Why the crystal structure of the element is such
lattice but not another?
Ðåôåðàò
Ãåííàä³ÿ Ô³ë³ïåíêî
Ãðîäíî
1996
"Why the crystal structure of the element is such
lattice but not another? How much electrons are placed in zone conductivity from
one the atom of lattice? "
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.Why
the crystal structure of the element is such lattice but not another? How much
electrons are placed in zone conductivity from one the atom of lattice?
This paper demonstrates that the metallic bond in the
densest packings (volume-centered and face-centered) between the centrally
selected 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, ie
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.
+ q
-q
à) á)
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
Ò = 1189K
7
9
face-centered
Iron
Fe
Ò = 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
Ò = 1135Ê
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
Ò = 1560Ê
2
8
body-centered
Dysprosium
Dy
-2,70
1
9
volume-centered
Dysprosium
Dy
Ò = 1657Ê
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
Ò = 2050Ê
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
