Lattice reference frame

Equation (8.29) implies that the fluxes are defined in the lattice reference frame. Inserting Eqn. (8.28) into Eqn. (8.29) and noting that, in principle, the forces Xk (k = 1,2,..., 9) can be varied independently, the following relations are obtained... 

The understanding and description of interdiffusion in alloys is based for about 60 years on the commonly accepted Darken s scheme. This scheme uses the following simple logic. The migration rate of atoms is different (tracer diffusion coefficients and the corresponding partial diffusion coefficients Dj, i = 1,2 are different). This results in different magnitude (and opposite direction) densities of components fluxes in a crystalline lattice reference frame ... 

Here is the vacancy relaxation time. The initial set of equations for diffusion fluxes in the lattice reference frame Equations 2.23-2.25, after having made some algebraic transformations, can be presented in the following form ... 

The above-mentioned vacancy concentration change makes a contribution into component fluxes in the crystal lattice reference frame [1,5] in the form ... 

All fluxes are written down in the lattice reference frame, Db, Dg, Dv are, respectively, the partial diffusivity of B, the tracer diffusivity of B, and the diffusivity of vacancies inside the nanoshell. In Equations 7.23 and 7.24, we have neglected Manningfs corrections (vacancy wind effect). It could be done but it would make the mathematical equations more cumbersome without a substantial change in the main results. Under the same assumption, the mentioned diffusivities are interrelated in the following way... 

Note that we use the fluxes in the lattice reference frame and intrinsic diffusivities (instead of interdiffusivity) according to our basic approximation of no Kirkendall shift. We neglect the correlation factors of Manning s vacancy wind terms [28] since they can change results only quantitatively. Both tracer diffusivities are proportional to vacancy concentration. Here we use one more approximation that we treat only one effective vacancy concentration, without distinguishing between sublattices in the compound. We take... 

A3. The difFusion fluxes of the main components and of the vacancies take into account the cross terms and are written in the lattice reference frame [2, 31]... 

First, as mentioned in the introduction, there is little place for vacancy sinks/sources in a nanovolume. Second, if all atoms of the lattice try to move along the radial direction, this shift would immediately generate a tangential deformation and corresponding stresses. Boundaries of the shell do move but not due to lattice shift - just some atoms leave one boundary and attach to another boundary. Therefore, we can write down the continuity equations (analog of second Pick s law) in the lattice reference frame as... 

The boundary conditions in Equations 7.126 and 7.127 would be sufficient for shrinkage of a single-component shell. In our case of a binary solution, we should have additional conditions. Boundary concentrations of the main components are not fixed (we do not have an analog of Equation 7.126 for A or B), but the conservation laws are valid, of course, implying the conditions on fluxes. The sum of the three fluxes is zero in the lattice reference frame (/V +Ja +Jb = 0). Thus, two fluxes are independent. This means that one should write down the flux balance equations at both moving boundaries for one of the main components, taking into account that fluxes outside the shell are equal to zero... 

Let us write down the equations for the fluxes in the a phase for A and B components in the lattice reference frame taking into account the nonequihbrium vacancy distribution and the effects of electric field [Ij. For simplicity, A and B are assumed to have (approximately) the same atomic volumes. If one neglects correlation effects (leading, in the case of random alloys, to Manning s corrections [21]), then... 

The vacancy flux in the lattice reference frame is expressed by the following equation ... 

From Eqn. (4.81), we see that if one adopts the lattice as the reference frame (which also is Pick s frame for constant molar volume), then... 

Here, 0, represents the phase of oscillator i, which is coupled with strength e to a set of nearest neighbors Ni in a one- or two-dimensional lattice. The natural frequencies u>i are fixed in time, uncorrelated and taken from a distribution p u>). A scaling of time and a transformation into a rotating reference frame can always be applied so that e = 1 and the ensemble mean frequency lJ is equal to zero. We refer to the variance = var(wi) of the random frequencies as the disorder of the medium. 

In the context of motion and flux, it is clear that the flux should be defined with respect to a reference frame (Chakraborty 1995). In crystalline silicates, because diffusion of oxygen and silicon can be much slower than that of other cations (with possible exception of Al), this can be achieved quite easily by using the fixed silicate lattice as a reference frame in which ions jump from site to site. This is the so-called lattice fixed frame, which commonly coincides with the laboratory frame. Note that the motion of a dilute isotope of oxygen (e.g. O) can still be treated in this frame. Pick s first law can readily be modified to take into account the variability of reference frames. 

The spin-lattice relaxation rate of a particular set of equivalent nuclei is the first-order time constant of the energy exchange process for those nuclei and we now shall use the rotating reference frame model to illustrate one of several possible methods whereby these time constants may be measured. Consider our ensemble of spins. When they are at thermal equilibrium with the lattice, their net magnetisation can be represented as a vector directed along the +z-axis (see... 

On the other hand, according to the law of conservation of matter, the vector sum of components fluxes in the laboratory reference frame (connected with one of the ends of an infinite, on diffusion scale, diffusion couple) must be equal to zero. According to Darken, the alloy provides fluxes balancing at the expense of lattice movement as a single whole at some certain velocity, u, which is measured by inert markers ( frozen in lattice) displacement. Correspondingly in the laboratory reference frame, components fluxes acquire a drift component (similar to Galilean velocity transformation equations) as... 

One may define the magnitude of the lattice drift rate without analyzing the diffusion micromechanisms, but just from the condition that the sum of fluxes equals zero in the laboratory reference frame ... 

If one accepts the classical interpretation to consider vacancy flux to cause the lattice displacement at a velocity u = jy, and resulting components fluxes in the laboratory reference frame to be defined according to the Galilean equation (Equation 2.2), we have... 

In the alternative case, there is no back stress at all, implying that aU possible stresses are relaxed immediately by the lattice shift In a usual polycrystalline body with large grains, it is described by a dislocation dimb as the source and sink of vacancies, and corresponding construction of extra planes in the region of atoms accumulation, and reconstruction of planes in the region of vacancy accumulation. Vacancy is at equilibrium everywhere in the sample as assumed in Darken s analysis of interdifiusion. Then in the laboratory reference frame... 

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