Journal Articles



The MOE Crystal Builder


A. Lin
Chemical Computing Group Inc.


Among the new application programs of the 1998.03 release of MOE is the Crystal Builder. The Crystal Builder is used to investigate, edit, and build crystals interactively. It offers full space group support, and has both 2D and 3D building capabilities. Graphical editing controls permit easy modification of both the unit cell parameters and the asymmetric unit. The unit cell can be replicated into a supercell that can in turn be used as the asymmetric unit in subsequent crystal building.

Contents

Overview

The Crystal Builder panel provides the following functionality:

  • Crystal build/unbuild: build/remove crystallographic unit cell
  • Space group selection: choose from 230 three-dimensional and 87 two-dimensional space groups
  • Unit cell report: display information about unit cell
  • Unit cell parameter (lattice constants) modification: dynamic unit cell editing using graphical controls
  • Asymmetric unit modification: re-position and re-orient asymmetric unit using graphical controls
  • Molecular shape retention: molecule shape retained when unit cell resized
  • Lattice dimensions and periodic box size specification: specify dimensions of "supercell" and set periodic box size.

Basic Crystal Concepts

Unit Cell

A crystal is an array of atoms packed together in a regular pattern. A unit cell of a pattern is a piece of the pattern which, when repeated through space without rotation and without gaps or overlaps, reconstructs the pattern to infinity. For filling space without holes, a unit cell must be either a parallelogram (in 2D) or a parallelepiped (in 3D).

The symmetries of a pattern determine the shape of the unit cell. For example, mirror symmetry requires a rectangular (in 2D) or tetragonal (in 3D) unit cell. There is an infinite number of possible unit cells for any pattern (e. g., a given unit cell can generate a family of unit cells by repeated doublings in size). By custom, the unit cell is chosen to be the smallest one that reveals the special geometry characteristic of the symmetry. Thus, although an oblique parallelogram can be used for a pattern with 4-fold symmetry in 2D, a square is preferred.

When a unit cell is repeatedly translated to fill all of 2D or 3D space, the vertices of all the unit cells in the filled space constitute a lattice. A lattice is an infinite array of regularly-spaced points. All points in the lattice have identical "environments" --- the view from every point in the lattice is identical to that from any other point in the lattice. The absolute positions of the points of a lattice, and hence the unit cell, are arbitrary with respect to a pattern.

Not all lattice points need coincide with unit cell vertices. Primitive unit cells use every lattice point as a unit cell vertex. Non-primitive unit cells, however, contain extra lattice points not at the corners.

A primitive unit cell contains exactly one lattice point. For example in 2D, each primitive unit cell joins four lattice points, each of which counts for 1/4 because every lattice point is shared among four unit cells. In 2D, a non-primitive unit cell has one additional lattice point exactly centered within it and is called a body-centered non-primitive unit cell. In 3D, non-primitive cells are of three kinds:

  • end-centered : an extra lattice point is centered in each of two opposing faces of the cell
  • face-centered : an extra lattice point is centered in every face of the cell
  • body-centered : an extra lattice point is centered in the exact middle of the cell

Although primitive unit cells are simpler than non-primitive unit cells, the non-primitive unit cell is preferred when its geometry is more favorable (simpler). For instance, a rectangular non-primitive cell would be chosen over a rhomboid primitive cell. In general, the unit cell used is the smallest one with the most regular geometry.

Lattice and Unit Cell Parameters

A lattice may be specified by two non-coincident vectors in 2D, and by three non-coplanar vectors in 3D. The vectors lie along the edges of the unit cell, and are labeled a, b, and (in 3D) c. The magnitude of the vectors is given by the dimensions of the unit cell in the real crystal under study.

The faces of the unit cell are labeled as follows:

  • A : edges defined by lattice vectors b and c
  • B : edges defined by lattice vectors a and c
  • C : edges defined by lattice vectors a and b

Similarly, the inter-facial angles of the unit cell are defined to be:

  • alpha : angle between edges b and c
  • beta : angle between edges a and c
  • gamma : angle between edges a and b

Lattice Systems: the 14 Bravais Lattices

Lattices can be classified into "systems", each system being characterized by the shape of its associated unit cell. In three dimensions, the lattices are categorized into seven crystal lattice "systems". Within several of these, lattices supporting non-primitive unit cells can be defined. The classification scheme yields a total of 14 possible lattices (called Bravais lattices).

The lattice symbols used for classification are as follows:

  • P : primitive
  • B : end-centered on B-face (convention for Monoclinic systems)
  • C : end-centered on C-face (convention for Orthorhombic systems)
  • I : body-centered (from German innenzentriertes)
  • F : face-centered
  • R : rhombohedral primitive

These symbols also appear in the Hermann-Mauguin International space group symbols, where they have the same meanings. For example, group 'Fmmm' is face-centered orthorhombic, and 'R3' is rhombohedral.

    System # of lattices in system Lattice symbols
    Triclinic (Anorthic) 1 P
    Monoclinic 2 P, B
    Orthorhombic 4 P, C, I, F
    Tetragonal 2 P, I
    Isometric (Cubic) 3 P, I, F
    Trigonal/Rhombohedral 1 P or R
    Hexagonal 1 P

In two dimensions, there are only four possible unit cell shapes and two possible lattice symbols:

    Cell shape Lattice symbol
    General parallelogram (rhomboid) p
    Rectangle p, c
    Square p
    Rhombus with 60 degree angle p

Crystal Builder

Unit Cell

The Crystal Builder constructs a unit cell at the origin of the MOE molecular coordinate system, and puts the unit cell b-axis on the Y-coordinate axis. The cell box is also drawn, with the axes labeled. The unit cell is a malleable unit: it can be modified and even removed. It can also be replicated into a "supercell" or "crystal lattice". Once converted into a supercell, the unit cell is lost (and therefore no longer editable).

