Band theory of a metal from the side of its crystal lattice.

 

  The main problem is that using X-rays, the types of crystal lattices of different metals were determined, and why they are such and not others is not yet known. For example, copper crystallizes in the fcc lattice, and iron in the bcc lattice, which, when heated, becomes fcc and this transition is used in heat treatment of steels.

 Usually in the literature, the metallic bond is described as carried out through the socialization of the outer electrons of the atoms and does not possess the property of directionality. Although there are attempts (see below) to explain the directional metal bond since the elements crystallize into a specific type of lattice.

 The main types of crystal lattices of metals are body-centered cubic; face-centered cubic; hexagonal close-packed. It is still impossible in the general case to deduce the crystal structure of a metal from the electronic structure of the atom from quantum-mechanical calculations, although, for example, Ganzhorn and Delinger pointed out a possible connection between the presence of a cubic body-centered lattice in the subgroups of titanium, vanadium, chromium and the presence of valence d-orbitals in the atoms of these metals.  It is easy to see that the four hybrid orbitals are directed along the four solid diagonals of the cube and are well suited for connecting each atom with its 8 neighbors in a body-centered cubic lattice. In this case, the remaining orbitals are directed to the centers of the unit cell faces and, possibly, can take part in the bond of the atom with its six second neighbors. The first coordination number (K.CH.1) \ "8 \" plus the second coordination number (K.CH.2) \ "6 \" in total is \ "14 \". Let us show that the metallic bond in the closest packing (HEC and FCC) between a centrally selected atom and its neighbors, in the general case, is presumably carried out through 9 (nine) directional bonds, in contrast to the number of neighbors equal to 12 (twelve) the first (coordination number) ... The second (K.P. 2 \ '' 6 \ '' in total is \ '' 18 \ ''.

 In the literature, there are many factors affecting crystallization, so I decided to remove them as much as possible, and the metal model in the article, let's say, is ideal, i.e. all atoms are the same (pure metal), crystal lattices without inclusions, without interstices, without defects, etc. Using the Hall effect and other data on properties, as well as calculations by Ashcroft and Mermin, for me the main factor determining the type of lattice turned out to be the outer electrons of the core of an atom or ion, which resulted from the transfer of some of the outer electrons to the conduction band. It turned out that the metallic bond is due not only to the socialization of electrons, but also to the outer electrons of the atomic cores, which determine the direction or type of the crystal lattice.

 How did I start to build models of ideal single crystals of pure metals? Ideal crystals for getting away from dependence on lattice defects, impurities and other inclusions. Using simple examples, we will show that one bond for a diamond at a density packing 34% and coordination number 4 account for 34%: 4 = 8.5%. The cubic primitive lattice has a packing density of 52% and coordination number 6 accounts for 52%: 6 = 8.66%. For a cubic body-centered lattice, the packing density is 68% and coordination number 8 accounts for 68%: 8 = 8.5%. For a cubic face-centered lattice, the packing density is 74% and the coordination number 12 is 74%: 12 = 6.16% (!!!), and if 74%: 9 = 8.22%. For a hexagonal lattice, the packing density is 74% and the coordination number 12 is 74%: 12 = 6.16%, and if 74%: 9 = 8.22%. (!!!) Obviously, these 8.66-8.22% carry some physical meaning. The remaining 26% are multiples of 8.66 and 100% hypothetical packing density is possible with 12 bonds. But is such a possibility real? The outer electrons of the last shell or subshells of the metal atom form the conduction band. The number of electrons in the conduction band affects the Hall constant, the compression ratio, etc. Let us construct a model of an element metal so that the remaining, after filling the conduction band, the outer electrons of the last shell or subshells of the atomic core in some way affect the structure of the crystal structure (for example: for the bcc lattice-8 "valence" electrons, and for HEC and FCC -12 or 9). As a result of studying the lattices of chemical elements, we can say that the bcc lattices of light elements are formed by 8 bond electrons, and heavy 14 electrons of the atomic core. FCC lattices are formed by 9 bond electrons for light elements and 15 for heavy ones.

