromagnetic ratios of the two nuclei, y and js -... 

The 13C NMR sensitivity can sometimes be a problem, but for the kind of samples studied here the effective concentration of monomer units is several molar which does not place excessive demands on present Fourier transform NMR spectrometers. In addition to the sensitivity of the chemical shift to structure (9), the relaxation of protonated carbons is dominated by dipole-dipole interaction with the attached proton (9). The dependence of the relaxation parameters T, or spin-lattice, and Tor spin-spin, on isotropic motional correlation time for a C-H unit is shown schematically in Figure 1. The T1 can be determined by standard pulse techniques (9), while the linewidth at half-height is often related to the T2. Another parameter which is related to the correlation time is the nuclear Overhauser enhancement factor, q. The value of this factor for 13C coupled to protons, varies from about 2 at short correlation times to 0.1 at long correlation... 

Figure 1. Schematic representation of dependence of the T, and Tt relaxation times on isotropic correlation time (tc) of motion for a C-H fragment assuming dipole-dipole relaxation and 7 T magnetic field.

Figure 4.9 illustrates time-gated imaging of rotational correlation time. Briefly, excitation by linearly polarized radiation will excite fluorophores with dipole components parallel to the excitation polarization axis and so the fluorescence emission will be anisotropically polarized immediately after excitation, with more emission polarized parallel than perpendicular to the polarization axis (r0). Subsequently, however, collisions with solvent molecules will tend to randomize the fluorophore orientations and the emission anistropy will decrease with time (r(t)). The characteristic timescale over which the fluorescence anisotropy decreases can be described (in the simplest case of a spherical molecule) by an exponential decay with a time constant, 6, which is the rotational correlation time and is approximately proportional to the local solvent viscosity and to the size of the fluorophore. Provided that... 

Another transport property of interfacial water which can be studied by MO techniques is the dipole relaxation time. This property is computed from the dipole moment correlation function, which measures the rate at which dipole moment autocorrelation is lost due to rotational motions in time (63). Larger values for the dipole relaxation time indicate slower rotational motions of the dipole... 

Fig. 4.1 a Typical time evolution of a given correlation function in a glass-forming system for different temperatures (T >T2>...>T ), b Molecular dynamics simulation results [105] for the time decay of different correlation functions in polyisoprene at 363 K normalized dynamic structure factor at the first static structure factor maximum solid thick line)y intermediate incoherent scattering function of the hydrogens solid thin line), dipole-dipole correlation function dashed line) and second order orientational correlation function of three different C-H bonds measurable by NMR dashed-dotted lines)... 

As stated in Section II.B of Chapter 2, the actual correlation time for electron-nuclear dipole-dipole relaxation, is dominated by the fastest process among proton exchange, rotation, and electron spin relaxation. It follows that if electron relaxation is the fastest process, the proton correlation time Xc is given by electron-spin relaxation times Tie, and the field dependence of proton relaxation rates allows us to obtain the electron relaxation times and their field dependence, thus providing information on electron relaxation mechanisms. If motions faster than electron relaxation dominate Xc, it is only possible to set lower limits for the electron relaxation time, but we learn about some aspects on the dynamics of the system. In the remainder of this section we will deal with systems where electron relaxation determines the correlation time. 

The failure of this model led to the application of motional descriptions involving several correlation times. The simplest of these, a two correlation time model, was developed by Woessner ( ) and suggested for macromolecular systems by Allerhand, Dodrell, and Glushko ( ). The model considers two motions modulating the dipole-dipole interaction anisotropic Internal rotation about an axis which also undergoes overall rotatory diffusion. This model can successfully account for the carbon-13 Ti and NOE values observed for the methyl carbons in PIB ( ). The methyl group is... 

The relationship between the spin relaxation rate and the correlation time is expressed by using the BPP equation for heteronuclear dipole-dipole interaction [32] ... 

The cross-relaxation rates between two spins can be experimentally measured in the laboratory (rotating frame They depend on interspin distance r and correlation time that modulates the dipole-dipole interaction [4] ... 

The classical dipole correlation function is symmetric in time, C(—t) = C(f), as may be seen from Eq. 5.59 by replacing x by x — t the classical scalar product in Eq. 5.59 is, of course, commutative. Classical line shapes are, therefore, symmetric, J(—. Furthermore, classical dipole autocorrelation functions are real. 

Strong intermolecular interactions such as hydrogen bonds or ion-dipole pairs restrict the motion of molecules and pertinent molecular segments. These interactions increase the correlation time zc and accelerate the 13C spin-lattice relaxation. Shorter 13C relaxation times can therefore also indicate the presence of such interactions. The Tj values of the C atoms of carboxylic acids, phenols, alcohols, and solvated molecular ions behave in this way. 

The temperature dependence of dipole-dipole relaxation is that of the correlation time, which is usually written in the form of an Arrhenius equation (3.30) ... 

Using single-frequency and noise-modulated resonance and off-resonance proton decoupling, 7] relaxation time measurements, relaxation reagents like Gd (fod)3 and specifically deuterated compounds, all the carbons in retinal isomers, the model compounds a-and /i-ionone, and vitamin A and its isomers [165, 555-557] were assigned. The olefinic ring carbons (C-5 and C-6) could be identified on the assumption that the 13C relaxation times are dominated by intramolecular dipole-dipole interactions with neighboring protons and that the same rotational correlation time characterizes the interactions for both carbons. Consequently the ratio of T/s for C-5 and C-6 can be estimated from eq. (5.1)... 

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. 

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