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Vacancies and interstitials in lead
Volume35A,number2
PHYSICS LETTERS
VACANCIES
AND INTERSTITIALS
17Mav1971
IN LEAD
A. SEEGER Max-Planck-Institut für Metallforschung, Inst itut für Physik, Stuttgart and Inst Lint 13 r theoretische und angewandte Physik der Universitdt Stuttgart, Stuttgart, Germany
Received 13 April 1971
It is shown that the experimental data on lead permit an unambiguous assignment of recovery stage III to the migration of self—interstitials and of recovery stage IV to monovacancy migration. In the continuing discussion [1] on the properties of the simple point-defects in f.c.c. metals Pb has sofar played a rather insignificant role, The present note purports to show that enough experimental data on lead are now available to permit definite conclusions to be drawn on the energies of migration of monovacancies (E~) and of self-interstitials (E’~) as well as on tne assignment of the stages observed in the recovery of the electrical resistivity after 3 Me V-electron irradiation. Self-diffusion and vacancy properties. From the temperature dependence of the tracer selfdiffusion coefficient, usually represented as DT(T)
=
D~’exp (- QSD/kT),
(1)
we have to extract information on monovacancies in~articular on the quantity Qi = E + Ety (Ei~= monovacancy formation energy)*. Experimental pairs for QSD and DTare: 1.13 eV, 1.47 cm2/s [3]; 1.07 eV, 0.43 c~2/s [4]; 1.05 eV, 0.28 cm2/s [5]; 1.115 eV, 1.2 cm2/s [6]; 1.07 eV, 0.33 cm2/s [7]; 1.11 eV, 0.89 cm2/s [8]. From the magnitude of the activation volume of self-diffusion ~VSD (average value between 476°Kand 600°K[4,7] ~~SD = 0.66ç~,where = atomic volume) and its temperature variation (weak increase with temperature [7]) we may conclude that the divacancy contribution to selfdiffusion is small but not negligible. We estimate that the mono-vacancy contribution to the tracer self-diffusion coefficient may be represented by
*
In analogy to the other f.c.c. metals and supported by high—t~mperatureX-ray and lattice parameter meas -
urements [2] we assume that vacancy—type defects are responsible for both the point—defect equilibrium concentration and self-diffusion,
2/s
D~v(T)= 0.15 exp(- 1.04 eV/kT) cm
(2)
with uncertainties of less than ±0.02eV in the activation energy and less than a factor of two in the preexponential factor. According to Feder and Nowick 121 the total vacancy concentration at the melting point (Tm=600.6°K) jS C~~(T~~) = 1.7x104. They report EYV = (0.49 ±0.10)eV. Recent positron annihilation experiments [9] yielded = (0.50±0.03) eV, (3) indicating that Feder and Nowick [2] underestimated the accuracy of their own experiments. Eq. (1) was obtained at vacancy concentrations that make a correction for divacancies unnecessary. Combining eq. (1) with C~(Tm), allowing for a 10% correction for divacancies, gives us for the monovacancy formation entropy ~ = (0.9 ±0.6)k. For the monovacancy migration energy one obtains EM = 0.54 eV with approximate limits 0.50 eV~YEM 0.58 eV. V/e see that lead constitutes anotli~rexample [10,111 for an f.c.c. metal with E~”V~EtV.
