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Damage kinetics in MeV gold ion – Irradiated crystalline quartz
Nuclear Instruments and Methods in Physics Research B 166±167 (2000) 31±34
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Damage kinetics in MeV gold ion ± Irradiated crystalline quartz S.M.M. Ramos
a,*
, C. Clerc b, B. Canut a, J. Chaumont b, M. Toulemonde c, H. Bernas
b
a
D epartement de Physique des Mat eriaux (UMR CNRS 5586) (URA 172), Universit e Claude Bernard, LYON I, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France b Centre de Spectrom etrie Nucl eaire et de Spectrom etrie de Masse (CNRS-IN2P3), 91405 Orsay Campus, France c CIRIL B.P. 5133, 14070 Caen Cedex 5, France
Abstract Damage creation in crystalline a-quartz under gold irradiation was studied at 1.0 and 5.5 MeV using the ARAMIS accelerator at CSNSM (Orsay). Although at these energies the total stopping powers are nearly equal (respectively, 4.20 and 4.46 keV nmÿ1 ), the electronic stopping power is only 1.23 keV nmÿ1 (25% of the total) at 1 MeV while it reaches 2.75 keV nmÿ1 (62% of the total) at 5.5 MeV. The electronic stopping power threshold for damage creation in a-quartz is about 1.8 keV/nm [1]. The experiment thus allows us to follow the damage production kinetics due to nuclear collisions (at 1 MeV) versus electronic collisions (at 5.5 MeV). The damage was determined by channeling Rutherford backscattering (RBS-C) using the 2 MV Van de Graa at DPM (Villeurbanne). Single ion impacts create damage when electronic stopping dominates, while several impacts are necessary to achieve complete damage when nuclear stopping dominates. Dierences in damage eciencies will be discussed. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Irradiation; Quartz; Disorder; Kinetics; Gold
1. Introduction In the last two decades many works were devoted to the irradiation eects in a-quartz crystals due to its manifold applications, especially in optical devices. The largest part of results obtained up to now can be divided in two great categories: in the ®rst one, the irradiations were performed at low energy (keV) and the defect creation is due
* Corresponding author. Tel.: +33-0-472-431-218; fax: +330-472-432-648. E-mail address: [email protected] (S.M.M. Ramos).
to the elastic collisions [2±4]. In the second one, the irradiations were performed at high energy (GeV) and the disorder creation, which occurs above a dE=dxe threshold depending on the target, is ascribed to the high density of energy deposited via electronic processes [5±9]. Both the damage morphology and the disorder kinetics in these two dierent energy ranges seem now relatively well characterized. However, when the nuclear and electronic contributions of the projectile slowing-down are of the same order of magnitude near the dE=dxe threshold, little or none results have been published about the radiation-induced damage. The purpose of this work is to investigate the defect creation kinetics in SiO2 quartz
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 0 7 3 5 - 1
32
S.M.M. Ramos et al. / Nucl. Instr. and Meth. in Phys. Res. B 166±167 (2000) 31±34
irradiated in this referred energy range. For such study the irradiated samples were characterized by Rutherford Backscattering Spectrometry in channeling geometry (RBS-C). 2. Experimental procedure a-quartz single crystals were irradiated at ARAMIS accelerator with 197 Au ions of two dierent energies: 1.0 and 5.5 MeV, respectively. All the irradiations were performed at room temperature, with ¯uences extending from 1012 to 1.2 ´ 1013 ions cmÿ2 . The main irradiation parameters, deduced from SRIM97 code [10] in conjunction with the relative velocities b v/c (where v is the velocity of the ions and c is the light velocity) are listed in Table 1. The lattice disorder a0 at the target surface was measured at every irradiation ¯uence by RBS-C analysis whose principle is to align an incident beam of light ions with respect to an axial direction of the single crystalline sample. The backscattering spectrum recorded in this channeling geometry is then compared with the one obtained without preferential alignment of the sample (``random'' spectrum), thus allowing to determine a0 . In the present work, RBS-C analyses were performed using a 2 MeV 4 He beam delivered by a Van de Graa accelerator. The analyzed sample was mounted on a three axis goniometer head oering an angle resettability better than 0.05°. The backscattered He ions were detected by a silicon implanted junction with an energy resolution of 13 keV. The detection angle in respect with the direction of the incident beam was set at 150°. This value results from a compromise between the sensitivity and the selectivity of the RBS-C analysis.
