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Diffusion of tin in α-iron
Acta mater. 48 (2000) 2925±2931 www.elsevier.com/locate/actamat
DIFFUSION OF TIN IN a-IRON D. N. TORRES 1, R. A. PEREZ 2 and F. DYMENT{ 2{ 1
Instituto de TecnologõÂ a ``Prof. J. A. Sabato'', CNEA±UNSAM, Avda. Libertador 8250, 1429 Buenos Aires, Argentina and 2Materials Department, Atomic Energy Commission (CNEA), Avda. Libertador 8250, 1429 Buenos Aires, Argentina (Received 31 January 2000; accepted 13 March 2000)
AbstractÐThe eect of the magnetic ordering on the diusion of Sn was investigated; for this purpose, the measurements were made in an extended temperature range (1163±673 K), using several techniques like serial sectioning, heavy ion Rutherford backscattering spectrometry (HIRBS) and Rutherford backscattering spectrometry (RBS) in superposed ranges of temperature. By spanning the temperature range with respect to the previous works, the Arrhenius plot was extended over 10 decades; then the variation of the slope at the Curie temperature and a soft-up curvature in the ferromagnetic region is now clear. The curvature observed is well ®tted by the model developed by Ruch et al. (J. Phys. Chem. Solids, 1976, 37, 649). The parameter that takes into account the in¯uence of the magnetic order in the vacancy mobility during the Sn diusion process is quite similar to that observed in Fe self-diusion; then, no magnetic interaction between the impurity and the matrix is postulated. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Diusion; Iron; Rutherford backscattering spectrometry (RBS); Tin; Magnetic properties
1. INTRODUCTION
The in¯uence of the magnetic ordering on diusion in iron was observed in a-Fe self-diusion by several authors [1±3]. Below the Curie temperature TC 1043 K), the self-diusion coecients have lower values than the extrapolated ones from the paramagnetic region and show a curvature in the Arrhenius plot. Although this in¯uence was already revealed by Borg and Birchenall [4] and Bungton et al. [5] at the beginning of the 1960s, complete measurements, in an extended temperature range, were only performed in the late 1970s, due to the need for techniques that allow the analysis of shallow penetration pro®les, like ion beam sputtering (IBS). Descriptions of such temperature dependence of the self-diusion coecient in the whole temperature range of a-Fe by a single equation have been carried out by several authors such as Ruch et al. [6], KucÏera [7] and Hettich et al. [1]. A similar deviation from the Arrhenius law was observed in the heterodiusion of Co [8±10], Ni [11] and Cr [9, 12] in a-Fe. In Ref. [9] it was shown that the in¯uence of the magnetic order eect on the diusion coecient is larger for Co and smaller
{ Also at Carrera del Investigador, CONICET. { To whom all correspondence should be addressed.
for Cr than that on self-diusion. The authors related this behavior to the impurity magnetic moment in the a-Fe matrix. Also KucÏera et al. [13] have applied their analysis to the impurity diusion of Co in a-Fe and Braun and Feller-Kniepmeier [14] have extended the analysis of KucÏera et al. to the diusion of Co, Cr, Cu and Zn in a-Fe above and below the Curie temperature. More recently, Klugkist and Herzig [15] studied Ti diusion in paramagnetic a-Fe and in a very short temperature range in the ferromagnetic phase, obtained also a reduction of the diusion coecients compared with the values extrapolated from the linear Arrhenius relationship obtained in the paramagnetic range. The present work deals with the diusion of Sn in a-Fe. Several authors have studied Sn diusion in a-Fe with dierent techniques and dierent Fe impurity contents, in short temperature ranges [16± 18]. None of these works alone shows a curvature in the Arrhenius plot. The goal of the present work is to investigate the eect of the magnetic ordering on Sn diusion; for this purpose, the measurements were made in an extended temperature range (1163±673 K), using several techniques like serial sectioning, heavy ion Rutherford backscattering spectrometry (HIRBS) and Rutherford backscattering spectrometry (RBS) in superposed ranges of temperature.
1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 0 7 4 - 4
2926
TORRES et al.: DIFFUSION OF TIN IN a-IRON
As mentioned above, several models were developed in order to explain the reduction in the diusion coecient in the ferromagnetic range of a-Fe. We have chosen that one which ®ts the curvature in the Arrhenius plot with the smallest number of parameters to be determined. Based on the work by Girifalco [19] on the use of a long range order parameter to describe the eects of ordering on diusion, Ruch et al. [6] have expressed the temperature dependence of the self-diusion coecient D of aFe as D T Dp0 expÿQp 1 as 2 =RT
1
where Dp0 and Q p are the frequency factor and the activation energy in the paramagnetic state, s is the ratio of the spontaneous magnetization at temperature T to that at 0 K (reduced magnetization) and has been experimentally determined by Crangle and Goodman [20]; a is a constant composed of two contributions: a
af am Qp
2
where af is the increment of the formation energy of a vacancy due to the magnetic transformation and am is the increment of the migration energy of a vacancy induced by the magnetic ordering. Ruch et al. [6] also quanti®ed the value of af for pure aFe: af 4:3 kJ=mol: In the application of this model to the heterodiffusion in a-Fe, Hirano and Iijima [9] made the assumptions that af and the spontaneous magnetization s remain constant. This is plausible if the diffusion occurs at in®nite dilution. Then, the identity of the diusion impurity is re¯ected in the increment of the migration energy of a vacancy induced by the magnetic order, am, when the vacancy has an impurity atom as nearest neighbor. 2. EXPERIMENTAL PROCEDURE
Samples of polycrystalline a-Fe of 99.99% purity were used. The impurity contents, as analyzed in the CNEA Chemistry Department, are shown in Table 1. The samples were discs of about 5 mm diameter and 3 mm thickness. The discs were mechanically polished and submitted to three to ®ve cycles of an annealing of 5 h at 1473 K followed by 6 days at 1163 K in order to increase the grain size. The ®nal size was around 500 mm and for the higher temperatures some triand bi-crystals were obtained. The introduction of Sn to obtain the diusion pairs was made in dierent ways, according to the measurement technique to be used. In order to cover the largest possible temperature range, serial sectioning was used for the highest temperatures and analysis with backscattered ion beams for intermediate (HIRBS) and low temperatures (RBS).
