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Dissociation of doubly charged alkali metal clusters.

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Ann. Physik 1 (1992) 270-280
Annalen
der Physik
0 Johann Ambrosius Barth 1992
Dissociation of doubly charged alkali metal clusters
J. M. Mpez', J. A. Alonso', F. Garcias2, and M. Barranco3
'Departamento de Fisica Teorica, Universidad de Valladolid, E-47011 Valladolid, Spain
2Departamento de Fisica, Universidad de las Idas Baleares, E-07071 Palma de Mallorca, Spain
3Departamento de Estructura y Constituyentes de la Materia, Facultad de Fisica. Universidad de
Barcelona, E-08028 Barcelona, Spain
Received 9 March 1992, accepted 24 March 1992
Abstract. We have studied the competition experimentally observed between fission and neutral atom
evaporation, as dissociation channels of excited doubly charged sodium clusters, using the Density Functional Theory and the jellium model. The fission barrier has been obtained from an Extended ThomasFermi calculation including density gradient corrections to the kinetic energy of the electronic cloud. We
discuss the influence of the coefficient of the density gradient term on the barrier height.
Keywords: Metals clusters; Dissociation; Fission barrier.
1 Introduction
Since the experimental discovery [ 11 that the electrostatic repulsion in isolated multiply
charged clusters Xg+ may lead them to fission into aggregates with smaller charges
(coulomb explosion)
the question of what is the critical size N, above which multiply charged clusters can be
observed has attracted a considerable theoretical and experimental interest.
A purely energetic criterion would lead to the conclusion that N, is the size below
which the sum of the ground state energies of the fragments is lower than the ground state
energy of the parent cluster (in other words, the heat of fission A Hf is negative below
N,) [2-61. However, multiply charged clusters with N smaller than the critical size
expected from the energy criterion have been observed (for a review of the literature see
Refs. [7- lo]), thus suggesting that some of those clusters may be stabilized against
coulomb explosion by large barriers. Although this means that a purely energetic
criterion is not adequate to predict N,, this criterion has afforded, nevertheless, some
interesting predictions. For instance, Ifiiguez et al. [3] have studied the fission of doubly
charged Sodium clusters ( N a g ) based on Kohn-Sham density functional calculations
[ l 11 using the spherical jellium model [12- 141. The energy criterion predicted that the
most favorable fission channels are dominated by electronic shell effects (or magic
number effects), known to control also other electronic and cohesive properties of simple
metal clusters. This prediction has now been confirmed by experiments.
J. M. Lopez et al., Dissociation of metal clusters
27 1
The fission behavior of simple metal clusters has been substantially clarified by a series
of remarkable experiments studying the dissociation of excited doubly charged alkaline
[ 15 - 181 and noble metal [19,20] clusters. These experiments have shown the competition
between fission and neutral monomer evaporation. Atom evaporation is dominant for
large clusters, but asymmetric fission becomes competitive as N decreases. The two
channels compete in the neighborhood of N, and X F becomes undetectable below the
critical value N,. The competition is qualitatively illustrated in Fig. 1. Brechignac et al.
[16] propose that an excited Na? (or K F ) cluster with large N preferently evaporates
a neutral atom because in this size range the fission barrier (F,) is larger than the
binding energy AH, of the neutral monomer. Consequently, when X$! clusters are
formed by atom evaporation from hot clusters of higher masses, N, is the size below
which the fission barrier becomes lower than the binding energy of the neutral monomer;
thus, doubly ionized clusters with N < N, dissociate into two charged fragments.
However, if the X k clusters are formed from cold neutral aggregates by a two-step
ionization process, metastable doubly charged clusters can exist below the critical size
defined above.
.. .
Fig. 1 Schematic representation
of the competition between the
fission and evaporation reactions.
The heats of fission and
evaporation are A Hf and A He
respectively. B , is the maximum
of the capture barrier for the
reaction NaA-, + Na'
Nac
and F, is the maximum of the
fission barrier. The fission barrier
is lower than the heat of
evaporation in the left panel and
larger in the right panel.