3. Results and discussion Typical RBS-C results are shown in Fig. 1, which presents the random (a) and aligned (b) spectra obtained in irradiated SiO2 quartz. For comparison the aligned spectrum of a pristine crystal (curve c) is also presented. By assuming that in the channeling conditions the incident ions are uniformly distributed across the channels (absence of ``¯ux peaking'' eect), the relative disorder a0 near the irradiated sample surface can be calculated by the following relation [11]: v ÿ vv ; 1 a0 1ÿv where v is the normalized-to-random Rutherford yield measured at the surface energy (here 1170
Table 1 Main features of the irradiations in SiO2 : incident energy (E), projected range (Rp ), incident electronic dE=dxe and nuclear dE=dxn stopping powers and relative velocities (b v=c) Energy (MeV)
Rp (lm)
dE=dxe (keV nmÿ1 )
dE=dX n (keV nmÿ1 )
b (%)
1.0 5.5
0.25 1.26
1.23 2.75
3.07 1.71
0.01 0.02
Fig. 1. RBS-C spectra of SiO2 quartz irradiated with 4 1012 Au cmÿ2 at two dierent energies: (A) 1.0 MeV and (B) 5.5 MeV. (a) and (b) random and channeled spectra, respectively, (c) channeled spectrum of a virgin sample. Analysis conditions: 2 MeV 4 He beam; detection angle of 150°.
S.M.M. Ramos et al. / Nucl. Instr. and Meth. in Phys. Res. B 166±167 (2000) 31±34
keV) of the silicon signal. vv is the same parameter measured on the pristine crystal (typically vv 3:0%). Fig. 2(a) shows, for irradiations at 1.0 MeV energy, the evolution of the relative disorder a0 versus the ¯uence /. The experimental curve follows a S-like behaviour. Such a kinetics cannot be described by a direct impact model [12], which was previously used to ®t the ¯uence evolution of a0 at large electronic stopping powers [7,13,14]. The disorder kinetics when the nuclear stopping power dominates suggests a more complicate scenario for the target amorphization, as for example, the ``nucleation and growth'' approach based on the Avrami formalism [15±18]. The model includes
Fig. 2. Fluence dependence of the relative disorder a0 , calculated by Eq. (1), in a-quartz irradiated at two dierent energies: (a) 1.0 MeV; (b) 5.5 MeV. The continuous curve corresponds to the best ®t of the experimental data by using Eq. (2) based on the AVRAMI model. The dashed lines are theoretical curves obtained for both n 1 and n 3:7 by using the same approach.
33
two adjustable parameters and can be summarized as follows: 1. Creation of isolated point defects. 2. Accommodation of the crystal to compensate the atomic displacements. This induces lattice strains. 3. Nucleation of amorphous islands once lattice strains exceed a critical value. 4. Ion beam induced growth of these islands. According to these hypotheses, the damage kinetics can be described by the following relation: a0 1 ÿ exp ÿ A/n ;
2
where / is the ¯uence, A a parameter which depends on the nucleation rate and the growth rate and n is an exponent related to the geometry of the growth (n 3±4 for a 3-dimensional growth, 1.5±2.5 for a 2-dimensional growth). In our experiments we found n 3:7 indicating 3-dimensional nucleation and growth. The present results are in good agreement with those obtained by Bolze for a-quartz irradiated with 50 keV Na-ions [2,4]. In addition, from this same ®tting we found A 1:88 10ÿ13 cm2 . This parameter has been interpretated as a ``damage cross-section'' due to a nuclear collisions contribution. The evolution of the relative disorder a0 versus the ¯uence / for samples irradiated at 5.5 MeV energy is displayed in Fig. 2(b). In this case, the electronic stopping power exceeds the threshold for damage creation in a-SiO2 by collective electronic excitations [7] it is higher than the nuclear stopping power. The best ®tting of our experimental results by using the Avrami approach (Fig. 2(b)) gives a exponent n 1:4 and a parameter A 1:29 10ÿ13 cm2 . Whereas these two values imply less nuclear contribution to the damage creation, the value of n 1:4 evidences that the damage kinetics is not solely governed by the electronic processes. The dashed curves in Fig. 2(b) correspond to the ®ttings obtained by using two ®xed n values (n 1 or n 3:7) and by keeping only one free parameter (A) in both cases. Neither of these ®ttings are suitable to account for the experimental data. This suggests that a linear combination of these two contributions could be used to ®t our results. However, for to determine accurately the respective weighing of one each
34
S.M.M. Ramos et al. / Nucl. Instr. and Meth. in Phys. Res. B 166±167 (2000) 31±34
contribution and the mechanisms involved in the damage kinetics, additional experimental data are required. For this purpose, a systematic investigation taking into account a larger energy range where the nuclear and electronic stopping power are of the same order of magnitude is now underway.
