Materials Transactions, Vol. 51, No. 10 (2010) pp. 1809 to 1813
`#2010 The Japan Institute of Metals
`
`Diffusion of Aluminum in

-Titanium
`
`Sung-Yul Lee1, Osamu Taguchi2;* and Yoshiaki Iijima3
`
`1Department of Marine Equipment Engineering, College of Engineering, Korea Maritime University, Pusan 606 791, Korea
`2Department of Materials Science and Engineering, Miyagi National College of Technology, Natori 981 1239, Japan
`3Department of Materials Science, Graduate School of Engineering, Tohoku University, Sendai 980 8579, Japan
`
`Interdiffusion coefficient ~DD in the

phase of Ti Al alloys has been determined by Matano’s method in the temperature range 1323 to
`1823 K with (pure Ti) (Ti 8.5 at% Al alloy), (pure Ti) (Ti 16.5 at% Al alloy) and (Ti 8.5 at% Al alloy) (Ti 16.5 at% Al alloy) couples. In the
`whole temperature range the value of ~DD increases gradually with increasing aluminum content. The Arrhenius plot of ~DD up to 6 at% Al shows an
`upward curvature similar to that recognized in the self diffusion in

Ti. The curvature becomes small with increasing aluminum content, and it
`is nearly linear in the concentration more than 10 at% Al. The activation energies for the impurity diffusion in

Ti are proportional to the square
`of radius of the diffusing atom. This suggests that the size effect is dominant in the impurity diffusion in

Ti.
`[doi:10.2320/matertrans.M2010225]
`
`(Received July 2, 2010; Accepted July 27, 2010; Published September 8, 2010)
`
`Keywords: aluminum diffusion in

titanium, diffusion in

titanium, curved Arrhenius plot, size effect
`
`1.
`
`Introduction
`
`It is well known that the IVb metals (Ti, Zr and Hf) and
`their alloys in the b.c.c. phase show a so-called anomalous
`diffusion behavior which can be characterized by a signifi-
`cant upward curvature in the Arrhenius plot of the diffusion
`coefficients.1) This behavior has been explained by the model
`of phonon-assisted diffusion jumps via monovacancies by
`Ko¨hler and Herzig.2,3) They have proposed a mechanism
`of temperature dependence of the self-diffusion coefficient
`taking into account a temperature-dependent migration
`energy of a vacancy on the basis of the experimental
`evidence of significant lattice softening of the longitudinal
`acoustic (LA) phonon in the h111i direction at the reduced
`wave vector 2/3, LA2/3h111i phonon, in

-Zr.4) The LA2/
`3h111i phonon mode in the b.c.c. lattice is directly related to
`the nearest-neighbor jump process of the diffusing atom,
`because its displacement pattern facilitates the promotion of
`the migrating central atom through the saddle point into the
`position of the vacancy. Therefore the softening of this
`phonon mode results in a reduction in the restoring forces in
`the h111i direction and in an overall decrease in the free
`energy GM of atomic migration. Furthermore, investigations
`on the phonon dispersion in IVb metals,

-Ti,5)

-Zr6) and

-
`Hf,7) have revealed a considerable softening of the transverse
`acoustic (TA) T1A1/2h110i phonon with decreasing temper-
`ature in addition to the significant softening of the LA2/
`3h111i phonon with negligible temperature dependence.
`These observations support strongly the mechanism by
`Ko¨hler and Herzig.2,3) Besides the atomic displacement in
`the jump direction the importance of
`the concomitant
`displacements of the triangular configuration of the saddle-
`point atoms has been pointed out.3) An opening motion of the
`saddle-point atoms perpendicular to the jump direction
`decreases the potential barrier for the jump. Such a ‘breath-
`ing’ is partly achieved by the LA2/3h111i phonon and partly
`by the T1A1/2h110i phonon.8,9) The degree of softening of
`
`*Present address: Professor Emeritus, Miyagi National College of
`Technology. Corresponding author, E mail: mniotaosm@yahoo.co.jp
`
`the LA2/3h111i and T1A1/2h110i phonons correlates with
`the activation energy for self-diffusion in the

