Neutral Sodium Clusters Research Articles

Particle Number Dependence of Size and Energy Quantities in Sodium Clusters

The effective radial electronic potentials for neutral sodium clusters, which were determined by Ekardt on the basis of the local density approximation and the jellium model, are parametrized by means of the (symmetrized) Woods-Saxon and "Wine-Bottle" symmetrized Woods-Saxon potentials with the aim of investigating the dependence of size and energy quantities on the cluster particle number. The potential parameters are determined by vari­ous least-squares fitting procedures. It is found that for the radius R of the above potentials, complex expressions are more appropriate than the stan­dard one R = r0N^{1/3} for relatively small values of N. Furthermore, N-power expansions are derived for those complex expressions of R, as well as for the r.m.s. radius of the potential. It is also found that improved results in these cases are obtained with an expression of the form R = r0N^{1/3}+b, which is still very simple. There is also investigated the variation of energy quan­tities, such as the single particle energies of the 1s and 1p states, the level spacing |E1p-E1s| and the average energy level spacing, with respect to the particle number N. Expressions for the first three of these quantities with N-dependent terms of the form aN^{2/3} + βΝ^{-1} give good results.

Evolution of the Structural and Electronic Properties of Medium-Sized Sodium Clusters: A Honeycomb-Like Na20 Cluster

Sodium is one of the best examples of a free-electron-like metal and of a certain technological interest. However, an unambiguous determination of the structural evolution of sodium clusters is challenging. Here, we performed an unbiased structure search among neutral and anionic sodium clusters in the medium size range of 10-25 atoms, using the Crystal structure AnaLYsis by Particle Swarm Optimization (CALYPSO) method. Geometries are determined by CALYPSO structure searches, followed by reoptimization of a large number of candidate structures. For most cluster sizes the simulated photoelectron spectra of the lowest-energy structures are in excellent agreement with the experimental data, indicating that the current ground-state structures are the true minima. The equilibrium geometries show that, for both neutral and anionic species, the structural evolution from bilayer structures to layered outsides with interior atoms occurs at n = 16. A novel unprecedented honeycomb-like structure of Na20 cluster with C3 symmetry is uncovered, which is more stable than the prior suggested structure based on pentagonal structural motifs.

Density functional investigations on structural and electronic properties of anionic and neutral sodium clusters NaN (N = 40–147): comparison with the experimental photoelectron spectra

Density functional calculations have been carried out to obtain low energy equilibrium geometries of anionic and neutral sodium clusters over a wide range of sizes 40 ≤ N ≤ 147, where N is the number of atoms. An exhaustive search for the low energy equilibrium geometries has been carried out. The density of states of the lowest energy geometries are compared with the experimental photoelectron spectra (Huber et al 2009 Phys. Rev. B 80 235425; Kostko et al 2007 Phys. Rev. Lett. 98 043401) for N > 41. The agreement between theory and experiment is good for almost all the clusters and the changes in the spectrum with size correlate very well with the changes in the shapes as observed in the evolutionary trend of the ground state geometries.

First-principles determination of the structure of NaN and ${\rm Na}_N^-$NaN− clusters with up to 80 atoms

We have performed an extensive computational search for the global minimum (GM) structures of both neutral and anionic sodium clusters with up to 80 atoms. The theoretical framework combines basin hopping unbiased optimizations based on a Gupta empirical potential (EP) and subsequent reoptimization of many candidate structures at the density functional theory level. An important technical point is that the candidates are selected based on cluster shape descriptors rather than the relative stabilities of the EP model. An explicit comparison of the electronic density of states of cluster anions to experimental photoemission spectra suggests that the correct GM structures have been identified for all but two sizes (N = 47 and 70). This comparison validates the accuracy of the proposed methodology. Furthermore, our GM structures either match or improve over the results of previous works for all sizes. Sodium clusters are seen to accommodate strain very efficiently because: (a) many structures are based on polyicosahedral packing; (b) others are based on Kasper polyhedra and show polytetrahedral order; (c) finally, some (N + 1)-atom structures are obtained by incorporating one adatom into the outermost atomic shell of a compact N-atom cluster, at the cost of increasing the bond strain. GM structures of neutrals and anions differ for most sizes. Cluster stabilities are analyzed and shown to be dominated by electron shell closing effects for the smaller clusters and by geometrical packing effects for the larger clusters. The critical size separating both regimes is around 55 atoms. Some implications for the melting behavior of sodium clusters are discussed.

