ATOMIC VOLUME AND CRYSTAL STRUCTURE OF ELEMENTS

S. Hübner, M. Trömel

Institut für Anorganische Chemie, Marie Curie-Str. 11, 60439 Frankfurt, Germany
e-mail: sven@solid.anorg.chemie.uni-frankfurt.de
http://solid.anorg.chemie.uni-frankfurt.de/~sven/

Keywords: atomic volume, dense structures, bond order, atomic valence, high pressure modifications

1 Dense Structures

Modifications of elements with close-packed structures (fcc or hcp) transform into each other almost without change of volume  [1] and the same holds good for transformations fcc  bcc and hcp  bcc. Structures which obey this rule are called 'dense'. V_D stands for their atomic volume. Density is defined as the number of particles per volume unit.

Volume rule: Phase transitions between dense structures occur, in first approximation,without a change of atomic volume

Dense structures are fcc, hcp (with ), bcc and the tetragonal body-centered structure with  (Pa, -Mn, In, all of them true metals) [2].

2 Non-Dense Structures

From the definition of V_D it follows that , where V_X is the atomic volume in any structure X. So we can define the volume factor f_X:

(1)

Here a structure X is considered 'non-dense' if f_X>1.01. To calculate the volume factor f_X for any structure X we have to assume in addition to the validity of the volume rule:
- The atomic valence W of an element is the same in the dense and in the non-dense modifications
- Bond order s depends on the distance r of each neighbouring atom according to

(2)

R₁ and b are bond order parameters. The sum of the bond orders around each atom equals its atomic valence W:

(3)

f_X then can be calculated by (see [1], [3]):

(4)

mi	=	multiplicity of the i-th atom
j(i)	=	geometrical coordination number [2]
rj(i)	=	distances from i to the j-th atom
C	=

Using equation (1) the atomic volume of the dense modification can be determined from any structure X of an element. The example of phoshorus shows a rather good agreement between the atomic volume of the hypothetical dense modification of phosphorus calculated from different observed structures with largely differing atomic volumes:

Pblack	18.97	13.84	1.371	[4]
PHittorf	21.79	13.65	1.596	[5]
P4	26.47	13.15	2.013	[6]

This result and some more [3] indicate the validity of equation (4) and the foregoing assumptions.

3 Atomic Volumes of High Pressure Modifications

The atomic volume of a dense modification V_D is expected to decrease smoothly with pressure. Fig. (1) shows the calculated V_D of phosphorus. V_D for normal pressure is extrapolated from this curve as 13.70(6)  in very good agreement with the average V_D from P_black and P_Hittorf. Figure 1: Observed atomic volumes V_X and calculated V_D of phosphorus depending on pressure

References