+
AHf <O
The latest experiments [17, 18, 20, 211 show that the most probable fission channels
are influenced by shell closing effects. More precisely, K$ was found [17, 181 as a
dominant fragment in the fission of K C and (Na KN-J2+. Also, Katakuse et al. [21]
have found Ag$ to be a dominant fragment in the fission of Ag? . Additionally, Ag;
is also a prominent fragment; notice that X: and X,+ are closed shell clusters with 2 and
8 valence electrons, respectively, when X is an alkali or noble metal. Finally, the
probability for more symmetric fission was observed to increase with cluster size.
In summary, two key ingredients for a theory attempting to explain the critical size for
the observability of multiply charged clusters are: (i) the consideration of the fission
barrier, and (ii) the introduction of electronic shell effects. In a recent paper [22] we have
calculated fission barriers for N a p clusters using an approximate version of the Density
Functional formalism, namely an extended Thomas-Fermi energy functional, and a
272
Ann. Physik l(1992)
model based on the jellium description of clusters [12- 141. This theory gave barrier
heights in rough agreement with those inferred from experiment [ 161and also showed that
there is no barrier for very small values of N. However, some additional points related
to that calculation remain to be studied. One of them is the sensitivity of the results to
the value of the numerical constant L in the expression for the density gradient
contribution to the kinetic energy functional in the extended Thomas-Fermi (ETF)
approximation (see Eq. (7) below). There is some discussion in the literature about the
most appropriate value for this numerical constant in different specific situations (see,
for instance, Ref. [23-251). Thus, our task in the present paper is to analyze the
sensitivity of the fission barrier (and then of N,) to the value taken for 1.Although the
ETF method gives useful insight into the fission barrier, it does not account for the shell
effects. A full Kohn-Sham treatment is needed to account for the shell effects. However,
this adds substantial computational difficulties because of the lack of spherical
symmetry of the system along the fission path. A full Kohn-Sham Molecular Dynamics
description of the fission of small doubly charged sodium clusters has been recently
presented by Barnett et al. [26, 271. This work goes beyond the jellium model and
accounts for the full granularity of the cluster using the pseudopotential description of
the electron-ion interaction. The results, however, are restricted to N 5 12 which is a size
range below the region of interest concerning the critical size (N, = 27 from experiments
for sodium). Since reaching this size region is a considerable demand for a Kohn-Sham
Molecular Dynamics simulation, we think that our ETF calculations for larger clusters
using a computationally less demanding method can provide complementary information on the fissionability of doubly charged clusters.
The layout of this paper is as follows. In Section I1 we describe the calculation of the
fission barrier in the Extended Thomas-Fermi formalism. In Section 111 we compare the
results obtained from different values of the gradient coefficient. Finally we draw our
conclusions in Section IV. A short preliminary account of this investigation has been
presented elsewhere [28].
2 The Extended Thomas-Fermi Formalism
In a previous paper [22] we have performed a calculation of the fission barrier for the
most asymmetric fission channel of N a c , that is,
With reference to Fig. 1 the maximum of the fission barrier, F,, can be expressed as the
sum
F,
=
B,
+ AHf.
(3)
B,,, is the maximum of the barrier for the opposite process
and AHf is the heat of fission, which can be written in terms of the energies of the
parent and product clusters involved in reaction (2):
273
J. M. Lopez et al., Dissociation of metal clusters
AHf
=
+ E(Na+)
E(Na&-,)
-
E(Nac) .
(5)
A Hr is negative for small N and positive for large N.
Density functional theory [29] and the jellium model [12- 141 have been employed to
calculate AHr and B,. In the spherical jellium model the positive charge of the ionic
background is homogeneously distributed over the volume of a sphere with the radius
R of the cluster. For the energies we have used an extended Thomas-Fermi functional,
namely
EM]
=
T
+
U,,
+
Uje + Ex,
+
Uj,
.
