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heteroatom microclusters
Volume 142, number 2,3
PHYSICS LETTERS A
MAGIC NUMBERS IN ALKALI/WETEROATOM
4 December 1989
MICROCLUSTERS
*
G.S. ANAGNOSTATOS Centerfor TheoreticalPhysics,Laboratoryfor NuclearScience and DepartmentofPhysics,MassachusettsInstituteof Technology,Cambridge,MA 02139, USA and Instituteof Nuclear Physics,DemokritosNationalResearch Centerfor PhysicalSciences, GR-15310 Aghia Paraskevi,Attiki,Greece Received 5 May 1989; revised manuscript received 7 August 1989; accepted for publication 28 September 1989 Communicated by D. Bloch
The magic numbers in alkali/heteroatom microclusters are theoretically studied and new experiments are proposed. These clusters X-Y, where X stands for alkali and Y for heteroatom, exhibit the magic numbers n=6 (8) 18 (20), 38, 56,88, ... based on equilibrium geometry and supported by all experimental data available to date.
In a previous publication [ 1 ] a unified explanation of magic numbers in small clusters for rare-gas and alkali atoms was presented. This unification comes from the consideration of the close-packing concept for two different types of softness of spheres standing for atoms. Specifically, if the spheres are soft the microclusters possess close-packing of spheres structure and exhibit magic numbers at n, = 1, 13, 55, 147, 309, 561, .... while if the spheres are hard they possess close-packing of shells structure and exhibit magic numbers at nz= 2, 8, 20, 40, 58, ... . In later publications based on the previous one, the magic numbers in mixed clusters were presented when the clusters were made up of two alkali [ 21 or of two rare gases [ 3 1. All these three publications are along the line that magic numbers could be interpreted via equilibrium geometry of atoms. From the other point of view, however, spherical shell models (jellium models) refer strictly to electron structure and assume negligible contribution of atomic geometric structure to total energy [ 43. While contribution of both reasons cannot be excluded, heteroatom probing experiments were designed to gauge the relative magnitude of electronic and geo* This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract No. DE-ACOZ76ER03069.
146
metric contributions to total energy [ 5-7 1. Specifically, alkali atoms were replaced by heteroatoms with more than one valence electron (and different atomic radius) to form the heterocluster X,Y. Then magic islands of enhanced thermodynamic stability determined purely by the number of valence electrons in the framework of the jellium model should be shifted with respect to the homoclusters by an amount appropriate to the total valence electron count. For these heteroclusters here we will offer a different explanation based on equilibrium geometries. The basic reference for our study is ref. [ 1 ] and the main points of the present model are demonstrated in fig. 1. The building units of the equilibrium geometry are the shells whose hard-sphere atoms, assumed at the vertices of the polyhedra presented, are in contact with hard spheres of the previous and next shell, but not in contact with spheres of the same shell (as happens in the case of rare-gas cluster structure; see refs. [ 1 ] and [ 3 ] ) . Each polyhedral shell is an equilibrium [ 81 polyhedron which permits an equilibrium of particles at its vertices for any sort of force among particles and, in particular here, minimizes their mutual repulsion. The relative orientation of polyhedra in the different shells is such that their sets of particles are in equilibrium with each other [ 81 and, as a consequence, their relevant polyhedral axes of symmetry coincide.
0375-9601/89/$03.50
0 Elsevier Science Publishers B.V. (North-Holland)
4 December 1989
PHYSICS LETTERS A
Volume 142, number 2,3 lctahedron
(b)
lcosahedron
(c.
lodecahedron
@I)
4%
i)
161
iexahedron
(12)
20)
[481
I lcosldodecahedron I
(dl:
lhomblc
I381
Tnacontah.
(e)
Fig. 1. Close-packing-of-shells equilibrium geometries: Nested equilibrium polyhedra standing for closed shells of hard-sphere atoms. Shells in contact: Octahedron with icosahedron and dcdecahedron, hexahedron with icosidodecahedron, and icosi$odecahedron with rhombic [email protected] heteroatom, at the center of the microcluster, is in contact with all six sphere-atoms of the octahedron.
