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Hemispheroidal quantum harmonic oscillator
Physics Letters A 372 (2008) 5448–5451
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Hemispheroidal quantum harmonic oscillator D.N. Poenaru a,b,∗ , R.A. Gherghescu a,b , A.V. Solov’yov a , W. Greiner a a b
Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany Horia Hulubei National Institute of Physics and Nuclear Engineering (IFIN-HH), RO-077125 Bucharest-Magurele, Romania
a r t i c l e
i n f o
Article history: Received 17 January 2008 Received in revised form 19 May 2008 Accepted 22 June 2008 Available online 27 June 2008 Communicated by R. Wu PACS: 03.65.Ge 31.10.+z 61.46.Bc
a b s t r a c t A deformed single-particle shell model is derived for a hemispheroidal potential well. Only the negative parity states of the Z ( z) component of the wave function are allowed, so new magic numbers are obtained. The influence of a term proportional to l2 in the Hamiltonian is investigated. The maximum degeneracy is reached at a superdeformed hemispheroidal prolate shape whose magic numbers are identical with those obtained at the spherical shape of the spheroidal harmonic oscillator. This remarkable property suggests an increased stability of such a distorted shape of deposited clusters when the planar surface remains opaque. © 2008 Elsevier B.V. All rights reserved.
Keywords: Solutions of wave equations: Bound states Theory of electronic structure Atomic clusters
The spheroidal harmonic oscillator has been used in various branches of Physics. The famous single-particle Nilsson model [1] is very successful in Nuclear Physics. Its variants [2–4] are of particular interest for atomic clusters [5]. Major spherical-shells N = 2, 8, 20, 40, 58, 92 have been found [2] in the mass spectra of sodium clusters of N atoms per cluster, and the Clemenger’s shell model [3] was able to explain this sequence of spherical magic numbers. We would like to write explicitly the analytical expressions for the energy levels of the spheroidal harmonic oscillator and to derive the corresponding solutions for a hemispheroidal harmonic oscillator which may be useful to study atomic clusters deposited on planar surfaces. The hemispheroidal quantum harmonic oscillator may be used as a reliable approximation of a realistic singleparticle shell model for small magic numbers, giving the input data for shell correction calculations [2,3,6–11]. The nanostructured coating of surfaces by cluster deposition is at present a rapidly growing field [12]. By using scanning probe microscopy (SPM) or atomic force microscopy (AFM) it is possible to observe how the shapes of such clusters are distorted after remaining on the surface. The final shape of some of the them, e.g. in Fig. 1 of Ref. [13], or in Figs. 3, 4 and 9 of Ref. [14], may be
*
Corresponding author at: Frankfurt Institute for Advanced Studies (FIAS), RuthMoufang-Str. 1, D-60438 Frankfurt am Main, Germany. E-mail address: d.poenaru@fias.physik.uni-frankfurt.de (D.N. Poenaru). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.06.054
approximated in the first order by a hemispheroid. This gives us a motivation to develop a shell model accepting a hemispheroidal shape as an equipotential surface of the chosen Hamiltonian. Another argument relies on the 2D measurement [15] of the strong magicity at the number equal to 6, which can be approximated as a limiting case of an extremely large oblate deformation in our model. Also one has to consider the possibility to solve the problem analytically within such an idealization. The advantage of an analytical relationship is not only a very short computational time but also an easier interpretation of the results. In all studies using a harmonic oscillator published since 1955, the maximum degeneracy of the quantum states was reached for a spherical shape, explaining the high stability of the doubly magic nuclei or of the metal clusters with spherical closed shells. Intuitively one would expect to arrive at a similar conclusion for the hemispheroidal shape: maximum degeneracy at zero deformation (hemisphere). To our surprise the maximum stability of the hemispheroidal quantum harmonic oscillator occurs at a superdeformed prolate shape (semiaxes ratio a/c = 1/2), a shape which is also the most stable one within the LDM [16]. Experimentalists have already started to determine the magic numbers of deposited clusters for the two-dimensional case [15]; they will soon begin on the 3D systems. For spheroidal equipotential surfaces, generated by a potential with cylindrical symmetry, the states of the valence electrons were found [3] by using an effective single-particle Hamiltonian with a potential
D.N. Poenaru et al. / Physics Letters A 372 (2008) 5448–5451
5449
Fig. 1. Energy levels in units of h¯ ω0 vs. the deformation parameter δ . Each level is labelled by n, n⊥ quantum numbers shown at the right-hand side, and is (2n⊥ + 2)-fold degenerate. LEFT: Six major shells of a spheroidal harmonic oscillator. RIGHT: Eight major shells of the hemispheroidal harmonic oscillator. n z = n − n⊥ = 1, 3, 5, . . . . The maximum degeneracy is reached at a superdeformed hemispheroidal prolate shape whose magic numbers are identical with those obtained at the spherical shape of the spheroidal harmonic oscillator. 2, 8, 20, 40, 70, 112, 168, . . . .
V =
M ω02 R 20
2
ρ
2
2+δ
2/ 3 +z
2−δ
2
2−δ
(1)
.
