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Cooling and heating of channelled ion beams
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 245 (2006) 19–21 www.elsevier.com/locate/nimb
Cooling and heating of channelled ion beams Christian Toepffer
*
Schonland Research Institute for Nuclear Sciences, University of the Witwatersrand, Johannesburg, South Africa Institut fu¨r Theoretische Physik II, Universita¨t Erlangen, Staudtstr. 7, D-91058 Erlangen, Germany Available online 27 December 2005
Abstract Experiments showing a transverse cooling or heating of channelled ions have been interpreted in terms of electron capture and loss processes between the projectile ions and the target. Reversibility is violated as the projectile captures electrons from occupied bound states and looses them to unoccupied weakly bound or continuum states. If, on the average, capture by the ion occurs at smaller distances from the target atoms than loss, then the transversal kinetic energy of the ion is reduced and vice versa. The impact parameter dependence of transition probabilities for capture and loss is determined by scale factors which depend in turn on the relative velocity and the binding energies of the transferred electrons in the projectile and in the crystal, respectively. The resulting rules for the appearance of cooling and heating in terms of the crystal atom and projectile ion species and the projectile velocity agree with experimental observations. Ó 2005 Elsevier B.V. All rights reserved. PACS: 34.70.+e; 34.20.Cf; 61.85.+p Keywords: Channelling; Charge transfer; Transverse cooling and heating
Measurements of the angular distribution of initially isotropic heavy ions after passage through Si crystals show a strong redistribution of flux. In some cases, an enhancement along the channelling directions has been observed (cooling), in other cases a reduction (heating) [1]. This seems to contradict the principle of detailed balance for the scattering of channelled ions which rests on the time reversibility of trajectories [2]. For an explanation of the experiments a new mechanism was proposed: cooling or heating of the transverse motion of the channelling ions due to the capture of electrons from crystal atoms or loss of electrons, respectively. As the transverse potential between the ion and the atomic rows of the crystal is repulsive and decreases with distance, the ion looses transverse energy if capture takes place at smaller distances from the row than loss and vice versa. This * Address: Institut fu¨r Theoretische Physik II, Universita¨t Erlangen, Staudtstr. 7, D-91058 Erlangen, Germany. Tel.: +49 9131 852 8461; fax: +49 9131 852 8907. E-mail address: toepff[email protected]
0168-583X/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.11.057
mechanism is schematically shown in Fig. 3 of [1]. Typical parameters were estimated from total cross-sections [3]. In agreement with the experiments, this model yields cooling for C ions and heating for heavier ions I and Au while Cu projectiles are an intermediate case. A further experiment showed that there is no heating of channelled ions at low velocities if the ion is lighter than the crystal atom [4]. In a later series of experiments on the channelling of Ag ions in Si, Pt and Ni crystals the correlation between shifts in the charge state distribution with cooling and heating was investigated [5]. For example, the proposed cooling mechanism implies for well-channelled ions a larger mean charge than for randomly moving ions, because the channelled ions are confined to larger impact parameters where the probability for electron capture decreases faster than the probability for electron loss. In these experiments, cooling was observed at high ion velocities and heating at low ion velocities. Such a transition cannot be explained in terms of effective impact parameters derived from total cross-sections for capture and loss. Instead the experiments have been
C. Toepffer / Nucl. Instr. and Meth. in Phys. Res. B 245 (2006) 19–21
simulated by classical trajectory Monte Carlo calculations of collisions of the projectile with two strings of N atoms each carrying n electrons (nNCTMC) [6]. The mean charge states for ions approaching a string are different from those which leave a string. Reversibility is violated as the projectile captures electrons from occupied bound states which have a larger binding energy than the unoccupied states in the target, to which the electrons are lost. In this paper, this will be illustrated in a transparent manner: as the channelling projectiles move on nearly straight trajectories between the crystal strings and are confined by the transverse potential to regions outside the atomic cores the transition amplitude can be calculated in the impact parameter Born approximation [7]. The projectile charge is Z2 and Z1 is the charge number of the crystal atomic nucleus. Atomic units (a.u.) will be employed throughout, i.e. impact parameters r are measured in units of the Bohr radius r0 = h2/(e 0 2m), the 2 projectile velocity v in units of v0 = e 0 / h c/137 and energies in units of e 0 4m/ h2, where e 0 2 = e2/(4pe0), with the elementary charge e, the electron mass m and the permittivity of the vacuum e0. Assuming hydrogenic wave functions the amplitude for a transition 1s M n, s is ð0Þ
