The crystal structure of Be29.5B81

With 19 inequivalent sites, and 110.5 atoms per unit cell, the crystal structure of Be30.5B81 is quite involved. The space group of the unit cell is P-62m. and the unit cell is hexagonal. As the images below will show, the crystal structure can be visualized as a hexagonal lattice where the contents of each hexagon is not the same.

The fundamental structure perpendicular to the c-axis starts with a honeycomb lattice. In each honeycomb cell of the lattice, a subunit is placed. There are three kinds of subunits, and near the vertices of the honeycomb lattice 4th subunit consisting of a chain of 4 atoms is placed.

Take a unit cell as follows. Picking one of the hexagons of the lattice, inset a subunit 1 as defined below. In the six adjoining hexagons, place alternately subunits 2a and 2b as shown below. This fundamental pattern, with subunit one surrounded by set of three each of subunits 2a and 2b, form most of the crystal structure. The atom chains, denoted unit 3 below, are placed close to the vertices of a honeycomb lattice.

The crystal structure can be visualized along the c-axis a as a decorated honeycomb lattice.

The subunits

The subunits described here also have links to VRML images of the units. The color scheme has Be as yellow in the structure described and red elsewhere in the crystal, and B as green in the subunit and blue elsewhere. The initial view should be close to the c-axis.

Subunit 1

This unit contains 11 Be and 21 B atoms (I have duplicated the 3 Be atoms at the top and bottom of the unit cell in this VRML).
Mine Xray x y z (n-fold)
Be 18 Be 12 0.124 0.248 1.000 (3)
Be 17 Be 11 0.000 0.000 0.893  
B 6 B 4 0.000 0.188 0.812 (6)
Be 13 Be8 0.238 0.118 0.642 (3)
B 12 B 8 0.118 0.238 0.642 (3)  
B 11 B 7 0.062 0.124 0.500 (3)
B 12 B 8 0.119 0.238 0.358 (3)
Be 13 Be 8 0.238 0.118 0.358 (3)
B 6 B 4 0.000 0.188 0.188 (6)
Be 17 Be 11 0.000 0.000 0.107  
Be 18 Be 12 0.124 0.248 0.000 (3)

Unit 2a

This unit, contains 5 Be and 24 B atoms, and is shown in this VRML file.

Mine Xray x y z (n-fold) about (2/3,1/3,z)
B 2 B 1 0.493 0.164 1.00 (6)
Be 19 Be 13 2/3 1/3 0.875  
B 5 B 3 0.552 0.105 0.847 (3)
B 8 B 5 0.506 0.210 0.693 (3)
B 10 B 6 0.458 0.228 0.595 (3)
Be 15 Be 9 0.560 0.440 0.500 (3)
B 10 B 6 0.458 0.228 0.405 (3)
B 8 B 5 0.506 0.210 0.307 (3)
B 5 B 3 0.552 0.105 0.153 (3)
Be 19 Be 13 2/3 1/3 0.125  
B 2 B 1 0.493 0.164 0.000 (6)

Unit 2b is same as unit 2a, except Be sites at z=0.125, 0.875 are absent. The coordinates are obtained by switching x and y in table for Unit 2a.

VRML

Subunit 3

Unit 3 (a vertical column of 4 atoms -- see VRML) sits roughly at the midpoint of the triangle whose vertices are at the centers of the Unit 1, Unit 2a and Unit 2b objects, or equivalently near the vertices of the hexagonal lattice whose openings contain the 1 and 2 subunits. There is a six-fold axis around (0,0,z) yielding six equivalent sites. This third unit could also be attached to the hexagonal corners of Unit 1, but it makes seeing Unit 1 difficult.

Mine Xray x y z
B 3 B 2 0.348 0.000 0.905
Be 16 Be 10 0.364 0.000 0.679
Be 16 Be 10 0.364 0.000 0.321
B 3 B 2 0.348 0.000 0.095

Be29 B81 corresponds to putting only Unit 2a's at the vertices of the honeycomb lattice. Be31 B81 corresponds to Unit 2a's at half of the honeycomb lattice and Unit 2b's on the other half. Be33 B81 would have all Unit 2b's on the honeycomb lattice