The heteroatom occupies the center of all those nested polyhedra and all spheres of the first shell are in contact with it. That is, the equilibrium geometry of hard-sphere atoms of alkali/heteroatom microclusters is made up of centered equilibrium polyhedral shells in contact and in equilibrium with each other. Thus, the equilibrium of particles does not only refer to the interactions of particles in the same shell, but also to the interactions of particles in the neighboring (inner and outer) shells (i.e., in all shells). That is, for the equilibrium of each particle all other particles in tbe cluster are taken into consideration. The six polyhedra employed in fig. 1 are the three reciprocal [ 9!] pairs hexahedron-octahedron, dodecahedron-icosahedron, and icosidodecahedronrhombic triacontahedron. That is, only three out of the nine existing [ 81 equilibrium polyhedra are not used in fig. 1,’namely the tetrahedron - which does not possess central symmetry, a fact which could disturb the equilibrium of the nested neighboring shells - and the reciprocal pair of the cuboctahedron and rhombic dodecahedron, where neither member possesses stable equilibrium [ 8 1. Thus,&% three polyhedra are excluded for reasons of stability. The order of polyhedra used in fig. 1 is consistent with closepacking, that is, it proceeds from the polyhedron with
the smaller number of vertices to the polyhedron with the larger number of vertices. The relative orientation of polyhedra follows also from the close-packing and leads to the maximum possible symmetry. That is, rays through the vertices of an inner polyhedron pass through vertices, or middles of edges, or centers of faces of the outer polyhedra. Finally, we may say that the kind of polyhedra employed in fig. 1, their order and relative orientation, lead to a unique closepacking structure for the small clusters of alkali/ heteroatom. The only difference between the present equilibrium geometry for heteroclusters and that of ref. [ 1 ] for alkali homoclusters is that the first shell, the zerohedron, in the latter (made up of two atom-spheres in contact) has been replaced here by the one sphere of the heteroatom. This omission of the zerohedron here is necessary according to the following reason. In ref. [ 1 ] the zerohedron was directed along a C3 axis of the octahedron and all six atom-spheres of this second shell were in contact with the two spheres of the first shell (in two sets of three). This simultaneous fulfillment of equilibrium geometry and of close-packing cannot be preserved here without the omission of the zcrohedron. With the zerohedron and equilibrium geometry present we cannot have close147
Volume 142, number 2,3
PHYSICS LETTERS A
packing (i.e., the sphere-atoms of the octahedron do not touch the sphere-atoms of the zerohedron), or with the zerohedron and close-packing present we cannot have equilibrium geometry (i.e., the regular octahedron is transformed into a trigonal antiprysm which is not an equilibrium polyhedron [ 8 ] ) . At the top of each block in fig. 1 presenting a polyhedron, the name of the polyhedron is given, while at the bottom of the block two specific numbers are shown. The first, in parentheses, presents the number of vertices for the particular polyhedron, while the second, in brackets, gives the cumulative number of vertices of all previous and that polyhedron. As becomes obvious from fig. 1, up to N=40 all previously formed polyhedral shells coexist in size and form. After N= 40, however, the last shell in N= 40, i.e., the dodecahedron (which does not possess stable equilibrium [ 81) does not coexist with the new coming shells. That is, when the icosidodecahedron is added out of the twenty vertices of the dodecahedron only eight vertices forming a cube remain whose spheres are all in contact with those of the icosidodecahedron which possesses a stable equilibrium by itself [ 8 ] (see ref. [ 11. ) As one can see from the numbers in brackets, the cumulative numbers that appear in the structure of fig. 1 (when successive shells are completed) are 6, 18, 38 (and 56, 88) and coincide with the experimentally observed magic numbers for the heteroclusters Na/Mg, Na/Ca, Na/Sr and Na/Yb shown in table 1. From the same table we see that the first experimental magic number for the heteroclusters Na/Zn, K/M& K/Zn, and K/Hg is 8 and not 6 as Table 1 Neutral abundance maxima in allcali/heteroatom cluster beams.
