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

ON THE PROBLEM OF CRYSTAL METALLIC LATTICE IG FILIPENKO www.belarus.netdiscoveryfilipenko sci.materials1999 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 3p. 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 1p. 20, the number of Z-electrons in the conductivitiy zone has been obtained by the authors, allegedly, on the basis of metals 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,

Z1 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 Halls 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. ITATIVE MEASUREMENT OF NUMBER OF ELECTRONS IN CONDUCTION BAND OF METAL - ELEMENT. EXINFLUENCING 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. ElementRH . 1010 cubic metres KZ numberZ kernel numberLattice typeNatriumNa-2,3018body-centeredMagnesi

umMg-0,9019volume-centeredAluminium OrAl-0,3829face-centeredAluminiumAl-0,38 112face-centeredPotassiumK-4,2018body-ce nteredCalciumCa-1,7819face-centeredCalci omCaT737K28body-centeredScandium OrSc-0,6729volume-centeredScandiumSc-0,6 7118volume-centeredTitaniumTi-2,4019volu me-centeredTitaniumTi-2,4039volume-cente redTitaniumTiT1158K48body-centeredVanadi umV0,7658body-centeredChromiumCr3,6368bo dy-centeredIron orFe8,0088body-centeredIronFe8,00214body -centeredIron orFeТ1189K79face-centeredIronFeТ1189K412 face-

centeredCobalt orCo3,6089volume-centeredCobaltCo3,60512 volume-centeredNickelNi-0,6019face-cente redCopper orCu-0,52118face-centeredCopperCu-0,5229 face-centeredZink orZn0,90218volume-centeredZinkZn0,9039vo lume-centeredRubidiumRb-5,9018body-cente redItriumY-1,2529volume-centeredZirconiu m orZr0,2139volume-centeredZirconiumZrТ113 5К48body-centeredNiobiumNb0,7258body-cen teredMolybde-numMo1,9168body-centeredRut heniumRu2279volume-centeredRhodium OrRh0,48512face-centeredRhodiumRh0,4889f ace-centeredPalladiumPd-6,8019face-

cente redSilver orAg-0,90118face-centeredSilverAg-0,9029 face-centeredCadmium orCd0,67218volume-centeredCadmiumCd0,673 9volume-centeredCaesiumCs-7,8018body-cen teredLanthanumLa-0,8029volume-centeredCe rium orCe1,9239face-centeredCeriumCe1,9219fac e-centeredPraseodymium orPr0,7149volume-centeredPraseodymiumPr0 ,7119volume-centeredNeodymium orNd0,9759volume-centeredNeodymiumNd0,97 19volume-centeredGadolinium orGd-0,9529volume-centeredGadoliniumGdT1 533K38body-centeredTerbium orTb-4,3019volume-centeredTerbiumTbТ1560

К28body-centeredDysprosiumDy-2,7019volum e-centeredDysprosiumDyТ1657К28body-cente redErbiumEr-0,34119volume-centeredThuliu mTu-1,8019volume-centeredYtterbium orYb3,7739face-centeredYtterbiumYb3,7719 face-centeredLuteciumLu-0,53529volume-ce nteredHafniumHf0,4339volume-centeredHafn iumHfТ2050К48body-centeredTantalumTa0,98 58body-centeredWolframW0,85668body-cente redRheniumRe3,1569volume-centeredOsmiumO s 0412volume centeredIridiumIr3,18512face-centeredPla tinumPt-0,19419face-centeredGold orAu-0,69118face-centeredGoldAu-0,6929fa ce-centeredThallium orTl0,24318volume-

centeredThalliumTl0,24 49volume-centeredLead Pb0,09418face-centeredLeadPb0,0959face-c entered 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 6,13rsao5 1010 dynecm2 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 34 n 13 Where n is the density of conductivity electrons. Table 1. Calculation according to Ashcroft and Mermine

ElementZrsaotheoreticalcalculatedCs15.62 1.541.43Cu12.6763.8134.3Ag13.0234.599.9A l32.0722876.0 Table 2. Calculation according to the models considered in this paper ElementZrsaotheoreticalcalculatedCs15.62 1.541.43Cu22.12202.3134.3Ag22.39111.099. 9Al22.40108.676.0 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