- Email: [email protected]
Uniform behavior of insulators irradiated by swift heavy ions
Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Uniform behavior of insulators irradiated by swift heavy ions G. Szenes ⇑ Department of Materials Physics, Eötvös University, P.O. Box 32, H-1518 Budapest, Hungary
a r t i c l e
i n f o
Article history: Received 31 July 2014 Received in revised form 21 January 2015 Accepted 21 January 2015 Available online xxxx Keywords: Swift heavy ion Thermal spike Spike temperature Tracks
a b s t r a c t Ion induced Re track radii are derived from a universal relation H(r) without involving any materials parameter apart from the melting point Tm. The effect is related to the formation of identical ion-induced temperature distributions in track forming insulators for hse i ¼ Se =N = constant, where Se, and N are the electronic stopping power and the atomic density. Based on H(r), an R2e hse i=ðT m T ir Þ plot is applied where the experimental curves coincide for various insulators without adjustable parameters (Tir – temperature of irradiation). The analysis extends to all track-forming insulators studied up until now. The application of the equilibrium value of Tm is justified in thermal spike calculations. The physical meaning of the condition hse i ¼ Se =N = constant is discussed. H(r) may be valid in those insulators as well in which tracks are not induced. The Fourier equation is not valid under spike conditions. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Recently, old ion-induced track data were analyzed again with quite unexpected result [1,2]. It was found that there is a simple relationship between track radii Re measured in different solids when a simple condition is fulfilled. The explanation of this result led to such conclusions which are irreconcilable with the present ideas about the governing processes. In the present paper we summarize the most important previous results, extend the analysis of experimental results and discuss some points which have not received sufficient attention previously.
2. Previous results When a swift heavy ion hits a solid it deposits a large amount of energy in a localized manner, which may induce irreversible changes. One of these possible changes is the formation of amorphous tracks along the trajectory of the projectile. The effect depends on the inelastic energy deposition which is characterized by the electronic stopping power Se. We found a simple relation between Re induced in different solids by different ions of different electronic stopping power Se and different specific ion energy E [1,2]. The only condition is hse i ¼ Se =N = constant where N is the number density of atoms. The quantitative relation is valid only for the initial values of Re which may be changed with time in some ⇑ Tel.: +36 1 372 2821; fax: +36 1 372 2811. E-mail address: [email protected]
solids. A typical result is shown in Fig. 1 [2]. The enveloping curve is given by
HðrÞ ¼
f hse i 3pkw
2
exp r 2 =w2 ;
ð1Þ
where k is the Boltzmann constant, r is the radial distance from the trajectory, w = 4.5 nm and the efficiency f = 0.4/0.17 at low (E < 2 MeV/nucleon, LO) or high (E > 8 MeV/nucleon, HI) ion velocities which is the consequence of the velocity effect [3]. Track radii can be obtained from the equation H(Re) = Tm Tir (Tm and Tir are the melting and irradiation temperatures). H(r) does not contain materials parameters (MPs), does not depend on the experimental parameters, and f has the same values for track forming insulators in the LO and HI ranges leading to different thermal energies e = fSe. Thus track radii are given by a universal function and MPs apart Tm have no effect on Re [1,2]. This result is irreconcilable with the traditional application of the heat flow equation under spike conditions. It was found that the observed relationship between track radii is the consequence of the formation of an identical temperature distribution in irradiated track forming insulators [1]. 3. Discussion Fig. 1 convincingly demonstrates the relationship between track radii showing that Tm alone determines Re, other MPs are indifferent. This conclusion is valid only for a single value of hse i which is given in the plot. However, here we show another simple and direct method which is free of this disadvantage. When the universal
http://dx.doi.org/10.1016/j.nimb.2015.01.046 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: G. Szenes, Uniform behavior of insulators irradiated by swift heavy ions, Nucl. Instr. Meth. B (2015), http://dx.doi.org/ 10.1016/j.nimb.2015.01.046
2
G. Szenes / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
