Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 1 Part 5 pps

5.4 Intermetallics 87 in Chap. 20. Some intermetallics are ordered up to their melting tempera- ture, others undergo order-disorder transitions in which an almost random arrangement of atoms is favoured at high temperatures. Such transitions occur, for example, between the β  and β phases of the Cu–Zn system or in Fe–Co. There are intermetallic phases with wide phase fields and others which exist as stoichiometric compounds. Examples for both types can even be found in the same binary alloy system. For example, the Laves phase in the Co-Nb system (approximate composition Co 2 Nb) exists over a composi- tion range of about 5 at. %, whereas the phase Co 7 Nb 2 is a line compound. Some intermetallics occur for certain stoichiometric compositions only. Oth- ers are observed for off-stoichiometric compositions. Some phases compensate off-stoichiometry by vacancies, others by antisite atoms. Thermal defect populations in intermetallics can be rather complex and we shall confine ourselves to a few remarks. Intermetallic compounds are physically very different from the ionic compounds considered in the previous section. Combination of various types of disorder are conceivable: vacancies and/or antisite defects on both sublattices can form in some intermetallics. As self-interstitials play no rˆole in thermal equilibrium for pure metals, it is reasonable to assume that this holds true also for intermetallics. To be specific, let us suppose a formula A x B y for the stoichiometric com- pound and that there is a single A sublattice and a single B sublattice. This is, for example, the case in intermetallics with the B2 and L1 2 structure (see Fig. 20.1). The basic structural elements of disorder are listed in Table 5.3. A first theoretical model for thermal disorder in a binary AB intermetal- lic with two sublattices was treated in the pioneering work of Wagner and Schottky [2]. Some of the more recent work on defect properties of inter- metallic compounds has been reviewed by Chang and Neumann [42] and Bakker [43]. In some binary AB intermetallics so-called triple defect disorder occurs. These intermetallics form V A defects on the A sublattice on the B rich side and A B antisites on the B sublattice on the A rich side of the stoichiometric composition. This is, for example, the case for some intermetallics with B2 structure where A = Ni, Co, Pd and B = Al, In, Some other inter- metallics also with B2 structure such as CuZn, AgCd, . can maintain high concentrations of vacancies on both sublattices. Table 5.3. Elements of disorder in intermetallic compounds A A = A atom on A sublattice B B = B atom on B sublattice V A = vacancy on A sublattice V B = vacancy on B sublattice B A = B antisite on A sublattice A B = A antisite on B sublattice 88 5 Point Defects in Crystals Triple defects (2V A + A B ), bound triple defects (V A A B V A )andvacancy pairs (V A V B ) have been suggested by Stolwijk et al. [46]. They can form according to the reactions V A + V B  2V A + A B    triple defect  V A A B V A    bound triple defect and V A + V B  V A V B  vacancy pair . (5.41) Very likely bound agglomerates are important in intermetallics for thermal disorder and diffusion in addition to single vacancies. In this context it is interesting to note that neither triple defects nor vacancy pairs disturb the stoichiometry of the compound. The physical understanding of the defect structure of intermetallics is still less complete compared with metallic elements. However, considerable progress has been achieved. Differential dilatometry (DD) and positron an- nihilation studies (PAS) performed on intermetallics of the Fe-Al, Ni-Al and Fe-Si systems have demonstrated that the total content of vacancy-type de- fects can be one to two orders of magnitude higher than in pure metals [44, 45]. The defect content depends strongly on composition and its temperature dependence can show deviations from simple Arrhenius behaviour. According to Schaefer et al. [44] and Hehenkamp [45] typical defect concentrations in these compounds near the solidus temperature can be as high as several percent. 