Coulomb barriers in the dissociation of doubly charged clusters

The barrier height for the most asymmetric fission decay of doubly charged sodium clusters (Na&'+) into singly ionized fragments has been computed with use of density-functional theory and the jellium model. We have found that the barrier is sizable for large or intermediate-size clusters, but vanishes for N~9. We have also computed the energy AH, needed to evaporate a neutral monomer from Na& +. For N +40, the barrier height is smaller than EH„and emission of a Na+ ion is the preferred decay channel of hot Naz + clusters. On the other hand, the barrier height is larger than AH, for N )40 and, in this case, monomer evaporation becomes competitive. The critical cluster size, N, =40, for the transition from one decay mode to the other is in reasonable agreement with the experimental result. Our calculations suggest that the mechanism for neutralmonomer evaporation is diA'erent from the one currently assumed.


I. INTRODUCTION
Multiply charged clusters X&q+ are usually observable only beyond a certain critical size N, (q) which depends on the charge q. ' It is commonly admitted that the criti- cal size can be interpreted as the size below which the Coulomb repulsion between the positive holes makes the cluster unstable against fragmentation into clusters with smaller charges.A purely energetic criterion has been proposed by several authors.
According to this cri- terion, N, (q) 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.The measured critical size is, however, often different for different ex- periments on clusters of a common element.This fact casts some doubts on a purely energetic criterion.Furthermore, unexpectedly small multiply charged clusters have sometimes been observed, ' suggesting that some of these clusters may be metastable, stabilized against fission by large barriers.Explicit calculations for doubly charged transition-metal dimers support this interpretation.Several dimers, such as Vz +, Crz +, Fe2 +, Nb2 +, metastable, although stabilized by large barriers.Some others, such as Ni2 +, Co2 +, Pdz +, and Pt2 +, are sim- ply unstable.
Recent experimental work by Brechignac et al. ' '" on the induced dissociation of mass-selected doubly charged alkali clusters has clarified substantially some points related to the observability of multiply charged clusters.
These authors have noticed that the critical size below which multiply charged clusters are not observable in mass spectra depends on the cluster-formation mechanism.First of all, one of their key results is that an "excited" Na& + (or K& +) cluster with large N preferen- tially evaporates a neutral atom and not a charged frag- ment Na + (p (N).The explanation that they propose is that in this size range the fission barrier is larger than the binding energy of the neutral monomer.This means that when Na& + clusters are formed by atom evapora- tion from hot clusters of higher masses, the critical size X, is the size below which the fission barrier becomes lower than the binding energy of the neutral monomer.
However, the same authors also point out that doubly charged clusters can also be formed from cold neutral clusters by a two-step ionization.If the ionization pro- cess is such that the cluster is maintained cold, with an internal excitation energy below the top of the Coulomb barrier, metastable doubly charged clusters can exist below the critical size defined above.Stressing the impor- tance of the Coulomb barrier is then a key point of the work of    1991 The American Physical Society Au& + was observed for X ~9 (in addition to Au3 + ).
The clusters were then fragmented by collisions with Kr atoms.The analysis of the fragmentation products re- vealed, in agreement with the results of Brechignac et al. ,   the competition between neutral-atom evaporation and fission (i.e. , decay into two charged fragments).Atom evaporation is dominant for large clusters but fission competes for N~18 and increases strongly as N decreases.For instance, the fissionability, given by the ratio of the fission and evaporation rates I &/I " increases by four orders of magnitude for ¹ ven clusters between N=18 and N=12, having a value of 20 -30 for N=12 (I &/I', is lower for ¹ddthan for X-even clusters be- cause of a reduced stability of clusters with an odd num- ber of electrons' ).Saunders finds that the experimental data on the fissionability are consistent with a liquid-drop model similar to the nuclear liquid-drop model.' Ac- cording to this model, the fission barrier becomes zero for N (6.This means that Au& + clusters with N &6 are predicted to be unstable against spontaneous fission.This critical number is in semiquantitative agreement with the N, =9 observed in the LMIS experiments. It is clear from the work of Brechignac et al. and of Saunders that a better knowledge of the fission barrier in multiply charged clusters of monovalent (alkali and no- ble) metals should provide a useful piece of information in the analysis of experiments concerning these clusters.