2+δ
We begin with this Hamiltonian and neglect for the moment an additional term proportional to (l2 − l2 n ). Then we investigate its influence at the end of the Letter. K.L. Clemenger introduced the deformation δ by expressing the dimensionless two semiaxes (in units of the radius of a sphere with the same volume, R 0 = r s N 1/3 , where r s is the Wigner–Seitz radius, 2.117 Å for Na [9,17]) as
a=
2−δ
1/ 3 c=
,
2+δ
2+δ
2/ 3 (2)
.
2−δ
We use dimensionless cylindrical coordinates ρ and z. Volume conservation leads to a2 c = 1. One can separate the variables in the Schrödinger equation, H Ψ = E Ψ . As a result the wave function [18,19] may be written as
Ψ (η, ξ, ϕ ) = ψnmr (η)Φm (ϕ ) Z nz (ξ ),
(3)
ρ /α ⊥ , m = where each component is orthonormalized, η = (n⊥ − 2i ) with i = 0, 1, . . . up to (n⊥ − 1)/2 for an odd n⊥ or to (n⊥ − 2)/2 for an even n⊥ , nr = (n⊥ − |m|)/2, α⊥ = h¯ / M ω⊥ . We are interested in the z-component R 20
2
2
2 Z n z (ξ ) = N n z e −ξ /2 H n z (ξ ),
Nnz =
√
(αz π
1 2n z n
z
!)1/2
(5)
n = E n /(¯hω0 ),
2
(2 − δ)1/3 (2 + δ)2/3
n+
δ
(7) (8)
and similar formulae for oblate deformations, δ < 0. The low lying energy levels for the six shells (main quantum number n = 0, 1, 2, 3, 4, 5) can be seen in the left-hand side of Fig. 1. Each level, labelled by n⊥ , n, may accommodate 2n⊥ + 2 particles. One n has 2 n⊥ =0 (n⊥ + 1) = (n + 1)(n + 2) atoms in a completely filled shell characterized byn, and the total number of states of the lown lying n + 1 shells is n=0 (n + 1)(n + 2) = (n + 1)(n + 2)(n + 3)/3 leading to the magic numbers 2, 8, 20, 40, 70, 112, 168, . . . for a spherical shape. Besides the important degeneracy at a spherical shape (δ = 0), one also has degeneracies at some superdeformed shapes, e.g. for prolate shapes at the ratio c /a = (2 + δ)/(2 − δ) = 2 i.e. δ = 2/3. More details may be found in Table 1. The first four shells can reproduce the experimental magic numbers 2, 8, 20, 40; in order to describe the other ones (58, 92) Clemenger introduced the term proportional to (l2 − l2 n ) in the Hamiltonian. Inspired by the experimental shapes we mentioned above we consider the simplest approximation as a hemispheroidal deposited cluster with the z axis perpendicular on the surface and the ρ axis in the surface plane. The surface equation is given by 2 2 2 ρ 2 = (a/c ) (c − z ) z 0,
0
E n = h¯ ω⊥ (n⊥ + 1) + h¯ ωz (n z + 1/2)
n =
nmin = (2 − δ)1/3 (2 + δ)2/3 m −
(4)
,
where ξ = R 0 z/αz , αz = h¯ / M ωz , and the main quantum number n = n⊥ + n z = 0, 1, 2, . . . . The parity of the Hermite polynomials H n z (ξ ) is given by (−1)n z . The eigenvalues are
or in units of h¯ ω0 ,
1 −3 , 2 2+δ 1 δ nmax = (2 − δ)1/3 (2 + δ)2/3 m − − 3 , 2 2−δ
4/ 3
3 2
n 1 . + δ n⊥ − + 2
4
(6)
For a prolate spheroid, δ > 0, at n⊥ = 0 the energy level decreases with deformation except for n = 0, but when n⊥ = n it increases. For a given prolate deformation and a maximum energy m , there are nmin closed shells and other levels for high-order shells up to nmax :
z < 0.
(9)
The radius of the hemisphere obtained for the deformation δs = 0 is R s . Volume conservation leads to R s = 21/3 R 0 . We shall give ρ , z, a, c in units of R s instead of R 0 , so that again a2 c = 1. The definition of the δs is the same as that of δ in Eq. (2), but now a and c are expressed in terms of R s instead of R 0 . The new potential well we have to consider is shown in the right-hand side of Fig. 2. The potential along the symmetry axis, V z ( z), has a wall of an infinitely large height at z = 0, and concerns only positive values of z, implying opacity of the surface:
Vz =
∞
M R 2s ω2z z2 /2
z 0, z > 0.
(10)
In this case the wave functions should vanish in the origin, where the potential wall is infinitely high, so that only negative parity Hermite polynomials (n z odd) should be taken into consideration.
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D.N. Poenaru et al. / Physics Letters A 372 (2008) 5448–5451
Table 1 Top: Deformed magic numbers of the spheroidal harmonic oscillator. Bottom: Deformed magic numbers of the hemispheroidal harmonic oscillator Oblate
Prolate
δ
a/c
Magic numbers
δ
a/c
Magic numbers
−0.8/3 − 0.4 −2/3 −1
17/13 1.5 2 3
2, 2, 2, 2,
8, 6, 6, 6,
18, 20, 34, 38, 58, 64, 92, 100, 136, 148, . . . 8, 14, 18, 28, 34, 48, 58, 76, 90, 114, 132, . . . 14, 26, 44, 68, 100, 140, . . . 12, 22, 36, 54, 78, 108, 144, . . .
