fulltext

Collisions between metal clusters and multiply charged ions

( 

)

Y. ABE(1), K. YABANA(2), P. BO˙ZEK(1)and T. TAZAWA(3) (

1

) YITP, Kyoto University - Kyoto 606-01, Japan (

2

) Graduate School of Science and Technology, Niigata Univ., Niigata 950-21, Japan (

3

) Department of Physics, Yamaguchi University - Yamaguchi 753, Japan

(ricevuto il 13 Ottobre 1997; approvato il 3 Novembre 1997)

Summary. — Metal clusters are electrically charged by collisions with charged ions through electron transfer reactions from the clusters to the ions. The reaction pro-cesses are studied by the time-dependent mean field theory with the usual Local Den-sity Approximation for the exchange-correlation potential. Preliminary results show the interesting feature that collisions with large impact parameters make ionized clus-ters with relatively low excitation, which gives important new information on the sta-bility of metal clusters against fission.

PACS 82.30.Fi – Ion-molecule, ion-ion, and charge-transfer reactions. PACS 36.40.Wa – Charged clusters.

PACS 01.30.Cc – Conference proceedings.

1. – Introduction

Since the discovery of the magic numbers in metallic clusters and their interpretation by the electronic shell structure, similarities between metallic clusters and atomic nuclei have been discussed intensively. The empirical approach with the charged liquid drop model is one of them concerning stability of the system against fission. In atomic nuclei, the fissility parameter which is the ratio between the Coulomb and the surface energies is the essential parameter characterizing the stability. Thus, the stability of metallic clusters is also discussed with this parameter, though there is an essential difference between the two systems, i.e. in the former, charge is uniformly distributed over the whole volume due to the charge symmetry of strong nucleon-nucleon interactions, whereas it is only on the surface in the latter. In metallic clusters experimental information on the stability ap-pears to be not yet precise enough. A powerful method of changing the fissility parameter, i.e. changing charge multiplicities of clusters is multiple photo-ionizations by the use of

( 

)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and

Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.

nuclear heavy ion collisions [5], the time-dependent mean field theory is a suitable one to apply practically. In electronic systems the local density approximation is known to work well widely. Actually, the time-dependent local density approximation (TDLDA) has been shown to be successful in describing the collective electronic excitations as a linear response to the weak external field [6]. It is thus natural to try to apply it to electronic motions in collision processes beyond the linear regime.

2. – Theoretical framework: TDLDA

Here we briefly recapitulate TDLDA applied to slow collision processes between charged ions and metallic clusters. The energy range considered is 10–100 keV. So, ve-locities of incident ions are of the same order as those of the conduction electrons in the cluster and thus only the electrons are excited during the collision, but not the ionic de-grees of freedom. Of course electronic excitation energies are expected to be transferred to the ionic degrees later on. We, therefore, assume the spherical jellium for the clusters. In TDLDA, the time-dependent single-particle wave functions for electrons are de-scribed by the time-dependent Kohn-Sham equation:

i



h @@t

i (

t

)=f, 

h

2 2

m

5 2 +

V

cluster(

~r

)+

e

2Z



(

~r

0

;t

) j

~r

,

~r

0 j d

~r

0 + (1) +



xc(



(

~r;t

))+

V

ion(

~r

,

~R

(

t

))g i (

t

)

with the time-dependent density of the electrons,



(

~r;t

)= X i j i (

~r;t

)j 2

_:

(2)

For the exchange-correlation potential



xc, we employ that of [7]. The potential

V

cluster (

~r

) denotes the potential by the positive ion charges of the clusters which is ap-proximated by the spherical jellium model as stated above. We also assume a small-sized spherical jellium for the potential

V

ionby the incident ion, which is useful for the stabi-lization of numerical calculations and is of course valid for distant collisions. The ion tra-jectory

~R

(

t

)is assumed to be a straight line with constant velocity,

~R

(

t

) =

~b

+

~vt

, since

deflection angles of the ion are expected to be extremely small.