An0 ðrÞ ¼ An0 ðrÞ þ corrections ¼
2 ðZ 1 Z 2 Þ v n3=2
5=2
r2 K 2 ðrxÞ þ corrections. x2
ð1Þ
Here Kp is a modified Bessel function and x is a scale factor which depends on the relative velocity v and the binding energies E1 and E2 in the crystal and the projectile, respectively v2 1 þ E2 þ E1 þ 2 ðE2 E1 Þ2 . ð2Þ v 4 The corrections, which are not explicitly displayed in Eq. (1) are sums of terms of the form (rx)pKp(rx) with p P n and similar expressions occur in the transition amplitudes Anlm(r) to states n, l > 0. All these corrections yield contributions which cancel each other in averaged quantities like the cross-section for capture from the nth major shell Z Z 1 2p X 1 2p ð0Þ 2 2 rn ¼ 2 drrjAnlm ðrÞj ¼ 2 drrAn0 ðrÞ n l;m 0 n 0 x2 ¼
28 pðZ 1 Z 2 Þ5 ¼ . 5n5 v2 x10
a quantitative example we consider the channeling of C5+ with v = 7.9 in Si [1]. The ratio of the transition probabilities p = jAj2 for loss from the K-shell in C4+ (E2 = 14.5) to the M-shell in Si (E1 = 0.471) and for capture from the K-shell in Si (E1 = 70.2) is shown in Fig. 1. The solid curve was calculated with Eq. (1), the dashed curve includes the corrections. The close agreement illustrates that the shape of the transition probabilities is already adequately described by A(0). Indeed all modified Bessel functions vanish exponentially like K2 for large arguments and for small arguments the dominant terms (rx)pKp(rx) are all inverted parabolas. The shape of the transition probabilities as a function of the impact parameter can thus be adequately described by Eq. (1) which is adopted as an effective model. This exhibits the dominant role of the scale factors (2) which have min1 1 imal values xmin ¼ ð2 maxðE1 ; E2 ÞÞ2 at vmin ¼ ð2jE2 E1 jÞ2 . The binding energies E1 of occupied target states from which electrons are captured are larger than those of unoccupied target states, to which electrons are lost. In fact as E1 ! 0 for the latter, these hardly matter in Eq. (2). The cross-over from heating to cooling with increasing velocity [5] observed for heavy, highly charged projectiles with large binding energies E2 can be understood in a transparent manner. The value of vmin for capture will be smaller than that for loss. At low velocities the scale factor for capture tends then to be smaller than the scale factor for loss. The probability for capture decreases slower with impact parameter than the probability for loss. This leads to heating, vice versa there is cooling at high velocity. Such a behaviour is found, for example, in experiments on the channelling of highly charged Ag ions in Si crystals [5]. The scale factors for capture by Ag17+ (E2 = 16.1) from the Si L-shell (E1 = 4.28) (dashed curve) and subsequent loss to the Si M-shell (E1 = 0.471) (solid curve) are shown in Fig. 2. At low energies one obtains xloss > xcapture, at high energies the situation is reversed, xcapture > xloss. The 3-d plot of ploss/pcapture as a function of the relative velocity v and the impact parameter r in Fig. 3 shows indeed a
1000 100
ð3Þ
Rather than by the details of the interior node structure the transitions are dominated by the exterior tails of the wave functions, which depend only on the binding energy. It is precisely these tails which the complicated atoms and ions share with the hydrogenic ones. This encourages to generalize equations (1) and (2) for the impact parameter dependent transition amplitude by employing binding energies taken from many-body calculations [8] for arbitrary atoms and ions. Here, for simplicity, the Rydberg formula for binding energies is modified by introducing shielding Z ! Z s and effective quantum numbers n ! n* [9]. As
10 ploss pcapture
20
1 0.1 0.01 0
0.2
0.4
0.6
0.8
1
r Fig. 1. Ratio of transition probabilities as function of the impact parameter for the channelling of C ions in Si at v = 7.9 (E = 1.5 MeV/u [1]). A C5+ captures an electron from the Si K-shell and the C4+ looses it to Si M-shell. The solid curve was calculated with Eq. (2), the dashed curve includes the corrections indicated in Eq. (1).