-phase of IVb
`metals.
`According to Ko¨hler and Herzig,2) GM in the anomalous
`b.c.c. metals is expressed by
`ð1Þ
`GM ¼ G0
`Mð1 T0=TÞ;
`M is the free energy of migration of a monovacancy
`where G0
`and T0 is the hypothetical lowest temperature for the metal to
`hold the b.c.c. structure. If the lattice is completely softened
`at T0, the frequency of LA2/3h111i would diminish to zero,
`then GM would diminish to zero. Thus the temperature
`dependence of the diffusion coefficient D in the anomalous
`b.c.c. metals should be expressed by
`ð2Þ
`MT0=RT 2Þ;
`D ¼ D0 expðQ=RTÞ expðG0
`where D0 and Q are the preexponential factor and the
`activation energy, respectively, for the monovacancy mech-
`anism. The extent of deviation from the linearity in the
`Arrhenius plot of the diffusion coefficient is represented by
`MT0=RT 2Þ in eq. (2). A similar equation to
`the term expðG0
`eq. (2) has been obtained by Sanchez and de Fontaine10,11) on
`the basis of the ! embryo model where the ! embryo in the
`transition from the

-phase to the !-phase has the same
`structure as the activated complex, and the ! embryo is also
`regarded as a lattice in significantly softened state of the
`LA2/3h111i phonon.
`In our previous studies on the impurity diffusion of
`transition elements (Cr and Pd),12) Ib elements (Cu,13) Ag13)
`and Au14)), IIIb elements (Ga and In)15) and IVb elements (Si,
`Ge and Sn)16) in

-Ti, eq. (2) has been confirmed to hold
`well, and it has been recognized that the activation energies
`for the impurity diffusion and self-diffusion2) in

-Ti are
`proportional
`to the square of the atomic radius of the
`diffusing atom. In the present work, diffusion behavior of IIIb
`element Al in

-Ti has been investigated. Although the
`diffusion coefficients of Al in

-Ti have been measured by
`Araki et al.17) and Ko¨ppers et al.,18) the diffusion parameters
`in eq. (2) have not been determined. In all the

-phase of
`dilute alloys of Ti with Cr, Pd, Cu, Ag, Au, Ga, In, Si, Ge
`
`Page 1 of 5
`
`Tianma Exhibit 1015
`
`

