Motional correlation time

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... 

Since the molecular motion correlation time, Xc, depends on the temperature, the T1 curves, plotted in semilogarithmic coordinates, go through a minimum, Ti mm (Figure 4.19) which can be expressed as... 

Lee et al. evolved a comprehensive analytical-theoretical treatment, based on the solution of the reorientational isotropic diffusion equation, for an ensemble of high-spin systems under motion. These authors developed an analytical expression for the slow-tumbling motional region that relates the orientational-motion correlation time t (in s), or the corresponding tumbling rate t, with the step separation bB, of the ESR fine structure of a quartet by Eq. 8,... 

Figure 14.1. Dependence of the spin-spin relaxation time (T2) and spin-lattice relaxation time (7)) on the motional correlation time xc. Approximate values of the expected line widths for small organic molecules, proteins, and lipids in a bilayer are indicated.

It is well known that 2D NOESY is an effective method to study the three-dimensional (3D) structure of large molecules, such as proteins which have long motional correlation times.70-71 Cross-dipolar interaction peaks in a NOESY spectrum rely on the cross-relaxation of the longitudinal magnetization during the mixing time. One can extract valuable information about intermolecular distances from the intensity of the NOESY cross-peaks. The appearance of... 

The criterion for such motional narrowing is Zl- r 1, where A is the static linewidth of the relevant species and ris its motional correlation time. "" We reasonably substitute the... 

An application of the saturation-recovery filter to the suppression of signal from rigid components in bisphenol-apoly(carbonate) is shown in Fig. 7.2.2 [Hanl]. The wideline solid-echo spectrum of the phenyl deuterons exhibits a range of broad and narrow components (a) as a result of a distribution of motional correlation times. The mobile components are characterized by a shorter T than the more rigid components. Consequently the rigid components can be suppressed by partial saturation. After application of the saturation-recovery filter the shape of the wideline spectrum is dominated by the narrow signal in the centre from the mobile ring deuterons (b). 

Fig. 7.2.2 [Hanl] Solid-echo wideline spectra of the ring deuterons of bisphenyl-a polycarbonate-d at 253 K. The phenyl rings undergo a 180° flip motion with a wide distribution of motional correlation times, (a) Spectrum with signals from fast and slow flipping rings.

Very slow motions (correlation times, tc>103s) may be studied via exchange methods. Such techniques are often two (or higher) dimensional. The strength of a particular nuclear spin interaction is monitored during the ti period of the experiment, usually by allowing transverse nuclear spin... 

Fig. 25. Random walk simulations for static 2H NMR powder lineshapes arising from a quadrupole echo 90°x-t-90°v-t-FID pulse sequence for the model of an isotropic 3° jump.36 (a) Jump correlation time, tj = 411 gs correlation time for the motion, xc = 100 ms, echo delays x as given in the figure. Dotted line is the spectrum for an isotropic random jump with xj = xc = 100 ms and an echo delay x — 200 gs. (b) Jump correlation times xj and motional correlation times xc as given in the figure, echo delay x = 100 gs.

More typically, relaxation data is analyzed in the form of a Ta versus temperature curve, where Ta denotes a relaxation time constant, derived from one of the experiments in Fig. 28. The motional process is either assumed to be known, or is a choice between a very restricted set. The knowledge of the form of the motional process and the spin interaction, which governs the relaxation then allows the correlation functions of Eq. (12) to be determined as a function of the motional correlation time, and this ultimately allows the determination of an equation for the characteristic relaxation time constant for the motional process as a function of its correlation time. The experimental Ta versus temperature curves are then fitted to the equation by varying the motional correlation time. The resulting variation in motional correlation time as a function of temperature then means that an activation energy for the motional process can be derived. 

If the form of the motion is not known, then two or three possible models are considered, with the appropriate equations for the relaxation time constant as a function of motional correlation time being derived. The experimental Ta versus temperature curve is then attempted to be reproduced by each of the equations, with some variation of correlation time, in the hope that only one equation provides a suitable fit and that the corresponding motional model then represents a true picture of the molecular motion in the sample. 

Fig. 29. Calculated central transition spectra48 for a single spin-3/2 nucleus undergoing (a) four-site tetrahedral jumps and (b) 12-site icosahedral jumps between equally populated sites in each case. The motional correlation times are given in the figure in seconds for each lineshape. The quadrupole coupling constant (axially symmetric) used in the simulations was 3 MHz.

Conventional X-ray diffraction measures a space and time average of the electronic density. Therefore, any dynamical disorder will be transformed into spatial disorder between positions whose probabilities are determined by the average time spent on each position. Certainly, one of the most tremendous advantages of NMR compared to X-ray diffraction is its ability to measure the occurrence of motion at different time-scales. Whether the motion correlation time is on the Larmor frequency scale, the linewidth scale or much slower (exchange NMR) will affect differently the NMR parameters like relaxation rates, apparent anisotropy and asymmetry of the interaction and ID or 2D lineshape. With suitable sequences, the motion correlation times and site probabilities as a function of an external parameter (temperature or pressure) can be explicitly measured. 

From the physical point of view, the motion correlation time tc alone carries precious informations concerning the microscopic times related to the motion as well as the activation energy, enthalpy or volume of the motion. The activation enthalpy A//a (activation volume A Fa) is a measure of the excess enthalpy (volume) needed to enable the given motion (translation, rotation, etc.). This is an important information to be correlated to the structure, and it may give important hints to distinguish different phases as well as... 

Fig. 6.2.2. Left Simulated NMR lineshapes that are averaged by various characteristic segmental motions. In the case of fast rotation, y represents the angle between the rotation axis and the C—bond. For a two-site jump, j3 denotes the angle between the C—bond in the two configurations, and the effective asymmetry parameter becomes 17 7 0. Right Calculated 2D exchange spectra for a two-site jump with /3 = 120° (top), and for continuous diffusion (bottom). The distribution functions P(/3) of the reorientation angle are shown, together with the contour maps of the corresponding spectra. All data are displayed on the reduced frequency scale in units of Cq, and mixing times are set equal to the motional correlation time r,..

In both simulated examples, the mixing time t, of the 2D exchange experiment was set equal to the motional correlation time Tc, which defines the jump rate or diffusion rate. In practice, it is possible to probe a wide range of motional timescales by choosing appropriate mixing times between about 500 [JUS to 20 s. 

The fact that, for each of the samples represented in Table 25.4, the T value measured at 260 MHz is greater than T measured at 200 MHz indicates that the motional correlation time associated with the motions responsible for proton spin-lattice relaxation is larger than (260 x 106 Hz)i.e., >3.8 X 10 s. The fact that Tj measurements (not presented here) reveal that T < T for each case represented in Table 25.4 shows that Tc > (200 X 10 Hz) , i.e., >5 x lO s. For the polysiloxane-immobilized diamine sample represented in Table 25.4, both TJ and T p (results not given here) are increased by treatment with HCl(aq), or Cd (aq). If one assumes that both treatments reduce mobility in these polysiloxane-immobilized amine systems (by introducing ionic interactions and complexation), the increases imply that Tc in treated samples is larger than about (40 x 10 Hz) i.e., >2.5 x 10 s. 

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