UWU/MRT/09/0015Mineral Resources & TechnologyFaculty of Science & technology

Crystallography MRT-111-1

Assignment-01

Table of Contents

1. Introduction 1

2. Crystal systems 2

3. Different crystal forms 8

4. Description of symmetry operations and symmetry elements 10

1 | P a g e

01. Introduction

Crystallography is the study about crystals with their crystal habits and how to change their crystal elements with external shape and their planes and ect. Crystallography groups are composed of 32 classes of symmetry derived from observations of the external crystal form.  From these 32 classes, 230 space groups are distinguishable using x-ray analysis.

Crystals are forming different structures and they have own structure. Crystals have two different types of packing systems. There are two types’ cubic close packing and hexagonal close packing. Crystals have another important is symmetry elements. There are three major symmetry elements.

1. Planes of symmetry – (m) 2. Centre of symmetry – (i)3. Rotational Axes – (A)

Also we can use symmetry operation for observe the symmetry elements. There are several symmetry operations can be used.

1. Reflection by a plane2. Rotation3. Inversion through a centre point4. Roto-inversion

2 | P a g e

02. Crystal Systems

There are six crystal systems; their symmetries are decreasing as above arrangement. There are six crystal systems and there diagrams.

1. isometric crystal system,

2. hexagonal crystal system,

3. tetragonal crystal system

4. orthorhombic crystal system,

5. monoclinic crystal system,

6. triclinic crystal system.

crystallographic axes are denoted by a (a1,a2,a3),b and c.

α is angle between axis b & c

β is angle between axis a & c

ϒ is angle between axis a & b

01. Isometric system (cubic)

In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three

3 | P a g e

Isometric

α ,β, ϒ=90 degrees

main varieties of these crystals, called simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc), plus a number of other variants listed below. Note that although the unit cell in these crystals is conventionally taken to be a cube, the primitive unit cell often is not. This is related to the fact that in most cubic crystal systems, there is more than one atom per cubic unit cell.

e.g.; pyrite

02. Hexagonal system

Hexagonal crystal family is one of the 6 crystal families. They are closely related and often confused with each other, but they are not the same. The hexagonal lattice system consists of just one Bravais lattice type: the hexagonal one. The hexagonal crystal system consists of the 7 point groups such that all their space groups have the hexagonal lattice as underlying lattice. The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system.

e.g.; Quarts

4 | P a g e

Hexagonal

α, ϒ, β=120 degrees

03.Tetragonal system

Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square base (a by a) and height (c, which is different from a).

There are two tetragonal Bravais lattices: the simple tetragonal (from stretching the simple-cubic lattice) and the centered tetragonal (from stretching either the face-centered or the body-centered cubic lattice).

e.g.; Zircon

04. Ortho-rhombic system

5 | P a g e

(a ≠ c)

Tetragonal

β, ϒ, α=90 degrees

Orthorhombic

α, β, ϒ =90 degrees

Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles. The three lattice vectors remain mutually orthogonal.

There are four orthorhombic Bravais lattices: simple orthorhombic, base-centered orthorhombic, body-centered orthorhombic, and face-centered orthorhombic.

e.g.; Danburite

05. Monoclinic system

6 | P a g e

Monoclinic

β=ϒ=90 degrees

α>90

Crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal length, as in the orthorhombic system. They form a rectangular prism with a parallelogram as its base. Hence two pairs of vectors are perpendicular, while the third pair makes an angle other than 90°.

Two monoclinic Bravais lattices exist: the primitive monoclinic and the centered monoclinic lattices, with layers with a rectangular and rhombic lattice, respectively.

e.g.; Orthoclase

06. Triclinic system

7 | P a g e

Triclinic α≠ β≠ ϒ

A crystal system is described by three basis vectors. In the triclinic system, the crystal is described by vectors of unequal length, as in the orthorhombic system. In addition, all three vectors are not mutually orthogonal

The triclinic lattice is the least symmetric of the 14 three-dimensional Bravais lattices. It has (itself) the minimum symmetry all lattices have: points of inversion at each lattice point and at 7 more points for each lattice point: at the midpoints of the edges and the faces, and at the center points. It is the only lattice type that itself has no mirror planes.

e.g.; Axinite

03. Different crystal forms

Crystal form is a group of similar crystal shape which has relation to symmetry elements. Crystals form divided to two groups there

open form close form.

Open form

Open form needs to combine with other form to enclose volume of space. The eighteen open-forms are those facet groupings that are related by symmetry, but do not completely enclose a volume of space. A crystal with open-form faces also requires some additional closed-form facets to complete a structure. Open-forms include:

Pedion , Pinacoid, Dome, Sphenoid , Pyramid, Prism

8 | P a g e

Close form

Close form has a enclose volume of a space. There are several crystal forms in the cubic crystal systems that are common in diamond, garnet, spinel and other "symmetrical" gemstones. These forms include:

Hexahedron (Pyrite) Octahedron (Diamond, Spinel) Tetrahedron (Tetrahedrite) Dodecahedron (Rhombic - Garnet) Hexoctahedron (Diamond)

04. Description of symmetry operations and symmetry elements

9 | P a g e

Symmetry ElementsMainly there are 3 symmetry elements.

1. Planes of symmetry – (m)Planes of symmetry can be also known as mirror planes or reflection planes. It denoted by using simple “m”.

2. Centre of symmetry – (i)Describe the point which the symmetry of object is occurred. Denoted by using simple “ i ”

3. Rotational Axes – (A)Same feature displayed one or more time during the rotating of 3600. It denoted by “A”.

Symmetry OperationsSymmetry operations are used to find if there are symmetry elements in

the considering objects. Several operations are used to find that.

1. Reflection by a planeThrough this operation, it can be found the planes of symmetry which the object has.

As an example:-

This object has 5 planes that can reflect the other part. So it has 5 mirror planes (5m).

2. RotationBy rotating through an axis for 3600 can find if there is a rotational axis for considering object.

As an example:-

10 | P a g e

This object has 3 axes that show same feature 4 times when it is rotating for 360 degrees. So it is said to be has Three 4-fold axis (3A4).

3. Inversion through a centre pointIf an object can be inversion through a centre point, that object said to be has a centre of symmetry.

This Object can be invert through a centre point. So it has a centre of symmetry (i).

4. Roto-inversionFor this operation it includes both Rotation and inversion.

When this object rotate for 1200 and invert for 1800 we can find the same feature which we could see before rotation. So it is a 3-fold roto inversion axis. we can find 6 axes like that. So we can say it is a Six 3-fold axis (6Ā3).

11 | P a g e