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Md. Hussain MonsurProfessorDept. of GeologyUniversity of Dhaka CRYSTALLOGRAPHY

Varieties of Beautiful Crystals

Crystals are so natural! So beautiful!

Crystals are so natural! So beautiful!

CRYSTALLOGRAPHY

Crystallographyis the branch of science which deals with the crystal: their development and growth, external form, internal arrangement and physical properties. The word "crystallography" derives from the Greek wordscrystallon. Combination of twoWords: Cold and congeal, means congealed by cold or"cold drop, frozen drop.

J. Kepler (1619) AstronomerRobert Hooke (1665) Inventor microscope Concept of crystals as regular arrangement of spherical particles.

Christian Huygens (1690): Studied Calcite crstal- Regular internal arrangement.

Nicolaus Stensen (1669): Law of constancy of angles between crystal faces.

A crystalline solid: atomic resolution image ofstrontium titanate. Brighter atoms arestrontiumand darker ones are titaniumCrystallographyis the experimental science of determining the arrangement of atoms in the crystalline solids.

Crystal structure of sodium chloride (table salt)

The Koh-I-Noor was mounted on the Peacock Throne, the Mughal throne of India. It is said that Shah Jahan, the ruler who commanded the building of the throne and that of the Taj Mahal was imprisoned by his son and he could only ever see the Taj Mahal again through the reflection of the diamond.Later, Shahs son, Aurangazeb brought the Koh-I-Noor to the Badshahi Mosque in Lahore. It was robbed from there by Nadir Shah who took the diamond to Persia in 1739, but the diamond found its way back to Punjab in 1813 after the deposed ruler of Afghanistan, Shuja Shah Durrani took it to India and made a deal to surrender the diamond in exchange for help in winning back the Afghan throne.The Brits came across the gem when they conquered Punjab in 1849, and Queen Victoria got it in 1851. The stone was then at 186 carats as before this point, the diamond was not cut

Legend says that the diamond is 5000 years old and was referred to in Sanskrit writings as the Syamantaka jewelRaja of Gwalior in the 13th centuryBabur documented (1526)Wt. before cutting 186 carat (37gm). After cutting (by Albert) 108.93 carats. Oval shape.Queen Elizabeth (later Queen Mother) wearing the Koh-I-Noor set in her crown on the balcony of Buckingham Palace, after the coronation of King George VI, with daughter Princess Elizabeth, now Queen Elizabeth II.

DIAMONDGRAPHITE

(1s)2(2s)2(2p)2Carbon-Carbon bond - Hybridization

sp3-Hybridization (Diamond)Sp2= Hybridization (graphite)Carbon atom

sp3-Hybridization (Diamond)Sp2= Hybridization (graphite)

DIAMOND (Multi faceted ball) DIAMOND ATOMIC STRUCTURE GRAPHITE ATOMIC STRUCTURE There are no covalent bonds between the layers and so the layers can easily slide over each other making graphite soft and slippery and a good lubricant.

DIAMOND ATOMIC STRUCTURE

VIDEO BONDING OF DIAMOND AND GRAPHITECarbon-Carbon bond - Hybridization

CARBON ALLOTROPES

DEFINITIONS

MINERAL: Mineral is a naturally occurring homogeneous solid having external form, regular arrangement of internal structure and a chemical formula.

CRYSTAL: A crystal is a body that is formed by the solidification of a chemical element, a compound, or a mixture and has a regularly repeating internal arrangement of its atoms and often external plane faces.

Congealed by cold. OldEnglishcristal"clearice,clearmineral, from Latincrystallus"crystal,ice,"fromGreekkrystallos,fromkryos"frost,"fromPIEroot*kru(s)-"hard,hardoutersurface"

CRYSTALLOGRAPHY: Crystallography is the branch of science which deals with crystals, their growth and development, external form and internal structures.