With this motivation in mind, we have undertaken a semiclassical calculation of the fission barrier of doubly charged sodium clusters.This calculation is presented in Sec.II.For this purpose we have used density-functional theory' and the jellium model of simple metal clusters.'   The calculated barrier is then compared to classical esti- mates.In Sec.III we discuss our results in relation to ex- periment.Finally, Sec.IV contains a summary.
This means that neutral-monomer evaporation cannot occur spontaneously if Naz + is in its ground state, but it is possible for a highly excited parent cluster.Which of the two reactions (evaporation or fission) occurs depends, as proposed by Brechignac et al. , " on the relative mag- nitudes of AH, and the fission barrier.A schematic rep- resentation of this competition is shown in Fig. 1, where the fission barrier is shown along the dissociation coordinate d (to be defined precisely below).The energy of the system formed by the two ionized fragments (Na& and Na+) at infinite separation is taken as zero of ener- gies.B (d) is the barrier for the opposite process of form- ing Naz + from Na~, + and Na+, and B is the max- imum of B (d).We can call B (d) the capture barrier and, although the present paper is concerned with fission, we often find it convenient to think in terms of B (d).The fission barrierthat is, the barrier for the emission of Na+ from Naz +is given by F(d) =B (d)+bH~, and the height of the fission barrier is F =B +AH&.The quantity B is positive.F is larger than B when AHI is positive, and smaller than B when AH& is nega- tive.According to Brechignac et ah., " evaporation be- comes preferred over fission when the condition II.MODEL FOR CLUSTER DISSOCIATION   In this section we want, first of all, to set up a model for the calculation of the Coulomb barrier and the heat of reaction of the completely asymmetric fission of a doubly charged sodium cluster, The heat of the reaction is defined in terms of the total ground-state energies of parent and fragments (at infinite separation): -AH& is negative for small N and positive for large N.A   negative value of AH& means that the reaction is exo- thermic.On the other hand, the calculation of the fission barrier requires modeling the process by which the ion Na+ leaves the remaining fragment.
In a second step we want to compare the probability for reaction (l) with that for the evaporation of a neutral monomer: Na~~Na~i ++Na .
The evaporation of a neutral monomer is endothermic- A. Jellium model for spherical clusters and heats of reaction Let us begin with the calculation of AH& and AH, .
These two quantities are defined in Eqs. ( 2) and (4), re- spectively.The energies of the clusters involved in these equations are obtained using the spherical-jellium model.This model ' assumes a background of positive charge (representing the ions) with spherical shape and constant density and a distribution of interacting valence electrons (one per atom in the case of Na clusters) with a density n (r) which is self-consistently calculated in the external potential provided by the positive background.The jellium radius R is related to the number N of atoms by R =N' r"where r, =4.0 is the Wigner-Seitz radius of metallic sodium.Notice that R remains unchanged when the cluster is ionized.
The ground-state electron density is obtained by minimization of the following extended-Thomas-Fermi (ETF) functional '' (we shall use Hartree atomic units throughout): The maximum values of these two barriers are indicated as B and F, respectively.The fission barrier is lower than the heat of the evaporation in the left panel and larger in the right panel. T U" is the classical Coulomb energy of the electrons, 3 / 3, n (r)n(r') UJ, gives the electron-jellium electrostatic interaction, UJ, = d rVlrn r (10) V~(r) being the jellium potential; and UJJ is the jellium Coulombic self-interaction.Finally, E", is the sum of the electronic exchange and correlation energies.In the local-density approximation (LDA), where p is the electron chemical potential.In the case of the spherical-jellium cluster model, Eq. ( 12) reduces to a one-dimensional radial equation.This case has been dis- cussed in detail previously ' (see also Sec.II B below).