relationship H(r) is valid for an insulator the plot R2e lnðf hse i= ðT m T ir ÞÞ is a straight line (Tm-parameter, f hse i-variable, whose slope is w2. Since H(r) does not contain MPs the lines for various insulators coincide. The plot in Fig. 1 is a special case of this procedure when Tm is a variable and f hse i = constant. In this paper, we apply this plot for tracks induced by low velocity ions in TeO2 [4], by cluster ions in Y3Fe5O12 [5], and by high velocity projectiles in NiFe2O4 [6]. There is a good agreement between the theoretical line and the experimental data in Fig. 2. The track data of all track-forming insulators analyzed in Ref. [1,2] also show similar good agreement. The plot confirms the validity of H(r) in a broad range of hse i. The track data which were used in Fig. 1 were obtained in independent experiments. In this case, one cannot expect that track radii be measured at the same value of hse i. Therefore, when drawing the plot in Fig. 1, the data were evaluated by linear interpolation between neighboring experimental values. This might increase the error of the estimated track radii. In Fig. 2 a new plot is used for the first time which allows the use of the original track data making any interpolation unnecessary. Besides the broad range of the hse i values this is also a great advantage. This method can be applied even when systematic studies have not been performed and only a single value of Re is known. In Fig. 2 we show such data for ZrSiO4 [7], MgAl2O4 [8], Al2O3 [8], KTiOPO4 [9], and Y2O3 [10] which were obtained in experiments with LO or HI projectiles. In the case of Al2O3 and MgAl2O4, Re values have been also measured in a range of Se but track re-crystallization reduced the initial track sizes. Therefore, we do not use those data. Recently, we analyzed the study of Aruga [11] and derived reliable estimate of Set which is the highest value of Se without inducing a track [8]. The method does not require the measurement of track sizes only the position of the amorphous–crystalline boundary must be known. Therefore the track re-crystallization has only a minor effect. New measurements by Khalfaoui et al. [12] and Skuratov et al. [13] confirm our estimate for Al2O3. In Fig. 2, the single experimental track data nicely fit to the line derived from the relation H(r) verifying its validity for these solids as well. We emphasize that the good agreement with the relation H(r) was achieved without individual adjustable parameters in a broad range of Se for a large number of different projectiles having energies in the range 0.02–20 MeV/nucleon and track radii were measured by different methods in different laboratories. Thus the reliability of the conclusions is without doubt.
Fig. 1. Variation of the track radii Re with the melting temperature Tm in various insulators; Tir is the irradiation temperature, k is the Boltzmann constant. The tracks were induced by ion beams with E<2 MeV/nucleon specific energy at hse i=3k ¼ 3:42 105 nm2K. The curve H(r) = TpL exp{r2/w2} is a fit with w = 4.45 ± 0.18 nm and TpL = 2024 ± 90 K [2].
Fig. 2. Variation of the track size with the atomic stopping power hse i; Re, f, and k are the track radius, the efficiency and the Boltzmann constant, w = 4.5 nm and Tm and Tir are the melting and the irradiation temperatures. Y3Fe5O12LO and Y3Fe5O12cl denote irradiation by monoatomic and C cluster ions. The line is drawn according to Eq. (1) with the slope w2.
Recently we finished reviewing all track data reported up until now. The validity of the H(r) relation for track formation has been verified for 18 insulators including the present results as well [1,2]. In another publication we complete this list with two other solids [14] and there is a number of studies which had to be ignored because they were not suitable for our analysis [14]. After analyzing all experimental track data we conclude that track-forming insulators respond identically to the irradiation by swift heavy ions. Track radii can be derived applying a universal relation H(r) which is free of MPs. The simple relationship between track radii shown in Figs. 1 and 2 is the direct consequence of this feature. Based on this result we assume that such behavior can be found in a large number of irradiated insulators. The presence of the melting point in the solution of H(r) for Re demonstrates that thermal processes have an important effect on Re. We arrived to the same conclusion when analyzed Set in various insulators and a linear dependence on Tm was observed [1]. Actually, Set is proportional to the energy necessary to increase the local temperature from Tir to Tm. Thus it is reasonable to assume that a melt is formed when the local temperature T > Tm and the amorphous track is formed in the fast cooling process of the melt. The ion-induced temperature increase is given by the DT(r,t) function and the time is set to t = 0 when it attains its highest value. It is reasonable to assume that Re is equal to the maximum radius of the melt or the maximum width of the radial temperature distribution DT(Re,0) at t = 0 [1]. Let us denote various MPs as X and assume that DT(r,t,X) is valid. It is a reasonable assumption that when DT = DT(r,t,X), the track radius Re also depends on X. When DT(r,t) does not depend on MPs it is evident that Re depends neither. This must be valid in the opposite direction as well: if MPs have no effect on Re, then DT(r,0) should not depend on MPs either. When this simple consideration is applied to our case, the result is rather unexpected. When track forming insulators are irradiated by swift heavy ions the induced temperature distributions DT(r,0) are identical when hse i = constant as they do not contain MPs. The H(Re) = DT(Re,0) = Tm Tir relation is valid for each track-forming insulator. Therefore, H(r) = DT(r,0) holds for many (melting) temperatures. Thus we claim that the individual DT(r,0) distributions are equal to the universal relation H(r). Here we refer again to Fig. 2. It is a highly extraordinary situation when the experimental data are compared with a universal function without fitting parameters. In this case, the agreement is a much stronger evidence than usually. Actually, the plot in Fig. 2 verifies that the formation of an identical ion-induced
