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Tracer diffusion of iron in α-Mn1±yS polycrystals
PERGAMON
Solid State Communications 117 (2001) 719±722
www.elsevier.com/locate/ssc
Tracer diffusion of iron in a-Mn1^yS polycrystals Jolanta Gilewicz-Wolter a,*, Marcin Wolter b a
Faculty of Physics and Nuclear Techniques, University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 KrakoÂw, Poland b Institute of Nuclear Physics, ul. Radzikowskiego 152, 31-342 KrakoÂw, Poland Received 13 October 2000; accepted 3 January 2001 by P. Dederichs
Abstract Tracer diffusion of iron in polycrystalline samples of a-MnS was studied. In the diffusion experiments the radioactive isotope of iron, 59Fe, was used. The dependence of diffusion coef®cient on sulfur activity and temperature was investigated. It was found that the dominant transport mechanism is the volume diffusion, which proceeds via point defects of a-MnS, and that the iron diffusion rate is larger than the self-diffusion of manganese in manganous sul®de. The activation energy for the vacancy mechanism of diffusion was calculated. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Semiconductors; C. Point defects PACS: 66.30; 61.72.J
1. Introduction Cr±Mn steels, due to their good mechanical properties at high temperatures, are frequently used in these conditions. It was found that during the sul®dation of Cr±Mn steel multicomponent and multiphase scales are formed. The main transport process during sul®dation is the outward diffusion of metal atoms. On the surface of the metallic core, a layer containing MnS is produced [1], so it is important to know the diffusion rate of iron in this sul®de and what diffusion mechanism is privileged. a-MnS has the rocksalt structure. Over the major part of the phase ®eld, corresponding to higher sulfur activities, there is a metal de®cient p-type semiconductor with the dominant defects being doubly ionized cation vacancies and electron holes (Mn12yS). At low sulfur activity, near the Mn/MnS phase boundary, this sul®de is a metal-excess n-type semiconductor with doubly ionized interstitial cations and quasi-free electrons as dominant defects (Mn11yS). The defect concentration in this sul®de is small, so defect±defect interactions can be neglected [2±4]. The self-diffusion of Mn in manganous sul®de was studied by means of the measurements of manganese sul®dation kinetics and its diffusional evaporation [4,5], as well * Corresponding author. Tel.: 148-12-617-2961; fax: 148-12634-0010.
as the measurements of tracer diffusion of manganese performed using the 54Mn radioisotope [6]. In this paper, studies of diffusion of radioactive isotope of iron, 59Fe, in a-MnS polycrystalline specimens are reported. The diffusion experiments were carried out at 1069, 1173 and 1258 K under various sulfur pressures. Also, the calculation method of diffusion coef®cients, different from the one usually used [7] is presented. Instead of ®tting the differential diffusion pro®le using the gaussian obtained from Fick's second law, the integral of this function is ®tted directly to the measured data points. The use of the above method improves the measurement precision and permits proper error handling. 2. Experimental procedure Manganous sul®de was obtained by sul®dation of spectrally pure manganese (Johnson and Matthey) at 1173 K in sulfur vapor at 5 £ 10 3 Pa. Under these conditions a compact a-MnS scale was formed, with grain size ranging from 0.4 to 1.8 mm. Rectangular sul®de samples of dimensions of (10 £ 10 £ 0.7) mm were prepared, polished to get a ¯at surface and degreased with acetone. The diffusion experiments were carried out by annealing the samples within evacuated and sealed quartz ampoules at a de®ned sulfur vapor pressure and temperature [6]. The sources of sulfur vapor were liquid sulfur (for higher
0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(01)00024-2
720
J. Gilewicz-Wolter, M. Wolter / Solid State Communications 117 (2001) 719±722
visible on the autoradiographs. Such a network was not observed. 3. Data analysis
Fig. 1. Diffusion pro®le obtained by the integral method (pS2 300 Pa, T 1173 K).
