- Email: [email protected]
Elastic and inelastic collision of low energy proton scattering from rare gas solids
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 256 (2007) 71–75 www.elsevier.com/locate/nimb
Elastic and inelastic collision of low energy proton scattering from rare gas solids Masahiko Kato b
a,*
, Ryutaro Souda
b
a Department of Quantum Engineering, Nagoya University, Aichi 464-8603, Japan Nanomaterials and Advanced Materials Laboratory, National Institute for Materials Science, Ibaraki 305-0044, Japan
Available online 20 February 2007
Abstract We have carried out the systematic measurements of energy spectrum for low energy (50–500 eV) proton scattering from rare gas solids of Ar, Kr and Xe. The observed energy spectra showed a multiple peak structure. The multiple peaks have been ascribed to the multiple inter-band electronic excitations, from valence to conduction band of the rare gas solid. In addition, each peak exhibited an asymmetric shape with a long tail toward the low energy side. In order to study these feature of spectrum, we develop a simple statistical model. This model involves three physics parameters: the average number of close collisions C, the electronic excitation probability p, and the asymmetry index a. From C and p, the electronic stopping power can be estimated, whereas from a, the nuclear stopping power can be estimated. Ó 2007 Elsevier B.V. All rights reserved. PACS: 34.10.+x; 34.50.Bw; 34.50.Dy Keywords: Low energy ion scattering; Energy spectrum shape; Insulators
1. Introduction The study about elastic and inelastic stopping process of energetic atoms in solids is still a current issue of discussion, from both fundamental and application point of view. Indeed, since the age of pioneers, Bethe, Bloch, Bohr, Fermi, etc, the stopping power has been extensively investigated. However, there still remain various problems to be answered. One of them is the stopping process of low energy projectiles for insulators. In the case of electronic stopping for insulators, several interesting studies about solid state effects are already available in literature [1–5]. However, further studies are obviously needed, in order to completely understand various aspects involved in the electronic excitation process in insulators. While, in the case of the nuclear stopping,
*
Corresponding author. Fax: +81 52 789 5155. E-mail address: [email protected] (M. Kato).
0168-583X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.11.092
the solid state effects must be expected for very slow projectiles, as the interatomic scattering potentials in solids are not identical with those for isolated atoms. This is particularly the case for ionic crystals. With these backgrounds in mind, we have carried out the systematic measurements of the energy spectrum, for low energy proton scattering from rare gas solids of Ar, Kr and Xe. These targets are a typical insulator with a large band gap, but not ionic crystals. Thus, we could minimize the uncertainty, arising from the unknown interatomic potentials. In the present paper, we provide a simple theory, and report briefly the application of this theory. 2. Experimental observations We have done ion scattering experiments by using low energy proton (50–500 eV) for the solids of rare gas, Ar, Kr and Xe, which were grown on a Pt substrate at a low temperature (20 K). The rare gas solids were sufficiently
72
M. Kato, R. Souda / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 71–75
H+, 400 eV
ENERGY (eV)
Ar (100 L) 300
320
340
360
H+, 400 eV
380
400
420
400
420
Kr(100 L)
EXPERIMENT
EXPERIMENT THEORY E g = 13.5 eV
INTENSITY (arb.)
INTENSITY (arb.)
THEORY α = 0.26 α =0
Paramter values E g = 15.0 eV p = 0.21 C = 2.3
p = 0.55 C = 2.1 α = 0.29
H+, 400 eV
Xe(100 L)
EXPERIMENT
320
340
360
380
400
420
ENERGY (eV) Fig. 1. The open circles show experimentally observed energy spectrum of H+ ions scattered from the solid film of Ar (100 L in exposure). Primary proton beam of 400 eV energy was incident at an angle of 20° (measured from the surface plane), and the scattering angle was 60°. The solid (dashed) line is the best fitting curve to the experimental observation, when the elastic collisions are taken into account (not taken into account).
THEORY E g = 11.2 eV
INTENSITY (arb.)