Table 1. Impurity contents in mg/g. ND: not detected Symbol C S N Al Mg Si Mn Cr Cu Ti Ag Ni Co Cd B Mo H O
mg/g 5825 < 10 240210 050 05 020 < 20 ND < 50 ND < 20 ND < 50 ND < 50 ND < 200 ND < 200 ND < 200 ND < 20 ND < 200 ND 622 3023
The diusion annealings were performed in sealed quartz tubes under a high pure argon atmosphere and carried out at speci®ed temperatures controlled within 20:5 K: Corrections for heat-up time were made when the total annealing time was R7200 s: Serial sectioning with a grinding machine: the basic method used in this case was to observe the diusion from a thin surface layer of a radiotracer, 113 Sn, that was purchased from New England Nuclear in the form of SnCl2 in HCl solution. A drop was dried onto the polished surface of the Fe samples. The appropriate solution of Fick's second law for this condition is equation (3). Analysis with backscattered ion beams: natural Sn was introduced in two dierent ways, by ion implantation or by evaporation on the sample surface. Some specimens (see Table 2) were implanted using the 500 kV ion implanter of the Instituto de FõÂ sica, Universidade Federal de Rio Grande do Sul (IFUFRGS), Porto Alegre, Brazil, with an implantation energy of 50 keV and 5 1015 at=cm 2 ¯uence. In the other specimens, Sn was evaporated using an ion gun system in a vacuum better than 10ÿ4 Pa; a 20 nm ®lm was measured by RBS after evaporation and before the diusion annealing. At low temperatures (see Table 2) the diusion pro®les were obtained using the RBS technique. A 2 MeV 2He ion beam from the IF-UFRGS TANDETROM accelerator was used. The backscattered a-particles were detected using a Si surface-barrier detector, located at a scattering angle of 1658. The electronic resolution of the system was better than 13 keV, which implies a depth resolution of around 10 nm. At higher temperatures HIRBS was used. The use of a heavier ion beam at higher energies increases the mass and depth resolution. The experiments were performed in the TANDAR accelerator of the Departamento de FõÂ sica±CNEA, using a 9F 38 MeV ion beam. The experimental set-up was
TORRES et al.: DIFFUSION OF TIN IN a-IRON
2927
Table 2. Diusion data for Sn in iron No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Temperature (K)
Annealing time (104 s)
Technique
Pair preparation
D (m2/s)
673 696 723 773 823 873 923 953 973 993 1000 1023 1048 1103 1163
648 1185 907.2 22.8 72 0.216 0.18 0.228 0.162 180.68 0.048 0.075 17.16 2.88 0.45
RBS RBS RBS RBS HIRBS RBS RBS HIRBS HIRBS Grinding HIRBS HIRBS Grinding Grinding Grinding
Implanted Implanted Implanted Implanted Implanted Evaporated Implanted Evaporated Evaporated Drop Implanted Implanted Drop Drop Drop
(2.020.5)10ÿ23 (4.021.0)10ÿ23 (1.020.25)10ÿ21 (1.520.4)10ÿ20 (9.522.4)10ÿ20 (1.120.3)10ÿ18 (8.022.0)10ÿ18 (4.021.0)10ÿ17 (7.121.7)10ÿ17 (9.520.3)10ÿ17 (2.120.5)10ÿ16 (5.721.4)10ÿ16 (2.520.07)10ÿ15 (8.320.2)10ÿ15 (2.120.06)10ÿ14
similar to the one described in Ref. [21], which implies a depth resolution of around 30 nm. In both cases, the conversion of the spectrum to a diusion pro®le was made according to the algorithm developed in Ref. [22] and the solution of Fick's second law is equation (3). 3. RESULTS
The diusion coecients obtained for Sn diusion in a-Fe are listed in Table 2 together with temperatures, annealing times and techniques used for preparing the diusion pairs and for penetration analysis. The experiments were performed in order to obtain a solution of Fick's equation given by a Gaussian function. Then, each diusion pro®le can be ®tted by C0 ÿ x ÿ x 0 2 C x p exp 3 4Dt pDt where C is the Sn concentration at depth x, C0 is the initial amount of Sn, D is the diusion coecient, t is the annealing time and x0 is zero when Sn is evaporated or dropped and about 10 nm (the depth where the implanted Sn peak is located) when the material is implanted. All the penetration pro®les (ln[C(x )] against x ÿ x 0 2 are displayed in Fig. 1. As can be observed, straight lines were obtained in all the cases, following equation (3). In the paramagnetic region, the D values follow an Arrhenius law, as can be observed in Fig. 2, with an activation energy Qp 190220 kJ=mol and pre-exponential factor Dp0 623 10 ÿ6 m 2 =s: In the ferromagnetic region a soft upward curvature can be observed. Previous results [16±18] are included for comparison. 4. DISCUSSION
Diusion studies of Sn in a-Fe were performed
previously, with dierent techniques, dierent kinds of Fe samples and in several short temperature ranges. As can be seen in Fig. 2, the agreement in the paramagnetic region with the work of Treheux et al. [16] is excellent. In the ferromagnetic region, in the higher temperature range (above 870 K), the agreement with Henesen et al. [17] again is excellent. In Ref. [17] radiotracer studies were done in coarse grains of pure Fe. The agreement is very good with Myers [18] who applied RBS to implanted polycrystalline Fe samples. Below 870 K, Henesen et al. [17] made Auger electron spectroscopy (AES) measurements of segregation kinetics to determine the diusion coecients in single crystals of Fe±4 wt% Sn; therefore the comparison is not straightforward. In the present study measurements were made in superposed ranges of temperatures when dierent techniques were used. The Arrhenius plot shows a variation of the slope at the Curie temperature and a soft upward curvature in the ferromagnetic region. It is possible to analyze this behavior in the frame of the model described in 1. Then, rewriting equation (1) as "
D T T ln Dp0
!# ÿ
Qp ÿ R
aQp 2 s T R
4
using the experimental values of s(T ) given by Ref. [20] and the present D values for the ferromagnetic region, the plot T ln D=Dp0 against s 2 ®ts a straight line quite well as observed in Fig. 3. Here Dp0 is taken from the results in the paramagnetic region. From the slope a 0:23220:023 is obtained. Using this a value we ®t the measured diusion coecients with equation (1); the agreement showed in Fig. 4 is remarkable. Q p can be recalculated from the intercept of Fig. 3 adjusted by a minimum square ®t, getting Qprecal 19022 kJ=mol, in excellent agreement with Q p obtained directly from the diusion coecients of the paramagnetic region.