~
By equating a2flVv?V exp[(S~+S~’~)/k] (a =4.95 A=lattice constant; correlation factor flV = 0.78) to the preexponential factor in (2) we obtain for the monovacancy jump frequency a vlVexp(S~/k)exp(-E~/kT) 13exp (-0.54 eV/kT)s~. 3x10
V1V =
=
=
(4)
Self -interstitials and radiation damage recovery. From recent electron [12]- andy [13]irradiation experiments on Pb the following conclusions may be drawn: (i) The electrical resistivity damage rate curve [12, fig. 1] shows that no free migration of a defect takes place below 4.6°K. 135
Volume 35A. number 2
PHYSICS
(ii) The electrical resistivity annealing curve 12. fig. 2! shows that no free migration takes place between 4.6°Kand about 150°K. In particular, there is no evidence for a low-temperature annealing stage showing the characteristics of stage ‘E in copper [iJ. (iii) The pronounced recovery stage at about l65°Kshows very clearly (in particular with respect to its position on a (T/Tm) - temperature scale and its shift with irradiation dose) the characteristics of stage III in other f.c.c. metals: its assignment to ‘D [l2j must therefore be considered incorrect, in agreement with recent dislocation pinning experiments 1131. (iv) Stage III cannot be due to monovacancy migration. According to eq. (4) during a 10 mm. anneal at 165°Ka monovacancy makes only one jump. Since there is no other low-temperature annealing stage to which the free migration of self-interstitials could be attributed, we have to ascribe stage III to interstitial migration with 0.4 eV. This is in excellent agreement with the assignments for the other f.c.c. metals, in particular for gold. In gold sofar also no lowtemperature annealing stage associated with free migration has been found [14-16]. (v) The low-temperature annealing stages (e.g., at 8°K, 15°K, 30°K. 50°K, 80°K)are to be attributed to the recombination of close interstitial-vacancy pairs. (vi) The final annealing stage at 280°K[12, fig. 2! has all the characteristics of stage IV ~ cluding the internal friction behaviour 113!. According to eq. (4), during 10 mm. at 280°Ka monovacancy performs 5X 106 jumps, so that the assignment of monovacancy migration to stage IV is most natural. In lead the interpretation of the principal annealing stages in terms of self-interstitial mlgration (stage III, EM 0.4 eV) and monovacancy migration (stage IV. ‘E~ 1=0.54 eV) is rather straight-forward, since, as in gold, a concentration dependent low-temperature annealing stage (which could he considered as an alter-
136
LETTERS
17 May 1971.
native for the present assignment to free migration of self-interstitials) does not exist, and since, unlike gold, the annealing stages III and IV (or the associated dislocation pinning stages) are so far apart and so large that their separation and identification presents no problem. In terms of the two-interstitial model 115,171 these results mean that in lead crowdion-type interstitials must be converted to stage-Ill interstitials before they can migrate to vacancies. The author is grateful to Professor A. T. Stewart, Dr. W. Triftshäuser and Dr. B. T. A. McKee for having made available the results of [9] before publication.
References
[1] A. Seeger. H. Schumacher, \V. Schilling and J. Diehi (eds.). Vacancies and interstitials in metals (North—Holland, 1970). [2[ B. Feder and A. S.Amsterdam Nowick, Phil. Mag. 15 (1967) 805 [3] H. A. Resing and N. H. Nachtrieb, J. Phys. Chern. Sol. 31 (1961) 40. [4] NIl. Nachtrieb. H. A. Busing and S. A.Rice. J.
Chem. I’hys. 31(1959) 135. [5] N. H. Nachtrieb and U.S. Handler, .1. Chem. Phys. 23 (1955) 1569. [6] B. Okkerse. Acta Met. 2 (1954) 551. [7] J. B. hudson and 14. E. Hoffman. Trans. Met. Soc. AIME 221 (1961) 761. ~] .J. W. 11111cr, Phys. Rev. 181 (1969) 1095. [9] ~V.Triftshttuser, B. T. A. McKee and A. T - Stewart. Tripartite
,June Meeting,
Winnipeg 1970. and
private communication. [10] A. Seeger, Phys. Letters 12 (1964) 176. [ii] II. Mehrer and A. Seeger. Phys. Stat. Sal. 35 (1969) 313. [12] C. Papastaikoudis, H. Ullmaier and R. H. Kernohan. Phys. Stat. Sol. (a) 2 (1970) K171. [13] D. Lena and K. LOcke, Crystal Lattice Defects 1 (1970) 297.
[14] A. Seeger, J. Phys. Soc. lap. 1.8, Suppl. III (1963) 260. 15] A. Seeger. ref. 1, p. 999-1014. [16] A. Seeger, Bad. Effects 2 (1970) 165. [17] W. Bauer. A. Seeger and A. Sosin, Phys. Letters 24A (1967) 195.