C. Toepffer / Nucl. Instr. and Meth. in Phys. Res. B 245 (2006) 19–21
11 10
x
9 8 7 6 0
10 v
5
20
15
Fig. 2. Scale factors x (2) for the capture (dashed curve) and the loss (solid curve) of electrons in the channelling of Ag17+ in Si as a function of velocity.
1
0.8
r 0.6
0.4
0.2
– 2.2 10 4
The present model supports the original suggestion which relates the observed cooling and heating of channelled ions to the transfer of electrons between the projectiles and the crystal atoms [1]. It validates this mechanism on the level of transition probabilities which are calculated in a manner which is explicitly dependent on the impact parameter. It turns out that the tails of the electronic wave functions are most important for the transfer processes. An effective model can therefore be formulated for the transition amplitudes (1) which involves scale factors (2) depending on the velocity and the binding energies of the states involved. This allows to extract the shapes of the probabilities for loss and capture and yields a transparent explanation for the transition from heating to cooling observed experimentally [5]. Acknowledgements
– 1.8 ploss lg – 2 [ pcapture ]
6
21
12
8 v
Fig. 3. Ratio of transition probabilities for loss and capture of electrons by Ag17+ ions channelling in Si as function of the impact parameter and the relative velocity.
warped surface so that at low velocities the ratio of the probabilities for loss and capture decreases with impact parameter corresponding to heating and vice versa at larger velocities. On the other hand, if the projectile is lighter than the crystal atom, the binding energies E2 tend to lie between the binding energies of the occupied and the unoccupied states in the target. At low velocities the scale factor for loss is smaller than that for capture and the probability for loss decreases slower than the probability for capture which explains the observed absence of heating for light projectiles at low velocities [1,4].
This work was supported by the BMBF (06 ER 128) and by an Ablett Fellowship at the University of the Witwatersrand in Johannesburg, South Africa. The author thanks S. Connell for his hospitality and for suggesting these investigations. He also acknowledges valuable discussions with R.H. Lemmer and B. Fricke. References [1] W. Assmann, H. Huber, S.A. Karamanian, F. Gru¨ner, H.D. Mieskes, J.U. Andersen, M. Posselt, B. Schmidt, Phys. Rev. Lett. 83 (1999) 1759. [2] J. Lindhard, Dan. Mat. Fys. Medd. 34 (1965) 14. [3] N. Bohr, J. Lindhard, Dan. Mat. Fys. Medd. 28 (1954) 7. [4] F. Gru¨ner, M. Schubert, W. Assmann, F. Bell, S. Karamanian, J.U. Andersen, Nucl. Instr. and Meth. B 193 (2002) 165. [5] M. Schubert, F. Gru¨ner, W. Assmann, F. Bell, A. Bergmaier, L. Goergens, O. Schmelmer, G. Dollinger, S. Karamanian, Nucl. Instr. and Meth. B 209 (2003) 224. [6] F. Gru¨ner, F. Bell, W. Assmann, M. Schubert, Phys. Rev. Lett. 93 (2004) 213201. [7] M.R.C. McDowell, J.P. Coleman, Introduction to the Theory of Ion– Atom Collisions, North-Holland, Amsterdam, 1970 (Chapters 4 and 8). [8] B. Fricke, J.H. Blanke, D. Heinemann, D.V. Schmieden, W. Eckstein. Available from:. [9] J.C. Slater, Quantum Theory of Atomic Structure, McGraw Hill, New York, 1960 (Chapter 15).