Crystals are found in three forms 1. Euhedral, 2. Subhedral, 3. Anhedral SOME DEFINITIONS

FACES Crystals are bounded by smooth plane surfaces (some varieties of diamond have curve faces), these are called Faces. a) Like faces & a) Unlike faces

2. EDGES The intersecting line of two adjacent faces is called Edge.3. ZONE AND ZONE AXIS4. INTERFACIAL ANGLE The interfacial angle between two crystal faces as the angle between lines that are perpendicular to the faces. Such a lines are called thepolesto the crystal face. The interfacial angle is the angle between two normal to two intersecting faces. The interfacial angles between corresponding faces of the same mineral will be the same. This is known as theLaw of constancy of interfacial angles,

5. SOLID ANGLES Three or more faces make a solid angle.

Space Lattice

A space lattice is an array of points showing how particles (atoms, ions or molecules) are arranged at different sites in three dimensional spaces. Crystals, of course, are made up of 3-dimensional arrays of atoms. Such 3-dimensional arrays are calledspace lattices.The ordered internal arrangement of atoms in a crystal structure is called aLATTICE.

Unit Cell

The unit cell may be defined as,the smallest repeating unit in space lattice which, when repeated over again, results in a crystal of the given substance. The unit cell may also be defined as the unit parallelepiped which is repeated throughout the crystal by translation along any lattice row.

Space Lattice and Unit Cell

CRYSTALLOGRAPHIC AND COORDINATE AXESCrystallographic AxesThe crystallographic axes are imaginary lines within the crystal lattice. These define a coordinate system within the crystal. 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. The INTERCEPTS are distances between the centre of the crystal (point of intersection of the crystallographic axes) and the points of intersection of the face and axes. PARAMETERS are the ratios of the intercepts. This is known as parameter system of Weiss.

Symmetry axes are equal to Coordinate axes. The angles between the axes are equal to

Alattice systemis generally identified as a set of lattices with the same shape according to the relative lengths of the cell edges (a,b,c) and the angles between them (,,).

Crystallographic axes Coincide with coordinate axesCrystallographic axesdo not coincide with coordinate axesCRYSTALLOGRAPHIC AND COORDINATE AXESLower Systems : Monoclinic, Triclinic and Orthorhombic. Intermediate Systems: Tetragonal and Hexagonal.Higher System: Isometric (Cubic).

Crystal familyLengthsAnglesCommon examplesIsometrica=b=c===90Garnet,halite,pyriteTetragonala=bc===90Rutile,zircon, andalusiteOrthorhombicabc===90Olivine, aragonite, orthopyroxenesHexagonala=bc==90, =120Quartz,calcite, tourmalineMonoclinicabc==90, 90Clinopyroxenes,orthoclase, gypsumTriclinicabc90Anorthite,albitekyanite

CRYSTAL SYSTEMS WITH UNIT CELLS

abc

abcabcabcabcabc

Video of crystal Lattice and Unit Cell

Video of crystal Lattice structures The Simple Cubic Lattice

A face is more commonly developed in a crystal if it intersects a larger number of lattice points.This is known as theBravais Law (1848).

Faces are more commonlyDevelops and 1 &2.

SymbolsP - Primitive: simple unit cellF - Face-centered: additional point in the center of each faceI - Body-centered: additional point in the center of the cellC - Centered: additional point in the center of each endR - Rhombohedral: Hexagonal class onlyAuguste Bravais 14 arrangement of space lattices(bornAug. 23, 1811,Annonay, Fr.diedMarch 30, 1863,Le Chesnay)

Isometric CellsThe F cell is very important because it is the pattern for cubic closest packing. There is no C cell because such a cell would not have cubic symmetry.

Tetragonal CellsA C cell would simply be a P cell with a smaller cross-section, as shown below. An F cell would reduce to a network of I cells.

Hexagonal CellsThe R cell is unique to hexagonal crystals. The two interior points divide the long diagonal of the cell in thirds. This is the only Bravais lattice with more than one interior point. A rhombohedron can be thought of as a cube distorted along one of its diagonals.

