Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 2 Part 4 docx

322 18 Diffusion of Interstitial Solutes in Metals Fig. 18.2. Diffusion of H in Pd, Ni and Fe according to Alefeld and V ¨ olkl [13] bility of H in Ni other methods such as Gorsky effect, NMR, and QENS have limited applicability for the Ni-H system. The Arrhenius parameters below the magnetic transition (Curie temperature T C = 627 K) seem to be only slightly different than above (see Fig. 18.2). Diffusion of hydrogen in iron has considerable importance due to the technical challenge of hydrogen embrittlement of iron and steels. The absolute values for H-diffusion in Fe are higher than those for Pd and Ni (see Fig 18.2). In spite of the technological interest and the large number of studies (for references see [19]), the scatter of the data is rather large especially below room temperature. Several reasons for this scatter have been discussed in the literature: surface effects, trapping of hydrogen by impurities, dislocations, grain boundaries, or precipitates, and the formation of molecular hydrogen in micropores, either already existing in the material or produced by excess loading (for references see [18]). Hydrogen in niobium has been studied over a very wide temperature region (see Fig. 18.3). Because of the high diffusivities and its relatively small changes with temperature the Gorsky technique could be used for both H and D in nearly the whole temperature range. Tritium diffusion has been studied in a more limited temperature range. Isotope effects of hydrogen diffusion in Nb are very evident from Fig. 18.3. Similar isotope effect studies are available for V and Ta (see Table 18.3). 18.2 Hydrogen Diffusion in Metals 323 Fig. 18.3. Diffusion of H, D, and T in Nb according to [13] 18.2.4 Non-Classical Isotope Effects Classical rate theory for the isotope effect in diffusion predicts, if many-body effects are neglected, for the ratio of the diffusivities [33, 34], D 1 D 2 =  m 2 m 1 , (18.8) where m 1 and m 2 are the isotope masses. If manybody effects are taken into account, Eq. (18.8) must be modified (see Chap. 9). Nevertheless, the isotope effect is still completely attributed to the pre-exponential factors. In the classical limit, the activation enthalpies are independent of the isotope masses. For several metal-hydrogen systems non-classical isotope effects have been reported. Figure 18.3 and Table 18.3 show that in group-V transition metals hydrogen (H) diffuses more rapidly than deuterium (D), and deuterium dif- fuses more rapidly than tritium (T). In addition, the activation enthalpies of hydrogen isotopes are different and hence the ratio of the diffusivities varies with temperature. Interestingly, the characteristic features of the dependence of the diffusiv- ity on the isotope mass are correlated with the structure of the host metal [13, 14]. In the bcc metals V, Nb and Ta, hydrogen diffuses faster than deuterium 324 18 Diffusion of Interstitial Solutes in Metals Table 18.3. Activation parameters for diffusion of H, D, and T in Nb, Ta and V according to Alefeld and V ¨ olkl [13] Isotope Parameters Nb Ta V H D 0 /m 2 s −1 5 × 10 −8 (T>273 K) 4.4 × 10 −8 2.9 × 10 −8 0.9 × 10 −8 (T<273 K) ∆H/eV 0.106 (T>273 K) 0.140 0.043 0.068 (T<223 K) D D 0 /m 2 s −1 5.4 × 10 −8 4.9 × 10 −8 3.7 × 10 −8 ∆H/eV 0.129 0.163 0.08 T D 0 /m 2 s −1 4.5 × 10 −8 ∆H/eV 0.125 in the whole temperature regime investigated. For the activation enthalpies ∆H H < ∆H D (18.9) is