b, Hf and b,H"which are the heats of the reactions ( 1) and (3), respectively, are easily obtained since in their calculation one only deals with spherical clusters.
where the first term is the exchange part and the second term uses Wigner's interpolation formula for the correla- tion energy.'  The variational electron density is obtained by solving the Euler-Lagrange equation associated with the func- tional (7):

B. Fission barrier
The simplest model for the barrier F(d) opposing the dissociation of Naz + into Naz, + plus Na+ is to take a purely Coulombic approximation for the capture barrier &(d): where d is the distance between the centers of the two fragments.
When d =R (Na~, + )+R (Na+ ), that is, close to the sum of the radii of the two fragments, Eq. ( 13) may not be a good approximation because of the penetration of the spilled-out electron density of the large fragment into the small one.
To obtain a better description of the fission barrier, we have used a deformed, self-consistent extended Thomas-Fermi model.We have considered as initial configuration (see top left panel of Fig. 2) that of a deformed cluster formed by two tangent jellium spheres, corresponding to clusters sizes N -1 and 1, respectively.For this configuration d is the sum of the radii of these two spheres.The other configurations along the dissociation coordinate have been obtained by increasing the separa- tion between the two jellium spheres.
For any given separation, including the initial one, the ground-state electron density and the ground-state energy of the system are obtained by minimizing the energy functional of Eq. ( 7).In this case, the Euler-Lagrange equation ( 12) is a partial dilferential equation that, due to the axial symmetry of the problem, can be solved in cylin- drical coordinates (r, z) To do so,.we have employed the so-called imaginary-time-step method.
We have used a Ar =hz=2 a.u.mesh size, i.e. , four times greater than the Ar that we have used in the spherical calcula- tions.' To make sure that the electron density is negligible at the mesh edge, we have carried out the calculation in a 30 X 60 a.u.(r, z) box.Technical details concerning FIG. 2. Electron-density contour plots at four separations (D=O, 1, 2, and 3 a.u. ) for the totally asymmetric fission chan- nel of Na»'+.D is the separation between the jellium edges.
The electron constant-density contour lines correspond, from outside to inside, to n = 0.5 X 10, 1 X 10, 2 X 10, 3 X 10 3.5 X 10,4X 10 and again 3.5 X 10 a.u.The jellium edges are represented by dashed lines.
the spatial discretization of Eq. ( 12) and the obtainment of the direct Coulomb potential can be found in Refs. 23  and 24.
We have tested the two-dimensional ETF code by performing some spherical calculations with it.We have checked that the deformed code is able to reproduce the total energy of the Na& + clusters obtained with the essentially exact spherical code to better than 0.5 -1%.
It takes less than 100 iterations of the two-dimensional code to stabilize the total energy of a given cluster within 10 'a.u.
Figure 2 shows electron-density contour plots at four separations along the reaction path for the totally asym- metric fission of Na27 Na27 ~Na~6 +Na In this figure the separations, measured by the distance D between the sharp surfaces of the jellium spheres, are D=O, 1, 2, and 3 a.u.The edges of the jellium spheres are represented by dashed lines and the electron density by the solid line contours.The electron density plotted in each case is the one that minimizes the total energy for the corresponding cluster configuration.
The electron constant-density contour lines correspond, from outside to inside, to n =0.5 X 10, 1 X 10, 2 X 10, 3 X 10 3.5X10, 4X10 and again 3.5X10 a.u.Notice that the density at the center of Na26+ does not corre- spond to the maximum value due to the density oscilla- tions at the jellium surface, clearly visible in all spherical ETF calculations (see, for instance, Ref. 19).The figure shows the appreciable polarization of the electron density of Na» + due to the presence of the Na+ ion.The polarization has a sizable effect on the height and shape of the fission barrier.The calculation correctly yields that, as the separation increases, the excess positive charge is shared between the two clusters, i.e. , the final products are singly charged.