pressures) and a powdered FeS2/FeS mixture (for lower pressures). The sulfur pressure was established by the equilibrium of sulfur vapor with sulfur sources at desired temperature. The values of sulfur pressure were calculated on the basis of thermochemical data [8] and assuming ideal gas behavior. In order to establish the required defect concentration throughout the specimen, all the samples were preannealed at the same temperature and sulfur vapor pressure as during the diffusion experiments, in which the samples were later used. The annealing temperature was controlled with the accuracy of 2 K using Pt±Pt10%Rh thermocouple. The polished and degreased surfaces of the specimens were coated with a carrier-free solution of 59FeCl3 in alcohol and dried under infrared radiation. The activity of 59Fe was about 0.6 MBq cm 22, which corresponds to about 0.4 ng cm 22 of Fe. Such a mass per unit surface area ful®lls the conditions of the thin layer geometry. Radioactive isotope of iron, 59Fe, emits the g-radiation of energies 1.1 and 1.29 MeV, so the radiation absorption in specimen material may be neglected. After annealing the ampoules were rapidly cooled. The edges of the specimen were cut to eliminate the additional radioactivity originating from the side of the sample. The 59Fe penetration pro®les were obtained by the serial sectioning technique. The samples were mounted in polyacrylic resin together with three steel spheres having a well-known diameter [6]. The spheres were ground together with the specimen. To calculate the thickness of the removed layer the diameters of the crosssections of the spheres were measured. Following the removal of each section the residual activity of a specimen was counted using the scintillation (NaJ(Tl)) counter. The statistical error was about 1%. In order to check, whether grain boundary diffusion prevails, autoradiographs of surfaces normal to the diffusion direction were taken from some samples at different penetration depths. In the case, when the grain boundary diffusion is signi®cant, the grain boundary network should be
The duration of diffusional annealing was chosen in such a way, that the penetration depth of the tracer was smaller than the thickness of the sample. This allows considering the discussed system in terms of approximation to a semi-in®nite solid. For these boundary conditions (semi-in®nite solid and thin layer geometry) the solution of Fick's second law gives the following equation [7,9]: ! z2 1 c z M exp 2 4DT t where c(z) is the concentration of the tracer at a distance z from the sample surface, t the annealing time, DT the tracer diffusion coef®cient and M the proportionality coef®cient. As the residual radioactivity is measured and the selfabsorption of the 59Fe radiation may be neglected, the counting rate after removing the layer of thickness x is proportional to the whole contents of 59Fe in the remaining sample: Z1 I x K c z dz 2 x
where I is the counting rate and K the proportionality coef®cient. The numerically integrated function is ®tted to the measured counting rates at different thickness of the removed material. In the ®tting procedure the following function is used: ! Z1 z 2 x0 2 I x A exp 2 dz 1 B 3 4´DT t x where the normalization A and DTt are the ®tted variables. Also B describing the residual constant counting rate is left free within the ®t. This background probably originates from the surface diffusion on the opposite side of the sample. The signi®cant contribution of grain boundary diffusion was not found. In order to include the uncertainty of the penetration depth measurements into the ®t, and therefore into the estimated DT error, the statistical errors of counting rate I are increased by the x uncertainty multiplied by the differential 2I=2xx : The total error used in the ®tting procedure is given by q DI 0 DI2 1 Dx 2I=2xx 2 : 4 The value of Dx (the same for every point) is chosen in such a way, that the x 2 of the ®t per degree of freedom is equal to one. Estimating an x uncertainty as described above authors assume, that the solution of Fick's second equation describes well the diffusion process and all the uncertainties are due to the statistical errors of x. The obtained estimation
J. Gilewicz-Wolter, M. Wolter / Solid State Communications 117 (2001) 719±722
Fig. 2. Diffusion pro®le obtained by the differential method (pS2 300 Pa, T 1173 K).
of Dx is later used to calculate the uncertainty of the tracer diffusion coef®cient DT. The estimated value of DTt is correlated with x0, which corresponds to the surface of the sample coated by the radioisotope. The x0 uncertainty is estimated to be about 10 mm, since the sample can be a bit deformed during the annealing. The uncertainty of x0 is the dominant source of DT error. 4. Results Fig. 1 shows the ®t for the diffusion of 59Fe in a-MnS at sulfur vapor pressure of 300 Pa at 1173 K. The error bars represent the corrected errors as shown in Eq. (4). Differentiating Eq. (3) the following formula is obtained: ! x 2 x0 2 dI=dx C exp 2 5 4DT t where C is the proportionality coef®cient. In this formula one can replace the differential dI=dx by the change of activity after removing a single layer divided by the layer thickness. Using this approach and plotting x 2 on the horizontal axis the linear dependence is obtained (see Fig. 2). In this method the spread of the data points is arti®cially increased Table 1 Diffusion coef®cients of 59Fe in Mn1^yS.
