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Splitting of the giant dipole resonance in deformed metal clusters
Physics Letters A 180 ( 1993) 453-455 North-Holland
PHYSICS LETTERS A
Splitting of the giant dipole resonance in deformed metal clusters Yu.A. M a l o v a n d D.F. Z a r e t s k y Russian Research Centre "Kurchatov Institute" Moscow 123182, Russian Federation
Received 8 June 1993; accepted for publication 22 July 1993 Communicated by V.M. Agranovich
The photoabsorption in small deformed metal clusters is considered. The valence electron spill-out is disregarded. It is shown that in the deformed clusters the giant dipole resonance splits into two separate resonances. This splitting is due to the difference of the oscillator frequencies for the electron motion in the effective field of the deformed clusters. The ratio of the cross-sections in the resonance maxima is obtained. The theoretical results are compared with the experimental data.
1. Introduction A number of papers devoted to the problem of photoabsorption in deformed metal clusters [ 1-3] have been recently published. The splitting of a photoabsorption resonance cross-section into two maxima was observed practically in all cases. It is reasonable to assume that this splitting arises due to the cluster deformation. It is clear that the plasma oscillator resonance cannot be split by the cluster deformation if the valence electrons are concentrated inside the cluster. Therefore in ref. [ 4 ] allowance was made for the valence electron tail outside the cluster ("spill-out effect"). But the author failed to explain the value o f the resonance cross-section for the deformed clusters. In ref. [ 5 ] another interpretation o f dipole resonances in metal clusters has been proposed. It is shown that there exist two types of valence electron oscillations in small metal clusters: giant dipole resonance and surface plasmons. The giant dipole resonance arises due to the particle-hole transition between the filled shell and an empty one. Surface plasmons may be observed in the clusters with overlapping shells. The results of calculations in terms o f the finite Fermi system theory are in good agreement with the experimental data [ 5]. In this paper the theory of giant dipole resonance presented in ref. [ 5 ] is applied to the interpretation o f the cross-section m a x i m u m splitting in deformed clusters. It is shown Elsevier Science Publishers B.V.
that the value o f the splitting is proportional to the cluster deformation. The ratio of the maximum crosssection values for two resonances corresponding to spheroidal clusters is calculated.
2. Basic equations Let us consider a spheroidal cluster with axes a and b. The deformation parameter fl is defined by the relation [ 6 ] a-b
fl=2a+b,
(1)
where a > b. If the cluster volume is the same as for a spherical one then we have a b E = R 3 ( R is the radius of the spherical cluster of the same volume). Using definition (1) and supposing that the deformation is small (fl << l ) we obtain a = R ( l + ]fl) ,
(2)
b=R(1-]fl)
(3)
.
In this cluster the intershell in the direction o f cluster perpendicular direction. In the giant dipole resonance to be oha =X/~3VF/R ,
transition energies differ deformation and in the the spherical cluster case frequency is known [ 5 ] (4) 453
Volume 180, number 6
PHYSICS LETTERS A
where E F is the Fermi e n e r g y , VF= 2X//2EF/m. To obtain the resonance frequency for the spheroidal cluster it is convenient to perform a coordinate transformation, x x ' = - a= x ( 1 - ] f l ) ,
"v'=
y
=y(l+~fl)
Z
z'= ~ :z(l+I/ff ) .
H=-
h2 ~mmA+ "~mtaM I ar2
2
02
for spherical
( r 2 = x 2 + y 2 + 2 2) ,
1_ /_t 0 m 2 3m
OV 2
1
- g 2i l l
tav=taz =taM(1 + lfl) -
(6)
(7)
Because of the difference between tax and coy= co: the giant dipole resonance splits into two components. The frequency between these two components, 8taM, is proportional to the deformed parameter,
8o~ =BtaM.
a(co) = A d ( t a - t a , ) + B d ( t a - t a y ) .
(9)
The coefficients A and B may be calculated using two sum rules, 2rt2e 2
o oo f ~
mc
N,
d t a = - 2 = 2 ~ P(O, ,
(10)
(11)
o
where N is the number o f valence electrons; P ( 0 ) is 454
(12)
2 llz2~ 2
a(ta)-
-N[d(ta-o&)+2d(ta-tay)]. 3 mc
(13)
co.,.=ta:-,ta~,~-iF~,z.
