Material Science and Engineering Assignment

MATERIAL SCIENCE AND ENGINEERING ASSIGNMENT

Assignment By -

Submitted to - Prof. Ranjit Singh

MATERIAL SCIENCE ASSIGNMENT

CRYSTAL SYSTEM AND CLASSES

Before studying about crystal system and classes we should first know what exactly are crystals! As Thomas B. Macaulay has said “Half knowledge is worse than ignorance”.

What is a crystal? What are crystal structures? How are they different from other substances? What is a crystal lattice?

So let’s talk about these briefly before we discuss our topic!

As we all know any solid substance is made up of many small atoms or particles bound together by a force of attraction. What differs crystals from other substances is that Crystals are the solids in which atoms are arranged in some regular repetition pattern in all direction.

Here’s a picture of a sodium chloride crystal. As you can see there’s an order and a definite arrangement between all the atoms. When we want to understand what do we mean by the crystal system or structure , we use lattice basically as the framework. A lattice is an ordered array of points describing the arrangement of particles that form a crystal. One important property of a lattice is that a lattice has same surroundings. If I take any two points in the lattice no matter how far they are they should look same from a particular direction. For example , you must have seen those tiles on the pavement they are put next to each other in an ordered way. This idea was proposed by Auguste Bravais that is why they are called Bravais lattice. He said there are seven crystal systems and fourteen crystal structures.

Now if I start putting atoms or molecules or ions on the lattice points then I’m building a crystal structure. A lattice point is the position in a crystal where the probability of finding an atom or an ion is highest.

In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three- dimensional space in matter.

The small particles which constitute these crystals are called Unit Cells. A Unit cell is a geometric pattern which repeats itself through the three dimensional pattern of solid. It generates the crystal lattice when repeated in space indefinitely. Unit cells are arranged like building blocks in crystals.

So before going to crystal system and classes we must know that , crystals are first divided into systems then they are divided into classes and then finally classes are divided into forms.

CRYSTAL SYSTEM AND CLASSES Major subdivisions include six divisions of system. Each system has some classes and every class has some forms. Due to different symmetries crystals have been classified into classes. There are only 32 possible combinations of symmetry operations, which define 32 crystal classes.

The first system is the ISOMETRIC OR CUBIC SYSTEM. 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. In this system all the three axes which is all the three crystallographic axes are of equal length and angle between them is 90 degree. They are orthogonal to each other and hence they are interchangeable. It has four axes of three fold symmetry. It has five classes.

This is a simple cubic lattice. This is how all the unit cells get arranged to form the crystal. The main varieties of these crystals :- 1)Simple cubic 2)Body centred cubic 3)Face centred cubic

Next is the TETROGONAL SYSTEM In tetragonal system , the two horizontal axis are equal and perpendicular to each other and both of the axes are perpendicular to the third axis but not equal to it. It can be small or it can be larger than the two horizontal axes.

There are two tetragonal 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). One might suppose stretching face-centered cubic would result in face-centered tetragonal, but the face-centered tetragonal is equivalent to the body-centered tetragonal, BCT (with a smaller lattice spacing). Tetragonal system has total seven number of classes.

It looks somewhat like this. The tetragonal unit cell has unique four fold axis symmetry or four fold axis of seven fold inversion.

Next is HEXAGONAL SYSTEM. Components of crystals in this system are located by reference to four axes— three of equal length and a fourth axis perpendicular to the plane of the other three. Hexagonal system is again divided into Hexagonal and trigonal divisions and it has total twelve classes.

If the atoms or atomic groups in the solid are represented by points and the points are connected by line segments, the resulting lattice will define the edges of an orderly stacking of blocks, or unit cells. The hexagonal unit cell is distinguished by the presence of a single line, called an axis of 6-fold symmetry, about which the cell can be rotated by either 60° or 120° without changing its appearance. In hexagonal system , among the four axes three axes are horizontal and at 60 degree to each other. The fourth axis is vertical and can be shorter or longer in length than the horizontal ones and angle between the horizontal axes and the vertical axis is 90 degree. It has unique six fold axis or inversion of seven classes and the other five classes have unique three fold axis or inversion.

Next is ORTHORHOMBIC SYSTEM

as you can see in the image. Crystals in this system are referred to have three mutually perpendicular axes that are unequal in length. 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 and height, such that a, b, and c are distinct. It has three classes and three axes of two fold symmetry.

Next we have MONOCLINIC SYSTEM. 35% of the minerals crystallises in this monoclinic system. Here all the three crystallographic axes are of unequal length. But the angle between a and b axes and b and c axes is 90 degree whereas the angle between a and c axes is not equal to 90 degree.