5.5 Semiconductors Covalent crystals such as diamond, Si, and Ge are more different from the defect point of view as one might expect from their chemical classification as group IV elements. Diamond is an electrical insulator, whose vacancies are mobile at high temperatures only. Si is a semiconductor which supports vacancies and self-interstitials as intrinsic defects. By contrast, Ge is a semi- conductor in which vacancies as intrinsic defects predominate like in the metallic group IV elements Sn and Pb. Because Si and Ge crystallise in the diamond structure with coordination number 4, the packing density is considerably lower than in metals. This holds true also for compound semiconductors. Most compound semiconduc- tors formed by group III and group V elements like GaAs crystallise in the zinc blende structure, which is closely related to the diamond structure. Semi- conductor crystals offer more space for self-interstitials than close-packed metal structures. Formation enthalpies of vacancies and self-interstitials in semiconductors are comparable. In Si, both self-interstitials and vacancies are present in thermal equilibrium and are important for self- and solute dif- fusion. In Ge, vacancies dominate in thermal equilibrium and appear to be the only diffusion-relevant defects (see Chap. 23 and [47, 50]). 5.5 Semiconductors 89 Semiconductors have in common that the thermal defect concentrations are orders of magnitude lower than in metals or ionic crystals. This is a con- sequence of the covalent bonding of semiconductors. Defect formation ener- gies in semiconductors are higher than in metals with comparable melting temperatures. Neither thermal expansion measurements nor positron annihi- lation studies have sufficient accuracy to detect the very low thermal defect concentrations. Point defects in semiconductors can be neutral and can occur in various electronic states. This is because point defects introduce energy levels into the band gap of a semiconductor. Whether a defect is neutral or ionised depends on the position of the Fermi level as illustrated schematically in Fig. 5.10. A wealth of detailed information about the electronic states of point defects in these materials has been obtained by a variety of spectroscopic means and has been compiled, e.g., by Schulz [14]. Let us consider vacancies and self-interstitials X ∈ (V,I) and suppose that both occur in various ionised states, which we denote by j ∈ (0, 1±, 2±, ). The total concentration of the defect X at thermal equilibrium can be written as C eq X = C eq X 0 + C eq X 1+ + C eq X 1− + C eq X 2+ + C eq X 2− + . (5.42) Whereas the equilibrium concentration of uncharged defects depends only on temperature (and pressure), the concentration of charged defects is ad- ditionally influenced by the position of the Fermi energy and hence by the doping level. If the Fermi level changes due to, e.g., background doping the concentration of charged defects will change as well. The densities of electrons, n, and of holes, p, are tied to the intrinsic carrier density, n i , via the law of mass action relation np = n 2 i . (5.43) Fig. 5.10. Electronic structure of semiconductors, with a defect with double ac- ceptor character (left) and donor character (right) 90 5 Point Defects in Crystals Then, Eq. (5.42) can be rewritten as C eq X = C eq X 0 + C eq X 1+ (n i ) n i n + C eq X 1− (n i ) n n i + C eq X 2+ (n i )  n i n  2 + C eq X 2− (n i )  n n i  2 + , (5.44) where C eq X j± (n i ) denotes the equilibrium concentration under intrinsic con- ditions for defect X with charge state j±. From Eq. (5.44) it is obvious that n-doping will enhance (decrease) the equilibrium