In solving eq. (1), we employ a finite difference method in three-dimensional Cartesian coordinates, following nuclear time-dependent Hartree-Fock theory [5]. To integrate time evolutions of the system, the time step is taken to bed

t=



h

=0

:

01eV

,1

Fig. 1. – Time evolution of the electron density atR z

=,5;0;5, and 25 ˚A from left to right for

Na40+Ar 8+

collision atE=80keV andb=14A.˚

3. – Preliminary results on Na₄₀+Ar8+collisions

We take the geometry of the reaction as follows: the reaction plane is the(

x;z

)-plane,

the impact parameter vector

~b

is parallel to the

x

-axis, and the incident ion velocity

~v

is parallel to the

z

-axis. The calculational space is the rectangular box of size:,16A˚

< x <

32 ˚A,,16A˚

< y <

16 ˚A and,20A˚

< z <

40 ˚A. The spatial meshsize is 0.8 ˚A, so the total

number of the grid points are about 200,000. We employ a boundary condition of vanishing electronic wave function. The absorptive boundary condition with the negative imaginary potential at the boundary region was checked and turned out not to change the results much.

Figure 1 shows the typical time evolution of the electronic density distribution in Na₄₀+Ar8+ collisions with the ion incident energy of 80keV and the impact parameter

of 14 ˚A (Note that the jellium radius of Na40is 7.1 ˚A). The density distributions in the

(

x;z

)-plane are shown for the instants when

R

z

= ,5

;

0

;

5and 25 ˚A, respectively. The

electronic flow is seen along the axis connecting the centers of the cluster and the ion. At the final stage

R

z

= 25A, the electronic density around the ion is seen to be largely˚

extended in space, indicating the formation of a highly excited ion like a hollow atom. These features are consistent with the picture of the classical overbarrier model (COM). In the COM, the electrons are considered to go over the saddle point of the poten-tial energy when the electron’s single-particle energy is higher than the potenpoten-tial energy at the saddle. At large impact parameters, only the electrons in the highest occupied or-bitals can be transferred to the ion, leaving the cluster in low excitation. The electrons transferred to the ion sit in the weakly bound orbitals whose energies are approximately degenerate to those in the cluster, and thus form a spatially extended ion with high exci-tation.

In fig. 2, we show the impact parameter dependence of the number of transferred electrons. We count the number of electrons belonging to the cluster and ion by inte-grating the electron density over a certain spatial region around the cluster and the ion, respectively. In practice, the integrated spatial area is inside a sphere of radius 10 ˚A. We see that the electron number removed from the cluster is systematically larger than the electron number captured by the incident ion. The difference between the two numbers represents the electron number which belongs neither to the cluster nor to the ion. They are probably scattered into the continuum, though we cannot say decisively in the present calculation.

0 2

8 12 16 20 24

Transferred Electron Number

Impact parameter [Å]

Fig. 2. – The number of electrons removed from the cluster and that trapped in the ion at the final stage are plotted for Na40

+Ar 8+

collision atE=80keV.

In order to compare the present results with the COM, first we explain our treatment of classical overbarrier model. We assume that the potential energy for the electron is approximately given by

V

(

;z

)=,

e

2

_q

_cluster(

t

) p



2+

z

2 ,

e

2

_q

_ion(

t

) p (

R

(

t

),

z

)2+



2 +

V

im(

;z

)

:

(3)

Here,



and

z

represent the cylindrical coordinates of the electron position, where the

z

axis is chosen to be the axis connecting the centers of the cluster and ion.

V

_im is an image potential consisting of the image charge in the cluster by the ion charge, and the electron self-image in the cluster.

q

cluster(

t

)and

q

ion(

t

)represent the charge numbers of

the cluster and ion. The single-particle orbitals of the cluster are Stark shifted by the ion Coulombic field and are also lowered by the ionization of the cluster as

E

f

(

t

)=

E

0 f

,

e

2

q

ion(

t

)

=R

(

t

),

e

2

q

cluster(

t

)

=R

cl

:

(4)

Here,

R

_cl is the radius of the cluster, and we choose the jellium radius plus 0.8 ˚A reflecting the electron spill-out effect. If

E

f

(

t

)is higher than the potential barrier height

at the saddle point of eq. (3), the electrons start to flow to the ion. The transferred elec-trons are then assumed to screen the ion charge.