Figures 3 -5 show the shape of the capture barrier for the three cases Na2O +, Na» +, and Na4o .The pure Coulomb barrier of Eq. (13) (dashed line) and the self- consistent ETF barrier (thick solid line) are shown as a function of the distance d between the centers of the two fragments, starting from an initial configuration in which the two positive jellium spheres are tangent.As is cus- tomary in similar problems, like that of nuclear e disin- tegration, the energies are referred to the value of the Na~, ++Na+ system at infinite separationthat is, B(d) goes to zero when the distance d goes to infinity.
The ground-state energy of Na& + is indicated in each case by a horizontal line on the left part of each figure.
The height of the fission barrier is then the difference be- tween the top of the capture barrier and the horizontal line representing the ground-state energy of Na& +.
The results of Figs. 3 -5 show that, for large separation between the fragments, the barrier is purely Coulombic.
However, the differences between the ETF and pure Coulombic barriers, due to spilled-out electron density in the ETF model, become evident for the relevant separa- tions around the maximum of the ETF barrier.This difference, nevertheless, becomes rather small as the size N increases.In this section we wish to consider the classical prob- lem of an isolated conducting sphere (representing a clus- ter Na~, +) with net charge Q in the presence of a point charge q.The force acting on the charge q can be written from Coulomb's law: .The meaning of the different lines and symbols is as in Fig. 3.
where R is the radius of the conducting sphere and d is the vector position of the charge q with respect to the center of the sphere.In the limit d ))R, the force reduces to the usual Coulomb's law for two small bodies.
But close to the sphere the force is modifIed because of the induced-charge distribution on the surface of the sphere.
From ( 15) we can compute the work made against the Coulomb forces when moving the charge q from infinite distance to a distance d from the center of the sphere: From this equation we have computed the classical bar- rier, represented in Figs. 3 -5 with a thin solid line.To be consistent with the other calculations, 8"""'"(d) has been plotted only for d ~d"where d, is the distance at which the two jellium spheres considered in Sec.IIB above are tangent.The second term on the right-hand side (r.h.s.) of Eq. ( 16) accounts for the polarization of Naz & as Na+ approaches it coming from infinity.This polarization effect cancels part of the Coulomb repulsion and, consequently, 8"""'(d) is below 8 '"" (d) in the figures.Also, as a consequence of the polarization of the large cluster, g"""" has a maximum.
This maximum occurs at a value of d very close to the corresponding maximum in 8 ".Actually, the polariza- tion of Na~, is also the reason for the maximum in 8 "(d).The maximum of 8'""" is, however, smaller.
This indicates that the classical model exaggerates the polarization effect.In the classical model an accumula- tion of negative charge is built on the surface region fac- ing the approaching Na+ ion, and a deficit of charge ap- pears on the opposite side of the cluster surface.Howev- er, the more realistic ETF calculations indicate that the charge deficit occurs in the interior of' the cluster.The inAuence on the barrier height of this unrealistic feature of the conducting-sphere model increases with the radius R, as Figs.Table I gives the ETP values of 8, EHf, and their sum F for four clusters investigated (1V= 10, 20, 27, and   40).In this range, 8 is remarkably constant, with a value close to 0.051 a.u.The value of AHf decreases fast for decreasing size of Naz + and it changes sign (it be- comes negative) at %=31 (see also Ref. 4).As a conse- quence, the height of the fission barrier, I', also de- creases fast for decreasing X.Our calculated F at FIG. 6. Fission-barrier height (F ) vs cluster size X.Only the points for N=10, 20, 27, and 40 correspond to actual calcu- lations and the line through them is only intended to guide the eye.The heat AH, for neutral-monomer evaporation is also plotted.
