On the problem of crystal metallic lattice in the densest packings of chemical elements (реферат)

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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

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.


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

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).


(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

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.


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 centeredIridium Ir +3,18 5 12 face-centeredPlatinum Pt -0,194 1 9 face-centeredGold or Au -0,69 1 18 face-centeredGold Au -0,69 2 9 face-centeredThallium or Tl +0,24 3 18 volume-centeredThallium Tl +0,24 4 9 volume-centeredLead Pb +0,09 4 18 face-centeredLead Pb +0,09 5 9 face-centeredWhere 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).ConclusionsIn 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, 1975Characteristics of elements. G.V. Samsonov. Moscow, 1976Grundzuge der Anorganischen Kristallchemie. Von. Dr. Heinz Krebs. Universitat Stuttgart, 1968Physics of metals. Y.G. Dorfman, I.K. Kikoin. Leningrad, 1933What affects crystals characteristics. G.G.Skidelsky. Engineer № 8, 1989Appendix 1Metallic 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 2Theoretical calculation of the uniform compression modulus (B).B = (6,13/(rs|ao))5* 1010 dyne/cm2Where B is the uniform compression modulusAo is the Bohr radiusrs – the radius of the sphere with the volume being equal to the volume falling at one conductivity electron.rs = (3/4 (n ) 1/3Where n is the density of conductivity electrons.Table 1. Calculation according to Ashcroft and MermineElement Z rs/ao theoretical calculatedCs 1 5.62 1.54 1.43Cu 1 2.67 63.8 134.3Ag 1 3.02 34.5 99.9Al 3 2.07 228 76.0Table 2. Calculation according to the models considered in this paperElement Z rs/ao theoretical calculatedCs 1 5.62 1.54 1.43Cu 2 2.12 202.3 134.3Ag 2 2.39 111.0 99.9Al 2 2.40 108.6 76.0Of 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, 1975GrodnoMarch 1996 G.G. Filipenko-q-e-e+q

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