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Magic numbers in alkali-halide microclusters
Volume 150, number 5,6,7
PHYSICS LETTERSA
12 November 1990
Magic numbers in alkali-halide microclusters -AG.S. A n a g n o s t a t o s Centerfor TheoreticalPhysics, Laboratoryfor Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA and Institute of Nuclear Physics, Demokritos National Research Centerfor PhysicalSciences, GR-15310 Aghia Paruskevi, Attiki, Greece
Received 13 March 1990; revised manuscript received 9 July 1990; accepted for publication 11 September 1990 Communicatedby D. Bloch
Close packing of soft spheres, each standing for the envelope of both ions in an alkali.halide molecule and forming nested octahedral shells, is proposed for the structure of alkali-halide microclusters (magic numbers: 6, 14, 18, 20, 24, 30, 32, 38, ...). Thus the structure for low photon energy of ionization correspondsto a metastable structure which is transformed into a nested icosahedral-shellground-state structure (magic numbers: 7, 10, 13, 17, 19, 25, ...).
1. Introduction
Studies of alkali-halide small clusters belong to the general category of those of ionic clusters [ 1 ], where the stable configurations are determined by numerically minimizing the total energy. In ref. [ 1 ], both rigid ion and polarizable ion models are used to describe the two particle interactions, while in another study [2 ] a simple shell model is used for the interionic forces. Because of the long-range forces between ions, close packed structures are not necessarily the most favorable as they are in rare gas microclusters. In both cases, however, cluster stability is mainly determined by the cluster's geometrical structure which assumes a variety of different equilibrium shapes [ 1 ]. Enhanced abundances of neutral alkali-halide small clusters (i.e. M,X,, where M stands for alkali and X for halide ions), which come from the secondary-ion mass spectrometry resulting in the stability of the ion alkali-halide microclusters (i.e. M , Z , _ 1 ) by using different preparation procedures, have been found [3,4] for n~=6, 14, 18, 20, 24, 30, 32, 38 ..... For *~ This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract No. DE-AC0276ER03069. Elsevier Science Publishers B.V. (North-Holland)
low ionization-photon energies, however, neutral clusters (e.g. for CsI) enhancements appear [5] at n2=7, 10, 13, 17, 19, 25, .... The existing explanation of the second series of magic numbers (n2) utilizes the pentagonal growth sequence [6,7 ] while the existing explanation of the first series (nt) employs cubic- and rectangular-like structures [3,4] (i.e., structures made up of small fractions of the rock salt lattice). The purpose of the present work is to give a different, common explanation of both series of magic numbers by using close packing of soft spheres forming nested polyhedra. Thus, the present approach, in a broader sense, resembles that of rare-gas microclusters and supports the view that microclusters of different elements could be understood, more or less, in a unified way. Here a somehow spherical-like shape of alkali-halide molecule ( M X ) is implied. The alkali-halide sphere-like molecule here stands for the envelope of both ions and demonstrates the lack of a specific orientation of the ion-dimer within this sphere. Perhaps, this sphere-envelope is defined by the continuous change of the aforementioned orientation in such a way that the whole sphere is occupied in different times. The basic reason for the different shapes assigned 303
Volume 150,number 5,6,7
PHYSICS LETTERSA
to the structure of alkali-halide microclusters, here and in previous publications [ 1,3,4,6,7 ] are the different building elements employed in the two cases. In the present work only one building element, the alkali-halide molecule considered as a soft sphere (in the way described above), is used, while in all previous publications two building elements, the alkali and the halide ions considered as separate entities, are employed. Thus, in the previous works on alkalihalides, like in their crystals, to each alkali or halide ion is assigned a different site of the structure, while in the present paper to each alkali-halide molecule (i.e. to one alkali and one halide ion together) is assigned a site of the structure. As a result the present structure has a somewhat spherical-like shape, while the previously proposed structures have shapes far from spherical ones. Differences in the shapes of alkali-halide microclusters in the literature, among other reasons, come from referring either to different sets of characteristic enhancements (e.g. refs. [6] and [7] refer to the n2 sequence of magic numbers, while refs. [ 3 ] and [4] refer to the n~ sequence) or from referring to the same sequence but different sizes of clusters (e.g. ref. [ 1 ] refers to sizes up to n = 12, while ref. [ 3 ] refers to sizes with n >I 14).