N=27 is 1.27 eV, in reasonable agreement with the value of 0.8 eV estimated by Brechignac et al. " A plot of F versus N indicates that the fission barrier vanishes for N=9.For this size and below, the doubly charged clus- ter spontaneously decays according to reaction (1).The heat of the evaporation reaction AH, is also given in Table I and plotted in Fig. 6.AH, is almost constant.Comparing the magnitudes of the fission barrier and AH, in Fig. 6, we observe that AH, is smaller than the fission barrier at large N and larger than it at low N.This is in agreement with the analysis of the experimental results of Brechignac et al. " and Saunders.' The crossing be- tween the two curves occurs at N=41.This number is to be compared with the critical number N, =27 measured by Brechignac:" Our calculations overestimate N, .This is not surprising considering the approximations made (jellium model, approximate kinetic-energy functional; see also further comments below).We stress, neverthe- less, that the physics behind the existence of N, is well reproduced by our calculations.
Our results then suggest the following interpretation of the so-called critical numbers for Coulomb explosion of doubly charged clusters: We can define a critical number N, (in our case, N, =41) such that fission is the preferred decay mode of excited clusters for N &N, because the fission barrier is smaller than AH, .On the other hand, evaporation of a neutral atom competes with fission above N, .This critical number, as indicated by Brechignac et al. , " should be observable when doubly charged clusters are formed by neutral monomer eva- poration from hot clusters of higher masses.The results of our calculations suggest, however, a picture different from the one used until now" for the process of mono- mer evaporation.
When the distance d between the two fragments is still small, we can consider the system as a supermolecule.As such, what we have calculated and plotted in Figs. 3 -5 is the minimum-energy curve of the supermolecule as a function of d.This implies that the barrier exists always (for a doubly charged cluster, we stress) and it must be overcome during the dissociation process, irrespective of the charge state of the small fragment after dissociation (Na+ or Na).In other words, there is no way to avoid surpassing the fission barrier even for neutral-monomer evaporation.We then suggest that, after passing the bar- rier and when the system is undergoing dissociation, im- pelled by the Coulomb repulsion between the singly- charged fragments, the state Na& i +Na appears sud- denly as an available channel [see Fig. 1(right panel)].This state can be realized when one electron of the super- molecule suddenly becomes localized around Na+.For this to occur, d must still be small enough for the electron distribution of Na~& to overlap with that of Na+.
Since at the same time the two fragments are moving away from each other, a neutral Na atom can then es- cape.The probability for the dissociating supermolecule to choose the evaporation channel should increase as the energy difference 5 V, p between the two dissociated states (Na~, + + Na+ and Na~, +Na) decreasesthat is, as N increases.This is supported by Saunders's experi-  ments' on the dissociation of Au& +.Notice that 6 Vip is just the difference between the second ionization potential of Na& & and the first ionization potential of neutral Na.Elucidating the details of the evaporation mecha- nism proposed here is beyond the capabilities of the present static-barrier calculations.
There is, on the other hand, a size rangein our case between 10 N 40where metastable doubly charged Na clusters may be observable if prepared by successive ionization of cold neutral clusters, under the condition that the doubly ionized cluster is left with an internal ex- citation energy below the top of the fission barrier.
Then we could define a second critical number N, (in our case, N, *=9), such that Na& + is unstable for N ~N, * because there is no fission barrier.It is likely that this picture also holds for other alkali-metal clusters but one should be careful in extrapolating it to other groups.For instance, according to the calculations by Reuse et al. , ' the fission barrier does not vanish for low N in Mg& +.
To put our results in a proper perspective, we stress that our model calculations have concentrated on the most asymmetric fission reactionthat is, the emission of a charged monomer.But, in principle, the whole range of possibilities, represented by the different values of p in the reaction Na~~Nap+ +Na (17) should be investigated.If we focus on the heat of reac- tion b, Hf(p), our ETF model predicts that p= 1 is the most favorable channel.However, other fission channels may lead to a more negative AHf if electronic shell- closing effects are taken into account; these are absent in our simple ETF calculations.Additionally, as stressed in the discussion of our results above, the preferred fission channel is controlled by the height of the fission barrier and not by the size of AHf.In fact, channels with p&1 are sometimes dominant: for example, emission of Au3+ in Saunders's experiments on Au& + clusters.'   So, a complete study of the fission barrier as a function of p should be performed for a complete understanding of the fission process.We plan to undertake this study in the near future.A preliminary investigation based on the purely Coulombic barrier of Eq. ( 13) leads to the follow- ing results: B '"" "is smaller for the emission of Na2+ compared with the emission of Na+; on the other hand, the heat of the reaction ( 17) is more positive (less nega- tive) for the emission of Na~.Adding these two quanti- ties, we find that the fission-barrier height is smaller for p= 1, that is, for Na+.These results should, however, be checked by a full ETF calculation of I' (d).
