Thank you for your assistance.
ON THE PROBLEM OF CRYSTAL METALLIC LATTICE IN THE DENSEST PACKINGS OF CHEMICAL ELEMENTS
Yours faithfully, Н.G FILIPENKА
The literature generally describes a metallic bond as the one formed by means of mutual bonds between atoms' exterior electrons and not possessing the directional properties. However, attempts have been made to explain directional metallic bonds, as a specific crystal metallic lattice.
This paper demonstrates that the metallic bond in the densest packings (volume-centered and face-centered) between the centrally elected atom and its neighbours in general is, probably, effected by 9 (nine) directional bonds, as opposed to the number of neighbours which equals 12 (twelve) (coordination number).
Probably, 3 (three) "foreign" atoms are present in the coordination number 12 stereometrically, and not for the reason of bond. This problem is to be solved experimentally.
At present, it is impossible, as a general case, to derive by means of quantum-mechanical calculations the crystalline structure of metal in relation to electronic structure of the atom. However, Hanzhorn and Dellinger indicated a possible relation between the presence of a cubical volume-centered lattice in subgroups of titanium, vanadium, chrome and availability in these metals of valent d-orbitals. It is easy to notice that the four hybrid orbitals are directed along the four physical diagonals of the cube and are well adjusted to binding each atom to its eight neighbours in the cubical volume-centered lattice, the remaining orbitals being directed towards the edge centers of the element cell and, possibly, participating in binding the atom to its six second neighbours /3/p. 99.
Let us try to consider relations between exterior electrons of the atom of a given element and structure of its crystal lattice, accounting for the necessity of directional bonds (chemistry) and availability of combined electrons (physics) responsible for galvanic and magnetic properties.
According to /1/p. 20, the number of Z-electrons in the conductivitiy zone has been obtained by the authors, allegedly, on the basis of metal's valency towards oxygen, hydrogen and is to be subject to doubt, as the experimental data of Hall and the uniform compression modulus are close to the theoretical values only for alkaline metals. The volume-centered lattice, Z=1 casts no doubt. The coordination number equals 8.
The exterior electrons of the final shell or subcoats in metal atoms form conductivity zone. The number of electrons in the conductivity zone effects Hall's constant, uniform compression ratio, etc.
Let us construct the model of metal - element so that external electrons of last layer or sublayers of atomic kernel, left after filling the conduction band, influenced somehow pattern of crystalline structure (for example: for the body-centred lattice - 8 valency electrons, and for volume-centered and face-centred lattices - 12 or 9).
ROUGH, QUALITATIVE MEASUREMENT OF NUMBER OF ELECTRONS IN CONDUCTION BAND OF METAL - ELEMENT. EXPLANATION OF FACTORS, INFLUENCING FORMATION OF TYPE OF MONOCRYSTAL MATRIX AND SIGN OF HALL CONSTANT.
(Algorithm of construction of model)
The measurements of the Hall field allow us to determine the sign of charge carriers in the conduction band. One of the remarkable features of the Hall effect is, however, that in some metals the Hall coefficient is positive, and thus carriers in them should, probably, have the charge, opposite to the electron charge /1/. At room temperature this holds true for the following: vanadium, chromium, manganese, iron, cobalt, zinc, circonium, niobium, molybdenum, ruthenium, rhodium, cadmium, cerium, praseodymium, neodymium, ytterbium, hafnium, tantalum, wolfram, rhenium, iridium, thallium, plumbum /2/. Solution to this enigma must be given by complete quantum - mechanical theory of solid body.
Roughly speaking, using the base cases of Born- Karman, let us consider a highly simplified case of one-dimensional conduction band. The first variant: a thin closed tube is completely filled with electrons but one. The diameter of the electron roughly equals the diameter of the tube. With such filling of the area at local movement of the electron an opposite movement of the site of the electron, absent in the tube, is observed, i.e. movement of non-negative sighting. The second variant: there is one electron in the tube - movement of only one charge is possible - that of the electron with a negative charge. These two opposite variants show, that the sighting of carriers, determined according to the Hall coefficient, to some extent, must depend on the filling of the conduction band with electrons. Figure 1.
Figure 1. Schematic representation of the conduction band of two different metals. (scale is not observed).
a) - the first variant;
b) - the second variant.