300
p = 0.84 C = 2.05 α = 0.18
300
thick (100 L in exposure). The angle of incidence was 20°, measured from the surface plane, and the laboratory scattering angle was 60°. The scattered H+ ions were detected by electrostatic analyzer, of which the energy resolution was about 2 eV. Since the target formed as an aggregation of small crystals, we did not observe any ion-channeling effects. We obtained the energy spectra of scattered proton, shown by the open circles of Figs. 1 and 2. Two typical features were generally observed, regardless of the primary energy of proton. Feature (1): the spectrum has multi peak structure, and the multi peaks appear by equidistance in the energy axis. This energy distance precisely coincides with the band gap energy, Eg, between valence and conduction band of the rare gas solid (Eg = 15, 13.5 and 11.2 eV for Ar, Kr and Xe, respectively). Therefore, this peak structure can be assigned to the inter-band electronic excitation in a close collision of proton with a target atom. The multiple peaks can be explained as the multiple inter-band excitations, in the collision sequence of proton with several target atoms. It should be noted that, the excitation of exciton is possible to occur [2], but there was no evidence for the exciton excitation in our measurements. Feature (2): each discrete peak exhibits asymmetric shape with a long tail toward the low energy side. These features of the energy spectra can be consistently explained by a simple theory. It is as follows.
320
340
360
380
ENERGY (eV) Fig. 2. The open circles show experimentally observed energy spectrum of H+ ions scattered from the solid thick film of Kr and Xe. Experimental conditions are the same as those in Fig. 1. The solid line is the best fitting curve to the experimental observation.
3. Theoretical model The probability of the inter-band electronic excitation to occur in a single close collision could be analyzed by using the modified Poisson distribution [6], F ðEÞ ¼ expðpCÞ
1 X ðpCÞj dðE E0 þ jEg Þ j! j¼0
expðCÞdðE E0 Þ;
ð1Þ
where p is the inter-band electronic excitation probability in a single close collision, and C is the average number of close collisions with target atoms. E0 is the primary energy of proton, and Eg is the band gap energy. From the best fitting of Eq. (1) to the experimental observations, we could determine the unknown parameter values of p and C simultaneously. The result obtained, by such an analysis, is shown by the dashed line of Fig. 1, in which the line spectra, i.e. d-functions of Eq. (1), have been replaced by the Gaussian functions with an appropriate variance. The dashed line
M. Kato, R. Souda / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 71–75
reproduces the multi peak structure fairly well, but the underestimate of the low energy tail is evident. We previously attributed this underestimation to the unpredictable background [6]. For insulators, the possible electronic excitation process is the inter-band electronic excitation or the exciton excitation. Because of a large excitation energy, these excitations cannot contribute to the continuous background of energy spectrum, but contribute only to the discrete spectrum. Therefore, the elastic collision must be responsible for the long tail of each discrete peak of spectrum. For the elastic collisions between ion and target atoms, we propose a theory, as follows. If the elastic collision is approximated by the binary collision between ion and target atoms, we can readily derive a transport equation, which should be satisfied by the energy distribution of scattering ion. It is written as d F ðE; xÞ ¼ dx
Z
T max
N 0
Z
drðE þ T Þ F ðE þ T ; xÞ dT dT
T max
N 0
drðEÞ F ðE; xÞ dT ; dT
ð2Þ
where F ðE; xÞ is the energy distribution of scattering ions, which have the kinetic energy E at a position x (along the ion trajectory). drðEÞ=dT is the differential cross-section for energy transfer T, in which the ion with energy E undergoes a binary collision. N is the number density of target atoms. Tmax is the maximum energy transfer, given 1M2 by T max ¼ ðM4MþM 2 E. 1 2Þ If the energy loss of ion is small, drðEÞ=dT may be approximated by its value for a representative energy E0, e.g. primary energy of ion. By this approximation, drðE0 Þ=dT does not depend on E, and moreover, the analytic solution of Eq. (2) can be written as [7] Z 1 1 i F ðEÞ ¼ ds exp Es 2ph 1 h Z T max drðE0 Þ i dT 1 exp T s ; exp N L dT h 0 ð3Þ is the average path length taken by scattering ions where L in the inside of target. With regard to drðE0 Þ=dT , several analytical expressions are available [8]. For simplicity, we assume the crosssection for the power potential, which was proposed by 0Þ Winterbon et al. [9]. It is given by drðE ¼ EmCTm1þm , where dT 0 Cm is a constant, depending on m. The value of m varies from 1/3 to 1, depending on the energy range of E0. For low energy ion scattering under consideration, Winterbon et al. recommended the use of m ¼ 1=3 and 4=3 2=3 1=3 C 1=3 ¼ 1:309 p2 aTF ð2Z 1 Z 2 e2 Þ ðM 1 =M 2 Þ . 