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Ionization of xenon Rydberg atoms at surfaces
Nuclear Instruments and Methods in Physics Research B 193 (2002) 403–407 www.elsevier.com/locate/nimb
Ionization of xenon Rydberg atoms at surfaces Z. Zhou, C. Oubre, S.B. Hill, P. Nordlander, F.B. Dunning
*
Department of Physics and Astronomy and the Rice Quantum Institute, Rice University, 6100 Main Street, Houston, TX 77005-1892, USA
Abstract Measurements of the ionization of the extreme red and blue states in high-lying Xe(n) Stark manifolds at a metal surface are reported. The data show that, despite their very different initial spatial characteristics, the extreme members of a given Stark manifold ionize at similar atom/surface separations. This is attributed to energy level shifts induced by the presence of the surface which lead to avoided crossings between states in adjacent n manifolds prior to ionization. These energy level shifts are examined using hydrogenic complex scaling theory. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 79.20.Rf; 34.50.Dy Keywords: Surface ionization; Atom/surface interaction dynamics
1. Introduction Because of their large physical size and weak binding, Rydberg atoms, i.e. atoms in which one electron is excited to a state of large principal quantum number n, are strongly perturbed by the presence of a nearby metal surface. Even relatively far from the surface, the motion of the excited electron is influenced by image charge interactions which distort the electronic wave functions and shift the atomic energy levels. Furthermore, ionization can occur through resonant tunneling of the excited electron into a vacant level in the metal which reduces the atomic lifetime and causes the states to become very broad near to the surface.
*
Corresponding author. Tel.: +1-713-348-3544; fax: +1-713348-4150. E-mail address: [email protected] (F.B. Dunning).
Measurements with Rydberg atoms can therefore provide a powerful probe of image charge effects and of charge transfer during atom/surface interactions. Theoretical studies have focussed on hydrogen Rydberg atoms, initially using perturbation methods [1] and, more recently, the complex scaling and time-dependent close-coupling techniques [2–4]. Theory shows that, as the surface is approached, the degeneracy of the (hydrogenic) levels is lifted through formation of hybridized ‘‘Stark-like’’ states. The electron probability density for some hybridized states is maximal towards the surface, others towards vacuum. The predicted tunneling rates vary widely from state to state and are many orders of magnitude greater for states oriented toward the surface. The predicted ionization distances for the lowest energy, i.e. redmost, Stark-like states for which the electron probability density is most strongly oriented toward
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 8 1 2 - 1
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Z. Zhou et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 403–407
the surface, are in good agreement with recent experimental results [5] obtained with xenon Rydberg atoms when initially exciting the redmost states in each Xe(n) Stark manifold. Here we report measurements of ionization distances for the blue-most Xe(n) Stark states for which the electron probability density is initially strongly oriented toward vacuum. Surprisingly, the ionization distance is observed to be similar to that for the red-most Stark states. This is attributed to energy level shifts caused by the presence of the surface that lead to avoided crossings between states in adjacent n-manifolds prior to ionization. These energy level shifts are examined using hydrogenic complex-scaling theory.
Excitation occurs in the presence of a relativelyweak dc field allowing selective excitation of ‘‘red’’ or ‘‘blue’’ Stark states. Immediately after the laser pulse, a strong pulsed electric field of 0.1 ls risetime and 10 ls duration is applied to establish the ion collection field. Ions that escape the surface are detected by a bell-mouthed electron multiplier. Arrival-time gating is used to discriminate against ions not formed in atom/surface interactions. If tunneling occurs at an atom/surface separation Zi , the minimum external field (in a.u.) that must be applied to prevent the ion striking the surface and being lost is [3] rffiffiffiffiffiffi2 1 T? Emin ðZi ; T? Þ ¼ ; ð1Þ þ 2Zi Zi
2. Experimental approach
where T? ¼ mv2? =2 is the kinetic energy perpendicular to the surface at the time of ionization. Thus, by measuring the ion signal as a function of applied field, Emin , and hence Zi , can be determined.