The order of electron movement will also be affected by the structure of the conductivity zone, as well as by the temperature, admixtures and defects. Magnetic quasi-particles, magnons, will have an impact on magnetic materials.
Since our reasoning is rough, we will further take into account only filling with electrons of the conductivity zone. Let us fill the conductivity zone with electrons in such a way that the external electrons of the atomic kernel affect the formation of a crystal lattice. Let us assume that after filling the conductivity zone, the number of the external electrons on the last shell of the atomic kernel is equal to the number of the neighbouring atoms (the coordination number) (5).
The coordination number for the volume-centered and face-centered densest packings are 12 and 18, whereas those for the body-centered lattice are 8 and 14 (3).
The below table is filled in compliance with the above judgements.
ElementRH . 1010
(cubic metres /K)Z
(number)Lattice typeNatriumNa-2,3018body-centeredMagnesiumMg-0,9019volume-centeredAluminium OrAl-0,3829face-centeredAluminiumAl-0,38112face-centeredPotassiumK-4,2018body-centeredCalciumCa-1,7819face-centeredCalciomCaT=737K28body-centeredScandium OrSc-0,6729volume-centeredScandiumSc-0,67118volume-centeredTitaniumTi-2,4019volume-centeredTitaniumTi-2,4039volume-centeredTitaniumTiT=1158K48body-centeredVanadiumV+0,7658body-centeredChromiumCr+3,6368body-centeredIron orFe+8,0088body-centeredIronFe+8,00214body-centeredIron orFeТ=1189K79face-centeredIronFeТ=1189K412face-centeredCobalt orCo+3,6089volume-centeredCobaltCo+3,60512volume-centeredNickelNi-0,6019face-centeredCopper orCu-0,52118face-centeredCopperCu-0,5229face-centeredZink orZn+0,90218volume-centeredZinkZn+0,9039volume-centeredRubidiumRb-5,9018body-centeredItriumY-1,2529volume-centeredZirconium orZr+0,2139volume-centeredZirconiumZrТ=1135К48body-centeredNiobiumNb+0,7258body-centeredMolybde-numMo+1,9168body-centeredRutheniumRu+2279volume-centeredRhodium OrRh+0,48512face-centeredRhodiumRh+0,4889face-centeredPalladiumPd-6,8019face-centeredSilver orAg-0,90118face-centeredSilverAg-0,9029face-centeredCadmium orCd+0,67218volume-centeredCadmiumCd+0,6739volume-centeredCaesiumCs-7,8018body-centeredLanthanumLa-0,8029volume-centeredCerium orCe+1,9239face-centeredCeriumCe+1,9219face-centeredPraseodymium orPr+0,7149volume-centeredPraseodymiumPr+0,7119volume-centeredNeodymium orNd+0,9759volume-centeredNeodymiumNd+0,9719volume-centeredGadolinium orGd-0,9529volume-centeredGadoliniumGdT=1533K38body-centeredTerbium orTb-4,3019volume-centeredTerbiumTbТ=1560К28body-centeredDysprosiumDy-2,7019volume-centeredDysprosiumDyТ=1657К28body-centeredErbiumEr-0,34119volume-centeredThuliumTu-1,8019volume-centeredYtterbium orYb+3,7739face-centeredYtterbiumYb+3,7719face-centeredLuteciumLu-0,53529volume-centeredHafniumHf+0,4339volume-centeredHafniumHfТ=2050К48body-centeredTantalumTa+0,9858body-centeredWolframW+0,85668body-centeredRheniumRe+3,1569volume-centeredOsmiumOs<0412volume centeredIridiumIr+3,18512face-centeredPlatinumPt-0,19419face-centeredGold orAu-0,69118face-centeredGoldAu-0,6929face-centeredThallium orTl+0,24318volume-centeredThalliumTl+0,2449volume-centeredLead Pb+0,09418face-centeredLeadPb+0,0959face-centered
Where Rh is the Halls constant (Halls coefficient)
Z is an assumed number of electrons released by one atom to the conductivity zone.
Z kernel is the number of external electrons of the atomic kernel on the last shell.
The lattice type is the type of the metal crystal structure at room temperature and, in some cases, at phase transition temperatures (1).