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Rotational diffusion frame

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... [Pg.103]

Another rotational diffusion model known as the anisotropic viscosity model156,157 is very similar to the above model, and its main feature is to diagonalize the rotational diffusion tensor in the L frame defined by the director. A similar (but not the same) expression as Eq. (71) is J R(r)co)... [Pg.105]

The overall tumbling of a protein molecule in solution is the dominant source of NH-bond reorientations with respect to the laboratory frame, and hence is the major contribution to 15N relaxation. Adequate treatment of this motion and its separation from the local motion is therefore critical for accurate analysis of protein dynamics in solution [46]. This task is not trivial because (i) the overall and internal dynamics could be coupled (e. g. in the presence of significant segmental motion), and (ii) the anisotropy of the overall rotational diffusion, reflecting the shape of the molecule, which in general case deviates from a perfect sphere, significantly complicates the analysis. Here we assume that the overall and local motions are independent of each other, and thus we will focus on the effect of the rotational overall anisotropy. [Pg.292]

P(Qol, t) is the conditional probability of the orientation being at time t, provided it was Qq a t time zero. The symbol — F is the rotational diffusion operator. In the simplest possible case, F then takes the form of the Laplace operator, acting on the Euler angles ( ml) specifying the orientation of the molecule-fixed frame with respect to the laboratory frame, multiplied with a rotational diffusion coefficient. Dr. Equation (44) then becomes identical to the isotropic rotational diffusion equation. The rotational diffusion coefficient is simply related to the rotational correlation time introduced earlier, by tr = 1I6Dr. [Pg.65]

Estimates of the rotational diffusivity may be made from MD calculations by fitting an exponential function to Legendre polynomials that express the decorrelation of a unit vector that is fixed in the methane coordinate frame (11). The rotational diffusivity was found to increase with concentration (as a result of sorbate-sorbate collisions which act to decorrelate the molecular orientation). The values are of the same order as those for liquid methane and are 2 orders of magnitude larger than those found by Jobic et al. (73) from a quasi-elastic neutron scattering study of methane in NaZSM-5. [Pg.29]

Molecular motions in low molecular weight molecules are rather complex, involving different types of motion such as rotational diffusion (isotropic or anisotropic torsional oscillations or reorientations), translational diffusion and random Brownian motion. The basic NMR theory concerning relaxation phenomena (spin-spin and spin-lattice relaxation times) and molecular dynamics, was derived assuming Brownian motion by Bloembergen, Purcell and Pound (BPP theory) 46). This theory was later modified by Solomon 46) and Kubo and Tomita48 an additional theory for spin-lattice relaxation times in the rotating frame was also developed 49>. [Pg.18]

Let us come back to the problem of rotational diffusion. To derive the equation of motion of (r), we have to remember first of all that the equations corresponding to the rotational part are written in the reference frame of the molecular principal axis. By making a transformation from this reference system to that of the laboratory, we have... [Pg.290]