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148
Metals (X Y)
Maxima (XY)
Heteroatom (Y ) configuration
NalMg Na/Ca Na/Sr Na/Ba Na/Zn Na/Eu Na/Yb
6-8,18 6, 18 6,16-18,38 6, 16 8, 18 6, 16 6, 18,38
.. 3s2 4s2 5S2 6s2 3d1°4s2 4F 6s’ 4F4 6s’
K/Mg KfZn K/HO
8, 18 8, 18 8, 19
5d’06s2
4 December 1989
previously discussed. This difference, however, can be easily understood as follows. According to fig. 1 the first shell is an octahedron and the second is an icosahedron, and their (most symmetric) relative orientation shown is such that each C4 axis of the first coincides with a C2 axis of the second. At n=6 we have the formation of the octahedron and at n = 8 we have two choices. The first is to start a new shell (partial tilling of the icosahedron) and the second is to rearrange all together the eight alkali atoms and form a cube (hexahedron ). The cube is an unstable [ 81 equilibrium polyhedron and, in general, cannot stand as a first shell in the structure of a microcluster. Here, however, the cube acquires additional stability (and reaches a stable equilibrium) due to the fact that its spheres are all in contact with the heteroatom. When even more atoms, however, are added into the cluster, the preferable structure is an interior octahedron (n= 6) and an exterior (partially or totally filled) icosahedron. Thus, n = 8 is a somehow “equally” good magic number like n= 6, but it is not an equally good shell, since it transforms itself later (when n > 8) into n= 6 core (octahedron) plus two atoms in the next shell. At this point two interesting comments should be made. First, we may expect n = 8 to be a magic number even for the cases which do not show up in table 1 (e.g., in the Na/Ca heterocluster, etc. ), or in other words both n = 6 and 8 to be magic numbers as in the heterocluster Na/Mg shown in table 1. Second, we cannot easily exclude the possibility where n = 8 remains as a core even up to the complete filling of the next shell (icosahedron possessing 12 atom-spheres). There is no reason of lack of equilibrium or closepacking to support such an exclusion. That is, we cannot exclude the appearance of a magic peak at n=20 in the mass spectrum of alkali/heteroatom microclusters even if this number has not appeared in the experiments up to now. The appearance of “magic” properties at n = 19 in the heterocluster K/ Hg, as shown in table 1 lends support to our prediction for a sort of magic number at n = 20 for all alkali/heteroatom microclusters. According to this prediction 19 implies a magic number at n = 20 where the one atom has escaped from the microcluster during the ionization procedure. The difference between 6 and 8, and 18 and 20 rests on the thermodynamic internal energy of the microcluster.
Volume 142, number 2,3
PHYSICS LETTERS A
Up to this point, our arguments about magicity of 6 and 8 (or 18 and 20) strictly refer to equilibrium alone and no consideration of sizes for the alkali atoms and heteroatom has been taken. It is significant, however, to consider that n = 8 (and thus n = 20) implies smaller interatomic distances than n = 6 (and n = 18 ) if the heteroatom is assumed to be the same. Thus, it is possible for some cases, for the n = 8 to lead to nearest interatomic distances corresponding to repulsion of the alkali atoms for a specific interatomic potential, a fact which would exclude n=8 and n = 20 from being magic numbers for these alkali/heteroatom microclusters. Perhaps some of the cases in table 1 where n = 8 does not show up as magic number belong to this category of microclusters. However, if n1=8 leads to nearest interatomic distances corresponding to attraction of the alkali atoms, then both n=6 and n= 8 could exist, but n= 8 is energetically more favored since these distances for n = 8 are closer to the distance at minimum potential value than those for n = 6. Thus, according to the present model, depending on the specific potential and distances involved, n = 6 or 8 or both could be the case, as indeed appears in table 1 for the experimental data. The relationship between heteroatom and alkali atom sizes, which allows one to understand that magic number 8 (and thus 20) can be excluded for some systems is the following, r,_,=2(R+r)/$cd,
(1)