1 2 3 4 5 6 7
T (K)
pS2 (Pa)
DT (cm 2 s 21)
1069 1173 1173 1173 1173 1173 1258
4.3 £ 10 3 4.3 £ 10 3 300.0 15.0 2.5 £ 10 29 9.0 £ 10 211 4.3 £ 10 3
(1.08 ^ 0.06) £ 10 29 (4.32 ^ 0.24) £ 10 29 (2.50 ^ 0.10) £ 10 29 (1.38 ^ 0.08) £ 10 29 (0.44 ^ 0.02) £ 10 29 (0.68 ^ 0.02) £ 10 29 (9.27 ^ 0.37) £ 10 29
721
Fig. 3. Dependence of the tracer diffusion coef®cient on temperature for the sulfur vapor pressure 4.3 £ 10 3 Pa (Arrhenius plot).
by dividing small DI by small layer thickness Dx: Also the neighbor points are correlated, since the same data points are used to calculate them. It is visible in Fig. 1, that the ®tted function describing the volume diffusion ®ts well to the data points. This indicates, that the predominant iron transport mechanism is the volume diffusion. The additional autoradiographic studies con®rm this conclusion: the blackening of the autoradiographs is uniform. Table 1 lists the obtained values of the tracer diffusion coef®cients of 59Fe in a-MnS. On the basis of obtained tracer diffusion coef®cients the activation energy of iron volume diffusion was calculated for the temperature range (1069±1258) K at sulfur vapor pressure of 4.3 £ 10 3 Pa, E 127 ^ 9 kJ mol21 (see Fig. 3). So, the diffusion coef®cient of 59Fe in a-MnS at higher sulfur vapor pressures may be described by the following equation: 127 kJ DT 4:36 £ 1024 p1=6 : 6 S2 exp 2 RT
5. Discussion As mentioned previously, the predominant point defects in a-MnS over nearly the whole homogeneity range are cation vacancies and electron holes. At very low sulfur activities the dominant point defects are interstitial cations and quasi-free electrons. Using KroÈger±Vink notation [10] defect equilibria in both parts of the phase ®eld can be described by the following equations. The ®rst one describes the formation of doubly ionized vacancies _ 1=2S2 g SS 1 V 00Mn 1 2h:
7
The second one describes the formation of doubly charged
722
J. Gilewicz-Wolter, M. Wolter / Solid State Communications 117 (2001) 719±722
of iron should occur by the same point defects as that of manganese. However, the rate of iron diffusion is higher than that of manganese. The reason of higher 59Fe diffusion coef®cient is probably due to the difference between radii of diffusing ions: the radius of iron ion is smaller than that of manganese [11]. The value of activation energy for vacancy mechanism of iron diffusion (127 ^ 9) kJ mol 21 is close to that of manganese diffusion (122 kJ mol 21). 6. Conclusions
Fig. 4. Dependence of the diffusion coef®cient on sulfur vapor pressure.
interstitial cations MnMn 1 SS Mn´´i 1 2e2 1 1=2S2 g :
8
Using Eqs. (7) and (8), the mass action law, and the simpli®ed electroneutrality conditions: _ 2V 00Mn h; Mn´´i 1=2e2
9 10
allows to formulate the dependence of the defect concentration on sulfur vapor pressure: V 00Mn const p1=6 S2 ;
11
21=6 Mn´´i const pS2 :
12
The relations (11) and (12) may be employed, if only one type of point defects is predominant and the defect concentration is suf®ciently low. It was found [6], that at relatively high sulfur activities the diffusion of 54Mn in a-MnS occurs via doubly ionized vacancies, while at very low sulfur activities, near the Mn/ MnS phase boundary, by an interstitial mechanism. Fig. 4 shows the dependence of tracer diffusion coef®cients of 59Fe in a-MnS on the sulfur vapor pressure. In this ®gure the, tracer diffusion coef®cients of 54Mn are also plotted. It is clearly seen that this dependence for both, 54Mn and 59Fe, is similar. At higher sulfur activities, the rate of diffusion increases with increasing sulfur pressure as predicted for metal de®cit Mn12yS, and at low pressures it decreases, which is in agreement with the defect structure model for metal excess Mn11yS sul®de. The slope of the line for higher sulfur activities is about 1/6, and for low sulfur activities the slope is about 21/6 (see Eqs. (11) and (12)). The diffusion
