structural characteristics of a crystal

Objectives:

The objectives of this experiment are to visualize various structural characteristics of a crystal structure, and get familiarized with X-ray diffraction, an important technique used for determining the crystal structure.

Specifically, after completion of this experiment a student is expected to

Know the difference between a Crystal - Lattice and Crystal Structure.

Create simple crystal structures (monatomic decoration of FCC and BCC lattices).
View the various physical characteristics of a crystal structure such as planes, voids, etc.,
as well as crystallographic-directions from different orientations, and develop a visual feel for these.
Measure the distance between atoms, angles between directions/planes.
Generate X-ray diffraction pattern using the software for a given crystal-structure. Extract
some preliminary information about a crystal structure from its X-ray pattern.

This experiment has been divided in two parts:

Visualization of crystal-structures and related features using CaRIne crystallography software.
Using X-ray diffraction pattern to determine crystal-structure.

Visualization of crystal-structures and related features using CaRIne crystallography software

Before we learn about the software, lets familiarize ourselves with some basic definitions/ information regarding crystals lattice/structure:

Crystal-Systems: The 3-dimentional space can be divided into smaller volume units such that each unit is made of three sets of planes. When one thinks of all such possible volume units that can be repeated onto each other while maintaining the same orientation as the starting unit volume, there are only seven such unit volumes that will span the entire space without leaving any gap between them. These 7 unit volumes are known as 7 crystal- systems. That are: (1) Triclinic, (2) Monoclinc, (3) Orthorhombic, (4) Tetragonal, (5) Cubic, (6) Rhombohedral, and (7) Hexagonal. A crystal-system is identified by the lengths of the 3 sides (viz. a, b, c) of the unit volume-space and the 3 angles ( , , ) between these three sides.

Note: For each of the above mentioned unit-cells there are 6 faces. But it is possible to divide 3-D into such volume-units where each unit is surrounded by more than 6 faces, and these units when placed onto each other while maintaining the relative orientation can span the entire space without leaving any gap between them.

Crystal-Lattice: A crystal-lattice is defined as a periodical arrangement of geometrical- points in 3-dimentional space in such a way that each point has identical environment around it. By identical environment it is meant that when the lattice is observed from different lattice-points (one point at a time) in a particular orientation, it looks identical from each of the points. A lattice is a mathematical concept; there are no atoms or molecules located at these points
Bravais-Lattice: When one looks for all the possible arrangements of geometrical-points in the above mentioned seven crystal-systems such that these points conform to the translation and orientation constraints of a Crystal-lattice, one finds there are total 14 such arrangements of points in 3-D. The special arrangements are identified as 14 Bravais Lattices.

Primitive unit-cell & Conventional unit-cell: The primitive-unit cell is the smallest volume which contains exactly one lattice point, and when it is translated by all the vectors in a Bravais-lattice (vectors connecting a lattice-point, selected as an origin, to all other lattice-points).

A conventional unit-cell is larger than primitive unit-cell containing more than one lattice points. It also satisfies the criteria of a unit-cell i.e. when it is translated by some sub-set of Bravais-lattice vectors, it spans the entire lattice. Generally the conventional unit-cells are used because they easily reveal the geometrical-symmetry of the lattice.

Basis/Motif: The basis/motif is the minimal unit which when placed at lattice points, gives rise to a physical crystal. The basis can be an atom, a molecule, an ion etc.
Crystal-Structure: A crystal-structure is a regular arrangement of a physical unit called basis/motif at all the points of a Bravais-Lattice.

Note, that when the basis/motif consists of single atomic unit, the resulting crystal- structures are said to be belonging to Bravais-lattice. This is because, when such a crystal-structure is observed from any atom it is made of, while maintaining the same viewing orientation every time, the crystal-structure looks identical. This condition also requires that each single atom must belong to the same element.

On the other hand, in general, when the basis/motif consists more than one atom; the resulting crystal-structure usually doesnt retain the characteristics of having-identical-view from-all-the-individual-atoms. And therefore, such crystal-structures are identified as non- Bravais lattice. Sometime these are also identified as lattice with a basis.

Crystallographic-planes and Miller Indices: Due to periodic arrangement of motifs/basis they can be visualized as if lying in a plane, and these are identified as crystallographic-planes.

Usually Miller indices are used to indicate the planes. If a plane intercepts the three axis at a1a, a2b and a3c distance from the origin, where a, b, and c are lattice parameters along the three axis. The corresponding Miller indices are calculated by:

Taking the reciprocal of the three numbers: a/a1a, b/a1b, c/a1c, which results into ratios: 1/a1, 1/a2, 1/a3
Finding the lowest common multiplier (LCM) for the obtained fractions suchthat multiplication of it with the three ratios gives a set of 3 integrals numbers.

For example a plane intersecting the axes at 2a, 5b, 2c will have Miller indices of 5,6,5. The Miller Indices of a plane are shown in parentheses without any comma separator. Therefore for the current plane the Miller indices are (565). These indices also represent all the other planes that are parallel to the plane for which we have calculated these indices.

Crystallographic direction: A direction in a crystal having the form of ua + vb + wc is said to have a crystallographic direction as [uvw]. It also represents all the other vectors that are parallel to the given vector. Here a, b, and c are vectors along the three crystal axis.
Voids: The vacant space between the atoms in a crystal structure is identified as interstitial voids. Two types of voids are, 1) Tetrahedral and 2) Octahedral void.

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