Реферат: On the problem of crystal metallic lattice in the densest packings of chemical elements

ON THE PROBLEM OF CRYSTAL METALLIC LATTICE IN THEDENSEST PACKINGS OF CHEMICAL ELEMENTS G.G FILIPENKOwww.belarus.net/discovery/filipenkosci.materials(1999)GrodnoAbstractThe literature generally describes a metallic bondas the one formed by means of mutual bonds between atoms' exterior electronsand not possessing the directional properties. However, attempts have been madeto explain directional metallic bonds, as a specific crystal metallic lattice.This paper demonstrates that the metallic bond inthe densest packings (volume-centered and face-centered) between the centrallyelected 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 arepresent in the coordination number 12 stereometrically, and not for the reasonof bond. This problem is to be solved experimentally.IntroductionAt present, it is impossible, as a general case, toderive by means of quantum-mechanical calculations the crystalline structure ofmetal in relation to electronic structure of the atom. However, Hanzhorn andDellinger indicated a possible relation between the presence of a cubicalvolume-centered lattice in subgroups of titanium, vanadium, chrome andavailability in these metals of valent d-orbitals. It is easy to notice thatthe four hybrid orbitals are directed along the four physical diagonals of thecube and are well adjusted to binding each atom to its eight neighbours in thecubical volume-centered lattice, the remaining orbitals being directed towardsthe edge centers of the element cell and, possibly, participating in bindingthe atom to its six second neighbours /3/p. 99.Let us try to consider relations between exteriorelectrons of the atom of a given element and structure of its crystal lattice,accounting for the necessity of directional bonds (chemistry) and availabilityof combined electrons (physics) responsible for galvanic and magneticproperties.According to /1/p. 20, the number of Z-electrons inthe conductivitiy zone has been obtained by the authors, allegedly, on thebasis of metal's valency towards oxygen, hydrogen and is to be subject todoubt, as the experimental data of Hall and the uniform compression modulus areclose to the theoretical values only for alkaline metals. The volume-centeredlattice, Z=1 casts no doubt. The coordination number equals 8.The exterior electrons of the final shell orsubcoats in metal atoms form conductivity zone. The number of electrons in the conductivity zoneeffects Hall's constant, uniform compression ratio, etc.

Letus constructthe 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 ofcrystalline 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 ELECTRONSIN CONDUCTION BAND OF METAL — ELEMENT. EXPLANATION OF FACTORS, INFLUENCINGFORMATION OF TYPE OF MONOCRYSTAL MATRIX AND SIGN OF HALL CONSTANT.

(Algorithmof construction of model)

Themeasurements of the Hall field allow us to determine the sign of chargecarriers in the conduction band. One of the remarkable features of the Halleffect is, however, that in some metals the Hall coefficient is positive, andthus carriers in them should, probably, have the charge, opposite to theelectron 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/. Solutionto this enigma must be given by complete quantum — mechanical theory of solidbody.

Roughlyspeaking, using the base cases of Born- Karman, let us consider a highly simplifiedcase of one-dimensional conduction band. The first variant: a thin closed tubeis completely filled with electrons but one. The diameter of the electronroughly equals the diameter of the tube. With such filling of the area at localmovement 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. Thesecond variant: there is one electron in the tube — movement of only one chargeis possible — that of the electron with a negative charge. These two oppositevariants show, that the sighting of carriers, determined according to the Hallcoefficient, to some extent, must depend on the filling of the conduction bandwith electrons. Figure 1.

-e

-q

а) б)

Figure 1. Schematic representation of the conduction band of twodifferent metals. (scale is not observed).

a) — the first variant;

b) — the second variant.

The order ofelectron movement will also be affected by the structure of the conductivityzone, as well as by the temperature, admixtures and defects. Magneticquasi-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 theconductivity zone. Let us fill the conductivity zone with electrons in such away that the external electrons of the atomic kernel affect the formation of acrystal lattice. Let us assume that after filling the conductivity zone, thenumber of the external electrons on the last shell of the atomic kernel isequal to the number of the neighbouring atoms (the coordination number) (5). The coordination number for thevolume-centered and face-centered densest packings are 12 and 18, whereas thosefor the body-centered lattice are 8 and 14 (3).