0.8/3 0.4 2/3 1
13/17 2/3 0.5 1/3
2, 2, 2, 4,
8, 20, 22, 42, 46, 76, 82, 124, 134, . . . 8, 10, 22, 26, 46, 54, 66, 84, 96, 114, 138, 156, . . . 4, 10, 16, 28, 40, 60, 80, 110, 140, . . . 12, 18, 24, 36, 48, 60, 80, 100, 120, 150, . . .
−0.8/3 − 0.4 −2/3 −1
17/13 1.5 2 3
2, 2, 2, 2,
6, 6, 6, 6,
12, 12, 12, 12,
0.8/3 0.4 2/3 1
13/17 2/3 0.5 1/3
2, 2, 2, 2,
6, 8, 8, 8,
22, 22, 20, 20,
26, 36, 32, 30,
36, 54, 48, 42,
42, 78, 68, 58,
56, 64, 82, 92, 114, 126, 154, . . . 108, 144, . . . 92, 122, 158, . . . 78, 102, 130, . . .
8, 14, 18, 28, 34, 48, 58, 76, 90, 114, 132, . . . 18, 20, 34, 38, 50, 58, 64, 80, 92, 100, . . . 20, 40, 70, 112, 168, . . . 10, 14, 22, 26, 46, 54, 66, 84, 96, 114, 138, 156, . . .
Fig. 2. LEFT: Harmonic oscillator potential V = V (ξ ), the wave functions Z n z = Z n z (ξ ) for n z = 0, 1, 2, 3, 4, 5 and the corresponding contributions to the total energy levels znz = E nz /¯hωz = (n z + 1/2) for spherical shapes, δ = 0. ξ = zR 0 / h¯ / M ωz . RIGHT: The similar functions for a hemispherical harmonic oscillator potential. Only negative parity states are retained which are vanishing at ξ = 0 where the potential wall is infinitely high.
From the energy levels given in the left-hand side of Fig. 1 we have to select only those corresponding to this condition. In this way the former lowest level with n = 0, n⊥ = 0 should be excluded. From the two levels with n = 1 we can retain the level with n⊥ = 0 i.e. n z = 1. This will be the lowest level for the hemispherical harmonic oscillator and will accommodate 2n⊥ + 2 = 2 atoms. From the three levels with n = 2 only the one with n z = n⊥ = 1 and 2n⊥ + 2 = 4 degeneracy is retained so that the first two magic numbers at spherical shape (δ = 0) are now 2 followed by 6, etc. Some deformed magic numbers may be found in Table 1 and as position of minima in Fig. 3. Each level, labelled by n⊥ , n, may accommodate 2n⊥ + 2 particles. When n is an odd number, one should only have even n⊥ in order to select the odd n z = n − n⊥ . The contribution of the shells with odd n to the hemispherical magic numbers will be n−1
(2n⊥ + 2) =
neven ⊥ =0
(n + 1)2 2
nodd ⊥ =1
(2n⊥ + 2) =
n(n + 2) 2
The orthonormalization condition of the Z ns z component of the wave function becomes
+∞ Z ns ( z) Z ns z ( z) dz = δnz n z
(13)
z
0
(11)
leading to the sequence 2, 8, 18 for n = 1, 3, 5. The contribution of the shells with even n to the hemispherical magic numbers will be n −1
Fig. 3. Variation of shell corrections with N for Na hemispheroidal clusters. Top: δ = 0. The hemispherical magic numbers are identical with those obtained at the oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, . . . . For a prolate superdeformed (δ = 2/3) shape the magic numbers are identical with those obtained at the spherical shape: 2, 8, 20, 40, 70, 112, 168, . . . . Other magic numbers are given in Table 1. Oblate and prolate shapes are considered on the left-hand side and right– hand side, respectively.
(12)
which gives the sequence 4, 12, 24 for n = 2, 4, 6. This should be interlaced with the preceding one so that the magic numbers will be 2, 2 + 4 = 6, 6 + 8 = 14, 14 + 12 = 26, 26 + 18 = 44, 44 + 24 = 68, as shown at the right-hand side of Fig. 1. Eq. (6) from the harmonic oscillator, in units of h¯ ω0 is still valid, but one should only allow the values of n and n⊥ for which n z = n − n⊥ 1 are odd numbers.
with n z = 1, 3, 5, . . . , n for odd n and n z = 1, 3√, 5, . . . , n − 1 for even n. Consequently the normalization factor is 2 times the preceding one from Eq. (4) Z ns z (ξ ) = Nnz =
√
2 2N n z e −ξ /2 H n z (ξ ),
1
. √ (αz π 2nz n z !)1/2
(14)
The shell gap for an atomic cluster [20] is given by h¯ ω0 ( N ) =
13.72 eV Å rs R s
2
1+
t r s N 1/ 3
−2 .
(15)
Since we consider solely monovalent elements, N in this equation is the number of atoms, and t denotes the electronic spillout for the neutral cluster [20].
D.N. Poenaru et al. / Physics Letters A 372 (2008) 5448–5451
Fig. 4. Hemispheroidal oscillator energy levels in units of h¯ ω0 vs. the deformation parameter δs obtained by taking into account the term proportional to (l2 − l2 n ) in the Hamiltonian.