observed (see Table 18.3). As a consequence, the ratio D H /D D for the group-V transition metals increases with decreasing temperature. For the fcc metals, the pre-exponential factors of H and D scale in accordance with Eq. (18.8) within the experimental errors. However, again the activation en- thalpies are mass-dependent, but in contrast to bcc metals one observes ∆H H > ∆H D . (18.10) This fact leads to an inverse isotope effect at lower temperatures. For example, in Pd below about 773 K deuterium diffuses faster than hydrogen. Because of the non-classical behaviour of hydrogen and the large temper- ature region over which diffusion coefficients were measured, it is by no means evident that the diffusion coefficient of hydrogen in metals should obey a lin- ear Arrhenius relation over the entire temperature range. The break observed in the Arrhenius relation for H in Nb (see Fig. 18.3) was first observed with the Gorsky effect method and confirmed independently by measurements of resistivity changes [35] and by QENS [36]. The deviation of hydrogen diffu- sion in Nb from an Arrhenius law at lower temperatures (Fig. 18.3) has also been observed in the tantalum-hydrogen system. It has been attributed to incoherent tunneling. References 1. A.D. Le Claire, Diffusion of C, N, and O in Metals Chap. 8 in: Diffusion in Solid Metals and Alloys, H. Mehrer (Vol.Ed.), Landolt-B¨ornstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Group III: Crystal and Solid State Physics, Vol. 26, Springer-Verlag, 1990, p. 471 References 325 2. A.S. Nowick, B.S. Berry, Anelastic Relaxation in Crystalline Solids, Academic Press, New York, (1972) 3. R.D. Batist, Mechanical Spectroscopy,in:Materials Science and Technology, Vol. 2B: Characterisation of Materials, R.W. Cahn, P. Haasen, E.J. Cramer (Eds.), VCH, Weinheim, 1994, p. 159 4. H. Kronm¨uller, Nachwirkung in Ferromagnetika, Springer Tracts in Natural Philosophy, Springer-Verlag (1968) 5. J. Philibert, Atom Movements – Diffusion and Mass Transport in Solids,Les Editions de Physique, Les Ulis, 1991 6. Th. Heumann, Diffusion in Metallen, Springer-Verlag, Berlin, 1992 7. A.J. Bosman, PhD Thesis, University of Amsterdam (1960) 8. D.C. Parris, R.B. McLellan, Acta Metall. 24, 523 (1976) 9. R.P. Smith, Acta Metall. 1, 576 (1953) 10. I.V. Belova, G.E. Murch, Philos. Mag. 36, 4515 (2006) 11. T. Graham, Phil. Trans. Roy. Soc. (London) 156, 399 (1866) 12. A. Sieverts, Z. Phys. Chem. 88, 451 (1914) 13. G. Alefeld, J. V¨olkl (Eds.), Hydrogen in Metals I – Application-oriented Prop- erties, Topics in Applied Physics, Vol. 28, Springer-Verlag, 1978 14. G. Alefeld, J. V¨olkl (Eds.), Hydrogen in Metals II – Basic Properties,Topics in Applied Physics, Vol. 29, Springer-Verlag, 1979 15. H. Wipf (Ed.), Hydrogen in Metals III – Properties and Applications,Topicsin Applied Physics Vol. 73, Springer-Verlag, 1997 16. H.K. Birnbaum, C.A. Wert, Ber. Bunsenges. Phys. Chem. 76, 806 (1972) 17. R. Hempelmann, J. Less Common Metals 101, 69 (1984) 18. J. V¨olkl,G.Alefeld,inDiffusion in Solids – Recent Developments, A.S. Nowick, J.J. Burton (Eds.) Academic Press, 1975, p. 232 19. G.V. Kidson, The Diffusion of H, D, and T in Solid Metals, Chap. 9 in: Dif- fusion in Solid Metals and Alloys, H. Mehrer (Vol. Ed.), Landolt-B¨ornstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Group III: Crystal and Solid State Physics, Vol. 26, Springer-Verlag, 1990, p. 504 20. H. Matzke, V.V. Rondinella, Diffusion in Carbides, Nitrides, Hydrides, and Borides,in:Diffusion in Semiconductors and Non-metallic Solids,D.L.Beke (Vol.Ed), Landolt-B¨ornstein, Numerical Data and Functional Relationships in Science and Technology, Vol. 33, Subvolume B1, Springer-Verlag, 1999 21. H. Z¨uchner, N. Boes, Ber. Bunsenges. Phys. Chem. 76, 783 (1972) 22. H. Kronm¨uller, Magnetic After-Effects of Hydrogen Isotopes in Ferromagnetic Metals and Alloys, Ch. 11 in [13] 23. R.B. McLellan, W.A. Oates, Acta Metall. 