In the present work xenon Rydberg atoms are directed at near grazing incidence onto the target surface. Ions formed by tunneling are attracted to the surface by the electric fields associated with their image charges. These fields are large and will rapidly accelerate an ion to the surface where it will be neutralized by an Auger process. To prevent this an electric field is applied perpendicular to the surface. Because the initial image charge field experienced by an ion, and thus the external field required to counteract it, depends on the distance from the surface at which ionization occurs, the ionization distance can be inferred from measurements of the surface ionization signal as a function of applied field. The present apparatus is described in detail elsewhere [5]. Xenon Rydberg atoms are created by photoexciting the 3 P0 atoms contained in a tightly collimated mixed Xe(3 P0:2 ) metastable atom beam that is produced by electron impact excitation of ground-state xenon atoms contained in a supersonic expansion. The atoms are excited close to the target surface using the crossed output of a UV-pumped CR899-21 C-47 dye laser. The laser beam polarization is selected to excite m ¼ 0 states. Experiments are conducted in a pulsed mode by forming the output of the laser into a train of pulses of 1 ls duration and 5 kHz repetition frequency using an A-O modulator.
3. Results and discussion Fig. 1 shows the applied field dependence of the surface ion signal observed when atoms initially prepared in the blue-most and red-most states in the Xe (n ¼ 17 and 20) Stark manifolds are incident at h 4° on a near atomically-flat Au(1 1 1) surface. The data are normalized to the initial number of Rydberg atoms created, which was determined by field ionization. (Note that for sufficiently large applied fields, the majority of these atoms are detected through surface ionization.) The sudden decrease in the surface ion signal at large fields is due to direct field ionization of the Rydberg atoms in vacuum before they reach the target surface. The observed onset in the surface ionization signal is similar for both the red-most and bluemost states in each Stark manifold despite their very different initial orientations. The critical threshold fields Emin correspond to ionization at an atom/surface separation Zi 4:5n2 a0 , where n2 a0 is the Bohr radius. This ionization distance is close to the value Zi 3:8n2 a0 predicted theoretically for
Z. Zhou et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 403–407
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Fig. 2. Calculated energy shifts for the (- - -) H(n ¼ 10) and (––) H(n ¼ 11) m ¼ 0 states as a function of atom/surface separation Z for the applied electric field strengths (in a.u.) indicated. Fig. 1. Applied field dependence of the surface ion signals measured when atoms initially prepared in the blue-most (d) and red-most ( ) states in the Xe (n ¼ 17 and 20) Stark manifolds are incident on a near atomically flat Au(1 1 1) surface.
the red-most, most strongly surface oriented, m ¼ 0 hydrogenic states [5]. Theory predicts, however, that the blue-most m ¼ 0 states will ionize at much smaller atom/surface separations, 1:2n2 a0 [6] which would result in a dramatic increase in the required threshold fields Emin . As evident from Fig. 1, no such increase is seen when initially exciting the blue-most Stark states. This apparent discrepancy can be explained by considering the Rydberg level structure in the presence of a nearby surface. Calculation of the energy shifts and broadening of atomic Rydberg states near a metal surface using the non-perturbative complex scaling technique has been discussed in several publications [3]. The calculated energy level shifts for the H(n ¼ 10) and H(n ¼ 11) m ¼ 0 states are shown in Fig. 2 as a function of atom/ surface separation Z for several values of applied field. The presence of the field breaks the degen-
eracy of the hydrogenic levels at large atom/surface separations but, for the range of fields considered in Figs. 2 and 3, the resultant energy level shifts are insufficient to cause states from adjacent n manifolds to overlap. As the surface is approached, however, surface-induced perturbations lead to further energy level shifts that result in level crossings. The atom/surface separations at which these crossings occur depend on the magnitude of the applied field. This is illustrated in Fig. 3 which shows the energies of the blue-most H(n ¼ 10) and red-most H(n ¼ 11) m ¼ 0 states as a function of atom/surface separation for three values of applied field E ¼ 2, 2.5 and 2:8 106 a.u. The level crossings are indicated by the circles. Surface ionization of the blue-most H(n ¼ 10) state is expected to occur at an atom/surface separation of 120 a.u. For ionization at this distance, a minimum applied field Emin of 1:7 105 a.u. would be required to detect the ion, which is larger than the field required for direct field ionization of the parent atoms. However, the data in Fig. 3 show that even in much smaller applied fields many crossings between levels in
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Z. Zhou et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 403–407
Fig. 3. Energies of the blue-most H(n ¼ 10) and red-most H(n ¼ 11) m ¼ 0 states as a function of atom/surface separation for applied fields of (- - -) 2 106 , (– – –) 2:5 106 and (––) 2:8 106 a.u.