2. The model
In fig. 1 the first five successive shells proposed here for the structure of alkali-halide microclusters are shown. These shells are made of close-packed spheres standing for atoms with 2, 3, 4, 5 and 6
t st
Shell
2 nd Shell
® (6)
[6] (ts)
3 rd
Q
[z4] (3e)
spheres at each edge of the five parallel nested regular octahedra (as shown). At the bottom of each block, the number of spheres accommodated by the relevant shell is given inside parentheses, while the cumulative number of spheres accommodated by all previous and that shell is also given inside square brackets. Numbers inside spheres at each shell stand for the number of vertices of the specific equilibrium polyhedron formed by the spheres characterized by the same number. Specifically, the number 6 inside the spheres at the comers of each octahedral shell denotes the number of vertices of the related octahedron. Also, the number 8 inside spheres denotes the number of middles of the faces of the (third) octahedral shell forming a cube, while the number 12 inside spheres of the second and fourth shell stands for the number of vertices of the cuboctahedron [ 9 ] formed by these spheres situated at the middles of the edges of the aforementioned shells. In general, numbered spheres are only those which when considered in groups of same numbers form equilibrium [ 10 ] polyhedra, like the tetrahedron, the octahedron, and the cuboctahedron mentioned earlier. Equilibrium polyhedron is that polyhedron whose sphere-atoms at its vertices are at equilibrium whatever the exact law of force among atoms may be [ 10 ]. Unnumbered spheres, of course, do not have the aforementioned property and acquire their stability in the shell by their close-packing with the numbered shell-equilibrated spheres. The physical significance between these two categories of spheres (numbered and unnumbered) is that numbered spheres have a priority in filling up a particular shell, 4 th
Shell
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5 th
Shell
Shell
Q
[62] (66)
[t281 1t02)
[z3o]
Fig. 1. First five successiveshells of close-packedsoft-sphere-likealkali-halide moleculesin the form of nested octahedra. All numbers shown in the figureare explainedin the text. 304
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that is, their positions are first filled by atoms leading to a symmetric partial filling of this shell [ 11 ]. The present packing-of-spheres structure resembles that packing of rare gases, but for the latter case the concentric nested polyhedral shells have icosahedral shape [ 11 ], while in the former case they have a tetrahedral shape. One should notice, however, by evaluating the radii of the polyhedral shells, that in both cases while spheres are in contact at the surface of each shell, these spheres overlap with the spheres of the previous and next shell. Hence, the spheres standing for atoms should be soft and not hard, thus permitting the necessary overlapping. It is worth mentioning that while the overlapping in the case of nested icosahedra is about 10% of the unit sphere radius, that for nested octahedra is about 30%, which implies that the sphere atoms in the latter case should be much softer than in the former case. Harder spheres for rare gases, however, is a rather reasonable assumption since their outermost electron shell is complete, leading to an inactive nature of their electronic structure.