where r&A is the smallest distance between two alkali atoms in the case of a cubic arrangement around the heteroatom, R and r are the radii of heteroatom and alkali atoms, respectively, and d is a parameter in the Lennard-Jones interatomic potential
(2) Relationships ( 1) and (2) obviously imply that, in such an arrangement, the force among nearest neighbor alkali atoms is repulsive, a fact which makes this arrangement impossible. Our table 1 provides a good sample (perhaps a complete list ) of all alkali/heteroatom experiments performed to date [ 5-7 1. However, it includes only probing cases to gauge the relative magnitudes of
4 December 1989
electronic structure and equilibrium geometry contributions to the total energy of the microclusters. Of course, heteroclusters where the heteroatom is also an alkali atom should be excluded from our conclusions. In these cases the heteroatom, if the homologous atoms are not too disparate in size, does not occupy the center of the cluster as here [ 51 and.different magic numbers are expected as seen in ref. [ 2 1. Experimental facts involving rare gas/heteroatom clusters support our predictions above [ 101. While we do not forget that, in general, rare gas microclusters differ more substantially in their equilibrium geometries [ 1,3] than alkali/heteroatom clusters, for a small number of rare-gas atoms, however, the equilibrium geometry resembles that of the alkali/heteroatom microclusters. Indeed, the unique stability of the CoArz microcluster has been attributed [ lo] to an octahedral arrangement of Ar atoms around a Co + core which is the same arrangement assumed by fig. 1 for the structure of magic number 6 here. The appearance of n = 8 instead of n = 6 as a magic number in some of the cases of table 1 can somehow be understood in the framework of the jellium model by making the modification of reversing the filling order of Id and 2s electron shells leading to closed shellsat n=2, 8, 10, 20... (s, lp, 2s, Id, ...) instead of the usual n=2, 8, 18, 20, ... (Is, lp, Id, 2s, ...) electrons [ 5-7 1. Such a reversal could be induced by a small depression in the center of the well which can be justified by the higher ionization potential of the heteroatom (compared to an alkali atom) assumed at the center of the cluster. However, the experimentally observed lower ionization potential of the more stable 10 electron system, e.g. K,Mg, versus the higher ionization potential of the less stable 8 electron system, e.g. K,Mg, cannot be understood within the jellium model. Thus, the experimental data falsify even the modified jellium model of metal clusters [ 6-71. Since electron shell closure and ionization-potential strength are electron-only calculations they should be consistent with each other and not inconsistent as previously stated. If, however, as in the present work, n = 8 is supported by equilibrium geometry and not by electron structure, there is no inconsistency if the ionization potential for n = 8 is lower than that for n = 6. In conclusion we may say that the magic numbers 149
Volume 142, number 2,3
PHYSICS LETTERS A
in alkali/heteroatom microclusters are 6 (a), 18 (20), 38, 56 and 88, as shown in fig. 1. These magic numbers are supported by all experimental data available to date [ 5-7 1. Additional experiments are recommended for a larger variety of heteroatoms. It is satisfying that the close-packing of shells concept applied earlier [ 1 ] for the alkali homoclusters is successfully applied here for the alkali/heteroatom microclusters. Thus, the unification of explanation presented in ref. [ 1] for the magic numbers in rare-gas and alkali small clusters is extended to the present case of heteroclusters as well. The equilibrium geometries for Li,Mg (n < 7), Li,Al (n < 4)) and Li,Be (n < 6 ) have been theoretically studied in refs. [ 111 and [ 121, respectively. For the magic number n=6 this geometry in both references is the same as here, i.e., it is that of an octahedron (see fig. 1). I appreciate the constructive comments on the present work made by Professor P. Jena of Virginia Commonwealth University, Department of Physics. I also express my sincere appreciation to Professors J.W. Negele and J. Goldstone of the MIT Depart-
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ment of Physics for their invitation to join the stimulating environment at their Center for Theoretical Physics during my sabbatical year.
