Brack, Matthias

Genzken, O.

English

University of Regensburg Publication Server

Selfconsistent calculation of electronic supershells in metal clusters

S E L F C O N S I S T E N T C A L C U L A T I O N O F E L E C T R O N I C S U P E R S H E L L S I N M E T A L C L U S T E R S 0 . Genzken and M . Brack Inst. f. Theor. Physik, Universitdi Regensburg, D-8400 Regensburg, Germany A b s t r a c t : We report on self consistent microscopic calculations of the electronic shell and supershell structure of sodium clusters w i th up to N ~ 3000 atoms. The spherical je l l ium model i n local density approximation is used and the Kohn-Sham equations are solved numerically. The finite temperature of the valence electrons is included by treating them as a canonical subsystem embedded i n the heat bath of the ions. In particular, we evaluate the total free energy F(N) and investigate its fluctuating part , the shell-correction energy 8F(N), as a function of temperature T and particle number N. We also discuss the second difference A2-F(JV) = F(N +1) + F(N — 1) — 2F(N) and its relation to the recently measured cluster mass abundance spectra. Meta l clusters provide a unique example for the study of shell effects i n finite fermion systems containing up to several thousand particles [1]. A selfconsistent microscopic description of such large systems is possible only i n the so-called self-consistent je l l ium model [2], where the interacting valence electrons move i n the field of a uniformly charged sphere ( ' jellium') representing the ions. We report here on recent Kohn-Sham calculations for sodium clusters wi th up to N ~ 3000 atoms at finite temperatures [3]. Nishioka et al. [4], using a phenomenological Woods-Saxon potential fitted to the microscopic potentials of Ekardt [2], have drawn attention to the 'supershell structure' i n the level density and the oscillating part of the total binding energy of clusters wi th N up to 4000: a pronounced beating pattern i n which the shell structure is enveloped by a slowly oscillating amplitude. This is, i n fact, a very general feature of discrete eigenmodes i n a cavity or i n any steep potential confining many particles to a l imited domain of space. Ba l ian and Bloch have shown i n their fundamental work [5] that the beating pattern of the level density i n an infinite square well is explained by the superposition of amplitudes associated to closed classical trajectories; they reproduced the approximate shape of the exact level density by summing up only the contributions from triangular and squared classical orbits. Extending this method to smooth potentials, Strutinsky et al. [6] were able to explain the gross shell structure of atomic nuclei using realistic deformed shell model potentials. Nucle i , however, are not big enough to exhibit the 'supershell ' beating of their shell structure. The experimental observation of supershell structure i n cluster expansion sources is inhibited by the fact that the clusters so produced have, at least init ial ly , a finite temperature which tends to reduce the shell effects [7,8,9]. Nevertheless, i n the newest sodium vapour expansion experiments of the Copenhagen - Orsay - Stuttgart collaboration [1], a supershell beating i n the mass abundance spectrum of sodium clusters has been put into evidence. We shall show here that the experimental results can be explained semi-quantitatively i n the selfconsistent je l l ium model, i f the effect of finite temperature on the valence electrons is properly included. We use the finite-temperature density functional theory i n the Kohn-Sham ( K S ) approach [10], employing the spherical je l l ium model and the local-density functional for exchange and correlations by Gunnarsson and Lundqvist [11]. The Wigner-Seitz radius of bulk sodium, ra = 3.96 a.u., is used; otherwise our calculations are com-pletely parameter free. We treat the valence electrons as a canonical ensemble i n the heat bath of the ions and minimize the