PHYSICS LETTERS
VACANCIES
AND INTERSTITIALS
17Mav1971
IN LEAD
A. SEEGER Max-Planck-Institut für Metallforschung, Inst itut für Physik, Stuttgart and Inst Lint 13 r theoretische und angewandte Physik der Universitdt Stuttgart, Stuttgart, Germany
Received 13 April 1971
It is shown that the experimental data on lead permit an unambiguous assignment of recovery stage III to the migration of self—interstitials and of recovery stage IV to monovacancy migration. In the continuing discussion [1] on the properties of the simple point-defects in f.c.c. metals Pb has sofar played a rather insignificant role, The present note purports to show that enough experimental data on lead are now available to permit definite conclusions to be drawn on the energies of migration of monovacancies (E~) and of self-interstitials (E’~) as well as on tne assignment of the stages observed in the recovery of the electrical resistivity after 3 Me V-electron irradiation. Self-diffusion and vacancy properties. From the temperature dependence of the tracer selfdiffusion coefficient, usually represented as DT(T)
=
D~’exp (- QSD/kT),
(1)
we have to extract information on monovacancies in~articular on the quantity Qi = E + Ety (Ei~= monovacancy formation energy)*. Experimental pairs for QSD and DTare: 1.13 eV, 1.47 cm2/s [3]; 1.07 eV, 0.43 c~2/s [4]; 1.05 eV, 0.28 cm2/s [5]; 1.115 eV, 1.2 cm2/s [6]; 1.07 eV, 0.33 cm2/s [7]; 1.11 eV, 0.89 cm2/s [8]. From the magnitude of the activation volume of self-diffusion ~VSD (average value between 476°Kand 600°K[4,7] ~~SD = 0.66ç~,where = atomic volume) and its temperature variation (weak increase with temperature [7]) we may conclude that the divacancy contribution to selfdiffusion is small but not negligible. We estimate that the mono-vacancy contribution to the tracer self-diffusion coefficient may be represented by
*
In analogy to the other f.c.c. metals and supported by high—t~mperatureX-ray and lattice parameter meas -
urements [2] we assume that vacancy—type defects are responsible for both the point—defect equilibrium concentration and self-diffusion,
2/s
D~v(T)= 0.15 exp(- 1.04 eV/kT) cm
(2)
with uncertainties of less than ±0.02eV in the activation energy and less than a factor of two in the preexponential factor. According to Feder and Nowick 121 the total vacancy concentration at the melting point (Tm=600.6°K) jS C~~(T~~) = 1.7x104. They report EYV = (0.49 ±0.10)eV. Recent positron annihilation experiments [9] yielded = (0.50±0.03) eV, (3) indicating that Feder and Nowick [2] underestimated the accuracy of their own experiments. Eq. (1) was obtained at vacancy concentrations that make a correction for divacancies unnecessary. Combining eq. (1) with C~(Tm), allowing for a 10% correction for divacancies, gives us for the monovacancy formation entropy ~ = (0.9 ±0.6)k. For the monovacancy migration energy one obtains EM = 0.54 eV with approximate limits 0.50 eV~YEM 0.58 eV. V/e see that lead constitutes anotli~rexample [10,111 for an f.c.c. metal with E~”V~EtV.
