Structure
Contents
Structure¶
A structure can be explicitly defined given its unit cell geometry and location of its sites along with the type of the atomic species residing there. Consider the cubic BaTiO3 structure shown below:
(This image was obtained via VESTA)
Here, the Ba atoms are depicted in green; Ti atom is depicted in blue, and the O atoms are colored red. As can be seen, the Ba atoms sit in the corners of the cube; Ti sits in the center and the O atoms are located on the center of each face.
The unit cell’s shape is a cube with edge length of 4.006 Å.
The atoms’ locations are:
Ba: (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1)
Ti: (1/2,1/2,1/2)
O: (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0), (1,1/2,1/2), (1/2,1,1/2), (1/2,1/2,1)
(Notice that we are using the so called “fractional coordinates”, i.e., using a system such that each lattice direction’s length is normalized to 1)
Due to the crystal lattice translations, we immediately recognize that all the 8 Ba atom sites are actually the same site in different unit cells, with only the (0,0,0) atom being within our reference unit cell (0≤{x,y,z}<1). Likewise, of the 6 O atom sites, only 3 of them being within our unit cell: (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0).
Hint
To deduce the number of each species in the unit cell, one can also visualize the atoms as solid balls (as shown in the figure) and then calculate the total portions of the “balls” inside the unit cell: for the 8 Ba atoms, each has 1/8 portion inside; for the Ti atom, it is completely inside; and for the 6 O atoms, we have half of each, so the totals are: Ba (\(8\times\tfrac{1}{8}=1\)), Ti (\(1\times1=1\)), O (\(6\times\tfrac{1}{2}=1\))
Therefore, we can completely define our structure using these sites, along with the lattice parameters \((a,b,c,\alpha,\beta,\gamma)\) information as:
a = b = c = 4.006 Å
α = β = γ = 90°
Ba 0, 0, 0
Ti 0.5, 0.5, 0.5
O1 0, 0.5, 0.5
O2 0.5, 0, 0.5
O3 0.5, 0.5, 0
As you might have noticed, the data above contains nothing about the symmetry information. In reality, it is easy to see that, all the Oxygen sites are equivalent and therefore can be related via symmetry operations. Our structure’s unit cell being a cube also implies high symmetry. In fact, the structure is compatible with the Pm3m (#221) space group which contains 48 symmetry operators (a high number of symmetry operators indicate a high symmetry setting). We can acquire the listing of these operators via the GENPOS tool:
The Wyckoff positions of this group can be obtained using the WYCKPOS tool:
Comparing our structure’s sites with the list above, we concur that:
Ba (0,0,0) : 1a
Ti (0.5,0.5,0.5) : 1b
O ((0,0.5,0.5) : 3c
(As the other elements of the orbit can be derived from one element of the orbit and using the symmetry operations, it is sufficient to pick any of the O sites as the representative)
Armed with this information, we can represent our structure using a more basic and informative entry:
Space Group: 221
a = b = c = 4.006 Å
α = β = γ = 90°
Ba 0, 0, 0
Ti 0.5, 0.5, 0.5
O 0, 0.5, 0.5
which essentialy makes up the foundation of our BCS format:
# Space Group ITA number
221
# Lattice parameters
4.006 4.006 4.006 90 90 90
# Number of independent atoms in the asymmetric unit
3
# [atom type] [number] [WP] [x] [y] [z]
Ba 1 1a 0.0 0.0 0.0
Ti 1 1b 0.5 0.5 0.5
O 1 3c 0.5 0.0 0.5
Warning
Even though being extensively used throughout the Bilbao Crystallographic Server tools, the BCS format is a local format. If you want to use your structure data in other programs, widely accepted CIF format will be more compatible. You can use the STRCONVERT tool to convert your structural data from and to a wide range of formats.
Hint
You can use the FINDSYM program integrated in the STRCONVERT tool to identify the symmetry of a structure given in triclinic (i.e., P1 (#1) or P1 (#2)) setting!
CIF Files¶
CIF file format is the standard format to exchange structure information between crystallography programs. A sample CIF file for the BaTiO3 structure is included here. It contains the space group information, lattice parameters, symmetry operators and the atomic sites in the asymmetric unit cell.