The shell correction energy, δ U [6], in Fig. 3 shows minima at the oblate and prolate magic numbers given in the lower part of Table 1. The striking result is that the maximum degeneracy is obtained at a superdeformed prolate shape (δs = 2/3). The magic numbers are those of spherical shape (δ = 0). This is the consequence of the fact that the dependence on n and n⊥ of eigenvalues in Eq. (6) for δ = 0 is given simply by n with all possible combinations of n⊥ and n z , while for δs = 2/3 one has n + n⊥ with only odd quantum numbers n z = 1, 3, . . . . In both cases the degeneracy associated to an energy level labelled by n, n⊥ is 2n⊥ + 2. In a similar way at δ = −1 we compare 3n − 2n⊥ with the hemispheroidal 6n − 2n⊥ at δs = −0.4, and at δ = −2/3 the term 2n − n⊥ with n for the hemisphere (δs = 0). By including a term proportional to (l2 − l2 n ) in the Hamiltonian H =−
h¯ 2 2M
+
M ω02 R 20
ρ2 a2
2
+
z2
c2
− h¯ ω0 U l2 − l2 n
(16)
with the dimensionless quantity U = 0.04 and l2 n = n(n + 3)/2 like in Ref. [3] for the spheroidal oscillator, we obtained again an analytical relationship for the energy levels
n =
En h¯ ω0
=
n⊥ + 1 a
+
n z + 1/2 c
−
U m2 4a4
+
U n(n + 3) 2
.
(17)
The terms proportional to U are both diagonal, the first one
−U m2 /(4a4 ) representing the contribution of the ˆl2z part of the angular momentum operator. The possible nondiagonal terms coming from (ˆl+ˆl− + ˆl−ˆl+ )/2 are not present since their contribution vanishes due to the selection rules excluding even values of the quantum number n z . As we mentioned above the quantum num-
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ber m = (n⊥ − 2i ) with i = 0, 1, . . . , so that for n⊥ = 0 one has m = 0—the energy level is not changed. When n⊥ = 1, m = ±1 and the energy is changed but the degeneracy of 4 remains unlifted. For n⊥ = 2, m = ±2, 0 so that one has a split leading to one level with m = 0 and the degeneracy 2, and another with m = ±2 and the degeneracy 4, etc. In this way we obtain the level scheme of Fig. 4 which is different from the scheme plotted in the righthand part of Fig. 1. Nevertheless, for the lower levels (say up to 10 closed shells), the sequence of the magic numbers at the maximum degeneracy, taking place at the superdeformed prolate shape δ = 2/3, remain the same: N = 2, 8, 20, 40, 70, 112, 168. Another remarkable fact is that for very large oblate deformations, leading to “pan-cake” shapes approximating a 2D situation, one of the magic number is 6, in agreement with the experiments performed by Chiu et al. [15]. In conclusion we expect that the present results will get further support from the experiments in which the ultrasensitive microscopy (e.g. SPM or AFM) is used. In particular for a large range of atomic numbers the increased stability of the prolate superdeformed hemispheroidal shapes may be seen for deposited clusters on planar surfaces with high opacity. Acknowledgements This work was partly supported by the network of excellence EXCELL of the European Commission, and by the Ministry of Education and Research, Bucharest within the “IDEI” programme. References [1] S.G. Nilsson, Det Kongelige Danske Videnskabernes Selskab, Dan. Mat. Fys. Medd. 27 (29) (1953). [2] W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, M.L. Cohen, Phys. Rev. Lett. 52 (1984) 2141; W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, M.L. Cohen, Phys. Rev. Lett. 53 (1984) 510, errata. [3] K.L. Clemenger, Phys. Rev. B 32 (1985) 1359. [4] S.M. Reimann, M. Brack, K. Hansen, Z. Phys. D 28 (1993) 235. [5] J.P. Connerade, A.V. Solov’yov, W. Greiner, Europhys. News 33 (2002) 200. [6] V.M. Strutinsky, Nucl. Phys. A 95 (1967) 420. [7] C. Yannouleas, U. Landman, Phys. Rev. B 48 (1993) 8376. [8] C. Bréchignac, et al., Phys. Rev. B 49 (1994) 2825. [9] C. Yannouleas, U. Landman, Phys. Rev. B 51 (1995) 1902. [10] T.P. Martin, Phys. Rep. 273 (1996) 199. [11] C. Yannouleas, U. Landman, Phys. Rev. Lett. 78 (1997) 1424. [12] J.V. Barth, G. Costantini, K. Kern, Nature 437 (2005) 671. [13] K. Seeger, R.E. Palmer, Appl. Phys. Lett. 74 (1999) 1627. [14] B. Bonanni, S. Cannistraro, J. Nanotechnology Online 1 (November 2005) 1, doi: 10.2240/azojono0105. [15] Y.-P. Chiu, L.-W. Huang, C.-M. Wei, C.-S. Chang, T.-T. Tsong, Phys. Rev. Lett. 97 (2006) 165504. [16] D.N. Poenaru, R.A. Gherghescu, A.V. Solov’yov, W. Greiner, Europhys. Lett. 79 (2007) 63001. [17] M. Brack, Phys. Rev. B 39 (1989) 3533. [18] A.J. Rassey, Phys. Rev. 109 (1958) 949. [19] D. Vautherin, Phys. Rev. C 7 (1973) 296. [20] K.L. Clemenger, Ph.D. Dissertation, University of California, Berkeley, 1985.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Hemispheroidal quantum harmonic oscillator D.N. Poenaru a,b,∗ , R.A. Gherghescu a,b , A.V. Solov’yov a , W. Greiner a a b
Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany Horia Hulubei National Institute of Physics and Nuclear Engineering (IFIN-HH), RO-077125 Bucharest-Magurele, Romania
a r t i c l e
i n f o
Article history: Received 17 January 2008 Received in revised form 19 May 2008 Accepted 22 June 2008 Available online 27 June 2008 Communicated by R. Wu PACS: 03.65.Ge 31.10.+z 61.46.Bc
a b s t r a c t A deformed single-particle shell model is derived for a hemispheroidal potential well. Only the negative parity states of the Z ( z) component of the wave function are allowed, so new magic numbers are obtained. The influence of a term proportional to l2 in the Hamiltonian is investigated. The maximum degeneracy is reached at a superdeformed hemispheroidal prolate shape whose magic numbers are identical with those obtained at the spherical shape of the spheroidal harmonic oscillator. This remarkable property suggests an increased stability of such a distorted shape of deposited clusters when the planar surface remains opaque. © 2008 Elsevier B.V. All rights reserved.