A further ingredient of the calculation that should be discussed is the value of the coefficient A, in the gradient term of Eq. ( 8).We have argued elsewhere that the original von Weizsacker value (X= 1) is the correct one for a good description of the electron density in the tail region of a finite system (atom, molecule, or cluster).However, from an empirical point of view, a value of X=0.5 has sometimes been found appropriate.' ' ' We have performed only a few exploratory calculations using X=0.5.In the case of emission of Na+ from Na2O +, B is almost identical to that given in Table I, which was cal- culated with A, =1.We then expect to find the same situ- ation for other values of N. With respect to the heats of reaction, however, we have found some differences.As a general trend, EHf (A, =0. 5) is a little more positive than AHf (A, = 1), and it changes from negative to positive at N=23 as X increases.This number is smaller than the corresponding one (N=31) for A, = l.But we expect the changes of the heat of evaporation to be similar to those in AHf and, consequntly, a very small net effect when comparing the fission and evaporation channels.

IV. SUMMARY
In this paper we have used an extended Thomas-Fermi method and the jellium model to calculate the fission barrier for the emission of a Na+ ion from a doubly charged sodium cluster.The height of this barrier decreases as the size N of the parent cluster decreases and it vanishes in our model for N=9.Simple descriptions of the fission barrier which neglect the spill-out of the electron density give correctly the order of magnitude, although these are not accurate enough (as compared with the ETF barrier) in the region of the maximum of the barrier.In making this statement, one should keep in mind that the theory used to calculate the ETF barrier contains itself some ap- proximate ingredients.We have also compared the barrier height to the energy needed to evaporate a neutral monomer.Neutral-monomer evaporation is the predicted preferred decay channel for hot clusters with large N whereas fission takes over for small N.This agrees with the experimental results of Brechignac et al. " and Saunders.' The transition between these two decay modes occurs at N, =41.This number is not far from that deduced from the experiments of Brechignac and collaborators considering the approximations introduced in our model.Between %=10 and N=40, metastable doubly charged clusters may be observable if careful ionization from cold neutral clusters leaves the charged clus- ter with an energy below the barrier maximum.

T
FIG.1.Schematic representation of the competition between the fission and evaporation reactions.The heats of fission and eva- poration are AHf and AH"respectively.The capture barrier for the reaction Na»++ Na+ -+Naz + is B (d), and the fission bar- FIG. 3. Calculated capture barrier B (d) for the reaction Nal9 +Na+~Na, o +.Dashed line, pure Coulomb approxi- mation; thick solid line, extended Thomas-Fermi barrier; thin solid line, classical conducting-sphere model ~Notice that the fission barrier F(d) is obtained by measuring B(d) from the ground-state energy of Na2O + [see Eq. (5)] which is indicated by a horizontal line on the left.
-sphere model for the 6ssion barrier FIG.4.Calculated capture barrier for the reaction Na26++Na+~Na».The meaning of the different lines and symbols is as in Fig.3.
FIG.5.Calculated capture barrier for the reaction Na39 +Na ~Na40 +.The meaning of the different lines and symbols is as in Fig.3. Brechignac et

TABLE I .
Calculated fission-barrier height (F ) and separat- ed components (B and heat of reaction EHf) for several Naz + clusters.The heat of evaporation of a neutral monomer, AH, is also given.All quantities are in atomic units (1 a.u.= 27.21eV).