GEOLOGY Module - INFLIBNET Centre

GEOLOGY

Paper: Crystallography and Mineralogy

Module: Laws of Crystallography and Crystal

System

Subject Geology

Paper No and Title Crystallography and Mineralogy

Module No and Title Laws of Crystallography and Crystal System

Module Tag Min IIIa

Principal Investigator Co-Principal Investigator Co-Principal Investigator

Prof. Talat Ahmad

Vice-Chancellor

Jamia Millia Islamia

Delhi

Prof. Devesh K Sinha

Department of Geology

University of Delhi

Delhi

Prof. P. P. Chakraborty


University of Delhi

Delhi

Paper Coordinator Content Writer Reviewer

Prof. Naresh C. Pant


University of Delhi

Delhi

Dr. Ashima Saikia


University of Delhi

Delhi

Prof. Santosh Kumar


Kumaun University

Nainital

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Table of Content

1. Learning outcomes

2. Introduction

3. Crystallographic Laws

3.1 Steno’s Law

3.2 The law of rational indices

3.3 Law of axial-ratio

3.4 Law of crystallographic axes

3.5 Law of constancy of symmetry

4. Crystal Axes and Six Crystal Systems

5. Elements of Symmetry

6. Forms

7. Structure and chemistry

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1. Learning outcomes

After studying this module, you shall be able to:

What is crystallography?

How to define a crystal? The different forms of crystal?

The geometric relationships, which defines the symmetries in a crystal?

2. Introduction- Crystallography and Crystal Systems

Crystallography: Science that deals with the chemical and physical properties,

structure, formation and applications of crystals.

Crystals: A crystal is a homogenous solid, which is formed by a 3-D repeating unit

(pattern of ions, atoms or molecules) with fixed distances.

Crystallography is a branch of mineralogy. Crystallography in simple meaning

means "the study of crystals". At one time, the word crystal referred only to quartz

crystal, but has taken on a broader definition, which includes all minerals with well-

expressed crystal shapes.

Crystallography is divided into three sections - geometrical, physical and chemical.

The latter two involve the relationships of the crystal form (geometrical) upon the

physical and chemical properties of any given mineral. For earth science students,

the following basic books on crystallography are recommended.

Manual of Mineral Science, 23rd Edition, by Cornelis Klein and Barbara

Dutrow 2008

Ford's Textbook of Mineralogy (4th edition, 1932).

A CRYSTAL is a regular polyhedral form, bounded by smooth faces, have a

definite a chemical compound on account of its interatomic forces.

A polyhedral form simply means a solid bounded by flat planes. "A chemical

http://www.amazon.com/s/ref=dp_byline_sr_book_1?ie=UTF8&text=Cornelis+Klein&search-alias=books&field-author=Cornelis+Klein&sort=relevancerank

http://www.amazon.com/s/ref=dp_byline_sr_book_2?ie=UTF8&text=Barbara+Dutrow&search-alias=books&field-author=Barbara+Dutrow&sort=relevancerank

http://www.amazon.com/s/ref=dp_byline_sr_book_2?ie=UTF8&text=Barbara+Dutrow&search-alias=books&field-author=Barbara+Dutrow&sort=relevancerank

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compound" tell us that all minerals are chemicals, just formed by and found in

nature. The last half of the definition tells us that a crystal normally forms during the

change of matter from liquid or gas to the solid state. In the liquid and gaseous state

of any compound, the atomic forces that bind the mass together in the solid-state are

not present. Therefore, we must first crystallize the compound before we can study

its geometry. Liquids and gases take on the shape of their container; solids take on

one of several regular geometric forms.

These forms may be subdivided, using geometry, into six systems.

3. Crystallographic Laws

Way back in 1669, Nicholas Steno, a Danish physician and natural scientist,

discovered one of these laws. By examination of numerous specimens of the same

mineral, he found that, when measured at the same temperature, the angles between

similar crystal faces remain constant regardless of the size or the shape of the

crystal. Therefore, whether the crystal grew under ideal conditions or not, if one

compares the angles between corresponding faces on various crystals of the same

mineral, the angle remains the same.