adjacent n manifolds will occur prior to reaching an atom/surface separation of 120 a.u. Indeed, the first crossings occur at atom/surface separations comparable to the ionization distances expected for the red-most hydrogenic m ¼ 0 states. (At high n, the required ion collection fields are such that states from adjacent Stark manifolds will overlap even far from the surface. However, a series of level crossings will again occur as the surface is approached.) In the case of xenon, non-hydrogenic effects such as core polarization couple states of the same m leading the appearance of avoided crossings wherever levels of the same m intersect [7]. The energy separations at such avoided crossings are strongly m dependent and are particularly large for m ¼ 0 states. The behavior of an atom as it approaches the surface is then governed by its behavior at these avoided crossings, in particular, whether these are traversed diabatically or adiabatically. The data in Fig. 1 can be understood if it is assumed that the avoided crossings are traversed adiabatically. Adiabatic passage of an excited atom initially in state jAi through an avoided crossing with state jBi results in the atom assum-
Fig. 4. Contour plots of the probability densities jw j2 and jwþ j2 associated with (w ) the blue-most H(n ¼ 10) and (wþ ) the red-most H(n ¼ 11) m ¼ 0 states for an atom/surface separation Z ¼ 670 a.u. and an applied field of 2:8 106 a.u. The contours correspond to 0.1, 1.0, 10 and 100 107 a.u.
ing the characteristics of state jBi. The consequences of this can be seen by considering the avoided crossing between the blue-most n ¼ 10 state (w ) and red-most n ¼ 11 state (wþ ) highlighted in Fig. 3. The associated electron probability densities jwþ j2 and jw j2 are shown in Fig. 4. The red-most n ¼ 11 state is strongly oriented toward the surface whereas the blue-most n ¼ 10 state is strongly oriented toward vacuum. Following an avoided crossing, therefore, the character of each initial atomic state will change dramatically. As an excited atom approaches a surface it will (typically) undergo a series of avoided crossings, the electron probability density oscillating between being oriented toward the surface and toward vacuum. In consequence, the electron probability densities associated with the extreme red and blue members of adjacent Stark manifolds will, on average, be similar. In this event, it is not surprising that both states are observed to ionize at similar atom/surface separations. Furthermore, because the ionization rate increases exponentially with the degree of overlap between the surface and the electronic wave function, ionization should occur at atom/surface separations characteristic of the red-most m ¼ 0 states, again in agreement with experimental observations.
Z. Zhou et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 403–407
The present work demonstrates that the effects of level crossings are important in excited atom/ surface interactions suggesting that a rich variety of dynamical behaviors is to be expected as the initial velocity and m-value of the incident atoms, and the applied electric field, are varied. Further experimental work to investigate this is underway. Calculations are being extended to xenon [8] to examine the level shifts and avoided crossings for this atom in detail.
Acknowledgements This work was supported by the NSF and the Robert A. Welch Foundation.
407
References [1] A.V. Chaplik, Zh. Eksp. Teor. Fiz. 54 (1968) 332 (Sov. Phys. J. Expt. Theor. Phys. 27 (1968) 178). [2] P. Nordlander, Phys. Rev. B 53 (1996) 4125. [3] P. Nordlander, F.B. Dunning, Phys. Rev. B 53 (1996) 8083; Nucl. Instrum. Meth. Phys. Res. Sect. B 125 (1997) 300. [4] P. K€ urpick, U. Thumm, U. Wille, Phys. Rev. A 57 (1998) 1920. [5] S.B. Hill, C.B. Haich, Z. Zhou, P. Nordlander, F.B. Dunning, Phys. Rev. Lett. 85 (2000) 5444. [6] J. Hanssen, C.F. Martin, P. Nordlander, Surf. Sci 423 (1999) L271. [7] See, for example, the article by D. Kleppner, M.G. Littman, M.L. Zimmerman, in: R.F. Stebbings, F.B. Dunning (Eds.), Rydberg States of Atoms and Molecules, Cambridge, London, 1983. [8] J. Braun, P. Nordlander, Surf. Sci. 448 (2000) L193.