21, 181 (1973) 24. R.M. Cotts, Ber. Bunsenges. Phys. Chem. 76, 760 (1972) 25. G. Majer, Die Methoden der Kernspinresonanz zum Studium der Diffusion von Wasserstoff in Metallen und intermetallischen Verbindungen, Cuvillier Verlag, G¨ottingen, 2000 26. P. Heitjans, S. Indris, M. Wilkening, Solid-State Diffusion and NMR,in:Diffu- sion Fundamentals – Leipzig 2005,J.K¨arger, F. Grinberg, P. Heitjans (Eds.), Leipziger Universit¨atsverlag 2005 27. R. Hempelmann, Quasielastic Neutron Scattering and Solid-State Diffusion, Oxford Science Publication, 2000 28. T. Springer, Z. Phys. Chem. NF 115, 317 (1979) 326 18 Diffusion of Interstitial Solutes in Metals 29. T. Springer, Diffusion Studies of Solids by Quasielastic Neutron Scattering,in: Diffusion in Condensed Matter,J.K¨arger, P. Heitjans, R. Haberlandt (Eds.), Vieweg-Verlag, 1998, p. 59 30. T. Springer, R.E. Lechner, Diffusion Studies of Solids by Quasielastic Neutron Scattering,in:Diffusion in Condensed Matter – Methods, Materials, Models,P. Heitjans, J. K¨arger (Eds.), Springer-Verlag, 2005 31. H. Zabel, Quasielastic Neutron Scattering: a Powerful Tool for Investigating Diffusion in Solids,in:Nontraditional Methods in Diffusion,G.E.Murch,H.K. Birnbaum, J.R. Cost (Eds.), The Metallurgical Society of AIME, Warrendale, 1984, p. 1 32. K. Sk¨old, Quasielastic Neutron Scattering Studies of Metal Hydrides,Ch.11 in [13] 33. C.Wert,C.Zener,Phys.Rev.76, 1169 (1949) 34. C. H Vineyard, J. Phys. Chem. Sol. 3 121 (1957) 35. H. Wipf, G. Alefeld, Phys. Stat. Sol. (a) 23, 175 (1974) 36. D. Richter, B. Alefeld, A. Heidemann, N. Wakabayashi, J. Phys. F: Metal Phys. 7, 569 (1977) 37. K. Yamakawa, M. Ege, B. Ludescher, M. Hirscher, H. Kronm¨uller, J. of Alloys and Compounds 321, 17 (2001) 38. M.Hirscher,H.Kronm¨uller, J. of the Less-common Metals 172, 658–670 (1991) 19 Diffusion in Dilute Substitutional Alloys Diffusion in alloys is more complex than self-diffusion in pure metals. In this chapter, we consider dilute substitutional binary alloys of metals A and B with the mole fraction of B atoms much smaller than that of A atoms. Then A is denoted as the solvent (or matrix ) and B is denoted as the solute.Dif- fusion in a dilute alloy has two aspects: solute diffusion and solvent diffusion. We consider at first solute diffusion at infinite dilution. This is often called impurity diffusion. Impurity diffusion implies concentrations of the solute less than 1 %. In practice, the sensitivity for detection of radioactive solutes en- ables one to study diffusion of impurities at very high dilution of less than 1 ppm. In very dilute substitutional alloys solute and solvent diffusion can be analysed in terms of vacancy-atom exchange rates. 