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.



                                                                          )
                  )

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            |=1189K   |7     |9     |face-centered|
|Iron                 |Fe            |=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            |=1135   |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            |=1560   |2     |8     |body-centered|
|Dysprosium           |Dy            |-2,70     |1     |9     |volume-center|
|                     |              |          |      |      |ed           |
|Dysprosium           |Dy            |=1657   |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            |=2050   |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 Halls constant (Halls 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 Halls  constant.  On
  the contrary the Halls constant is  negative  for  the  elements
  which have released one or  two  electrons  to  the  conductivity
  zone, which doesnt 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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"On the problem of crystal metallic lattice in the densest packings of chemical elements "