Assuming a free electronic gas motion around the saddle point of the potential, the rate of the transferred electrons may be calculated as

, d

N

d

t

= 2 (2





h

)3 Z d

x

d

y

d3

p

1 2 j

p

z j

m

 

E

f (

t

),

V

(

;z

B ),

p

2 2

m



;

(5)

0 2 4 6 8 10 8 12 16 20 24

Transferred Electron Number

Impact parameter [Å] COM 20keV 80keV 320keV TDLDA 20keV 80keV 320keV

Fig. 3. – The numbers of transferred electron in Na40+Ar 8+

collision. Crosses are obtained by TDLDA and curves by COM.

where

z

B is the

z

-value of the saddle point. In evaluating the integral, we employ a

quadratic approximation for

V

(

;z

)around the saddle in transverse (



)direction. For

the Fermi energy of the neutral cluster,

E

0

f

, we use the ionization potential in the jellium model. The charge numbers

q

cluster(

t

)and

q

ion(

t

)are set to 1 and 8 initially for the case of

the collision of a neutral cluster and a8+ion. They change in time according to the rate

of eq. (5).

In fig. 3, we show the number of transferred electrons (the electrons removed from the cluster) calculated by the TDLDA for Na40+Ar8+for incident energies

E

=20

;

80

;

320

keV, compared with those by the COM. As is seen, the overall features such as the impact parameter dependence and the incident energy dependence in the TDLDA is nicely re-produced by the COM. As the incident ion energy gets higher, the number of transferred electrons decreases in both TDLDA and COM except for small impact parameters. This may be understood from eq. (5). Namely, the rate of electron transfer in eq. (5) does not depend on the incident energy. However, the time period of the collision available for electron transfer is shorter at high incident energy, and thus the number of transferred electrons decreases.

For the region of small impact parameters, the number of transferred electrons in TDLDA is larger than that of COM. Furthermore, the number of transferred electrons at higher incident energy is larger than that at a lower incident energy. Such incident energy dependence cannot be explained in the COM. It implies that the TDLDA includes the dynamics beyond the COM and may be applicable to the cases where the COM may not be successful.

Finally we discuss the excitation energy of the highly charged clusters produced in the final stage of calculation. They are calculated by the following procedure: We first calcu-late the total energy of the cluster in the final stage of calculation. We then calcucalcu-late the ground state energy of the cluster for the same charge state (usually fractional charge), and obtain the excitation energy as the difference between them.

In the left panel of fig. 4, we show the excitation energy of the cluster in the Na₄₀+Ar8+ collision for three different ion incident energies. The excitation energy of

0 10

0 2 4 6 8 10

Excitation Energy [eV]

Cluster Charge

0 10

0 2 4 6 8 10

Cluster Charge

Fig. 4. – The excitation energy of the ionized clusters in the final stage of calculation for Na40 +Ar

8+

collision atE =20;80and 320 keV (left). Cases with Na

40and Na138 +Ar

8+

atE =80keV are

compared (right).

the cluster is plotted against the final charge state of the cluster. Higher charge states correspond to smaller impact parameters. It may be found that the cluster’s excitation energy is actually very low, less than 10 eV up to

q

=4+. It is also interesting to see that

clusters with lower excitation energy are obtainable with higher incident energy of the ion for high charge states. In the right panel of fig. 4, the excitation energy of the cluster in a Na138+Ar8+collision at

E

=80keV is shown and compared with the case of Na40. The

excitation energy is found to be nearly the same for the same charge state. This means that the temperature, or the average excitation energy per vibrational degree of freedom, may be lower in the larger clusters.

4. – Remarks

The production of highly charged clusters in the low excitation by collisions with ions is confirmed in the TDLDA calculation. It was found that the collision at higher incident ion energy is superior in producing charged clusters in low excitation. Also it was found that the excitation energy is almost independent of size for the same charge state. This means that highly charged clusters prepared by the collision method have lower temperatures for larger sizes. These features are consistent with the picture of the classical overbarrier model.

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