Frequency jumps

The jump frequency is related in an exponential fashion to the free energy of activation between the ground state and the saddle point ... 

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] ... 

This relationship makes it possible to calculate the maximum ionic conductivity of solid electrolytes. Assuming that the mobile ions are moving with thermal velocity v without resting and oscillating at any lattice site, this results in a jump frequency... 

Jahn-Teller distortions 309 ff Japanese separators 264, 267 Joule effect, heat losses 13 jump frequency, solid electrolytes 532 Jungner nickel cadmium batteries 22... 

Referring to the Pick Equations of diffusion, let us now reexamine 1/T, the jump frequency, so as to relate diffusion processes to lattice vibration processes. The reciinrocal of the time of stay is the jump frequency and is related to the diffusion coefficient by ... 

The nearly-identical /, 1 rates of 115In and 75As from 300-100 K were observed to vary as T2 over the entire range [317]. However, the addition of trace Cu (11 and 25 ppm) increased the rates above 500 K, and allowed a mean Cu jump frequency and activation barrier of 0.64 eV to be deduced. 

The atoms in a crystal are vibrating continually with frequency v, which is usually taken to have a value of about 1013 Hz at room temperature. It is reasonable to suppose that the number of attempts at a jump, sometimes called the attempt frequency, will be equal to the frequency with which the atom is vibrating. The number of successful jumps that an atom will make per second, the jump frequency T, will be equal to the attempt frequency v, multiplied by the probability of a successful move, that is,... 

If the process is random, the jump frequency is independent of direction and we can set T12 equal to r2i- In addition, when a large number of jumps are taken into account, the number of jumps in either direction will be equal, so ... 

This energy assists the motion to the right in Figure 3.9 and would oppose it in returning to the left. Using Vm as the minimum volume occupied by Avogadro s number of molecules enables us to write the jump frequency as... 

In other words, in unit time the atom makes/(on the order of 10 " s ) attempts to surmount the barrier, each time with the probability of the exponent. Thus, the quantity p is also known as the jump frequency,... 

The effect of atomic motion in the solid state on nuclear resonance line width is illustrated by the behavior of Na resonance from NaCl as a function of temperature 97). In Fig. 9 is shown the variation of the Na line width with temperature for pure NaCl and NaCl doped with an atomic fraction concentration of 6 X 10 of CdCU. As discussed in Section II,A,2 the low-temperature, rigid-lattice line width will narrow when the frequency of motion of the nuclei under observation equals the line width expressed in sec.-. The number of vacancies present should be equal to the concentration of divalent impurities and the jump frequency of Na+ is the product of the atomic vacancy concentration and the vacancy jump frequency... 

The diffusion of a liquid or gas through a membrane is similar to the diffusion in liquid systems. The mechanism for each case involves a transfer of a small molecule to a hole in the liquid or membrane. For diffusion, the jumping frequency from hole to hole is dependent on the activation energy, k), for this jump, which is dependent on the size and shape of the diffusing molecule and the size of the holes in the membrane, i.e., the free volume. For permeation, the activation energy is on the order of 2(F40 kJ/mol. The values of Fp are somewhat higher. 

Thus, microscopically, the diffusion coefficient may be interpreted as one-sixth of the jumping distance squared times the overall jumping frequency. Since / is of the order 3 x 10 ° m (interatomic distance in a lattice), the jumping frequency can be roughly estimated from D. For D m /s such as Mg diffusion in... 

For an anisotropic crystal, jumping frequencies in different directions may not be the same, and, hence, D along each crystallographic direction may be different. The relation between D and jumping frequency may be written as follows ... 

Electrodic reactions are interfacial electron-transfer reactions, with the special circumstance that the electron transfer occurs between the solution and an electronic conductor, or vice versa. In Section 4.2.17 it was pointed out that the jump frequency of an ion is given by... 

A new set of flow characteristics gradually emerges as the concentration of polymer becomes large. The solution viscosity loses its direct dependence on solvent viscosity and comes to depend on the product of two parameters a friction factor C which is controlled solely by local features such as the free volume (or alternatively the segmental jump frequency), and a structure factor F which is controlled by the large scale structure and configuration of the chains (16) ... 

Eyring et al. (226) examine the entanglement problem from a somewhat different point of view, the activated complex theory of liquid viscosity. In a monomeric liquid the molecules move by random jumps from one equilibrium position to another. The jump frequency is controlled by an activation barrier between neighboring sites. According to activated complex theory, a shear stress lowers the barrier in the direction of the stress and raises it in the opposite direction, producing a bias in jump frequency and a net flow of molecules in the stress direction. For low stresses, the expression for viscosity in a monomeric system is ... 

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