3. Results and discussion
According to the present model the completion of the shells in fig. 1 leads to magic numbers of the alkali-halide microclusters. That is, the cumulative numbers shown in brackets in the figure, i.e., 6, 24, 62, 128 and 230 are considered as magic numbers. Indeed, the first three of them coincide with three of the known series of magic numbers, while for the remaining two larger shell closures there are not as yet available experimental values for their evaluation. The experimental magic numbers which do not correspond to closures of shells in fig. 1 can be considered as semi-magic numbers here and are explained as follows, in the framework of the present model. n = 14: It is made up of the 6 spheres of the first octahedral shell plus 8 spheres, forming a curve, situated at the middles of the faces [ 9 ] of that shell, i.e. 14 = 6 + 8. When more spheres are added, however, these 8 spheres and the extra spheres form part of the next shell. Thus 14 is a magic number, but not a shell, since it does not remain as a core for the larger clusters. n = 18: It comes from the symmetric partial filling of the second shell, i.e. 1 8 = 6 + 12, where 6 is the
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cumulative number of atoms accommodated by the first shell and 12 is the number of vertices of the cuboctahedron as identified in the previous section. n = 24: It is understood as 22 = 6 + 12 + 4, where 6 and 12 as mentioned earlier, and 4 is the number of vertices of a square formed by four selected spheres (in a central plane) of the octahedron in the second shell. n=30: In a similar way, 3 0 = 2 4 + 6 , where 24 is the cumulative number of spheres of the first two shells and 6 comes from the symmetric partial filling of the third shell in the form of an octahedron as noted in the figure, n=32: Similarly, 3 2 = 2 4 + 8 , where now 8 is a cube instead of a octahedron constitutes the symmetric partial filling of the third shell, n = 38: Now 38 = 24 + 6 + 8, where now both the octahedron (6) and the cube (8) are filled from the third shell. Thus, all experimentally observed magic numbers of the series n~ have been interpreted by the present model. Moreover, some additional magic numbers can be derived by symmetric partial filling of the third, fourth and fifth shells as follows. n=68=62+6; n=70=62+8; n = 7 4 = 6 2 + 12; n = 7 6 = 6 2 + 6 + 8 ; n = 8 0 = 6 2 + 6 + 12; n = 8 2 = 6 2 + 8+12; n = 8 8 = 6 2 + 6 + 8 + 1 2 ; n=134=128+6; n = 136= 128+8 and n = 142= 1 2 8 + 6 + 8 , where the notations are considered obvious and refer directly to the figure, except the number 8 which corresponds to the number of vertices of the cube formed by the centers of faces of the octahedron [ 9 ] either of the fourth or fifth shell. Also, the following, less probable, magic numbers can be derived from the third and fourth shells of fig. 1: n = 4 8 = 6 2 - 8 - 6 ; n=54=62-8; n=56=62-6; n=110=128-12-6; n=116=128-12; n=122= 1 2 8 - 6 , and n = 2 2 4 = 2 3 0 - 6 , where all numbers involved come directly from the figure and the minus sign stands for symmetric partial incompletion of the relevant shell. Some additional enhancements (not explained above), however, have been reported in ref. [3] for NanCl+_~ at n=40, 50, 60, and 72. These numbers are reported only for warm clusters and not for cold ones [ 3 ] and perhaps are related to their neighboring magic numbers (explained above) at n = 38, 48, 62, and 72 by some mechanism following the multiphoton ionization procedure. This procedure of cluster production, in principle, can generate inten305
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sity anomalies by resonance enhancement or by fragmentation [ 3 ]. For the explanation of the second series of magic numbers n2 we use the following reasoning. The magic numbers 7, 13, 19 and 25 could be seen as coming from fig. 1 (utilized for the explanation of the first series of magic numbers nl ) by considering an additional central sphere. Specifically, n=7=(6)+1; n=13=(6+6)+1; n=19= ( 6 + 12) + 1, and n = 2 5 = (24) + 1, where the numbers in parenthesis (i.e. 6, 6, 12, and 24) come directly from the figure and have been discussed earlier in detail. The structure of the last three of these magic numbers (i.e. 13, 19, and 25), however, can be transformed into more stable structures familiar from the rare-gas microclusters, where spheres are closely packed in nested icosahedral shells [ 11 ]. That is, there is a competition between two different structures for each of these three magic numbers depending on the specific forces and sphere sizes involved. Specifically, n = 13 = 12 + 1; n = 19 = 13 + 6, and n = 2 5 = 1 3 + 1 2 , where 13 is the number of spheres accommodated by the first icosahedral shell around a central sphere, and 6 and 12 stand for the spheres at the vertices of an oetahedron and of an icosahedron, respectively, coming from the symmetric partial fillings of the second icosahedral shell (see fig. 1 of ref. [ 11 ] ). These icosahedral packing structures for n = 13, 19 and 25 stand for their ground state, while the octahedral packing structures discussed above stand for their metastable structures. The explanation of the remaining magic numbers in the second series n2, i.e. of the numbers 10 and 17, is as follows, n = 17 = ( 1 + 12) + 4, where 1 + 12 = 13 corresponds to the icosahedral structure, as explained earlier, and 4 to the number of spheres at the vertices of a tetragon formed by four vertices in a plane of the octahedron shown as symmetric partial filling of the second shell in fig. 1 of ref. [ 11 ]. n = 10 = 6 + 4, where 6 is the number of spheres in the first shell of the present fig. 1 and 4 the number of spheres at the vertices of a tetragon derived now by four vertices in a plane of the octahedron shown in the second shell of fig. 1. This structure proposed for n = 10 is the only one among those for the numbers in the series n2 without a central sphere. Planar (i.e., tetragonal) partial filling, as for the numbers 306
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n = 17 and 10 above, is also an equilibrium structure according to ref. [ 10 ]. The following predictions can be made for metastable structures heavier than 25. n=31 = ( 2 5 ) + 6 ; n = 3 3 = ( 2 5 ) + 8 , and n = 3 9 = ( 2 5 ) + 6 + 8 , where 25 corresponds to the completion of the second shell in fig. 1 when a central sphere is included, and 6 and 8 as shown in the third shell of the figure. None of these structures has a counterpart icosahedral-packing structure. Thus, for these metastable structures there are not corresponding ground state structures.