concentration of neg- atively (positively) charged defects. Correspondingly, p-doping will enhance (decrease) the equilibrium concentration positively (negatively) charged de- fects. Furthermore, the ratio n/n i varies with temperature because the intrinsic carrier density according to n i =  N c eff N v eff exp  − E g 2k B T  (5.45) increases with increasing temperature. N c eff and N v eff denote the effective den- sities of states in the conduction and valence band, respectively. The values of n i at different temperatures are determined mainly by the band gap en- ergy E g of the semiconductor. For a given background doping concentration the ratio n/n i will be large at low temperatures and approaches unity at high temperatures. Then, the semiconductor reaches intrinsic conditions. The band gap energy is characteristic for a given semiconductor. It increases in the sequence Ge (0.67 eV), Si (1.14 eV), GaAs (1.43 eV). The intrinsic carrier density at a fixed temperature is highest for Ge and lowest for GaAs. Thus, doping effects on the concentration of charged defects are most prominent for GaAs and less pronounced for the elemental semiconductors. Let us consider as an example a defect X which introduces a single X 1− and a double X 2− acceptor state with energy levels E X 1− and E X 2− above the valence band edge. Then, the ratios between charged and uncharged defect populations in thermal equilibrium are given by C eq X 1− C eq X 0 = 1 g X 1− exp  E f − E X 1− k B T  , C eq X 2− C eq X 0 = 1 g X 2− exp  2E f − E X 2− − E X 1− k B T  , (5.46) where E f denotes the position of the Fermi level. The degeneracy factors g X 1− and g X 2− take into account the spin degeneracy of the defect and the degeneracy of the valence band. The total concentration of point defects in thermal equilibrium for the present example is given by C eq X = C eq X 0  1+ C eq X 1− C eq X 0 + C eq X 2− C eq X 0  . (5.47) References 91 Diffusion in semiconductors is affected by doping since defects in various charge states can act as diffusion-vehicles. Diffusion experiments are usually carried out at temperatures between the melting temperature T m and about 0.6 T m . As the intrinsic carrier density increases with increasing temperature, doping effects in diffusion are more pronounced at the low temperature end of this interval. One can distinguish two types of doping effects: – Background doping is due to a homogeneous distribution of donor or ac- ceptor atoms, that are introduced during the process of crystal growing. Background doping is relevant for diffusion experiments, when at the dif- fusion temperature the carrier density exceeds the intrinsic density. – Self-doping is relevant for diffusion experiments of donor or acceptor ele- ments. If the in-diffused dopant concentration exceeds either the intrinsic carrier density or the available background doping, complex diffusion pro- files can arise. References 1. J.I. Frenkel, Z. Physik 35, 652 (1926) 2. C. Wagner, W. Schottky, Z. Physik. Chem. B 11, 163 (1931) 3. C.P. Flynn, Point Defects and Diffusion, Clarendon Press, Oxford, 1972 4. A.M. Stoneham, Theory of Defects in Solids, Clarendon Press, Oxford, 1975 5. F. Agullo-Lopez, C.R.A. Catlow, P. Townsend, Point Defects in Materials, Academic Press, London, 1988 6. H.J. Wollenberger, Point Defects,in:Physical Metallurgy, R.W. Cahn, P. Haasen (Eds.), North-Holland Publishing Company, 1983, p. 1139 7. L.W. Barr, A.B. Lidiard, Defects in Ionic Crystals,in:Physical Chemistry – An Advanced Treatise, Academic Press, New York, Vol. X, 1970 8. R.G. Fuller, Ionic Conductivity including Self-diffusion,in:Point Defects in Solids, J. M. Crawford Jr., L.M. Slifkin (Eds.), Plenum Press, 1972, p. 103 9. A. Seeger, D. Schumacher, W. Schilling, J. Diehl (Eds.), Vacancies and Inter- stitials in Metals, North-Holland Publishing Company, Amsterdam, 1970 10. N.L. Peterson, R.W. Siegel (Eds.), Properties of Atomic Defects