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

G.G FILIPENKO

www.belarus.net/discovery/filipenko

sci.materials(1999)

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|

| | | | | |ed |

|Scandium |Sc |-0,67 |1 |18 |volume-center|

| | | | | |ed |

|Titanium |Ti |-2,40 |1 |9 |volume-center|

| | | | | |ed |

|Titanium |Ti |-2,40 |3 |9 |volume-center|

| | | | | |ed |

|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|

| | | | | |ed |

|Cobalt |Co |+3,60 |5 |12 |volume-center|

| | | | | |ed |

|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|

| | | | | |ed |

|Zink |Zn |+0,90 |3 |9 |volume-center|

| | | | | |ed |

|Rubidium |Rb |-5,90 |1 |8 |body-centered|

|Itrium |Y |-1,25 |2 |9 |volume-center|

| | | | | |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

Ao 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 Mermine

|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

G.G. Filipenko

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