0Þ By assuming drðE ¼ EmCTm1þm , the integration with respect dT 0 to T in the exponential function of Eq. (3) can be easily calculated, and it is found to be
drðE0 Þ i 1 exp T s dT dT h 0 iT max a ln 1 þ s ; h
N L
Z
73
T max
ð4Þ
where a, referred to as asymmetry index, has been introduced, and it is defined by a NL
Cm : ð1 mÞEm0 T mmax
ð5Þ
Substitution of Eq. (4) into Eq. (3) yields F ðEÞ ¼
1 1 1 expðE=T max Þ: CðaÞ T amax E1a
ð6Þ
This function is known as the gamma distribution. Furthermore, we have to take into account the uncertainty in energy, due to three reasons:energy resolution of detector, valence and conduction bandwidths. This uncertainty is considered by a Gaussian function with variance r2. Convoluting Eq. (6) with this Gaussian function, we finally obtain ðEE0 Þ2 1 E E0 2 F ðEÞ ¼ pffiffiffiffiffiffiffiffiffiffi e 4r Da : ð7Þ r 2pr2 where Da(x) is a parabolic cylinder function. This function reduces to a Gaussian function, when a = 0. Each discrete spectrum, d-functions of Eq. (1), should be replaced by Eq. (7) with appropriate E0 and r. Let us examine the physical implication of a. As the mean value of Gamma distribution is given by aT max , the nuclear stopping cross-section Sn can be defined by aT max Indeed, by using Eq. (5), we can readily confirm that /(N LÞ. is identical with Sn, that was derived by the aT max /(N LÞ power potential of Winterbon et al. Although the identity, Sn ¼
a T max ; NL
ð8Þ
is proven only for the case of power potentials, we strongly think that this relation holds generally. While, the electronic stopping cross-section Se is defined by Se ¼
C p Eg : NL
ð9Þ
This identity is justified, as the mean value of the modified Poisson distribution, Eq. (1), is C p Eg . 4. Results The solid line in the Fig. 1 is the best fitting curve to the experimental observation by Eq. (1) with Eq. (7), instead of d-functions. For the case of Ar, E0 = 400 eV, the asymmetry index a is 0.26. For each discrete peak, j of Eq. (1), E0 of Eq. (7) is replaced by E0 jEg , variance r2j is assumed to be r2j ¼ r20 þ jðD2v þ D2c Þ, where r0 is the energy resolution of detector (2 eV), Dv is the valence band width (1.0 eV), and Dc is the conduction band width (2.7 eV). These
M. Kato, R. Souda / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 71–75
assumptions have been already justified [6,10]. p and C are 0.21 and 2.3, respectively. A little discrepancy is still observed, however, the overall structure is excellently reproduced, in comparison to the dashed line (a = 0). Substituting the known values of N, C 1=3 , T max and also the measured value of a into Eq. (5), is estimated as 8 nm for the case of Ar, E0 ¼ 400 eV. L The results obtained by the same fitting procedure are shown in Fig. 2, for Kr and Xe. The parameter values are shown in the inset of Fig. 2. Such a fitting procedure has been systematically applied to our experimental observations. Then, we have obtained Fig. 3, in which a, p and C are shown as a function of primary energy of proton. The parameter values, in particular, of p, is slightly different from those obtained in [6]. This is a result from the use of more accurate fitting functions, in which the contribution from multiple elastic collisions is properly treated. The solids lines in the middle panel, p, are the theoretical prediction by quantum dynamPRIMARY ENERGY OF PROTON (eV) 0.4
0
100
200
300
400
500
600
α
0.3
0.2
Ar Kr Xe
0.1
0
Xe 0.8
Kr
p
0.6 0.4
Ar
0.2 0
3.5
Stopping Crosssection (10-15eVcm2)
74
Ar Kr Xe LS(Kr)
3
2.5
Xe
2
∝ E0.5 1.5
Kr 1
0.5
Ar 0 0
100
200
300
400
500
600
ENERGY (eV) Fig. 4. The electric stopping cross-section, calculated by Eq. (9), is shown together with the Lindhart–Scharff stopping cross-section for Kr. The Lindhart–Scharff cross-sections for Ar and Xe are not appreciably different from that for Kr.