4. Concluding remarks The present interpretation of magic numbers in alkali-halide microclusters is completely different from previous explanations based on cubic-like [ 4 ] or rectangular [3 ] or pentagonal-growth sequence [6] structures. While the present explanation is unique for all magic numbers in these microclusters, each of the previous explanations stands only for a set of these numbers. In all previous explanations [3,4,6], each ion (alkali or halide) occupies a different site in the structure resembling, more or less, the bulk structure of alkali-halide crystals, while in the present explanation each alkali-halide molecule is presented by a cloudy sphere standing for the envelope of both ions considered in an arbitrary and changeable relative orientation inside this sphere. The present interpretation, based on close packing of soft sphere-like molecules leading to nested octahedral shells (see fig. 1 ) in some way supports the view that microclusters of different elements could be understood, more or less, in a unified way [ 1115]. Some predictions for larger magic numbers are also made here.
Acknowledgement I express my sincere appreciation to Professor J.W. Negele and J. Goldstone of the MIT Department of Physics for their invitation to join the stimulating environment at their Center for Theoretical Physics during my sabbatical year, and to Professors P. Jena,
Volume 150, number 5,6,7
PHYSICS LETTERS A
B.K. R a o a n d S.N. K h a n n a o f the V i r g i n i a C o m m o n w e a l t h U n i v e r s i t y , D e p a r t m e n t o f Phys, for m a n y v a l u a b l e d i s c u s s i o n s o n the physics o f microclusters.
References [ 1] T.P. Martin, J. Chem. Phys. 69 (1978) 2036. [ 2 ] D.O. Welch, O.W. Lazareth, G.J. Dienes and R.D. Hatcher, J. Chem. Phys. 68 (1978) 2159. [3 ] R. Pflaum, K. Sattler and E. Rechnagel, Chem. Phys. Lett. 138 (1987) 8. [4] J.E. Campana, T.M. Barlak, R.J. Colton, J.J. DeCorpo, J.R. Wyatt and B.I. Dunlap. Phys. Rev. Lett. 47 ( 1981 ) 1046.
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[5] R. Pflaum, K. Sattler and E. Rechnagel, Phys. Rev. B 33 (1986) 1522. [6 ] M.R. Hoare and P. Pal, Adv. Phys. 20 ( 1971 ) 161. [7]J.C. Philips, in: Proc. Int. Syrup. on The physics and chemistry of small clusters, eds. P. Jena, B.K. Rao and S.N. Khanna, Richmond, VA, 1986, p. 249. [ 8 ] R. Pflaum and E. Rechnagel, Z. Phys. D 12 (1989) 249. [9] H.S.M. Coxeter, Regular polytopes, 3rd Ed. (Macmillan, New York, 1973). [ 10] J. Leech, Math. Gaz. 41 (1957) 81. [ 11 ] G.S. Anagnostatos, Phys. Lett. A 124 (1987) 58. [ 12] G.S. Anagnostatos, Phys. Lett. A 128 (1988) 266. [ 13 ] G.S. Anagnostatos, Phys. Lett. A 133 ( 1988 ) 419. [ 14] G.S. Anagnostatos, Phys. Lett. A 142 (1989) 146. [ 15 ] G.S. Anagnostatos, Phys. Lett. A 143 (1990) 332.