19.1 Diffusion of Impurities Impurity diffusion is a topic of diffusion research to which much scientific work has been devoted. We consider at first ‘normal’ behaviour of substitutional impurities, which is illustrated for the solvent silver. Similar behaviour is observed for the other noble metals, for hexagonal Zn and Cd, and for Ni. There are exceptions from ‘normal’ impurity diffusion. A prominent ex- ample is the slow diffusion of transition elements in the trivalent solvent aluminium. Another example is the very fast diffusion of impurities in so- called ‘open’ metals. Lead is the most famous open metal, for which very fast impurity diffusion has been observed. The rapid diffusion of Au in Pb was discovered by the diffusion pioneer Roberts-Austen in 1896 (see Chap. 1). It is beyond the scope of this book to give a comprehensive overview of impurity diffusion. Instead, we refer the reader to the chapter of Le Claire and Neumann in the data collection edited by the present author [1] and to the review by Neumann and Tuijn [2]. 19.1.1 ‘Normal’ Impurity Diffusion Figure 19.1 shows an Arrhenius diagram for diffusion of many substitutional foreign atoms in a Ag matrix together with self-diffusion of Ag. A comparison 328 19 Diffusion in Dilute Substitutional Alloys Fig. 19.1. Diffusion of substitutional impurities in Ag and self-diffusion of Ag (dashed line). Diffusion parameters from [1, 2] of impurity diffusion and self-diffusion coefficients, D 2 and D, reveals the following features of ‘normal’ diffusion of substitutional impurities: – The diffusivities of impurities, D 2 , lie in a relatively narrow band around self-diffusivities, D, of the solvent atoms. The following limits apply in the temperature range between the melting temperature T m and about 2/3 T m : 1/100 ≤ D 2 /D ≤ 100 . (19.1) – The pre-exponential factors mostly lie in the interval 0.1 < D 0 (solute) D 0 (self ) < 10 . (19.2) – The activation enthalpies of impurity and self-diffusion, ∆H 2 and ∆H, are not much different: 0.75 < ∆H 2 ∆H < 1.25 . (19.3) 19.1 Diffusion of Impurities 329 Substitutional impurities in other fcc metals (Cu, Au, Ni) and in the hcp met- als (Zn, Cd) behave similarly as in Ag (for references see the chapter of Le Claire and Neumann in the data collection [1]). Like self-diffusion of the solvent, diffusion of substitutional impurities occurs via the vacancy mecha- nism. For impurity diffusion in a very dilute alloy it is justified to assume that solute atoms are isolated, i.e. interaction with other solute atoms (formation of solute pairs, triplets, etc.) is negligible. The theory of vacancy-mediated diffusion of substitutional impurities takes into account three aspects: 1. The formation of solute-vacancy pairs: we remind the reader of the Lomer equation introduced in Chap. 5, which shows that the proba- bility of a vacancy occupying a nearest-neighbour site of a substitutional impurity is given by p = C eq 1V exp  G B k B T  =exp  S F 1V − S B k B  exp  − H F 1V − H B k B T  (19.4) where G B = H B −TS B is the Gibbs free energy of solute-vacancy inter- action, C eq 1V the atomic fraction of vacancies in the pure matrix in thermal equilibrium. H F 1V and S F 1V denote the formation enthalpy and entropy of a monovacancy. For G B > 0 the interaction is attractive, for G B < 0it is repulsive. 2. In contrast to the case of self-diffusion in the pure matrix, for impurity diffusion it is necessary to consider several atom-vacancy exchange rates. Five types of exchanges, between vacancy, impurity and host atoms have been introduced in Lidiards ‘five-frequency model’ (see Chap. 7). 