Please cite this article in press as: G. Szenes, Uniform behavior of insulators irradiated by swift heavy ions, Nucl. Instr. Meth. B (2015), http://dx.doi.org/ 10.1016/j.nimb.2015.01.046
G. Szenes / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
temperature distribution given by H(r) is a reality, it is valid in a broad range of hse i, the value of hset i is predicted correctly, and w = 4.5 nm is a universal constant which does not depend on MPs, and on the experimental parameters: the kind of the projectile, hse i and E. When DT(r,t) is formed T > Tm and T < Tm are equivalent. The plot in Fig. 1 does not indicate that the heat of fusion might have a significant effect. Therefore, it is reasonable to assume that if H(r) = DT(r,0) for T > Tm, this is valid for a range of temperature when T < Tm as well [1]. The above considerations show that the universal relation H(r) is not necessarily related to the track formation as tracks are not formed at T < Tm. Consequently, the spike temperature can be described correctly by H(r) in some of those insulators as well where amorphous tracks are not formed though Tm is exceeded in the spike. Thus the universal relation H(r) may have an even more general validity. However, presently, the experimental verification of this assumption is problematic, as there are no suitable methods for the measurement of the temperature in the spike when tracks are not formed. hse i = Se/N is a key parameter of the effect. However it would be incautious to identify this parameter as the mean deposited inelastic energy per atom according to the units. In reality, N = 1022– 1023 atoms/cm3 while only n = vN atoms are within the volume v where the deposited inelastic energy is transferred to thermal energy. According to Eq. (1) the spike is confined approximately into a cylinder with a diameter 20 nm while the temperature at its surface is about 0.01Tp (Tp–peak temperature). Thus n is a relatively small number and the physical meaning of the condition hse i = constant is not obvious. A simple solution to this problem is given by Eq. (1) where the peak temperature Tpahse i while f, w and k are constants. Thus the condition hse i = constant is required in Fig. 1 because this ensures that the peak temperatures of the H(r) distribution agree in various insulators. If this is not fulfilled the relation between track radii shown in Fig. 1 is not possible. In thermal spike models the melting point Tm is a key parameter. In all models, the equilibrium values of Tm are used which is a rather rough approximation. Klaumünzer criticized this approach from a thermodynamical point of view and claimed that within the conditions of the irradiation by energetic ions superheating must occur that cannot be ignored. In the present and previous studies track radii Re and the equilibrium values of Tm are compared in various solids. The results in Figs. 1 and 2 unambiguously show a clear functional relationship between these two quantities and this justifies that Tm is a suitable parameter for describing the effect. Although this seems to be in contradiction with the effect of superheating, nevertheless, if this were not true the smooth curve connecting the raw data could not be possible. The results can be explained if the real temperature of phase transition at high heating rates scales approximately with Tm. This is in agreement with theoretical considerations [15] and the estimates of Klaumünzer as well [16]. Thus the plots in Figs. 1 and 2 and other similar plots confirm that the equilibrium values of Tm can be applied rightly in the analysis of the thermal spikes induced by swift heavy ions. However, this may lead to some systematic error. When Eq. (1) is applied for the analysis only the ratio f/(Tm Tir) can be estimated from the experimental data. The efficiency values f = 0.4/0.17 were derived using the equilibrium values of Tm, thus these are underestimated values. When reliable estimates will be known for the superheating temperatures the efficiency values can be simply corrected. This would not modify any of our conclusions. We showed that track radii can be evaluated by using the universal relation H(r) without any information on the MPs. This is the direct consequence of the formation of identical temperature distributions in insulators when hse i = constant. This is a significant contradiction with the usual analyses where the propagation of the
3
