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Electronic shell structure of laser-warmed Na clusters
Volume 186, number I
CHEMICAL PHYSICS LETTERS
1 November 1991
Electronic shell structure of laser-warmed Na clusters T.P. Martin a, S. Bjramholm b, J. Borggreen b, C. Brechignac ‘, Ph. Cahuzac c, IL Hansen b and J. Pedersen b a Max-Planck-lnstitut ftir Festkiirperfrschung,
Heisenbergstrasse
I, W- 7000 Stuttgart 80, Germany
b The Niels Bohr Institute, University of Copenhagen, DK-4000 Roskilde, Denmark ’ Laboratoire AimP Cotton, CNRS II, Biitiment 505, 91405 Orsay Cedex. France
Received 19 April 1991; in final form 16 May 1991
Mass spectra are reported for large sodium clusters warmed by a continuous laser beam prior to ionization. During the 1 ms warming period, the clusters lose more than IO%of their mass by single-atom evaporation. The resulting size distribution reveals what appears to be electronic shell structure for clusters containing up to 2500 atoms.
1. Introduction The one-particle eigenstates of fermions moving in a spherically symmetric potential are characterized by a well-defined angular momentum. In large systems, the degeneracy of these so-called subshells is considerable, 2( 21+ 1). For certain forms of the radial dependence of the potential, there can be a further condensation of subshells into highly degenerate shells. However, there exist only a few known cases for which this shell degeneracy is exact, e.g. the hydrogen atom and the spherical harmonic oscillator. More often, shells appear as an approximate grouping of subshells on an energy scale, as in atomic nuclei. In fact, the earliest model developed for nuclei [ 1,2] describes very nicely the electronic structure of metal clusters [ 3- 191 containing hundreds of electrons. In this paper, we present evidence for electronic shell formation in sodium clusters containing up to 2500 valence electrons.
2. Experimental The technique we have used to study shell structure in metal clusters is photoionization time-of-flight (TOF) mass spectrometry. The mass spectrometer has a mass range of 600000 amu and a mass reso-
lution of up to 20000. The cluster source is a lowpressure, rare-gas, condensation cell. Sodium vapor was quenched in cold He gas having a pressure of about 1 mbar. Clusters condensed out of the quenched vapor were transported by the gas stream through a nozzle and through two chambers of intermediate pressure into a high-vacuum chamber. The size distribution of the clusters could be controlled by varying the oven-to-nozzle distance, the He gas pressure, and the oven temperature. The clusters were photoionized with a 0.4 t.tJ, 2x 1 mm, 15 ns, 308 nm (4.0 eV) excimer laser pulse. The wellresolved mass spectra showed the sodium clusters to be free of oxygen and other impurities. The clusters were warmed prior to ionization with a continuous Ar ion laser beam running parallel to the neutral cluster beam. The laser light entered the ionization chamber through a heated window, passed through the ionization volume, through 2.2 and 3.0 mm diameter skimmer apertures, through a 3.0 mm diameter nozzle, through the oven chamber and finally excited through a second window where the laser intensity was recorded. Short-wavelength light was found to warm much more efficiently. Using the 458 nm (2.71 eV) laser line, we found 10 mW sufficient to alter appreciably the neutral size distribution.
