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|
|Scandium |Sc |-0,67 |1 |18 |volume-center|
|Titanium |Ti |-2,40 |1 |9 |volume-center|
|Titanium |Ti |-2,40 |3 |9 |volume-center|
|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|
|Cobalt |Co |+3,60 |5 |12 |volume-center|
|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|
|Zink |Zn |+0,90 |3 |9 |volume-center|
|Rubidium |Rb |-5,90 |1 |8 |body-centered|
|Itrium |Y |-1,25 |2 |9 |volume-center|
|Zirconium or |Zr |+0,21 |3 |9 |volume-center|
|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|
|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|
|Cadmium |Cd |+0,67 |3 |9 |volume-center|
|Caesium |Cs |-7,80 |1 |8 |body-centered|
|Lanthanum |La |-0,80 |2 |9 |volume-center|
|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|
|Praseodymium |Pr |+0,71 |1 |9 |volume-center|
|Neodymium or |Nd |+0,97 |5 |9 |volume-center|
|Neodymium |Nd |+0,97 |1 |9 |volume-center|
|Gadolinium or |Gd |-0,95 |2 |9 |volume-center|
|Gadolinium |Gd |T=1533K |3 |8 |body-centered|
|Terbium or |Tb |-4,30 |1 |9 |volume-center|
|Terbium |Tb |Т=1560К |2 |8 |body-centered|
|Dysprosium |Dy |-2,70 |1 |9 |volume-center|
|Dysprosium |Dy |Т=1657К |2 |8 |body-centered|
|Erbium |Er |-0,341 |1 |9 |volume-center|
|Thulium |Tu |-1,80 |1 |9 |volume-center|
|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|
|Hafnium |Hf |+0,43 |3 |9 |volume-center|
|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|
|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-center|
|Thallium |Tl |+0,24 |4 |9 |volume-center|
|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.
[pic]
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.
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
аo 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 (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
|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.
Solid state physics. N.W. Ashcroft, N.D. Mermin. Cornell
March 1996
Н.G. Filipenkа
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