As a result expression ( 13) takes the form
a(ta) -
2 rte 2 3 mc
CG × N ((o)_O)x)2..]_F2 +2 (o)_~y)2+F2-) (14) It is necessary to point out that the cross-section values in the maxima do not depend in first approximation on the deformation parameter ft.
(8)
An expression similar to (8) was also obtained in the case of the nuclear photoeffect [ 7 ]. Let us now define the photoabsorption crosssection for the deformed clusters. For the resonance width equal to zero, according to (7) we may write down
i a(ta) do)=
10 EF
tax ~tax - iE,.,
02
3mflOz 2
3 NR z
Actually the particle-hole excitations have a finite relaxation width, due to their interaction with more complex states (for example, two particles-two holes states). Therefore we have to introduce complex frequencies,
should be added. Now the transition frequencies for the deformed cluster can be easily defined, tax = taM ( l
P(O) =
Using (12) and substituting (9) into expressions (10) and ( 11 ) we arrive at
the term H'= ~mflOx2
the polarization operator for zero frequency (co = 0 ). The expression for the polarization operator P ( 0 ) is found to be [5]
(5)
Then to the known Hamiltonian clusters,
20 September 1993
3. Discussion of results. Conclusion It is shown that the giant dipole resonance frequency in deformed clusters splits into two components. Therefore the resonance cross-section has two maxima. The results o f photoabsorption measurements in deformed Na clusters were presented in ref. [ 1 ]. Two maxima were observed in the photoabsorption cross-section which can be ascribed to the splitting of the giant dipole resonance. The experimental values o f the first and second resonances are F x = 0.21 eV and F~.= 0.35 eV, respectively. Substituting these values into (14) we obtain the m a x i m u m values of the cross-section, 0 " ( m X ) x ( t a ) = 6 X l 0 -16 c m 2, and O'~a)x(co) = 11 X 10 -16 cm 2 for the second. These values are in good agreement with the experimental data.
Volume 180, number 6
PHYSICS LETTERS A
Recently in ref. [2 ] the p h o t o a b s o r p t i o n crosssection for N a has been investigated. It was also observed that there are two m a x i m a in the photoabsorption cross-section with widths F x = 0 . 2 eV and F y = 0 . 4 eV and the m a x i m u m cross-sections are t r x = 7 × 1 0 -16 cm 2 a n d a y = 7 . 2 × 1 0 t6 cm 2 respectively. These values are also in agreement with the experimental data. Experimental d a t a on the p h o t o a b s o r p t i o n crosssection in Ag [ 3 ] are also available. There exist two m a x i m a having widths F x = 0.39 eV and F y = 0.77 eV and the corresponding cross-sections are in a reasonable agreement with the experiment. In conclusion we would like to recall that the giant dipole resonance splitting is p r o p o r t i o n a l to the cluster deformation. Therefore when the value o f &oM for a cluster with a closed shell is known, then according to ( 8 ) the d e f o r m a t i o n p a r a m e t e r fl is
p=&oM/o~. We would also like to stress that the results ob-
20 September 1993
tained in the quasiclassical a p p r o x i m a t i o n do not d e p e n d on the particular type o f effective field inside the cluster [ 5 ].
References [ 1] K. Selby, V. Kresin, J. Masui, M. Vollmer, W.A. de Heer, A. Seheidemann and W.D. Knight, Phys. Rev. B 43 (1991) 4565. [2] C. Brechignac, Ph. Cahurac, F. Carlier, M. de Frutos and J. Leygnier, Chem. Phys. Lett. 189 (1992) 28. [3]J. Tiggesbaumker, L. Koller, H.O. Lutz and K.H. MeiwesBroer, in: Proc. 88th WE-Heraeus Seminar on Nuclear physics concepts in the study of atomic cluster physics (Springer, Berlin, 1991 ) p. 247. [4] V.V. Kresin, Phys. Rev. B 45 (1992) 14321. [ 5 ] Yu.A. Malov and D.F. Zaretsky, Phys. Lett. A 177 ( 1993 ) 379. [6]A.B. Migdal, Qualitative methods in quantum theory (Moscow, 1975). [7] A.A. Lushnikov and D.F. Zaretsky, Nucl. Phys. 66 (1965) 35.