`

`1810
`
`S. Y. Lee, O. Taguchi and Y.lijima
`
`time has been corrected taking account of the amount of
`diffusion occurred during heating the specimen from room
`temperature to the diffusion temperature.” For diffusion
`above 1673 K, the alumina tube containing the couple was
`sealed in double tubes of quartz, keeping the inner quartz
`tube from crushing by adjusting the pressure of Ar gas
`betweenthe inner and outer tubes. The diffusion temperature
`was controlled to within +1 K at 1323 1473 K and to within
`+3K at 1673 1823K.
`After the diffusion, the couple was cut to parallel to the
`diffusion direction, and the cut surface was polished on a buff
`with fine alumina paste to examine the concentration-
`penetration profiles with an electron probe microanalyzer.
`The concentration of Al was determined by using the ZAF
`method. The interdiffusion coefficient was calculated as a
`function of solute concentration by Matano’s method.”
`
`3. Results and Discussion
`
`shows the concentration dependence of the
`Figure 1
`interdiffusion coefficient D determined in the temperature
`range 1323 to 1823K with (pure Ti)-(Ti-8.5 at% Al alloy)
`couple and in the temperature range 1423 to 1673K with
`(pure Ti)-(Ti-16.5 at% Al alloy) couple and in the temper-
`ature range 1473 to 1573K with (Ti-8.5 at% Al alloy)-(Ti-
`16.5 at% Al alloy) couple. The interdiffusion coefficients
`shown in the present work include the experimental error of
`10 to 15%. Interdiffusion coefficients determined for differ-
`ent diffusion times are showndistinguishably from each other
`by different marks. As shown in Fig. 1, at each temperature D
`is independent of diffusion time and D increases almost
`linearly with increasing Al content. According to Darken’s
`relation, ) the extrapolated value of D to theinfinite dilution
`of Al can be regarded as the impurity diffusion coefficient
`
`and Sn, the interdiffusion coefficient D is independent of
`concentration of solute.!2"!© On the other hand, D in the B-
`phase upto 2 at% Al of Ti-Al alloys increases with increasing
`Al content.'” Thenit is interesting to examine whether the
`curvature of the Arrhenius plot of D depends on the Al
`content in the f-phase.
`In the present work, interdiffusion experiments with the
`couples of pure Ti and the f-phase Ti-Al alloys containing
`8.5 and 16.5 at% Al have been made. The impurity diffusion
`coefficient of Al in £-Ti has been determined by applying
`Darken’s relation,’ i.e.
`the extrapolated value of the
`interdiffusion coefficient to the infinite dilution of the solute
`can be regarded as the impurity diffusion coefficient of the
`solute in f-Ti. This method is effective to avoid some
`troubles such as chemical reaction on the surface of reactive
`Ti specimen with radiotracer diffusion experiments. Further-
`more, this is especially useful in determining the impurity
`diffusion coefficient of element, such as Al for which the
`tracer diffusion experiment with a radioactive isotope is not
`easy.
`
`2. Experimental Procedure
`
`Pure rods 12mmin diameter and 100 mm in length were
`machined from a Ti bar 160mm in diameter supplied by
`Kobe Steel Ltd. The main impurities in this material were
`0.046 mass% Fe, 0.0032 mass% N and 0.0041 mass% O. The
`rods were polished chemically, sealed in quartz tube with
`high-purity Ar gas and then annealed at 1373 K for 172.8ks
`(2 days) to cause grain growth. Aluminarings (higher than
`99% purity) were fitted at both ends of the rod to prevent
`reaction with the quartz tube. The resultant grain size was
`about 3 mm. The rod was cut to make disc specimens 5 mm in
`thickness. To obtain a fully flat surface, the specimen was set
`in a stainless steel holder 50mm in diameter, ground on
`abrasive papers and polished on a buff with fine alumina
`paste.
`Buttons of Ti-8.5 and 16.5 at% Al alloys were made by Ar
`arc melting the pure Ti block with Al blocks of 99.999%
`purity. To homogenize the buttons,
`the arc melting was
`repeated a few times. Finally, the buttons were cast into a
`water-cooled copper boat to make a rod ingot 10mm in
`diameter and 80mm in length. The resultant grain size in
`alloy rods after the same grain growth treatment as described
`above was about 2 mm.Thealloy rods were cut to make disc
`specimens 5 mm in thickness. The cut surface of the alloy
`specimen was ground and polished in the same way as the
`pure Ti specimen.
`To make the semi-infinite interdiffusion couple the pure Ti
`and the alloy discs were put in a stainless steel holder with
`two screws, pressed by the screws, wrapped with a V foil,
`surrounded by Ti sponges and then diffusion welded by
`heating at 1073 K for 3.6ks in a stream of high-purity Ar gas.
`After the diffusion welding, the couple was removed from the
`holder and put into alumina tube; then the alumina tube
`containing the couple was sealed in a quartz tube with the
`high-purity Ar gas. By putting the quartz tube in a furnace,
`diffusion annealing was carried out at temperatures in the
`range from 1323 to 1823 K for between 3.6 and 691 ks (1 hto
`80d). At the temperatures higher than 1773 K, the diffusion
`
`Page 2 of 5
`
`10° O @ O (Ti}(Ti-8.5 at.% Al) couple
`
`
`
`Interdiffusoncoefficient,D/m*s”
`
`10%
`
`A A (Ti)-(16.5 at% Al) couple
`VW (Ti-8.5 at.% Al)-(16.5 at.% Al) couple
`
`0
`
`2
`
`4
`
`6
`
`8
`
`10
`
`12
`
`14
`
`16
`
`18
`
`20
`
`Concentrationof Al / at.%
`
`1 Concentration dependence ofinterdiffusion coefficient D in Ti Al
`Fig.
`alloys.
`
`

