Isotropic random motion

Here, tr and tj are the correlation times for the diffusional rotation and the isotropic random motion, respectively. 0r is the angle between the C—H internuclear vector and the z axis. 

Unfortunately, we do not know Tj in Eq. 10.10. If we did, we could calculate the H—H distance. It turns out that on cooling the sample, T passes through a minimum value. Equation 10.10 predicts that this should happen when = 0.62/(1). Since we know o), we can calculate Tc at the minimum and so estimate the H—H distance directly. A number of precautions need to be taken because rotation of the H2 about the M—(H2) bond reduces the relaxation rate, and certain metals, notably Re, Nb, V, Mn, Co, and Ta, cause a substantial, but easily calculable, dipole-dipole relaxation of attached protons because both 7 and I are high. We also assume isotropic (random) motion of the molecule, which is not the case for such systems as IrH5L2 and Cp ReH5, where the MH unit has a low moment of inertia and so spins rapidly. 

The temperature dependence of the spin-lattice relaxation rate Ef is usually analyzed using the simple Bloembergen, Purcell and Pound (BPP) model, which assumes a single correlation time describing non-correlated isotropic random motions. The spin-lattice relaxation expressed in terms of the spectral density function J(a>) evaluated at the NMR Larmor frequencies (Do and 2(Oo is... 

Ionic transport in solid electrolytes and electrodes may also be treated by the statistical process of successive jumps between the various accessible sites of the lattice. For random motion in a three-dimensional isotropic crystal, the diffusivity is related to the jump distance r and the jump frequency v by [3] ... 

Fig. 1. Schematic representation of the NMR absorption of a carbonyl functionality a) Single crystal with two different orientations b) Polycrystallinc sample (contributions from the random distribution of orientations, chemical shift anisotropy, CSA) c) In solution (random motion of the molecules yields the isotropic average chemical shift) (Reproduced by permission of The Royal Society, London).

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.

In contrast, Howarth [12] derived J,(w) for the 3t model corresponding to p = 3, where three independent motions are assumed to be superposed for the overall motion of the C—H vector as shown in Fig. 3.6. Namely, the C—H vector undergoes diffusional rotation about the Z axis in the Oi frame, whereas the zi axis librates within a cone whose axis is parallel to the Z2 axis in the O2 frame. Moreover, the Z2 axis undergoes the isotropic random reorientation in the laboratory frame. Although an empirical approximation was made in the previous calculation, we obtained the following equations by the exact mathematical derivation [8-10] ... 

Assuming the particles undergo Brownian motion, the velocity is by definition an isotropic random variable described by a real Gaussian process of zero mean. Hence all the odd-ordered velocity correlation functions are zero, and the factorization property for a real Gaussian process can be used to determine the even ordered velocity correlations in terms of the second order velocity correlation function. Noting in particular that... 

Liquid crystal most crystalline substances melt to give clear, isotropic liquids in which all the order of the crystalline state has been replaced by random motion of die molecules in the liquid state. 

We choose the initial conditions to be /o(x, 0) = 1 for jc <0 and /o(Jt, 0) = 0 for X > 0. This initial condition describes, for example, a territory divided into an invaded zone, x < 0, and a noninvaded zone, x > 0, separated by a frontier at X = 0. If particles disperse according to an isotropic random walk with KPP kinetics, this initial condition turns into a front propagating from left to right, i.e., the invasion starts. Since the particle jumps are isotropic, the reaction is responsible for the motion of the front from left to right. It is the reaction process that starts and maintains a successful invasion. A bias to the left in the random walk will hinder the invasion. Therefore we expect that the critical reaction rate is given by a balance between the factor favoring the invasion, the reaction process, and the factor opposing the invasion, the bias in the transport process. 

The derivations given in Section 9.5.2 refer to small, isotropic, randomly distributed molecules that move independently of one another, e.g., in a vacuum. The total intensity of scattered light is here given as the sum of the intensities scattered by the individual molecules. In liquids, the Brownian motions of the molecules are not independent of each other. Because of intermolecular interference, the measured total intensity of scattered light is less than the sum of the individual intensities. 

As mentioned in Section 2.2.5, in liquid isotropic samples the molecules typically execute fast and random motions so that the anisotropic components of spin... 

In flexible polymers, the chains tend to form random coils and the local conformations allow isotropic rotational motions of the residues by a combination of angular fluctuations and conformational transitimis [14, 38]. Stiff macromolecules with flexible side groups, however, lack conformadmial freedom within the backbone. 

Diffusion can be defined as the process in which the transfer of matter is due to random motion. The mathematics of diffusion can be found in Crank s book [23], in which different diffusion processes are analyzed. Let s assume an isotropic and homogeneous medium in which the diffusivity D is constant and the rate of transfer of matter (ions, atom or naolecules) is described by Pick s first law of diffusion. Despite that diffusion may be treated as a threeHlimensional process, it may be assumed that it occurs in isotropic media. 

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