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PHYSICS LETTERSA
12 November 1990
Magic numbers in alkali-halide microclusters -AG.S. A n a g n o s t a t o s Centerfor TheoreticalPhysics, Laboratoryfor Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA and Institute of Nuclear Physics, Demokritos National Research Centerfor PhysicalSciences, GR-15310 Aghia Paruskevi, Attiki, Greece
Received 13 March 1990; revised manuscript received 9 July 1990; accepted for publication 11 September 1990 Communicatedby D. Bloch
Close packing of soft spheres, each standing for the envelope of both ions in an alkali.halide molecule and forming nested octahedral shells, is proposed for the structure of alkali-halide microclusters (magic numbers: 6, 14, 18, 20, 24, 30, 32, 38, ...). Thus the structure for low photon energy of ionization correspondsto a metastable structure which is transformed into a nested icosahedral-shellground-state structure (magic numbers: 7, 10, 13, 17, 19, 25, ...).
1. Introduction
Studies of alkali-halide small clusters belong to the general category of those of ionic clusters [ 1 ], where the stable configurations are determined by numerically minimizing the total energy. In ref. [ 1 ], both rigid ion and polarizable ion models are used to describe the two particle interactions, while in another study [2 ] a simple shell model is used for the interionic forces. Because of the long-range forces between ions, close packed structures are not necessarily the most favorable as they are in rare gas microclusters. In both cases, however, cluster stability is mainly determined by the cluster's geometrical structure which assumes a variety of different equilibrium shapes [ 1 ]. Enhanced abundances of neutral alkali-halide small clusters (i.e. M,X,, where M stands for alkali and X for halide ions), which come from the secondary-ion mass spectrometry resulting in the stability of the ion alkali-halide microclusters (i.e. M , Z , _ 1 ) by using different preparation procedures, have been found [3,4] for n~=6, 14, 18, 20, 24, 30, 32, 38 ..... For *~ This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract No. DE-AC0276ER03069. Elsevier Science Publishers B.V. (North-Holland)
low ionization-photon energies, however, neutral clusters (e.g. for CsI) enhancements appear [5] at n2=7, 10, 13, 17, 19, 25, .... The existing explanation of the second series of magic numbers (n2) utilizes the pentagonal growth sequence [6,7 ] while the existing explanation of the first series (nt) employs cubic- and rectangular-like structures [3,4] (i.e., structures made up of small fractions of the rock salt lattice). The purpose of the present work is to give a different, common explanation of both series of magic numbers by using close packing of soft spheres forming nested polyhedra. Thus, the present approach, in a broader sense, resembles that of rare-gas microclusters and supports the view that microclusters of different elements could be understood, more or less, in a unified way. Here a somehow spherical-like shape of alkali-halide molecule ( M X ) is implied. The alkali-halide sphere-like molecule here stands for the envelope of both ions and demonstrates the lack of a specific orientation of the ion-dimer within this sphere. Perhaps, this sphere-envelope is defined by the continuous change of the aforementioned orientation in such a way that the whole sphere is occupied in different times. The basic reason for the different shapes assigned 303
Volume 150,number 5,6,7
PHYSICS LETTERSA
to the structure of alkali-halide microclusters, here and in previous publications [ 1,3,4,6,7 ] are the different building elements employed in the two cases. In the present work only one building element, the alkali-halide molecule considered as a soft sphere (in the way described above), is used, while in all previous publications two building elements, the alkali and the halide ions considered as separate entities, are employed. Thus, in the previous works on alkalihalides, like in their crystals, to each alkali or halide ion is assigned a different site of the structure, while in the present paper to each alkali-halide molecule (i.e. to one alkali and one halide ion together) is assigned a site of the structure. As a result the present structure has a somewhat spherical-like shape, while the previously proposed structures have shapes far from spherical ones. Differences in the shapes of alkali-halide microclusters in the literature, among other reasons, come from referring either to different sets of characteristic enhancements (e.g. refs. [6] and [7] refer to the n2 sequence of magic numbers, while refs. [ 3 ] and [4] refer to the n~ sequence) or from referring to the same sequence but different sizes of clusters (e.g. ref. [ 1 ] refers to sizes up to n = 12, while ref. [ 3 ] refers to sizes with n >I 14).