Effect of Electronic and Geometric Shell Closures on the Stability of Neutral and Anionic TiNan (n = 1−13) Clusters

Theoretical studies on the electronic structure and stability of neutral and anionic Ti-doped sodium clusters have been carried out to examine the role of electronic and geometric shells on the physical and chemical behavior. Our studies delineate the effect of the filling of atomic electronic states, electronic shells in a two- and three-dimensional confined electronic gas, and the completion of geometrical shells on the energetic stability and chemical reactivity. In particular, a TiNa9 cluster is shown to be marked by a filled 1S2 1P6 super shell, and a half-filled atomic electronic d-level on the Ti atom, with the composite closure leading to a magnetic superatom.

Photoelectron spectra of anionic sodium clusters from time-dependent density-functional theory in real time

We calculate the excitation energies of small neutral sodium clusters in the framework of time-dependent density-functional theory. In the presented calculations, we extract these energies from the power spectra of the dipole and quadrupole signals that result from a real-time and real-space propagation. For comparison with measured photoelectron spectra, we use the ionic configurations of the corresponding single-charged anions. Our calculations clearly improve on earlier results for photoelectron spectra obtained from static Kohn-Sham eigenvalues.

Density functional calculation of photoabsorption in metal clusters using an exchange‐correlation potential with correct asymptotic behavior

AbstractThe exchange‐correlation potential with the correct −1/r asymptotic behavior proposed by Parr and Ghosh (Phys Rev A 1995, 51, 3564) is used here to study the linear response to external fields of (1) neutral and charged sodium clusters, and (2) doped clusters of the type NanPb (n = 4, 6, 16). Our results for the photoabsorption cross section are compared with experiments, when available, and with previous density functional theory calculations using local and nonlocal functionals. The calculated static polarizabilities and plasmon frequencies are in better agreement with experiments than were previous calculations. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Simulation of mass spectra produced in collisions ofhighly charged ions with sodium clusters

Neutral sodium clusters containing up to several hundred atoms are multi-ionized in low-energy collisions with Xe28+ projectiles (560 keV). The time-of-flight mass spectra ofsingly and multiply charged sodium clusters, produced in singlecollisions, are measured with a high-resolution reflectron-typemass spectrometer. They are compared with theoretical spectrabased on Monte Carlo simulations. The comparison allows one toevaluate the energy transferred to the cluster during theionizing collision and suggests for high cluster charges amulti-fragmentation process, where nearly all charges areemitted creating a low-charged residual ion.

An inelastic threshold in electron - alkali cluster collisions

An analysis of integral cross sections for slow electron collisions with neutral sodium clusters and nanoparticles reveals that, in addition to an effective negative ion formation channel, there exists a strong inelastic threshold-type process which appears above a collision energy of 1-1.3 eV. We show that it can be plausibly associated with the onset of direct electron-induced cluster fragmentation. This result highlights the importance of understanding the dynamics of electron-vibrational energy transfer in nanoclusters, including the relative probability of direct vs. statistical energy transfer.

Negative-ion formation in collisions of low-energy electrons with neutral sodium clusters

Electron collisions with neutral sodium clusters in the 0\char21{}3 eV energy range have been studied in a crossed-beam geometry, and direct evidence for the formation of ${\mathrm{Na}}_{n}^{\ensuremath{-}}$ anions has been obtained. The dependence of ion formation efficiency on the electron energy is well represented by the Langevin cross section, suggesting that the electrons are captured by the polarization field of the cluster. This is consistent with the highly polarizable nature of alkali-metal clusters.