The analysis presented above shows that the dominant transport mechanism of iron in a-MnS is volume diffusion. At higher sulfur activities diffusion occurs by cationic vacancies, while at low sulfur pressures, close to the aMnS dissociation pressure, an interstitial mechanism dominates. Also the diffusion rates of iron were compared with those of manganese. It was shown that the rate of iron diffusion is higher. The activation energy for the vacancy mechanism of iron diffusion is close to that of manganese for the same range of sulfur pressure. Acknowledgements This work was partially supported by the State Committee for Scienti®c Research. Authors wish to thank Prof. A. ZieÃba and Dr L. Pytlik for their remarks and fruitful discussions. References [1] J. èaskawiec, Z. ZÇurek, M. Danielewski, M. HetmanÂczyk, A. Iwaniak, Materiaøy 6 OgoÂlnopolskiej Konferencji Korozyjnej, Materiaøy a SÂrodowisko. CzeËstochowa, Ochrona przed korozjaË, wydanie specjalne, 1999, p. 544. [2] H.J. Rau, J. Phys. Chem. Solids 39 (1978) 339. [3] M. Danielewski, S. Mrowec, Solid State Ionics 17 (1985) 29. [4] M. Danielewski, S. Mrowec, Solid State Ionics 17 (1985) 41. [5] F.A. Elrefaie, W.W. Smeltzer, Oxid. Met. 16 (1981) 267. [6] J. Gilewicz-Wolter, M. Danielewski, S. Mrowec, Phys. Rev. B 56 (1997) 8695. [7] D. Shaw (Ed.), Atomic Diffusion in Semiconductors, Plenum Press, London, 1973. [8] J. Barin, O. Knacke, Thermochemical Properties of Inorganic Substances and Suppl., Springer, Berlin, 1977. [9] S.J. Rothman, The Measurements of Tracer Diffusion Coef®cients in Solids, in: G.E. Murch, A.S. Nowick (Eds.), Diffusion in Crystalline Solids, Academic Press, New York, 1984. [10] F. KroÈger, The Chemistry of Imperfect Crystals, NorthHolland, Amsterdam, 1964. [11] T. Penkala, Zarys krystalogra®i, PWN, Warsaw, 1972.
Solid State Communications 117 (2001) 719±722
www.elsevier.com/locate/ssc
Tracer diffusion of iron in a-Mn1^yS polycrystals Jolanta Gilewicz-Wolter a,*, Marcin Wolter b a
Faculty of Physics and Nuclear Techniques, University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 KrakoÂw, Poland b Institute of Nuclear Physics, ul. Radzikowskiego 152, 31-342 KrakoÂw, Poland Received 13 October 2000; accepted 3 January 2001 by P. Dederichs
Abstract Tracer diffusion of iron in polycrystalline samples of a-MnS was studied. In the diffusion experiments the radioactive isotope of iron, 59Fe, was used. The dependence of diffusion coef®cient on sulfur activity and temperature was investigated. It was found that the dominant transport mechanism is the volume diffusion, which proceeds via point defects of a-MnS, and that the iron diffusion rate is larger than the self-diffusion of manganese in manganous sul®de. The activation energy for the vacancy mechanism of diffusion was calculated. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Semiconductors; C. Point defects PACS: 66.30; 61.72.J
1. Introduction Cr±Mn steels, due to their good mechanical properties at high temperatures, are frequently used in these conditions. It was found that during the sul®dation of Cr±Mn steel multicomponent and multiphase scales are formed. The main transport process during sul®dation is the outward diffusion of metal atoms. On the surface of the metallic core, a layer containing MnS is produced [1], so it is important to know the diffusion rate of iron in this sul®de and what diffusion mechanism is privileged. a-MnS has the rocksalt structure. Over the major part of the phase ®eld, corresponding to higher sulfur activities, there is a metal de®cient p-type semiconductor with the dominant defects being doubly ionized cation vacancies and electron holes (Mn12yS). At low sulfur activity, near the Mn/MnS phase boundary, this sul®de is a metal-excess n-type semiconductor with doubly ionized interstitial cations and quasi-free electrons as dominant defects (Mn11yS). The defect concentration in this sul®de is small, so defect±defect interactions can be neglected [2±4]. The self-diffusion of Mn in manganous sul®de was studied by means of the measurements of manganese sul®dation kinetics and its diffusional evaporation [4,5], as well * Corresponding author. Tel.: 148-12-617-2961; fax: 148-12634-0010.