ical calculations, in which we employed adiabatic coupling approximation to the conduction band of target and H-1s state. The theoretical prediction well agrees with the experimental observations. We here note that the adiabatic potentials necessary for quantum dynamical calculations were determined by molecular orbital calculations [6]. The electric stopping cross-section is shown in Fig. 4 together with the Lindhart–Scharff electronic stopping cross-section [8] for Kr. For rare gas solids with a large band gap, electronic stopping cross-section is significantly different from the Lindhart–Scharff formula in both magnitude and energy dependence. The reason can be attributed to the solid state effect, which is not taken into account in the Lindhart–Scharff theory. Indeed, this theory dose not tale into account the presence of electronic band gap. The energy dependence of electronic stopping power, Fig. 4, may be explained by the band gap of the rare gas solids.
2.5
5. Conclusion
C
2
Xe
1.5
Ar
Kr
1
Kr
Ar
0.5
Xe
0
0
100 200 300 400 500 600 PRIMARY ENERGY OF PROTON (eV)
Fig. 3. The a index, the electronic excitation probability p and the average number of close collision C are shown as a function of primary energy of proton. The solids lines in the middle panel, p, are the theoretical prediction.
We observed the energy spectra of low energy proton scattered from the rare gas solids. The spectra exhibit the discrete and asymmetric structure. From such energy spectra, one can study the electronic and nuclear stopping process simultaneously, by using the statistical theory developed in this paper. References [1] K. Eder, D. Semrad, P. Bauer, Phy. Rev. Lett. 79 (1997) 4112. [2] P. Roncin, J. Villete, J.P. Atanas, H. Khemlishe, Phys. Rev. Lett. 83 (1999) 864.
M. Kato, R. Souda / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 71–75 [3] H. Winter, C. Auth, A. Mertens, Nucl. Instr. and Meth. B 164&165 (2000) 559. [4] J.A. Scheer, P. Wurz, W. Heiland, Nucl. Instr. and Meth. B 212 (2003) 291. [5] J. Stockel, T. Suta, F. Ditori, H.P. Winter, F. Aumayer, Phys. Rev. Lett. 93 (2004) 263201. [6] M. Kato, R. Souda, Nucl. Instr. and Meth. B 203 (2003) 89. [7] S. Shindo, Surf. Sci. 159 (1985) 283.
75
[8] M. Nastasi, J.W. Mayer, J.K. Hirvonen, Chapter 5 of Ion-Solid Interactions: Fundamentals and applications, Cambridge University Press, 1996. [9] K.B. Winterbon, P. Sigmund, J.B. Sanders, Mat. Fys. Medd. Danske. Vid. Selsk. 37 (1970) 14. [10] N. Schwentner, E.-E. Kock, J. Jortner, Electronic Excitations Condensed Rare Gasses, Springer-Verlag, Berlin, 1985.