455
PHYSICS LETTERS A
Splitting of the giant dipole resonance in deformed metal clusters Yu.A. M a l o v a n d D.F. Z a r e t s k y Russian Research Centre "Kurchatov Institute" Moscow 123182, Russian Federation
Received 8 June 1993; accepted for publication 22 July 1993 Communicated by V.M. Agranovich
The photoabsorption in small deformed metal clusters is considered. The valence electron spill-out is disregarded. It is shown that in the deformed clusters the giant dipole resonance splits into two separate resonances. This splitting is due to the difference of the oscillator frequencies for the electron motion in the effective field of the deformed clusters. The ratio of the cross-sections in the resonance maxima is obtained. The theoretical results are compared with the experimental data.
1. Introduction A number of papers devoted to the problem of photoabsorption in deformed metal clusters [ 1-3] have been recently published. The splitting of a photoabsorption resonance cross-section into two maxima was observed practically in all cases. It is reasonable to assume that this splitting arises due to the cluster deformation. It is clear that the plasma oscillator resonance cannot be split by the cluster deformation if the valence electrons are concentrated inside the cluster. Therefore in ref. [ 4 ] allowance was made for the valence electron tail outside the cluster ("spill-out effect"). But the author failed to explain the value o f the resonance cross-section for the deformed clusters. In ref. [ 5 ] another interpretation o f dipole resonances in metal clusters has been proposed. It is shown that there exist two types of valence electron oscillations in small metal clusters: giant dipole resonance and surface plasmons. The giant dipole resonance arises due to the particle-hole transition between the filled shell and an empty one. Surface plasmons may be observed in the clusters with overlapping shells. The results of calculations in terms o f the finite Fermi system theory are in good agreement with the experimental data [ 5]. In this paper the theory of giant dipole resonance presented in ref. [ 5 ] is applied to the interpretation o f the cross-section m a x i m u m splitting in deformed clusters. It is shown Elsevier Science Publishers B.V.
that the value o f the splitting is proportional to the cluster deformation. The ratio of the maximum crosssection values for two resonances corresponding to spheroidal clusters is calculated.
2. Basic equations Let us consider a spheroidal cluster with axes a and b. The deformation parameter fl is defined by the relation [ 6 ] a-b
fl=2a+b,
(1)
where a > b. If the cluster volume is the same as for a spherical one then we have a b E = R 3 ( R is the radius of the spherical cluster of the same volume). Using definition (1) and supposing that the deformation is small (fl << l ) we obtain a = R ( l + ]fl) ,
(2)
b=R(1-]fl)
(3)
.
In this cluster the intershell in the direction o f cluster perpendicular direction. In the giant dipole resonance to be oha =X/~3VF/R ,
transition energies differ deformation and in the the spherical cluster case frequency is known [ 5 ] (4) 453
Volume 180, number 6
PHYSICS LETTERS A
where E F is the Fermi e n e r g y , VF= 2X//2EF/m. To obtain the resonance frequency for the spheroidal cluster it is convenient to perform a coordinate transformation, x x ' = - a= x ( 1 - ] f l ) ,
"v'=
y
=y(l+~fl)
Z
z'= ~ :z(l+I/ff ) .
H=-
h2 ~mmA+ "~mtaM I ar2
2
02
for spherical
( r 2 = x 2 + y 2 + 2 2) ,
1_ /_t 0 m 2 3m
OV 2
1
- g 2i l l
tav=taz =taM(1 + lfl) -
(6)
(7)
Because of the difference between tax and coy= co: the giant dipole resonance splits into two components. The frequency between these two components, 8taM, is proportional to the deformed parameter,
8o~ =BtaM.
a(co) = A d ( t a - t a , ) + B d ( t a - t a y ) .
(9)
The coefficients A and B may be calculated using two sum rules, 2rt2e 2
o oo f ~
mc
N,
d t a = - 2 = 2 ~ P(O, ,
(10)
(11)
o
where N is the number o f valence electrons; P ( 0 ) is 454
(12)
2 llz2~ 2
a(ta)-
-N[d(ta-o&)+2d(ta-tay)]. 3 mc
(13)
co.,.=ta:-,ta~,~-iF~,z.