Keywords: Solutions of wave equations: Bound states Theory of electronic structure Atomic clusters
The spheroidal harmonic oscillator has been used in various branches of Physics. The famous single-particle Nilsson model [1] is very successful in Nuclear Physics. Its variants [2–4] are of particular interest for atomic clusters [5]. Major spherical-shells N = 2, 8, 20, 40, 58, 92 have been found [2] in the mass spectra of sodium clusters of N atoms per cluster, and the Clemenger’s shell model [3] was able to explain this sequence of spherical magic numbers. We would like to write explicitly the analytical expressions for the energy levels of the spheroidal harmonic oscillator and to derive the corresponding solutions for a hemispheroidal harmonic oscillator which may be useful to study atomic clusters deposited on planar surfaces. The hemispheroidal quantum harmonic oscillator may be used as a reliable approximation of a realistic singleparticle shell model for small magic numbers, giving the input data for shell correction calculations [2,3,6–11]. The nanostructured coating of surfaces by cluster deposition is at present a rapidly growing field [12]. By using scanning probe microscopy (SPM) or atomic force microscopy (AFM) it is possible to observe how the shapes of such clusters are distorted after remaining on the surface. The final shape of some of the them, e.g. in Fig. 1 of Ref. [13], or in Figs. 3, 4 and 9 of Ref. [14], may be
*
Corresponding author at: Frankfurt Institute for Advanced Studies (FIAS), RuthMoufang-Str. 1, D-60438 Frankfurt am Main, Germany. E-mail address: d.poenaru@fias.physik.uni-frankfurt.de (D.N. Poenaru). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.06.054
approximated in the first order by a hemispheroid. This gives us a motivation to develop a shell model accepting a hemispheroidal shape as an equipotential surface of the chosen Hamiltonian. Another argument relies on the 2D measurement [15] of the strong magicity at the number equal to 6, which can be approximated as a limiting case of an extremely large oblate deformation in our model. Also one has to consider the possibility to solve the problem analytically within such an idealization. The advantage of an analytical relationship is not only a very short computational time but also an easier interpretation of the results. In all studies using a harmonic oscillator published since 1955, the maximum degeneracy of the quantum states was reached for a spherical shape, explaining the high stability of the doubly magic nuclei or of the metal clusters with spherical closed shells. Intuitively one would expect to arrive at a similar conclusion for the hemispheroidal shape: maximum degeneracy at zero deformation (hemisphere). To our surprise the maximum stability of the hemispheroidal quantum harmonic oscillator occurs at a superdeformed prolate shape (semiaxes ratio a/c = 1/2), a shape which is also the most stable one within the LDM [16]. Experimentalists have already started to determine the magic numbers of deposited clusters for the two-dimensional case [15]; they will soon begin on the 3D systems. For spheroidal equipotential surfaces, generated by a potential with cylindrical symmetry, the states of the valence electrons were found [3] by using an effective single-particle Hamiltonian with a potential
D.N. Poenaru et al. / Physics Letters A 372 (2008) 5448–5451
5449
Fig. 1. Energy levels in units of h¯ ω0 vs. the deformation parameter δ . Each level is labelled by n, n⊥ quantum numbers shown at the right-hand side, and is (2n⊥ + 2)-fold degenerate. LEFT: Six major shells of a spheroidal harmonic oscillator. RIGHT: Eight major shells of the hemispheroidal harmonic oscillator. n z = n − n⊥ = 1, 3, 5, . . . . The maximum degeneracy is reached at a superdeformed hemispheroidal prolate shape whose magic numbers are identical with those obtained at the spherical shape of the spheroidal harmonic oscillator. 2, 8, 20, 40, 70, 112, 168, . . . .
V =
M ω02 R 20
2
ρ
2
2+δ
2/ 3 +z
2−δ
2
2−δ
(1)
.
2+δ
We begin with this Hamiltonian and neglect for the moment an additional term proportional to (l2 − l2 n ). Then we investigate its influence at the end of the Letter. K.L. Clemenger introduced the deformation δ by expressing the dimensionless two semiaxes (in units of the radius of a sphere with the same volume, R 0 = r s N 1/3 , where r s is the Wigner–Seitz radius, 2.117 Å for Na [9,17]) as
a=
2−δ
1/ 3 c=
,
2+δ
2+δ
2/ 3 (2)
.