Although he did not know why this was true (x-rays not yet been discovered) we

now know that this is so because studies of the atomic structure of any mineral

indicates that the structure of mineral is a close set of given limits or geometric

relationships.

3.1 Steno’s Law called the CONSTANCY OF INTERFACIA ANGLES

The law of the constancy of interfacial angles (also called the 'first law of

crystallography') states that the angles between the crystal faces of a given

species are constant, whatever the lateral extension of these faces and the origin

of the crystal, and are characteristic of that species (Fig.1). It paved the way for

Haüy's law of rational indices.

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Fig. 1 Cross section along prism and rhombohedral faces indicating constancy of

interfacial angles even if the quartz crystals is flattened along one axis.

Contact Goniometer: Consists of a printed protractor to which is attached an

arm swiveling plastic that is pivoted at the center and with a hairline mark that

can be read against the scale (see figure below). The goniometer is held with the

straight edge of the protractor in contact with one face, the straight edge of the

plastic strip in contact with the other face and with the plane surface of the

protractor and the strip perpendicular to both crystal faces. Two values of the

interfacial angle, which total 180o, can be read from the protractor. One is the

internal angle; the other is the external angle.

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3.2 The law of rational indices

States that the intercepts, OP, OQ, OR, of the natural faces of a crystal form with

the unit-cell axes a, b, c (see Figure 2) are inversely proportional to prime

integers, h, k, l. They are called the Miller indices of the face. They are usually

small because the corresponding lattice planes are among the densest and have

therefore a high interplanar spacing and low indices (source: dictionary of

Crystallography).

Two crystals of the same substance may differ considerably in appearance that in

number, size and shape of the individual faces. In order to describe the external

form of crystals, a mathematical method of relating plane; to certain imaginary

lines in space is used. The position of any plane can be uniquely fixed by the

intercepts it makes on the axes of reference. The ratio of the distances from the

origin, at which the crystal face cuts the crystallographic axes, is known as the

‘parameter of a crystal face’.

In the above given figure, let OX, OY, OZ represents the crystallographic axes

and ABC is a crystal face making intercepts of 'OA' on, 'OX', 'OB' on OY and

'OC' on 'OZ'. The parameters of the face ABC are given by the ratio of OA, OB,

and OC. It is convenient to take the relative intercepts of this face as standard

length for the purpose of representing the position of any other face such as

http://reference.iucr.org/dictionary/Miller_indices

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DEF. In this case OD=OA, 0E=20B, OF= ½OC. Therefore, the parameters are of

DEF with reference to the standard face ABC.

The reciprocals of the parameters are known as indices. According to the

crystallographic notation by Miller a law- has been established, which states that

"the intercepts that any face makes on the crystallographic axes are either infinite

or small rational multiples of the intercepts made by the unit form". Hence, the

ratio between the intercepts on the axes of different faces on a crystal can always

be expressed by rational numbers as 1: 2, 1: 3, 1: 4.

3.3 Law of axial-ratio

This law states that 'the ratio between the lengths of the axes of the crystals of a

given substance is- constant. This ratio is termed as 'axial-ratio'. Axial-ratio,

which is the ratio of the lengths of the crystallographic, expressed in terms of

one of the horizontal, axes, usually, '6'-axis, as unity.

Haüy (1784, 1801) deduced the law of rational indices from the observation of

the stacking laws required to build the natural faces of crystals by piling up

elementary blocks. E.g. cubes to construct the {110} faces of the rhomb-

dodecahedron observed in garnets or rhombohedrons to construct the

scalenohedron of calcite or {210} faces of the pentagon-dodecahedron observed

in pyrite.

Figure 3

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3.4 Law of crystallographic axes

The positions of the crystallographic axes are more or less fixed by symmetry of

the crystals, for in most crystals they are symmetry axes or normal to symmetry

plane. It has been observed that "crystals of a given mineral can be referred to

the same set of crystallographic axes".