4. Conclusion In this work, we have measured signi®cant lattice disorder in SiO2 single crystals submitted to 197 Au irradiations. The energy used were chosen in order to follow the damage production kinetics due to nuclear collisions (at 1 MeV) versus electronic collisions (at 5.5 MeV). The sigmoõdal shape exhibited by the disorder evolution versus the irradiation ¯uence suggests that the disorder kinetics can be described by a nucleation and growth approach where two adjustable parameters are included. These two parameters are a damage cross-section (A) and an exponent n which is associated to the geometry of the growth. Fittings of our experimental data indicate a correlation between the exponent n and the damage process (nuclear collisions or electronic excitations). In this work, we have found n 3:7 and A 1:88 10ÿ13 cm2 when the damage results only from nuclear collisions. In the regime, where the electronic excitations become predominant in the defect creation process we have found n 1:4 and A 1:29 10ÿ13 cm2 . Although in this last case, the electronic stopping power is higher than the electronic stopping power threshold for damage
creation in a-quartz, the n value evidences that the damage kinetics is not solely governed by the electronic processes. References [1] J.F. Ziegler, J.P. Biersack, U. Littmark, in: Stopping Power and Ranges of Ions in Matter, Vol. I, Pergamon, New York, 1985. [2] F. Harsmeier, W. Bolse, J. Appl. Phys. 83 (1998) 4049. [3] F. Harsmeier, W. Bolse, Mater. Sci. Forum 248±249 (1997) 319. [4] W. Bolse, Nucl. Instr. and Meth. B 148 (1999) 83. [5] R.L. Fleischer, P.B. Brice, R.M. Walker, in: Nuclear Tracks in Solids, University of California, Berkeley, 1975. [6] T.A. Tombrello, Nucl. Instr. and Meth. B 2 (1984) 555. [7] A. Meftah, F. Brisard, J.M. Constantini, E. Dooryhee, M. Hage-Ali, M. Hervieu, J.P. Stoquert, F. Studer, M. Toulemonde, Phys. Rev. B 49 (1994) 12457. [8] M. Toulemonde, E. Balanzat, S. Bouard, J.J. Grob, M. Hage-Ali, J.P. Stoquert, Nucl. Instr. and Meth. B 46 (1990) 64. [9] F. Jollet, J.P. Duraud, C. Noguera, Radiat. E. Def. Solids J. 110 (1989) 185. [10] J.P. Biersack, L.G. Haggmark, Nucl. Instr. and Meth. 174 (1980) 257. [11] W.K. Chu, J.W. Mayer, M.A. Nicolet, in: Backscattering Spectrometry, Academic Press, New York, 1978, p. 257. [12] J.F. Gibbons, Proc. IEEE 60 (1972) 1062. [13] B. Canut, A. Benyagoub, G. Marest, A. Meftah, N. Moncore, S.M.M. Ramos, F. Studer, P. Thevenard, M. Toulemonde, Phys. Rev. B 51 (1995) 12194. [14] B. Canut, S.M.M. Ramos, R. Brenier, P. Thevenard, J.L. Loubet, M. Toulemonde, Nucl. Instr. and Meth. B 107 (1996) 194. [15] M. Avrami, J. Chem. Phys. 7 (1939) 1103. [16] M. Avrami, J. Chem. Phys. 8 (1940) 212. [17] M. Avrami, J. Chem. Phys. 9 (1941) 177. [18] W.A. Johnson, R.F. Melh, Trans. Am. Instr. Min. Metal. Pet. Engrg. 135 (1939) 416.