   Then I began to fill the conduction band with external electrons. One of the remarkable features of the Hall effect is, however, that in some metals the Hall coefficient is positive, and therefore the carriers in them should apparently have a charge opposite to the charge electron. This property required clarification. Option one: a thin closed tube, completely filled with electrons except one. With such a filling of the zone, with the local movement of an electron, the opposite movement of the \ "place \" of the electron, which has not filled the tube, is observed, that is, the movement of a non-negative charge. Option two: there is one electron in the tube, therefore, only one charge, a negatively charged electron, can move. It can be seen from these two extreme variants that the sign of the carriers determined by the Hall coefficient should, to some extent, depend on the filling of the conduction band with electrons. Let us fill the conduction band with electrons so that the outer electrons of the atomic cores influence on the formation of a type of crystallization lattice. Let us assume that the number of external bond electrons on the last shell of the atomic core, after filling the conduction band, is equal to the number of neighboring atoms (coordination number) in the crystal lattice. It turned out that the metallic bond is due not only to the socialization of electrons, but also to the outer electrons of the atomic cores, which determine the direction or type of the crystal lattice. Let's try to connect the outer electrons of an atom of a given element with the structure of its crystal lattice, taking into account the need for directed bonds (chemistry) and the presence of socialized electrons (physics) responsible for the galvanomagnetic properties.

 see the main part of the work on pages (in Russian and English) https://natureofchemicalelements.blogspot.com 

I consider the main achievement of my work that the real first coordination number for atoms in single crystals of pure metals (fcc and geocrystalline lattices) was determined equal to 9. This number was deduced from the physical and chemical properties of crystals. About bond electrons in single crystals of metals, which determine the type of crystal lattice. For potassium, sodium, rubidium, cesium in the conduction band, 1 electron and 8 bond electrons each - the Hall constant is negative (in the conduction band, one electron from an atom), the type of bcc lattice ... each selected atom has 8 neighbors in the crystal lattice ... Nickel, copper, silver, platinum, palladium and gold have an fcc lattice ... crystallization requires 15 bond electrons from an atom ... let's look at nickel as an example 1s2 2s2 2p6 3s2 3p6 3d8 4s2 external electrons in total 16 (3p6 3d8 4s2) one went into the conduction band 15 entered into communication with neighboring atoms ... this one electron from the conduction band is checked by the Hall constant, if it is negative, then there are 1-2 electrons in the conduction band, and if it is positive, then more. Magnesium 2 electrons are bonded to the nucleus, 9 bond electrons (GEK) and one electron in the conduction band - Hall constant is negative, aluminum 2 electrons are bonded to the nucleus, 9 bond electrons (FCC) and two electrons in the conduction band - Hall constant is negative.

 Let us summarize the results of the work. According to my constructions, for almost all metals, conduction electrons (their number), bond electrons, which mainly determine the type of crystal lattice and electrons associated with the nucleus, are determined, possibly with small errors. In metal crystals, atoms are united not only by the socialization of conduction electrons, but also by bond electrons, which were revealed in my work. In other words, the valencies of atoms in single crystals of some metals can be 15, 14, 9, 8 and probably less. In alloys, the valences of these atoms can most likely change downward. For some single crystal elements, I may be mistaken in counting bond electrons, which affect the formation of a particular type of crystal lattice. However, it seems to me that such a pattern exists.

  Consider the most refractory and hardest metal, tungsten. The electronic configuration of its atom is [Xe] 4f 14 5d 4 6s 2. Of the 20 outer electrons, 14 are needed for bonds with neighboring atoms, and since the Hall constant is positive and equal to approximately unity, there are 2 electrons in the conduction band of tungsten. This means that 4 electrons from f or from d remain associated with the nucleus. But in order to be the hardest, tungsten needs to have both many bond electrons (14) and many electrons in the conduction band (6) for a strong metallic bond. Therefore, verification by experiment is required. 

  Band theory of a metal from the side of its crystal lattice. The conduction band, the valence electron band of the bond between atoms and the zone of the nucleus with the rest of the electrons. 

  Henadzi Filipenka