Crystallography Reports, Vol. 50, No. 5, 2005, pp. 715–720. Translated from Kristallografiya, Vol. 50, No. 5, 2005, pp. 775–781. Original Russian Text Copyright © 2005 by VeremeÏchik. CRYSTALLOGRAPHIC SYMMETRY Electrons of l Shells of Free Atoms As a Regular System of Points on a Sphere: III. Factors of Self-Organization of Electrons in the Periodic Table T. F. VeremeÏchik Shubnikov Institute of Crystallography, Russian Academy of Sciences, LeninskiÏ pr. 59, Moscow, 119333 Russia e-mail: [email protected] Received March 23, 2005 Abstract—The factors determining the self-organization of the electron system of an atom at different levels of the periodic table are considered. Specifically, these factors are the isotropy and three-dimensional nature of space and the indistinguishability of electrons. The concept of a simplex is used, whose vertices correspond to a regular system of particles (minimum in number for a given space) in the state of the global minimum of the system’s potential. These factors implement the principle of simplicity (small number of particles) and hierar- chy in the periodic table of elements. The global minimum of the potential of s, p, d, and f shells is reached in odd-dimensional spaces. In a three-dimensional space, such a minimum is reached for d and f shells, in contrast to s and p shells, through shell mixing. © 2005 Pleiades Publishing, Inc. STATEMENT OF THE PROBLEM ond (third) and fourth (fifth) periods is ten elements. The difference between the fourth (fifth) and sixth (sev- In was shown in [1] that the maxima of the electron- enth) periods is 14 elements. The number of elements density probability in the l shells of an atom (s, p, d, and in the first period suggests that the formation of groups f shells, well-known in quantum mechanics) are mod- of particles begins with the simplest group consisting of eled by vertices of antiprisms. This result made it pos- two particles. sible to explain a number of structural and physical properties of crystals [2]. Furthermore, we will use for In the three-dimensional (3D) isotropic space, the brevity the term “electron” instead of “maximum of centrosymmetrical potential U(r), invariant with electron-density probability.” In this study, we consider respect to all rotations around fixed axes and inversion, the factors implementing in the periodic table the prin- is the most symmetrical. From regular closed systems ciple of simplicity (minimum number of particles) and {Ai, N}, where N is the number of indistinguishable hierarchy in self-organization of electrons at different Coulomb particles, the simplest stable system in the levels of the periodic table: l shells, periods, and the field of the potential U(r) is apparently a dimer {A1, N = number of periods of different types. The specific fea- 2}, where N is exactly the number of elements in the tures of antiprisms are analyzed in the context of the first period. Groups with a small number of particles principle of simplicity and the existence of a global remain in the attractive field at larger radii r than the minimum in the most stable systems of particles (which groups with a large number of particles. Therefore, the are listed in the periodic table). To describe the crystal- systems {Ai} with different N will be spatially sepa- lographic model of electron shells in more detail, we rated. compare it with the quantum model and the structures of coordination polyhedra and clusters. The equivalent vertices of a regular simplex—the simplest polyhedron in space [3]—correspond to the simplest regular system of particles and the global min- STABLE SYSTEMS OF COULOMB PARTICLES imum of its potential. The number of points equal to the IN THE PERIODIC TABLE number of simplex fix a sphere in this space. Therefore, The difference in the numbers of elements in the the number of particles in the system {Ai + 1} will be periods of the periodic table indicates the existence of equal to the number of particles in the system {Ai} plus stable groups of discrete particles and gives their num- four particles fixing a new sphere. The sequence of val- bers in groups. The first period contains two elements. ues of N in these systems forms an infinite series: 2, 6, The difference between the first and second (third) peri- 10, 14, …. The periodic table limits this series to the ods is eight elements. The difference between the sec- first four terms. 