2−δ
We use dimensionless cylindrical coordinates ρ and z. Volume conservation leads to a2 c = 1. One can separate the variables in the Schrödinger equation, H Ψ = E Ψ . As a result the wave function [18,19] may be written as
Ψ (η, ξ, ϕ ) = ψnmr (η)Φm (ϕ ) Z nz (ξ ),
(3)
ρ /α ⊥ , m = where each component is orthonormalized, η = (n⊥ − 2i ) with i = 0, 1, . . . up to (n⊥ − 1)/2 for an odd n⊥ or to (n⊥ − 2)/2 for an even n⊥ , nr = (n⊥ − |m|)/2, α⊥ = h¯ / M ω⊥ . We are interested in the z-component R 20
2
2
2 Z n z (ξ ) = N n z e −ξ /2 H n z (ξ ),
Nnz =
√
(αz π
1 2n z n
z
!)1/2
(5)
n = E n /(¯hω0 ),
2
(2 − δ)1/3 (2 + δ)2/3
n+
δ
(7) (8)
and similar formulae for oblate deformations, δ < 0. The low lying energy levels for the six shells (main quantum number n = 0, 1, 2, 3, 4, 5) can be seen in the left-hand side of Fig. 1. Each level, labelled by n⊥ , n, may accommodate 2n⊥ + 2 particles. One n has 2 n⊥ =0 (n⊥ + 1) = (n + 1)(n + 2) atoms in a completely filled shell characterized byn, and the total number of states of the lown lying n + 1 shells is n=0 (n + 1)(n + 2) = (n + 1)(n + 2)(n + 3)/3 leading to the magic numbers 2, 8, 20, 40, 70, 112, 168, . . . for a spherical shape. Besides the important degeneracy at a spherical shape (δ = 0), one also has degeneracies at some superdeformed shapes, e.g. for prolate shapes at the ratio c /a = (2 + δ)/(2 − δ) = 2 i.e. δ = 2/3. More details may be found in Table 1. The first four shells can reproduce the experimental magic numbers 2, 8, 20, 40; in order to describe the other ones (58, 92) Clemenger introduced the term proportional to (l2 − l2 n ) in the Hamiltonian. Inspired by the experimental shapes we mentioned above we consider the simplest approximation as a hemispheroidal deposited cluster with the z axis perpendicular on the surface and the ρ axis in the surface plane. The surface equation is given by 2 2 2 ρ 2 = (a/c ) (c − z ) z 0,
0
E n = h¯ ω⊥ (n⊥ + 1) + h¯ ωz (n z + 1/2)
n =
nmin = (2 − δ)1/3 (2 + δ)2/3 m −
(4)
,
where ξ = R 0 z/αz , αz = h¯ / M ωz , and the main quantum number n = n⊥ + n z = 0, 1, 2, . . . . The parity of the Hermite polynomials H n z (ξ ) is given by (−1)n z . The eigenvalues are
or in units of h¯ ω0 ,
1 −3 , 2 2+δ 1 δ nmax = (2 − δ)1/3 (2 + δ)2/3 m − − 3 , 2 2−δ
4/ 3
3 2
n 1 . + δ n⊥ − + 2
4
(6)
For a prolate spheroid, δ > 0, at n⊥ = 0 the energy level decreases with deformation except for n = 0, but when n⊥ = n it increases. For a given prolate deformation and a maximum energy m , there are nmin closed shells and other levels for high-order shells up to nmax :
z < 0.
(9)
The radius of the hemisphere obtained for the deformation δs = 0 is R s . Volume conservation leads to R s = 21/3 R 0 . We shall give ρ , z, a, c in units of R s instead of R 0 , so that again a2 c = 1. The definition of the δs is the same as that of δ in Eq. (2), but now a and c are expressed in terms of R s instead of R 0 . The new potential well we have to consider is shown in the right-hand side of Fig. 2. The potential along the symmetry axis, V z ( z), has a wall of an infinitely large height at z = 0, and concerns only positive values of z, implying opacity of the surface:
Vz =
∞
M R 2s ω2z z2 /2
z 0, z > 0.
(10)
In this case the wave functions should vanish in the origin, where the potential wall is infinitely high, so that only negative parity Hermite polynomials (n z odd) should be taken into consideration.
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D.N. Poenaru et al. / Physics Letters A 372 (2008) 5448–5451
Table 1 Top: Deformed magic numbers of the spheroidal harmonic oscillator. Bottom: Deformed magic numbers of the hemispheroidal harmonic oscillator Oblate
Prolate
δ
a/c
Magic numbers
δ
a/c
Magic numbers
−0.8/3 − 0.4 −2/3 −1
17/13 1.5 2 3
2, 2, 2, 2,
8, 6, 6, 6,
18, 20, 34, 38, 58, 64, 92, 100, 136, 148, . . . 8, 14, 18, 28, 34, 48, 58, 76, 90, 114, 132, . . . 14, 26, 44, 68, 100, 140, . . . 12, 22, 36, 54, 78, 108, 144, . . .