3.5 Law of constancy of symmetry

According to this law, all crystals of a substance have the same elements of

symmetry i.e. plane of symmetry, axis of symmetry and center of symmetry.

4. Crystal Axes and Six Crystal Systems

In order to obtain insight into the solid geometry of a crystal. We need describe the

crystallographic axes. Since crystals are dealt in three dimensions, we need to have

three axes.

Let’s for simplicity make them all equal and at right angles to each other. This is

the simplest case to consider. The axes pass through the center of the crystal. We can

describe the intersection of any given face with these 3 axes. The axes are called a, b

and c. The interaxial angles as , and (Fig. 3).

The crystallographic axes are imaginary lines that we can draw within the crystal

lattice. These will define a coordinate system within the crystal. For 3-dimensional

space lattices we need 3 or in some cases 4 crystallographic axes that define

directions within the crystal lattices. Depending on the symmetry of the lattice, the

directions may or may not be perpendicular to one another, and the divisions along

the coordinate axes may or may not be equal along the axes.

Below figures shows the right hand convention rule name each axes of any crystal

system.

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Crystal Axes and six crystal System

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Symmetry: repetitive arrangement of features (faces, corners and edges) of a crystal

around imaginary lines, points or planes. Reflects internal ordering of atoms in the

mineral structure.

Motif: the fundamental part of a symmetric design that, when repeated creates the

whole pattern.

Operation: some act that produces the motif to create the pattern.

Operations include, (a) elements of symmetry; (b) translations; (c) glide planes; (d)

screw axes.

Element: an operation located at a particular point in space.

Unit Cell: The smallest unit of a structure that can be indefinitely repeated to

generate the whole structure.

Irrespective of the external form (Euhedral, Subhedral, or Anhedral) the properties

and symmetry of every crystal can be condensed into the study of one single unit

cell.

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Repetition of unit cell creates a motif (as shown in example below)

In the above example, the unit cell is a cube.

The arrangement and stacking differs between shapes.

5. Elements of Symmetry

Elements of symmetry identified in the unit cell will be present in the crystal

1. Elements without translation - Mirror (reflection)

2. Center of symmetry (inversion)

3. Rotation

4. Glide

These are all referred to as symmetry operation.

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Elements of symmetry: Types:

Axes of rotation (1, 2, 3, 4 or 6): If during the rotation of a crystal around an

axis, one of the faces repeats itself two or more times, the crystal is said to have

an axis of symmetry. Symmetry axes may be two fold (diagonal) if a face is

repeated twice every 360°, three fold (trigonal) if it is repeated three times, four

fold (tetragonal) if it is repeated four times, or six fold (hexagonal) if that face is

repeated 6 times. Figure below shows these relations.

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Symmetry operations found in crystals that map a lattice into self-coincidence,

for instance, rotations. Not all possible rotations can exist in a crystal, for

instance, we do not see 7-fold rotations. It is only the rotations that can map a

lattice into self-coincidence, those with turn angles of 0, 60, 90, 120, 180° that

are possible. We call these symmetry operations 1-fold, 6-fold, 4-fold, 3-fold and

2-fold rotations, respectively. The name comes from the fact that a n-fold

rotation has a turn angle of (360/n)°.

These rotation are defined such that a counter-clockwise rotation is considered

positive (right-hand rule). A clockwise rotation is designated as an inverse n-fold

rotation and is denoted n-1. This is because an n-fold rotation followed by an

inverse n-fold rotation takes you back to where you started, i.e. n n-1 = 1.

The set of all symmetry operations that map an object into self-coincidence,

while not moving the origin, is called its point group. The translation symmetry

operation is not an element of a point group, because it does not leave any point

fixed.

Based upon the 1-, 2-, 3-, 4-, and 6-fold rotations described above, we can define

5 different point groups, generated by 1-fold, 2-fold, 3-fold, 4-fold, and 6-fold

rotations. A 5-fold, 7-fold and other symmetries are not possible because one

cannot fill space with 5 - sided OR 7- sided as shown in below figure.

