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On the problem of crystal metallic lattice in the densest packings of chemical elements
Другое : On the problem of crystal metallic lattice in the densest packings of chemical elements
On the problem of crystal metallic lattice in the densest packings of chemical elements
ON THE PROBLEM OF CRYSTAL METALLIC LATTICE IN THE
DENSEST PACKINGS OF CHEMICAL ELEMENTS
G.G FILIPENKO
www.belarus.net/discovery/filipenko
sci.materials(1999)
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
|
<0
|
4
|
12
|
volume centered
|
|
Iridium
|
Ir
|
+3,18
|
5
|
12
|
face-centered
|
|
Platinum
|
Pt
|
-0,194
|
1
|
9
|
face-centered
|
|
Gold or
|
Au
|
-0,69
|
1
|
18
|
face-centered
|
|
Gold
|
Au
|
-0,69
|
2
|
9
|
face-centered
|
|
Thallium or
|
Tl
|
+0,24
|
3
|
18
|
volume-centered
|
|
Thallium
|
Tl
|
+0,24
|
4
|
9
|
volume-centered
|
|
Lead
|
Pb
|
+0,09
|
4
|
18
|
face-centered
|
|
Lead
|
Pb
|
+0,09
|
5
|
9
|
face-centered
|
Where Rh is the Hall’s constant (Hall’s
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).
Conclusions
In spite of the rough reasoning the table shows that
the greater number of electrons gives the atom of the element to the
conductivity zone, the more positive is the Hall’s constant. On the contrary
the Hall’s constant is negative for the elements which have released one or two
electrons to the conductivity zone, which doesn’t contradict to the conclusions
of Payerls. A relationship is also seen between the conductivity electrons (Z)
and valency electrons (Z kernel) stipulating the crystal structure.
The
phase transition of the element from one lattice to another can be explained by
the transfer of one of the external electrons of the atomic kernel to the metal
conductivity zone or its return from the conductivity zone to the external
shell of the kernel under the influence of external factors (pressure,
temperature).
We tried to unravel the puzzle, but
instead we received a new puzzle which provides a good explanation for the
physico-chemical properties of the elements. This is the “coordination number”
9 (nine) for the face-centered and volume-centered lattices.
This frequent occurrence of the
number 9 in the table suggests that the densest packings have been studied
insufficiently.
Using
the method of inverse reading from experimental values for the uniform
compression towards the theoretical calculations and the formulae of Arkshoft
and Mermin (1) to determine the Z value, we can verify its good agreement with
the data listed in Table 1.
The metallic bond seems to be due to
both socialized electrons and “valency” ones – the electrons of the atomic
kernel.
Literature:
1)
Solid state physics. N.W. Ashcroft, N.D. Mermin. Cornell University,
1975
2)
Characteristics of elements. G.V. Samsonov. Moscow,
1976
3)
Grundzuge der Anorganischen Kristallchemie. Von. Dr. Heinz Krebs.
Universitat Stuttgart, 1968
4)
Physics of metals. Y.G. Dorfman, I.K. Kikoin.
Leningrad, 1933
5)
What affects crystals characteristics.
G.G.Skidelsky. Engineer № 8, 1989
Appendix 1
Metallic Bond in Densest Packing (Volume-centered
and face-centered)
It follows from the speculations on the
number of direct bonds ( or pseudobonds, since there is a conductivity zone
between the neighbouring metal atoms) being equal to nine according to the
number of external electrons of the atomic kernel for densest packings that
similar to body-centered lattice (eight neighbouring atoms in the first
coordination sphere). Volume-centered and face-centered lattices in the first
coordination sphere should have nine atoms whereas we actually have 12 ones. But
the presence of nine neighbouring atoms, bound to any central atom has
indirectly been confirmed by the experimental data of Hall and the uniform
compression modulus (and from the experiments on the Gaase van Alfen effect the
oscillation number is a multiple of nine.
Consequently, differences from other atoms in the coordination
sphere should presumably be sought among three atoms out of 6 atoms located in
the hexagon. Fig.1,1. d, e shows coordination spheres in the densest hexagonal
and cubic packings.
Fig.1.1. Dense Packing.
It should be noted that in the hexagonal packing, the triangles of
upper and lower bases are unindirectional, whereas in the hexagonal packing
they are not unindirectional.
Literature:
Introduction into physical chemistry and chrystal
chemistry of semi-conductors. B.F. Ormont. Moscow, 1968.
Appendix 2
Theoretical calculation of the uniform compression
modulus (B).
B = (6,13/(rs|ao))5*
1010 dyne/cm2
Where B is the
uniform compression modulus
Ao
is the Bohr radius
rs
– the radius of the sphere with the volume being equal to the volume falling at
one conductivity electron.
rs
= (3/4 pn ) 1/3
Where
n is the density of conductivity electrons.
Table
1. Calculation according to Ashcroft and Mermine
|
Element
|
Z
|
rs/ao
|
theoretical
|
calculated
|
|
Cs
|
1
|
5.62
|
1.54
|
1.43
|
|
Cu
|
1
|
2.67
|
63.8
|
134.3
|
|
Ag
|
1
|
3.02
|
34.5
|
99.9
|
|
Al
|
3
|
2.07
|
228
|
76.0
|
|