Nuclear Instruments and Methods in Physics Research B 245 (2006) 19–21 www.elsevier.com/locate/nimb
Cooling and heating of channelled ion beams Christian Toepffer
*
Schonland Research Institute for Nuclear Sciences, University of the Witwatersrand, Johannesburg, South Africa Institut fu¨r Theoretische Physik II, Universita¨t Erlangen, Staudtstr. 7, D-91058 Erlangen, Germany Available online 27 December 2005
Abstract Experiments showing a transverse cooling or heating of channelled ions have been interpreted in terms of electron capture and loss processes between the projectile ions and the target. Reversibility is violated as the projectile captures electrons from occupied bound states and looses them to unoccupied weakly bound or continuum states. If, on the average, capture by the ion occurs at smaller distances from the target atoms than loss, then the transversal kinetic energy of the ion is reduced and vice versa. The impact parameter dependence of transition probabilities for capture and loss is determined by scale factors which depend in turn on the relative velocity and the binding energies of the transferred electrons in the projectile and in the crystal, respectively. The resulting rules for the appearance of cooling and heating in terms of the crystal atom and projectile ion species and the projectile velocity agree with experimental observations. Ó 2005 Elsevier B.V. All rights reserved. PACS: 34.70.+e; 34.20.Cf; 61.85.+p Keywords: Channelling; Charge transfer; Transverse cooling and heating
Measurements of the angular distribution of initially isotropic heavy ions after passage through Si crystals show a strong redistribution of flux. In some cases, an enhancement along the channelling directions has been observed (cooling), in other cases a reduction (heating) [1]. This seems to contradict the principle of detailed balance for the scattering of channelled ions which rests on the time reversibility of trajectories [2]. For an explanation of the experiments a new mechanism was proposed: cooling or heating of the transverse motion of the channelling ions due to the capture of electrons from crystal atoms or loss of electrons, respectively. As the transverse potential between the ion and the atomic rows of the crystal is repulsive and decreases with distance, the ion looses transverse energy if capture takes place at smaller distances from the row than loss and vice versa. This * Address: Institut fu¨r Theoretische Physik II, Universita¨t Erlangen, Staudtstr. 7, D-91058 Erlangen, Germany. Tel.: +49 9131 852 8461; fax: +49 9131 852 8907. E-mail address: toepff[email protected]
0168-583X/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.11.057
mechanism is schematically shown in Fig. 3 of [1]. Typical parameters were estimated from total cross-sections [3]. In agreement with the experiments, this model yields cooling for C ions and heating for heavier ions I and Au while Cu projectiles are an intermediate case. A further experiment showed that there is no heating of channelled ions at low velocities if the ion is lighter than the crystal atom [4]. In a later series of experiments on the channelling of Ag ions in Si, Pt and Ni crystals the correlation between shifts in the charge state distribution with cooling and heating was investigated [5]. For example, the proposed cooling mechanism implies for well-channelled ions a larger mean charge than for randomly moving ions, because the channelled ions are confined to larger impact parameters where the probability for electron capture decreases faster than the probability for electron loss. In these experiments, cooling was observed at high ion velocities and heating at low ion velocities. Such a transition cannot be explained in terms of effective impact parameters derived from total cross-sections for capture and loss. Instead the experiments have been
C. Toepffer / Nucl. Instr. and Meth. in Phys. Res. B 245 (2006) 19–21
simulated by classical trajectory Monte Carlo calculations of collisions of the projectile with two strings of N atoms each carrying n electrons (nNCTMC) [6]. The mean charge states for ions approaching a string are different from those which leave a string. Reversibility is violated as the projectile captures electrons from occupied bound states which have a larger binding energy than the unoccupied states in the target, to which the electrons are lost. In this paper, this will be illustrated in a transparent manner: as the channelling projectiles move on nearly straight trajectories between the crystal strings and are confined by the transverse potential to regions outside the atomic cores the transition amplitude can be calculated in the impact parameter Born approximation [7]. The projectile charge is Z2 and Z1 is the charge number of the crystal atomic nucleus. Atomic units (a.u.) will be employed throughout, i.e. impact parameters r are measured in units of the Bohr radius r0 = h2/(e 0 2m), the 2 projectile velocity v in units of v0 = e 0 / h c/137 and energies in units of e 0 4m/ h2, where e 0 2 = e2/(4pe0), with the elementary charge e, the electron mass m and the permittivity of the vacuum e0. Assuming hydrogenic wave functions the amplitude for a transition 1s M n, s is ð0Þ