in Metals, North-Holland Publishing Company, Amsterdam, 1978 11. C. Abromeit, H. Wollenberger (Eds.), Vacancies and Interstitials in Metals and Alloys; also: Materials Science Forum 15–18, 1987 12. O. Kanert, J M. Spaeth (Eds.), Defects in Insulating Materials, World Scien- tific Publ. Comp., Ltd., Singapore, 1993 13. H. Ullmaier (Vol. Ed.), Atomic Defects in Metals, Landolt-B¨ornstein, New Se- ries, Group III: Crystal and Solid State Physics, Vol. 25, Springer-Verlag, Berlin and Heidelberg, 1991 14. M. Schulz, Landolt-B¨ornstein, New Series, Group III: Crystal and Solid State Physics, Vol. 22: Semiconductors, Subvolume B: Impurities and Defects in Group IV Elements and III-V Compounds, M. Schulz (Ed.), Springer-Verlag, 1989 92 5 Point Defects in Crystals 15. N.A. Stolwijk, Landolt-B¨ornstein, New Series, Group III: Crystal and Solid State Physics, Vol. 22: Semiconductors, Subvolume B: Impurities and Defects in Group IV Elements and III-V Compounds, M. Schulz (Ed.), Springer-Verlag, 1989, p. 439 16. N.A. Stolwijk, H. Bracht, Landolt-B¨ornstein, New Series, Group III: Condensed Matter, Vol. 41: Semiconductors, Subvolume A2: Impurities and Defects in Group IV Elements, IV-IV and III-V Compounds, M. Schulz (Ed.), Springer- Verlag, 2002, p. 382 17. H. Bracht, N.A. Stolwijk, Landolt-B¨ornstein, New Series, Group III: Condensed Matter, Vol. 41: Semiconductors, Subvolume A2: Impurities and Defects in Group IV Elements, IV-IV and III-V Compounds, M. Schulz (Ed.), Springer- Verlag, 2002, p. 77 18. F. Beni`ere, Diffusion in Alkali and Alkaline Earth Halides,in:Diffusion in Semiconductors and Non-metallic Solids, Landolt-B¨ornstein, New Series, Group III: Condensed Matter, Vol. 33, Subvolume B1, D.L. Beke (Vol.Ed.). Springer-Verlag, 1999 19. G. Erdelyi, Diffusion in Miscelaneous Ionic Materials,in:Diffusion in Semi- conductors and Non-metallic Solids, Landolt-B¨ornstein, New Series, Group III: Condensed Matter, Vol. 33, Subvolume B1, D.L. Beke (Vol.Ed.). Springer- Verlag, 1999 20. A. Seeger, H. Mehrer, Analysis of Self-diffusion and Equilibrium Measurements, in: Vacancies and Interstitials in Metals,A.Seeger,D.Schumacher,J.Diehl and W. Schilling (Eds.), North-Holland Publishing Company, Amsterdam, 1970, p. 1 21. R. Feder, A.S. Nowick, Phys. Rev. 109, 1959 (1958) 22. R.O. Simmons, R.W. Balluffi, Phys. Rev. 117, 52 (1960); Phys. Rev. 119, 600 (1960); Phys. Rev. 129, 1533 (1963); Phys. Rev. 125, 862 (1962) 23. R.O. Simmons, R.W. Balluffi, Phys. Rev. 125, 862 (1962) 24. A. Seeger, J. Phys. F. Metal Phys. 3, 248 1973) 25. R.O. Simmons, R.W. Balluffi, Phys. Rev. 117, 52 (1960) 26. G. Bianchi, D. Mallejac, Ch. Janot, G. Champier, Compt. Rend. Acad. Science, Paris 263 1404 (1966) 27. B. von Guerard, H. Peisl, R. Sizmann, Appl. Phys. 3, 37 (1973) 28. H E. Schaefer, R. Gugelmeier, M. Schmolz, A. Seeger, J. Materials Science Forum 15–18, 111 (1987) 29. M. Doyama, R.R. Hasiguti, Cryst. Lattice Defects 4, 139 (1973) 30. P. Hautoj¨arvi, Materials Science Forum 15–18, 81 (1987) 31. H E. Schaefer, W. Stuck, F. Banhart, W. Bauer, Materials Science Forum 15–18, 117 (1987) 32. W.M. Lomer, Vacancies and other Point Defects in Metals and Alloys, Institute of Metals, 1958 33. J. E. Dorn, J.B. Mitchell, Acta Metall. 14, 71 (1966) 34. G. Berces, I. Kovacs, Philos. Mag. 15, 883 (1983) 35. C. Tubandt, E. Lorenz, Z. Phys. Chem. 87, 513, 543 (1914) 36. J. Maier, Physical Chemistry of Ionic Solids – Ions and Electrons in Solids, John Wiley & Sons, Ltd, 2004 37. A.R. Allnatt, A.B. Lidiard, Atomic Transport in Solids, Cambridge University Press, 1993 38. R. Friauf, Phys. Rev. 105, 843 (1957) References 93 39. R. Friauf, J. Appl. Phys. Supp. 33, 494 (1962) 40. J.H. Westbrook, Structural Intermetallics, R. Dariola, J.J. Lewendowski, C.T. Liu, P.L. Martin, D.B. Miracle, M.V. Nathal (Eds.), Warrendale, PA, TMS, 1993 41. G. Sauthoff, Intermetallics, VCH Verlagsgesellschaft, Weinheim, 1995 42. Y.A. Chang, J.P. Neumann, Progr. Solid