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

| | | | | |ed |

|Zirconium or |Zr |+0,21 |3 |9 |volume-center|

| | | | | |ed |

|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|

| | | | | |ed |

|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|

| | | | | |ed |

|Cadmium |Cd |+0,67 |3 |9 |volume-center|

| | | | | |ed |

|Caesium |Cs |-7,80 |1 |8 |body-centered|

|Lanthanum |La |-0,80 |2 |9 |volume-center|

| | | | | |ed |

|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|

| | | | | |ed |

|Praseodymium |Pr |+0,71 |1 |9 |volume-center|

| | | | | |ed |

|Neodymium or |Nd |+0,97 |5 |9 |volume-center|

| | | | | |ed |

|Neodymium |Nd |+0,97 |1 |9 |volume-center|

| | | | | |ed |

|Gadolinium or |Gd |-0,95 |2 |9 |volume-center|

| | | | | |ed |

|Gadolinium |Gd |T=1533K |3 |8 |body-centered|

|Terbium or |Tb |-4,30 |1 |9 |volume-center|

| | | | | |ed |

|Terbium |Tb |Т=1560К |2 |8 |body-centered|

|Dysprosium |Dy |-2,70 |1 |9 |volume-center|

| | | | | |ed |

|Dysprosium |Dy |Т=1657К |2 |8 |body-centered|

|Erbium |Er |-0,341 |1 |9 |volume-center|

| | | | | |ed |

|Thulium |Tu |-1,80 |1 |9 |volume-center|

| | | | | |ed |

|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|

| | | | | |ed |

|Hafnium |Hf |+0,43 |3 |9 |volume-center|

| | | | | |ed |

|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|

| | | | | |ed |

|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|

| | | | | |ed |

|Thallium |Tl |+0,24 |4 |9 |volume-center|

| | | | | |ed |

|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.

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

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

|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

Grodno

March 1996

Н.G. Filipenkа

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