as the measurements of tracer diffusion of manganese performed using the 54Mn radioisotope [6]. In this paper, studies of diffusion of radioactive isotope of iron, 59Fe, in a-MnS polycrystalline specimens are reported. The diffusion experiments were carried out at 1069, 1173 and 1258 K under various sulfur pressures. Also, the calculation method of diffusion coef®cients, different from the one usually used [7] is presented. Instead of ®tting the differential diffusion pro®le using the gaussian obtained from Fick's second law, the integral of this function is ®tted directly to the measured data points. The use of the above method improves the measurement precision and permits proper error handling. 2. Experimental procedure Manganous sul®de was obtained by sul®dation of spectrally pure manganese (Johnson and Matthey) at 1173 K in sulfur vapor at 5 £ 10 3 Pa. Under these conditions a compact a-MnS scale was formed, with grain size ranging from 0.4 to 1.8 mm. Rectangular sul®de samples of dimensions of (10 £ 10 £ 0.7) mm were prepared, polished to get a ¯at surface and degreased with acetone. The diffusion experiments were carried out by annealing the samples within evacuated and sealed quartz ampoules at a de®ned sulfur vapor pressure and temperature [6]. The sources of sulfur vapor were liquid sulfur (for higher
0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(01)00024-2
720
J. Gilewicz-Wolter, M. Wolter / Solid State Communications 117 (2001) 719±722
visible on the autoradiographs. Such a network was not observed. 3. Data analysis
Fig. 1. Diffusion pro®le obtained by the integral method (pS2 300 Pa, T 1173 K).
pressures) and a powdered FeS2/FeS mixture (for lower pressures). The sulfur pressure was established by the equilibrium of sulfur vapor with sulfur sources at desired temperature. The values of sulfur pressure were calculated on the basis of thermochemical data [8] and assuming ideal gas behavior. In order to establish the required defect concentration throughout the specimen, all the samples were preannealed at the same temperature and sulfur vapor pressure as during the diffusion experiments, in which the samples were later used. The annealing temperature was controlled with the accuracy of 2 K using Pt±Pt10%Rh thermocouple. The polished and degreased surfaces of the specimens were coated with a carrier-free solution of 59FeCl3 in alcohol and dried under infrared radiation. The activity of 59Fe was about 0.6 MBq cm 22, which corresponds to about 0.4 ng cm 22 of Fe. Such a mass per unit surface area ful®lls the conditions of the thin layer geometry. Radioactive isotope of iron, 59Fe, emits the g-radiation of energies 1.1 and 1.29 MeV, so the radiation absorption in specimen material may be neglected. After annealing the ampoules were rapidly cooled. The edges of the specimen were cut to eliminate the additional radioactivity originating from the side of the sample. The 59Fe penetration pro®les were obtained by the serial sectioning technique. The samples were mounted in polyacrylic resin together with three steel spheres having a well-known diameter [6]. The spheres were ground together with the specimen. To calculate the thickness of the removed layer the diameters of the crosssections of the spheres were measured. Following the removal of each section the residual activity of a specimen was counted using the scintillation (NaJ(Tl)) counter. The statistical error was about 1%. In order to check, whether grain boundary diffusion prevails, autoradiographs of surfaces normal to the diffusion direction were taken from some samples at different penetration depths. In the case, when the grain boundary diffusion is signi®cant, the grain boundary network should be
The duration of diffusional annealing was chosen in such a way, that the penetration depth of the tracer was smaller than the thickness of the sample. This allows considering the discussed system in terms of approximation to a semi-in®nite solid. For these boundary conditions (semi-in®nite solid and thin layer geometry) the solution of Fick's second law gives the following equation [7,9]: ! z2 1 c z M exp 2 4DT t where c(z) is the concentration of the tracer at a distance z from the sample surface, t the annealing time, DT the tracer diffusion coef®cient and M the proportionality coef®cient. As the residual radioactivity is measured and the selfabsorption of the 59Fe radiation may be neglected, the counting rate after removing the layer of thickness x is proportional to the whole contents of 59Fe in the remaining sample: Z1 I x K c z dz 2 x
where I is the counting rate and K the proportionality coef®cient. The numerically integrated function is ®tted to the measured counting rates at different thickness of the removed material. In the ®tting procedure the following function is used: ! Z1 z 2 x0 2 I x A exp 2 dz 1 B 3 4´DT t x where the normalization A and DTt are the ®tted variables. Also B describing the residual constant counting rate is left free within the ®t. This background probably originates from the surface diffusion on the opposite side of the sample. The signi®cant contribution of grain boundary diffusion was not found. In order to include the uncertainty of the penetration depth measurements into the ®t, and therefore into the estimated DT error, the statistical errors of counting rate I are increased by the x uncertainty multiplied by the differential 2I=2xx : The total error used in the ®tting procedure is given by q DI 0 DI2 1 Dx 2I=2xx 2 : 4 The value of Dx (the same for every point) is chosen in such a way, that the x 2 of the ®t per degree of freedom is equal to one. Estimating an x uncertainty as described above authors assume, that the solution of Fick's second equation describes well the diffusion process and all the uncertainties are due to the statistical errors of x. The obtained estimation
J. Gilewicz-Wolter, M. Wolter / Solid State Communications 117 (2001) 719±722
Fig. 2. Diffusion pro®le obtained by the differential method (pS2 300 Pa, T 1173 K).