www.elsevier.nl/locate/nimb
Damage kinetics in MeV gold ion ± Irradiated crystalline quartz S.M.M. Ramos
a,*
, C. Clerc b, B. Canut a, J. Chaumont b, M. Toulemonde c, H. Bernas
b
a
D epartement de Physique des Mat eriaux (UMR CNRS 5586) (URA 172), Universit e Claude Bernard, LYON I, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France b Centre de Spectrom etrie Nucl eaire et de Spectrom etrie de Masse (CNRS-IN2P3), 91405 Orsay Campus, France c CIRIL B.P. 5133, 14070 Caen Cedex 5, France
Abstract Damage creation in crystalline a-quartz under gold irradiation was studied at 1.0 and 5.5 MeV using the ARAMIS accelerator at CSNSM (Orsay). Although at these energies the total stopping powers are nearly equal (respectively, 4.20 and 4.46 keV nmÿ1 ), the electronic stopping power is only 1.23 keV nmÿ1 (25% of the total) at 1 MeV while it reaches 2.75 keV nmÿ1 (62% of the total) at 5.5 MeV. The electronic stopping power threshold for damage creation in a-quartz is about 1.8 keV/nm [1]. The experiment thus allows us to follow the damage production kinetics due to nuclear collisions (at 1 MeV) versus electronic collisions (at 5.5 MeV). The damage was determined by channeling Rutherford backscattering (RBS-C) using the 2 MV Van de Graa at DPM (Villeurbanne). Single ion impacts create damage when electronic stopping dominates, while several impacts are necessary to achieve complete damage when nuclear stopping dominates. Dierences in damage eciencies will be discussed. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Irradiation; Quartz; Disorder; Kinetics; Gold
1. Introduction In the last two decades many works were devoted to the irradiation eects in a-quartz crystals due to its manifold applications, especially in optical devices. The largest part of results obtained up to now can be divided in two great categories: in the ®rst one, the irradiations were performed at low energy (keV) and the defect creation is due
* Corresponding author. Tel.: +33-0-472-431-218; fax: +330-472-432-648. E-mail address: [email protected] (S.M.M. Ramos).
to the elastic collisions [2±4]. In the second one, the irradiations were performed at high energy (GeV) and the disorder creation, which occurs above a dE=dxe threshold depending on the target, is ascribed to the high density of energy deposited via electronic processes [5±9]. Both the damage morphology and the disorder kinetics in these two dierent energy ranges seem now relatively well characterized. However, when the nuclear and electronic contributions of the projectile slowing-down are of the same order of magnitude near the dE=dxe threshold, little or none results have been published about the radiation-induced damage. The purpose of this work is to investigate the defect creation kinetics in SiO2 quartz
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 0 7 3 5 - 1
32
S.M.M. Ramos et al. / Nucl. Instr. and Meth. in Phys. Res. B 166±167 (2000) 31±34
irradiated in this referred energy range. For such study the irradiated samples were characterized by Rutherford Backscattering Spectrometry in channeling geometry (RBS-C). 2. Experimental procedure a-quartz single crystals were irradiated at ARAMIS accelerator with 197 Au ions of two dierent energies: 1.0 and 5.5 MeV, respectively. All the irradiations were performed at room temperature, with ¯uences extending from 1012 to 1.2 ´ 1013 ions cmÿ2 . The main irradiation parameters, deduced from SRIM97 code [10] in conjunction with the relative velocities b v/c (where v is the velocity of the ions and c is the light velocity) are listed in Table 1. The lattice disorder a0 at the target surface was measured at every irradiation ¯uence by RBS-C analysis whose principle is to align an incident beam of light ions with respect to an axial direction of the single crystalline sample. The backscattering spectrum recorded in this channeling geometry is then compared with the one obtained without preferential alignment of the sample (``random'' spectrum), thus allowing to determine a0 . In the present work, RBS-C analyses were performed using a 2 MeV 4 He beam delivered by a Van de Graa accelerator. The analyzed sample was mounted on a three axis goniometer head oering an angle resettability better than 0.05°. The backscattered He ions were detected by a silicon implanted junction with an energy resolution of 13 keV. The detection angle in respect with the direction of the incident beam was set at 150°. This value results from a compromise between the sensitivity and the selectivity of the RBS-C analysis.