References [l] G.S. Anagnostatos, Phys. Lett. A 124 (1987) 85. [2] G.S. Anagnostatos, Phys. Lett. A 128 (1988) 266. [3] G.S. Anagnostatos, Phys. Lett. A 133 (1988) 419. [ 41 W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chow and M.L. Cohen, Phys. Rev. Lett. 52 (1984) 2141. [ 51M.M. Kappes, P. Radi, M. Schiir and E. Schumacher, Chem. Phys.Lett. 119 (1985) 11. [ 6 ] M.M. Kappes, M. Schiir, C. Yeretzian, U. Heiz, A. Vayloyan and E. Schumacher, in: Proc. Int. Symp. on The physics and chemistry of small clusters, eds. P. Jena, B.K. Rao and S.N. Khanna (Wiley, New York, 1987). [7] M.M. Kappes, Chem. Rev. 88 (1988) 369. [8] J. Leech, Math. Gaz. 41 (1957) 81. [ 91 H.S.M. Coxeter, Regular polytopes, 3rd Ed. (Macmillan, New York, 1973). [IO] D. Lessen and P.J. Brucat, Chem. Phys. Lett. 149 (1988) 10. [ 111 B.K. Rao and P. Jena, Phys. Rev. B 37 ( 1988) 2867. [ 12 ] J. Kouteckjl and P. Fantucci, 2. Phys. D 3 ( 1986) 147.
PHYSICS LETTERS A
MAGIC NUMBERS IN ALKALI/WETEROATOM
4 December 1989
MICROCLUSTERS
*
G.S. ANAGNOSTATOS Centerfor TheoreticalPhysics,Laboratoryfor NuclearScience and DepartmentofPhysics,MassachusettsInstituteof Technology,Cambridge,MA 02139, USA and Instituteof Nuclear Physics,DemokritosNationalResearch Centerfor PhysicalSciences, GR-15310 Aghia Paraskevi,Attiki,Greece Received 5 May 1989; revised manuscript received 7 August 1989; accepted for publication 28 September 1989 Communicated by D. Bloch
The magic numbers in alkali/heteroatom microclusters are theoretically studied and new experiments are proposed. These clusters X-Y, where X stands for alkali and Y for heteroatom, exhibit the magic numbers n=6 (8) 18 (20), 38, 56,88, ... based on equilibrium geometry and supported by all experimental data available to date.
In a previous publication [ 1 ] a unified explanation of magic numbers in small clusters for rare-gas and alkali atoms was presented. This unification comes from the consideration of the close-packing concept for two different types of softness of spheres standing for atoms. Specifically, if the spheres are soft the microclusters possess close-packing of spheres structure and exhibit magic numbers at n, = 1, 13, 55, 147, 309, 561, .... while if the spheres are hard they possess close-packing of shells structure and exhibit magic numbers at nz= 2, 8, 20, 40, 58, ... . In later publications based on the previous one, the magic numbers in mixed clusters were presented when the clusters were made up of two alkali [ 21 or of two rare gases [ 3 1. All these three publications are along the line that magic numbers could be interpreted via equilibrium geometry of atoms. From the other point of view, however, spherical shell models (jellium models) refer strictly to electron structure and assume negligible contribution of atomic geometric structure to total energy [ 43. While contribution of both reasons cannot be excluded, heteroatom probing experiments were designed to gauge the relative magnitude of electronic and geo* This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract No. DE-ACOZ76ER03069.
146
metric contributions to total energy [ 5-7 1. Specifically, alkali atoms were replaced by heteroatoms with more than one valence electron (and different atomic radius) to form the heterocluster X,Y. Then magic islands of enhanced thermodynamic stability determined purely by the number of valence electrons in the framework of the jellium model should be shifted with respect to the homoclusters by an amount appropriate to the total valence electron count. For these heteroclusters here we will offer a different explanation based on equilibrium geometries. The basic reference for our study is ref. [ 1 ] and the main points of the present model are demonstrated in fig. 1. The building units of the equilibrium geometry are the shells whose hard-sphere atoms, assumed at the vertices of the polyhedra presented, are in contact with hard spheres of the previous and next shell, but not in contact with spheres of the same shell (as happens in the case of rare-gas cluster structure; see refs. [ 1 ] and [ 3 ] ) . Each polyhedral shell is an equilibrium [ 81 polyhedron which permits an equilibrium of particles at its vertices for any sort of force among particles and, in particular here, minimizes their mutual repulsion. The relative orientation of polyhedra in the different shells is such that their sets of particles are in equilibrium with each other [ 81 and, as a consequence, their relevant polyhedral axes of symmetry coincide.