It has three classes and one axis of two fold symmetry.

The last one is TRICLINIC SYSTEM.

In this system neither the length of the axes are equal nor the angles are 90 degree. This is the system with the minimum amount of symmetry and it does’nt have any axis or plane of symmetry. It has two classes.

So this was my brief idea and summary on the different crystal systems and classes. We have been given Isometric and Tetragonal systems to discuss in detail which will be done by my team members.

ISOMETRIC (CUBIC) SYSTEM

Definition: All those crystals that can be referred to three crystallographic axes, which are-

● essentially equal in length, ● at right angles to each other, and ● mutually interchangeable,

are said to belong to the isomeric or cubic system.

It has got the following symmetry:

(a) Axes of Symmetry:

13 in all,

3 are axes of four-fold symmetry;

4 are axes of three-fold symmetry;

6 are axes of two-fold symmetry.

The three axes of four-fold symmetry are chosen as the crystallographic axes.

(b) Planes of Symmetry:

9 in all.

3 planes of symmetry are at right angles to each other and are termed the principal (axial) planes;

6 planes of symmetry are diagonal in position and bisect the angles between the principal planes.

Forms:

Following are the forms that commonly develop in the crystals belonging to Isometric System: i. Cube:

A form bounded by six similar square faces, each of which is parallel to two of three crystallographic axes and meets the third axis. Symbol(100).

ii. Octahedron:

A form bounded by eight similar faces, each of the shape of an equilateral triangle, each meeting the three crystallographic axes at equal distances. Symbol- (111)

iii. Dodecahedron:

It is a form with twelve similar faces each of which is parallel to one of the three crystallographic axes and meets the other two at equal distances. Symbol(101). iv. Trisoctahedron (hhl):

A form of twenty four (24) faces; each face meeting two axes at unit length and to the third at greater than unity. Faces occur in eight groups of three each.

. v. Trapezohedron (hll):

A form of twenty four (24) faces each face meeting one axes at unit length and to the other two at greater than unity. Each face is a trapezium. vi. Tetra-Hexahedron (hol):

Twenty four (24) faces; each face is parallel to one axis and meets other two at unequal distance which are simple multiple of each other; faces occur in six groups of four each.

vii. Hexaocatahedron (hkl):

Forty eight (48) faces; each face meets the three axis at unequal distances.

Other Classes:

Isometric system comprises five symmetry classes in all.

Beside the normal class, following three classes are also represented among the minerals:

A. Pyritohedral Class (Pyrite Type):

(a) Symmetry:

7 Axes of symmetry, of which, 3 are axial axes of two-fold symmetry,

4 are diagonal axes of two fold symmetry.

3 Planes of symmetry. Centre of symmetry

(b) Forms: Pyritohedron and Diploid are two typical forms of this symmetry class. Pyritohedron is a hemihedral form having twelve faces and a general symbol (hko)

Diploid is a closed form of twenty-four faces that typically occur in pairs (hence the name) and have a symbol (hki)

B. Tetrahedral Class (Tetrahedrite Type):

(a) Symmetry:

7 axes of symmetry (as in pyrite type),

6 planes of symmetry,

no center of symmetry. (b) Forms:

Most typical form of this class is a four sided solid in which each face is an equilateral triangle. It is termed tetrahedron. It has a general symbol of (111).

C. Plagiohedral Class (Cuprite Type):

(a) Symmetry:

13 axes of symmetry (as in normal class) No planes of symmetry. No Centre of symmetry.

(b) Forms:

Icositedraherons, each of 24 faces, with a symbol (hkl) and commonly enantiomorphous in character are typical forms of this class.

Examples of Isometric Minerals:

A vast number of common minerals crystallize in isometric system.

Following are few examples: i. Galena ii. Pyrite

CUBIC UNIT CELL

Unit cell is the smallest portion of a crystal lattice which, when repeated in different direction , generates the entire lattice . Unit cells can be broadly divided into two categories: ● Primitive unit cells ● Centred unit cells Primitive unit cell When constituent particle are present on the corner positions of a unit cell, it is called primitive unit cell. Centred unit cells When a unit cell contains one or more particles present at position other than corners in addition to those at corners , it is called a centred unit cell Centred unit cell are of three types: ● Body centred unit cell: such a unit cell contains one constituent particle at its body centre besides the one that are at its corners

● Face centre unit cell: such a unit cell contains one constituent particle present at centre at each face, beside the one that are at its corners.