2928
TORRES et al.: DIFFUSION OF TIN IN a-IRON
Taking, as mentioned before, the increment of the vacancy formation energy due to the magnetic order of pure Fe af 4:3 kJ=mol, the increment of the migration energy am is Qprecal a ÿ af 4024 kJ=mol: This value is similar to the one
obtained for self-diusion (35 kJ/mol) [9]. When the diusion of magnetic impurities was studied, a dierent behavior was observed. The case of Co diffusion [8, 10] is an interesting one. Colbalt is known as the only solute element that enhances the
Fig. 1. Diusion pro®les obtained: (a) from RBS experiments; (b) from HIRBS experiments; (c) from grinder sectioning experiments.
TORRES et al.: DIFFUSION OF TIN IN a-IRON
2929
Fig. 3. Plot of T lnD T =D0 vs s 2 for Sn diusion in ferromagnetic a-Fe.
Fig. 2. Arrhenius plot for Sn diusion in a-Fe. Full circles correspond to the value listed in Table 2. When the error associated with a given point is lower than the circle size, no error bar is displayed.
magnetic moment of the Fe matrix; thus, the eect of the magnetic transition on the diusion of Co in a-Fe is expected to be larger than that on self-diusion. In fact the diusion coecients obtained are about one half of the self-diusion in Fe in the ferromagnetic state and above the critical temperature 1043 K and until 1184 K, the Arrhenius plot does not show linearity. Hirano and Iijima [9] applied equation (1) to the available data on Co diusion in a-Fe, using for Dp0 the self-diusion value and the experimental value of s(T ) [20] for pure Fe; they obtained a 0:23: From this value it is possible to estimate am as 54.5 kJ/mol, which is a higher value than the one for self-diusion. They proposed that a local magnetic ®eld around a Co atom in the Fe matrix is responsible for this increment in the am value. Following this idea, and assuming that no magnetic moment is associated with a Sn atom, there is no reason for a change in the am value when Sn is diusing in a-Fe. From the magnetic order point of view, a Sn atom in a substitutional position in the Fe matrix is like a site with no magnetic moment, then we can think of it as a ``magnetic vacant site''. To support this idea, lets rewrite equation (1) as " !# 1 kT D T s T ÿ 1 p ln : a Q Dp0 2
experimental measurements [20] and the diamonds correspond to equation (5) applied to self-diusion. The pointed line represent the values obtained from Co diusion data; as can be seen, s 2 is larger than that in self-diusion in the whole temperature range and it remains dierent from zero above the Curie temperature. This means that the local magnetic ®eld around the Co atoms remains even above the Curie temperature. Finally the plus symbol represents the s 2 values calculated from the present work. They ®t the experimental magnetization data as well as the self-diusion ones and become zero at TC. This behavior indicates no interaction of Sn atoms with the Fe magnetic matrix, as proposed. That idea was also supported by performing ab initio calculations using the local spin density ap-
5
The reduced magnetization can be calculated from the diusion coecients. This is shown in Fig. 5. For comparison, the full line represents the
Fig. 4. Diusion coecients of Sn in a-Fe. The full line is the ®t with equation (1).