`

`Diffusion of Aluminum in f Titanium
`
`1811
`
`5x10"
`
`4. =
`
`_ 2
`
`10° 5x10"
`
`Interdiffusoncoefficient,D/m*s" 3
`
`Table 1 Diffusion coefficient of Al in 6 Ti.
`
`Temperature/K
`1823
`1773
`1723
`1673
`1623
`1573
`1523
`1473
`1423
`1373
`1323
`
`Diffusion coefficient/m?-s~!
`(8.68 + 1.02) x 10-!?
`(5.91 + 0.50) x 10-!?
`(4.26 + 0.29) x 1071?
`(3.214 0.42) x 10-'?
`(1.94 + 0.47) x 107!?
`(1.33 + 0.19) x 107!?
`(8.98 + 0.13) x 10-3
`(5.76 + 0.12) x 10-3
`(3.61 + 0.63) x 10-3
`(2.124 0.31) x 10-8
`(1.3140.15) x 10-8
`
`10°
`
`4 =
`
`—_ o,
`
`
`
`Diffusioncoefficient,D/m?s” 5
`
`Tm — self diff. of Ti in B-Ti
`Impurity diff. of Al in B-Ti
`@ Present work
`O Araki et al.
`
`A Ko6ppers etal.
`
`x)
`
`5.0
`
`55
`
`6.0
`6.5
`7.0
`75
`Temperature, T'/ 10*K'
`
`8.0
`
`Fig. 3 Temperature dependence of interdiffusion coefficient D.
`
`Table 2 Diffusion parameters, Dp, Q and GoMTp.
`Concentration
`of Al/at%
`0
`2
`6
`10
`12
`
`Do/m-s“!
`(3.03475) x 10-4
`(3.9143) x 10-5
`(5.31723) x 10°
`(7.0247) x 10-7:
`(7.46428) x 10-7
`
`Q/kJ-mol-!
`331.6 + 16.3
`275.3 28.7
`220.84 43.6
`163.9 + 2.7
`162.3£3.2
`
`GoMT)/MJ-mol-'K
`125.6+11.9
`82.1422.
`42.3 + 33.5
`
`4
`
`5
`
`6
`
`7
`
`8
`
`9
`
`10
`
`Temperature, T'/ 10*K™
`
`Fig. 2 Temperature dependence of impurity diffusion coefficients of Al
`and self diffusion coefficient in 6 Ti.
`
`Da of Al in £-Ti. The values of Da; determined in this way,
`using the linear fitting function, are listed in Table 1.
`Figure 2 shows the temperature dependence of the impu-
`rity diffusion coefficient Da; obtained by the present work
`along with those by Araki et al.'”) and Képperset al.'®) The
`value of Da; by the latter are taken from the figure given by
`Mishin and Herzig.'®) The temperature dependence of Dai
`obtained by these three groups showsexcellent agreement
`each other and can be expressed bya single line as follows;
`Dai = (3.03 + 2.85/—1.47) x 10 *
`x exp(—331.6 + 16.3kJ-mol !/RT)
`(3)
`x exp(125.6 + 11.9MJ-mol !/RT*) m?-s |
`Asshownin Fig. 2, the value of Da) is about one half of the
`self-diffusion coefficient Dy; of Ti in the whole temperature
`range of the f-phase. The softening of the LA2/3(111) and
`T,A1/2(110) phonons increases the diffusion coefficient D1
`of Al by the factor exp(125.6 MJ-mol
`1 /RT?). The values of
`this factor at the melting temperature of Ti (J, = 1943 K)
`and the f—qa transformation temperature of Ti
`(Tg ¢ =
`
`1155K) are 54.7 and 8.3 x 10*, respectively. The corre-
`spondingvalues for self-diffusion in Ti are 61 and 1.1 x 10°,
`respectively.
`Figure 3 shows the temperature dependence of the
`interdiffusion coefficient D at 0, 2, 6, 10 and 12 at% Al. At
`2 and 6at% Al the temperature dependence of D shows a
`similar upward curvature to those of Da; and Dy;. At 6 at% Al
`the curvature in Fig. 3 is small. At 10 and 12at% Al the
`Arrhenius plots show almost linearity. Then, the diffusion
`parameters in eq. (2) are calculated for each concentration
`of Al and listed in Table 2. The activation energy Q
`decreases with increasing Al content. Furthermore, the term
`exp(GoMT/RT”) which represents the extent of deviation
`from the linearity in Arrhenius plot decreases also with
`increasing Al content and it becomesnearly zero at 10 at%
`Al. This suggests that the phonon softening in the £-phase of
`Ti-Al alloy becomes weak with increasing Al content.
`It is interesting to examinetherelationship between GoMTp
`and Q for diffusion in £-Ti. Since Q is the sum of enthalpy
`Ho* of formation and the enthalpy HoM of migration of a
`monovacancy, Go™Tp can be written as
`GoMT> = (HoM — TSo™)To
`(4)
`= (Q — Ho")To — ToSo“T,
`where SoM is the entropy of migration of a monovacancy.
`According to Sanchez,” the term TySp™ is negligible
`
`Page 3 of 5
`
`