Ionization of xenon Rydberg atoms at surfaces Z. Zhou, C. Oubre, S.B. Hill, P. Nordlander, F.B. Dunning
*
Department of Physics and Astronomy and the Rice Quantum Institute, Rice University, 6100 Main Street, Houston, TX 77005-1892, USA
Abstract Measurements of the ionization of the extreme red and blue states in high-lying Xe(n) Stark manifolds at a metal surface are reported. The data show that, despite their very different initial spatial characteristics, the extreme members of a given Stark manifold ionize at similar atom/surface separations. This is attributed to energy level shifts induced by the presence of the surface which lead to avoided crossings between states in adjacent n manifolds prior to ionization. These energy level shifts are examined using hydrogenic complex scaling theory. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 79.20.Rf; 34.50.Dy Keywords: Surface ionization; Atom/surface interaction dynamics
1. Introduction Because of their large physical size and weak binding, Rydberg atoms, i.e. atoms in which one electron is excited to a state of large principal quantum number n, are strongly perturbed by the presence of a nearby metal surface. Even relatively far from the surface, the motion of the excited electron is influenced by image charge interactions which distort the electronic wave functions and shift the atomic energy levels. Furthermore, ionization can occur through resonant tunneling of the excited electron into a vacant level in the metal which reduces the atomic lifetime and causes the states to become very broad near to the surface.
*
Corresponding author. Tel.: +1-713-348-3544; fax: +1-713348-4150. E-mail address: [email protected] (F.B. Dunning).
Measurements with Rydberg atoms can therefore provide a powerful probe of image charge effects and of charge transfer during atom/surface interactions. Theoretical studies have focussed on hydrogen Rydberg atoms, initially using perturbation methods [1] and, more recently, the complex scaling and time-dependent close-coupling techniques [2–4]. Theory shows that, as the surface is approached, the degeneracy of the (hydrogenic) levels is lifted through formation of hybridized ‘‘Stark-like’’ states. The electron probability density for some hybridized states is maximal towards the surface, others towards vacuum. The predicted tunneling rates vary widely from state to state and are many orders of magnitude greater for states oriented toward the surface. The predicted ionization distances for the lowest energy, i.e. redmost, Stark-like states for which the electron probability density is most strongly oriented toward
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 8 1 2 - 1
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Z. Zhou et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 403–407
the surface, are in good agreement with recent experimental results [5] obtained with xenon Rydberg atoms when initially exciting the redmost states in each Xe(n) Stark manifold. Here we report measurements of ionization distances for the blue-most Xe(n) Stark states for which the electron probability density is initially strongly oriented toward vacuum. Surprisingly, the ionization distance is observed to be similar to that for the red-most Stark states. This is attributed to energy level shifts caused by the presence of the surface that lead to avoided crossings between states in adjacent n-manifolds prior to ionization. These energy level shifts are examined using hydrogenic complex-scaling theory.
Excitation occurs in the presence of a relativelyweak dc field allowing selective excitation of ‘‘red’’ or ‘‘blue’’ Stark states. Immediately after the laser pulse, a strong pulsed electric field of 0.1 ls risetime and 10 ls duration is applied to establish the ion collection field. Ions that escape the surface are detected by a bell-mouthed electron multiplier. Arrival-time gating is used to discriminate against ions not formed in atom/surface interactions. If tunneling occurs at an atom/surface separation Zi , the minimum external field (in a.u.) that must be applied to prevent the ion striking the surface and being lost is [3] rffiffiffiffiffiffi2 1 T? Emin ðZi ; T? Þ ¼ ; ð1Þ þ 2Zi Zi
2. Experimental approach
where T? ¼ mv2? =2 is the kinetic energy perpendicular to the surface at the time of ionization. Thus, by measuring the ion signal as a function of applied field, Emin , and hence Zi , can be determined.