0009-2614/91/S 03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
53
Volume 186, number 1
CHEMICAL PHYSICS LETTERS
3. Results Electronic shell structure can be observed experimentally in several ways. Perhaps the most easily understood method is to (i) measure the ionization potentials for each cluster size. Electrons in newly opened shells are less tightly bound, i.e. have lower ionization energies. However, considerable experimental effort is required to measure the ionization energy of even a single cluster. A complete photoionization spectrum must be obtained and very often an appropriate source of tunable light is simply not available. It is much easier to observe shell closings in photoionization TOF mass spectra. This can be done in either one of two ways. (ii) If mass spectra are recorded using light with energy near threshold ionization, a shell is announced as an abrupt increase in cluster ion intensity. (iii) If mass spectra are recorded with light well above threshold, the neutral size distribution can be sampled. Here it is generally found [ 3,151 that a shell is announced by a decreased ion intensity. Fig. 1 shows mass spectra of laser-warmed Na
800- Threshold
1 November 1991
clusters obtained with two different wavelengths of the ionizing light, i.e. by using methods (ii) and (iii). Photons having an energy of 3.1 eV are very close to the size-dependent ionization threshold of the sodium clusters investigated. The corresponding mass spectrum is characterized by steps. For example, the step at about (Na)rO occurs because smaller cluster can hardly be ionized with 3.1 eV photons, but larger clusters, with a lower threshold energy, allow themselves to be ionized. The steps in the mass spectra obtained using threshold ionization reflect sudden changes in the ionization energy of clusters as a function of size. These features are relatively sharp and easily interpreted. The mass spectra obtained with ionizing photons having energy well above threshold are quite different from the spectrum just discussed. Without the warming laser, the mass spectra are without structure, i.e. the size distribution of the cold clusters emerging from our source is smooth, fig. 2. If the warming laser is turned on, we obtain not steps but peaks as seen at the bottom of figs. 1 and 2. We believe these peaks reflect the neutral size distribution
Ionization
Size Distribution
0
400
Number
of Atoms
Fig. 1, Mass spectra of (Na),, clusters obtained using ionizing light near the ionization threshold (top) and well above the ionization threshold (bottom ). In both cases, the neutral cluster beam was heated with 2.54 and 2.41 eV laser light.
54
Id00
Number
2000
3000
of Atoms, n
Fig. 2. Mass spectrum of (Na), clusters using 4.0 eV ionizing light. The top spectrum shows the size distribution of cold clusters produced in the source; the bottom spectrum, after heating with 2.54 eV laser light.
CHEMICAL PHYSICS LETTERS
Volume 186, number 1
of the laser-warmed clusters. It appears that it is usually possible to correlate a falling edge of the size distribution with a step in the threshold ionization spectrum. Because of this correlation, we will characterize mass spectra obtained using excimer light by the number of atoms at steep negative slopes. A &ore extended mass spectrum of laser-warmed sodium clusters obtained with 4.0 eV ionizing photons is shown in the top of fig. 3. This spectrum has been smoothed with a spline function extending over onehundred 16 ns time channels. Notice that the structure observed does not occur at equal intervals on a scale linear in mass. In order to present this structure in a form more convenient for analysis, the data have been processed in the following way: First, the raw data are averaged with a spline function extending over 20000 time channels or about 500 Na atom masses. The result is a smooth envelope curve containing no structure. Second, the raw data are av-
0.’ Number
of Atoms,
n
Fig. 3. Mass spectrum of (Na), clusters using 4.0 eV ionizing light and (458 nm) 2.7 1 eV continuous axial warming light having an intensity of 500 mW/cm-‘. The spectrum has been smoothed over one-hundred 16 ns time channels (top). In order to emphasize the shellstructure, an envelope function (obtained by smoothing over 20000 time channels) is subtracted from a structured mass spectrum (smoothed over 1500 time channels). The difference is shown in the bottom spectrum.
I November 1991
eraged with a spline over 1500 channels or about 35 Na atom masses. Finally, the two averages are subtracted. The result is shown in the bottom of fig. 3. Five independent measurements were made under the same experimental conditions. The positions, relative heights and widths of features in the mass spectra were well reproducible.