~
By equating a2flVv?V exp[(S~+S~’~)/k] (a =4.95 A=lattice constant; correlation factor flV = 0.78) to the preexponential factor in (2) we obtain for the monovacancy jump frequency a vlVexp(S~/k)exp(-E~/kT) 13exp (-0.54 eV/kT)s~. 3x10
V1V =
=
=
(4)
Self -interstitials and radiation damage recovery. From recent electron [12]- andy [13]irradiation experiments on Pb the following conclusions may be drawn: (i) The electrical resistivity damage rate curve [12, fig. 1] shows that no free migration of a defect takes place below 4.6°K. 135
Volume 35A. number 2
PHYSICS
(ii) The electrical resistivity annealing curve 12. fig. 2! shows that no free migration takes place between 4.6°Kand about 150°K. In particular, there is no evidence for a low-temperature annealing stage showing the characteristics of stage ‘E in copper [iJ. (iii) The pronounced recovery stage at about l65°Kshows very clearly (in particular with respect to its position on a (T/Tm) - temperature scale and its shift with irradiation dose) the characteristics of stage III in other f.c.c. metals: its assignment to ‘D [l2j must therefore be considered incorrect, in agreement with recent dislocation pinning experiments 1131. (iv) Stage III cannot be due to monovacancy migration. According to eq. (4) during a 10 mm. anneal at 165°Ka monovacancy makes only one jump. Since there is no other low-temperature annealing stage to which the free migration of self-interstitials could be attributed, we have to ascribe stage III to interstitial migration with 0.4 eV. This is in excellent agreement with the assignments for the other f.c.c. metals, in particular for gold. In gold sofar also no lowtemperature annealing stage associated with free migration has been found [14-16]. (v) The low-temperature annealing stages (e.g., at 8°K, 15°K, 30°K. 50°K, 80°K)are to be attributed to the recombination of close interstitial-vacancy pairs. (vi) The final annealing stage at 280°K[12, fig. 2! has all the characteristics of stage IV ~ cluding the internal friction behaviour 113!. According to eq. (4), during 10 mm. at 280°Ka monovacancy performs 5X 106 jumps, so that the assignment of monovacancy migration to stage IV is most natural. In lead the interpretation of the principal annealing stages in terms of self-interstitial mlgration (stage III, EM 0.4 eV) and monovacancy migration (stage IV. ‘E~ 1=0.54 eV) is rather straight-forward, since, as in gold, a concentration dependent low-temperature annealing stage (which could he considered as an alter-
136
LETTERS
17 May 1971.
native for the present assignment to free migration of self-interstitials) does not exist, and since, unlike gold, the annealing stages III and IV (or the associated dislocation pinning stages) are so far apart and so large that their separation and identification presents no problem. In terms of the two-interstitial model 115,171 these results mean that in lead crowdion-type interstitials must be converted to stage-Ill interstitials before they can migrate to vacancies. The author is grateful to Professor A. T. Stewart, Dr. W. Triftshäuser and Dr. B. T. A. McKee for having made available the results of [9] before publication.
References
[1] A. Seeger. H. Schumacher, \V. Schilling and J. Diehi (eds.). Vacancies and interstitials in metals (North—Holland, 1970). [2[ B. Feder and A. S.Amsterdam Nowick, Phil. Mag. 15 (1967) 805 [3] H. A. Resing and N. H. Nachtrieb, J. Phys. Chern. Sol. 31 (1961) 40. [4] NIl. Nachtrieb. H. A. Busing and S. A.Rice. J.
Chem. I’hys. 31(1959) 135. [5] N. H. Nachtrieb and U.S. Handler, .1. Chem. Phys. 23 (1955) 1569. [6] B. Okkerse. Acta Met. 2 (1954) 551. [7] J. B. hudson and 14. E. Hoffman. Trans. Met. Soc. AIME 221 (1961) 761. ~] .J. W. 11111cr, Phys. Rev. 181 (1969) 1095. [9] ~V.Triftshttuser, B. T. A. McKee and A. T - Stewart. Tripartite
,June Meeting,
Winnipeg 1970. and
private communication. [10] A. Seeger, Phys. Letters 12 (1964) 176. [ii] II. Mehrer and A. Seeger. Phys. Stat. Sal. 35 (1969) 313. [12] C. Papastaikoudis, H. Ullmaier and R. H. Kernohan. Phys. Stat. Sol. (a) 2 (1970) K171. [13] D. Lena and K. LOcke, Crystal Lattice Defects 1 (1970) 297.
[14] A. Seeger, J. Phys. Soc. lap. 1.8, Suppl. III (1963) 260. 15] A. Seeger. ref. 1, p. 999-1014. [16] A. Seeger, Bad. Effects 2 (1970) 165. [17] W. Bauer. A. Seeger and A. Sosin, Phys. Letters 24A (1967) 195.