deposited energy is considered as some kind of heat conduction problem. The existence of the universal H(r) relation is in disagreement with the Fourier equation even in the case of a most simple model as the heat flow equation cannot lead to an identical solution H(r) for various insulators for mathematical reasons. The origin of the universal constant w = 4.5 nm cannot be explained by this way either. Thus we found that the Fourier equation is not valid under spike condition, because the relation between track radii shown in Fig. 1 and the scaling feature demonstrated in Fig. 2 cannot be explained by assuming the controlling role of the heat equation. An uncertainty arises from the fact that good agreement with experiments can be achieved by one-two fitting parameters even in this case [17]. The reader may have an impression that the application of the Fourier equation may lead to a correct description. However, such method may be successful only for the analysis of experiments in a single solid when the relation between Re values and the scaling property shown in Figs. 1 and 2 are hidden. The ion-induced temperature distributions are specific for each solid when calculated by using the Fourier equation, therefore, they are incorrect in this case. These are not small errors but large deviations which are increased additionally by the fact that f = 1 is applied for the efficiency [17] while it is only f = 0.17/0.4 for HI and LO irradiations in Eq. (1). This difference in the efficiency parameter necessarily leads to a large distortion of the calculated temperature distributions as the thermal energy of the spike e = fSe is about six times exceeds the value derived from Eq. (1) for HI irradiations. Various ion-induced effects: sputtering, ion-beam mixing, swelling, track formation, hillock formation are controlled by thermal processes. Therefore, the understanding of these effects requires the reliable knowledge of the local temperature. However, this is not warranted when DT(r,t) is calculated with a method based on the application of the Fourier equation. The universal relation H(r) characterizes the localized energy deposition only in the central part with a Gaussian width w = 4.5 nm. Thus the thermal effects with the identical behavior of insulators is restricted to this area. On the other hand, the related energy is less than 20% of Se in the case of HI irradiations. Thus there may proceed simultaneously numerous effects whose uniform features do not follow from this study. 4. Conclusions Track formation is analyzed in ZrSiO4, KTiOPO4, Y2O3, Al2O3 and MgAl2O4 in which only a single value of Re is known (for the last two solids only Set is reliable with Re = 0). An R2e f hse i=ðT m T ir Þ plot without adjustable parameters is applied which coincides for all solids with the common slope w2. The good agreement with the theoretical curve verifies unambiguously the validity of H(r) in a broad range of hse i. The analysis of all track data reported up until now in insulators led to a similar conclusion. It is shown that identical ion-induced temperature distributions DT(r,0) = H(r) are formed in insulators when hse i = constant. The validity of the Fourier equation under spike conditions is doubted as it has no such uniform solution for the ion-induced temperatures. The meaning of the condition hse i = constant is explained. The universal relationship H(r) may also be valid in solids where amorphous tracks are not formed. The simple and unambiguous functional relation between track radii Re and the equilibrium value of Tm justifies that Tm is rightly used in spike conditions induced by swift heavy ions. References [1] G. Szenes, Nucl. Instrum. Meth. Phys. Res. B 280 (2012) 88. [2] G. Szenes, Nucl. Instrum. Meth. Phys. Res. B 312 (2013) 118.
Please cite this article in press as: G. Szenes, Uniform behavior of insulators irradiated by swift heavy ions, Nucl. Instr. Meth. B (2015), http://dx.doi.org/ 10.1016/j.nimb.2015.01.046
4
G. Szenes / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
[3] A. Meftah, F. Brisard, J.M. Costantini, M. Hage-Ali, J.P. Stoquert, F. Studer, M. Toulemonde, Phys. Rev. B 48 (1993) 920. [4] G. Szenes, F. Pászti, Á. Péter, A.I. Popov, Nucl. Instr. Meth. B 166–167 (2000) 949. [5] J. Jensen, A. Dunlop, S. Della-Negra, M. Toulemonde, Nucl. Instr. Meth. B 146 (1998) 412. [6] M. Toulemonde, F. Studer, Solid State Phenom. 30–31 (1993) 477. [7] L.A. Bursill, G. Braunshausen, Philos. Mag. A 62 (1990) 395. [8] G. Szenes, J. Nucl. Mat. 336 (2005) 81. [9] Th. Opfermann, Th. Höche, S. Klaumünzer, W. Wesch, Nucl. Instrum. Meth. B 166–167 (2000) 954. [10] S. Hemon et al., Eur. Phys. J. B 19 (2001) 517. [11] T. Aruga, Y. Katano, T. Ohmichi, S. Okayasu, Y. Kazumata, S. Jitsukawa, Nucl. Instrum. Meth. B 197 (2002) 94.
[12] N. Khalfaoui, J.P. Stoquert, F. Haas, C. Trautmann, A. Meftah, M. Toulemonde, Nucl. Instrum. Meth. B 286 (2012) 247. [13] V.A. Skuratov, J. O’Connell, N.S. Kirilkin, J. Neethling, Nucl. Instrum. Meth. B 326 (2014) 223. [14] G. Szenes, Rad. Eff. Def. Solids, in press. [15] S.N. Luo, T.J. Ahrens, T. Cagin A., W.A. Goddard III, D.C. Swift, Phys. Rev. B 68 (2003) 134206. [16] S. Klaumünzer, Mat. Fys. Med. 5 (2006) 293. [17] M. Toulemonde, W. Assmann, C. Dufour, A. Meftah, F. Studer, C. Trautmann, Mat. Fys. Med. 5 (2006) 263.