4. Discussion The clusters in this experiment have been warmed with a continuous laser beam running parallel to the neutral cluster beam. But what is implied by “warming”? If we assume that the RRK theory of unimolecular decay can be applied to-these clusters, then it is possible to calculate the fate of a typical 500-atom cluster as it moves from the nozzle to the detector. Thus, it leaves the nozzle with the temperature of the He carrier gas ( z 100 K) traveling at a velocity of about 350 m/s. During its 1 ms flight to the ionization volume, it undergoes no further collisions but does begin to absorb photons. We do not really know the absorption cross section of this cluster at the warming laser wavelength (458 nm). However, I A’/ atom is a typical upper limit for smaller clusters. It can be expected that the cross section will be clustersize dependent. This size dependence will be reflected in the final mass distribution. The cluster absorbs the first 25 photons without evaporating any atoms, gaining an excess energy of about 70 eV and reaching a temperature of about 500 K. This all takes place in the first 450 KS.The temperature of the cluster remains rather constant for the last half of its journey to the ionization volume. It continues to absorb photons, of course, but after each absorption it evaporates 2 or 3 atoms returning to its original temperature before absorbing the next photon. It loses, on the average, a total of 80 atoms, i.e. 16% of its original mass. It appears that this repeated heating and cooling through the “critical temperature for evaporation” on this time scale favors the evolution of a size distribution with relatively strong peaks near sizes corresponding to closed electronic shells. The photon energy (4.0 eV) of the ionizing laser has been chosen so that it is well above the ionization threshold (3.0 eV) of the sodium clusters investigated. The excess energy ( 1 eV) is suf’fkient to 55
Volume186,number 1
CHEMICALPHYSICSLETTERS
cause only one atom to evaporate. This is a negligible loss on the mass scale we will be considering. For this reason, we believe that the magic numbers obtained reflect variations in the size distribution of the neutral clusters induced by the warming laser. The concept of shells can be associated with a characteristic length. Each time the radius of a cluster increases by one unit of this characteristic length, a new shell has been added. A good rough test of whether or not shell structure has been observed can IX quickly carried out by plotting the shell index as a function of the radius or n’13. If the points fall on a straight line, the data are consistent with shell formation. That this is indeed the case here can be seen in fig. 4. However, an even better tit can be obtained using two straight lines with a break between shells 13 and 14. This too can be interpreted in an interesting way. It has been suggested [ 20-221 that shell structure might periodically appear and disappear with increasing cluster size. Such a supershell structure can be understood as a beating pattern created by the interference of two nearly equal periodic contribu-
1 November 199I
tions. Quantum mechanically, the contributions can be described as arising from competing energy quantum numbers. Classically, the contributions can be described as arising from two closed electron trajectories within 3 spherical cavity. One trajectory is triangular, the other square.
5. Concluding remarks The size distribution of sodium clusters produced in our source is completely smooth. However, if the clusters are heated with a continuous laser beam, atoms boil off, resulting in a structured mass spectrum, the features of which occur at equal intervals on an n ‘I3 scale. For this reason, we associate the features with electronic shell structure in clusters containing up to 2500 electrons. Our data are consistent with (but offer no conclusive evidence for) a supershell minimum at about 800 electrons. More conclusive data on supershells are presently being obtained in the Niels Bohr Institute using a modified experimental approach [ 2 3 1.
Acknowledgement This work was supported in part by the Danish Natural Science Research Council.
References
Fig. 4. The electronic shell closing falls approximately on a straight line if plotted on an n ‘I3 scale. An even better fit is obtained using two straight lines with a break between shells I3 and 14. Such a break or “phase change” would be an indication of super-shell
56
[l] M.G. Mayer, Phys. Rev. 75 ( 1949) 1969L. [2]0. Haxel, J.H.D. Jensen and H.E. Suess, Phys. Rev. 75 (1949) 1766L. [ 31 W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou and M.L. Cohen, Phys. Rev. Letters 52 (1984) 2141. [4] M.M. Kappes, R.W. Kunz and E. Schumacher, Chem. Phys. Letters91 (1982) 413. [5] I. Katakuse, I, Ichihara, Y. Fujita, T. Matsuo, T. Sakurai and T. Matsuda, Intern. J. Mass Spectrom. Ion Processes 67 (1985) 229. (61 C. Brechignac, Ph. Cahuzac and J.-Ph. Roux, Chem. Phys. Letters 127 (I 986) 445. [7] W. Begemann, S. Dreihbfer, K.H. Meiwes-Broer and H.O. Lutz, 2. Physik D 3 (1986) 183. [S] W.A. Saunders, K. Clemenger, W.A. de Heer and W.D. Knight, Phys. Rev. B 32 (1986) 1366.