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Center (n or i): An inversion (i) produces an inverted object through an inversion

center. If ones draws lines from every point (as shown below) on the object through

the inversion center and out an equal distance on the other side.

For every point or face on one side of the center of symmetry, there is similar point

or face at an equal distance on the opposite side of the center.

Mirror Planes or Reflection (m): When one or more faces are the mirror images of

each other, the crystal is said to have a plane of symmetry). Motifs related to each

other by mirror planes are known as “enantiomorphs”. Reflection across a “mirror

plane” reproduces a motif.

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We now have 6 unique 2-D symmetry operations:

1 2 3 4 6 m

- Rotations are congruent operations reproductions are identical

- Inversion and reflection are enantiomorphic operations reproductions are

“opposite-handed”.

Combinations of symmetry elements are also possible. To create a complete analysis

of symmetry about a point in space, we must try all possible combinations of these

symmetry elements.

In the interest of clarity and ease of illustration, See more 2-D examples

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3-fold rotation axis with a mirror creates point group 3m & 6–fold rotation axis with

mirror creates point group 6mm

The original 6 elements plus the 4 combinations creates 10 possible 2-D Point

Groups:

1 2 3 4 6 m 2mm 3m 4mm 6mm

Any 2-D pattern of objects surrounding a point must conform to one of these above

groups

3D Symmetry

Rotoinversion (1, 2 , 3 , 4 or 6 ): When two similar faces are repeated 2, 3, 4 or 6

times when the crystal is rotated 360° around an axis, but in such a way that these

faces appear inverted. Therefore, if the face is repeated 2 times during a full rotation,

the axis is known as a 2-fold rotary inversion axis, 3 times 3-fold rotary

inversion, etc. Figure below shows the types of rotary inversion axes. Note that axes

of rotary inversion can also produce “enantiomorphs”.

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Types of symmetry possible in Minerals

-1, 2, 3, 4, 6: proper rotations

-m: mirror planes

-1 or i: center of symmetry or inversion

- 3: bar 3 rotoinversion

-4: bar 4 rotoinversion

-6: bar 6 rotoinversion

PLUS OTHER COMBINATION OF ROTATION, MIRROR

These can be combined in 32 ways to make crystal shapes

The 32 3-D Point Groups

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6. Forms

The term form is used to indicate general outward appearance

In Crystallography, external shape is denoted by the word habit, whereas form is

used in a special and restricted sense. Thus, forms consists of group of crystal faces,

all of which have the same relation to element of symmetry and display the same

chemical and physical properties.

The number of faces that belongs to a form is determined by the symmetry of the

crystal class.

Miller Indices enclosed in parenthesis as (hkl) or (010) indicate crystal face

Miller Indices enclosed in braces as {hkl} or {010} indicate form symbols

In each crystal, there is form, faces of which intersect all the crystallographic axes at

different lengths – general form {hkl}.

All other forms are called as special forms

The concept of a general form can also be related to the symmetry elements of a

specific crystal class.

An (hkl) face will not be parallel or perpendicular to single crystal symmetry

elements regardless of the crystal class, whereas special form consists of faces that

are parallel or perpendicular any Symmetry elements in the crystal class.

Nomenclature of crystallographic forms was initially proposed by Groth, 1895 and

later modified by A. F. Rogers in 1935 – It recognizes 48 forms of which 32 are the

general forms of 32 crystal classes. 10 are special, closed forms of isometric system

and 6 special open forms (prisms- hex and tetragonal)

Different scheme: Fedrov Institute of Leningrad, 1925: 47 instead of 48 forms

(Dihedrons).

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Refer to figures below for each forms

1. Pedion (Monohedron): A single face comprising form

2. Pinacoid(Parallehedron): An open form made up of two parallel faces

3. Dome (Dihedron): Two non-parallel faces symmetric w.r.t a mirror plane

(m).