State Chem. 14, 221 (1982) 43. H. Bakker, Tracer Diffusion in Concentrated Alloys,in:Diffusion in Crystalline Solids, G.E. Murch, A.S. Nowick (Eds.), Academic Press, Inc., 1984, p. 189 44. H E. Schaefer, K. Badura-Gergen, Defect and Diffusion Forum 143–147, 193 (1997) 45. Th. Hehenkamp, J. Phys. Chem. Solids 55, 907 (1994) 46. N.A. Stolwijk, M. van Gend, H. Bakker, Philos. Mag. A 42, 783 (1980) 47. W. Frank, U. G¨osele, H. Mehrer, A. Seeger, Diffusion in Silicon and Germa- nium,in:Diffusion in Crystalline Solids, G.E. Murch, A.S. Nowick (Eds.), Academic Press, Inc., 1984, p. 64 48. K. Girgis, Structure of Intermetallic Compounds,in:Physical Metallurgy, R.W. Cahn, P. Haasen (Eds.), North-Holland Physics Publishing, 1983, p. 219 49. M. Schulz (Vol Ed.), Impurities and Defects in Group IV Elements, IV-IV and III-V Compounds, Landolt-B¨ornstein, Group III, Vol. 41, Subvolume A2, Springer-Verlag, 2002 50. T.Y. Tan, U. G¨osele, Diffusion in Semiconductors,in:Diffusion in Con- densed Matter – Methods, Materials, Models,P.Heitjans,J.K¨arger (Eds.), Springer-Verlag, 2005 6 Diffusion Mechanisms Any theory of atom diffusion in solids should start with a discussion of dif- fusion mechanisms. We must answer the question: ‘How does this particular atom move from here to there?’ In crystalline solids, it is possible to describe diffusion mechanisms in simple terms. The crystal lattice restricts the posi- tions and the migration paths of atoms and allows a simple description of each specific atom displacements. This contrasts with a gas, where random distribution and displacements of atoms are assumed, and with liquids and amorphous solids, which are neither really random nor really ordered. In this chapter, we catalogue some basic atomic mechanisms which give rise to diffusion in solids. As discussed in Chap. 4, the hopping motion of atoms is an universal feature of diffusion processes in solids. Furthermore, we have seen that the diffusivity is determined by jump rates and jump distances. The detailed features of the atomic jump process depend on various factors such as crystal structure, size and chemical nature of the diffusing atom, and whether diffusion is mediated by defects or not. In some cases atomic jump processes are completely random, in others correlation between subsequent jumps is involved. Correlation effects are important whenever the atomic jump probabilities depend on the direction of the previous atom jump. If jumps are mediated by atomic defects, correlation effects always arise. The present chapter thus provides also the basis for a discussion of correlation effects in solid-state diffusion in Chap. 7. 6.1 Interstitial Mechanism Solute atoms which are considerably smaller than the solvent atoms are in- corporated on interstitial sites of the host lattice thus forming an interstitial solid solution. Interstitial sites are defined by the geometry of the host lat- tice. In fcc and bcc lattices, for example, interstitial solutes occupy octahedral and/or tetrahedral interstitial sites (Fig. 6.1). An interstitial solute can diffuse by jumping from one interstitial site to one of its neighbouring sites as shown in Fig. 6.2. Then the solute is said to diffuse by an interstitial mechanism. To look at this process more closely, we consider the atomic movements during a jump. The interstitial starts from an equilibrium position, reaches the saddle-point configuration where maximum lattice straining occurs, and 96 6 Diffusion Mechanisms Fig. 6.1. Octahedral and tetrahedral interstitial sites in the bcc (left)andfcc (right) lattice Fig. 6.2. Direct interstitial mechanism of diffusion settles again on an adjacent interstitial site. In the saddle-point configuration neighbouring matrix atoms must move aside to let the solute atom through. When the jump is completed, no permanent displacment of the matrix atoms remains. Conceptually, this is the simplest diffusion