of Dx is later used to calculate the uncertainty of the tracer diffusion coef®cient DT. The estimated value of DTt is correlated with x0, which corresponds to the surface of the sample coated by the radioisotope. The x0 uncertainty is estimated to be about 10 mm, since the sample can be a bit deformed during the annealing. The uncertainty of x0 is the dominant source of DT error. 4. Results Fig. 1 shows the ®t for the diffusion of 59Fe in a-MnS at sulfur vapor pressure of 300 Pa at 1173 K. The error bars represent the corrected errors as shown in Eq. (4). Differentiating Eq. (3) the following formula is obtained: ! x 2 x0 2 dI=dx C exp 2 5 4DT t where C is the proportionality coef®cient. In this formula one can replace the differential dI=dx by the change of activity after removing a single layer divided by the layer thickness. Using this approach and plotting x 2 on the horizontal axis the linear dependence is obtained (see Fig. 2). In this method the spread of the data points is arti®cially increased Table 1 Diffusion coef®cients of 59Fe in Mn1^yS.
1 2 3 4 5 6 7
T (K)
pS2 (Pa)
DT (cm 2 s 21)
1069 1173 1173 1173 1173 1173 1258
4.3 £ 10 3 4.3 £ 10 3 300.0 15.0 2.5 £ 10 29 9.0 £ 10 211 4.3 £ 10 3
(1.08 ^ 0.06) £ 10 29 (4.32 ^ 0.24) £ 10 29 (2.50 ^ 0.10) £ 10 29 (1.38 ^ 0.08) £ 10 29 (0.44 ^ 0.02) £ 10 29 (0.68 ^ 0.02) £ 10 29 (9.27 ^ 0.37) £ 10 29
721
Fig. 3. Dependence of the tracer diffusion coef®cient on temperature for the sulfur vapor pressure 4.3 £ 10 3 Pa (Arrhenius plot).
by dividing small DI by small layer thickness Dx: Also the neighbor points are correlated, since the same data points are used to calculate them. It is visible in Fig. 1, that the ®tted function describing the volume diffusion ®ts well to the data points. This indicates, that the predominant iron transport mechanism is the volume diffusion. The additional autoradiographic studies con®rm this conclusion: the blackening of the autoradiographs is uniform. Table 1 lists the obtained values of the tracer diffusion coef®cients of 59Fe in a-MnS. On the basis of obtained tracer diffusion coef®cients the activation energy of iron volume diffusion was calculated for the temperature range (1069±1258) K at sulfur vapor pressure of 4.3 £ 10 3 Pa, E 127 ^ 9 kJ mol21 (see Fig. 3). So, the diffusion coef®cient of 59Fe in a-MnS at higher sulfur vapor pressures may be described by the following equation: 127 kJ DT 4:36 £ 1024 p1=6 : 6 S2 exp 2 RT
5. Discussion As mentioned previously, the predominant point defects in a-MnS over nearly the whole homogeneity range are cation vacancies and electron holes. At very low sulfur activities the dominant point defects are interstitial cations and quasi-free electrons. Using KroÈger±Vink notation [10] defect equilibria in both parts of the phase ®eld can be described by the following equations. The ®rst one describes the formation of doubly ionized vacancies _ 1=2S2 g SS 1 V 00Mn 1 2h:
7
The second one describes the formation of doubly charged
722
J. Gilewicz-Wolter, M. Wolter / Solid State Communications 117 (2001) 719±722
of iron should occur by the same point defects as that of manganese. However, the rate of iron diffusion is higher than that of manganese. The reason of higher 59Fe diffusion coef®cient is probably due to the difference between radii of diffusing ions: the radius of iron ion is smaller than that of manganese [11]. The value of activation energy for vacancy mechanism of iron diffusion (127 ^ 9) kJ mol 21 is close to that of manganese diffusion (122 kJ mol 21). 6. Conclusions
Fig. 4. Dependence of the diffusion coef®cient on sulfur vapor pressure.