`

`S. Y. Lee, O. Taguchi and Y. Iijima
`
`200
`
`500
`
`1.0
`
`15
`
`2.0
`
`25
`
`3.0
`
`r?/107°m?
`
`1812
`
`180
`
`160
`
`140
`
`120
`
`100
`
`T,/MJmol"K
`
`Gy g -20
`
`100 150 200 250 300 350 400 450 500
`
`Q/kJmol'
`
`Fig. 4 Correlation between GoMTp and Q.
`
`Fig. 5 Correlation between activation energy Q for diffusion against
`square r? of radius of diffusing atoms in f Ti.
`
`A(T, p)e? is significant, and the normal monovacancy
`mechanism operates for the diffusion; then
`G(T > Tw) © Ae’ = GoM = Ho — TS“.
`
`(7)
`
`compared with HpM (= Q — Ho"). Because we are concerned
`with B-Ti, Tg .(= 1155 K) < T < Ty(= 1943 K), and Tpis
`estimated to be 610K as described below. Thus the second
`term of the right-handside in eq. (4) is much smaller than the
`first term. Then Go™Tp increases with increasing activation
`energy Q. As shown in Fig. 4, this is recognized by the
`experimental results of Sc,29 V,24) Ta,2>) W?®and Zr”)
`From theoretical considerations,”it has been supported that
`includingresults of present authors!?~!®andalinear relation
`Tg « > Tw > To. In the process of formation of the w-phase
`of the h.c.p. structure from b.c.c. f-Ti by the lattice
`Go™Tp (MJ-mol
`'K) = 0.610[Q (kJ-mol !) — 129]
`(5)
`displacementin the (111) direction, the intrinsic strain due
`to an activated configuration contributing to the transforma-
`tion to the w-phase comes about 3.5% in the direction [121]
`in the £-phase. However,if the atomic radius of one of the
`two atoms located on the unit cell of the w-phase is larger by
`1.7% than that of the other atom, the intrinsic strain in the
`q@-phase must be just cancelled out, and ¢ becomes unity.
`Furthermore, in the activated configuration the b.c.c. struc-
`ture becomes unstable, and the migration energy of an
`activated atom to the nearest-neighbor vacancy becomes a
`maximum. If the radius r of the diffusing impurity atom is
`smaller than 1.017 times the radius r,, of the solvent atom, the
`parameter ¢€ can be represented by
`
`indicating that Ho" does not depend on the
`is obtained,
`element. The value of HoF is estimated to be 129 +14
`kJ-mol
`! by extrapolating GpM7p to zero in Fig. 4. From the
`slope of the straight line in Fig. 4, Tp is estimated to be
`(610 + 26) K. This means that the activation energy Q for
`the impurity diffusion in £-Ti is controlled by the value of
`Ho™rather than by Ho". The magnitude of Ho should be
`connected with the ‘breathing’ motion of the saddle-point
`atoms andthus correlated with the size of diffusing atom.
`Now, we examine Ho™ for impurity diffusion in A-Ti.
`According to Sanchez and de Fontaine,!”the free energy G,,
`of formation of the w embryo (or the activated complex), is
`correlated with the transforming parameter ©, from the f-
`phase to the w-phase by
`G. = A(T, p)e” — B(T,pe’,
`
`(6)
`
`e€=r/1.017 ra(r < 1.017 rp)
`
`(8)
`
`and, for the impurity atom larger than 1.0177,
`
`e=1-—(r— 1.017 rg)/1.017 ra(r = 1.0177)
`
`~=—9)
`
`where A(T, p) and B(T, p) are functions of the temperature T
`and pressure p. The transforming parameter ¢ is proportional
`From eqs. (7) (9), it can be concluded that the migration
`enthalpy of an impurityin the direction [121] is proportional
`to the atomic displacement
`in the f-phase lattice which
`participates in the formation of the w embryo.”® ¢ =0in
`to the square of the radius of the impurity atom. As the
`coordination number in the b.c.c. structure of the A-Ti is
`the b.c.c. structure, but ¢ = | in the transformed lattice (@-
`phase). The driving force for the transformation expressed by
`eight, the atomic radius of the metal of different structure
`the second term B(T,p)e? is negligible at high temperatures
`from the b.c.c. is converted for a coordination number of
`eight by the conversion relation.” As shown in Fig.5,
`but increases with decreasing temperature. On the other hand,
`for r < 1.017rm, Q increases linearly with increasing r7,
`in a temperature range much higher than the highest
`temperature T,, for the w-phase to exist only the first term
`although the value of Q for W is much higherthose for the
`
`Page 4 of 5
`
`

`

`Diffusion of Aluminum in

Titanium
`
`1813
`
`other elements and, for r  1:017 rm, Q decreases with
`M
`increasing r2. This is consistent with the prediction that H0
`(¼ Q H0
`F) is proportional to r2, as deduced from eqs. (7)
`(9). Thus, it must be emphasized that the size effect is
`dominant in the impurity diffusion in