Nuclear Instruments and Methods in Physics Research B 256 (2007) 71–75 www.elsevier.com/locate/nimb
Elastic and inelastic collision of low energy proton scattering from rare gas solids Masahiko Kato b
a,*
, Ryutaro Souda
b
a Department of Quantum Engineering, Nagoya University, Aichi 464-8603, Japan Nanomaterials and Advanced Materials Laboratory, National Institute for Materials Science, Ibaraki 305-0044, Japan
Available online 20 February 2007
Abstract We have carried out the systematic measurements of energy spectrum for low energy (50–500 eV) proton scattering from rare gas solids of Ar, Kr and Xe. The observed energy spectra showed a multiple peak structure. The multiple peaks have been ascribed to the multiple inter-band electronic excitations, from valence to conduction band of the rare gas solid. In addition, each peak exhibited an asymmetric shape with a long tail toward the low energy side. In order to study these feature of spectrum, we develop a simple statistical model. This model involves three physics parameters: the average number of close collisions C, the electronic excitation probability p, and the asymmetry index a. From C and p, the electronic stopping power can be estimated, whereas from a, the nuclear stopping power can be estimated. Ó 2007 Elsevier B.V. All rights reserved. PACS: 34.10.+x; 34.50.Bw; 34.50.Dy Keywords: Low energy ion scattering; Energy spectrum shape; Insulators
1. Introduction The study about elastic and inelastic stopping process of energetic atoms in solids is still a current issue of discussion, from both fundamental and application point of view. Indeed, since the age of pioneers, Bethe, Bloch, Bohr, Fermi, etc, the stopping power has been extensively investigated. However, there still remain various problems to be answered. One of them is the stopping process of low energy projectiles for insulators. In the case of electronic stopping for insulators, several interesting studies about solid state effects are already available in literature [1–5]. However, further studies are obviously needed, in order to completely understand various aspects involved in the electronic excitation process in insulators. While, in the case of the nuclear stopping,
*
Corresponding author. Fax: +81 52 789 5155. E-mail address: [email protected] (M. Kato).
0168-583X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.11.092
the solid state effects must be expected for very slow projectiles, as the interatomic scattering potentials in solids are not identical with those for isolated atoms. This is particularly the case for ionic crystals. With these backgrounds in mind, we have carried out the systematic measurements of the energy spectrum, for low energy proton scattering from rare gas solids of Ar, Kr and Xe. These targets are a typical insulator with a large band gap, but not ionic crystals. Thus, we could minimize the uncertainty, arising from the unknown interatomic potentials. In the present paper, we provide a simple theory, and report briefly the application of this theory. 2. Experimental observations We have done ion scattering experiments by using low energy proton (50–500 eV) for the solids of rare gas, Ar, Kr and Xe, which were grown on a Pt substrate at a low temperature (20 K). The rare gas solids were sufficiently
72
M. Kato, R. Souda / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 71–75
H+, 400 eV
ENERGY (eV)
Ar (100 L) 300
320
340
360
H+, 400 eV
380
400
420
400
420
Kr(100 L)
EXPERIMENT
EXPERIMENT THEORY E g = 13.5 eV
INTENSITY (arb.)
INTENSITY (arb.)
THEORY α = 0.26 α =0
Paramter values E g = 15.0 eV p = 0.21 C = 2.3
p = 0.55 C = 2.1 α = 0.29
H+, 400 eV
Xe(100 L)
EXPERIMENT
320
340
360
380
400
420
ENERGY (eV) Fig. 1. The open circles show experimentally observed energy spectrum of H+ ions scattered from the solid film of Ar (100 L in exposure). Primary proton beam of 400 eV energy was incident at an angle of 20° (measured from the surface plane), and the scattering angle was 60°. The solid (dashed) line is the best fitting curve to the experimental observation, when the elastic collisions are taken into account (not taken into account).
THEORY E g = 11.2 eV
INTENSITY (arb.)