interstitial cations MnMn 1 SS Mn´´i 1 2e2 1 1=2S2 g :
8
Using Eqs. (7) and (8), the mass action law, and the simpli®ed electroneutrality conditions: _ 2V 00Mn h; Mn´´i 1=2e2
9 10
allows to formulate the dependence of the defect concentration on sulfur vapor pressure: V 00Mn const p1=6 S2 ;
11
21=6 Mn´´i const pS2 :
12
The relations (11) and (12) may be employed, if only one type of point defects is predominant and the defect concentration is suf®ciently low. It was found [6], that at relatively high sulfur activities the diffusion of 54Mn in a-MnS occurs via doubly ionized vacancies, while at very low sulfur activities, near the Mn/ MnS phase boundary, by an interstitial mechanism. Fig. 4 shows the dependence of tracer diffusion coef®cients of 59Fe in a-MnS on the sulfur vapor pressure. In this ®gure the, tracer diffusion coef®cients of 54Mn are also plotted. It is clearly seen that this dependence for both, 54Mn and 59Fe, is similar. At higher sulfur activities, the rate of diffusion increases with increasing sulfur pressure as predicted for metal de®cit Mn12yS, and at low pressures it decreases, which is in agreement with the defect structure model for metal excess Mn11yS sul®de. The slope of the line for higher sulfur activities is about 1/6, and for low sulfur activities the slope is about 21/6 (see Eqs. (11) and (12)). The diffusion
The analysis presented above shows that the dominant transport mechanism of iron in a-MnS is volume diffusion. At higher sulfur activities diffusion occurs by cationic vacancies, while at low sulfur pressures, close to the aMnS dissociation pressure, an interstitial mechanism dominates. Also the diffusion rates of iron were compared with those of manganese. It was shown that the rate of iron diffusion is higher. The activation energy for the vacancy mechanism of iron diffusion is close to that of manganese for the same range of sulfur pressure. Acknowledgements This work was partially supported by the State Committee for Scienti®c Research. Authors wish to thank Prof. A. ZieÃba and Dr L. Pytlik for their remarks and fruitful discussions. References [1] J. èaskawiec, Z. ZÇurek, M. Danielewski, M. HetmanÂczyk, A. Iwaniak, Materiaøy 6 OgoÂlnopolskiej Konferencji Korozyjnej, Materiaøy a SÂrodowisko. CzeËstochowa, Ochrona przed korozjaË, wydanie specjalne, 1999, p. 544. [2] H.J. Rau, J. Phys. Chem. Solids 39 (1978) 339. [3] M. Danielewski, S. Mrowec, Solid State Ionics 17 (1985) 29. [4] M. Danielewski, S. Mrowec, Solid State Ionics 17 (1985) 41. [5] F.A. Elrefaie, W.W. Smeltzer, Oxid. Met. 16 (1981) 267. [6] J. Gilewicz-Wolter, M. Danielewski, S. Mrowec, Phys. Rev. B 56 (1997) 8695. [7] D. Shaw (Ed.), Atomic Diffusion in Semiconductors, Plenum Press, London, 1973. [8] J. Barin, O. Knacke, Thermochemical Properties of Inorganic Substances and Suppl., Springer, Berlin, 1977. [9] S.J. Rothman, The Measurements of Tracer Diffusion Coef®cients in Solids, in: G.E. Murch, A.S. Nowick (Eds.), Diffusion in Crystalline Solids, Academic Press, New York, 1984. [10] F. KroÈger, The Chemistry of Imperfect Crystals, NorthHolland, Amsterdam, 1964. [11] T. Penkala, Zarys krystalogra®i, PWN, Warsaw, 1972.