In the present work xenon Rydberg atoms are directed at near grazing incidence onto the target surface. Ions formed by tunneling are attracted to the surface by the electric fields associated with their image charges. These fields are large and will rapidly accelerate an ion to the surface where it will be neutralized by an Auger process. To prevent this an electric field is applied perpendicular to the surface. Because the initial image charge field experienced by an ion, and thus the external field required to counteract it, depends on the distance from the surface at which ionization occurs, the ionization distance can be inferred from measurements of the surface ionization signal as a function of applied field. The present apparatus is described in detail elsewhere [5]. Xenon Rydberg atoms are created by photoexciting the 3 P0 atoms contained in a tightly collimated mixed Xe(3 P0:2 ) metastable atom beam that is produced by electron impact excitation of ground-state xenon atoms contained in a supersonic expansion. The atoms are excited close to the target surface using the crossed output of a UV-pumped CR899-21 C-47 dye laser. The laser beam polarization is selected to excite m ¼ 0 states. Experiments are conducted in a pulsed mode by forming the output of the laser into a train of pulses of 1 ls duration and 5 kHz repetition frequency using an A-O modulator.
3. Results and discussion Fig. 1 shows the applied field dependence of the surface ion signal observed when atoms initially prepared in the blue-most and red-most states in the Xe (n ¼ 17 and 20) Stark manifolds are incident at h 4° on a near atomically-flat Au(1 1 1) surface. The data are normalized to the initial number of Rydberg atoms created, which was determined by field ionization. (Note that for sufficiently large applied fields, the majority of these atoms are detected through surface ionization.) The sudden decrease in the surface ion signal at large fields is due to direct field ionization of the Rydberg atoms in vacuum before they reach the target surface. The observed onset in the surface ionization signal is similar for both the red-most and bluemost states in each Stark manifold despite their very different initial orientations. The critical threshold fields Emin correspond to ionization at an atom/surface separation Zi 4:5n2 a0 , where n2 a0 is the Bohr radius. This ionization distance is close to the value Zi 3:8n2 a0 predicted theoretically for
Z. Zhou et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 403–407
405
Fig. 2. Calculated energy shifts for the (- - -) H(n ¼ 10) and (––) H(n ¼ 11) m ¼ 0 states as a function of atom/surface separation Z for the applied electric field strengths (in a.u.) indicated. Fig. 1. Applied field dependence of the surface ion signals measured when atoms initially prepared in the blue-most (d) and red-most ( ) states in the Xe (n ¼ 17 and 20) Stark manifolds are incident on a near atomically flat Au(1 1 1) surface.
the red-most, most strongly surface oriented, m ¼ 0 hydrogenic states [5]. Theory predicts, however, that the blue-most m ¼ 0 states will ionize at much smaller atom/surface separations, 1:2n2 a0 [6] which would result in a dramatic increase in the required threshold fields Emin . As evident from Fig. 1, no such increase is seen when initially exciting the blue-most Stark states. This apparent discrepancy can be explained by considering the Rydberg level structure in the presence of a nearby surface. Calculation of the energy shifts and broadening of atomic Rydberg states near a metal surface using the non-perturbative complex scaling technique has been discussed in several publications [3]. The calculated energy level shifts for the H(n ¼ 10) and H(n ¼ 11) m ¼ 0 states are shown in Fig. 2 as a function of atom/ surface separation Z for several values of applied field. The presence of the field breaks the degen-
eracy of the hydrogenic levels at large atom/surface separations but, for the range of fields considered in Figs. 2 and 3, the resultant energy level shifts are insufficient to cause states from adjacent n manifolds to overlap. As the surface is approached, however, surface-induced perturbations lead to further energy level shifts that result in level crossings. The atom/surface separations at which these crossings occur depend on the magnitude of the applied field. This is illustrated in Fig. 3 which shows the energies of the blue-most H(n ¼ 10) and red-most H(n ¼ 11) m ¼ 0 states as a function of atom/surface separation for three values of applied field E ¼ 2, 2.5 and 2:8 106 a.u. The level crossings are indicated by the circles. Surface ionization of the blue-most H(n ¼ 10) state is expected to occur at an atom/surface separation of 120 a.u. For ionization at this distance, a minimum applied field Emin of 1:7 105 a.u. would be required to detect the ion, which is larger than the field required for direct field ionization of the parent atoms. However, the data in Fig. 3 show that even in much smaller applied fields many crossings between levels in
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Z. Zhou et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 403–407
Fig. 3. Energies of the blue-most H(n ¼ 10) and red-most H(n ¼ 11) m ¼ 0 states as a function of atom/surface separation for applied fields of (- - -) 2 106 , (– – –) 2:5 106 and (––) 2:8 106 a.u.
