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Articles and Publication Physics Physical chemistry 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
© HENADZI R.
FILIPENKA,
Contact: filipenko@tut.by
http://home.ural.ru/~filip
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.
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:
- Solid state physics. N.W. Ashcroft, N.D. Mermin. Cornell
University, 1975
- Characteristics of elements. G.V. Samsonov. Moscow, 1976
- Grundzuge der Anorganischen Kristallchemie. Von. Dr. Heinz
Krebs. Universitat Stuttgart, 1968
- Physics of metals. Y.G. Dorfman, I.K. Kikoin. Leningrad,
1933
- What affects crystals characteristics. G.G.Skidelsky.’’
Engineer’’ № 8, 1989,Moscow.
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
а o
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 p
n ) 1/3
Where n is the density of conductivity
electrons.
Table 1. Calculation according to Ashcroft and
Mermin
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 |
Table 2. Calculation according to the models
considered in this paper
Element |
Z |
rs/ao |
theoretical |
calculated |
Cs |
1 |
5.62 |
1.54 |
1.43 |
Cu |
2 |
2.12 |
202.3 |
134.3 |
Ag |
2 |
2.39 |
111.0 |
99.9 |
Al |
2 |
2.40 |
108.6 |
76.0 |
Of course, the pressure of free electrons
gases alone does not fully determine the compressive strenth of the metal,
nevertheless in the second calculation instance the theoretical uniform
compression modulus lies closer to the experimental one (approximated the
experimental one) this approach (approximation) being one-sided. The second
factor the effect of “valency” or external electrons of the atomic kernel,
governing the crystal lattice is evidently required to be taken into
consideration.
Literature:
-
Solid state physics. N.W. Ashcroft, N.D.
Mermin. Cornell University, 1975
Publishing date: May 19, 2003
Source: SciTecLibrary.ru
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