300
p = 0.84 C = 2.05 α = 0.18
300
thick (100 L in exposure). The angle of incidence was 20°, measured from the surface plane, and the laboratory scattering angle was 60°. The scattered H+ ions were detected by electrostatic analyzer, of which the energy resolution was about 2 eV. Since the target formed as an aggregation of small crystals, we did not observe any ion-channeling effects. We obtained the energy spectra of scattered proton, shown by the open circles of Figs. 1 and 2. Two typical features were generally observed, regardless of the primary energy of proton. Feature (1): the spectrum has multi peak structure, and the multi peaks appear by equidistance in the energy axis. This energy distance precisely coincides with the band gap energy, Eg, between valence and conduction band of the rare gas solid (Eg = 15, 13.5 and 11.2 eV for Ar, Kr and Xe, respectively). Therefore, this peak structure can be assigned to the inter-band electronic excitation in a close collision of proton with a target atom. The multiple peaks can be explained as the multiple inter-band excitations, in the collision sequence of proton with several target atoms. It should be noted that, the excitation of exciton is possible to occur [2], but there was no evidence for the exciton excitation in our measurements. Feature (2): each discrete peak exhibits asymmetric shape with a long tail toward the low energy side. These features of the energy spectra can be consistently explained by a simple theory. It is as follows.
320
340
360
380
ENERGY (eV) Fig. 2. The open circles show experimentally observed energy spectrum of H+ ions scattered from the solid thick film of Kr and Xe. Experimental conditions are the same as those in Fig. 1. The solid line is the best fitting curve to the experimental observation.
3. Theoretical model The probability of the inter-band electronic excitation to occur in a single close collision could be analyzed by using the modified Poisson distribution [6], F ðEÞ ¼ expðpCÞ
1 X ðpCÞj dðE E0 þ jEg Þ j! j¼0
expðCÞdðE E0 Þ;
ð1Þ
where p is the inter-band electronic excitation probability in a single close collision, and C is the average number of close collisions with target atoms. E0 is the primary energy of proton, and Eg is the band gap energy. From the best fitting of Eq. (1) to the experimental observations, we could determine the unknown parameter values of p and C simultaneously. The result obtained, by such an analysis, is shown by the dashed line of Fig. 1, in which the line spectra, i.e. d-functions of Eq. (1), have been replaced by the Gaussian functions with an appropriate variance. The dashed line
M. Kato, R. Souda / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 71–75
reproduces the multi peak structure fairly well, but the underestimate of the low energy tail is evident. We previously attributed this underestimation to the unpredictable background [6]. For insulators, the possible electronic excitation process is the inter-band electronic excitation or the exciton excitation. Because of a large excitation energy, these excitations cannot contribute to the continuous background of energy spectrum, but contribute only to the discrete spectrum. Therefore, the elastic collision must be responsible for the long tail of each discrete peak of spectrum. For the elastic collisions between ion and target atoms, we propose a theory, as follows. If the elastic collision is approximated by the binary collision between ion and target atoms, we can readily derive a transport equation, which should be satisfied by the energy distribution of scattering ion. It is written as d F ðE; xÞ ¼ dx
Z
T max
N 0
Z
drðE þ T Þ F ðE þ T ; xÞ dT dT
T max
N 0
drðEÞ F ðE; xÞ dT ; dT
ð2Þ
where F ðE; xÞ is the energy distribution of scattering ions, which have the kinetic energy E at a position x (along the ion trajectory). drðEÞ=dT is the differential cross-section for energy transfer T, in which the ion with energy E undergoes a binary collision. N is the number density of target atoms. Tmax is the maximum energy transfer, given 1M2 by T max ¼ ðM4MþM 2 E. 1 2Þ If the energy loss of ion is small, drðEÞ=dT may be approximated by its value for a representative energy E0, e.g. primary energy of ion. By this approximation, drðE0 Þ=dT does not depend on E, and moreover, the analytic solution of Eq. (2) can be written as [7] Z 1 1 i F ðEÞ ¼ ds exp Es 2ph 1 h Z T max drðE0 Þ i dT 1 exp T s ; exp N L dT h 0 ð3Þ is the average path length taken by scattering ions where L in the inside of target. With regard to drðE0 Þ=dT , several analytical expressions are available [8]. For simplicity, we assume the crosssection for the power potential, which was proposed by 0Þ Winterbon et al. [9]. It is given by drðE ¼ EmCTm1þm , where dT 0 Cm is a constant, depending on m. The value of m varies from 1/3 to 1, depending on the energy range of E0. For low energy ion scattering under consideration, Winterbon et al. recommended the use of m ¼ 1=3 and 4=3 2=3 1=3 C 1=3 ¼ 1:309 p2 aTF ð2Z 1 Z 2 e2 Þ ðM 1 =M 2 Þ . 