The Crystal Builder's "current" unit cell comprises the unit cell specification, the asymmetric unit, and symmetry images. When a crystal unit cell is being built and edited, other atoms not implicated in the crystal may exist concurrently in the system.

The unit cell is specified by six editable parameters: the lengths A, B, and C of the cell edges (in Angstroms), and the angles alpha, beta, and gamma between the faces of the unit cell (in degrees). Graphical editing controls provide an easy way to change these parameters. The unit cell is dynamically updated in the rendering window.

Changing space group can change the unit cell --- different lattice systems place different restrictions on the values that the unit cell parameters can take on. For instance, the Tetragonal lattice system imposes the requirement that all unit cell inter-facial angles be 90 degrees, and that edge lengths A and B be equal. The Triclinic lattice system imposes no restrictions on either edge lengths or angles. Geometric requirements, however, may restrict the values of the inter-facial angles (e. g., the sum of the three angles may not exceed 360 degrees).

Unit Cell Report

Information about the current cell is displayed in the Crystal Builder panel, and is updated dynamically as edits to the unit cell are made. The reported information is:

  • Full Hermann-Mauguin crystallographic space group symbol and International Tables of X-Ray Crystallography space group number
  • Crystallographic point group
  • Bravais lattice system
  • Cell volume (Ang3)
  • Cell density (grams/cc)

Cell density is calculated from the asymmetric unit associated with the unit cell.

Asymmetric Unit

The asymmetric unit is a set of atoms that is replicated to symmetrically equivalent positions in the unit cell. The number of generated copies ("images") and their positions is dictated by the space group. Image atoms remain "virtual" entities until a supercell is built. The images are colored cyan to differentiate them from the asymmetric unit, and the Crystal Builder destroys and creates them as needed.

When unit cell parameters are changed, the position of the asymmetric unit is automatically adjusted to preserve its displacement, specified in fractions of cell edge lengths, from the unit cell origin. This results in a corresponding adjustment of the position of each image of the asymmetric unit. Deletions from the asymmetric unit are also immediately reflected in the symmetry images.

Atoms can be added to an asymmetric unit only when it is empty. The asymmetric unit is empty when the Crystal Builder is first opened, when a supercell is built, or when the asymmetric unit is cleared (the unit cell is "restarted"). The asymmetric unit is "registered" to the unit cell when Make unit cell is pressed, and all selected atoms constitute the asymmetric unit.

Clearing the asymmetric unit associated with the current unit cell causes all symmetry images to disappear, and effectively unbuilds the crystal.

Even though the asymmetric unit may be empty, the current unit cell still exists, and can be edited and have its space group set. This permits for pre-setting the unit cell box, and then positioning the atoms of the asymmetric unit. After the asymmetric unit is registered to the unit cell, the individual atoms or molecules can be re-positioned with respect to the other atoms in the asymmetric unit by moving them in MOE's main rendering area.

In addition to manual re-positioning in the main rendering area, the asymmetric unit can be translated and rotated using graphical editing controls in the Crystal Builder panel. Translation and rotation are performed on the asymmetric unit as a whole with respect to the center of mass of the entire asymmetric unit.

Space Groups

The Crystal Builder provides full 2D and 3D space group support:

  • the 230 three-dimensional space groups,
  • the 63 two-sided layer (2D) groups, and
  • the 24 two-sided band (2D) groups.

In band groups, the singular axis is along the b-axis. We have extended the expression of planar (2D) groups to 3D molecules by interpreting two-sidedness as the conformation of the asymmetric unit above and below the slice of the plane. Otherwise, for films of atoms rather than molecules, the two-sided plane groups reduce to the 17 neutral layer and the 7 neutral band groups. "Neutral" means identical from both above and below the plane.

Building a Crystal Lattice

The unit cell can be replicated into a "supercell" or "lattice". Superposed atoms are eliminated, and the images of the asymmetric unit become "real" (the Crystal Builder can no longer delete them). The asymmetric unit is cleared, which permits the supercell to be registered immediately as the asymmetric unit in further crystal building.

The crystal supercell size is specified in integral numbers of unit cells. In the 2D space groups, the lattice can only be constructed in the plane. The final dimensions of the lattice are (in Angstroms):

  • in a : A * supercell size in a
  • in b : B * supercell size in b
  • in c : C * supercell size in c

The MOE periodic boundary conditions can be enabled, with the periodic box size set to the dimensions of the supercell.

Future directions

Planned extensions to the Crystal Builder reading and writing standard crystallographic database files, auto-bonding, drawing Miller planes, and crystal cleaving (faceting) using Miller indices. Other additional features under consideration polyhedral representation and support for different visualization modes of the unit cell.

References

International Tables of X-Ray Crystallography, Vol. I, Symmetry Groups, Henry, Norman F.M., and Kathleen Lonsdale, Ed., The Kynoch Press, Birmingham, 1952.

Smith, Joseph V., Geometrical and Structural Crystallography, John Wiley & Sons, Inc., New York, 1982.

Phillips, F.C., An Introduction to Crystallography, 4th ed., Oliver and Boyd, Edinburgh, 1971.