-Ti.
`
`4. Conclusions
`
`The present experimental results show the Arrhenius plot
`of the impurity diffusion coefficient of Al in

-Ti in the
`temperature range 1323 1823 K exhibit an upward curvature.
`This can be explained by a monovacancy mechanism with
`a temperature-dependent migration energy due to softening
`of the LA2/3h111i and T1A1/2h110i phonons in the

-Ti.
`Further,
`the weak curvature of the Arrhenius plots of
`interdiffusion coefficient in the

-phase of Ti-Al alloys less
`than 6 at% Al has been observed. The activation energy for
`the impurity diffusion and self-diffusion in

-Ti is propor-
`tional to the square of the radius of the diffusing atom.
`
`REFERENCES
`
`1) A. D. Le Claire: Diffusion in B.C.C. Metals, (ASM, Metals Park, Ohio
`1965) p. 3.
`2) U. Kohler and Ch. Herzig: Phys. Stat. Sol. (b) 144 (1987) 243.
`3) U. Kohler and Ch. Herzig: Phil. Mag. A 58 (1988) 769.
`4) C. Stassis, J. Zaresky and N. Wakabayashi: Phys. Rev. Lett. 41 (1978)
`1726.
`5) W. Petry, A. Heiming, J. Trampenau, M. Alba, C. Herzig, H. R.
`Schober and G. Vogl: Phys. Rev. B 43 (1991) 10933.
`
`6) A. Heiming, W. Petry, J. Trampenau, M. Alba, C. Herzig, H. R.
`Schober and G. Vogl: Phys. Rev. B 43 (1991) 10948.
`7) J. Trampenau, A. Heiming, W. Petry, M. Alba, C. Herzig, W. Miekeley
`and H. R. Schober: Phys. Rev. B 43 (1991) 10963.
`8) Chr. Herzig: Ber. Bunsenges. Phys. Chem. 93 (1989) 1247.
`9) W. Petry, A. Heiming, C. Herzig and J. Trampenau: Defect Diffusion
`Forum 75 (1991) 211.
`10) J. M. Sanchez and D. de Fontaine: Phys. Rev. Lett. 35 (1975) 227.
`11) J. M. Sanchez and D. de Fontaine: Acta Metall. 26 (1978) 1083.
`12) S. Y. Lee, Y. Iijima and K. Hirano: Mater. Trans., JIM 32 (1991) 451
`456.
`13) S. Y. Lee, Y. Iijima, O. Taguchi and K. Hirano: J. Japan Inst. Metals
`54 (1990) 502 508.
`14) S. Y. Lee, Y. Iijima and K. Hirano: Defect Diffusion Forum 95–98
`(1993) 623.
`15) S. Y. Lee, Y. Iijima and K. Hirano: Phys. Stat. Sol. (a) 136 (1993) 311.
`16) Y. Iijima, S. Y. Lee and K. Hirano: Phil. Mag. A 68 (1993) 901.
`17) H. Araki, T. Yamane, Y. Minamino, S. Saji, Y. Hana and S. B. Jung:
`Metall. Mater. Trans. 25A (1994) 874.
`18) Y. Mishin and C. Herzig: Acta Mater. 48 (2000) 589 623.
`19) L. S. Darken: Trans. AIME 175 (1948) 148.
`20) S. J. Rothman: Diffusion in Crystalline Solids, ed. by G. E. Murch and
`A. S. Nowick, (New York, Academic Press, Inc. 1984) p. 20.
`21) C. Matano: Japan J. Phys. 8 (1933) 109.
`22) J. M. Sanchez: Phil. Mag. A 43 (1981) 1407.
`23) J. Askill and G. B. Gibbs: Phys. Stat. Sol. 11 (1965) 557.
`24) J. F. Murdock, T. S. Lundy and E. E. Stansbury: Acta Metall. 12 (1964)
`1033.
`25) J. Askill: Phys. Stat. Sol. 16 (1966) K63.
`26) Y. Minamino, H. Araki, T. Yamane, S. Ogino, S. Saji and Y.
`Miyamoto: Defect Diffusion Forum 95–98 (1993) 685.
`I. Thibon, D. Ansel and T. Gloriant: J. Alloy. Compd. 470 (2009) 127.
`27)
`28) H. E. Cook: Acta Metal. 21 (1973) 1431.
`29) W. B. Pearson: The Crystal Chemistry and Physics of Metals and
`Alloys, (New York, Wiley 1972) p. 135.
`
`Page 5 of 5
`
`