Please cite this article in press as: G. Szenes, Uniform behavior of insulators irradiated by swift heavy ions, Nucl. Instr. Meth. B (2015), http://dx.doi.org/ 10.1016/j.nimb.2015.01.046
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Uniform behavior of insulators irradiated by swift heavy ions G. Szenes ⇑ Department of Materials Physics, Eötvös University, P.O. Box 32, H-1518 Budapest, Hungary
a r t i c l e
i n f o
Article history: Received 31 July 2014 Received in revised form 21 January 2015 Accepted 21 January 2015 Available online xxxx Keywords: Swift heavy ion Thermal spike Spike temperature Tracks
a b s t r a c t Ion induced Re track radii are derived from a universal relation H(r) without involving any materials parameter apart from the melting point Tm. The effect is related to the formation of identical ion-induced temperature distributions in track forming insulators for hse i ¼ Se =N = constant, where Se, and N are the electronic stopping power and the atomic density. Based on H(r), an R2e hse i=ðT m T ir Þ plot is applied where the experimental curves coincide for various insulators without adjustable parameters (Tir – temperature of irradiation). The analysis extends to all track-forming insulators studied up until now. The application of the equilibrium value of Tm is justified in thermal spike calculations. The physical meaning of the condition hse i ¼ Se =N = constant is discussed. H(r) may be valid in those insulators as well in which tracks are not induced. The Fourier equation is not valid under spike conditions. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Recently, old ion-induced track data were analyzed again with quite unexpected result [1,2]. It was found that there is a simple relationship between track radii Re measured in different solids when a simple condition is fulfilled. The explanation of this result led to such conclusions which are irreconcilable with the present ideas about the governing processes. In the present paper we summarize the most important previous results, extend the analysis of experimental results and discuss some points which have not received sufficient attention previously.
2. Previous results When a swift heavy ion hits a solid it deposits a large amount of energy in a localized manner, which may induce irreversible changes. One of these possible changes is the formation of amorphous tracks along the trajectory of the projectile. The effect depends on the inelastic energy deposition which is characterized by the electronic stopping power Se. We found a simple relation between Re induced in different solids by different ions of different electronic stopping power Se and different specific ion energy E [1,2]. The only condition is hse i ¼ Se =N = constant where N is the number density of atoms. The quantitative relation is valid only for the initial values of Re which may be changed with time in some ⇑ Tel.: +36 1 372 2821; fax: +36 1 372 2811. E-mail address: [email protected]
solids. A typical result is shown in Fig. 1 [2]. The enveloping curve is given by
HðrÞ ¼
f hse i 3pkw
2
exp r 2 =w2 ;
ð1Þ
where k is the Boltzmann constant, r is the radial distance from the trajectory, w = 4.5 nm and the efficiency f = 0.4/0.17 at low (E < 2 MeV/nucleon, LO) or high (E > 8 MeV/nucleon, HI) ion velocities which is the consequence of the velocity effect [3]. Track radii can be obtained from the equation H(Re) = Tm Tir (Tm and Tir are the melting and irradiation temperatures). H(r) does not contain materials parameters (MPs), does not depend on the experimental parameters, and f has the same values for track forming insulators in the LO and HI ranges leading to different thermal energies e = fSe. Thus track radii are given by a universal function and MPs apart Tm have no effect on Re [1,2]. This result is irreconcilable with the traditional application of the heat flow equation under spike conditions. It was found that the observed relationship between track radii is the consequence of the formation of an identical temperature distribution in irradiated track forming insulators [1]. 3. Discussion Fig. 1 convincingly demonstrates the relationship between track radii showing that Tm alone determines Re, other MPs are indifferent. This conclusion is valid only for a single value of hse i which is given in the plot. However, here we show another simple and direct method which is free of this disadvantage. When the universal
http://dx.doi.org/10.1016/j.nimb.2015.01.046 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: G. Szenes, Uniform behavior of insulators irradiated by swift heavy ions, Nucl. Instr. Meth. B (2015), http://dx.doi.org/ 10.1016/j.nimb.2015.01.046
2
G. Szenes / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