1063-7745/05/5005-0715 $26.00 © 2005 Pleiades Publishing, Inc. 716 VEREMEÎCHIK (a)(b) (c) (d) Fig. 1. Antiprisms in 3D space (octahedra in (a) 1-, (b) 3-, (c) 5-, and (d) 7-dimensional spaces), whose vertices model stable systems of 2, 6, 10, and 14 equivalent electrons of l shells in the periodic table. We can suggest that the limitation of the number of tron, which is an integer in h units. Therefore, a definite the most stable systems {Ai} to four is also related to value of the projection of l on an isolated axis m1 = 2l + 1 the dimension factor. The ends of the radii of the is juxtaposed to each vertex of the antiprism base. One spheres of the l shells in this set in 3D space fix the total of the antiprism bases is juxtaposed to the spin-up space sphere of 32 particles—the simplest and the most stable and the other base is juxtaposed to the spin-down space sum of the l shells. The number of shells limits also the [1]. The antiprisms are composed of N/2 dimers, which number of types of periods in the periodic table: the are transformed into each other in inversion, rotation, first period corresponds to the filling of the s shell; the and inversion rotation operations. Therefore, another second and third periods correspond to the filling of the analogy can be used: as well as a dimer, an electron pair s and p shells; the fourth and fifth periods correspond to with a compensated spin serves as a forming pair the filling of the s, p, and d shells; and the sixth and sev- (closed or ESR pair in quantum theory [8, 9]). enth periods correspond to the filling of the s, p, d, and f shells. Owing to the central symmetry of the potential U(r) EQUIVALENCE OF PARTICLES IN l SHELLS and the correspondence of the minimum of repulsion of AND OCTAHEDRA OF ODD-DIMENSIONAL identical Coulomb particles to inversion (the latter gen- SPACES erally leads to larger distances between particles than a The bases of a dimer and of trigonal, pentagonal, reflection in plane or a rotation around a twofold axis), and heptagonal antiprisms are regular simplices of the systems {Ai} should be centrosymmetric. zero-, two-, four-, and six-dimensional spaces. Anti- From the bodies whose vertices correspond to stable prisms, as pairs of regular simplices sharing an inver- systems {Ai} on a sphere, i.e., 5 regular Platonic bodies sion center, are at the same time octahedra of odd- (all faces, edges, and vertices are identical) and three dimensional spaces with dimensions of 1, 3, 5, and 7. groups of semiregular bodies (equivalent vertices and The odd dimensionality of the octahedra is caused edges but not all faces identical), specifically, a set of by the fact that inversion performed k times is equiva- 14 Archimedean bodies and 2 infinite series of prisms lent to unit operation at even k and single inversion at and antiprisms [4, 5], it is the series of antiprisms that odd k. Therefore, if a straight line is an odd-fold inver- exhibit central symmetry and satisfy the conditions sion axis, it is simultaneously a rotary axis of the same {A1, N = 2}, {Ai + 1, Ni + 1 = Ni + 4} [1] (Fig. 1). order and its singularity is an inversion center. For If the arrangement of particles somewhere corre- example, the presence of a fivefold axis and a center of sponds to the lowest energy, they will be arranged in the symmetry in a pentagonal antiprism is a necessary and same way in another place [6]. Thus, {Ai} systems and sufficient condition for the existence of a fivefold inver- their sequence will be repeated with an increase in the sion axis. However, the reverse statement is valid only number of particles N and the radius r. for odd-fold axes [10]. Thus, absolute indistinguish- The arrangement of particles in a regular system of ability of all particles occurs when the number of verti- points is related to the optimization of their potential ces in antiprism bases is odd. and implemented under the action of the potential gra- In the quantum model l shells, the coordinate space dient since the gradient becomes zero only at points of is odd-dimensional since at integer l the value of m1 = a regular system [7]. 2l + 1 is odd. Even dimension of m1 corresponds to the In the quantum model of atom, the potential of its j coupling, at which the spin–orbit interaction in a par- many-electron system is constructed in the 2(2l + 1)- ticle is stronger than the interaction of the particle with dimensional space since each value of the potential cor- the centrosymmetric field. Then, the total momentum responds to a certain distribution of electrons over 2l + of the particle j = l + 1/2 and the number of equivalent 1 vectors of coordinate space and two vectors (±1/2) of particles N = 2j + 1 acquire the values of the entire spin space. Here, l is the orbital momentum of an elec- series of even numbers. It can be seen easily that the CRYSTALLOGRAPHY REPORTS Vol.