0Þ By assuming drðE ¼ EmCTm1þm , the integration with respect dT 0 to T in the exponential function of Eq. (3) can be easily calculated, and it is found to be
drðE0 Þ i 1 exp T s dT dT h 0 iT max a ln 1 þ s ; h
N L
Z
73
T max
ð4Þ
where a, referred to as asymmetry index, has been introduced, and it is defined by a NL
Cm : ð1 mÞEm0 T mmax
ð5Þ
Substitution of Eq. (4) into Eq. (3) yields F ðEÞ ¼
1 1 1 expðE=T max Þ: CðaÞ T amax E1a
ð6Þ
This function is known as the gamma distribution. Furthermore, we have to take into account the uncertainty in energy, due to three reasons:energy resolution of detector, valence and conduction bandwidths. This uncertainty is considered by a Gaussian function with variance r2. Convoluting Eq. (6) with this Gaussian function, we finally obtain ðEE0 Þ2 1 E E0 2 F ðEÞ ¼ pffiffiffiffiffiffiffiffiffiffi e 4r Da : ð7Þ r 2pr2 where Da(x) is a parabolic cylinder function. This function reduces to a Gaussian function, when a = 0. Each discrete spectrum, d-functions of Eq. (1), should be replaced by Eq. (7) with appropriate E0 and r. Let us examine the physical implication of a. As the mean value of Gamma distribution is given by aT max , the nuclear stopping cross-section Sn can be defined by aT max Indeed, by using Eq. (5), we can readily confirm that /(N LÞ. is identical with Sn, that was derived by the aT max /(N LÞ power potential of Winterbon et al. Although the identity, Sn ¼
a T max ; NL
ð8Þ
is proven only for the case of power potentials, we strongly think that this relation holds generally. While, the electronic stopping cross-section Se is defined by Se ¼
C p Eg : NL
ð9Þ
This identity is justified, as the mean value of the modified Poisson distribution, Eq. (1), is C p Eg . 4. Results The solid line in the Fig. 1 is the best fitting curve to the experimental observation by Eq. (1) with Eq. (7), instead of d-functions. For the case of Ar, E0 = 400 eV, the asymmetry index a is 0.26. For each discrete peak, j of Eq. (1), E0 of Eq. (7) is replaced by E0 jEg , variance r2j is assumed to be r2j ¼ r20 þ jðD2v þ D2c Þ, where r0 is the energy resolution of detector (2 eV), Dv is the valence band width (1.0 eV), and Dc is the conduction band width (2.7 eV). These
M. Kato, R. Souda / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 71–75
assumptions have been already justified [6,10]. p and C are 0.21 and 2.3, respectively. A little discrepancy is still observed, however, the overall structure is excellently reproduced, in comparison to the dashed line (a = 0). Substituting the known values of N, C 1=3 , T max and also the measured value of a into Eq. (5), is estimated as 8 nm for the case of Ar, E0 ¼ 400 eV. L The results obtained by the same fitting procedure are shown in Fig. 2, for Kr and Xe. The parameter values are shown in the inset of Fig. 2. Such a fitting procedure has been systematically applied to our experimental observations. Then, we have obtained Fig. 3, in which a, p and C are shown as a function of primary energy of proton. The parameter values, in particular, of p, is slightly different from those obtained in [6]. This is a result from the use of more accurate fitting functions, in which the contribution from multiple elastic collisions is properly treated. The solids lines in the middle panel, p, are the theoretical prediction by quantum dynamPRIMARY ENERGY OF PROTON (eV) 0.4
0
100
200
300
400
500
600
α
0.3
0.2
Ar Kr Xe
0.1
0
Xe 0.8
Kr
p
0.6 0.4
Ar
0.2 0
3.5
Stopping Crosssection (10-15eVcm2)
74
Ar Kr Xe LS(Kr)
3
2.5
Xe
2
∝ E0.5 1.5
Kr 1
0.5
Ar 0 0
100
200
300
400
500
600
ENERGY (eV) Fig. 4. The electric stopping cross-section, calculated by Eq. (9), is shown together with the Lindhart–Scharff stopping cross-section for Kr. The Lindhart–Scharff cross-sections for Ar and Xe are not appreciably different from that for Kr.