3. Results and discussion Typical RBS-C results are shown in Fig. 1, which presents the random (a) and aligned (b) spectra obtained in irradiated SiO2 quartz. For comparison the aligned spectrum of a pristine crystal (curve c) is also presented. By assuming that in the channeling conditions the incident ions are uniformly distributed across the channels (absence of ``¯ux peaking'' eect), the relative disorder a0 near the irradiated sample surface can be calculated by the following relation [11]: v ÿ vv ; 1 a0 1ÿv where v is the normalized-to-random Rutherford yield measured at the surface energy (here 1170
Table 1 Main features of the irradiations in SiO2 : incident energy (E), projected range (Rp ), incident electronic dE=dxe and nuclear dE=dxn stopping powers and relative velocities (b v=c) Energy (MeV)
Rp (lm)
dE=dxe (keV nmÿ1 )
dE=dX n (keV nmÿ1 )
b (%)
1.0 5.5
0.25 1.26
1.23 2.75
3.07 1.71
0.01 0.02
Fig. 1. RBS-C spectra of SiO2 quartz irradiated with 4 1012 Au cmÿ2 at two dierent energies: (A) 1.0 MeV and (B) 5.5 MeV. (a) and (b) random and channeled spectra, respectively, (c) channeled spectrum of a virgin sample. Analysis conditions: 2 MeV 4 He beam; detection angle of 150°.
S.M.M. Ramos et al. / Nucl. Instr. and Meth. in Phys. Res. B 166±167 (2000) 31±34
keV) of the silicon signal. vv is the same parameter measured on the pristine crystal (typically vv 3:0%). Fig. 2(a) shows, for irradiations at 1.0 MeV energy, the evolution of the relative disorder a0 versus the ¯uence /. The experimental curve follows a S-like behaviour. Such a kinetics cannot be described by a direct impact model [12], which was previously used to ®t the ¯uence evolution of a0 at large electronic stopping powers [7,13,14]. The disorder kinetics when the nuclear stopping power dominates suggests a more complicate scenario for the target amorphization, as for example, the ``nucleation and growth'' approach based on the Avrami formalism [15±18]. The model includes
Fig. 2. Fluence dependence of the relative disorder a0 , calculated by Eq. (1), in a-quartz irradiated at two dierent energies: (a) 1.0 MeV; (b) 5.5 MeV. The continuous curve corresponds to the best ®t of the experimental data by using Eq. (2) based on the AVRAMI model. The dashed lines are theoretical curves obtained for both n 1 and n 3:7 by using the same approach.
33
two adjustable parameters and can be summarized as follows: 1. Creation of isolated point defects. 2. Accommodation of the crystal to compensate the atomic displacements. This induces lattice strains. 3. Nucleation of amorphous islands once lattice strains exceed a critical value. 4. Ion beam induced growth of these islands. According to these hypotheses, the damage kinetics can be described by the following relation: a0 1 ÿ exp ÿ A/n ;
2
where / is the ¯uence, A a parameter which depends on the nucleation rate and the growth rate and n is an exponent related to the geometry of the growth (n 3±4 for a 3-dimensional growth, 1.5±2.5 for a 2-dimensional growth). In our experiments we found n 3:7 indicating 3-dimensional nucleation and growth. The present results are in good agreement with those obtained by Bolze for a-quartz irradiated with 50 keV Na-ions [2,4]. In addition, from this same ®tting we found A 1:88 10ÿ13 cm2 . This parameter has been interpretated as a ``damage cross-section'' due to a nuclear collisions contribution. The evolution of the relative disorder a0 versus the ¯uence / for samples irradiated at 5.5 MeV energy is displayed in Fig. 2(b). In this case, the electronic stopping power exceeds the threshold for damage creation in a-SiO2 by collective electronic excitations [7] it is higher than the nuclear stopping power. The best ®tting of our experimental results by using the Avrami approach (Fig. 2(b)) gives a exponent n 1:4 and a parameter A 1:29 10ÿ13 cm2 . Whereas these two values imply less nuclear contribution to the damage creation, the value of n 1:4 evidences that the damage kinetics is not solely governed by the electronic processes. The dashed curves in Fig. 2(b) correspond to the ®ttings obtained by using two ®xed n values (n 1 or n 3:7) and by keeping only one free parameter (A) in both cases. Neither of these ®ttings are suitable to account for the experimental data. This suggests that a linear combination of these two contributions could be used to ®t our results. However, for to determine accurately the respective weighing of one each
34
S.M.M. Ramos et al. / Nucl. Instr. and Meth. in Phys. Res. B 166±167 (2000) 31±34
contribution and the mechanisms involved in the damage kinetics, additional experimental data are required. For this purpose, a systematic investigation taking into account a larger energy range where the nuclear and electronic stopping power are of the same order of magnitude is now underway.