0.8/3 0.4 2/3 1
13/17 2/3 0.5 1/3
2, 2, 2, 4,
8, 20, 22, 42, 46, 76, 82, 124, 134, . . . 8, 10, 22, 26, 46, 54, 66, 84, 96, 114, 138, 156, . . . 4, 10, 16, 28, 40, 60, 80, 110, 140, . . . 12, 18, 24, 36, 48, 60, 80, 100, 120, 150, . . .
−0.8/3 − 0.4 −2/3 −1
17/13 1.5 2 3
2, 2, 2, 2,
6, 6, 6, 6,
12, 12, 12, 12,
0.8/3 0.4 2/3 1
13/17 2/3 0.5 1/3
2, 2, 2, 2,
6, 8, 8, 8,
22, 22, 20, 20,
26, 36, 32, 30,
36, 54, 48, 42,
42, 78, 68, 58,
56, 64, 82, 92, 114, 126, 154, . . . 108, 144, . . . 92, 122, 158, . . . 78, 102, 130, . . .
8, 14, 18, 28, 34, 48, 58, 76, 90, 114, 132, . . . 18, 20, 34, 38, 50, 58, 64, 80, 92, 100, . . . 20, 40, 70, 112, 168, . . . 10, 14, 22, 26, 46, 54, 66, 84, 96, 114, 138, 156, . . .
Fig. 2. LEFT: Harmonic oscillator potential V = V (ξ ), the wave functions Z n z = Z n z (ξ ) for n z = 0, 1, 2, 3, 4, 5 and the corresponding contributions to the total energy levels znz = E nz /¯hωz = (n z + 1/2) for spherical shapes, δ = 0. ξ = zR 0 / h¯ / M ωz . RIGHT: The similar functions for a hemispherical harmonic oscillator potential. Only negative parity states are retained which are vanishing at ξ = 0 where the potential wall is infinitely high.
From the energy levels given in the left-hand side of Fig. 1 we have to select only those corresponding to this condition. In this way the former lowest level with n = 0, n⊥ = 0 should be excluded. From the two levels with n = 1 we can retain the level with n⊥ = 0 i.e. n z = 1. This will be the lowest level for the hemispherical harmonic oscillator and will accommodate 2n⊥ + 2 = 2 atoms. From the three levels with n = 2 only the one with n z = n⊥ = 1 and 2n⊥ + 2 = 4 degeneracy is retained so that the first two magic numbers at spherical shape (δ = 0) are now 2 followed by 6, etc. Some deformed magic numbers may be found in Table 1 and as position of minima in Fig. 3. Each level, labelled by n⊥ , n, may accommodate 2n⊥ + 2 particles. When n is an odd number, one should only have even n⊥ in order to select the odd n z = n − n⊥ . The contribution of the shells with odd n to the hemispherical magic numbers will be n−1
(2n⊥ + 2) =
neven ⊥ =0
(n + 1)2 2
nodd ⊥ =1
(2n⊥ + 2) =
n(n + 2) 2
The orthonormalization condition of the Z ns z component of the wave function becomes
+∞ Z ns ( z) Z ns z ( z) dz = δnz n z
(13)
z
0
(11)
leading to the sequence 2, 8, 18 for n = 1, 3, 5. The contribution of the shells with even n to the hemispherical magic numbers will be n −1
Fig. 3. Variation of shell corrections with N for Na hemispheroidal clusters. Top: δ = 0. The hemispherical magic numbers are identical with those obtained at the oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, . . . . For a prolate superdeformed (δ = 2/3) shape the magic numbers are identical with those obtained at the spherical shape: 2, 8, 20, 40, 70, 112, 168, . . . . Other magic numbers are given in Table 1. Oblate and prolate shapes are considered on the left-hand side and right– hand side, respectively.
(12)
which gives the sequence 4, 12, 24 for n = 2, 4, 6. This should be interlaced with the preceding one so that the magic numbers will be 2, 2 + 4 = 6, 6 + 8 = 14, 14 + 12 = 26, 26 + 18 = 44, 44 + 24 = 68, as shown at the right-hand side of Fig. 1. Eq. (6) from the harmonic oscillator, in units of h¯ ω0 is still valid, but one should only allow the values of n and n⊥ for which n z = n − n⊥ 1 are odd numbers.
with n z = 1, 3, 5, . . . , n for odd n and n z = 1, 3√, 5, . . . , n − 1 for even n. Consequently the normalization factor is 2 times the preceding one from Eq. (4) Z ns z (ξ ) = Nnz =
√
2 2N n z e −ξ /2 H n z (ξ ),
1
. √ (αz π 2nz n z !)1/2
(14)
The shell gap for an atomic cluster [20] is given by h¯ ω0 ( N ) =
13.72 eV Å rs R s
2
1+
t r s N 1/ 3
−2 .
(15)
Since we consider solely monovalent elements, N in this equation is the number of atoms, and t denotes the electronic spillout for the neutral cluster [20].
D.N. Poenaru et al. / Physics Letters A 372 (2008) 5448–5451
Fig. 4. Hemispheroidal oscillator energy levels in units of h¯ ω0 vs. the deformation parameter δs obtained by taking into account the term proportional to (l2 − l2 n ) in the Hamiltonian.