Volume 186, number 1
CHEMICAL PHYSICS LETTERS
[9]T. Bergmann, H. Limberger and T.P. Martin, Phys. Rev. Letters60 (1988) 1767. [lo] J.L. Martins and R. Car, J. Buttet, Surface Sci. 106 (1981) 265. [ 111W. Ekardt, Ber. Bunsenges. Physik. Chem. 88 ( i9g4) 289. 1121K. Clemenger, Phys. Rev. B 32 (1985) 1359. [ 131Y. Ishii, S. Ohnishi and S. Sugano, Phys. Rev. B 33 (1986) 5271. [ 141T. Bergmann and T.P. Martin, I. Chem. Phys. 90 (1989) 2848. [ 151S. Bjermholm,J. Borggreen, 0. Echt, K. Hansen, J. Pedersen and H.D. Rasmussen, Phys. Rev. Letters 65 ( 1990) 1627. [ 161H. GGhlich,T. Lange, T. Bergmann and T.P. Martin, Phys. Rev. Letters 65 (1990) 748.
1 November 1991
[ I7 ] T.P. Mar&in,T. Bergmann, H. Gijhlich and T. Lange, Chem. Phys. Letters 172 ( 1990) 209. [ 181J.L. Persson, R.L. Whetten, H.-P. Cheng and R.S. Berry, to be published. [ 191E.C. Honea, M.L. Homer, J.L. Persson and R.L. Whetten, Chem. Phys. Letters 171 (1990) 147. [ 201 R. Balian and C. Bloch, Ann. Physik 69 ( 197I ) 76. [21] A. Bohr and B.R. Mottelson, Nuclear structure (Benjamin, New York, 1975). [22] H. Nishioka, K. Hansen and B.R. Mottelson, Phys. Rev. B 42 (1990) 9377. [23] I. Pedersen, S. Bjernholm, J. Borggreen, K. Hansen, T.P. Martin and H.D. Rasmussen, to bc published.
57
CHEMICAL PHYSICS LETTERS
1 November 1991
Electronic shell structure of laser-warmed Na clusters T.P. Martin a, S. Bjramholm b, J. Borggreen b, C. Brechignac ‘, Ph. Cahuzac c, IL Hansen b and J. Pedersen b a Max-Planck-lnstitut ftir Festkiirperfrschung,
Heisenbergstrasse
I, W- 7000 Stuttgart 80, Germany
b The Niels Bohr Institute, University of Copenhagen, DK-4000 Roskilde, Denmark ’ Laboratoire AimP Cotton, CNRS II, Biitiment 505, 91405 Orsay Cedex. France
Received 19 April 1991; in final form 16 May 1991
Mass spectra are reported for large sodium clusters warmed by a continuous laser beam prior to ionization. During the 1 ms warming period, the clusters lose more than IO%of their mass by single-atom evaporation. The resulting size distribution reveals what appears to be electronic shell structure for clusters containing up to 2500 atoms.
1. Introduction The one-particle eigenstates of fermions moving in a spherically symmetric potential are characterized by a well-defined angular momentum. In large systems, the degeneracy of these so-called subshells is considerable, 2( 21+ 1). For certain forms of the radial dependence of the potential, there can be a further condensation of subshells into highly degenerate shells. However, there exist only a few known cases for which this shell degeneracy is exact, e.g. the hydrogen atom and the spherical harmonic oscillator. More often, shells appear as an approximate grouping of subshells on an energy scale, as in atomic nuclei. In fact, the earliest model developed for nuclei [ 1,2] describes very nicely the electronic structure of metal clusters [ 3- 191 containing hundreds of electrons. In this paper, we present evidence for electronic shell formation in sodium clusters containing up to 2500 valence electrons.
2. Experimental The technique we have used to study shell structure in metal clusters is photoionization time-of-flight (TOF) mass spectrometry. The mass spectrometer has a mass range of 600000 amu and a mass reso-
lution of up to 20000. The cluster source is a lowpressure, rare-gas, condensation cell. Sodium vapor was quenched in cold He gas having a pressure of about 1 mbar. Clusters condensed out of the quenched vapor were transported by the gas stream through a nozzle and through two chambers of intermediate pressure into a high-vacuum chamber. The size distribution of the clusters could be controlled by varying the oven-to-nozzle distance, the He gas pressure, and the oven temperature. The clusters were photoionized with a 0.4 t.tJ, 2x 1 mm, 15 ns, 308 nm (4.0 eV) excimer laser pulse. The wellresolved mass spectra showed the sodium clusters to be free of oxygen and other impurities. The clusters were warmed prior to ionization with a continuous Ar ion laser beam running parallel to the neutral cluster beam. The laser light entered the ionization chamber through a heated window, passed through the ionization volume, through 2.2 and 3.0 mm diameter skimmer apertures, through a 3.0 mm diameter nozzle, through the oven chamber and finally excited through a second window where the laser intensity was recorded. Short-wavelength light was found to warm much more efficiently. Using the 458 nm (2.71 eV) laser line, we found 10 mW sufficient to alter appreciably the neutral size distribution.
0009-2614/91/S 03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
53
Volume 186, number 1
CHEMICAL PHYSICS LETTERS
3. Results Electronic shell structure can be observed experimentally in several ways. Perhaps the most easily understood method is to (i) measure the ionization potentials for each cluster size. Electrons in newly opened shells are less tightly bound, i.e. have lower ionization energies. However, considerable experimental effort is required to measure the ionization energy of even a single cluster. A complete photoionization spectrum must be obtained and very often an appropriate source of tunable light is simply not available. It is much easier to observe shell closings in photoionization TOF mass spectra. This can be done in either one of two ways. (ii) If mass spectra are recorded using light with energy near threshold ionization, a shell is announced as an abrupt increase in cluster ion intensity. (iii) If mass spectra are recorded with light well above threshold, the neutral size distribution can be sampled. Here it is generally found [ 3,151 that a shell is announced by a decreased ion intensity. Fig. 1 shows mass spectra of laser-warmed Na
800- Threshold
1 November 1991
clusters obtained with two different wavelengths of the ionizing light, i.e. by using methods (ii) and (iii). Photons having an energy of 3.1 eV are very close to the size-dependent ionization threshold of the sodium clusters investigated. The corresponding mass spectrum is characterized by steps. For example, the step at about (Na)rO occurs because smaller cluster can hardly be ionized with 3.1 eV photons, but larger clusters, with a lower threshold energy, allow themselves to be ionized. The steps in the mass spectra obtained using threshold ionization reflect sudden changes in the ionization energy of clusters as a function of size. These features are relatively sharp and easily interpreted. The mass spectra obtained with ionizing photons having energy well above threshold are quite different from the spectrum just discussed. Without the warming laser, the mass spectra are without structure, i.e. the size distribution of the cold clusters emerging from our source is smooth, fig. 2. If the warming laser is turned on, we obtain not steps but peaks as seen at the bottom of figs. 1 and 2. We believe these peaks reflect the neutral size distribution
Ionization
Size Distribution
0
400
Number
of Atoms
Fig. 1, Mass spectra of (Na),, clusters obtained using ionizing light near the ionization threshold (top) and well above the ionization threshold (bottom ). In both cases, the neutral cluster beam was heated with 2.54 and 2.41 eV laser light.
54
Id00
Number
2000
3000
of Atoms, n
Fig. 2. Mass spectrum of (Na), clusters using 4.0 eV ionizing light. The top spectrum shows the size distribution of cold clusters produced in the source; the bottom spectrum, after heating with 2.54 eV laser light.
CHEMICAL PHYSICS LETTERS
Volume 186, number 1
of the laser-warmed clusters. It appears that it is usually possible to correlate a falling edge of the size distribution with a step in the threshold ionization spectrum. Because of this correlation, we will characterize mass spectra obtained using excimer light by the number of atoms at steep negative slopes. A &ore extended mass spectrum of laser-warmed sodium clusters obtained with 4.0 eV ionizing photons is shown in the top of fig. 3. This spectrum has been smoothed with a spline function extending over onehundred 16 ns time channels. Notice that the structure observed does not occur at equal intervals on a scale linear in mass. In order to present this structure in a form more convenient for analysis, the data have been processed in the following way: First, the raw data are averaged with a spline function extending over 20000 time channels or about 500 Na atom masses. The result is a smooth envelope curve containing no structure. Second, the raw data are av-
0.’ Number
of Atoms,
n
Fig. 3. Mass spectrum of (Na), clusters using 4.0 eV ionizing light and (458 nm) 2.7 1 eV continuous axial warming light having an intensity of 500 mW/cm-‘. The spectrum has been smoothed over one-hundred 16 ns time channels (top). In order to emphasize the shellstructure, an envelope function (obtained by smoothing over 20000 time channels) is subtracted from a structured mass spectrum (smoothed over 1500 time channels). The difference is shown in the bottom spectrum.
I November 1991
eraged with a spline over 1500 channels or about 35 Na atom masses. Finally, the two averages are subtracted. The result is shown in the bottom of fig. 3. Five independent measurements were made under the same experimental conditions. The positions, relative heights and widths of features in the mass spectra were well reproducible.
4. Discussion The clusters in this experiment have been warmed with a continuous laser beam running parallel to the neutral cluster beam. But what is implied by “warming”? If we assume that the RRK theory of unimolecular decay can be applied to-these clusters, then it is possible to calculate the fate of a typical 500-atom cluster as it moves from the nozzle to the detector. Thus, it leaves the nozzle with the temperature of the He carrier gas ( z 100 K) traveling at a velocity of about 350 m/s. During its 1 ms flight to the ionization volume, it undergoes no further collisions but does begin to absorb photons. We do not really know the absorption cross section of this cluster at the warming laser wavelength (458 nm). However, I A’/ atom is a typical upper limit for smaller clusters. It can be expected that the cross section will be clustersize dependent. This size dependence will be reflected in the final mass distribution. The cluster absorbs the first 25 photons without evaporating any atoms, gaining an excess energy of about 70 eV and reaching a temperature of about 500 K. This all takes place in the first 450 KS.The temperature of the cluster remains rather constant for the last half of its journey to the ionization volume. It continues to absorb photons, of course, but after each absorption it evaporates 2 or 3 atoms returning to its original temperature before absorbing the next photon. It loses, on the average, a total of 80 atoms, i.e. 16% of its original mass. It appears that this repeated heating and cooling through the “critical temperature for evaporation” on this time scale favors the evolution of a size distribution with relatively strong peaks near sizes corresponding to closed electronic shells. The photon energy (4.0 eV) of the ionizing laser has been chosen so that it is well above the ionization threshold (3.0 eV) of the sodium clusters investigated. The excess energy ( 1 eV) is suf’fkient to 55
Volume186,number 1
CHEMICALPHYSICSLETTERS
cause only one atom to evaporate. This is a negligible loss on the mass scale we will be considering. For this reason, we believe that the magic numbers obtained reflect variations in the size distribution of the neutral clusters induced by the warming laser. The concept of shells can be associated with a characteristic length. Each time the radius of a cluster increases by one unit of this characteristic length, a new shell has been added. A good rough test of whether or not shell structure has been observed can IX quickly carried out by plotting the shell index as a function of the radius or n’13. If the points fall on a straight line, the data are consistent with shell formation. That this is indeed the case here can be seen in fig. 4. However, an even better tit can be obtained using two straight lines with a break between shells 13 and 14. This too can be interpreted in an interesting way. It has been suggested [ 20-221 that shell structure might periodically appear and disappear with increasing cluster size. Such a supershell structure can be understood as a beating pattern created by the interference of two nearly equal periodic contribu-
1 November 199I
tions. Quantum mechanically, the contributions can be described as arising from competing energy quantum numbers. Classically, the contributions can be described as arising from two closed electron trajectories within 3 spherical cavity. One trajectory is triangular, the other square.
5. Concluding remarks The size distribution of sodium clusters produced in our source is completely smooth. However, if the clusters are heated with a continuous laser beam, atoms boil off, resulting in a structured mass spectrum, the features of which occur at equal intervals on an n ‘I3 scale. For this reason, we associate the features with electronic shell structure in clusters containing up to 2500 electrons. Our data are consistent with (but offer no conclusive evidence for) a supershell minimum at about 800 electrons. More conclusive data on supershells are presently being obtained in the Niels Bohr Institute using a modified experimental approach [ 2 3 1.
Acknowledgement This work was supported in part by the Danish Natural Science Research Council.
References
Fig. 4. The electronic shell closing falls approximately on a straight line if plotted on an n ‘I3 scale. An even better fit is obtained using two straight lines with a break between shells I3 and 14. Such a break or “phase change” would be an indication of super-shell
56
[l] M.G. Mayer, Phys. Rev. 75 ( 1949) 1969L. [2]0. Haxel, J.H.D. Jensen and H.E. Suess, Phys. Rev. 75 (1949) 1766L. [ 31 W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou and M.L. Cohen, Phys. Rev. Letters 52 (1984) 2141. [4] M.M. Kappes, R.W. Kunz and E. Schumacher, Chem. Phys. Letters91 (1982) 413. [5] I. Katakuse, I, Ichihara, Y. Fujita, T. Matsuo, T. Sakurai and T. Matsuda, Intern. J. Mass Spectrom. Ion Processes 67 (1985) 229. (61 C. Brechignac, Ph. Cahuzac and J.-Ph. Roux, Chem. Phys. Letters 127 (I 986) 445. [7] W. Begemann, S. Dreihbfer, K.H. Meiwes-Broer and H.O. Lutz, 2. Physik D 3 (1986) 183. [S] W.A. Saunders, K. Clemenger, W.A. de Heer and W.D. Knight, Phys. Rev. B 32 (1986) 1366.
Volume 186, number 1
CHEMICAL PHYSICS LETTERS
[9]T. Bergmann, H. Limberger and T.P. Martin, Phys. Rev. Letters60 (1988) 1767. [lo] J.L. Martins and R. Car, J. Buttet, Surface Sci. 106 (1981) 265. [ 111W. Ekardt, Ber. Bunsenges. Physik. Chem. 88 ( i9g4) 289. 1121K. Clemenger, Phys. Rev. B 32 (1985) 1359. [ 131Y. Ishii, S. Ohnishi and S. Sugano, Phys. Rev. B 33 (1986) 5271. [ 141T. Bergmann and T.P. Martin, I. Chem. Phys. 90 (1989) 2848. [ 151S. Bjermholm,J. Borggreen, 0. Echt, K. Hansen, J. Pedersen and H.D. Rasmussen, Phys. Rev. Letters 65 ( 1990) 1627. [ 161H. GGhlich,T. Lange, T. Bergmann and T.P. Martin, Phys. Rev. Letters 65 (1990) 748.
1 November 1991
[ I7 ] T.P. Mar&in,T. Bergmann, H. Gijhlich and T. Lange, Chem. Phys. Letters 172 ( 1990) 209. [ 181J.L. Persson, R.L. Whetten, H.-P. Cheng and R.S. Berry, to be published. [ 191E.C. Honea, M.L. Homer, J.L. Persson and R.L. Whetten, Chem. Phys. Letters 171 (1990) 147. [ 201 R. Balian and C. Bloch, Ann. Physik 69 ( 197I ) 76. [21] A. Bohr and B.R. Mottelson, Nuclear structure (Benjamin, New York, 1975). [22] H. Nishioka, K. Hansen and B.R. Mottelson, Phys. Rev. B 42 (1990) 9377. [23] I. Pedersen, S. Bjernholm, J. Borggreen, K. Hansen, T.P. Martin and H.D. Rasmussen, to bc published.
57