UM-VRL: Miller Indices in VRML

Miller Indices in VRML

The Miller Indices are used in crystallography to characterize planes within a crystal structure. The orientation of these planes is important, e.g., in semiconductor processing. This page provides two interactive VRML applications that allow to explore, visualize, and understand the geometric properties of these planes.

In 1850, August Bravais established the theory of space lattices, which presents that the internal structure of a crystal is a regular, cyclical and indefinite distribution of some atoms in space. This atom network is called a crystal lattice. There are indefinite families of parallel planes in a crystal lattice. Miller Indices (usually expressed by hkl) are a symbolic vector representation for the orientation of theses crystal planes.

To determine the Miller Indices (hkl) of a plane, one must take the following steps:

Choose an atom as the origin of the coordinate system and three crystallographic directions
Determine the intercepts of the plane along each of the three crystallographic directions
Take the reciprocals of the intercepts to obtain the values for hkl
If this results in fractions, then multiply each value by the denominator of the smallest fraction

In our VRML applications, the Miller Indices hkl are directly specified by the user via selectors for each of the three values. The corresponding intercepts are calculated and their values as well as the resulting plane are displayed.

Miller Indices on a Simple Cubic Lattice

110 plane
(enlarge)

020 plane
(enlarge)

Load the VRML Model for a Cubic Lattice (32K)

Miller Indices on a Silicon Lattice Cell

The Silicon diamond lattice consists of two interleaving face-centered cubic (fcc) cells displaced by 1/4 of the lattice constant. The density of atoms in different Miller Indices planes determines properties related to etching or oxidization.

111 plane and atoms on the plane
(enlarge)

110 plane and atoms on the plane
(enlarge)

Load the VRML Model for a Silicon Lattice Cell (52K)

Links to other Miller Indices pages on the Web

Miller Indices by Alan J. Jircitano, Pennsylvania State University

Miller Index Notation by Roger Nix, Queen Mary University of London

Crystallography - Miller Indices by David Jessey, California State Polytechnic University - Pomona

Miller Indices by Paul Schroeder, University of Georgia

Surface Explorer by Klaus Hermann and Fritz Rammer, Fritz-Haber-Institute, Germany

Miller Indices on Silicon wafers by ASTEP (Advanced Software for Teaching), includes VRML models

Miniaturization Science - Bulk Micromachining (pdf file) by Marc Madou, University of California, Irvine

Introduction to Semiconductor Materials and Crystal Structures (pdf file) by Meikei Ieong

Credits: The two VRML applications were developed by Bao Congxi, Visiting Professor at the Virtual Reality Laboratory from Shanghai Jioa Tong University. Advise was provided by Pran Mukherjee, Alvin Tessmer, and Klaus-Peter Beier.

Last update: June 15, 2003, kpb

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