The below table is filled in compliance with theabove judgements.

Element

RH.1010

(cubic metres /K)

(number)

Z kernel

(number)

Lattice type

Natrium

-2,30

body-centered

Magnesium

-0,90

volume-centered

Aluminium Or

-0,38

face-centered

Aluminium

-0,38

face-centered

Potassium

-4,20

body-centered

Calcium

-1,78

face-centered

Calciom

T=737K

body-centered

ScandiumOr

-0,67

volume-centered

Scandium

-0,67

volume-centered

Titanium

-2,40

volume-centered

Titanium

-2,40

volume-centered

Titanium

T=1158K

body-centered

Vanadium

+0,76

body-centered

Chromium

+3,63

body-centered

Iron or

+8,00

body-centered

Iron

+8,00

body-centered

Iron or

Т=1189K

face-centered

Iron

Т=1189K

face-centered

Cobalt or

+3,60

volume-centered

Cobalt

+3,60

volume-centered

Nickel

-0,60

face-centered

Copper or

-0,52

face-centered

Copper

-0,52

face-centered

Zink or

+0,90

volume-centered

Zink

+0,90

volume-centered

Rubidium

-5,90

body-centered

Itrium

-1,25

volume-centered

Zirconium or

+0,21

volume-centered

Zirconium

Т=1135К

body-centered

Niobium

+0,72

body-centered

Molybde-num

+1,91

body-centered

Ruthenium

+22

volume-centered

RhodiumOr

+0,48

face-centered

Rhodium

+0,48

face-centered

Palladium

-6,80

face-centered

Silver or

-0,90

face-centered

Silver

-0,90

face-centered

Cadmium or

+0,67

volume-centered

Cadmium

+0,67

volume-centered

Caesium

-7,80

body-centered

Lanthanum

-0,80

volume-centered

Cerium or

+1,92

face-centered

Cerium

+1,92

face-centered

Praseodymium or

+0,71

volume-centered

Praseodymium

+0,71

volume-centered

Neodymium or

+0,97

volume-centered

Neodymium

+0,97

volume-centered

Gadolinium or

-0,95

volume-centered

Gadolinium

T=1533K

body-centered

Terbium or

-4,30

volume-centered

Terbium

Т=1560К

body-centered

Dysprosium

-2,70

volume-centered

Dysprosium

Т=1657К

body-centered

Erbium

-0,341

volume-centered

Thulium

-1,80

volume-centered

Ytterbium or

+3,77

face-centered

Ytterbium

+3,77

face-centered

Lutecium

-0,535

volume-centered

Hafnium

+0,43

volume-centered

Hafnium

Т=2050К

body-centered

Tantalum

+0,98

body-centered

Wolfram

+0,856

body-centered

Rhenium

+3,15

volume-centered

Osmium

volume centered

Iridium

+3,18

face-centered

Platinum

-0,194

face-centered

Gold or

-0,69

face-centered

Gold

-0,69

face-centered

Thallium or

+0,24

volume-centered

Thallium

+0,24

volume-centered

Lead

+0,09

face-centered

Lead

+0,09

face-centered

Where Rh is the Hall’s constant (Hall’s coefficient)

Z is an assumed number ofelectrons released by one atom to the conductivity zone.

Z kernel is the number of external electrons of theatomic kernel on the last shell.

The lattice type is the type of the metal crystalstructure at room temperature and, in some cases, at phase transitiontemperatures (1).

Conclusions

In spite of the rough reasoning the table shows that the greater numberof electrons gives the atom of the element to the conductivity zone, the morepositive is the Hall’s constant. On the contrary the Hall’s constant isnegative for the elements which have released one or two electrons to theconductivity zone, which doesn’t contradict to the conclusions of Payerls. Arelationship is also seen between the conductivity electrons (Z) and valencyelectrons (Z kernel) stipulating the crystal structure.

