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. 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