ON THE PROBLEM OF CRYSTAL METALLIC LATTICE IN THE DENSEST PACKINGS OF
CHEMICAL ELEMENTS
G.G FILIPENKO
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
(cubic metres /K) Z
(number) Z kernel
(number) Lattice type
Natrium Na -2,30 1 8 body-centered
Magnesium Mg -0,90 1 9 volume-centered
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-centered
Scandium Sc -0,67 1 18 volume-centered
Titanium Ti -2,40 1 9 volume-centered
Titanium Ti -2,40 3 9 volume-centered
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-centered
Cobalt Co +3,60 5 12 volume-centered
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-centered
Zink Zn +0,90 3 9 volume-centered
Rubidium Rb -5,90 1 8 body-centered
Itrium Y -1,25 2 9 volume-centered
Zirconium or Zr +0,21 3 9 volume-centered
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-centered
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-centered
Cadmium Cd +0,67 3 9 volume-centered
Caesium Cs -7,80 1 8 body-centered
Lanthanum La -0,80 2 9 volume-centered
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-centered
Praseodymium Pr +0,71 1 9 volume-centered
Neodymium or Nd +0,97 5 9 volume-centered
Neodymium Nd +0,97 1 9 volume-centered
Gadolinium or Gd -0,95 2 9 volume-centered
Gadolinium Gd T=1533K 3 8 body-centered
Terbium or Tb -4,30 1 9 volume-centered
Terbium Tb Т=1560К 2 8 body-centered
Dysprosium Dy -2,70 1 9 volume-centered
Dysprosium Dy Т=1657К 2 8 body-centered
Erbium Er -0,341 1 9 volume-centered
Thulium Tu -1,80 1 9 volume-centered
Ytterbium or Yb +3,77 3 9 face-centered
Ytterbium Yb +3,77 1 9 face-centered
Lutecium Lu -0,535 2 9 volume-centered
Hafnium Hf +0,43 3 9 volume-centered
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-centered
Osmium Os
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