(6)
T is the electron kinetic energy, given as a sum of the local Thomas-Fermi term and the
lowest order gradient correction (Hartree atomic units are used through the paper unless
explicitly stated)
T[p]
=
[-
J d3r 3
(3n2)2’3
p5’3
+A
1-
(vP)2 ,
8
10
P
(7)
Uee is the classical coulomb energy of the electrons, Uj, gives the electron-jellium
electrostatic interaction, Ex, is the sum of the exchange and correlation energies in a
Local Density approximation (Wigner’s interpolation formula [30] was used for the correlation energy), and finally Ujj is the self-interaction of the positive jellium background. We used A = 1 in Ref. [22] because this is the value originally proposed by Von
Weizsacher [3 11. The ground state density is obtained by self-consistently solving the
Euler-Lagrange equation associated with the energy functional (6)
with appropriate boundary conditions (see Refs. 22 and 32 for details). ,D in Eq. (8) is the
chemical potential.
Using the spherical jellium model for the parent and product clusters Na? , Nab-,
and Na’, the computation of AHfis an easy task [4, 221. In a similar way, the neutral
monomer binding energy
AHe
=
E(Na?-,)
+ E(Na)
-
E(Na?),
(9)
which corresponds to the evaporation reaction
is also calculated from the energies of spherical clusters N a g , Na?-, and Na.
In contradistinction, the computation of the barrier B, requires the evaluation of the
electron density, and the corresponding energy, for the deformed cluster undergoing
fission. For this purpose we have used a deformed, fully self-consistent ETF model.
Ideally, we would like to continuously deform the cluster until it splits into two pieces
of the desired sizes which then fly away. In practice we have only taken several snap-shots
along the fission path. The first snap-shot (excluding the initial spherical configuration
of N a c ) is modelled as a deformed cluster composed of N-2 electrons moving in the
274
Ann. Physik 1 (1992)
mean-field created by two tangent jellium background spheres corresponding to cluster
sizes N-I and 1 respectively. The other snap-shots along the dissociation path to the final
state Nab-, + N a + have been obtained by increasing the separation between the two
jellium spheres representing the emerging fragments.
3 Results and discussion
The fission barrier F, for the reaction (2) was calculated, using the method described
above, for several clusters: Na:;, Nag;, Nag? and Nai$ [22]. A comparison with the
monomer evaporation energy A H e is shown in Fig. 2 (see upper curves for 1 = 1). It
predicts that F, becomes smaller than AH, below the critical size N, = 40. The
experiments of BrCchignac and coworkers [ 16, 171 indicate that evaporation of a neutral
monomer becomes dominant for N 1 3 1, whereas fission dominates for N < N, = 27.
Both processes compete around the critical size.
:
v
2,
P
W
0.06
-
0.05
-
AHJA -1)
0.04 -
0.03 0.02 0.01
-
/
I
Fig. 2 ETF fission barrier height
F, corresponding to reaction (2)
of the text and heat of monomer
evaporation A H , versus cluster
size for two values of the
At the time when the calculation of the fission barrier for reaction (2) was performed a
clear picture of the experimental situation concerning the most favorable fission channels
did not exist. The only evidence was that asymmetric fission was much more probable
than symmetric fission for sizes around N, [16, 171. This was later verified by a
calculation [33] of the barrier for the symmetric fission of Naiof
Na$t
+
Na&
+ Nag,
(1 1)
using the same ETF formalism and a similar description of the fissioning cluster in terms
of the two-jellium-spheres model. This calculation predicted a fission barrier for reaction
(1 1 ) approximately twice as large as the barrier for reaction (2) with N = 40.
A basic ingredient in the ETF calculation of the fission barrier presented above is the
coefficient 1 of the density gradient term in Eq. (7). A value 1 = 1 was proposed originally
by Von Weizsacker [311. This value is the correct one for a good description of the electron
density in the tail region of a finite system (atom, molecule or cluster) [23]. Other values
have also been proposed [24,25,29,34]. I = 1 /9 is the value arising in a series expansion
275
J. M. Lopez et al., Dissociation of metal clusters
of the kinetic energy T[p] in the gradients of the density [29]. From an empirical point
of view, I = 0.5 has been found to be appropriate for describing some properties of
simple metal clusters [34].