2930
TORRES et al.: DIFFUSION OF TIN IN a-IRON
atom is quite similar to pure Fe, whereas in the Co case, the magnetic moment of the neighboring Fe atom is increased. In the future, it will be interesting to measure the diusion of dierent non-magnetic impurities in aFe in such an extended temperature range, in order to con®rm if the quantity aQ remains constant. It will be interesting too, to know if this trend is followed by a constant value of the magnetic moment of the nearest neighbor Fe, performed using the precedent ab initio calculation. 5. CONCLUSIONS
Fig. 5. Temperature dependence of s 2 obtained from the diusion coecient of Sn in a-Fe (plus symbols) in comparison with those from self-diusion (diamonds) and Co diusion in a-Fe (dotted line). The full line corresponds to the experimental values obtained in Ref. [20].
proximation (LSDA) in three-dimensional periodic systems in a unit cell containing 16 atoms in the b.c.c. structure of a-Fe. The magnetic moment was calculated for the whole cell and for the individual atoms. Then, one of the Fe atoms was replaced by an impurity, Sn or Co, and the results were compared. The WIEN97 code, an implementation of the linearized augmented plane wave method, based on density functional theory, was used for this purpose [23]. No relaxation was allowed in the unit cell, it was always kept at the experimental lattice constant. The three calculations, Fe (for comparison), Co and Sn, are shown in Table 3. The columns show in order the magnetic moment (in Bohr magnetons): of the whole cell, at the impurity and at the nearest neighbor Fe atom (both inside the mun tin radius). Whereas the Co impurity increases the cell magnetic moment by increasing the polarization of the neighboring Fe atoms (even when the Co moment itself is smaller than that of Fe), the Sn as impurity decreases the cell magnetic moment. On the other hand, the magnetic moment at the Sn atom is almost zero, as expected, and the corresponding magnetic moment of the neighboring Fe Table 3. Magnetic moment (in Bohr magnetons) for dierent impurities embedded in a-Fe Impurity Fe Co Sn
M (cell)
M (impurity)
M (neighbor)
34.8 35.7 33.3
2.17 1.71 ÿ0.04
2.17 2.29 2.14
Diusion of Sn in a-Fe was studied in an extended temperature range (673±1163 K) and over almost 10 decades using dierent techniques. The agreement between them in the superposed temperature range is excellent. A deviation from the Arrhenius law was observed in the ferromagnetic region. The curvature observed is well ®tted by the model developed by Ruch et al. The parameter that takes into account the in¯uence of the magnetic order in the vacancy mobility during the Sn diusion process is quite similar to that observed in Fe self-diusion; then, no magnetic interaction between the Sn impurity and the matrix is postulated. AcknowledgementsÐThe authors wish to thank Dr M. Behar for his collaboration with the RBS measurements, to Drs G. GarcõÂ a BermuÂdez and D. Abriola for their collaboration with the HIRBS measurements, to Lic. S. Balart for her collaboration with the serial sectioning measurements and to Dra. M. Weissmann who made the ab initio calculations. This work was partly supported by the Grant PIP 4071 from CONICET and the Grant PICT'97 No. 12-00000-00152 from the Agencia Nacional de PromocioÂn CientõÂ ®ca y TecnoloÂgica. REFERENCES 1. Hettich, G., Mehrer, H. and Maier, K., Scripta metall., 1977, 11, 795. 2. Lubbehusen, M. and Mehrer, H., Acta metall. mater., 1990, 38, 283. 3. Iijima, Y., Kimura, K. and Hirano, K., Acta metall., 1988, 36, 2811. 4. Borg, R. J. and Birchenall, C. E., Trans. metall. Soc. A.I.M.E., 1960, 218, 980. 5. Bungton, F. S., Hirano, K. and Cohen, M., Acta metall., 1961, 9, 434. 6. Ruch, L., Sain, D. R., Yeh, H. L. and Girifalco, L. A., J. Phys. Chem. Solids, 1976, 37, 649. 7. KucÏera, J., Czech. J. Phys. B, 1979, 29, 797. 8. Mehrer, H., HoÈpfel, D. and Hettich, G., in DIMETA82: Diusion in Metals and Alloys, ed. F. J. Kedves and D. L. Beke. Trans. Tech. Publications, Aedermannsdorf, 1983, p. 360. 9. Hirano, K. and Iijima, Y., Defect Diusion Forum, 1989, 66±69, 1039. 10. Iijima, Y., Kimura, K., Lee, C. and Hirano, K., Mater. Trans., JIM, 1993, 34, 20. 11. Cermak, J., LuÈbbehusen, M. and Mehrer, H., Z. Metallk., 1989, 80, 213. 12. Lee, C., Iijima, Y., Hiratani, T. and Hirano, K., Mater. Trans., JIM, 1990, 31, 255.
TORRES et al.: DIFFUSION OF TIN IN a-IRON 13. KucÏera, J., KozaÂk, L. and Mehrer, H., Physica status solidi (a), 1984, 81, 497. 14. Braun, R. and Feller-Kniepmeier, M., Scripta metall., 1986, 20, 7. 15. Klugkist, P. and Herzig, Ch., Physica status solidi (a), 1995, 148, 413. 16. Treheux, D., Marchive, D., Delagrange, J. and Guiraldenq, C. R., Acad. Sci. Paris, SeÂrie C, 1972, 274, 1260. 17. Henesen, K., Keller, H. and Viefhaus, H., Scripta metall., 1984, 18, 1319. 18. Myers, S. M., in Nontraditional Methods in Diusion, ed. G. E. Murch, H. K. Birnbaum and J. R. Cost. The Metallurgical Society of AIME, 1983, p. 137.
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19. Girifalco, L. A., J. Phys. Chem. Solids, 1962, 23, 1171. 20. Crangle, J. and Goodman, G. M., Proc. R. Soc. Lond. A, 1971, 321, 477. 21. PeÂrez, R. A. and Dyment, F., Appl. Phys. A, 1999, 68, 667. 22. PeÂrez, R. A., BermuÂdez, G. G., Abriola, D., Dyment, F. and Somacal, H., Defect Diusion Forum, 1997, 143±147, 1335. 23. Blaha, P., Schwartz, K. and Luitz, J., WIEN97, Vienna University of Technology, Vienna 1997. (Improved and updated UNIX version of the original copyrighted WIEN code, which was published in Blaha, P., Schwartz, K., Sorantin, P. and Trickey, S. B., Comput. Phys. Commun., 1990, 59, 399.)