The shell correction energy, δ U [6], in Fig. 3 shows minima at the oblate and prolate magic numbers given in the lower part of Table 1. The striking result is that the maximum degeneracy is obtained at a superdeformed prolate shape (δs = 2/3). The magic numbers are those of spherical shape (δ = 0). This is the consequence of the fact that the dependence on n and n⊥ of eigenvalues in Eq. (6) for δ = 0 is given simply by n with all possible combinations of n⊥ and n z , while for δs = 2/3 one has n + n⊥ with only odd quantum numbers n z = 1, 3, . . . . In both cases the degeneracy associated to an energy level labelled by n, n⊥ is 2n⊥ + 2. In a similar way at δ = −1 we compare 3n − 2n⊥ with the hemispheroidal 6n − 2n⊥ at δs = −0.4, and at δ = −2/3 the term 2n − n⊥ with n for the hemisphere (δs = 0). By including a term proportional to (l2 − l2 n ) in the Hamiltonian H =−
h¯ 2 2M
+
M ω02 R 20
ρ2 a2
2
+
z2
c2
− h¯ ω0 U l2 − l2 n
(16)
with the dimensionless quantity U = 0.04 and l2 n = n(n + 3)/2 like in Ref. [3] for the spheroidal oscillator, we obtained again an analytical relationship for the energy levels
n =
En h¯ ω0
=
n⊥ + 1 a
+
n z + 1/2 c
−
U m2 4a4
+
U n(n + 3) 2
.
(17)
The terms proportional to U are both diagonal, the first one
−U m2 /(4a4 ) representing the contribution of the ˆl2z part of the angular momentum operator. The possible nondiagonal terms coming from (ˆl+ˆl− + ˆl−ˆl+ )/2 are not present since their contribution vanishes due to the selection rules excluding even values of the quantum number n z . As we mentioned above the quantum num-
5451
ber m = (n⊥ − 2i ) with i = 0, 1, . . . , so that for n⊥ = 0 one has m = 0—the energy level is not changed. When n⊥ = 1, m = ±1 and the energy is changed but the degeneracy of 4 remains unlifted. For n⊥ = 2, m = ±2, 0 so that one has a split leading to one level with m = 0 and the degeneracy 2, and another with m = ±2 and the degeneracy 4, etc. In this way we obtain the level scheme of Fig. 4 which is different from the scheme plotted in the righthand part of Fig. 1. Nevertheless, for the lower levels (say up to 10 closed shells), the sequence of the magic numbers at the maximum degeneracy, taking place at the superdeformed prolate shape δ = 2/3, remain the same: N = 2, 8, 20, 40, 70, 112, 168. Another remarkable fact is that for very large oblate deformations, leading to “pan-cake” shapes approximating a 2D situation, one of the magic number is 6, in agreement with the experiments performed by Chiu et al. [15]. In conclusion we expect that the present results will get further support from the experiments in which the ultrasensitive microscopy (e.g. SPM or AFM) is used. In particular for a large range of atomic numbers the increased stability of the prolate superdeformed hemispheroidal shapes may be seen for deposited clusters on planar surfaces with high opacity. Acknowledgements This work was partly supported by the network of excellence EXCELL of the European Commission, and by the Ministry of Education and Research, Bucharest within the “IDEI” programme. References [1] S.G. Nilsson, Det Kongelige Danske Videnskabernes Selskab, Dan. Mat. Fys. Medd. 27 (29) (1953). [2] W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, M.L. Cohen, Phys. Rev. Lett. 52 (1984) 2141; W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, M.L. Cohen, Phys. Rev. Lett. 53 (1984) 510, errata. [3] K.L. Clemenger, Phys. Rev. B 32 (1985) 1359. [4] S.M. Reimann, M. Brack, K. Hansen, Z. Phys. D 28 (1993) 235. [5] J.P. Connerade, A.V. Solov’yov, W. Greiner, Europhys. News 33 (2002) 200. [6] V.M. Strutinsky, Nucl. Phys. A 95 (1967) 420. [7] C. Yannouleas, U. Landman, Phys. Rev. B 48 (1993) 8376. [8] C. Bréchignac, et al., Phys. Rev. B 49 (1994) 2825. [9] C. Yannouleas, U. Landman, Phys. Rev. B 51 (1995) 1902. [10] T.P. Martin, Phys. Rep. 273 (1996) 199. [11] C. Yannouleas, U. Landman, Phys. Rev. Lett. 78 (1997) 1424. [12] J.V. Barth, G. Costantini, K. Kern, Nature 437 (2005) 671. [13] K. Seeger, R.E. Palmer, Appl. Phys. Lett. 74 (1999) 1627. [14] B. Bonanni, S. Cannistraro, J. Nanotechnology Online 1 (November 2005) 1, doi: 10.2240/azojono0105. [15] Y.-P. Chiu, L.-W. Huang, C.-M. Wei, C.-S. Chang, T.-T. Tsong, Phys. Rev. Lett. 97 (2006) 165504. [16] D.N. Poenaru, R.A. Gherghescu, A.V. Solov’yov, W. Greiner, Europhys. Lett. 79 (2007) 63001. [17] M. Brack, Phys. Rev. B 39 (1989) 3533. [18] A.J. Rassey, Phys. Rev. 109 (1958) 949. [19] D. Vautherin, Phys. Rev. C 7 (1973) 296. [20] K.L. Clemenger, Ph.D. Dissertation, University of California, Berkeley, 1985.