2. The model
In fig. 1 the first five successive shells proposed here for the structure of alkali-halide microclusters are shown. These shells are made of close-packed spheres standing for atoms with 2, 3, 4, 5 and 6
t st
Shell
2 nd Shell
® (6)
[6] (ts)
3 rd
Q
[z4] (3e)
spheres at each edge of the five parallel nested regular octahedra (as shown). At the bottom of each block, the number of spheres accommodated by the relevant shell is given inside parentheses, while the cumulative number of spheres accommodated by all previous and that shell is also given inside square brackets. Numbers inside spheres at each shell stand for the number of vertices of the specific equilibrium polyhedron formed by the spheres characterized by the same number. Specifically, the number 6 inside the spheres at the comers of each octahedral shell denotes the number of vertices of the related octahedron. Also, the number 8 inside spheres denotes the number of middles of the faces of the (third) octahedral shell forming a cube, while the number 12 inside spheres of the second and fourth shell stands for the number of vertices of the cuboctahedron [ 9 ] formed by these spheres situated at the middles of the edges of the aforementioned shells. In general, numbered spheres are only those which when considered in groups of same numbers form equilibrium [ 10 ] polyhedra, like the tetrahedron, the octahedron, and the cuboctahedron mentioned earlier. Equilibrium polyhedron is that polyhedron whose sphere-atoms at its vertices are at equilibrium whatever the exact law of force among atoms may be [ 10 ]. Unnumbered spheres, of course, do not have the aforementioned property and acquire their stability in the shell by their close-packing with the numbered shell-equilibrated spheres. The physical significance between these two categories of spheres (numbered and unnumbered) is that numbered spheres have a priority in filling up a particular shell, 4 th
Shell
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5 th
Shell
Shell
Q
[62] (66)
[t281 1t02)
[z3o]
Fig. 1. First five successiveshells of close-packedsoft-sphere-likealkali-halide moleculesin the form of nested octahedra. All numbers shown in the figureare explainedin the text. 304
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that is, their positions are first filled by atoms leading to a symmetric partial filling of this shell [ 11 ]. The present packing-of-spheres structure resembles that packing of rare gases, but for the latter case the concentric nested polyhedral shells have icosahedral shape [ 11 ], while in the former case they have a tetrahedral shape. One should notice, however, by evaluating the radii of the polyhedral shells, that in both cases while spheres are in contact at the surface of each shell, these spheres overlap with the spheres of the previous and next shell. Hence, the spheres standing for atoms should be soft and not hard, thus permitting the necessary overlapping. It is worth mentioning that while the overlapping in the case of nested icosahedra is about 10% of the unit sphere radius, that for nested octahedra is about 30%, which implies that the sphere atoms in the latter case should be much softer than in the former case. Harder spheres for rare gases, however, is a rather reasonable assumption since their outermost electron shell is complete, leading to an inactive nature of their electronic structure.