0375-9601/89/$03.50
0 Elsevier Science Publishers B.V. (North-Holland)
4 December 1989
PHYSICS LETTERS A
Volume 142, number 2,3 lctahedron
(b)
lcosahedron
(c.
lodecahedron
@I)
4%
i)
161
iexahedron
(12)
20)
[481
I lcosldodecahedron I
(dl:
lhomblc
I381
Tnacontah.
(e)
Fig. 1. Close-packing-of-shells equilibrium geometries: Nested equilibrium polyhedra standing for closed shells of hard-sphere atoms. Shells in contact: Octahedron with icosahedron and dcdecahedron, hexahedron with icosidodecahedron, and icosi$odecahedron with rhombic [email protected] heteroatom, at the center of the microcluster, is in contact with all six sphere-atoms of the octahedron.
The heteroatom occupies the center of all those nested polyhedra and all spheres of the first shell are in contact with it. That is, the equilibrium geometry of hard-sphere atoms of alkali/heteroatom microclusters is made up of centered equilibrium polyhedral shells in contact and in equilibrium with each other. Thus, the equilibrium of particles does not only refer to the interactions of particles in the same shell, but also to the interactions of particles in the neighboring (inner and outer) shells (i.e., in all shells). That is, for the equilibrium of each particle all other particles in tbe cluster are taken into consideration. The six polyhedra employed in fig. 1 are the three reciprocal [ 9!] pairs hexahedron-octahedron, dodecahedron-icosahedron, and icosidodecahedronrhombic triacontahedron. That is, only three out of the nine existing [ 81 equilibrium polyhedra are not used in fig. 1,’namely the tetrahedron - which does not possess central symmetry, a fact which could disturb the equilibrium of the nested neighboring shells - and the reciprocal pair of the cuboctahedron and rhombic dodecahedron, where neither member possesses stable equilibrium [ 8 1. Thus,&% three polyhedra are excluded for reasons of stability. The order of polyhedra used in fig. 1 is consistent with closepacking, that is, it proceeds from the polyhedron with
the smaller number of vertices to the polyhedron with the larger number of vertices. The relative orientation of polyhedra follows also from the close-packing and leads to the maximum possible symmetry. That is, rays through the vertices of an inner polyhedron pass through vertices, or middles of edges, or centers of faces of the outer polyhedra. Finally, we may say that the kind of polyhedra employed in fig. 1, their order and relative orientation, lead to a unique closepacking structure for the small clusters of alkali/ heteroatom. The only difference between the present equilibrium geometry for heteroclusters and that of ref. [ 1 ] for alkali homoclusters is that the first shell, the zerohedron, in the latter (made up of two atom-spheres in contact) has been replaced here by the one sphere of the heteroatom. This omission of the zerohedron here is necessary according to the following reason. In ref. [ 1 ] the zerohedron was directed along a C3 axis of the octahedron and all six atom-spheres of this second shell were in contact with the two spheres of the first shell (in two sets of three). This simultaneous fulfillment of equilibrium geometry and of close-packing cannot be preserved here without the omission of the zcrohedron. With the zerohedron and equilibrium geometry present we cannot have close147
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PHYSICS LETTERS A
packing (i.e., the sphere-atoms of the octahedron do not touch the sphere-atoms of the zerohedron), or with the zerohedron and close-packing present we cannot have equilibrium geometry (i.e., the regular octahedron is transformed into a trigonal antiprysm which is not an equilibrium polyhedron [ 8 ] ) . At the top of each block in fig. 1 presenting a polyhedron, the name of the polyhedron is given, while at the bottom of the block two specific numbers are shown. The first, in parentheses, presents the number of vertices for the particular polyhedron, while the second, in brackets, gives the cumulative number of vertices of all previous and that polyhedron. As becomes obvious from fig. 1, up to N=40 all previously formed polyhedral shells coexist in size and form. After N= 40, however, the last shell in N= 40, i.e., the dodecahedron (which does not possess stable equilibrium [ 81) does not coexist with the new coming shells. That is, when the icosidodecahedron is added out of the twenty vertices of the dodecahedron only eight vertices forming a cube remain whose spheres are all in contact with those of the icosidodecahedron which possesses a stable equilibrium by itself [ 8 ] (see ref. [ 11. ) As one can see from the numbers in brackets, the cumulative numbers that appear in the structure of fig. 1 (when successive shells are completed) are 6, 18, 38 (and 56, 88) and coincide with the experimentally observed magic numbers for the heteroclusters Na/Mg, Na/Ca, Na/Sr and Na/Yb shown in table 1. From the same table we see that the first experimental magic number for the heteroclusters Na/Zn, K/M& K/Zn, and K/Hg is 8 and not 6 as Table 1 Neutral abundance maxima in allcali/heteroatom cluster beams.
