Mineralogy Carleton College Winter 2003

Lattice and its properties Lattice: An imaginary 3-D framework, that can be referenced to a network of regularly spaced points each of which represents the position of a motif.

Lattice and its properties line lattice plane lattice space lattice –unit cell –primitive and non-primitive cells

Lattice and its properties I can generate a lattice line from a lattice point by translating my lattice point with a vector (a)

Lattice and its properties I can generate a lattice line from a lattice point by translating my lattice point with a vector (a)

Lattice and its properties I can generate a lattice line from a lattice point by translating my lattice point with a vector (a)

Lattice and its properties I can generate a lattice line from a lattice point by translating my lattice point with a vector (a)

Lattice and its properties Plane lattice: by introducing another vector b, that is not in the same direction as a, I can produce a plane lattice

Lattice and its properties Space lattice, by introducing another vector c, which is not in the same plane as a and b, I can generate a space lattice

Unit Cell The smallest representative unit of structure which when repeated in 3-D gives the whole crystal.

Structure: Nearly all minerals are crystalline solids composed of atoms or ions held in an orderly, 3-D array by inter atomic forces. Such array of atoms are called crystal structure and are characterized by periodic duplication of any grouping of atoms along any line through the structure. In other wards the ordered arrangement of atoms or group of atoms within crystalline substance.

Unit Cell How to choose a Unit cell from plane lattice?

Choice of a Unit Cell

Look at this pattern, it is produced by simple translations. There are several possible choices for the Unit Cell.

Choice of a Unit Cell

A lattice point occurs where the corners of four cells meet, and therefore, 1/4 point per corner lies in a give cell (1/4 * 4=1)

Choice of a Unit Cell Unit Cells that include one lattice point, such as A, and B are called primitive Cells. Unit Cell C is Non- primitive.

Choice of a Unit Cell Many different cells containing a single lattice point may be chosen.

Choice of a Unit Cell How do you chose the Unit Cell? –To keep the translations short –To provide as highly specialized a lattice geometry as possible –To have the cell shape comparable with the shape of the crystal

Symmetry of a Lattice: Lets see what symmetry exist in a lattice for a moment and we will come back to Unit Cell

Elements of symmetry operations: Symmetry operations: Movements performed on an object such that when completed, the object looks the same as when you started. –These include:

Elements of symmetry: –Translation –Reflection –Rotation –Inversion –Roto-inversion –Roto-reflection –Glide –screw axis

Elements of symmetry: What elements of repetition exist? –Translation

Elements of symmetry: What elements of repetition exist? –Reflection/Mirror Mirror plane: plane passed through object such that the images on opposite sides of the plane are mirror images of one another

Elements of symmetry: What elements of repetition exist? –Reflection

Elements of symmetry: What elements of repetition exist? –Rotation Rotation Axis - An axis through the object, around which the object is rotated such that the original "motif" (or appearance) is repeated a specific number of times during 360 degrees

Elements of symmetry: What elements of repetition exist? –Rotation

Elements of symmetry: What elements of repetition exist? –Rotation of 90 degrees will give me..

Elements of symmetry: What elements of repetition exist? –Rotation of 90 degrees will give me..

Elements of symmetry: What elements of repetition exist? –Rotation of 90 degrees will give me..

Elements of symmetry: What elements of repetition exist? –Rotation of 90 degrees will give me..

Elements of symmetry: What elements of repetition exist? –Rotation of 90 degrees will give me..

Elements of symmetry: What elements of repetition exist? –Here is a different unit cell

Elements of symmetry: What elements of repetition exist? –Here is a different unit cell

Elements of symmetry: What elements of repetition exist? –Rotation of 60 degrees gives me another motif

Elements of symmetry: What elements of repetition exist? –Rotation 1 axis360 degrees 2 axes180 degrees 3 axes120 degrees 4 axes90 degrees 6 axes60 degrees

Elements of symmetry: What elements of repetition exist? –Inversion

Elements of symmetry: What elements of repetition exist? –Roto-inversion first a rotation, then an inversion of 180 degrees

Elements of symmetry: What elements of repetition exist? –Roto-reflection

Elements of symmetry: What elements of repetition exist? –Glide

Elements of symmetry: What elements of repetition exist? –Glide

Elements of symmetry: What elements of repetition exist? –Glide

Elements of symmetry: What elements of repetition exist? –Glide

Elements of symmetry: What elements of repetition exist? –Glide

Elements of symmetry: What elements of repetition exist? –Glide

Elements of symmetry: What elements of repetition exist? –Glide

Elements of symmetry: What elements of repetition exist? –screw axis This include translation and rotation together

Screw Axis

Unit Cell Unit Cell parameters – a, b, c – ( sides) – , ,  – angles

Unit Cell Unit Cell parameters – a, b, c – sides – , ,  – angles

Translation Symmetry A translation is simply moving an object in some direction (a, b, c) without a rotation. Hence a point (x, y, z) is translated to the point (x+a, y+b, z+c).

Translation Symmetry Crystalline materials have structures with translational symmetry. The unit cell of the crystal contains the smallest atomic group that is needed to define the structure under repetition.

Translational Nets in 2-D There are five different ways to translate a point in two- dimensions. Here is the first simple net.

Translational Nets in 2-D There are five different ways to translate a point in two- dimensions. Here is the second simple net.

Translational Nets in 2-D There are five different ways to translate a point in two- dimensions. Here is the third simple net.

Translational Nets in 2-D There are five different ways to translate a point in two- dimensions. Here is the fourth simple net.

Translational Nets in 2-D There are five different ways to translate a point in two- dimensions. Here is the fifth simple net.

Translational Nets in 2-D (cont.) The diamond net can also be defined in terms of a “centered rectangular net” with a1 = a2 and g = 90degrees.