DIFFUSION OF TIN IN a-IRON D. N. TORRES 1, R. A. PEREZ 2 and F. DYMENT{ 2{ 1
Instituto de TecnologõÂ a ``Prof. J. A. Sabato'', CNEA±UNSAM, Avda. Libertador 8250, 1429 Buenos Aires, Argentina and 2Materials Department, Atomic Energy Commission (CNEA), Avda. Libertador 8250, 1429 Buenos Aires, Argentina (Received 31 January 2000; accepted 13 March 2000)
AbstractÐThe eect of the magnetic ordering on the diusion of Sn was investigated; for this purpose, the measurements were made in an extended temperature range (1163±673 K), using several techniques like serial sectioning, heavy ion Rutherford backscattering spectrometry (HIRBS) and Rutherford backscattering spectrometry (RBS) in superposed ranges of temperature. By spanning the temperature range with respect to the previous works, the Arrhenius plot was extended over 10 decades; then the variation of the slope at the Curie temperature and a soft-up curvature in the ferromagnetic region is now clear. The curvature observed is well ®tted by the model developed by Ruch et al. (J. Phys. Chem. Solids, 1976, 37, 649). The parameter that takes into account the in¯uence of the magnetic order in the vacancy mobility during the Sn diusion process is quite similar to that observed in Fe self-diusion; then, no magnetic interaction between the impurity and the matrix is postulated. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Diusion; Iron; Rutherford backscattering spectrometry (RBS); Tin; Magnetic properties
1. INTRODUCTION
The in¯uence of the magnetic ordering on diusion in iron was observed in a-Fe self-diusion by several authors [1±3]. Below the Curie temperature TC 1043 K), the self-diusion coecients have lower values than the extrapolated ones from the paramagnetic region and show a curvature in the Arrhenius plot. Although this in¯uence was already revealed by Borg and Birchenall [4] and Bungton et al. [5] at the beginning of the 1960s, complete measurements, in an extended temperature range, were only performed in the late 1970s, due to the need for techniques that allow the analysis of shallow penetration pro®les, like ion beam sputtering (IBS). Descriptions of such temperature dependence of the self-diusion coecient in the whole temperature range of a-Fe by a single equation have been carried out by several authors such as Ruch et al. [6], KucÏera [7] and Hettich et al. [1]. A similar deviation from the Arrhenius law was observed in the heterodiusion of Co [8±10], Ni [11] and Cr [9, 12] in a-Fe. In Ref. [9] it was shown that the in¯uence of the magnetic order eect on the diusion coecient is larger for Co and smaller
{ Also at Carrera del Investigador, CONICET. { To whom all correspondence should be addressed.
for Cr than that on self-diusion. The authors related this behavior to the impurity magnetic moment in the a-Fe matrix. Also KucÏera et al. [13] have applied their analysis to the impurity diusion of Co in a-Fe and Braun and Feller-Kniepmeier [14] have extended the analysis of KucÏera et al. to the diusion of Co, Cr, Cu and Zn in a-Fe above and below the Curie temperature. More recently, Klugkist and Herzig [15] studied Ti diusion in paramagnetic a-Fe and in a very short temperature range in the ferromagnetic phase, obtained also a reduction of the diusion coecients compared with the values extrapolated from the linear Arrhenius relationship obtained in the paramagnetic range. The present work deals with the diusion of Sn in a-Fe. Several authors have studied Sn diusion in a-Fe with dierent techniques and dierent Fe impurity contents, in short temperature ranges [16± 18]. None of these works alone shows a curvature in the Arrhenius plot. The goal of the present work is to investigate the eect of the magnetic ordering on Sn diusion; for this purpose, the measurements were made in an extended temperature range (1163±673 K), using several techniques like serial sectioning, heavy ion Rutherford backscattering spectrometry (HIRBS) and Rutherford backscattering spectrometry (RBS) in superposed ranges of temperature.
1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 0 7 4 - 4
2926
TORRES et al.: DIFFUSION OF TIN IN a-IRON
As mentioned above, several models were developed in order to explain the reduction in the diusion coecient in the ferromagnetic range of a-Fe. We have chosen that one which ®ts the curvature in the Arrhenius plot with the smallest number of parameters to be determined. Based on the work by Girifalco [19] on the use of a long range order parameter to describe the eects of ordering on diusion, Ruch et al. [6] have expressed the temperature dependence of the self-diusion coecient D of aFe as D T Dp0 expÿQp 1 as 2 =RT
1
where Dp0 and Q p are the frequency factor and the activation energy in the paramagnetic state, s is the ratio of the spontaneous magnetization at temperature T to that at 0 K (reduced magnetization) and has been experimentally determined by Crangle and Goodman [20]; a is a constant composed of two contributions: a
af am Qp
2
where af is the increment of the formation energy of a vacancy due to the magnetic transformation and am is the increment of the migration energy of a vacancy induced by the magnetic ordering. Ruch et al. [6] also quanti®ed the value of af for pure aFe: af 4:3 kJ=mol: In the application of this model to the heterodiffusion in a-Fe, Hirano and Iijima [9] made the assumptions that af and the spontaneous magnetization s remain constant. This is plausible if the diffusion occurs at in®nite dilution. Then, the identity of the diusion impurity is re¯ected in the increment of the migration energy of a vacancy induced by the magnetic order, am, when the vacancy has an impurity atom as nearest neighbor. 2. EXPERIMENTAL PROCEDURE
Samples of polycrystalline a-Fe of 99.99% purity were used. The impurity contents, as analyzed in the CNEA Chemistry Department, are shown in Table 1. The samples were discs of about 5 mm diameter and 3 mm thickness. The discs were mechanically polished and submitted to three to ®ve cycles of an annealing of 5 h at 1473 K followed by 6 days at 1163 K in order to increase the grain size. The ®nal size was around 500 mm and for the higher temperatures some triand bi-crystals were obtained. The introduction of Sn to obtain the diusion pairs was made in dierent ways, according to the measurement technique to be used. In order to cover the largest possible temperature range, serial sectioning was used for the highest temperatures and analysis with backscattered ion beams for intermediate (HIRBS) and low temperatures (RBS).
