bravais - Bravais lattices

Usage

Synopsis

Description

Crystal System	Centering	Vector Constraints	Angle Constraints
Triclinic	P	no constraints	any not of higher symmetries
Monoclinic	P,C	no constraints	alpha = gamma = 90 ≠ beta
Orthorhombic	P,C,F,I	a ≠ b ≠ c	alpha = beta = gamma = 90
Tetragonal	P,I	a = b ≠ c	alpha = beta = gamma = 90
Trigonal	P	a = b = c	alpha = beta = gamma ≠ 90
Hexagonal	P	a = b, c	alpha = beta = 90, gamma = 120
Cubic	P,F,I	a = b = c	alpha = beta = gamma = 90

Centering Types:

• P - Primitive (Simple) centering on corners (default)
• C - Base (also A or B)
• F - Face centering
• I - Body centering

Synonyms:

• Rhombohedral = Trigonal
• sc = Cubic P
• fcc = Cubic F
• bcc = Cubic I

Options

-h

program help

-H

additional help

-v <v,n>

vector lengths, non-zero, in form a,b,c (default: calculated), optional fourth number, vectors taken to root n

-a <angs>

angles in the form alpha,beta,gamma. Ignored for Orthorhombic, Tetragonal, and Cubic. For Hexagonal, any non-90 position may be 120. Otherwise, if not supplied then random angles are chosen. Angles cannot be zero or 180. Angles may be negative values. alpha + beta + gamma must be less than 360. Each angle must be less than or equal to the sum of the other two angles

-g <grid>

cell grid array, one or three positive integers separated by commas

• a - automatic, of sufficient size for radius (-G required)
• one integer - NxNxN grid. (default: calculated)
• three integers - IxJxK grid.

Combinations for grid centre:

• even,even,even - on cell corner
• odd,odd,odd - in cell body
• even,odd,even - on cell mid-edge
• odd,even,odd - on face center

-G <type>

automatic grid center type (type 8 invalid for cell centering = P):

• p - corner
• i - body
• f - face
• e - mid-edge
• 8 - eighth cell

-r <c,n>

radius, c is radius taken to optional root n (n = 2 is sqrt), or l - max insphere radius, or s - min insphere radius (default)

-P <xyz>

radius to lattice point x_val,y_val,z_val

-q <vecs>

centre offset, in form a_val,b_val,c_val (default: none)

-s <s,n>

create struts, s is strut length taken to optional root n, use multiple -s parameters for multiple struts

-u

add cell struts, added to cubic grid before transformation

-d <verts>

output dual of lattice based on primitive vectors

• c - use primitive vectors based on centering type
• four integers - primitive vectors are determined by four vertex numbers given by non negative integers. The first vertex number is the radial point and the next three vertices are the primitive vectors

-y <lim>

minimum distance for unique vertex locations as exponent 1e-lim (default: 12 giving 1e-12)

-o <file>

write output to file, if this option is not used the program writes to standard output

-l

display the list of lattices

-L

list unique radial distances of points from centre (and offset)

-S

list every possible strut value

-C <opt>

convex hull options

• c - convex hull only
• i - keep interior
• a - add original grid

-D <opt>

Voronoi (a.k.a Dirichlet) cells (Brillouin zones for duals)

• a - add to lattice
• c - cells only

-K <file>

use convex polyhedron in off file for container (uses radius)

-R

R hexagonal grid and struts (cell centering = P only)

• o - overlay
• f - fill

-O

translate centroid of final product to origin

-V <col>

vertex colour, in form R,G,B,A (values 0.0-1.0, or 0-255)

-E <col>

edge colour (including struts), in form R,G,B,A (values 0.0-1.0, or 0-255)

-F <col>

face colour (if convex hull), s - by symmetry using indexes, S - by symmetry using values

-T

face transparency, valid range from 0 (invisible) to 255 (opaque)

-Z <col>

add centroid vertex to final product in color col

Examples

   bravais fcc | antiview -v b

   bravais bcc -c s -r 4 -s 3,2 | antiview

   bravais bcc -g 3 -D c | antiview

   bravais bcc -g 3 -D a | antiview -v 0.1

   bravais bcc -d c | antiview -v b

   bravais triclinic -a 60,70,80 -v 1,1.5,2 -u | antiview

Notes

Definition of a Bravais Lattice: (partly from http://en.wikipedia.org/wiki/Bravais_lattice )

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation operations. A crystal is made up of one or more atoms (the basis) which is repeated at each lattice point. The crystal then looks the same when viewed from any of the lattice points. In all, there are 14 possible Bravais lattices that fill three-dimensional space.

