Atomic jump frequency

Kinetics deals with many-particle systems (thermodynamic ensembles). The properties measured as a function of time depend on the scale of observation, and this scale is chosen in relation to the question we wish to ask. The smaller the scale, the more inhomogeneous and fluctuating the homogeneous systems appear to be. For example, we describe the activated atomic jump frequency v as... 

Solution. Using a torsion pendulum, find the anelastic relaxation time, r, by measuring the frequency of the Debye peak, cup, and applying the relation cupr = 1. Having r, the relationship between r and the C atom jump frequency F is found by using the procedure to find this relationship for the split-dumbbell interstitial point defects in Exercise 8.5. Assume the stress cycle shown in Fig. 8.16 and consider the anelastic relaxation that occurs just after the stress is removed. A C atom in a type 1 site can jump into two possible nearest-neighbor type 2 sites or two possible type 3 sites. Therefore,... 

Shewmon 1963), where r is the atomic jump frequency for uncorrelated jumps, and d is the atomic jump distance. For diffusion driven by the vacancy mechanism, r is given by... 

What is the atomic jump frequency for Si self-diffusion at 1,000°C assuming that the atomic jump distance is 0.235 nm ... 

The activated complex K for the activated complex The Debye atomic jump frequency, 10 Hz The potential gradient per atom Avogadro s number... 

Here Tc is the dipolar correlation time, Av is the observed linewidth, and Avq is the rigid lattice linewidth measured at a temperature low enough that diffusion effects are no longer detectable. For Aptemperature dependence of Ap and Tc is normally set equal to the atomic jump time on the grounds that an atomic jump from one site to another alters the dipolar interaction between neighboring spins substantially. Since the atomic jump frequency is expected to obey an Arrhenius relation [Zener (1952)], Tc should also show such a dependence on temperature. The n values obtained from line-narrowing measurements have usually been fit to the relation... 

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,... 

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... 

Vq is the frequency of the small oscillation, and AG and AS are, respectively, the difference in Gibbs free energy and entropy of the adatom at the saddle point and the equilibrium adsorption site. Ed is the activation energy of surface diffusion, or the barrier height of the atomic jumps. 

The factor R may be called a diffusion anisotropy factor for the surface. For diffusion of a W on the W (110), Tsong Casanova find a diffusion anisotropy factor of 1.88 from a set of data taken at 299 K, and of 2.14 from a set of data taken at 309 K. The average is 2.01, which agrees with the theoretical value, 2, to within 0.5%. A more detailed general analysis has since then been reported,137 and diffusion anisotropy on the W (110) surface has also been observed in field emission experiments.138 It should be noted, however, that the same ratio can be expected if an adatom jumps instead in the [001] and [110] directions with an equal frequency. Thus a measurement of surface diffusion anisotropy factor alone does not establish uniquely the atomic jump directions. The atomic jump directions can, of course, be established from a measurement of the displacement distribution function in two directions as discussed in the last section. Such measurements can only be done with atomic resolution field ion microscopy. 

An observation of motion of single atoms and single atomic clusters with STEM was reported by Isaacson et al,192 They observed atomic jumps of single uranium atoms on a very thin carbon film of —15 A thickness or less. Coupled motion of two to three atoms could also be seen. As the temperature of the thin film could not be controlled, no Arrhenius plot could be obtained. Instead, the Debye frequency , kTIh, was used to calculate the activation energy of surface diffusion, as is also sometimes done in field ion microscopy. That the atomic jumps were not induced by electron bombardment was checked by observing the atomic hopping frequencies as a function of the electron beam intensity. 

For very dilute solid solutions of B in A, the basic physics of diffusional mixing is the same as for (A, A ). An encounter between VA and BA is necessary to render the B atoms mobile. But B will alter the jump frequencies of V in its surroundings and therefore numerical values of the correlation factor and cross coefficient are different from those of tracer A diffusion. Since the jump frequency changes also involve solvent A atoms, in addition to fB, the numerical value of fA must be reconsidered (see next section). 

In a similar way we find jp p+]- Note that vy/p x is the (activated) jump frequency for the exchange of i with other component atoms (e.g., B, if / = A). Let us express Eqn. (5.101) in terms of volume concentrations and assume that the concentration differences between adjacent planes are small enough so that vp Wp = vp/p, is a valid assumption. If d is the distance between the planes, then... 

There are two arenas for describing diffusion in materials, macroscopic and microscopic. Theories of macroscopic diffusion provide a framework to understand particle fluxes and concentration profiles in terms of phenomenological coefficients and driving forces. Microscopic diffusion theories provide a framework to understand the physical basis of the phenomenological coefficients in terms of atomic mechanisms and particle jump frequencies. 

The quantity Vm, given by Eq. 7.26, is the difference between the volume of the system in an activated state and a well state. This volume difference is generally termed the activation volume for migration and is a positive quantity because of the atomic squeezing and resulting expansion of the system that occurs in the activated state. The activation volume can be measured experimentally by measuring the pressure dependence of the jump frequency, T. Find an expression for the pressure dependence of F and describe how it can be used to determine Vm. 

The total jump frequency for a given C atom is T = 4T, and therefore... 

In Section 3.1.1, self-diffusion was analyzed by studying the diffusion of radioactive tracer atoms, which were isotopes of the inert host atoms, thereby eliminating any chemical differences. Possible effects of a small difference between the masses of the two species were not considered. However, this difference has been found to have a small effect, which is known as the isotope effect. Differences in atomic masses result in differences of atomic vibrational frequencies, and as a result, the heavier isotope generally diffuses more slowly than the lighter. This effect can—if migration is approximated as a single-particle process—be predicted from the mass differences and Eq. 7.14. If mi and m2 are the atomic masses of two isotopes of the same component, Eqs. 7.13 and 7.52 predict the jump-rate ratio,... 

The effect is demonstrated in a simple manner in two dimensions in Fig. 8.10, which shows an isolated solute atom with a vacancy occupying a nearest-neighbor site [4], Three jump frequencies are considered the intrinsic host-vacancy jump rate, the solute-vacancy jump rate, and the jump rate for a vacancy... 

Under equilibrium conditions in a stressed b.c.c. Fe crystal, interstitial C atoms are generally unequally distributed among the three types of sites identified in Fig. 8.86. This occurs because the C atoms in sites 1, 2, and 3 in Fig. 8.86 expand the crystal preferentially along the x, y, and 2 directions, respectively. These directions are oriented differently in the stress field, and the C atoms in the various types of sites therefore have different interaction energies with the stress field. In the absence of applied stress, this effect does not exist and all sites are populated equally. In Exercise 8.22 it was shown that when the stress on an equilibrated specimen is suddenly released, the relaxation time for the nonuniformly distributed C atoms to achieve a random distribution, t, is t = 2/(3r), where T is the total jump frequency of a C atom in the unstressed crystal. 

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