Then, our main intention here is to study the sensitivity of the fission barrier to the
value of I by comparing results obtained with I = 1 and I = 0.5. The barriers for the
most asymmetric fission channel of NaZ;, that is,
Nai; - N a &
+ Na'
(12)
are given in Fig. 3. The capture barrier B, for I = 0.5 is smaller than for I = 1. In fact
B(d), where d is the distance between the centers of the two fragments, deviates
substantially from a pure coulombic barrier (Bcou'omb( d ) = e 2 / d )for small distances in
the case I = 0.5. On the other hand, the heat of fission AHfis larger for I = 0.5. The
first of these two effects dominates and the fission barrier F, is lower for A = 0.5. A
similar situation occurs for other cluster sizes and the results are given in Tab. 1. In
conclusion, the fission barrier F, depends sensitively on A and the barrier for the
effective value A = 0.5 is lower (by 0.4 eV - 0.5 eV) than the barrier obtained using I =
1 [35].
0.06
0.04
1
?
9
>
Fig. 3 ETF barrier for the fission process Na" --t Na;,
+ Na' versus the seoaration between the centers of the
two fragments. Energies are measured with respect to the
energy of the fragments (E(Nai,) + E ( N a + ) ) at infinite
separation. Thin line: pure coulomb approximation; thick
solid line: barrier for I = 1; dashed line: barrier for I =
0.5. Notice that the fission barrier F(d) has to be
measured from the ground state energy of Na;:, given by
the horizontal lines at the left. The separation d ,
corresponds to the situation when the positive jellium
backgrounds of the two fragments are just touching each
other.
0.02-
P
;
-0.0O2
-F- - J;
10
,
20
30
,
,
40
50
Distance (a.u.1
The large reduction of the fission barrier which occurs when the value of A is lowered
from 1 to 0.5 (illustrated in Fig. 3 for the most asymmetric fission channel of NaZJ) is
a general property, that is, it also occurs for other fission channels and for other clusters.
In spite of this reduction, the electron densities of the compound system are rather
similar. We show this in figures 4-6 for the case of the symmetric fission of Nazi
(Eq. (ll)), for a separation of 4 a.u. between the sharp jellium surfaces (close to the
maximum of B(d)). Fig. 4 gives the equidensity contours in a plane containing the
centers of the two fissioning fragments. The results on the upper part correspond to
I = 1 and those on the lower part to I = 0.5. Fig. 5 is a three-dimensional representation
Ann. Physik 1 (1992)
276
Table 1 Calculated ETF fission barrier height (F,) and separated components (B, and A HI) for the
most asymmetric fission channel of N a p . All energies are in eV.
L
I = 1
=
0.5
N
Bm
A%
FIn
B,
A H.f
Fm
20
21
40
1.39
1.39
1.41
-0.41
-0.1 I
0.22
0.98
1.28
1.63
0.65
0.60
0.68
-0.14
0.16
0.52
0.5 1
0.77
1.20
x= 1
Fig. 4 Contours of equal electron density in a
plane containing the centers of the two
fragments for the symmetric fission of N a z
calculated from the ETF functional with two
values of 1. The separation between the edges
of the two positive backgrounds is 4 a. u. The
contour lines correspond, from outside to
inside, t o p = 0.4, 0.8, 1.2, 1.6, 2, 2.4, 2.8, 3.2,
3.6 and 4 x w3a.u.