August Bravais (1811-1863), a French naval officer, adventurer, and physicist taught a course in applied mathematics for astronomy in the faculty of sciences in Lyon from 1840. He served as the Chair of Physics, Ecole Polytechnique between 1845 and 1856. He is best remembered for pointing out in 1845, that there are 14 unique Bravais lattices in three dimensional crystalline systems, adjusting the previously result (15 lattices) by Moritz Ludwig Frankenheim obtained three years before.

A German Crystallographer, Frankenheim (1801-1869) is noted as the first to enumerate the 32 crystal classes. And he also solved the symmetry systems of the 7 crystal systems but this work went completely unnoticed at the time.

There is a bit of mystery surrounding what Frankenheim had as the 15th lattice. Even today, in some texts the Hexagonal lattice with two interior points is shown in the Trigonal class. But these two lattices use the same set of points and it is thought that it was this duplication that was eliminated by Bravais. However, in Bravais' paper, there is no mention of Frankenheim or the enumeration of lattices he presented.

In this program, the Hexagonal cells and Trigonal cells can be seen together by using the -R parameter.

Note that End Centered Cubic (would be Cubic C) does not exist but can be produced by Tetragonal P that has cells of dimensions a,b,c = 1,1,sqrt(2)

Face Centered Cubic (Cubic F or FCC) is duplicated in Body Centered Tetragonal (Tetragonal I) of dimensions a,b,c = 1,1,sqrt(2). However, the FCC embodied would be of higher symmetry than the Tetragonal crystal system is allowed.

Similarly, Trigonal at 90 degrees (improper) is SC. Trigonal at 60 degrees is FCC and Trigonal at acos(-1/3) or 109.47122063449... degrees is BCC.

Also there is no provision for Face Centered Tetragonal (would be Tetragonal F) or Base Centered Tetragonal (would be Tetragonal C). These would be embodied in Body Centered Tetragonal (Tetragonal I) and Simple Tetragonal (Tetragonal P) respectively. This is true at any proportion other than a,b,c = 1,1,sqrt(2)

In Hexagonal, Orthorhombic C can be seen to occur. When Hexagonal of a=b=c is produced, then Base Centered Tetragonal (would be Tetragonal C) occurs.

Bravais lattices will fall into the following symmetries

Crystal System	Possible Symmetries†
Triclinic	C1 Ci
Monoclinic	C2 Cs C2h
Orthorhombic	D2 C2v D2h
Tetragonal	C4 S4 C4h D4 C4v D2d D4h
Trigonal	C3 S6 D3 C3v D3d
Hexagonal	C6 C3h C6h D6 C6v D3h D6h
Cubic	T Th O Td Oh

†There are 32 possible symmetry types - note there are no 5 fold symmetries.

Of the symbols used for cell centering:

• P - stands for Primitive. It is a cube depicted by a vertex at eight corners
• C - stands for having a point filled in on the C side of the primitive cell this is described in some texts as Base Centering. C is most commonly used. A or B means use the A or B sides instead. Just a rotation of C.
• F - stands for Face Centering and fills all three, A, B and C sides.
• I - (from German: innenzentriert, meaning Body Centered) is the primitive cell with one point filled in the center of the cell.

The term Isometric is sometimes used for Cubic. Also allowable in this program.

Monoclinic is defined in this program with angles alpha = gamma = 90 ≠ beta as is found in the first volume of International Tables for Crystallography.

Rhombohedral cells are a special case of the trigonal class and can be seen by using the -R parameter

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