3. The impurity correlation factor is then no longer a constant depending on the lattice geometry as in the case of self-diffusion. It depends on the various jump rates of the vacancy (see Chap. 7). From the atomistic description developed in Chap. 8, we get for the impurity diffusion coefficient of vacancy-mediated diffusion in cubic Bravais lattices D 2 = f 2 a 2 ω 2 p = f 2 a 2 ω 2 C eq 1V exp  G B k B T  , (19.5) where a is the lattice parameter, f 2 the impurity correlation factor, and ω 2 the vacancy-impurity exchange rate. To be specific, we consider in the following fcc solvents. As discussed in Chap. 7, within the framework of the ‘five-frequency model’ the correlation factor can be written as [3] f 2 = ω 1 +(7/2)F 3 ω 3 ω 2 + ω 1 +(7/2)F 3 ω 3 . (19.6) The various jumps rates in an fcc lattice are illustrated in Fig. 7.4. For con- venience we remind the reader of their meaning: ω 1 is the rotation rate of 330 19 Diffusion in Dilute Substitutional Alloys the solute-vacancy complex, ω 3 and ω 4 denote rates of its dissociation or as- sociation. The escape probability F 3 is a function of the ratio ω/ω 4 ,where ω denotes the vacancy jump rate in the pure matrix. It is also useful to remember that in detailed thermal equilibrium according to ω 4 ω 3 =exp  G B k B T  (19.7) the dissociation and association rates are related to the Gibbs free energy of binding of the vacancy-impurity complex, G B = H B −TS B . Equation (19.5) can then be recast to give D 2 = f 2 a 2 ν 0 exp  S F 1V − S B + S M 2 k B  exp  − H F 1V − H B + H M 2 k B T  , (19.8) where H B and S B denote the binding enthalpy and entropy of the vacancy- impurity complex and H M 2 and S M 2 the enthalpy and entropy of the vacancy- impurity exchange jump ω 2 . Thus, the activation enthalpy of impurity diffu- sion is given by ∆H 2 = H F 1V − H B + H M 2 + C, (19.9) where C = −k B ∂ ln f 2 ∂(1/T ) . (19.10) This term describes the temperature dependence of the impurity correlation factor 1 . It is evident from Eqs. (19.5), (19.7), and Chap. 17 that the ratio of the diffusion coefficients of impurity and self-diffusion can be written as D 2 D = f 2 f ω 2 ω ω 4 ω 3 . (19.11) This expression shows that the diffusion coefficient of a substitutional im- purity differs from the self-diffusion coefficient of the pure solvent for three reasons, namely: (i) correlation effects, because f 2 = f, (ii) differences in the atom-vacancy exchange rates between impurity and solvent atoms, be- cause ω 2 = ω, and (iii) interaction between impurity and vacancy, because ω 4 /ω 3 =1orG B =0. Since solute and solvent atoms are located on the same lattice and since the diffusion of both is mediated by vacancies, the rather small diffusivity dis- persion (see Fig. 19.1) is not too surprising. It reflects the high efficiency of screening of point charges in some metals, which normally limits the vacancy- impurity interaction enthalpy H B to values between 0.1 and 0.3 eV. Such val- ues are small relative to the vacancy formation enthalpies (see Chap. 5). Using 1 Sometimes in the literature C is defined with the opposite sign. 19.1 Diffusion of Impurities 331 ∆H = H F 1V + H M 1V , we get for the difference of the activation enthalpies be- tween impurity and self-diffusion ∆Q ≡ ∆H 2 − ∆H = −H B +(H M 2 − H M 1V )+C. (19.12) A useful theoretical approach is the so-called electrostatic model,which associates ∆Q with the valence difference between solute and solvent. Rel- atively positive impurities, generally those of higher nominal valence than the solvent, tend to attract vacancies and, hence, to diffuse more rapidly and with lower activation enthalpies than self-diffusion. In the electrostatic