On the problem of crystal metallic lattice in the densest packings of chemical elements valency.The electrons in zone of conductivity.

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

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                        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.

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/(r_s|a_o))⁵* 10¹⁰ dyne/cm²

Where B is the uniform compression modulus

а_o is the Bohr radius

r_s – the radius of the sphere with the volume being equal to the volume falling at one conductivity electron.

r_s = (3/4 pn ) ^1/3

Where n is the density of conductivity electrons.

Table 1. Calculation according to Ashcroft and Mermin

Table 2. Calculation according to the models considered in this paper

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

Grodno

March 1996                                                                  Н.G. Filipenkа