Orthorhombic CellsThe orthorhombic class is the only one with all four types of Bravais lattice

Monoclinic and Triclinic CellsMonoclinic F or I cells could also be represented as C cells. Any other triclinic cell can also be represented as a P cell.MonoclinicMonoclinicTriclinic

Bravis 14 Different Types Of Space Lattices

Symmetry is a special characteristic of crystal. Crystals are subdivided into 32 classes on the basis of symmetry elements. Two figures (parts) or two bodies (parts) are said to be symmetrical when they coincide if they are matched. The transformation or matching of two symmetrical bodies are called symmetry operation and the object by which two bodies or figures become symmetrical is called Element of Symmetry. Hence, symmetry operation is spatial transformation (rotations, reflections and inversions)SYMMETRYPlane of symmetry (P)2. Axis of symmetry (A)3. Rotation-reflection Axis ( )4. Cetre of symmetry (C)Symmetry Elements

Equilateral TriangleIsosceles TriangleScalene Triangle

(No symmetry)Reflection Symmetry(Plane of Symmetry-P)

Plane of symmetry (P)

Plane of Symmetry (P)

Plane of symmetry (P)

Rotational Symmetry

A shape has Rotational Symmetry when it still looks the same after a rotation. Sometimes a figure turns into congruent position when it rotates about an axis. The line about which a figure rotates and turns into congruent position is called Axis of Symmetry. The angle of rotation by which a figure turns into congruent position is called Elementary angle of rotation. The number of turning of congruent position through a complete rotation is called Fold of an axis.

Axis of Symmetry (A)There may be:Two Fold Symmetry Axis (A2)Three Fold Symmetry Axis (A3)Four Fold Symmetry Axis (A4)Six Fold Symmetry Axis (A6)Axis of Rotary Inversion (A42 & A63)Five Fold Symmetry Axis and more than Six Fold Symmetry axes can not axist in crystals

Axis of Symmetry (A)

Axis of Symmetry (A)

Axis of Symmetry (A)

Axis of Symmetry (A)

Centre of symmetry (C)

CRYSTALLOGRAPHIC AND COORDINATE AXESCrystallographic AxesThe crystallographic axes are imaginary lines within the crystal lattice. These define a coordinate system within the crystal. 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. The intercept, made by the unit cell on three crystallographic axes are called PARAMETERS.

Symmetry axes are equal to Coordinate axes. The angles between the axes are equal to

Step 2: Specify the intercepts in fractional co-ordinatesCo-ordinates are converted to fractional co-ordinates by dividing by the respective cell-dimension - for example, a point (x,y,z) in a unit cell of dimensionsaxbxchas fractional co-ordinates of (x/a,y/b,z/c). In the case of a cubic unit cell each co-ordinate will simply be divided by the cubic cell constant ,a. This givesFractional Intercepts : a/a,/a,/ai.e.1 ,,Step 3: Take the reciprocals of the fractional interceptsThis final manipulation generates the Miller Indices which (by convention) should then be specified without being separated by any commas or other symbols. The Miller Indices are also enclosed within standard brackets (.) when one is specifying a unique surface such as that being considered here.The reciprocals of 1 andare 1 and 0 respectively, thus yieldingMiller Indices : (100)So the surface/plane illustrated is the (100) plane of the cubic crystal.Step 1: Identify the intercepts on the x- , y- and z- axes.In this case the intercept on the x-axis is atx=a( at the point (a,0,0) ), but the surface is parallel to they- andz-axes - strictly therefore there is no intercept on these two axes but we shall consider the intercept to be at infinity () for the special case where the plane is parallel to an axis. The intercepts on thex- ,y- andz-axes are thusIntercepts : a,,procedure OfDETERMInationOF MILLER INDICES

procedure OF DETERMINe OF MILLER INDICES

Solution: 1.Since the plane passes through the existing origin the new origin must be selected at the corner of adjust unit cell. 2.As related to new origin the following intercepts (in terms of lattice parameters a, b, and c) with x, y, z axes can be referred: (plane is // to x-axis), -1, 1/2 3.The reciprocal of these numbers are: 0, -1 and 2 and they are already integer! 4.Thus the Miller indices of the consider plane are: (0-12)

1. The (110) surfaceAssignmentIntercepts : a,a,Fractional intercepts : 1 , 1 ,Miller Indices : (110)

2. The (111) surfaceAssignmentIntercepts : a,a,aFractional intercepts : 1 , 1 , 1Miller Indices : (111)

3. The (210) surface

AssignmentIntercepts : a,a,Fractional intercepts : , 1 ,Miller Indices : (210)

Exercises

MILLER INDEX NOTATIONThe law of rational indices states that the intercepts,OP,OQ,OR, of the natural faces of a crystal form with the unit-cell axesa,b,c(see Figure 1) are inversely proportional to prime integers,h,k,l. They are called theMILLAR INDICES (hkl) of the face. They are usually small because the corresponding lattice planes are among the densest and have therefore a high inter-planar spacing and low indices. The Miller indices of the Planes ABC',ABC,ABC"AA"BB"are(112), (111),(221),(110),respectively. These planes haveAB, or, as common zone axis.