mechanism. It is also de- noted as the direct interstitial mechanism. It has to be distinguished from the interstitialcy mechanism discussed below, which is also denoted as the indi- rect interstitial mechanism. We note that no defect is necessary to mediate direct interstitial jumps, no defect concentration term enters the diffusiv- ity and no defect formation energy contributes to the activation energy of diffusion. Since the interstitial atom does not need to ‘wait’ for a defect to perform a jump, diffusion coefficients for atoms migrating by the direct in- terstitial mechanism tend to be fairly high. This mechanism is relevant for diffusion of small foreign atoms such as H, C, N, and O in metals and other materials. Small atoms fit in interstitial sites and in jumping do not greatly displace the solvent atoms from their normal lattice sites. 6.2 Collective Mechanisms 97 Fig. 6.3. Direct exchange and ring diffusion mechanism 6.2 Collective Mechanisms Solute atoms similar in size to the host atoms usually form substitutional solid solutions. The diffusion of substitutional solutes and of solvent atoms themselves requires a mechanism different from interstitial diffusion. In the 1930s it was suggested that self- and substitutional solute diffusion in metals occurs by a direct exchange of neighbouring atoms (Fig. 6.3), in which two atoms move simultaneously. In a close-packed lattice this mechanism requires large distortions to squeeze the atoms through. This entails a high activation barrier and makes this process energetically unfavourable. Theoret- ical calculations of the activation enthalpy for self-diffusion of Cu performed by Huntington et al. in the 1940s [1, 2], which were confirmed later by more sophisticated theoretical approaches, led to the conclusion that direct exchange at least in close-packed structures was not a likely mechanism. The so-called ring mechanism of diffusion was proposed for crystalline solids by the American metallurgist Jeffries [3] already in the 1920s and advocated by Zener in the 1950s [4]. The ring mechanism corresponds to a rotation of 3 (or more) atoms as a group by one atom distance. The required lattice distortions are not as great as in a direct exchange. Ring versions of atomic exchanges have lower activation energies but increase the amount of collective atomic motion, which makes this more complex mechanism unlikely for most crystalline substances. Direct exchange and ring mechanismshaveincommonthatlatticede- fects are not involved. The observation of the so-called Kirkendall effect in alloys by Kirkendall and coworkers [5, 6] during the 1940s had an im- portant impact on the field (see also Chaps. 1 and 10). The Kirkendall effect showed that the self-diffusivities of atoms in a substitutional binary alloy dif- fuse at different rates. Neither the direct exchange nor the ring mechanism can explain this observation. As a consequence, the ideas of direct or ring exchanges were abandoned in the diffusion literature. It became evident that [...]... 20 05 a 13 H Teichler, J Non-cryst Solids 293, 339 (20 01) 14 S Voss, S Divinski, A.W Imre, H Mehrer, J.N Mundy, Solid State Ionics 17 6, 13 83 (20 05) ; and: A.W Imre, S Voss, H Staesche, M.D Ingram, K Funke, H Mehrer, J Phys Chem B, 11 1, 53 01 53 07 (2007) 15 N.L Peterson, J Nucl Materials 69–70, 3 (19 78) 16 H Mehrer, J Nucl Materials 69–70, 38 (19 78) 17 F.C Frank, D Turnbull, Phys Rev 10 4, 617 (19 56 ) 18 ... (19 24) C Zener, Acta Cryst 3, 346 (19 50 ) E.O Kirkendall, Trans AIME 14 7, 10 4 (19 42) A.D Smigelskas, E.O Kirkendall, Trans AIME 17 1, 13 0 (19 47) C Tuijn, Defect and Diffusion Forum 14 3 14 7, 11 (19 97) 10 4 6 Diffusion Mechanisms 8 A Seeger, in: Ultra-High-Purity Metals, K Abiko, K Hirokawa, S Takaki (Eds.), The Japan Institute of Metals, Sendai, 19 95, p 27 9 A Seeger, Defect and Diffusion Forum 14 3 14 7, 21. .. Table 7 .1 Return probability of a vacancy, π0 , and mean number of tracer jumps in a tracer-vacancy encounter, nenc , for various lattices according to Allnatt and Lidiard [30] Lattice π0 nenc linear chain square planar diamond imple cubic bcc fcc 1 1 0.4423 0.34 05 0.2822 0. 