relationship H(r) is valid for an insulator the plot R2e lnðf hse i= ðT m T ir ÞÞ is a straight line (Tm-parameter, f hse i-variable, whose slope is w2. Since H(r) does not contain MPs the lines for various insulators coincide. The plot in Fig. 1 is a special case of this procedure when Tm is a variable and f hse i = constant. In this paper, we apply this plot for tracks induced by low velocity ions in TeO2 [4], by cluster ions in Y3Fe5O12 [5], and by high velocity projectiles in NiFe2O4 [6]. There is a good agreement between the theoretical line and the experimental data in Fig. 2. The track data of all track-forming insulators analyzed in Ref. [1,2] also show similar good agreement. The plot confirms the validity of H(r) in a broad range of hse i. The track data which were used in Fig. 1 were obtained in independent experiments. In this case, one cannot expect that track radii be measured at the same value of hse i. Therefore, when drawing the plot in Fig. 1, the data were evaluated by linear interpolation between neighboring experimental values. This might increase the error of the estimated track radii. In Fig. 2 a new plot is used for the first time which allows the use of the original track data making any interpolation unnecessary. Besides the broad range of the hse i values this is also a great advantage. This method can be applied even when systematic studies have not been performed and only a single value of Re is known. In Fig. 2 we show such data for ZrSiO4 [7], MgAl2O4 [8], Al2O3 [8], KTiOPO4 [9], and Y2O3 [10] which were obtained in experiments with LO or HI projectiles. In the case of Al2O3 and MgAl2O4, Re values have been also measured in a range of Se but track re-crystallization reduced the initial track sizes. Therefore, we do not use those data. Recently, we analyzed the study of Aruga [11] and derived reliable estimate of Set which is the highest value of Se without inducing a track [8]. The method does not require the measurement of track sizes only the position of the amorphous–crystalline boundary must be known. Therefore the track re-crystallization has only a minor effect. New measurements by Khalfaoui et al. [12] and Skuratov et al. [13] confirm our estimate for Al2O3. In Fig. 2, the single experimental track data nicely fit to the line derived from the relation H(r) verifying its validity for these solids as well. We emphasize that the good agreement with the relation H(r) was achieved without individual adjustable parameters in a broad range of Se for a large number of different projectiles having energies in the range 0.02–20 MeV/nucleon and track radii were measured by different methods in different laboratories. Thus the reliability of the conclusions is without doubt.
Fig. 1. Variation of the track radii Re with the melting temperature Tm in various insulators; Tir is the irradiation temperature, k is the Boltzmann constant. The tracks were induced by ion beams with E<2 MeV/nucleon specific energy at hse i=3k ¼ 3:42 105 nm2K. The curve H(r) = TpL exp{r2/w2} is a fit with w = 4.45 ± 0.18 nm and TpL = 2024 ± 90 K [2].
Fig. 2. Variation of the track size with the atomic stopping power hse i; Re, f, and k are the track radius, the efficiency and the Boltzmann constant, w = 4.5 nm and Tm and Tir are the melting and the irradiation temperatures. Y3Fe5O12LO and Y3Fe5O12cl denote irradiation by monoatomic and C cluster ions. The line is drawn according to Eq. (1) with the slope w2.
Recently we finished reviewing all track data reported up until now. The validity of the H(r) relation for track formation has been verified for 18 insulators including the present results as well [1,2]. In another publication we complete this list with two other solids [14] and there is a number of studies which had to be ignored because they were not suitable for our analysis [14]. After analyzing all experimental track data we conclude that track-forming insulators respond identically to the irradiation by swift heavy ions. Track radii can be derived applying a universal relation H(r) which is free of MPs. The simple relationship between track radii shown in Figs. 1 and 2 is the direct consequence of this feature. Based on this result we assume that such behavior can be found in a large number of irradiated insulators. The presence of the melting point in the solution of H(r) for Re demonstrates that thermal processes have an important effect on Re. We arrived to the same conclusion when analyzed Set in various insulators and a linear dependence on Tm was observed [1]. Actually, Set is proportional to the energy necessary to increase the local temperature from Tir to Tm. Thus it is reasonable to assume that a melt is formed when the local temperature T > Tm and the amorphous track is formed in the fast cooling process of the melt. The ion-induced temperature increase is given by the DT(r,t) function and the time is set to t = 0 when it attains its highest value. It is reasonable to assume that Re is equal to the maximum radius of the melt or the maximum width of the radial temperature distribution DT(Re,0) at t = 0 [1]. Let us denote various MPs as X and assume that DT(r,t,X) is valid. It is a reasonable assumption that when DT = DT(r,t,X), the track radius Re also depends on X. When DT(r,t) does not depend on MPs it is evident that Re depends neither. This must be valid in the opposite direction as well: if MPs have no effect on Re, then DT(r,0) should not depend on MPs either. When this simple consideration is applied to our case, the result is rather unexpected. When track forming insulators are irradiated by swift heavy ions the induced temperature distributions DT(r,0) are identical when hse i = constant as they do not contain MPs. The H(Re) = DT(Re,0) = Tm Tir relation is valid for each track-forming insulator. Therefore, H(r) = DT(r,0) holds for many (melting) temperatures. Thus we claim that the individual DT(r,0) distributions are equal to the universal relation H(r). Here we refer again to Fig. 2. It is a highly extraordinary situation when the experimental data are compared with a universal function without fitting parameters. In this case, the agreement is a much stronger evidence than usually. Actually, the plot in Fig. 2 verifies that the formation of an identical ion-induced
