Diffusion vacancy formation energy

By definition, the rate at which the tracer atom is displaced by a surface vacancy is the product of the vacancy density at the site next to the tracer times the rate at which vacancies exchange with the tracer atom. For the case where the interaction between the tracer atom and the vacancy is negligible, the activation energy obtained from the temperature dependence of the total displacement rate equals the sum of the vacancy formation energy EF and the vacancy diffusion barrier ED. When the measurements are performed with finite temporal resolution and if there is an interaction present between the vacancy and the indium atom, this simple picture changes. 

The importance of the measurements that we have presented so far for the diffusion of embedded tracer atoms becomes evident when we now use these measurements and the model discussed in Section 3 to evaluate the invisible mobility of the Cu atoms in a Cu(00 1) terrace. The results presented in Section 2 imply that not just the tracer atom, but all atoms in the surface are continuously moving. From the tracer diffusion measurements of In/Cu(0 0 1) we have established that the sum of the vacancy formation energy and the vacancy diffusion barrier in the clean Cu(0 01) surface is equal to 717 meV. For the case of self-diffusion in the Cu(0 01) surface we can use this number with the simplest model that we discussed in Section 3.2, i.e. all atoms are equal and no interaction between the vacancy and the tracer atom. In doing so we find a room temperature hop rate for the self-diffusion of Cu atoms in a Cu(00 1) terrace of v = 0.48 s-1. In other words, every terrace Cu atom is displaced by a vacancy, on average, about once per two seconds at room temperature and about 200times/sec at 100 °C. We illustrate this motion by plotting the calculated average displacement rate of Cu terrace atoms vs. 1 /kT in Fig. 14. 

The class of creep mechanisms of interest here are those that are mediated by stress-biased diffusion. If we are to consider the vacancy flux in a given grain within a material that is subjected to an applied stress, it is argued that the vacancy formation energy differs in different parts of the grain, and hence that there should be a gradient in the vacancy concentration leading to an associated flux. This... 

Equation (7.9) is only valid if AH , is much larger than the average energy of the system. In most cases for solids, that is the case. For example, the average energy of the atoms in a solid is on the order of kT which at room temperature is R50.025eV and at 1000 C is 0.11 eV. Typical activation energies for diffusion, vacancy formation, etc., are on the order of a few electron volts. So all is well. 

The Nb vacancy formation energy for NbCo.750 is 60.49 kcal/mol according to the vapor pressure data (2), which is in line with the 76.0 kcal/mol Q value found in NbCo.766. The Nb self-diffusion coefficients for NbCosas. NbCo.s34, and NbCo.766 are essentially independent of the C/Nb ratio in the temperature range 2370-2660 K and are expressed as... 

Equation 7.6 shows that the concentration of neutral vacancies increases exponentially with increasing temperature. For typical vacancy formation energies the concentration of the defects is very small, even at high temperatures. This is why the diffusivities of atoms in Si and other highly covalent semiconductors are low even at very high fractions of their melting points. 

Note that the smallest h can be is one for [I] 0 and the largest is two for [I] nj. Thus the maximum effect of an electric field on diffusivity is a factor of two. This stands in stark contrast to the effect of Fermi energy on vacancy formation energy and clustering, which can both change diffusivities by orders of magnitude. 

Self-Diffusion by the Interstitialcy Mechanism. If their formation energy is not too large, the equilibrium population of self-interstitials may be large enough to contribute to the self-diffusivity. In this case, the self-diffusivity is similar to that for self-diffusion via the vacancy mechanism (Eq. 8.19) with the vacancy formation and migration energies replaced by corresponding self-interstitial quantities. The... 

Two alternative approaches exist. The first one involves significantly lowering the temperature to values where the diffusion of vacancies can be observed with a technique like STM. At lower temperatures a surface vacancy can then be artificially created by ion bombardment or direct removal of an atom by the tip. This approach has been applied successfully to several semiconductor surfaces [29-31]. For metal surfaces, although vacancy creation at a step by direct tip manipulation of the surface has been demonstrated [32], to our knowledge, no studies have been published where the diffusion of artificially created vacancies in a terrace has successfully been measured. The second approach involves the addition of small amounts of appropriate impurities that serve as tracer atoms in the first layer of the surface [20-24]. The presence and passage of a surface vacancy is indirectly revealed by the motion of these embedded atoms. If one seeks to measure both the formation energy and the diffusion barrier of surface vacancies explicitly, a combination of these two approaches is needed. 

Finally, activation barriers have been determined for the diffusion of a vacancy in the MgO(lOO) surface, between sub-surface and surface layers, and close to surface defects [69,95]. As expected, the barrier is nearly twice lower in the first case than in the second one. Moreover, it is of the order of. 20 to. 25 times the formation energy, a quite reasonable ratio, in view of similar results on metal surfaces [257]. 

TiAl-base alloys are in the range 160-180 GPa which is only 10-20% lower than that of the superalloys (see Table 2). Recently, it has been found by ab initio calculations that deviations from stoichiometry are due to accommodated antistructure atoms, i.e. constitutional disorder, instead of vacancies in the sublattices, and that the concentration of thermal vacancies is comparatively low because of the high formation energy (Fu and Yoo, 1993). The self-diffusion of Ti in TiAl has been studied (Kroll etal., 1992). 

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