Table 1. Impurity contents in mg/g. ND: not detected Symbol C S N Al Mg Si Mn Cr Cu Ti Ag Ni Co Cd B Mo H O
mg/g 5825 < 10 240210 050 05 020 < 20 ND < 50 ND < 20 ND < 50 ND < 50 ND < 200 ND < 200 ND < 200 ND < 20 ND < 200 ND 622 3023
The diusion annealings were performed in sealed quartz tubes under a high pure argon atmosphere and carried out at speci®ed temperatures controlled within 20:5 K: Corrections for heat-up time were made when the total annealing time was R7200 s: Serial sectioning with a grinding machine: the basic method used in this case was to observe the diusion from a thin surface layer of a radiotracer, 113 Sn, that was purchased from New England Nuclear in the form of SnCl2 in HCl solution. A drop was dried onto the polished surface of the Fe samples. The appropriate solution of Fick's second law for this condition is equation (3). Analysis with backscattered ion beams: natural Sn was introduced in two dierent ways, by ion implantation or by evaporation on the sample surface. Some specimens (see Table 2) were implanted using the 500 kV ion implanter of the Instituto de FõÂ sica, Universidade Federal de Rio Grande do Sul (IFUFRGS), Porto Alegre, Brazil, with an implantation energy of 50 keV and 5 1015 at=cm 2 ¯uence. In the other specimens, Sn was evaporated using an ion gun system in a vacuum better than 10ÿ4 Pa; a 20 nm ®lm was measured by RBS after evaporation and before the diusion annealing. At low temperatures (see Table 2) the diusion pro®les were obtained using the RBS technique. A 2 MeV 2He ion beam from the IF-UFRGS TANDETROM accelerator was used. The backscattered a-particles were detected using a Si surface-barrier detector, located at a scattering angle of 1658. The electronic resolution of the system was better than 13 keV, which implies a depth resolution of around 10 nm. At higher temperatures HIRBS was used. The use of a heavier ion beam at higher energies increases the mass and depth resolution. The experiments were performed in the TANDAR accelerator of the Departamento de FõÂ sica±CNEA, using a 9F 38 MeV ion beam. The experimental set-up was
TORRES et al.: DIFFUSION OF TIN IN a-IRON
2927
Table 2. Diusion data for Sn in iron No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Temperature (K)
Annealing time (104 s)
Technique
Pair preparation
D (m2/s)
673 696 723 773 823 873 923 953 973 993 1000 1023 1048 1103 1163
648 1185 907.2 22.8 72 0.216 0.18 0.228 0.162 180.68 0.048 0.075 17.16 2.88 0.45
RBS RBS RBS RBS HIRBS RBS RBS HIRBS HIRBS Grinding HIRBS HIRBS Grinding Grinding Grinding
Implanted Implanted Implanted Implanted Implanted Evaporated Implanted Evaporated Evaporated Drop Implanted Implanted Drop Drop Drop
(2.020.5)10ÿ23 (4.021.0)10ÿ23 (1.020.25)10ÿ21 (1.520.4)10ÿ20 (9.522.4)10ÿ20 (1.120.3)10ÿ18 (8.022.0)10ÿ18 (4.021.0)10ÿ17 (7.121.7)10ÿ17 (9.520.3)10ÿ17 (2.120.5)10ÿ16 (5.721.4)10ÿ16 (2.520.07)10ÿ15 (8.320.2)10ÿ15 (2.120.06)10ÿ14
similar to the one described in Ref. [21], which implies a depth resolution of around 30 nm. In both cases, the conversion of the spectrum to a diusion pro®le was made according to the algorithm developed in Ref. [22] and the solution of Fick's second law is equation (3). 3. RESULTS
The diusion coecients obtained for Sn diusion in a-Fe are listed in Table 2 together with temperatures, annealing times and techniques used for preparing the diusion pairs and for penetration analysis. The experiments were performed in order to obtain a solution of Fick's equation given by a Gaussian function. Then, each diusion pro®le can be ®tted by C0 ÿ x ÿ x 0 2 C x p exp 3 4Dt pDt where C is the Sn concentration at depth x, C0 is the initial amount of Sn, D is the diusion coecient, t is the annealing time and x0 is zero when Sn is evaporated or dropped and about 10 nm (the depth where the implanted Sn peak is located) when the material is implanted. All the penetration pro®les (ln[C(x )] against x ÿ x 0 2 are displayed in Fig. 1. As can be observed, straight lines were obtained in all the cases, following equation (3). In the paramagnetic region, the D values follow an Arrhenius law, as can be observed in Fig. 2, with an activation energy Qp 190220 kJ=mol and pre-exponential factor Dp0 623 10 ÿ6 m 2 =s: In the ferromagnetic region a soft upward curvature can be observed. Previous results [16±18] are included for comparison. 4. DISCUSSION
Diusion studies of Sn in a-Fe were performed
previously, with dierent techniques, dierent kinds of Fe samples and in several short temperature ranges. As can be seen in Fig. 2, the agreement in the paramagnetic region with the work of Treheux et al. [16] is excellent. In the ferromagnetic region, in the higher temperature range (above 870 K), the agreement with Henesen et al. [17] again is excellent. In Ref. [17] radiotracer studies were done in coarse grains of pure Fe. The agreement is very good with Myers [18] who applied RBS to implanted polycrystalline Fe samples. Below 870 K, Henesen et al. [17] made Auger electron spectroscopy (AES) measurements of segregation kinetics to determine the diusion coecients in single crystals of Fe±4 wt% Sn; therefore the comparison is not straightforward. In the present study measurements were made in superposed ranges of temperatures when dierent techniques were used. The Arrhenius plot shows a variation of the slope at the Curie temperature and a soft upward curvature in the ferromagnetic region. It is possible to analyze this behavior in the frame of the model described in 1. Then, rewriting equation (1) as "
D T T ln Dp0
!# ÿ
Qp ÿ R
aQp 2 s T R
4
using the experimental values of s(T ) given by Ref. [20] and the present D values for the ferromagnetic region, the plot T ln D=Dp0 against s 2 ®ts a straight line quite well as observed in Fig. 3. Here Dp0 is taken from the results in the paramagnetic region. From the slope a 0:23220:023 is obtained. Using this a value we ®t the measured diusion coecients with equation (1); the agreement showed in Fig. 4 is remarkable. Q p can be recalculated from the intercept of Fig. 3 adjusted by a minimum square ®t, getting Qprecal 19022 kJ=mol, in excellent agreement with Q p obtained directly from the diusion coecients of the paramagnetic region.