Helmholtz free energy F(N) = E(N) — TS(N) of a cluster w i th N atoms, where E is its total internal energy and S the entropy of the electrons at a given temperature T . (See Ref. [9] for details and, i n particular, for a fast algorithm for the exact calculation of the canonical part i t ion function.) A canonical treatment wi th exactly conserved particle number N is important here since we investigate quantities like A2F(N) = F(N + 1) + F(N - 1) - 2F(N) which are very sensitive to temperature effects through the entropy part —TS: the large degeneracies of the spherical magic shells lead to large entropies even at small tem-peratures. The quantity A2E(N) = A2F(N)(T = 0) has often been taken as a measure for the stabil ity of the cluster: since it represents the curvature of the total binding energy as a function of TV, it is particularly large for the 'magic' systems which have a strongly negative shell correction. It has furthermore been argued [7] that i f the evaporation process, which takes place immediately after the adiabatic expansion, is responsible for enhancing the most stable clusters i n the final mass yie ld , A2F(N) should be proportional to — A i In Jyv, where IN is the fluctuating part of the observed mass yield. In Refs. [8,9] it has been shown that A2F(N) decreases very fast for N ~ 100 - 400 and becomes practically zero for JV > 500 already at T> 400 K . Therefore, the smoothing effect of a finite temperature on quantities like A2F(N) is very crucial for the observability of shell structure and, i n particular, the supershells i n large metal clusters. The temperature dependence of shell structure has been well studied i n nuclei, both schematically [12] and i n selfconsistent Hartree-Fock calculations [13]. In a schematic harmonic oscillator approximation, the amplitude of the shell-correction 6F — and thus also of the quantity A2F(N) in which the average energies pract i -cally cancel — is found to go like SF(T) = SF(0) r / ( S i n h r ) wi th r = 2ir2T/hw. Expanding for large temperatures and using hu oc TV"" 1 / 3 , this gives a temperature suppression factor oc exp(—JV 1 / 3 ) . Pedersen et al. [1] therefore mult ipl ied the loga-r i thmic derivatives of the mass yields IN by y/N exp(c iV 1 / 3 ) , where c is a constant containing an effective temperature, and the root factor compensates the decrease of the shell-correction at T = 0 with increasing N [12]. In the resulting plot, magic shell closures wi th N up to ~ 2720 and a beating of the shell oscillations can clearly be seen (see also F i g . 2 below). A s an example of our theoretical results, we show i n F i g . 1 the free energy shell-correction 6F(N) versus i V 1 / 3 at the three temperatures T = 0 K , 400 K and 600 K . [Hereby we simply used SF(N) = F(N) — F(N) w i th a l iquid drop model type expansion for the average free energy, F(N) = e\>N + aaN2^3 + acN1^3, determining as and ac at each temperature by a simple eye fit such that 6F(N) is oscillating around zero. The bulk energy is fixed at its theoretical value e& = -2.2567 eV.] The salient feature of the curves i n F i g . 1 is the supershell beating of the otherwise quite regular shell structure. The T = 0 curve is very similar to that obtained by Nishioka et al. [4] for a phenomenological Woods-Saxon potential; note that the present results are fully selfconsistent. The amplitude of the shell effects is clearly reduced wi th increasing temperature. In Figure 2, we have reproduced the relevant figure from the experimental analysis of Pedersen et al. [1] and compare it to our theoretical results. Here the negative second difference —A2F(N) is shown, multipl ied by the same enhancement factor (with the value of c readjusted by ~ 10%). In spite of the simplyfing assumptions underlying the identification of — A2F(N) w i th A i l n [ 7 , 9 ] , the agreement of the two curves is str iking. Th is demonstrates that the finite temperature of the valence electrons which alone contribute to the quantities shown here — the ionic parts of the free energies practically cancel i n the differences A2F(N) and A i In IN — plays an essential role i n the mass yields and can be correctly taken into account i n selfconsistent K S calculations even i n the simple je l l ium model. M u c h remains, however, to be understood - i n particular the value of the factor c which, using the above harmonic oscillator estimates, is too large for the estimated temperatures [1,7] of ~ 400 — 500 K . A more realistic study of the evaporation mechanism and, more generally, a non-equilibrium treatment of the ions' dynamics would be desirable to this a im. We are grateful to S. Bj0rnholm for many enlightening discussions and a continu-ing encouragement, and to K . Hansen for important contributions at the early stages of our investigations. References [1] J . Pedersen, S. Bj0rnholm, J . Borggreen, K . Hansen, T . P . M a r t i n and H . D . Rasmussen, Nature 3 5 3 , 733 (1991); T . P. M a r t i n , S. B j0rnholm, J . Borggreen, C. Brechignac, P h . Cahuzac, K . Hansen and J . Pedersen, Chem. Phys . Lett . 186 , 53 (1991); see also the talks of S. Bj0rnholm and T . P . M a r t i n , these Proceedings. [2] W . Ekardt , Phys . Rev. B 29 , 1558 (1984). [3] O. Genzken and M . Brack, Phys . Rev. Lett . 76 , 3286 (1991). [4] H . Nishioka, K . Hansen, and B . R . Mottelson, Phys . Rev. B 42 , 9377 (1990). [5] R . B a l i a n , C . B loch , A n n . Phys . (N .Y . ) 69 , 76 (1971). [6] V . M . Strutinsky, A . G . Magner, S. R . Ofengenden, and T . D0ssing, Z. Phys . A 283 , 269 (1977). [7] S. B j0rnholm, J . Borggreen, O. Echt , K . Hansen, J . Pedersen, H . D . Rasmussen, Z. Phys . D 19, 47 (1991). [8] M . Brack, O. Genzken, and K . Hansen, Z. Phys. D 19, 51 (1991). [9] M . Brack, O. Genzken, and K . Hansen, Z. Phys. D 2 1 , 65 (1991). [10] For a review on finite-temperature density functional theory, see U . G u p t a and A . K . Rajagopal , Physics Reports 87 , 259 (1982); for its justification for canonical systems, see R . Evans, A d v . i n Phys . 28 , 143 (1979). [11] O. Gunnarsson and B . I. Lundqvist , Phys. Rev. B 13 , 4274 (1976). [12] A . Bohr , and B . M . Mottelson, Nuclear Structure / / ( B e n j a m i n , 1975). [13] See, e.g., M . Brack and P. Quentin, Nuc l . Phys. A 3 6 1 , 35 (1981), and earlier references quoted there. > •SL o G O - 2 i—i—i—i—r — i — i — i — i — i — i — i — i — i — i — i — r T = 600K i . . * i « « » I i — i i l i — 10 12 14 > 3 0 u_ C O - 2 - i 1 i i i 1 i I - 1 1 — i — i — I — r -T= 400 K, 10 12 14 - 2 - 4 - 6 1 -- i 1 1 1 1 1 1 1 1 1 1 1 1 1 — i 1 r -T = OK C M m oo ^ co v - .j- 1 rX CM CM m o o m co m T - co CM I "1 CM O O ' ° 0 M3 ^ T - ^ v o C O CM CM m CM • i I i i i — l — i — 10 12 14 N 1/3 F i g u r e 1: Free energy shell-correction SF(N) = F(N) - F(N) for spherical Na clusters versus N1^3 for three different temperatures T , obtained in selfconsistent K S calculations [3]. L D M parameters used at T = 0 K : a , = 0.6259, a c = 0.2041; at T = 400 K : a, = 0.5918, a c = 0.3796; and at T = 600 K : a, = 0.5755, ac = 0.4204 (all in eV). Numbers near the bottom are the magic numbers of filled major spherical electronic shells. 10 r 8 -6 -y— Z cn 4 -2 | ^2 o 0 exp -2 X Q -4 ; z -6 -c -8 --10 --12 L 100 CLUSTER SIZE, N 500 1000 2000 7 8 9 10 11 12 13 14 15 m CL X X < i 10 - 2 0 r r - 3 0 -- 4 0 i ' i ' i ' i i i ' i i i i i ' i T= 600K • i i i i 8 N 1/3 10 12 > i i 14 F i g u r e 2: Upper part: Relative variation < A j In IN >K0 in experimental Na cluster abundance IN versus TV 1 / 3 (Pedersen et al. [1]; see this reference for the details). Lower part: Negative second difference — ^ F^N) of free energy obtained in selfconsistent K S calculations [3] at T = 600 K . Both quantities are enhanced by a iV-dependent factor to compensate for the temperature suppression (see text). 

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Selfconsistent calculation of electronic supershells in metal clusters

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