As a result expression ( 13) takes the form
a(ta) -
2 rte 2 3 mc
CG × N ((o)_O)x)2..]_F2 +2 (o)_~y)2+F2-) (14) It is necessary to point out that the cross-section values in the maxima do not depend in first approximation on the deformation parameter ft.
(8)
An expression similar to (8) was also obtained in the case of the nuclear photoeffect [ 7 ]. Let us now define the photoabsorption crosssection for the deformed clusters. For the resonance width equal to zero, according to (7) we may write down
i a(ta) do)=
10 EF
tax ~tax - iE,.,
02
3mflOz 2
3 NR z
Actually the particle-hole excitations have a finite relaxation width, due to their interaction with more complex states (for example, two particles-two holes states). Therefore we have to introduce complex frequencies,
should be added. Now the transition frequencies for the deformed cluster can be easily defined, tax = taM ( l
P(O) =
Using (12) and substituting (9) into expressions (10) and ( 11 ) we arrive at
the term H'= ~mflOx2
the polarization operator for zero frequency (co = 0 ). The expression for the polarization operator P ( 0 ) is found to be [5]
(5)
Then to the known Hamiltonian clusters,
20 September 1993
3. Discussion of results. Conclusion It is shown that the giant dipole resonance frequency in deformed clusters splits into two components. Therefore the resonance cross-section has two maxima. The results o f photoabsorption measurements in deformed Na clusters were presented in ref. [ 1 ]. Two maxima were observed in the photoabsorption cross-section which can be ascribed to the splitting of the giant dipole resonance. The experimental values o f the first and second resonances are F x = 0.21 eV and F~.= 0.35 eV, respectively. Substituting these values into (14) we obtain the m a x i m u m values of the cross-section, 0 " ( m X ) x ( t a ) = 6 X l 0 -16 c m 2, and O'~a)x(co) = 11 X 10 -16 cm 2 for the second. These values are in good agreement with the experimental data.
Volume 180, number 6
PHYSICS LETTERS A
Recently in ref. [2 ] the p h o t o a b s o r p t i o n crosssection for N a has been investigated. It was also observed that there are two m a x i m a in the photoabsorption cross-section with widths F x = 0 . 2 eV and F y = 0 . 4 eV and the m a x i m u m cross-sections are t r x = 7 × 1 0 -16 cm 2 a n d a y = 7 . 2 × 1 0 t6 cm 2 respectively. These values are also in agreement with the experimental data. Experimental d a t a on the p h o t o a b s o r p t i o n crosssection in Ag [ 3 ] are also available. There exist two m a x i m a having widths F x = 0.39 eV and F y = 0.77 eV and the corresponding cross-sections are in a reasonable agreement with the experiment. In conclusion we would like to recall that the giant dipole resonance splitting is p r o p o r t i o n a l to the cluster deformation. Therefore when the value o f &oM for a cluster with a closed shell is known, then according to ( 8 ) the d e f o r m a t i o n p a r a m e t e r fl is
p=&oM/o~. We would also like to stress that the results ob-
20 September 1993
tained in the quasiclassical a p p r o x i m a t i o n do not d e p e n d on the particular type o f effective field inside the cluster [ 5 ].
References [ 1] K. Selby, V. Kresin, J. Masui, M. Vollmer, W.A. de Heer, A. Seheidemann and W.D. Knight, Phys. Rev. B 43 (1991) 4565. [2] C. Brechignac, Ph. Cahurac, F. Carlier, M. de Frutos and J. Leygnier, Chem. Phys. Lett. 189 (1992) 28. [3]J. Tiggesbaumker, L. Koller, H.O. Lutz and K.H. MeiwesBroer, in: Proc. 88th WE-Heraeus Seminar on Nuclear physics concepts in the study of atomic cluster physics (Springer, Berlin, 1991 ) p. 247. [4] V.V. Kresin, Phys. Rev. B 45 (1992) 14321. [ 5 ] Yu.A. Malov and D.F. Zaretsky, Phys. Lett. A 177 ( 1993 ) 379. [6]A.B. Migdal, Qualitative methods in quantum theory (Moscow, 1975). [7] A.A. Lushnikov and D.F. Zaretsky, Nucl. Phys. 66 (1965) 35.
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