model of Lazarus [4] refined by Le Claire [5], it is assumed that the excess charge ∆Ze (e = electron charge) is responsible for ∆Q. Vacancy and im- purity are considered to behave as point charges −Ze and Z 2 e, respectively. In the Thomas-Fermi approximation, the excess charge of the impurity gives rise to a perturbing potential V (r)=α ∆Ze r exp(−qr) . (19.13) This equation describes a screened Coulomb potential with a screening radius 1/q, which is independent of ∆Z and can be calculated from the Fermi energy of the host. α is a dimensionless screening parameter, which depends on ∆Z [5]. In the electrostatic model, the interaction enthalpy H B is equal to the electrostatic energy V (d) of the vacancy located at a nearest-neighbour distance, d, of the impurity. For the calculation of the differences in the migration enthalpies in the second and third term of Eq. (19.12), Le Claire describes the saddle-point configuration by two ‘half-vacancies’ located on two neighbouring sites. Each half-vacancy carries a charge Ze/2atadistance 11 16 d [5]. In a number of solvents, the quantity ∆Q varies indeed monotonically with the valence difference ∆Z = Z 2 − Z (Z 2 = valence of impurity, Z= valence of solvent). Impurities with Z 2 >Z(Z 2 <Z) diffuse faster (slower) than self-diffusion, the pertaining activation enthalpies fulfill ∆Q<0(∆Q>0), and the impurity diffusion coefficients increase with ∆Z. This behaviour is observed for the following solvents: noble metals and group-IIB hexagonal metals Zn and Cd (for references see [1]). Calculations of ∆Q based on the electrostatic model yield good agreement with experiments for impurity dif- fusion in these metals. For transition metal solutes in noble metals and for other solvents such as the alkali metals, the divalent magnesium, and the trivalent aluminium the values of ∆Q calculated on the basis of a screened Coulomb potential are at variance with the experimentel values [5, 6]. There are several reasons for the failure of the electrostatic model: (i) The choice of a screened Coulomb potential may not be appropriate for certain solute-solvent combinations. A self-consistent potential has an oscillatory form with so-called Friedel oscillations. The vacancy-solute [...]... Le Claire, G Neumann, Diffusion of Impurities in Solid Metallic Elements, Chap 3 in: Diffusion in Solid Metals and Alloys, H Mehrer (Vol Ed.), References 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 339 Landolt-B¨rnstein, Numerical Data and Functional Relationships in Science o and Technology, New Series Vol III /26 , Springer-Verlag, (1990) G Neumann, C Tuijn, Impurity... diffusivities of about 10−19 m2 s 20 .2 Influence of Order-Disorder Transitions 345 Fig 20 .3 Self-diffusion of 64 Cu and 65 Zn in CuZn according to Kuper et al [17] Diffusion studies have been extended by Iijima and coworkers [20 , 21 ] down to 2 × 10 21 m2 s−1 The ratio DF e /DCo exeeds unity by not more than 20 % above about 940 K corresponding to diffusivities around 10 20 m2 s−1 This ratio becomes slightly... Howard and Manning [31] have deduced the following expression ω 4 χ1 ω 1 2 + 14 4 ω f ω3 f 2 ω 3 χ1 ω 1 + 14 exp = −18 + 4 f ω f ω b1 = −18 + GB kb T (19 . 24 ) Here f is the tracer correlation factor of self-diffusion in the pure solvent (only ω-jumps) The quantities χ1 and 2 are partial correlation factors tabulated in [31] Equation (19 . 24 ) is based on the assumption that the jump rates are independent... > ω) Using Eqs (19.5), (19.6), and (19 . 