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148
Metals (X Y)
Maxima (XY)
Heteroatom (Y ) configuration
NalMg Na/Ca Na/Sr Na/Ba Na/Zn Na/Eu Na/Yb
6-8,18 6, 18 6,16-18,38 6, 16 8, 18 6, 16 6, 18,38
.. 3s2 4s2 5S2 6s2 3d1°4s2 4F 6s’ 4F4 6s’
K/Mg KfZn K/HO
8, 18 8, 18 8, 19
5d’06s2
4 December 1989
previously discussed. This difference, however, can be easily understood as follows. According to fig. 1 the first shell is an octahedron and the second is an icosahedron, and their (most symmetric) relative orientation shown is such that each C4 axis of the first coincides with a C2 axis of the second. At n=6 we have the formation of the octahedron and at n = 8 we have two choices. The first is to start a new shell (partial tilling of the icosahedron) and the second is to rearrange all together the eight alkali atoms and form a cube (hexahedron ). The cube is an unstable [ 81 equilibrium polyhedron and, in general, cannot stand as a first shell in the structure of a microcluster. Here, however, the cube acquires additional stability (and reaches a stable equilibrium) due to the fact that its spheres are all in contact with the heteroatom. When even more atoms, however, are added into the cluster, the preferable structure is an interior octahedron (n= 6) and an exterior (partially or totally filled) icosahedron. Thus, n = 8 is a somehow “equally” good magic number like n= 6, but it is not an equally good shell, since it transforms itself later (when n > 8) into n= 6 core (octahedron) plus two atoms in the next shell. At this point two interesting comments should be made. First, we may expect n = 8 to be a magic number even for the cases which do not show up in table 1 (e.g., in the Na/Ca heterocluster, etc. ), or in other words both n = 6 and 8 to be magic numbers as in the heterocluster Na/Mg shown in table 1. Second, we cannot easily exclude the possibility where n = 8 remains as a core even up to the complete filling of the next shell (icosahedron possessing 12 atom-spheres). There is no reason of lack of equilibrium or closepacking to support such an exclusion. That is, we cannot exclude the appearance of a magic peak at n=20 in the mass spectrum of alkali/heteroatom microclusters even if this number has not appeared in the experiments up to now. The appearance of “magic” properties at n = 19 in the heterocluster K/ Hg, as shown in table 1 lends support to our prediction for a sort of magic number at n = 20 for all alkali/heteroatom microclusters. According to this prediction 19 implies a magic number at n = 20 where the one atom has escaped from the microcluster during the ionization procedure. The difference between 6 and 8, and 18 and 20 rests on the thermodynamic internal energy of the microcluster.