An0 ðrÞ ¼ An0 ðrÞ þ corrections ¼
2 ðZ 1 Z 2 Þ v n3=2
5=2
r2 K 2 ðrxÞ þ corrections. x2
ð1Þ
Here Kp is a modified Bessel function and x is a scale factor which depends on the relative velocity v and the binding energies E1 and E2 in the crystal and the projectile, respectively v2 1 þ E2 þ E1 þ 2 ðE2 E1 Þ2 . ð2Þ v 4 The corrections, which are not explicitly displayed in Eq. (1) are sums of terms of the form (rx)pKp(rx) with p P n and similar expressions occur in the transition amplitudes Anlm(r) to states n, l > 0. All these corrections yield contributions which cancel each other in averaged quantities like the cross-section for capture from the nth major shell Z Z 1 2p X 1 2p ð0Þ 2 2 rn ¼ 2 drrjAnlm ðrÞj ¼ 2 drrAn0 ðrÞ n l;m 0 n 0 x2 ¼
28 pðZ 1 Z 2 Þ5 ¼ . 5n5 v2 x10
a quantitative example we consider the channeling of C5+ with v = 7.9 in Si [1]. The ratio of the transition probabilities p = jAj2 for loss from the K-shell in C4+ (E2 = 14.5) to the M-shell in Si (E1 = 0.471) and for capture from the K-shell in Si (E1 = 70.2) is shown in Fig. 1. The solid curve was calculated with Eq. (1), the dashed curve includes the corrections. The close agreement illustrates that the shape of the transition probabilities is already adequately described by A(0). Indeed all modified Bessel functions vanish exponentially like K2 for large arguments and for small arguments the dominant terms (rx)pKp(rx) are all inverted parabolas. The shape of the transition probabilities as a function of the impact parameter can thus be adequately described by Eq. (1) which is adopted as an effective model. This exhibits the dominant role of the scale factors (2) which have min1 1 imal values xmin ¼ ð2 maxðE1 ; E2 ÞÞ2 at vmin ¼ ð2jE2 E1 jÞ2 . The binding energies E1 of occupied target states from which electrons are captured are larger than those of unoccupied target states, to which electrons are lost. In fact as E1 ! 0 for the latter, these hardly matter in Eq. (2). The cross-over from heating to cooling with increasing velocity [5] observed for heavy, highly charged projectiles with large binding energies E2 can be understood in a transparent manner. The value of vmin for capture will be smaller than that for loss. At low velocities the scale factor for capture tends then to be smaller than the scale factor for loss. The probability for capture decreases slower with impact parameter than the probability for loss. This leads to heating, vice versa there is cooling at high velocity. Such a behaviour is found, for example, in experiments on the channelling of highly charged Ag ions in Si crystals [5]. The scale factors for capture by Ag17+ (E2 = 16.1) from the Si L-shell (E1 = 4.28) (dashed curve) and subsequent loss to the Si M-shell (E1 = 0.471) (solid curve) are shown in Fig. 2. At low energies one obtains xloss > xcapture, at high energies the situation is reversed, xcapture > xloss. The 3-d plot of ploss/pcapture as a function of the relative velocity v and the impact parameter r in Fig. 3 shows indeed a
1000 100
ð3Þ
Rather than by the details of the interior node structure the transitions are dominated by the exterior tails of the wave functions, which depend only on the binding energy. It is precisely these tails which the complicated atoms and ions share with the hydrogenic ones. This encourages to generalize equations (1) and (2) for the impact parameter dependent transition amplitude by employing binding energies taken from many-body calculations [8] for arbitrary atoms and ions. Here, for simplicity, the Rydberg formula for binding energies is modified by introducing shielding Z ! Z s and effective quantum numbers n ! n* [9]. As
10 ploss pcapture
20
1 0.1 0.01 0
0.2
0.4
0.6
0.8
1
r Fig. 1. Ratio of transition probabilities as function of the impact parameter for the channelling of C ions in Si at v = 7.9 (E = 1.5 MeV/u [1]). A C5+ captures an electron from the Si K-shell and the C4+ looses it to Si M-shell. The solid curve was calculated with Eq. (2), the dashed curve includes the corrections indicated in Eq. (1).