The phase transition of the element from one lattice to another canbe explained by the transfer of one of the external electrons of the atomickernel to the metal conductivity zone or its return from the conductivity zoneto the external shell of the kernel under the influence of external factors(pressure, temperature).

We tried to unravel the puzzle, but instead wereceived a new puzzle which provides a good explanation for thephysico-chemical properties of the elements. This is the “coordination number”9 (nine) for the face-centered and volume-centered lattices.

This frequentoccurrence of the number 9 in the table suggests that the densest packings havebeen studied insufficiently.

Using the method of inverse reading from experimental values for theuniform compression towards the theoretical calculations and the formulae ofArkshoft and Mermin (1) to determine the Z value, we can verify its goodagreement with the data listed in Table 1.

The metallic bond seems to be due to both socializedelectrons and “valency” ones – the electrons of the atomic kernel.

Literature:

1)

Solid state physics. N.W.Ashcroft, N.D. Mermin. Cornell University, 1975

2)

Characteristicsof elements. G.V. Samsonov. Moscow, 1976

3)

Grundzuge der AnorganischenKristallchemie. Von. Dr. Heinz Krebs. Universitat Stuttgart, 1968

4)

Physics ofmetals. Y.G. Dorfman, I.K. Kikoin. Leningrad, 1933

5)

What affectscrystals 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 ( orpseudobonds, since there is a conductivity zone between the neighbouring metalatoms) being equal to nine according to the number of external electrons of theatomic kernel for densest packings that similar to body-centered lattice (eightneighbouring atoms in the first coordination sphere). Volume-centered andface-centered lattices in the first coordination sphere should have nine atomswhereas we actually have 12 ones. But the presence of nine neighbouring atoms,bound to any central atom has indirectly been confirmed by the experimentaldata of Hall and the uniform compression modulus (and from the experiments onthe Gaase van Alfen effect the oscillation number is a multiple of nine.

Consequently, differences from other atoms in the coordinationsphere should presumably be sought among three atoms out of 6 atoms located inthe hexagon. Fig.1,1. d, e shows coordination spheres in the densest hexagonaland cubic packings.

Fig.1.1. Dense Packing.

It should be noted that in the hexagonal packing, the triangles ofupper and lower bases are unindirectional, whereas in the hexagonal packingthey are not unindirectional.

Literature:

Introduction into physical chemistry and chrystalchemistry of semi-conductors. B.F.Ormont. Moscow, 1968.

Appendix 2

Theoreticalcalculation of the uniform compression modulus (B).

B = (6,13/(rs|ao))5*1010 dyne/cm2

Where B is the uniformcompression modulus

Aois the Bohr radius

rs– the radius of the sphere with the volume being equal to the volume falling atone conductivity electron.

rs= (3/4 p

n ) 1/3

Wheren is the density of conductivity electrons.

Table 1. Calculation according to Ashcroft and Mermine

Element

rs/ao

theoretical

calculated

5.62

1.54

1.43

2.67

63.8

134.3

3.02

34.5

99.9

2.07

228

76.0

Table 2. Calculation according to the models considered inthis paper

Element

rs/ao

theoretical

calculated

5.62

1.54

1.43

2.12

202.3

134.3

2.39

111.0

99.9

2.40

108.6

76.0

Of course,the pressure of free electrons gases alone does not fully determine thecompressive strenth of the metal, nevertheless in the second calculationinstance the theoretical uniform compression modulus lies closer to theexperimental one (approximated theexperimental one) this approach (approximation) being one-sided. The secondfactor the effect of “valency” or external electrons of the atomic kernel,governing the crystal lattice is evidently required to be taken intoconsideration.

Literature:

Solid statephysics. N.W. Ashcroft, N.D. Mermin. Cornell University, 1975

Grodno

March1996 G.G.Filipenko

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