Fragmentation of small sodium clusters

Fragmentation of small sodium clusters was studied by performing both ab initio and classical molecular dynamics simulations. In ab initio calculations at 1200–2400 K, neutral sodium clusters with 10 and 13 atoms ejected both monomers and dimers. The observed behavior is in agreement with previous calculations stating that the electronic shell oscillations diminish strongly as a function of temperature. The fragmentation rates obtained with the ab initio method are consistent with the Kelvin equation for the equilibrium vapor pressure of small clusters. The differences between the results obtained using different models reflect the differences between the dissociation energies calculated correspondingly.

Measuring cluster temperatures via kinetic-energy release

We use photodissociation of neutral sodium clusters to measure the recoil pattern of the fragmentation products. The time dependence of the depletion curves reflects the kinetic-energy release that depends on the internal energy, or the temperature, of the cluster. The fragmentation process is treated parallel to the Weisskopf statistical theory of the disintegration of compound nuclei. From the kinetic energies of evaporation, we are able to determine the temperatures of both the daughter and mother clusters. For Na20 the temperature is found to be T5440665 K, which is in good agreement with an estimate based on the evaporative ensemble model. The measured temperature is slightly larger than that found for large ionic clusters. Branching ratios for the evaporation of sodium dimers and monomers from hot neutral clusters are estimated. @S1050-2947~99!01301-3#

Microscopic description of the surface dipole plasmon in largeNaNclusters(950≲N≲12050)

Fully microscopic random phase approximation/local density approximation calculations of the dipole plasmon for very large neutral and charged sodium clusters ${\mathrm{Na}}_{N}^{Z+}$ in the size range $950\ensuremath{\lesssim}N\ensuremath{\lesssim}12050$ are presented. Sixty different sizes are considered altogether, which allows for an in-depth investigation of the asymptotic behavior of both the width and the position of the plasmon.

Long-range van der Waals forces between alkali clusters and atoms

We study the long-range attractive force between neutral sodium clusters Nan (2⩽n⩽20) and the alkali atoms Li, Na, and K. Absolute cross sections for the scattering of a beam of clusters by atomic vapor are measured, and are shown to arise from the van der Waals dispersive interaction V=−C6/r6. These cross sections are extremely large (up to thousands of Å2 in the center-of-mass frame). Their magnitudes are in good agreement with predictions based on the London theory of dispersion forces; the large strength of the interaction is a consequence of the high cluster polarizabilities. In addition, we evaluate the contribution of the higher-order potential term −C8/r8 to the scattering cross section and show that it can become quite significant for collisions involving large clusters (n∼102–103).

Nonradiative electronic deexcitation time scales in metal clusters

The lifetimes due to Auger-electron emission for a hole on a deep electronic shell of neutral and charged sodium clusters are studied for different sizes. We consider spherical clusters and calculate the Auger-transition probabilities using the energy levels and wave functions calculated in the local-density approximation. We obtain that Auger emission processes are energetically not allowed for neutral and positively charged sodium clusters. In general, the Auger probabilities in small ${\mathrm{Na}}_{N}^{\ensuremath{-}}$ clusters are remarkably different from the atomic ones and exhibit a rich size dependence. The Auger decay times of most of the cluster sizes studied are orders of magnitude larger than in atoms and might be comparable with typical fragmentation times.