Please cite this article in press as: G. Szenes, Uniform behavior of insulators irradiated by swift heavy ions, Nucl. Instr. Meth. B (2015), http://dx.doi.org/ 10.1016/j.nimb.2015.01.046
G. Szenes / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
temperature distribution given by H(r) is a reality, it is valid in a broad range of hse i, the value of hset i is predicted correctly, and w = 4.5 nm is a universal constant which does not depend on MPs, and on the experimental parameters: the kind of the projectile, hse i and E. When DT(r,t) is formed T > Tm and T < Tm are equivalent. The plot in Fig. 1 does not indicate that the heat of fusion might have a significant effect. Therefore, it is reasonable to assume that if H(r) = DT(r,0) for T > Tm, this is valid for a range of temperature when T < Tm as well [1]. The above considerations show that the universal relation H(r) is not necessarily related to the track formation as tracks are not formed at T < Tm. Consequently, the spike temperature can be described correctly by H(r) in some of those insulators as well where amorphous tracks are not formed though Tm is exceeded in the spike. Thus the universal relation H(r) may have an even more general validity. However, presently, the experimental verification of this assumption is problematic, as there are no suitable methods for the measurement of the temperature in the spike when tracks are not formed. hse i = Se/N is a key parameter of the effect. However it would be incautious to identify this parameter as the mean deposited inelastic energy per atom according to the units. In reality, N = 1022– 1023 atoms/cm3 while only n = vN atoms are within the volume v where the deposited inelastic energy is transferred to thermal energy. According to Eq. (1) the spike is confined approximately into a cylinder with a diameter 20 nm while the temperature at its surface is about 0.01Tp (Tp–peak temperature). Thus n is a relatively small number and the physical meaning of the condition hse i = constant is not obvious. A simple solution to this problem is given by Eq. (1) where the peak temperature Tpahse i while f, w and k are constants. Thus the condition hse i = constant is required in Fig. 1 because this ensures that the peak temperatures of the H(r) distribution agree in various insulators. If this is not fulfilled the relation between track radii shown in Fig. 1 is not possible. In thermal spike models the melting point Tm is a key parameter. In all models, the equilibrium values of Tm are used which is a rather rough approximation. Klaumünzer criticized this approach from a thermodynamical point of view and claimed that within the conditions of the irradiation by energetic ions superheating must occur that cannot be ignored. In the present and previous studies track radii Re and the equilibrium values of Tm are compared in various solids. The results in Figs. 1 and 2 unambiguously show a clear functional relationship between these two quantities and this justifies that Tm is a suitable parameter for describing the effect. Although this seems to be in contradiction with the effect of superheating, nevertheless, if this were not true the smooth curve connecting the raw data could not be possible. The results can be explained if the real temperature of phase transition at high heating rates scales approximately with Tm. This is in agreement with theoretical considerations [15] and the estimates of Klaumünzer as well [16]. Thus the plots in Figs. 1 and 2 and other similar plots confirm that the equilibrium values of Tm can be applied rightly in the analysis of the thermal spikes induced by swift heavy ions. However, this may lead to some systematic error. When Eq. (1) is applied for the analysis only the ratio f/(Tm Tir) can be estimated from the experimental data. The efficiency values f = 0.4/0.17 were derived using the equilibrium values of Tm, thus these are underestimated values. When reliable estimates will be known for the superheating temperatures the efficiency values can be simply corrected. This would not modify any of our conclusions. We showed that track radii can be evaluated by using the universal relation H(r) without any information on the MPs. This is the direct consequence of the formation of identical temperature distributions in insulators when hse i = constant. This is a significant contradiction with the usual analyses where the propagation of the
3