Exercises

55

procedure OF DETERMInation OF MILLER INDICES(Lecture)

procedure OF DETERMInation OF MILLER INDICES(Lecture)

LAW OF RATIONAL INDICESThe intercepts, made by a unit cell on three crystallographic axes are called parameters. The parameters are denoted by small letters, a, b and c. Rational Indices are reciprocal of the parameters. Any crystal face in space can be represented by three whole numbers, if the intercept, made by a unit cell on three crystallographic axes are taken as unit of measurement (Law of Rational Indices).

The law of rational indices was deduced by Hay (1784, 1801)

DETERMINATION OF MILLERS INDICESSome exercises

DETERMINATION OF MILLERS INDICESSome exercises

DETERMINATION OF MILLERS INDICESSome exercises

MILLAR INDICES FOR HEXAGONAL AND TRIGONAL SYSTEMS

+a1 +a2 +a3

LAW OF RATIONAL INDICES: Any crystal face in space can be represented by three whole numbers, If the crystallographic axes are taken as coordinate axes and if the intercepts make by a unit cell on crystallographic axes are taken as the unit of measurement.LAW OF CONSTANCY OF INTERFACIAL ANGLES: The angles between corresponding faces of the same mineral will be the same. This is known as theLaw of constancy of interfacial angles,BRAVAIS LAW: A face is more commonly developed in a crystal if it intersects a larger number of lattice points.This is known as theBravais Law. SOME IMPORTANT LAWS

STEREOGRAPHIC PROJECTION

CRYSTAL FORMS

ACRYSTAL FORMisa set of crystal like faces that are related to each other by symmetry, i.e. a set of like faces in a crystal make a specific form. There are debates in writing symbols of face and form. To avoid confusions, we shall represent the face and form symbols in the following manner: For example, in the case of a Cube,Face : 001 - indicate top face of a cube, without bracket.Form: (001)6 - Symbol of top face of a cube, with bracket. Number 6 means the forms is composed of six like faces. General symbol (hkl)n h is less than l less than k, n is the number of faces in a form.

The number of faces in a form depends on the symmetry of the crystal.

The 48 Special Crystal FormsAny group of crystal faces related by the same symmetry is called aform. There are 47 or 48 crystal forms depending on the classification used.

There are two kinds of forms in crystals1. Open form (17 or 18)2. Closed forms (30)

Open forms are those groups of like faces all related by symmetry that do not completely enclose a volume of space.

Closed forms are those groups of like faces all related by symmetry that completely enclose a volume of space.

Crystals are bounded by SIMPLE FORMS (all are like Faces) and COMBINATION (bounded by Like and Unlike Faces).

Pedion: A single face unrelated to any other by symmetry. Pinacoid: A pair of parallel faces related by mirror plane or twofold symmetry axis. Dihedron: A pair of intersecting faces related by mirror plane or twofold symmetry axis. Some crystallographers distinguish betweendomes(pairs of intersecting faces related by mirror plane) andsphenoids(pairs of intersecting faces related by twofold symmetry axis). Pyramid: A set of faces related by symmetry and meeting at a common point. All are Open formSIMPLE FORMS OF LOWER SYSTEMSTriclinic, Monoclinic and Orthorhombic Systems

Simple forms of the intermediate System(Tetragonal and Hexagonal Systems)3-, 4- and 6-Fold Prisms

A collection of faces all are parallel to a symmetry axis. All are open.Prismatic (Gypsum)

Simple forms of the intermediate System(Tetragonal and Hexagonal Systems)3-, 4- and 6-Fold Pyramids

PYRAMIDSA Pyramid: A set of triangular like faces intersecting at a point on a symmetry axis. All are open. The base of the pyramid would be a pedion.