C. Toepffer / Nucl. Instr. and Meth. in Phys. Res. B 245 (2006) 19–21
11 10
x
9 8 7 6 0
10 v
5
20
15
Fig. 2. Scale factors x (2) for the capture (dashed curve) and the loss (solid curve) of electrons in the channelling of Ag17+ in Si as a function of velocity.
1
0.8
r 0.6
0.4
0.2
– 2.2 10 4
The present model supports the original suggestion which relates the observed cooling and heating of channelled ions to the transfer of electrons between the projectiles and the crystal atoms [1]. It validates this mechanism on the level of transition probabilities which are calculated in a manner which is explicitly dependent on the impact parameter. It turns out that the tails of the electronic wave functions are most important for the transfer processes. An effective model can therefore be formulated for the transition amplitudes (1) which involves scale factors (2) depending on the velocity and the binding energies of the states involved. This allows to extract the shapes of the probabilities for loss and capture and yields a transparent explanation for the transition from heating to cooling observed experimentally [5]. Acknowledgements
– 1.8 ploss lg – 2 [ pcapture ]
6
21
12
8 v
Fig. 3. Ratio of transition probabilities for loss and capture of electrons by Ag17+ ions channelling in Si as function of the impact parameter and the relative velocity.
warped surface so that at low velocities the ratio of the probabilities for loss and capture decreases with impact parameter corresponding to heating and vice versa at larger velocities. On the other hand, if the projectile is lighter than the crystal atom, the binding energies E2 tend to lie between the binding energies of the occupied and the unoccupied states in the target. At low velocities the scale factor for loss is smaller than that for capture and the probability for loss decreases slower than the probability for capture which explains the observed absence of heating for light projectiles at low velocities [1,4].
This work was supported by the BMBF (06 ER 128) and by an Ablett Fellowship at the University of the Witwatersrand in Johannesburg, South Africa. The author thanks S. Connell for his hospitality and for suggesting these investigations. He also acknowledges valuable discussions with R.H. Lemmer and B. Fricke. References [1] W. Assmann, H. Huber, S.A. Karamanian, F. Gru¨ner, H.D. Mieskes, J.U. Andersen, M. Posselt, B. Schmidt, Phys. Rev. Lett. 83 (1999) 1759. [2] J. Lindhard, Dan. Mat. Fys. Medd. 34 (1965) 14. [3] N. Bohr, J. Lindhard, Dan. Mat. Fys. Medd. 28 (1954) 7. [4] F. Gru¨ner, M. Schubert, W. Assmann, F. Bell, S. Karamanian, J.U. Andersen, Nucl. Instr. and Meth. B 193 (2002) 165. [5] M. Schubert, F. Gru¨ner, W. Assmann, F. Bell, A. Bergmaier, L. Goergens, O. Schmelmer, G. Dollinger, S. Karamanian, Nucl. Instr. and Meth. B 209 (2003) 224. [6] F. Gru¨ner, F. Bell, W. Assmann, M. Schubert, Phys. Rev. Lett. 93 (2004) 213201. [7] M.R.C. McDowell, J.P. Coleman, Introduction to the Theory of Ion– Atom Collisions, North-Holland, Amsterdam, 1970 (Chapters 4 and 8). [8] B. Fricke, J.H. Blanke, D. Heinemann, D.V. Schmieden, W. Eckstein. Available from: