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Collective excitation in closed-shell potassium cluster ions
Volume 164, number 4
CHEMICAL PHYSICS LETTERS
15 December 1989
COLLECTIVE EXCITATION IN CLOSED-SHELL POTASSIUM CLUSTER IONS C. BRECHIGNAC, Ph. CAHUZAC, F. CARLIER and J. LEYGNIER LaboratoireAim6
Cotton, B&iment 505, Campur &Orsay, 91405 Orsay Cedex, France
Received 19 July 1989; in final form 15 September I989
Photoabsorption spectra of mass-selected potassium cluster ions Kg’ and K;I were obtained by photoevaporation spectroscopy in the range 0.6 to 4.7 eV. A giant resonance due to the collective dipole oscillation of the valence electrons dominates the photoabsorption spectra. For these spherical clusters, experimental evidence for the Lorentzian shape ofthe resonance is obtained. A blue-shift and line narrowing are observed for K:I. The dependence of damping on cluster size is discussed.
1. Introduction The photoexcitation spectroscopy of small metal particles (diameters in the range 2- 150 nm) has been investigated for a considerable time either on supported clusters [ l-51 or on free particles [ 61. The Mie solution [ 71 for scattering and absorption of light by a single homogeneous sphere provides a good description of the optical properties of isolated spherical particles. An extended Mie theory for both prolate and oblate ellipsoids of revolution has been derived by Gans [ 8 ] and applied to small prolate spheroids of silver embedded in gelatin [ 11. For very small clusters, quantum size effects are predicted. The absorption spectra should exhibit discrete structure and the familiar Mie sphere resonance is expected to be considerably broadened as a consequence of Landau damping in the discrete system [ 9 1. Up to now there have been very few studies on the photoabsorption of metal microclusters (less than 1 nm in diameter). The absorption spectra of trimers show the presence of electronic states [ 10, 1 1 ] with well-resolved vibrational progressions [ 121 which can be understood by the conventional molecular picture. On the other hand, recent photoabsorption cross sections of small neutral sodium clusters Na, (4 < n < 20) have been obtained by Knight’s group and the strong dependence of the cross section on wavelength and cluster size was interpreted as a surface plasma resonance [ 131. However, no complete photoabsorp-
tion curve has so far been experimentally determined for small free clusters and the detailed shapes and widths of the resonances are still rather uncertain. In particular, it is an open question as to how important the contribution to the linewidth due to the direct plasmon decay is. Moreover the main problem of the giant resonance in small systems either in nuclear physics [ 14,151 or in atomic cluster physics [ 9,161 is to explain their nature and find out the principal factors which determine their positions and widths. Two different approaches are used in theoretical investigations of the giant multipole resonances. The single particle approach in which the giant resonance is a coherent superposition of a definite number of particle-like and hole-like excitations and the collective approach involving collective electron motion. In this respect obtaining the photoexcitation spectra of cluster ions is of considerable interest. The collective effects found in neutral sodium clusters should also exist in cluster ions and the question as to the extend the symmetry of the electron gas governs the symmetry of jellium cluster may be answered. This paper presents photoexcitation spectra of mass-selected K,+ and K& using photoevaporation spectroscopy in an extended photon energy domain from 0.6 to 4.7 eV. In this energy range a single giant resonance dominates the photoabsorption spectra. Its Lorentzian shape is analyzed in terms of collective excitation of the valence electrons. Since the res433
Volume 164, number 4
CHEMICAL PHYSICS LETTERS
15 December 1989
onance profile is related to the cluster geometry [ I ] the spherical average shape of Kg and K2: with respectively 8 and 20 electrons is clearly observed. A blue-shift and narrowing as the cluster size increases is also demonstrated.
2. Experimental The experimental apparatus has been described elsewhere [ 171. An unseeded adiabatic expansion through a 0.1 mm diameter conical nozzle produces neutral potassium clusters which are ionized by a nitrogen laser focused between the first two plates of an ion-focusing and acceleration system. After acceleration and focusing the ions enter a drift region where they spatially resolve into individual mass packets. The ion packets pass between two electricfield plates which deflect all ions from the beam path. When the packet of interest arrives at the deflection plates, the field momentarily pulses off to allow the selected packet to proceed further. The mass-selected cluster ions KT enter a decelerating-accelerating grid system where they are excited by a second pulsed laser beam crossing the cluster beam at right angles. As shown previously, increases in the internal energy of the cluster ion are rapidly redistributed among the numerous vibrational modes of the cluster leading to sequential evaporative cooling [ 181. After the residence time in the exciting region ( z 500 ns), the overall result is K,+ +hv+Ki_,+pK where p is roughly proportional to the photon energy [ 171. Parent and fragment ions are then accelerated before entering a second drift region which again spatially disperses parent and fragments prior to their detection by an electron multiplier. A typical fragmentation pattern of KA at hv=2 eV is given in fig. 1. It should be mentioned that the efficiency of the photoevaporation depends critically on how well the laser beam matches the selected cluster packets. In our case the selected ion packets do not totally overlap the laser beam and only part of the selected parent ions is covered by the laser. At a given photon energy the ratio J=Zr/(I,+Ir) is measured versus the laser fluence. IF and ZPare the integrals of the ion signal of the fragment and parent, respectively. Experimental data are fitted by the expression S=A [ 1 -exp( -@r) ] where A corresponds to the 434
Fig. 1. Photofragmentation spectrum of K:, at h v = 2 eV. Upper trace: low laser fluence. Lower trace: high laser fluence.
proportion of selected parent ions which interact with the laser, u is the photoevaporation cross section and @r is the number of photons per square centimeter per pulse. For each cluster size and each photon energy, two parameters .4 and u are deduced from the best fit as shown in fig. 2.
3. Results and discussion Cross sections have been measured for K$ and KZ: at photon energies from 0.6 to 4.7 eV. The photon frequencies have been obtained either via a tunable dye laser pumped by Nd-YAG or by the pump laser frequencies shifted by the stimulated Raman effect in a hydrogen cell. For both clusters the spec-
Volume 164, number 4
CHEMICAL PHYSICS LETTERS
15 December 1989
ing the restoring force is the well-known plasma excitation [ 19 1. For a spherical particle with a radius smaller than the photon wavelength the photoabsorption cross section is given by ( I), where
Fig. 2. Photofragmentation ratio 6=1,/(&f&) of K& at hv=2.04 eV versus laser flux density. Crosses represent experimental measurements which are well fitted by the expression A [ 1 -exp( -uF~‘,, ] (solid line). pL is the mean laser flux density. a is proportional to the cross section:a=u/hufS. f is the laser repetition rate, S is the laser beam surface ofthe interacting region, hv is the photon energy, o the photoabsorption cross section.
tra are dominated by a single giant resonance which is well fitted by a Lorentzian (hu)2r2 (T(hV)=umax [(hu)2-(hvo)2]2+(hv)2~2’
(I)
where ummax is the maximum cross section, hu, the resonance energy and r the width of the resonance. The fitted parameters are K,f:
o,,,,,,=26A2,
hvo=1.93eV,
f =0.22 eV , K&: 0,,,,,,=88i2,
hvo=1.98eV,
rzO.16 eV. They show an increase of the maximum cross section, a blue-shift, and a narrowing of the K$ resonance compared to the Kc resonance. Two different approaches can be used to interpret these results. From a classical point of view, it is customary to assume that the physical properties of small metallic particles can be predicted by a Drude model [ 191. In this respect the giant resonance is considered to be due to the collective excitation of the valence electrons. The displacement by an electromagnetic field of the electron gas, as a whole, with respect to the fixed positive background of the ions provid-
umax=4rcNe2fm,cr,
(2)
v. = (1/2x) (Ne2/m,a)‘!2,
(3)
N is the number of electrons, cx the static polarizability [20]. For ellipsoidal particles, at most three frequencies can be observed depending on the pplarizabilities along the three axes [ 1,8,13], and the photoabsorption must present two or three resonances. Recent results on free sodium neutral clusters support the prediction of a spherical shape for Na, and an ellipsoidal one for Nas [ 131. In our case the observation of a single resonance for K,+ and K$ demonstrates the spherical shape of these clusters. Since they respectively have an 8- and 20-electron closed shell we can conclude that the symmetry of the cluster is governed by the symmetry of the electron gas. From eq. (2) we calculated the maximum cross section assuming the experimental values r$& = 25 A2 for Kt and a& = 87 .&*for K&. This shows very good agreement between the experimental and the calculated values, verifying the sum rule. The resonance frequencies can be evaluated from eq. (3). Since the mean value of the polarizability decreases as the cluster size increases: a blue-shift is expected for the single resonance as the cluster size increases. Our data support this behavior. However, polarizabilities have not yet been measured for cluster ions; only polarizabilities of neutral clusters are available [ 2 1] _The calculated resonance energies for Kg and K10 are respectively hvo( Ks) = 1.97 eV and hvo(K2,,)=2.04 eV which are close to, but higher than, the experimental values. Moreover for positively charged clusters having the same number of electrons, K,+ and K$ ,the number of ions exceeds by one unit the number of electrons and the restoring force is increased. The corresponding resonance energies must be higher for charged clusters than for the neutral. The collective plasma excitation of the cluster based on the idea that the collective motion of the electrons against the ions plays the dominant role in the formation of giant resonances correctly predicts the positions of these resonances as well as the rel435
Volume 164, number 4
(r
CHEMICAL PHYSICS LETTERS
(AZ) 25 _
20 _
154
7-
d 60 50 40 301
K+
-I
21
1
tIi i 7
2
3
4 hi,
W)
Fig.3. Photoevaporation
cross section for Kg’ and K&.
atively large photoevaporation cross sections. The resonance widths are fundamental in understanding the decay of the collective excitation. In the classical size effect theory [ 91 the width r of the resonance is given by r=r,
+iiV,/R
, in eV ,
(4)
where rs is the plasmon width of the bulk, VFis the Fermi velocity and R the sphere radius. The first term r, is the broadening due to the scattering of electrons in the bulk. The second term is the broadening due to surface cluster scattering. The electrons easily lose energy by bouncing inelastically from the cluster surface. For clusters comprising less than 1000 atoms, the surface scattering rate considerably exceeds the bulk scattering rate. For potassium the bulk value is r, = 0.1 eV whereas the calculated total broadening is 1.1 and 0.9 eV for Kg’ and K2:, respectively. These values are in total disagreement with our experimental plasmon widths for free clusters. It has been 436
I5 December I989
noted that if the classical law (4) turns out to be incapable of reproducing the observed resonance widths of small free clusters, it is well adapted to reproducing the observed resonance widths of supported clusters [ 2,4]. One may ask if the size-dependent broadening of the plasma resonance in supported clusters is due to intrinsic broadening of the surface scattering or to the coupling between cluster and substrate. From our results it is clear that the scattering of the electrons off the surface does not contribute significantly to the broadening effect. In a second approach the giant resonance can be calculated in the random phase approximation (RPA) [ 221 as a coherent superposition of a detinite number of particle-hole excitations. The coefficients of this superposition are determined by the interaction in the particle-hole channel. The position of the giant resonance is determined by the transition probability spread over an energy range. The RPA approach has proven to be extremely useful in describing giant resonances in nuclei [ 141. It has been used recently to calculate the excitation spectra of small metal clusters [ 23-25 ] _ In this model the width of the resonance is given by the coupling between the particle-hole excitations and the vibration of the ion cores. The width turns out to be much smaller than the width given by eq. (4) and in better agreement with the experimental results. In this respect the narrowing of the K& resonance as compared to the&! resonance seems to indicate stronger coupling for small clusters than for larger ones. However, the temperature of the cluster may also be a source of broadening. In our case the temperature of K$ is less than the temperature of Kg’ [ 18 1.
References [ I] D.C. Skillman and CR. Berry, J. Chem. Phys. 48 ( 1968) 3297. [2] L. Genzel, T.P. Martin and U. Kreibig, Z. Physik B 21 (1975) 339. [ 31 Y. Borensztein, P. de Andres, R. Monreal, T. Lopez-Rios and F. Flares, Phys. Rev. B 33 (1986) 2828. [ 41 M. Acheche, C. Colliex and P. Trebbia, Scanning Electron Microscopy 1 (I 986) 25. [ 51 W. Hoheisel, K. Jungmann, M. Wollmer, R. Weidenauer and F. Trager, Phys. Rev. Letters 60 ( 1988) 1649. [ 61 D.M. Mann and H.P. Broida, J. Appl. Phys. 44 ( 1973) 4950; J.D. Eversole and H.P. Broida, Phys. Rev. B 15 ( 1977) 1644.
Volume 164,number 4
CHEMICALPHYSICSLETTERS
[ 71Cl.Mie, Ann. Physik 25 (1908) 377. [8] R. Gans, Ann. Physik 47 (1915) 270. [9] D.M. Wood and N.W. Ashcroft, Phys. Rev. B 25 (1982) 6255. [ 101A. Herrmann, S. Leutwyler,E. Schumacherand L. Waste, Helv. Chim. Acta 61 (1978) 453. [ 1llM.D. Morse, J.B. Hopkins, P.R.R. Langridge-Smith and R.E. Smalley,I. Chem. Phys. 79 (1983) 5316. [ 121Cl. Delacretaz, E.R. Grant, R. Whetten, L. Waste and J. Zwanziger,Phys. Rev. Letters 56 (1986) 2598; M. Brayer, G. Delacretaz, P. Labastie, J.P. Wolf and L. Waste, Phys. Rev. Letters 57 (1986) 1851. [ 13] W. de Heer, K Selby, V. Kresin, I. Masui, M. Vollmer, A. Chatelain and W.D. Knight, Phys. Rev. Letters 59 ( 1987) 1805; K. Selby,M. Vollmer, J. Masui, V. Kresin, W. de Heer and W.D. Knight, Proceedings of the 4th International Symposiumon Small Particles and Inorganic Clusters, cds. C. Chapon, M.F. Gilet and C. Henry, Aixen Provence (July 1988) Z. Physik D, to be published.
15 December I989
[ 141G.F. Bertsch, P.F. Bortignon and R.A. Broglia,Rev. Mod. Phys. 55 (1983) 287. [ 151G.F. Fdipov, V.S. Vasilevskii, S.P. Kruchinin and L.L. Choposkii, Soviet J. Nucl. Phys. 43 (1986) 536. [ 161J.H. Parksand Sk McDonald,Phys.Rev.Letters62 (1989) 2301. [ 171C. Bmchignac,Ph. Cahuzacand J.Ph. Roux, J. Chem. Phys. 88 (1988) 3022. [ 181C. Brechignac,Ph. Cahuzac, J. Leygnier and J. Weiner, J. Chem. Phys. 90 ( 1989) 1492. [ 19] N.W.Ashcroft and N.D. Me&n, Solidstate physics (Holt, Rinehart and Winston, New York, 1976). [20] W. Ekardt, Phys. Rev. B 31 (1985) 6360. [ 211W.D. Knight, K. Clemenger,W.A.de Heer and W. Sanders, Phys. Rev. B 31 (1985) 2539. [22] 0. Bohigas, A.N. Lane and J. Martorell, Phys. Rept. 51 (1979)267. [ 231V. Kresin, Phys. Rev. B 38 ( 1988) 3741. [ 241M. Brack,Phys. Rev. B 39 (1989) 3533. [ 251C. Yannouleous,M. Brack, R.A. Brogliaand P.F. Bortigon, Phys. Rev. Letters, to be published.
437
CHEMICAL PHYSICS LETTERS
15 December 1989
COLLECTIVE EXCITATION IN CLOSED-SHELL POTASSIUM CLUSTER IONS C. BRECHIGNAC, Ph. CAHUZAC, F. CARLIER and J. LEYGNIER LaboratoireAim6
Cotton, B&iment 505, Campur &Orsay, 91405 Orsay Cedex, France
Received 19 July 1989; in final form 15 September I989
Photoabsorption spectra of mass-selected potassium cluster ions Kg’ and K;I were obtained by photoevaporation spectroscopy in the range 0.6 to 4.7 eV. A giant resonance due to the collective dipole oscillation of the valence electrons dominates the photoabsorption spectra. For these spherical clusters, experimental evidence for the Lorentzian shape ofthe resonance is obtained. A blue-shift and line narrowing are observed for K:I. The dependence of damping on cluster size is discussed.
1. Introduction The photoexcitation spectroscopy of small metal particles (diameters in the range 2- 150 nm) has been investigated for a considerable time either on supported clusters [ l-51 or on free particles [ 61. The Mie solution [ 71 for scattering and absorption of light by a single homogeneous sphere provides a good description of the optical properties of isolated spherical particles. An extended Mie theory for both prolate and oblate ellipsoids of revolution has been derived by Gans [ 8 ] and applied to small prolate spheroids of silver embedded in gelatin [ 11. For very small clusters, quantum size effects are predicted. The absorption spectra should exhibit discrete structure and the familiar Mie sphere resonance is expected to be considerably broadened as a consequence of Landau damping in the discrete system [ 9 1. Up to now there have been very few studies on the photoabsorption of metal microclusters (less than 1 nm in diameter). The absorption spectra of trimers show the presence of electronic states [ 10, 1 1 ] with well-resolved vibrational progressions [ 121 which can be understood by the conventional molecular picture. On the other hand, recent photoabsorption cross sections of small neutral sodium clusters Na, (4 < n < 20) have been obtained by Knight’s group and the strong dependence of the cross section on wavelength and cluster size was interpreted as a surface plasma resonance [ 131. However, no complete photoabsorp-
tion curve has so far been experimentally determined for small free clusters and the detailed shapes and widths of the resonances are still rather uncertain. In particular, it is an open question as to how important the contribution to the linewidth due to the direct plasmon decay is. Moreover the main problem of the giant resonance in small systems either in nuclear physics [ 14,151 or in atomic cluster physics [ 9,161 is to explain their nature and find out the principal factors which determine their positions and widths. Two different approaches are used in theoretical investigations of the giant multipole resonances. The single particle approach in which the giant resonance is a coherent superposition of a definite number of particle-like and hole-like excitations and the collective approach involving collective electron motion. In this respect obtaining the photoexcitation spectra of cluster ions is of considerable interest. The collective effects found in neutral sodium clusters should also exist in cluster ions and the question as to the extend the symmetry of the electron gas governs the symmetry of jellium cluster may be answered. This paper presents photoexcitation spectra of mass-selected K,+ and K& using photoevaporation spectroscopy in an extended photon energy domain from 0.6 to 4.7 eV. In this energy range a single giant resonance dominates the photoabsorption spectra. Its Lorentzian shape is analyzed in terms of collective excitation of the valence electrons. Since the res433
Volume 164, number 4
CHEMICAL PHYSICS LETTERS
15 December 1989
onance profile is related to the cluster geometry [ I ] the spherical average shape of Kg and K2: with respectively 8 and 20 electrons is clearly observed. A blue-shift and narrowing as the cluster size increases is also demonstrated.
2. Experimental The experimental apparatus has been described elsewhere [ 171. An unseeded adiabatic expansion through a 0.1 mm diameter conical nozzle produces neutral potassium clusters which are ionized by a nitrogen laser focused between the first two plates of an ion-focusing and acceleration system. After acceleration and focusing the ions enter a drift region where they spatially resolve into individual mass packets. The ion packets pass between two electricfield plates which deflect all ions from the beam path. When the packet of interest arrives at the deflection plates, the field momentarily pulses off to allow the selected packet to proceed further. The mass-selected cluster ions KT enter a decelerating-accelerating grid system where they are excited by a second pulsed laser beam crossing the cluster beam at right angles. As shown previously, increases in the internal energy of the cluster ion are rapidly redistributed among the numerous vibrational modes of the cluster leading to sequential evaporative cooling [ 181. After the residence time in the exciting region ( z 500 ns), the overall result is K,+ +hv+Ki_,+pK where p is roughly proportional to the photon energy [ 171. Parent and fragment ions are then accelerated before entering a second drift region which again spatially disperses parent and fragments prior to their detection by an electron multiplier. A typical fragmentation pattern of KA at hv=2 eV is given in fig. 1. It should be mentioned that the efficiency of the photoevaporation depends critically on how well the laser beam matches the selected cluster packets. In our case the selected ion packets do not totally overlap the laser beam and only part of the selected parent ions is covered by the laser. At a given photon energy the ratio J=Zr/(I,+Ir) is measured versus the laser fluence. IF and ZPare the integrals of the ion signal of the fragment and parent, respectively. Experimental data are fitted by the expression S=A [ 1 -exp( -@r) ] where A corresponds to the 434
Fig. 1. Photofragmentation spectrum of K:, at h v = 2 eV. Upper trace: low laser fluence. Lower trace: high laser fluence.
proportion of selected parent ions which interact with the laser, u is the photoevaporation cross section and @r is the number of photons per square centimeter per pulse. For each cluster size and each photon energy, two parameters .4 and u are deduced from the best fit as shown in fig. 2.
3. Results and discussion Cross sections have been measured for K$ and KZ: at photon energies from 0.6 to 4.7 eV. The photon frequencies have been obtained either via a tunable dye laser pumped by Nd-YAG or by the pump laser frequencies shifted by the stimulated Raman effect in a hydrogen cell. For both clusters the spec-
Volume 164, number 4
CHEMICAL PHYSICS LETTERS
15 December 1989
ing the restoring force is the well-known plasma excitation [ 19 1. For a spherical particle with a radius smaller than the photon wavelength the photoabsorption cross section is given by ( I), where
Fig. 2. Photofragmentation ratio 6=1,/(&f&) of K& at hv=2.04 eV versus laser flux density. Crosses represent experimental measurements which are well fitted by the expression A [ 1 -exp( -uF~‘,, ] (solid line). pL is the mean laser flux density. a is proportional to the cross section:a=u/hufS. f is the laser repetition rate, S is the laser beam surface ofthe interacting region, hv is the photon energy, o the photoabsorption cross section.
tra are dominated by a single giant resonance which is well fitted by a Lorentzian (hu)2r2 (T(hV)=umax [(hu)2-(hvo)2]2+(hv)2~2’
(I)
where ummax is the maximum cross section, hu, the resonance energy and r the width of the resonance. The fitted parameters are K,f:
o,,,,,,=26A2,
hvo=1.93eV,
f =0.22 eV , K&: 0,,,,,,=88i2,
hvo=1.98eV,
rzO.16 eV. They show an increase of the maximum cross section, a blue-shift, and a narrowing of the K$ resonance compared to the Kc resonance. Two different approaches can be used to interpret these results. From a classical point of view, it is customary to assume that the physical properties of small metallic particles can be predicted by a Drude model [ 191. In this respect the giant resonance is considered to be due to the collective excitation of the valence electrons. The displacement by an electromagnetic field of the electron gas, as a whole, with respect to the fixed positive background of the ions provid-
umax=4rcNe2fm,cr,
(2)
v. = (1/2x) (Ne2/m,a)‘!2,
(3)
N is the number of electrons, cx the static polarizability [20]. For ellipsoidal particles, at most three frequencies can be observed depending on the pplarizabilities along the three axes [ 1,8,13], and the photoabsorption must present two or three resonances. Recent results on free sodium neutral clusters support the prediction of a spherical shape for Na, and an ellipsoidal one for Nas [ 131. In our case the observation of a single resonance for K,+ and K$ demonstrates the spherical shape of these clusters. Since they respectively have an 8- and 20-electron closed shell we can conclude that the symmetry of the cluster is governed by the symmetry of the electron gas. From eq. (2) we calculated the maximum cross section assuming the experimental values r$& = 25 A2 for Kt and a& = 87 .&*for K&. This shows very good agreement between the experimental and the calculated values, verifying the sum rule. The resonance frequencies can be evaluated from eq. (3). Since the mean value of the polarizability decreases as the cluster size increases: a blue-shift is expected for the single resonance as the cluster size increases. Our data support this behavior. However, polarizabilities have not yet been measured for cluster ions; only polarizabilities of neutral clusters are available [ 2 1] _The calculated resonance energies for Kg and K10 are respectively hvo( Ks) = 1.97 eV and hvo(K2,,)=2.04 eV which are close to, but higher than, the experimental values. Moreover for positively charged clusters having the same number of electrons, K,+ and K$ ,the number of ions exceeds by one unit the number of electrons and the restoring force is increased. The corresponding resonance energies must be higher for charged clusters than for the neutral. The collective plasma excitation of the cluster based on the idea that the collective motion of the electrons against the ions plays the dominant role in the formation of giant resonances correctly predicts the positions of these resonances as well as the rel435
Volume 164, number 4
(r
CHEMICAL PHYSICS LETTERS
(AZ) 25 _
20 _
154
7-
d 60 50 40 301
K+
-I
21
1
tIi i 7
2
3
4 hi,
W)
Fig.3. Photoevaporation
cross section for Kg’ and K&.
atively large photoevaporation cross sections. The resonance widths are fundamental in understanding the decay of the collective excitation. In the classical size effect theory [ 91 the width r of the resonance is given by r=r,
+iiV,/R
, in eV ,
(4)
where rs is the plasmon width of the bulk, VFis the Fermi velocity and R the sphere radius. The first term r, is the broadening due to the scattering of electrons in the bulk. The second term is the broadening due to surface cluster scattering. The electrons easily lose energy by bouncing inelastically from the cluster surface. For clusters comprising less than 1000 atoms, the surface scattering rate considerably exceeds the bulk scattering rate. For potassium the bulk value is r, = 0.1 eV whereas the calculated total broadening is 1.1 and 0.9 eV for Kg’ and K2:, respectively. These values are in total disagreement with our experimental plasmon widths for free clusters. It has been 436
I5 December I989
noted that if the classical law (4) turns out to be incapable of reproducing the observed resonance widths of small free clusters, it is well adapted to reproducing the observed resonance widths of supported clusters [ 2,4]. One may ask if the size-dependent broadening of the plasma resonance in supported clusters is due to intrinsic broadening of the surface scattering or to the coupling between cluster and substrate. From our results it is clear that the scattering of the electrons off the surface does not contribute significantly to the broadening effect. In a second approach the giant resonance can be calculated in the random phase approximation (RPA) [ 221 as a coherent superposition of a detinite number of particle-hole excitations. The coefficients of this superposition are determined by the interaction in the particle-hole channel. The position of the giant resonance is determined by the transition probability spread over an energy range. The RPA approach has proven to be extremely useful in describing giant resonances in nuclei [ 141. It has been used recently to calculate the excitation spectra of small metal clusters [ 23-25 ] _ In this model the width of the resonance is given by the coupling between the particle-hole excitations and the vibration of the ion cores. The width turns out to be much smaller than the width given by eq. (4) and in better agreement with the experimental results. In this respect the narrowing of the K& resonance as compared to the&! resonance seems to indicate stronger coupling for small clusters than for larger ones. However, the temperature of the cluster may also be a source of broadening. In our case the temperature of K$ is less than the temperature of Kg’ [ 18 1.
References [ I] D.C. Skillman and CR. Berry, J. Chem. Phys. 48 ( 1968) 3297. [2] L. Genzel, T.P. Martin and U. Kreibig, Z. Physik B 21 (1975) 339. [ 31 Y. Borensztein, P. de Andres, R. Monreal, T. Lopez-Rios and F. Flares, Phys. Rev. B 33 (1986) 2828. [ 41 M. Acheche, C. Colliex and P. Trebbia, Scanning Electron Microscopy 1 (I 986) 25. [ 51 W. Hoheisel, K. Jungmann, M. Wollmer, R. Weidenauer and F. Trager, Phys. Rev. Letters 60 ( 1988) 1649. [ 61 D.M. Mann and H.P. Broida, J. Appl. Phys. 44 ( 1973) 4950; J.D. Eversole and H.P. Broida, Phys. Rev. B 15 ( 1977) 1644.
Volume 164,number 4
CHEMICALPHYSICSLETTERS
[ 71Cl.Mie, Ann. Physik 25 (1908) 377. [8] R. Gans, Ann. Physik 47 (1915) 270. [9] D.M. Wood and N.W. Ashcroft, Phys. Rev. B 25 (1982) 6255. [ 101A. Herrmann, S. Leutwyler,E. Schumacherand L. Waste, Helv. Chim. Acta 61 (1978) 453. [ 1llM.D. Morse, J.B. Hopkins, P.R.R. Langridge-Smith and R.E. Smalley,I. Chem. Phys. 79 (1983) 5316. [ 121Cl. Delacretaz, E.R. Grant, R. Whetten, L. Waste and J. Zwanziger,Phys. Rev. Letters 56 (1986) 2598; M. Brayer, G. Delacretaz, P. Labastie, J.P. Wolf and L. Waste, Phys. Rev. Letters 57 (1986) 1851. [ 13] W. de Heer, K Selby, V. Kresin, I. Masui, M. Vollmer, A. Chatelain and W.D. Knight, Phys. Rev. Letters 59 ( 1987) 1805; K. Selby,M. Vollmer, J. Masui, V. Kresin, W. de Heer and W.D. Knight, Proceedings of the 4th International Symposiumon Small Particles and Inorganic Clusters, cds. C. Chapon, M.F. Gilet and C. Henry, Aixen Provence (July 1988) Z. Physik D, to be published.
15 December I989
[ 141G.F. Bertsch, P.F. Bortignon and R.A. Broglia,Rev. Mod. Phys. 55 (1983) 287. [ 151G.F. Fdipov, V.S. Vasilevskii, S.P. Kruchinin and L.L. Choposkii, Soviet J. Nucl. Phys. 43 (1986) 536. [ 161J.H. Parksand Sk McDonald,Phys.Rev.Letters62 (1989) 2301. [ 171C. Bmchignac,Ph. Cahuzacand J.Ph. Roux, J. Chem. Phys. 88 (1988) 3022. [ 181C. Brechignac,Ph. Cahuzac, J. Leygnier and J. Weiner, J. Chem. Phys. 90 ( 1989) 1492. [ 19] N.W.Ashcroft and N.D. Me&n, Solidstate physics (Holt, Rinehart and Winston, New York, 1976). [20] W. Ekardt, Phys. Rev. B 31 (1985) 6360. [ 211W.D. Knight, K. Clemenger,W.A.de Heer and W. Sanders, Phys. Rev. B 31 (1985) 2539. [22] 0. Bohigas, A.N. Lane and J. Martorell, Phys. Rept. 51 (1979)267. [ 231V. Kresin, Phys. Rev. B 38 ( 1988) 3741. [ 241M. Brack,Phys. Rev. B 39 (1989) 3533. [ 251C. Yannouleous,M. Brack, R.A. Brogliaand P.F. Bortigon, Phys. Rev. Letters, to be published.
437