24 ), b1 can be expressed as a function ∗ ∗ of the ratio DB (0)/DA (0): b1 = −18 + 4 ∗ DB (0) f 4 1 (ω1 /ω3 ) + 14 2 ∗ DA (0) 1 − f2 f (4 1 /ω + 14F3 ) (19 .25 ) 338 19 Diffusion in Dilute Substitutional Alloys Fig 19.5 Tracer diffusion of 113 Sn and 195 Au in dilute Au-Sn solid- solution alloys according to Herzig and Heumann [29 ] ∗ ∗ This expression shows that faster... Turnbull, Phys Rev 1 04, 617 (1956) U G¨sele, W Frank, A Seeger, Appl Phys A 23 , 361 (1980) o N.A Stolwijk, G B¨sker, J P¨pping, Defect and Diffusion Forum 1 94 199, o o 687 (20 01) H Bracht, N.A Stolwijk, H Mehrer, Phys Rev B 52, 165 42 (1995) Chr Herzig, Th Heumann, Z f¨r Naturforschung 27 a, 1109 (19 72) u A.B Lidiard, Philos Mag 5, 1171 (1960) R.E Howard, J.R Manning, Phys Rev 1 54, 56 12 (1967) I.V Belova,... lattice In Fig 20 .1, A atoms occupy white and grey sites, B atoms occupy black sites D03 order is observed, for example, in Fe3 Si Fig 20 .1 Ideally ordered structures of some cubic intermetallics: B2 (left), D03 (middle), L 12 (right) Fig 20 .2 Ideally ordered structures of titanium aluminides: L10 (left), D019 (right) 20 .1 General Remarks 343 over a wide temperature range, in Fe3 Al below about 825 K, and... Switzerland, 20 02 u J.R Manning, Diffusion Kinetics for Atoms in Crystals, van Nostrand Comp., Princeton, 1968 D Lazarus, Phys Rev 93, 973 (19 54) A.D Le Claire, Philos Mag 7, 141 (19 62) J Philibert, Atom Movements – Diffusion and Mass Transport in Solids, Les Editions de Physique, Les Ulis, 1991 R.M Cotts, Ber Bunsenges Phys Chem 76, 760 (19 72) Th Heumann, Diffusion in Metallen, Springer-Verlag, Berlin, 19 92 G... enthalpies obey the following sequence ∆HD03 > ∆HB2 > ∆HA2 (20 .2) The activation enthalpy is highest for the structure with the highest degree of order and lowest for the disordered structure The effect of ordering transitions is, however, less pronounced in Fe3 Al than in CuZn  346 20 Diffusion in Binary Intermetallics 20 .3 B2 Intermetallics Intermetallics with B2 structure form one of the largest group of... on intermetallics concerns B2 phases (see Table 20 .1) Figure 20 .3 reveals a fairly general feature of self-diffusion in B2 intermetallics We recognise that Zn in β-brass diffuses only slightly faster than Cu and that the ratio DZn /DCu never exceeds 2. 3 [17] For equiatomic FeCo, the ratio DF e /DCo is always close to unity [18 21 ] This type of ‘cou- Fig 20 .4 Self-diffusion in B2 structure intermetallics... obey the inequality (20 .1) ∆HB2 > ∆HA2 Figure 20 .3 also shows that occurrence of order impedes the diffusion of both components in a similar way The Fe-Co system undergoes an order-disorder transition from a disordered A2 high-temperature phase to an ordered B2 phase at lower temperatures At the equiatomic composition the transformation temperature is 1003 K Similar effects of the B2-A2 transition as in . Ta V H D 0 /m 2 s −1 5 × 10 −8 (T> ;27 3 K) 4. 4 × 10 −8 2. 9 × 10 −8 0.9 × 10 −8 (T< ;27 3 K) ∆H/eV 0.106 (T> ;27 3 K) 0. 140 0. 043 0.068 (T< ;22 3 K) D D 0 /m 2 s −1 5 .4 × 10 −8 4. 9 × 10 −8 3.7. (19.6), and (19 . 24 ), b 1 can be expressed as a function of the ratio D ∗ B (0)/D ∗ A (0): b 1 = −18 + 4 D ∗ B (0) D ∗ A (0) f 1 − f 2 4 1 (ω 1 /ω 3 )+ 14 2 f (4 1 /ω +14F 3 ) . (19 .25 ) 338 19 Diffusion. deduced the following expression b 1 = −18 + ω 4 ω  4 χ 1 f ω 1 ω 3 + 14 χ 2 f  = −18 +  4 χ 1 f ω 1 ω + 14 χ 2 f ω 3 ω  exp  G B k b T  . (19 . 24 ) Here f is the tracer correlation factor of