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TORRES et al.: DIFFUSION OF TIN IN a-IRON
Taking, as mentioned before, the increment of the vacancy formation energy due to the magnetic order of pure Fe af 4:3 kJ=mol, the increment of the migration energy am is Qprecal a ÿ af 4024 kJ=mol: This value is similar to the one
obtained for self-diusion (35 kJ/mol) [9]. When the diusion of magnetic impurities was studied, a dierent behavior was observed. The case of Co diffusion [8, 10] is an interesting one. Colbalt is known as the only solute element that enhances the
Fig. 1. Diusion pro®les obtained: (a) from RBS experiments; (b) from HIRBS experiments; (c) from grinder sectioning experiments.
TORRES et al.: DIFFUSION OF TIN IN a-IRON
2929
Fig. 3. Plot of T lnD T =D0 vs s 2 for Sn diusion in ferromagnetic a-Fe.
Fig. 2. Arrhenius plot for Sn diusion in a-Fe. Full circles correspond to the value listed in Table 2. When the error associated with a given point is lower than the circle size, no error bar is displayed.
magnetic moment of the Fe matrix; thus, the eect of the magnetic transition on the diusion of Co in a-Fe is expected to be larger than that on self-diusion. In fact the diusion coecients obtained are about one half of the self-diusion in Fe in the ferromagnetic state and above the critical temperature 1043 K and until 1184 K, the Arrhenius plot does not show linearity. Hirano and Iijima [9] applied equation (1) to the available data on Co diusion in a-Fe, using for Dp0 the self-diusion value and the experimental value of s(T ) [20] for pure Fe; they obtained a 0:23: From this value it is possible to estimate am as 54.5 kJ/mol, which is a higher value than the one for self-diusion. They proposed that a local magnetic ®eld around a Co atom in the Fe matrix is responsible for this increment in the am value. Following this idea, and assuming that no magnetic moment is associated with a Sn atom, there is no reason for a change in the am value when Sn is diusing in a-Fe. From the magnetic order point of view, a Sn atom in a substitutional position in the Fe matrix is like a site with no magnetic moment, then we can think of it as a ``magnetic vacant site''. To support this idea, lets rewrite equation (1) as " !# 1 kT D T s T ÿ 1 p ln : a Q Dp0 2
experimental measurements [20] and the diamonds correspond to equation (5) applied to self-diusion. The pointed line represent the values obtained from Co diusion data; as can be seen, s 2 is larger than that in self-diusion in the whole temperature range and it remains dierent from zero above the Curie temperature. This means that the local magnetic ®eld around the Co atoms remains even above the Curie temperature. Finally the plus symbol represents the s 2 values calculated from the present work. They ®t the experimental magnetization data as well as the self-diusion ones and become zero at TC. This behavior indicates no interaction of Sn atoms with the Fe magnetic matrix, as proposed. That idea was also supported by performing ab initio calculations using the local spin density ap-
5
The reduced magnetization can be calculated from the diusion coecients. This is shown in Fig. 5. For comparison, the full line represents the
Fig. 4. Diusion coecients of Sn in a-Fe. The full line is the ®t with equation (1).
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TORRES et al.: DIFFUSION OF TIN IN a-IRON
atom is quite similar to pure Fe, whereas in the Co case, the magnetic moment of the neighboring Fe atom is increased. In the future, it will be interesting to measure the diusion of dierent non-magnetic impurities in aFe in such an extended temperature range, in order to con®rm if the quantity aQ remains constant. It will be interesting too, to know if this trend is followed by a constant value of the magnetic moment of the nearest neighbor Fe, performed using the precedent ab initio calculation. 5. CONCLUSIONS
Fig. 5. Temperature dependence of s 2 obtained from the diusion coecient of Sn in a-Fe (plus symbols) in comparison with those from self-diusion (diamonds) and Co diusion in a-Fe (dotted line). The full line corresponds to the experimental values obtained in Ref. [20].