3. Results and discussion
According to the present model the completion of the shells in fig. 1 leads to magic numbers of the alkali-halide microclusters. That is, the cumulative numbers shown in brackets in the figure, i.e., 6, 24, 62, 128 and 230 are considered as magic numbers. Indeed, the first three of them coincide with three of the known series of magic numbers, while for the remaining two larger shell closures there are not as yet available experimental values for their evaluation. The experimental magic numbers which do not correspond to closures of shells in fig. 1 can be considered as semi-magic numbers here and are explained as follows, in the framework of the present model. n = 14: It is made up of the 6 spheres of the first octahedral shell plus 8 spheres, forming a curve, situated at the middles of the faces [ 9 ] of that shell, i.e. 14 = 6 + 8. When more spheres are added, however, these 8 spheres and the extra spheres form part of the next shell. Thus 14 is a magic number, but not a shell, since it does not remain as a core for the larger clusters. n = 18: It comes from the symmetric partial filling of the second shell, i.e. 1 8 = 6 + 12, where 6 is the
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cumulative number of atoms accommodated by the first shell and 12 is the number of vertices of the cuboctahedron as identified in the previous section. n = 24: It is understood as 22 = 6 + 12 + 4, where 6 and 12 as mentioned earlier, and 4 is the number of vertices of a square formed by four selected spheres (in a central plane) of the octahedron in the second shell. n=30: In a similar way, 3 0 = 2 4 + 6 , where 24 is the cumulative number of spheres of the first two shells and 6 comes from the symmetric partial filling of the third shell in the form of an octahedron as noted in the figure, n=32: Similarly, 3 2 = 2 4 + 8 , where now 8 is a cube instead of a octahedron constitutes the symmetric partial filling of the third shell, n = 38: Now 38 = 24 + 6 + 8, where now both the octahedron (6) and the cube (8) are filled from the third shell. Thus, all experimentally observed magic numbers of the series n~ have been interpreted by the present model. Moreover, some additional magic numbers can be derived by symmetric partial filling of the third, fourth and fifth shells as follows. n=68=62+6; n=70=62+8; n = 7 4 = 6 2 + 12; n = 7 6 = 6 2 + 6 + 8 ; n = 8 0 = 6 2 + 6 + 12; n = 8 2 = 6 2 + 8+12; n = 8 8 = 6 2 + 6 + 8 + 1 2 ; n=134=128+6; n = 136= 128+8 and n = 142= 1 2 8 + 6 + 8 , where the notations are considered obvious and refer directly to the figure, except the number 8 which corresponds to the number of vertices of the cube formed by the centers of faces of the octahedron [ 9 ] either of the fourth or fifth shell. Also, the following, less probable, magic numbers can be derived from the third and fourth shells of fig. 1: n = 4 8 = 6 2 - 8 - 6 ; n=54=62-8; n=56=62-6; n=110=128-12-6; n=116=128-12; n=122= 1 2 8 - 6 , and n = 2 2 4 = 2 3 0 - 6 , where all numbers involved come directly from the figure and the minus sign stands for symmetric partial incompletion of the relevant shell. Some additional enhancements (not explained above), however, have been reported in ref. [3] for NanCl+_~ at n=40, 50, 60, and 72. These numbers are reported only for warm clusters and not for cold ones [ 3 ] and perhaps are related to their neighboring magic numbers (explained above) at n = 38, 48, 62, and 72 by some mechanism following the multiphoton ionization procedure. This procedure of cluster production, in principle, can generate inten305
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sity anomalies by resonance enhancement or by fragmentation [ 3 ]. For the explanation of the second series of magic numbers n2 we use the following reasoning. The magic numbers 7, 13, 19 and 25 could be seen as coming from fig. 1 (utilized for the explanation of the first series of magic numbers nl ) by considering an additional central sphere. Specifically, n=7=(6)+1; n=13=(6+6)+1; n=19= ( 6 + 12) + 1, and n = 2 5 = (24) + 1, where the numbers in parenthesis (i.e. 6, 6, 12, and 24) come directly from the figure and have been discussed earlier in detail. The structure of the last three of these magic numbers (i.e. 13, 19, and 25), however, can be transformed into more stable structures familiar from the rare-gas microclusters, where spheres are closely packed in nested icosahedral shells [ 11 ]. That is, there is a competition between two different structures for each of these three magic numbers depending on the specific forces and sphere sizes involved. Specifically, n = 13 = 12 + 1; n = 19 = 13 + 6, and n = 2 5 = 1 3 + 1 2 , where 13 is the number of spheres accommodated by the first icosahedral shell around a central sphere, and 6 and 12 stand for the spheres at the vertices of an oetahedron and of an icosahedron, respectively, coming from the symmetric partial fillings of the second icosahedral shell (see fig. 1 of ref. [ 11 ] ). These icosahedral packing structures for n = 13, 19 and 25 stand for their ground state, while the octahedral packing structures discussed above stand for their metastable structures. The explanation of the remaining magic numbers in the second series n2, i.e. of the numbers 10 and 17, is as follows, n = 17 = ( 1 + 12) + 4, where 1 + 12 = 13 corresponds to the icosahedral structure, as explained earlier, and 4 to the number of spheres at the vertices of a tetragon formed by four vertices in a plane of the octahedron shown as symmetric partial filling of the second shell in fig. 1 of ref. [ 11 ]. n = 10 = 6 + 4, where 6 is the number of spheres in the first shell of the present fig. 1 and 4 the number of spheres at the vertices of a tetragon derived now by four vertices in a plane of the octahedron shown in the second shell of fig. 1. This structure proposed for n = 10 is the only one among those for the numbers in the series n2 without a central sphere. Planar (i.e., tetragonal) partial filling, as for the numbers 306
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n = 17 and 10 above, is also an equilibrium structure according to ref. [ 10 ]. The following predictions can be made for metastable structures heavier than 25. n=31 = ( 2 5 ) + 6 ; n = 3 3 = ( 2 5 ) + 8 , and n = 3 9 = ( 2 5 ) + 6 + 8 , where 25 corresponds to the completion of the second shell in fig. 1 when a central sphere is included, and 6 and 8 as shown in the third shell of the figure. None of these structures has a counterpart icosahedral-packing structure. Thus, for these metastable structures there are not corresponding ground state structures.