Simple forms of the intermediate System(Tetragonal and Hexagonal Systems)3-, 4- and 6-Fold Bipyramids

(Beryl)

Disphenoid: A solid with four congruent triangle faces, like a distorted tetrahedron. Midpoints of edges are twofold symmetry axes. In the tetragonal disphenoid the faces are isoceles triangles and a fourfold inversion axis joins the midpoints of the bases of the isoceles triangles.Scalenohedron: A solid made up of scalene triangle faces (all sides unequal)Trapezohedron: A solid made of trapezia (irregular quadrilaterals)Rhombohedron: A solid with six congruent parallelogram faces. Can be considered a cube distorted along one of its diagonal three-fold symmetry axesSimple forms of the intermediate System(Tetragonal and Hexagonal Systems)Scalenohedra and Trapezohedra

Six square faces. Eachintersects one cryst. Axisand parallel to other two. Form symbols: (001)68 equilateral triangularfaces. Each face cutsall cryst. axes at equaldistance. symbol: (111)812 rhombo shapedfaces. Each face cutstwo cryst. axes at equalDistance and parallel to third. symbol: (011)1224 isosceles traiangularfaces. Each face cuts two cryst. axes at unequal lengths and parallel to third. symbol (okl,012)2424 trapezoid faces. Each face cuts two cryst. axes at equallengths and third a smaller length. symbol: (hhl, 112)2424 isosceles traiangular faces. Each face cuts two cryst. axes at equal lengths and third at greater length. symbol: (hll, 122)2424 scalene traiangular faces. Each face cuts all cryst. axes At unequal lengths. Symbol (hkl,012)48SIMPLE FORMSNORMAL CLASSCUBIC SYSTEMHigher System, CUBIC

Tetrahedron: Four equilateral triangle faces (111)Trapezohedral Tristetrahedron: 12 kite-shaped faces (hll)Trigonal Tristetrahedron: 12 isoceles triangle faces (hhl). Like an tetrahedron with a low triangular pyramid built on each face.Hextetrahedron: 24 triangular faces (hkl) The general form.Simple forms of higher System(Cubic Systems)Hextetrahedral Forms

SIMPLE FORMS OF HIGHER SYSTEM(Isometric System)Tetartoidal, Gyroidal and Diploidal Forms Tetartoid: The general form for symmetry class 233. 12 congruent irregular pentagonal faces. The name comes from a Greek root for one-fourth because only a quarter of the 48 faces for full isometric symmetry are present.Gyroid: The general form for symmetry class 432. 24 congruent irregular pentagonal faces.Diploid: The general form for symmetry class 2/m3*. 24 congruent irregular quadrilateral faces. The name comes from a Latin root for half, because half of the 48 faces for full isometric symmetry are present.Pyritohedron:Special form (hk0) of symmetry class 2/m3*. Faces are each perpendicular to a mirror plane, reducing the number of faces to 12 pentagonal faces. Although this superficially looks like the Platonic solid with 12 regular pentagon faces, these faces are not regular.

EXERCISES&Examples

Carl's Gold Figures A schematic diagram relating the various form types within the holohedral isometric crystal class

Isometric-This system comprises crystals with three axes, all perpendicular to one another and all equal in length.Basicwooden modelHalite (salt)

Basicwooden modelApophylliteTetragonal-This system comprises crystals with three axes, all perpendicular to one another; two are of equal length.

Orthorhombic-This system comprises crystals with three mutually perpendicular axes, all of different lengths.

Golden TopazBasicwooden model

Monoclinic-This system comprises crystals with three axes of unequal lengths, two of which are oblique (that is, not perpendicular) to one another, but both of which are perpendicular to the third

Gypsum

Basicwooden model

Triclinic-This system comprises crystals with three axes, all unequal in length and oblique to one another

OrthoclaseBasicwooden model

99

Hexagonal-This system comprises crystals with four axes. Three of these axes are in a single plane, symmetrically spaced, and of equal length. The fourth axis is perpendicular to the other three.

Sapphire

Basicwooden model