4. Conclusion In this work, we have measured signi®cant lattice disorder in SiO2 single crystals submitted to 197 Au irradiations. The energy used were chosen in order to follow the damage production kinetics due to nuclear collisions (at 1 MeV) versus electronic collisions (at 5.5 MeV). The sigmoõdal shape exhibited by the disorder evolution versus the irradiation ¯uence suggests that the disorder kinetics can be described by a nucleation and growth approach where two adjustable parameters are included. These two parameters are a damage cross-section (A) and an exponent n which is associated to the geometry of the growth. Fittings of our experimental data indicate a correlation between the exponent n and the damage process (nuclear collisions or electronic excitations). In this work, we have found n 3:7 and A 1:88 10ÿ13 cm2 when the damage results only from nuclear collisions. In the regime, where the electronic excitations become predominant in the defect creation process we have found n 1:4 and A 1:29 10ÿ13 cm2 . Although in this last case, the electronic stopping power is higher than the electronic stopping power threshold for damage
creation in a-quartz, the n value evidences that the damage kinetics is not solely governed by the electronic processes. References [1] J.F. Ziegler, J.P. Biersack, U. Littmark, in: Stopping Power and Ranges of Ions in Matter, Vol. I, Pergamon, New York, 1985. [2] F. Harsmeier, W. Bolse, J. Appl. Phys. 83 (1998) 4049. [3] F. Harsmeier, W. Bolse, Mater. Sci. Forum 248±249 (1997) 319. [4] W. Bolse, Nucl. Instr. and Meth. B 148 (1999) 83. [5] R.L. Fleischer, P.B. Brice, R.M. Walker, in: Nuclear Tracks in Solids, University of California, Berkeley, 1975. [6] T.A. Tombrello, Nucl. Instr. and Meth. B 2 (1984) 555. [7] A. Meftah, F. Brisard, J.M. Constantini, E. Dooryhee, M. Hage-Ali, M. Hervieu, J.P. Stoquert, F. Studer, M. Toulemonde, Phys. Rev. B 49 (1994) 12457. [8] M. Toulemonde, E. Balanzat, S. Bouard, J.J. Grob, M. Hage-Ali, J.P. Stoquert, Nucl. Instr. and Meth. B 46 (1990) 64. [9] F. Jollet, J.P. Duraud, C. Noguera, Radiat. E. Def. Solids J. 110 (1989) 185. [10] J.P. Biersack, L.G. Haggmark, Nucl. Instr. and Meth. 174 (1980) 257. [11] W.K. Chu, J.W. Mayer, M.A. Nicolet, in: Backscattering Spectrometry, Academic Press, New York, 1978, p. 257. [12] J.F. Gibbons, Proc. IEEE 60 (1972) 1062. [13] B. Canut, A. Benyagoub, G. Marest, A. Meftah, N. Moncore, S.M.M. Ramos, F. Studer, P. Thevenard, M. Toulemonde, Phys. Rev. B 51 (1995) 12194. [14] B. Canut, S.M.M. Ramos, R. Brenier, P. Thevenard, J.L. Loubet, M. Toulemonde, Nucl. Instr. and Meth. B 107 (1996) 194. [15] M. Avrami, J. Chem. Phys. 7 (1939) 1103. [16] M. Avrami, J. Chem. Phys. 8 (1940) 212. [17] M. Avrami, J. Chem. Phys. 9 (1941) 177. [18] W.A. Johnson, R.F. Melh, Trans. Am. Instr. Min. Metal. Pet. Engrg. 135 (1939) 416.