4. Sphenoid (Dihedron): Two non-parallel faces symmetric w.r.t a 2-fold

rotation axis

5. Prism: An open form composed of 3,4,6,8 or 12 faces, all of which are

parallel to the same axis

6. Pyramid: An open form composed of 3,4,6,8 or 12 non parallel faces that

meet at a point

7. Dipyramid: A closed form having 6, 8, 12, 16 or 24 faces.

8. Trapezohedron: A closed form that has 6,8 or 12 faces in all, with 3,4 or 6

upper faces offset with 3,4 or 6 faces lower faces. These faces are the result

of 3, 4 or 6 fold axis combined with perpendicular 2-fold axes.

There is Isometric trapezohedron (tetragon-trisoctahedron) – 24 face form

9. Scalenohedron: A closed form with 8 or 12 faces grouped in symmetrical

pairs. In the tetragonal scalenohedron, (rhombic scalenohedron) pairs of

upper faces are related by an axis of 4-fold rotoinverstion to pairs of lower

faces. The 12 faces of hexagonal scalenohedron display three pairs of upper

faces and three pairs of lower faces in alternating positions. The pairs are

related by the center of symmetry; coexist with a 3-fold axis of

rotoinverstion.

10. Rhombohedron: A closed form composed of six faces of which three faces at

the top alternates with three faces at the bottom, the two sets of faces being

offset by 60o. Only seen in point groups.

11. Disphenoid: A closed form consisting of two upper faces that alternate with

two lower faces, offset by 90o.

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Names of 15 different Isometric Forms:

Groth and Rogers No. of Faces Internationally recommended

name (After FedrovInsituture)

1. Cube 6 Hexahedron

2. Octahedron 8 Octahedron

3. Dodecahedron(rhombic) 12 Rhom-dodecahedron

4. Tetrahexahedron 24 Tetrahexahedron

5. Trapezohedron 24 Tetragon-Trioctahedron

6. Trisoctahedron 24 Tetragon-Trioctahedron

7. Hexaoctahedron 48 Hexaoctahedron

8. Tetrahedron 4 Tetrahedron

9. Tristetrahedron 12 Trigon-Tetrahedron

10. Deltoid dodecahedron 12 Tetragon-tritetrahedron

11. Hexatetrahedron 24 Hexatetrahedron

12. Gyroid 24 Pentagon-trioctahedron

13. Pyritohedron 12 Dihexahedron

14. Diploid 24 Di-dodecahedron

15. Tetartoid 12 Pentagon-tristetrahedron

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Multiple Choice Questions-

1. On the basis of length and interfacial angles of a unit cell, the crystal have been

divided into

(a) Six crystal system

(b) Seven crystal system

(c) Eight crystal system

(d) Nine crystal system

2. Choose the incorrect combination of rotation and reflection symmetry

(a) 3mm

(b) 3m

(c) 4mm

(d) 2mm

3. If the exterior angle measured between (001) (111) = 45°. What would be

exterior angle between (010) (111)

(a) 40°

(b) 45°

(c) 135°

(d) 140°

4. What symmetry will be produced if the perpendicular mirror plane to c-axis is

removed in 4/m 2/m 2/m symmetry

(a) 422

(b) 4/m

(c) 4mm

(d) 4m2

5. The isometric system is characterized by

(a) 3 crystallographic axes of 4-fold symmetry

(b) 4 crystallographic axes of 4-fold symmetry

(c) 3 crystallographic axes of 3-fold symmetry

(d) 3 crystallographic axes of 2-fold symmetry

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Suggested Readings:

1. Klien, Cornelis and Dutrow, Barbara, (2008). Manual of Mineral Sciences

(Manual of Mineralogy), 23rd Edn. John Wiley & Sons, New York. ISBN:

0471721573, 978-0471721574.

2. Ford, William E., (2006). Dana's textbook of mineralogy (with extended

treatise crystallography & physical mineralogy, 4th Edn., CBS Publishers and

Distributors Pvt. Ltd., New Delhi. ISBN: 81239080910, 978-81239080909.

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