Fig. 9. Rotational diffusion of molecules as detected in pulsed fluorescence experiments. r(t) anisotrophy emitted by a symmetric rotor after pulsed excitation with polarized light L. pa and ge absorption and emission vectors with coordinates 1, 2 and 0 (molecular frame). P2 polarizer, Az and Ay analyzers in z and y directions of the laboratory frame 47)... Fig. 9. Rotational diffusion of molecules as detected in pulsed fluorescence experiments. r(t) anisotrophy emitted by a symmetric rotor after pulsed excitation with polarized light L. pa and ge absorption and emission vectors with coordinates 1, 2 and 0 (molecular frame). P2 polarizer, Az and Ay analyzers in z and y directions of the laboratory frame 47)...
Fig. 2. Reference frames and sets of Euler angles required for the description of rotational diffusion and sample spinning in a general NMR experiment. The magnetic tensor system (top, left) is characterized by a diagonal chemical shift tensor <7. The sample system (centre, left) is defined by the axis z of sample spinning while the laboratory system is determined by the direction of the external magnetic field. For a transformation from the magnetic tensor system into the sample system, a full set of three Euler angles 3>, 0, and W is needed. For the transformation from the sample system into the laboratory system, only two Euler angles a and /3 are required as the external magnetic field is assumed to be of rotational symmetry. Fig. 2. Reference frames and sets of Euler angles required for the description of rotational diffusion and sample spinning in a general NMR experiment. The magnetic tensor system (top, left) is characterized by a diagonal chemical shift tensor <7. The sample system (centre, left) is defined by the axis z of sample spinning while the laboratory system is determined by the direction of the external magnetic field. For a transformation from the magnetic tensor system into the sample system, a full set of three Euler angles 3>, 0, and W is needed. For the transformation from the sample system into the laboratory system, only two Euler angles a and /3 are required as the external magnetic field is assumed to be of rotational symmetry.
Similar interpretations of the crystal size of zinc salts of ethylene ionomers were obtained in discussions about H spin-diffusion with ]H spin-lattice relaxation times in the laboratory (Tf3) and rotating (T ) frames. Table 3... [Pg.10]

The frame (B) was chosen such that the rotational diffusion tensor is diagonal. In general, the polarizability tensor a will not be diagonal in the same body fixed frame that diagonalizes. In the special case when a and are simultaneously diagonalized in the frame (B) that is, when the molecule is a true symmetric top, then aij(B) = 0 for i = j. Referring back to Eq. (7.4.1) we see that in this eventuality azz(B) = a and axx(B) = ccyy(B) = a , and... [Pg.128]

Fig. 6. (a) Different coordinate systems (laboratory L, director D, and magnetic m) nsed to define motion parameters for a nitroxide spin label, (b) Diffusion rotation angles used to define the magnetic axes relative to the diffusion axes. Note that the reference system for these angles is the diffusion frame, whereas the reference system is the magnetic (g) frame for the magnetic tilt angles (cf. Fig. 3). [Pg.61]

The orienting potential is expressed as a function of the polar angles (0,( )) of the director in the rotational diffusion axis frame. It is most conveniently included in the SLE equation by expanding it in a series of spherical harmonic functions as follows ... [Pg.64]

FIGURE 6.1. A schematic illustration of the coordinate systems used in the evaluation of physical motions in liquid crystals, (a) The rotational diffusion tensor Dr is used to define the molecular frame, (b) n B is used. Both h(f) and Dr are referred to as the space-fixed frame. [Pg.135]

The second term in the above expression represents a cross-term between the two types of motion, but is zero except when rriL = 0. Unless it is necessary to calculate Jo (a ), or the spin-spin relaxation time, the overall correlation functions will be approximated by linear combinations of the products of the correlation functions for each motion [i.e., retain only the first term in Eq. (8.10)]. To discuss the superimposed rotations model, it is assumed that internal rotations about different C-C bonds are independent and use additional coordinate frames to carry out successive transformations from the local a frame to the molecule-fixed frame. Free rotational diffusion will be used to describe each bond rotation in the following section. [Pg.218]

First, a rigid subunit of 5CB is chosen to define the molecular frame (Xm, 1m, Zm)- This subunit should be chosen so that, to a good approximation, the reorientation of this reference axis system relative to the laboratory frame is independent of the internal motions. The rotational diffusion tensor of the whole molecule is supposed to be diagonal in this molecular frame. As a result, the small step rotational diffusion model (Section 7.2.2) may be used to account for the reorientation of the whole molecule. The internal rotation axis Zj) linking the th fragment (CjH2) and j — l)th fragment (Cj iH2) is used to define the Z axis of the jth subunit, while its Y axis is taken to be perpendicular to the Zj and Zj i axes (see Fig. 8.2). The correlation functions for the deuterons on the may be calculated... [Pg.218]


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