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Up to this point, our arguments about magicity of 6 and 8 (or 18 and 20) strictly refer to equilibrium alone and no consideration of sizes for the alkali atoms and heteroatom has been taken. It is significant, however, to consider that n = 8 (and thus n = 20) implies smaller interatomic distances than n = 6 (and n = 18 ) if the heteroatom is assumed to be the same. Thus, it is possible for some cases, for the n = 8 to lead to nearest interatomic distances corresponding to repulsion of the alkali atoms for a specific interatomic potential, a fact which would exclude n=8 and n = 20 from being magic numbers for these alkali/heteroatom microclusters. Perhaps some of the cases in table 1 where n = 8 does not show up as magic number belong to this category of microclusters. However, if n1=8 leads to nearest interatomic distances corresponding to attraction of the alkali atoms, then both n=6 and n= 8 could exist, but n= 8 is energetically more favored since these distances for n = 8 are closer to the distance at minimum potential value than those for n = 6. Thus, according to the present model, depending on the specific potential and distances involved, n = 6 or 8 or both could be the case, as indeed appears in table 1 for the experimental data. The relationship between heteroatom and alkali atom sizes, which allows one to understand that magic number 8 (and thus 20) can be excluded for some systems is the following, r,_,=2(R+r)/$cd,
(1)
where r&A is the smallest distance between two alkali atoms in the case of a cubic arrangement around the heteroatom, R and r are the radii of heteroatom and alkali atoms, respectively, and d is a parameter in the Lennard-Jones interatomic potential
(2) Relationships ( 1) and (2) obviously imply that, in such an arrangement, the force among nearest neighbor alkali atoms is repulsive, a fact which makes this arrangement impossible. Our table 1 provides a good sample (perhaps a complete list ) of all alkali/heteroatom experiments performed to date [ 5-7 1. However, it includes only probing cases to gauge the relative magnitudes of
4 December 1989
electronic structure and equilibrium geometry contributions to the total energy of the microclusters. Of course, heteroclusters where the heteroatom is also an alkali atom should be excluded from our conclusions. In these cases the heteroatom, if the homologous atoms are not too disparate in size, does not occupy the center of the cluster as here [ 51 and.different magic numbers are expected as seen in ref. [ 2 1. Experimental facts involving rare gas/heteroatom clusters support our predictions above [ 101. While we do not forget that, in general, rare gas microclusters differ more substantially in their equilibrium geometries [ 1,3] than alkali/heteroatom clusters, for a small number of rare-gas atoms, however, the equilibrium geometry resembles that of the alkali/heteroatom microclusters. Indeed, the unique stability of the CoArz microcluster has been attributed [ lo] to an octahedral arrangement of Ar atoms around a Co + core which is the same arrangement assumed by fig. 1 for the structure of magic number 6 here. The appearance of n = 8 instead of n = 6 as a magic number in some of the cases of table 1 can somehow be understood in the framework of the jellium model by making the modification of reversing the filling order of Id and 2s electron shells leading to closed shellsat n=2, 8, 10, 20... (s, lp, 2s, Id, ...) instead of the usual n=2, 8, 18, 20, ... (Is, lp, Id, 2s, ...) electrons [ 5-7 1. Such a reversal could be induced by a small depression in the center of the well which can be justified by the higher ionization potential of the heteroatom (compared to an alkali atom) assumed at the center of the cluster. However, the experimentally observed lower ionization potential of the more stable 10 electron system, e.g. K,Mg, versus the higher ionization potential of the less stable 8 electron system, e.g. K,Mg, cannot be understood within the jellium model. Thus, the experimental data falsify even the modified jellium model of metal clusters [ 6-71. Since electron shell closure and ionization-potential strength are electron-only calculations they should be consistent with each other and not inconsistent as previously stated. If, however, as in the present work, n = 8 is supported by equilibrium geometry and not by electron structure, there is no inconsistency if the ionization potential for n = 8 is lower than that for n = 6. In conclusion we may say that the magic numbers 149
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in alkali/heteroatom microclusters are 6 (a), 18 (20), 38, 56 and 88, as shown in fig. 1. These magic numbers are supported by all experimental data available to date [ 5-7 1. Additional experiments are recommended for a larger variety of heteroatoms. It is satisfying that the close-packing of shells concept applied earlier [ 1 ] for the alkali homoclusters is successfully applied here for the alkali/heteroatom microclusters. Thus, the unification of explanation presented in ref. [ 1] for the magic numbers in rare-gas and alkali small clusters is extended to the present case of heteroclusters as well. The equilibrium geometries for Li,Mg (n < 7), Li,Al (n < 4)) and Li,Be (n < 6 ) have been theoretically studied in refs. [ 111 and [ 121, respectively. For the magic number n=6 this geometry in both references is the same as here, i.e., it is that of an octahedron (see fig. 1). I appreciate the constructive comments on the present work made by Professor P. Jena of Virginia Commonwealth University, Department of Physics. I also express my sincere appreciation to Professors J.W. Negele and J. Goldstone of the MIT Depart-
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ment of Physics for their invitation to join the stimulating environment at their Center for Theoretical Physics during my sabbatical year.
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