Fig. 5 Three dimensional representation of the
results shown in Fig. 4.
of the same results. Finally, in Fig. 6 we compare the electron densities along a straight
line joining the centers of the two fragments. The electron density in the central region
of each fragment is very close to the value in the (bulk) macroscopic metal @bulk =
0.0037 a.u.), especially in the case of A = 0.5. We can observe the small surface
oscillation at the jellium edge. Both electron densities are rather similar, although the
decay is a little bit more abrupt for A = 0.5. This can also be seen in Fig. 4 where the
equidensity contours are more closely packed for I = 0.5. In spite of the apparent similar
electron densities, the values of B(d) at this fragment separation are very different: B =
0.048 a. u. for I = 1 and B = 0.026 a. u. for I = 0.5. The same conclusion follows from
J. M. Lopez et al., Dissociation of metal clusters
,
0.004
277
I
r
I
I
-
c 0.003 0
2
v
A
c
:0.002 C
0.001
Fig. 6 Electron density profiles
along the line joining the centers
of the two fragments for the same
configuration of Figs. 4 and 5.
-
-A-0.5
-15.0
-10.0
-5.0
0.0
5.0
10.0
15.0
2 (a. u.)
a semiclassical calculation of the fission barriers in the nuclear case [36]: when A
decreases, the surface becomes sharper and the fission barrier decreases.
The lowering of F, when A decreases influences the critical size N, at which F,
becomes smaller than AH,, that is, the size at which fission via reaction (2) is more
favorable than monomer evaporation. Fig. 2 shows A He and F, as a function of size for
the two values A = 1 and A = 0.5. We can observe that also A He decreases for L = 0.5
and the A He and F, curves intersect at N, = 36 in this case. This number is in better
agreement with the critical size observed by Brtchignac (N, = 27) [ 161 than the value N,
= 40 we had previously obtained using A = 1 [22].
We think that the curves with A = 0.5 describe the competition between fission and
monomer evaporation better. The value of AH, (A = 0.5) is 1 . 1 eV in the size range
studied. This is in perfect agreement with the experimental cohesive energy (per atom)
in the bulk metal [37]. On the other hand A He (A = 1) overestimates the experimental
cohesive energy by 0.5 eV. For this reason it is natural that N, improves a little with the
use of A = 0.5.
It is not surprising that we predict a critical number, N, = 36, a little different from
the experimentally observed one, N, = 27. First of all we have made several approximations concerning the energy functional and the cluster structure. On the other hand it is
reasonable that our calculation overestimates the observed N,. The theoretical N, could
only be observed if, in the experiment, the energy of the parent cluster is only just enough
to surpass the lowest of F, or A He. But, even in the region where F, < A He, the cluster
can sometimes choose the evaporation channel if its excitation energy is equal to A He
or larger. This will be the normal situation in a typical experiment and one can then expect
to observe evaporation below the theoretical value of N,.
For the fission of K? and ( K N - Na)2+ the experimental situation has been recently
clarified [18]: the two-electron ion Kj+ is the preferred fission channel. The same occurs
for A g g , where in addition the eight electron fragment Ag; is also a probable fission
channel. This is a shell effect wich cannot be accounted for by our ETF method. Instead,
the ETF method gives reaction (2) as the most favorable fission channel. In contrast to
the potassium case, the experimental situation concerning the fission behavior of N a g
clusters is rather complex. The experiments [15, 161 point out to a n o n trivial competition
278
Ann. Physik 1 (1992)
between the Na:, N a + and other fission channels. For the sake of completeness let us
record here the critical number N, if we just compare the fission barrier for the reaction
with the energy of monomer evaporation. Using again A = 0.5 we obtain N, = 20. The
experimental critical number N, = 27 is halfway between the two theoretical numbers
N:3) = 20 and N:') = 36 obtained by considering reactions (1 3) and (2), respectively, as
the fission channels competing with monomer evaporation. With the notation Ni3)and
N:') we have explicitly indicated that the critical numbers are obtained by separate
consideration of the trimer ion and monomer ion as fission channels.