deposited energy is considered as some kind of heat conduction problem. The existence of the universal H(r) relation is in disagreement with the Fourier equation even in the case of a most simple model as the heat flow equation cannot lead to an identical solution H(r) for various insulators for mathematical reasons. The origin of the universal constant w = 4.5 nm cannot be explained by this way either. Thus we found that the Fourier equation is not valid under spike condition, because the relation between track radii shown in Fig. 1 and the scaling feature demonstrated in Fig. 2 cannot be explained by assuming the controlling role of the heat equation. An uncertainty arises from the fact that good agreement with experiments can be achieved by one-two fitting parameters even in this case [17]. The reader may have an impression that the application of the Fourier equation may lead to a correct description. However, such method may be successful only for the analysis of experiments in a single solid when the relation between Re values and the scaling property shown in Figs. 1 and 2 are hidden. The ion-induced temperature distributions are specific for each solid when calculated by using the Fourier equation, therefore, they are incorrect in this case. These are not small errors but large deviations which are increased additionally by the fact that f = 1 is applied for the efficiency [17] while it is only f = 0.17/0.4 for HI and LO irradiations in Eq. (1). This difference in the efficiency parameter necessarily leads to a large distortion of the calculated temperature distributions as the thermal energy of the spike e = fSe is about six times exceeds the value derived from Eq. (1) for HI irradiations. Various ion-induced effects: sputtering, ion-beam mixing, swelling, track formation, hillock formation are controlled by thermal processes. Therefore, the understanding of these effects requires the reliable knowledge of the local temperature. However, this is not warranted when DT(r,t) is calculated with a method based on the application of the Fourier equation. The universal relation H(r) characterizes the localized energy deposition only in the central part with a Gaussian width w = 4.5 nm. Thus the thermal effects with the identical behavior of insulators is restricted to this area. On the other hand, the related energy is less than 20% of Se in the case of HI irradiations. Thus there may proceed simultaneously numerous effects whose uniform features do not follow from this study. 4. Conclusions Track formation is analyzed in ZrSiO4, KTiOPO4, Y2O3, Al2O3 and MgAl2O4 in which only a single value of Re is known (for the last two solids only Set is reliable with Re = 0). An R2e f hse i=ðT m T ir Þ plot without adjustable parameters is applied which coincides for all solids with the common slope w2. The good agreement with the theoretical curve verifies unambiguously the validity of H(r) in a broad range of hse i. The analysis of all track data reported up until now in insulators led to a similar conclusion. It is shown that identical ion-induced temperature distributions DT(r,0) = H(r) are formed in insulators when hse i = constant. The validity of the Fourier equation under spike conditions is doubted as it has no such uniform solution for the ion-induced temperatures. The meaning of the condition hse i = constant is explained. The universal relationship H(r) may also be valid in solids where amorphous tracks are not formed. The simple and unambiguous functional relation between track radii Re and the equilibrium value of Tm justifies that Tm is rightly used in spike conditions induced by swift heavy ions. References [1] G. Szenes, Nucl. Instrum. Meth. Phys. Res. B 280 (2012) 88. [2] G. Szenes, Nucl. Instrum. Meth. Phys. Res. B 312 (2013) 118.
Please cite this article in press as: G. Szenes, Uniform behavior of insulators irradiated by swift heavy ions, Nucl. Instr. Meth. B (2015), http://dx.doi.org/ 10.1016/j.nimb.2015.01.046
4
G. Szenes / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
[3] A. Meftah, F. Brisard, J.M. Costantini, M. Hage-Ali, J.P. Stoquert, F. Studer, M. Toulemonde, Phys. Rev. B 48 (1993) 920. [4] G. Szenes, F. Pászti, Á. Péter, A.I. Popov, Nucl. Instr. Meth. B 166–167 (2000) 949. [5] J. Jensen, A. Dunlop, S. Della-Negra, M. Toulemonde, Nucl. Instr. Meth. B 146 (1998) 412. [6] M. Toulemonde, F. Studer, Solid State Phenom. 30–31 (1993) 477. [7] L.A. Bursill, G. Braunshausen, Philos. Mag. A 62 (1990) 395. [8] G. Szenes, J. Nucl. Mat. 336 (2005) 81. [9] Th. Opfermann, Th. Höche, S. Klaumünzer, W. Wesch, Nucl. Instrum. Meth. B 166–167 (2000) 954. [10] S. Hemon et al., Eur. Phys. J. B 19 (2001) 517. [11] T. Aruga, Y. Katano, T. Ohmichi, S. Okayasu, Y. Kazumata, S. Jitsukawa, Nucl. Instrum. Meth. B 197 (2002) 94.
[12] N. Khalfaoui, J.P. Stoquert, F. Haas, C. Trautmann, A. Meftah, M. Toulemonde, Nucl. Instrum. Meth. B 286 (2012) 247. [13] V.A. Skuratov, J. O’Connell, N.S. Kirilkin, J. Neethling, Nucl. Instrum. Meth. B 326 (2014) 223. [14] G. Szenes, Rad. Eff. Def. Solids, in press. [15] S.N. Luo, T.J. Ahrens, T. Cagin A., W.A. Goddard III, D.C. Swift, Phys. Rev. B 68 (2003) 134206. [16] S. Klaumünzer, Mat. Fys. Med. 5 (2006) 293. [17] M. Toulemonde, W. Assmann, C. Dufour, A. Meftah, F. Studer, C. Trautmann, Mat. Fys. Med. 5 (2006) 263.
Please cite this article in press as: G. Szenes, Uniform behavior of insulators irradiated by swift heavy ions, Nucl. Instr. Meth. B (2015), http://dx.doi.org/ 10.1016/j.nimb.2015.01.046