proximation (LSDA) in three-dimensional periodic systems in a unit cell containing 16 atoms in the b.c.c. structure of a-Fe. The magnetic moment was calculated for the whole cell and for the individual atoms. Then, one of the Fe atoms was replaced by an impurity, Sn or Co, and the results were compared. The WIEN97 code, an implementation of the linearized augmented plane wave method, based on density functional theory, was used for this purpose [23]. No relaxation was allowed in the unit cell, it was always kept at the experimental lattice constant. The three calculations, Fe (for comparison), Co and Sn, are shown in Table 3. The columns show in order the magnetic moment (in Bohr magnetons): of the whole cell, at the impurity and at the nearest neighbor Fe atom (both inside the mun tin radius). Whereas the Co impurity increases the cell magnetic moment by increasing the polarization of the neighboring Fe atoms (even when the Co moment itself is smaller than that of Fe), the Sn as impurity decreases the cell magnetic moment. On the other hand, the magnetic moment at the Sn atom is almost zero, as expected, and the corresponding magnetic moment of the neighboring Fe Table 3. Magnetic moment (in Bohr magnetons) for dierent impurities embedded in a-Fe Impurity Fe Co Sn
M (cell)
M (impurity)
M (neighbor)
34.8 35.7 33.3
2.17 1.71 ÿ0.04
2.17 2.29 2.14
Diusion of Sn in a-Fe was studied in an extended temperature range (673±1163 K) and over almost 10 decades using dierent techniques. The agreement between them in the superposed temperature range is excellent. A deviation from the Arrhenius law was observed in the ferromagnetic region. The curvature observed is well ®tted by the model developed by Ruch et al. The parameter that takes into account the in¯uence of the magnetic order in the vacancy mobility during the Sn diusion process is quite similar to that observed in Fe self-diusion; then, no magnetic interaction between the Sn impurity and the matrix is postulated. AcknowledgementsÐThe authors wish to thank Dr M. Behar for his collaboration with the RBS measurements, to Drs G. GarcõÂ a BermuÂdez and D. Abriola for their collaboration with the HIRBS measurements, to Lic. S. Balart for her collaboration with the serial sectioning measurements and to Dra. M. Weissmann who made the ab initio calculations. This work was partly supported by the Grant PIP 4071 from CONICET and the Grant PICT'97 No. 12-00000-00152 from the Agencia Nacional de PromocioÂn CientõÂ ®ca y TecnoloÂgica. REFERENCES 1. Hettich, G., Mehrer, H. and Maier, K., Scripta metall., 1977, 11, 795. 2. Lubbehusen, M. and Mehrer, H., Acta metall. mater., 1990, 38, 283. 3. Iijima, Y., Kimura, K. and Hirano, K., Acta metall., 1988, 36, 2811. 4. Borg, R. J. and Birchenall, C. E., Trans. metall. Soc. A.I.M.E., 1960, 218, 980. 5. Bungton, F. S., Hirano, K. and Cohen, M., Acta metall., 1961, 9, 434. 6. Ruch, L., Sain, D. R., Yeh, H. L. and Girifalco, L. A., J. Phys. Chem. Solids, 1976, 37, 649. 7. KucÏera, J., Czech. J. Phys. B, 1979, 29, 797. 8. Mehrer, H., HoÈpfel, D. and Hettich, G., in DIMETA82: Diusion in Metals and Alloys, ed. F. J. Kedves and D. L. Beke. Trans. Tech. Publications, Aedermannsdorf, 1983, p. 360. 9. Hirano, K. and Iijima, Y., Defect Diusion Forum, 1989, 66±69, 1039. 10. Iijima, Y., Kimura, K., Lee, C. and Hirano, K., Mater. Trans., JIM, 1993, 34, 20. 11. Cermak, J., LuÈbbehusen, M. and Mehrer, H., Z. Metallk., 1989, 80, 213. 12. Lee, C., Iijima, Y., Hiratani, T. and Hirano, K., Mater. Trans., JIM, 1990, 31, 255.
TORRES et al.: DIFFUSION OF TIN IN a-IRON 13. KucÏera, J., KozaÂk, L. and Mehrer, H., Physica status solidi (a), 1984, 81, 497. 14. Braun, R. and Feller-Kniepmeier, M., Scripta metall., 1986, 20, 7. 15. Klugkist, P. and Herzig, Ch., Physica status solidi (a), 1995, 148, 413. 16. Treheux, D., Marchive, D., Delagrange, J. and Guiraldenq, C. R., Acad. Sci. Paris, SeÂrie C, 1972, 274, 1260. 17. Henesen, K., Keller, H. and Viefhaus, H., Scripta metall., 1984, 18, 1319. 18. Myers, S. M., in Nontraditional Methods in Diusion, ed. G. E. Murch, H. K. Birnbaum and J. R. Cost. The Metallurgical Society of AIME, 1983, p. 137.
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19. Girifalco, L. A., J. Phys. Chem. Solids, 1962, 23, 1171. 20. Crangle, J. and Goodman, G. M., Proc. R. Soc. Lond. A, 1971, 321, 477. 21. PeÂrez, R. A. and Dyment, F., Appl. Phys. A, 1999, 68, 667. 22. PeÂrez, R. A., BermuÂdez, G. G., Abriola, D., Dyment, F. and Somacal, H., Defect Diusion Forum, 1997, 143±147, 1335. 23. Blaha, P., Schwartz, K. and Luitz, J., WIEN97, Vienna University of Technology, Vienna 1997. (Improved and updated UNIX version of the original copyrighted WIEN code, which was published in Blaha, P., Schwartz, K., Sorantin, P. and Trickey, S. B., Comput. Phys. Commun., 1990, 59, 399.)