To properly account for electronic shell effects a Kohn-Sham (KS) calculation of the
fission barrier is needed. This is a high priority in our plans for the near future. A
preliminary idea of the expected changes can be obtained by using the Kohn-Sham
method to evaluate the heat of fission A Hr, but still keeping the ETF result for B, (see
Eq. (3)). Performing a KS calculation for AHr is easy, since only consideration of
spherical clusters is needed. The results obtained by this simple approach for the fission
of Na;; are given in Fig. 7. Here we have plotted the barriers for the trimer ion and
006 -
-
0 . 0 4 .-
3
0.02
-
,,'
r
01
Q,
C
W
'.,NC-
I
L
0
I
-
I
F
i
h
-0.02
t
10
20 30 40
Distance (0.u.)
50
Fig. 7 Barriers for the fission of Na:.
Dashed curve:
Na channel in both ETF and mixed ETF-KS theories.
Continuous upper curve: Na; channel is ETF theory.
Continuous lower curve: Na; channel in mixed ETF-KS
theory.
+
monomer ion channels for both pure ETF and mixed ETF-KS theories (the ETF
calculations use A = 0.5). The mixed ETF-KS calculation predicts that the emission of
Na: is more favorable than the emission of Na' since the fission barrier is smaller for
the former case. The zero of energies in this figure is the sum [E(Na&) + E(Na+)] with
the fragments at infinite separation. More precisely [E(Na&) + E(Na+)lETFis taken as
energy zero of the ETF curves and [E(Na&) + E(Na+)lKsis the zero of energies for the
mixed ETF-KS curves. This is the reason why the capture barrier B(d) for N a + in both
ETF and ETF-KS theories is described by a common curve (the corresponding fission
barrier heights F, for the emission of Na + are, however, slightly different). Plotting the
results in this way has the virtue of clearly displaying how the introduction of shell effects
leads to a lowering of the barrier of the trimer ion channel relative to that of the Na2+
channel for Na!;.
Although this simple calculation rightly exposes the competition between different
fission channels, one cannot take the results too far. By performing similar calculations
J. M. Lopez et al., Dissociation of metal clusters
279
for other sizes we have verified that the mixed ETF-KS calculation is not accurate enough
to explain quantitatively the experimental details concerning the competition between
channels. In summary, a full Kohn-Sham calculation of both B, and A H,is needed for
this purpose.
4 Summary and comments
We have studied the influence of the gradient coefficient A on the fission barrier of doubly
charged clusters obtained from an ETF calculation and have found that, when I
decreases, the diffusivity of the electron density also decreases and the fission barrier
height F, of Na$ is reduced. When we use A = 0.5 the comparison of F, for the Na+
channel with the heat of monomer evaporation leads to the critical value N, = 36 which
has to be compared with the observed value NFP = 27. Part of the discrepancy can be
ascribed to the approximations made in our model. However, we argue that in a normal
experimental situation the parent clusters have an excitation energy above threshold, and
this leads to observe reduced values of N,.
Concerning other fission channels, the ETF model predicts N, = 20 for Na3+
emission (in competition with neutral monomer evaporation), but the corresponding
fission barriers are higher than for Na+ emission. Only the introduction of shell effects
can make trimer ion emission (or other fission channels) more favorable in some cases.
As an attempt to show this qualitatively, we have compared the fission barriers for Na'
and Na$ emission in the case of Naf; in a mixted ETF-KS model.
In conclusion, we think that our ETF model is able to explain the gross features of the
competition between fission and neutral monomer evaporation in the dissociation of
doubly charged Sodium clusters. To go beyond it, we must improve the calculation in two
ways. On the one hand, our parametrization of the fission path in terms of two jellium
spheres should be substituted by a more general one in which the doubly charged parent
cluster is continuously deformed until it breaks in two pieces. On the other hand, a KohnSham calculation of the fission barrier has to be carried out. The Kohn-Sham calculation
is the only way to treat properly the nontrivial competition between different fission
channels. Work along both lines is in progress.
This work has been suported by DGICYT, Grants PB 89-032and PB 89-0352-C02-01. We thank A. Rubio
and A. Maiianes for helpful comments.
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