ical calculations, in which we employed adiabatic coupling approximation to the conduction band of target and H-1s state. The theoretical prediction well agrees with the experimental observations. We here note that the adiabatic potentials necessary for quantum dynamical calculations were determined by molecular orbital calculations [6]. The electric stopping cross-section is shown in Fig. 4 together with the Lindhart–Scharff electronic stopping cross-section [8] for Kr. For rare gas solids with a large band gap, electronic stopping cross-section is significantly different from the Lindhart–Scharff formula in both magnitude and energy dependence. The reason can be attributed to the solid state effect, which is not taken into account in the Lindhart–Scharff theory. Indeed, this theory dose not tale into account the presence of electronic band gap. The energy dependence of electronic stopping power, Fig. 4, may be explained by the band gap of the rare gas solids.
2.5
5. Conclusion
C
2
Xe
1.5
Ar
Kr
1
Kr
Ar
0.5
Xe
0
0
100 200 300 400 500 600 PRIMARY ENERGY OF PROTON (eV)
Fig. 3. The a index, the electronic excitation probability p and the average number of close collision C are shown as a function of primary energy of proton. The solids lines in the middle panel, p, are the theoretical prediction.
We observed the energy spectra of low energy proton scattered from the rare gas solids. The spectra exhibit the discrete and asymmetric structure. From such energy spectra, one can study the electronic and nuclear stopping process simultaneously, by using the statistical theory developed in this paper. References [1] K. Eder, D. Semrad, P. Bauer, Phy. Rev. Lett. 79 (1997) 4112. [2] P. Roncin, J. Villete, J.P. Atanas, H. Khemlishe, Phys. Rev. Lett. 83 (1999) 864.
M. Kato, R. Souda / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 71–75 [3] H. Winter, C. Auth, A. Mertens, Nucl. Instr. and Meth. B 164&165 (2000) 559. [4] J.A. Scheer, P. Wurz, W. Heiland, Nucl. Instr. and Meth. B 212 (2003) 291. [5] J. Stockel, T. Suta, F. Ditori, H.P. Winter, F. Aumayer, Phys. Rev. Lett. 93 (2004) 263201. [6] M. Kato, R. Souda, Nucl. Instr. and Meth. B 203 (2003) 89. [7] S. Shindo, Surf. Sci. 159 (1985) 283.
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[8] M. Nastasi, J.W. Mayer, J.K. Hirvonen, Chapter 5 of Ion-Solid Interactions: Fundamentals and applications, Cambridge University Press, 1996. [9] K.B. Winterbon, P. Sigmund, J.B. Sanders, Mat. Fys. Medd. Danske. Vid. Selsk. 37 (1970) 14. [10] N. Schwentner, E.-E. Kock, J. Jortner, Electronic Excitations Condensed Rare Gasses, Springer-Verlag, Berlin, 1985.