Strong long-range forces betweenC60and Na atoms and microclusters

We describe a measurement of the long-range van der Waals interaction $V=\ensuremath{-}{C}_{6}{/r}^{6}$ between free neutral sodium atoms and clusters $({\mathrm{Na}}_{n},1<~n<~20)$ and ${\mathrm{C}}_{60}$ fullerenes. The ${C}_{6}$ coefficients were derived from absolute integral cross sections for low-energy scattering of a supersonic beam of metal particles by ${\mathrm{C}}_{60}$ vapor. Their values are extremely high, ranging from $\ensuremath{\sim}{10}^{4}\mathrm{a}.\mathrm{u}.$ up to more than ${10}^{5}\mathrm{a}.\mathrm{u}.,$ corresponding to center-of-mass scattering cross sections of up to $\ensuremath{\sim}6000{\mathrm{\AA{}}}^{2}$ (which exceeds the hard-sphere dimensions of the particles by a factor of 20). These experimental values of ${C}_{6}$ are in excellent quantitative agreement with the predictions of the London theory of dispersion forces based on atom and cluster spectral properties. The strength of the long-range potential is due to high cluster polarizabilities.

Molecular dynamics study of neutral and multiply charged sodium clusters

The fission mechanism of multiply charged sodium clusters was investigated with molecular dynamics and a new potential featuring a local density approximation second-moment approach. We show that the critical si which 2+, 3+ and 4+ charged clusters undergo fission due to Coulomb forces depends strongly on the cluster temperature. The smallest critical size occur for cold clusters. Master curves predicting upper bounds to the cluster temperature at the fissioning critical size are provided. A scaling of the cluster ion critical size with a power law of the total charge is used to predict the energy barrier that sodium clusters need to surmount to undergo fission.

Particle number dependence of size and energy quantities in sodium clusters

The effective radial electronic potentials for neutral sodium clusters, which were determined by Ekardt on the basis of the local density approximation and the jellium model, are parameterized by means of the (symmetrized) Woods-Saxon and “wine-bottle”-symmetrized Woods-Saxon potentials with the aim of investigating the dependence of size and energy quantities on the cluster particle number. The potential parameters are determined by various least-squares fitting procedures. It is found that for the radius R of the above potentials, complex expressions are more appropriate than the standard one R = r 0N 1 3 for relatively small values of N. Furthermore, N-power expansions are derived for those complex expressions of R, as well as for the rms radius of the potential. It is also found that improved results in these cases are obtained with an expression of the form R=R 0N 1 3 + b , which is still very simple. There is also investigated the variation of energy quantities, such as the single-particle energies of the 1s and 1p states, the level spacing | E 1p − E 1s| and the average energy-level spacing, with respect to the particle number N. Expressions for the first three of these quantities with N-dependent terms of the form aN t-2 3 + βN −1 give good results. Finally, the total energy per particle is considered with the aim of deriving its asymptotic expression for large values of the particle number.

Potential energy surfaces of sodium clusters

We have calculated multidimensional Born- Oppenheimer energy surfaces of singly charged and neutral sodium clusters with quadrupole, octupole, and hexadecapole deformed shapes in a particle range from 8 ≤ N ≤ 58. We use the local-density approximation (LDA) and solve the Kohn-Sham equations on a cylindrical mesh for axially symmetric shapes. Employing the structure-averaged jellium model (SAJM), we ascertain that the correct empirical bulk properties and surface tension are reproduced. Besides a pronounced isomerism in the β2/β4 plane we also find super-deformed shapes. We compare the PES data with shape transitions deduced from experimental splittings of the dipole. photoabsorption cross sections. The influence of large octupole moments reverts the scheme of prolate-oblate shape transitions above the filled 2p-shell (N = 42, 44) which is wrongly predicted in spheroidal models.

Ionization and fragmentation of metallic clusters (Na n) induced by ion impact

Neutral sodium clusters (Na n , n ≤ 800) have been ionized in collisions with different ion beams (H +, O 5+, Ar 8+) at collision energies between 1 and 19 keV per charge. From an analysis of the measured mass / charge spectrum critical sizes for multiply charged clusters are deduced and compared with those obtained by laser-ionization. The dominant ionization mechanism depends on the charge state of the projectile. For highly charged ions the multi-electron capture process leads to the formation of relatively “cold” multiply charged clusters; in the proton case multiple ionization is discussed in terms of electronic excitation followed by electron evaporation on a sub-nanosecond time scale.