253 6 ∞ ∞ 1. 7929 1. 51 6 4 1. 3932 1. 3447 bcc, octahedral interstices bcc, tetrahedral interstices 0.4287 0.47 65 1. 750 4 1. 910 2 dependence... τ1V , on a particular ¯ lattice site is given by 1 τ1V = ¯ (7 . 15 ) Zω1V eq In Chap 5 we have seen that for metals the site fraction of vacancies, C1V , is a very small number Even close to the melting temperature it never exceeds 10 −3 to 10 −4 ; in semiconductors the site fractions of point defects in equilibrium are even much smaller Due to its Arrhenius type temperature 11 0 7 Correlation in Solid- State. .. Phys 23, 3 61 (19 80) o 19 T.Y Tan, U G¨sele, Diffusion in Semiconductors, Ch 4 in: Diffusion in Cono densed Matter – Methods, Materials, Models, P Heitjans, J K¨rger (Eds.), a Springer-Verlag, 20 05 20 W Frank, U G¨sele, H Mehrer and A Seeger, in: Diffusion in Crystalline o Solids, G.E Murch, A.S Nowick (Eds.), Academic Press, 19 84, p 63 21 H Bracht, N.A Stolwijk, H Mehrer, Phys Rev B 52 , 16 54 2 (19 95) 7 Correlation... fraction of vacancies, C1V The total number of atomic (and tracer) jumps per unit time ( jump rate Γtot ), is proportional to this probability and to the rate of vacancy-atom exchanges, ω1V : eq Γtot = ZC1V ω1V (7 .13 ) eq The explicit form of C1V ω1V is given by Eq (6.2) The mean residence time of the tracer atom, τ , on a particular lattice site is ¯ τ= ¯ 1 1 = eq Γtot Zω1V C1V (7 .14 ) On the other hand,... crystal, C1V , is given by Eq (5 .11 ), which we repeat for convenience: eq C1V = exp − GF 1V kB T = exp F S1V kB exp − F H1V kB T (6 .1) F F GF is the Gibbs free energy of vacancy formation S1V and H1V denote the 1V formation entropy and the formation enthalpy of a monovacancy, respectively Typical values for the site fraction of vacancies near the melting temperature of metallic elements lie between 10 −4... metallic elements lie between 10 −4 and 10 −3 From Eqs (4. 31) and (6 .1) we get for the exchange jump rate Γ of a vacancy-mediated jump of a matrix atom to a particular neighbouring site eq Γ = ω1V C1V = ν 0 exp F M S1V + S1V kB exp − F M H1V + H1V kB T (6.2) ω1V denotes the exchange rate between an atom and a vacancy and ν 0 the M M pertinent attempt frequency H1V and S1V denote the migration enthalpy and... equations – apart from a few (but interesting) special cases – can only be obtained by numerical methods [20, 21] These solutions also explain unusual (non-Fickian) shapes of concentrationdistance profiles observed for hybrid diffusers Details are discussed in Part IV of this book References 1 2 3 4 5 6 7 H.B Huntington, F Seitz, Phys Rev 61, 3 15 (19 42) H.B Huntington, Phys Rev 61, 3 25 (19 42) Z Jeffries,... square displacement of an ensemble of N such atoms, where R2 = 1 N N R2 k (7.2) k =1 It is assumed that the net displacement of the k th atom is the result of a large number of nk jumps with microscopic jump vectors ri (i =1, k), so that nk ri Rk = i =1 (7.3) k Thus R2 = 1 N nk N 2 ri k =1 i =1 + k 2 N N ⎛ nk 1 nk −i ⎝ i =1 k =1 ⎞ r i r i+j ⎠ j =1 (7.4) k For simplicity, we restrict our discussion to cases . Amsterdam, 19 70, p. 1 21. R. Feder, A.S. Nowick, Phys. Rev. 10 9, 19 59 (19 58 ) 22. R.O. Simmons, R.W. Balluffi, Phys. Rev. 11 7, 52 (19 60); Phys. Rev. 11 9, 600 (19 60); Phys. Rev. 12 9, 15 33 (19 63); Phys Funke, H.Mehrer,J.Phys.Chem.B ,11 1, 53 01 53 07 (2007) 15 . N.L. Peterson, J. Nucl. Materials 69–70, 3 (19 78) 16 . H. Mehrer, J. Nucl. Materials 69–70, 38 (19 78) 17 . F.C. Frank, D. Turnbull, Phys. Rev. 10 4, 617 (19 56 ) 18 3, 346 (19 50 ) 5. E.O. Kirkendall, Trans. AIME 14 7, 10 4 (19 42) 6. A.D. Smigelskas, E.O. Kirkendall, Trans. AIME 17 1, 13 0 (19 47) 7. C. Tuijn, Defect and Diffusion Forum 14 3 14 7, 11 (19 97) 10 4 6 Diffusion