adjacent n manifolds will occur prior to reaching an atom/surface separation of 120 a.u. Indeed, the first crossings occur at atom/surface separations comparable to the ionization distances expected for the red-most hydrogenic m ¼ 0 states. (At high n, the required ion collection fields are such that states from adjacent Stark manifolds will overlap even far from the surface. However, a series of level crossings will again occur as the surface is approached.) In the case of xenon, non-hydrogenic effects such as core polarization couple states of the same m leading the appearance of avoided crossings wherever levels of the same m intersect [7]. The energy separations at such avoided crossings are strongly m dependent and are particularly large for m ¼ 0 states. The behavior of an atom as it approaches the surface is then governed by its behavior at these avoided crossings, in particular, whether these are traversed diabatically or adiabatically. The data in Fig. 1 can be understood if it is assumed that the avoided crossings are traversed adiabatically. Adiabatic passage of an excited atom initially in state jAi through an avoided crossing with state jBi results in the atom assum-
Fig. 4. Contour plots of the probability densities jw j2 and jwþ j2 associated with (w ) the blue-most H(n ¼ 10) and (wþ ) the red-most H(n ¼ 11) m ¼ 0 states for an atom/surface separation Z ¼ 670 a.u. and an applied field of 2:8 106 a.u. The contours correspond to 0.1, 1.0, 10 and 100 107 a.u.
ing the characteristics of state jBi. The consequences of this can be seen by considering the avoided crossing between the blue-most n ¼ 10 state (w ) and red-most n ¼ 11 state (wþ ) highlighted in Fig. 3. The associated electron probability densities jwþ j2 and jw j2 are shown in Fig. 4. The red-most n ¼ 11 state is strongly oriented toward the surface whereas the blue-most n ¼ 10 state is strongly oriented toward vacuum. Following an avoided crossing, therefore, the character of each initial atomic state will change dramatically. As an excited atom approaches a surface it will (typically) undergo a series of avoided crossings, the electron probability density oscillating between being oriented toward the surface and toward vacuum. In consequence, the electron probability densities associated with the extreme red and blue members of adjacent Stark manifolds will, on average, be similar. In this event, it is not surprising that both states are observed to ionize at similar atom/surface separations. Furthermore, because the ionization rate increases exponentially with the degree of overlap between the surface and the electronic wave function, ionization should occur at atom/surface separations characteristic of the red-most m ¼ 0 states, again in agreement with experimental observations.
Z. Zhou et al. / Nucl. Instr. and Meth. in Phys. Res. B 193 (2002) 403–407
The present work demonstrates that the effects of level crossings are important in excited atom/ surface interactions suggesting that a rich variety of dynamical behaviors is to be expected as the initial velocity and m-value of the incident atoms, and the applied electric field, are varied. Further experimental work to investigate this is underway. Calculations are being extended to xenon [8] to examine the level shifts and avoided crossings for this atom in detail.
Acknowledgements This work was supported by the NSF and the Robert A. Welch Foundation.
407
References [1] A.V. Chaplik, Zh. Eksp. Teor. Fiz. 54 (1968) 332 (Sov. Phys. J. Expt. Theor. Phys. 27 (1968) 178). [2] P. Nordlander, Phys. Rev. B 53 (1996) 4125. [3] P. Nordlander, F.B. Dunning, Phys. Rev. B 53 (1996) 8083; Nucl. Instrum. Meth. Phys. Res. Sect. B 125 (1997) 300. [4] P. K€ urpick, U. Thumm, U. Wille, Phys. Rev. A 57 (1998) 1920. [5] S.B. Hill, C.B. Haich, Z. Zhou, P. Nordlander, F.B. Dunning, Phys. Rev. Lett. 85 (2000) 5444. [6] J. Hanssen, C.F. Martin, P. Nordlander, Surf. Sci 423 (1999) L271. [7] See, for example, the article by D. Kleppner, M.G. Littman, M.L. Zimmerman, in: R.F. Stebbings, F.B. Dunning (Eds.), Rydberg States of Atoms and Molecules, Cambridge, London, 1983. [8] J. Braun, P. Nordlander, Surf. Sci. 448 (2000) L193.