4. Concluding remarks The present interpretation of magic numbers in alkali-halide microclusters is completely different from previous explanations based on cubic-like [ 4 ] or rectangular [3 ] or pentagonal-growth sequence [6] structures. While the present explanation is unique for all magic numbers in these microclusters, each of the previous explanations stands only for a set of these numbers. In all previous explanations [3,4,6], each ion (alkali or halide) occupies a different site in the structure resembling, more or less, the bulk structure of alkali-halide crystals, while in the present explanation each alkali-halide molecule is presented by a cloudy sphere standing for the envelope of both ions considered in an arbitrary and changeable relative orientation inside this sphere. The present interpretation, based on close packing of soft sphere-like molecules leading to nested octahedral shells (see fig. 1 ) in some way supports the view that microclusters of different elements could be understood, more or less, in a unified way [ 1115]. Some predictions for larger magic numbers are also made here.
Acknowledgement I express my sincere appreciation to Professor J.W. Negele and J. Goldstone of the MIT Department of Physics for their invitation to join the stimulating environment at their Center for Theoretical Physics during my sabbatical year, and to Professors P. Jena,
Volume 150, number 5,6,7
PHYSICS LETTERS A
B.K. R a o a n d S.N. K h a n n a o f the V i r g i n i a C o m m o n w e a l t h U n i v e r s i t y , D e p a r t m e n t o f Phys, for m a n y v a l u a b l e d i s c u s s i o n s o n the physics o f microclusters.
References [ 1] T.P. Martin, J. Chem. Phys. 69 (1978) 2036. [ 2 ] D.O. Welch, O.W. Lazareth, G.J. Dienes and R.D. Hatcher, J. Chem. Phys. 68 (1978) 2159. [3 ] R. Pflaum, K. Sattler and E. Rechnagel, Chem. Phys. Lett. 138 (1987) 8. [4] J.E. Campana, T.M. Barlak, R.J. Colton, J.J. DeCorpo, J.R. Wyatt and B.I. Dunlap. Phys. Rev. Lett. 47 ( 1981 ) 1046.
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[5] R. Pflaum, K. Sattler and E. Rechnagel, Phys. Rev. B 33 (1986) 1522. [6 ] M.R. Hoare and P. Pal, Adv. Phys. 20 ( 1971 ) 161. [7]J.C. Philips, in: Proc. Int. Syrup. on The physics and chemistry of small clusters, eds. P. Jena, B.K. Rao and S.N. Khanna, Richmond, VA, 1986, p. 249. [ 8 ] R. Pflaum and E. Rechnagel, Z. Phys. D 12 (1989) 249. [9] H.S.M. Coxeter, Regular polytopes, 3rd Ed. (Macmillan, New York, 1973). [ 10] J. Leech, Math. Gaz. 41 (1957) 81. [ 11 ] G.S. Anagnostatos, Phys. Lett. A 124 (1987) 58. [ 12] G.S. Anagnostatos, Phys. Lett. A 128 (1988) 266. [ 13 ] G.S. Anagnostatos, Phys. Lett. A 133 ( 1988 ) 419. [ 14] G.S. Anagnostatos, Phys. Lett. A 142 (1989) 146. [ 15 ] G.S. Anagnostatos, Phys. Lett. A 143 (1990) 332.
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