Dear Sirs.
Thank you for your assistance.
ON THE PROBLEM OF CRYSTAL METALLIC LATTICE IN THE DENSEST PACKINGS OF
CHEMICAL ELEMENTS
Yours faithfully, Н.G FILIPENKА
http://home.ural.ru/~filip/
Grodno
Abstract
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.
Introduction
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.
а) б)
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.
|Element | |RH . 1010 |Z |Z |Lattice type |
| | |(cubic |(numbe|kernel| |
| | |metres /K)|r) | | |
| | | | |(numbe| |
| | | | |r) | |
|Natrium |Na |-2,30 |1 |8 |body-centered|
|Magnesium |Mg |-0,90 |1 |9 |volume-center|
| | | | | |ed |
|Aluminium Or |Al |-0,38 |2 |9 |face-centered|
|Aluminium |Al |-0,38 |1 |12 |face-centered|
|Potassium |K |-4,20 |1 |8 |body-centered|
|Calcium |Ca |-1,78 |1 |9 |face-centered|
|Calciom |Ca |T=737K |2 |8 |body-centered|
|Scandium Or |Sc |-0,67 |2 |9 |volume-center|
| | | | | |ed |
|Scandium |Sc |-0,67 |1 |18 |volume-center|
| | | | | |ed |
|Titanium |Ti |-2,40 |1 |9 |volume-center|
| | | | | |ed |
|Titanium |Ti |-2,40 |3 |9 |volume-center|
| | | | | |ed |
|Titanium |Ti |T=1158K |4 |8 |body-centered|
|Vanadium |V |+0,76 |5 |8 |body-centered|
|Chromium |Cr |+3,63 |6 |8 |body-centered|
|Iron or |Fe |+8,00 |8 |8 |body-centered|
|Iron |Fe |+8,00 |2 |14 |body-centered|
|Iron or |Fe |Т=1189K |7 |9 |face-centered|
|Iron |Fe |Т=1189K |4 |12 |face-centered|
|Cobalt or |Co |+3,60 |8 |9 |volume-center|
| | | | | |ed |
|Cobalt |Co |+3,60 |5 |12 |volume-center|
| | | | | |ed |
|Nickel |Ni |-0,60 |1 |9 |face-centered|
|Copper or |Cu |-0,52 |1 |18 |face-centered|
|Copper |Cu |-0,52 |2 |9 |face-centered|
|Zink or |Zn |+0,90 |2 |18 |volume-center|
| | | | | |ed |
|Zink |Zn |+0,90 |3 |9 |volume-center|
| | | | | |ed |
|Rubidium |Rb |-5,90 |1 |8 |body-centered|
|Itrium |Y |-1,25 |2 |9 |volume-center|
| | | | | |ed |
|Zirconium or |Zr |+0,21 |3 |9 |volume-center|
| | | | | |ed |
|Zirconium |Zr |Т=1135К |4 |8 |body-centered|
|Niobium |Nb |+0,72 |5 |8 |body-centered|
|Molybde-num |Mo |+1,91 |6 |8 |body-centered|
|Ruthenium |Ru |+22 |7 |9 |volume-center|
| | | | | |ed |
|Rhodium Or |Rh |+0,48 |5 |12 |face-centered|
|Rhodium |Rh |+0,48 |8 |9 |face-centered|
|Palladium |Pd |-6,80 |1 |9 |face-centered|
|Silver or |Ag |-0,90 |1 |18 |face-centered|
|Silver |Ag |-0,90 |2 |9 |face-centered|
|Cadmium or |Cd |+0,67 |2 |18 |volume-center|
| | | | | |ed |
|Cadmium |Cd |+0,67 |3 |9 |volume-center|
| | | | | |ed |
|Caesium |Cs |-7,80 |1 |8 |body-centered|
|Lanthanum |La |-0,80 |2 |9 |volume-center|
| | | | | |ed |
|Cerium or |Ce |+1,92 |3 |9 |face-centered|
|Cerium |Ce |+1,92 |1 |9 |face-centered|
|Praseodymium or |Pr |+0,71 |4 |9 |volume-center|
| | | | | |ed |
|Praseodymium |Pr |+0,71 |1 |9 |volume-center|
| | | | | |ed |
|Neodymium or |Nd |+0,97 |5 |9 |volume-center|
| | | | | |ed |
|Neodymium |Nd |+0,97 |1 |9 |volume-center|
| | | | | |ed |
|Gadolinium or |Gd |-0,95 |2 |9 |volume-center|
| | | | | |ed |
|Gadolinium |Gd |T=1533K |3 |8 |body-centered|
|Terbium or |Tb |-4,30 |1 |9 |volume-center|
| | | | | |ed |
|Terbium |Tb |Т=1560К |2 |8 |body-centered|
|Dysprosium |Dy |-2,70 |1 |9 |volume-center|
| | | | | |ed |
|Dysprosium |Dy |Т=1657К |2 |8 |body-centered|
|Erbium |Er |-0,341 |1 |9 |volume-center|
| | | | | |ed |
|Thulium |Tu |-1,80 |1 |9 |volume-center|
| | | | | |ed |
|Ytterbium or |Yb |+3,77 |3 |9 |face-centered|
|Ytterbium |Yb |+3,77 |1 |9 |face-centered|
|Lutecium |Lu |-0,535 |2 |9 |volume-center|
| | | | | |ed |
|Hafnium |Hf |+0,43 |3 |9 |volume-center|
| | | | | |ed |
|Hafnium |Hf |Т=2050К |4 |8 |body-centered|
|Tantalum |Ta |+0,98 |5 |8 |body-centered|
|Wolfram |W |+0,856 |6 |8 |body-centered|
|Rhenium |Re |+3,15 |6 |9 |volume-center|
| | | | | |ed |
|Osmium |Os |