● End centred unit cell: In such unit cell, one constituent particle is present at the centre of any two opposite faces beside the one present at its corners. Some important terms:

▪ Number of effective atom in a unit cell

▪ Coordination number (C.N) =Number of nearest neighbour of particle in unit cell.

▪ Packing Fraction

(P.F)=

▪ Voids = Vacant space between the constituent particles in a closed packed structure. There are basically two types of void

(a) Octahedral void: The void surrounded by six sphere sitting at the corners of regular octahedron is called octahedral void.

(b) Tetrahedral void: The void surrounded by four spheres sitting at the corner of regular tetrahedron is called a tetrahedral void.

NOTE: If in a cubic unit cell =n Number of octahedral void in a unit cell =n Number of tetrahedral void in a unit cell=2n

PRIMITIVE CUBIC UNIT CELL

It has atoms only at its corners. Each atom at a corner is shared between eight adjacent unit cell, four unit cell in the same layer and four unit cell in upper (or lower ) layer.

●

● C.N=6 ● P.F=54.4%

● Relation between edge length of cube and radius of sphere , where Let edge length of cube= a Radius of sphere =r o a=2r BODY CENTRED CUBIC UNIT CELL

A body centred cubic unit cell has atom at each of it corners and also one atom at its body centred.

●

● C.N=8 ● P.F=68%

● Relation between edge length of cube and radius of sphere , where Let edge length of cube= a Radius of sphere =r o √

FACE CENTRED CUBIC UNIT CELL

It contains atom at all the corners and at the centre of all the face of cube. Each atom located at the face centre is shared between two adjacent unit cells.

●

● C.N=12 ● P.F=74%

● Relation between edge length of cube and radius of sphere , where Let edge length of cube= a o Radius of sphere =r , √

TETRAGONAL CRYSTAL SYSTEM: In crystallography, the tetragonal crystal system is one of the 7 crystal systems. 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).

BRAVAIS LATTICES: 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). One might suppose stretching face-centered cubic would result in face-centered tetragonal, but the face-centered tetragonal is equivalent to the body-centered tetragonal, BCT (with a smaller lattice spacing). BCT is considered more fundamental, and therefore this is the standard terminology..[1] Primitive Body-centered Bravais lattice tetragonal tetragonal

Pearson symbol tP Ti

Unit cell

The Unique Symmetry Element of the Tetragonal Crystal System: The single 4-fold axis of rotational symmetry is unique to the tetragonal crystal system.The only other crystal system with a 4-fold axis is the Isometric System, which often has three of them mutually perpendicular to each other. The Isometric system always has four 3-fold axes of rotational symmetry. Finding only one 4-fold axis and no 3-fold axes assures that one is observing a tetragonal crystal. CRYSTALLOGRAPHIC AXES: The symmetry of the Tetragonal System is applied to a set of 3 axes with two of the axes equal in length and the third axis either longer or shorter than the other two. The three axes are mutually perpendicular to each other. The two axes, equal in length are labelled a1 and a2. The third axis is labelled c and is usually viewed as the vertical axis.

The top end of the c-axis is designated as positive, c+, and the bottom end as negative,c-. One of the a-axes is set horizontally from front to back with a1+ as the front end and a1- as the rear. The a2 axis is horizontal from a2+ on the right to a2- on the left. Note the conventions on the Axial Diagram.

GENERAL AND SPECIAL FORMS: There are two types of forms, general forms and special forms.

Any form which is not a general form is a special form. Most often, the general form is the form for which the crystal class is named. A general form has the maximum number of faces of any form in its crystal class. Special forms are not unique to a particular class but may appear in several classes.

The general form intercepts all axes, each at a different axial unit distance. Thus each index of the Miller indices for a general form must have a different value, and not be zero. It can be {123}, {341}, {352}, or something similar. The general form is not always expressed on the crystal.

Forms in the Tetragonal System: Click any image to enlarge it.

Pyramids &

Dipyramids Tetragonal Tetragonal Ditetragon Dipyramid 1st Pyramid 2nd al Order Order Positive Dipyramid

Prisms &

Diprisms Tetragonal Tetragonal Ditetragon Prism 1st Prism 2nd al Prism Order Order Trapezohedron s; Disphenoid;

& Tetragonal Tetragonal Tetragonal Scalenohedron Trapezohedro Trapezohedro Tetragonal Scalenohedro n Left n, Right Disphenoid n Positive

Pinacoid &

Pedion Tetragonal Tetragonal Pinacoid Pedion, basal

1st, 2nd, & 3rd Order Forms in the

Tetragonal Crystal System: Prism faces of the 1st order form, {hh0}/{110}, intercept both a-axes (see diagram at right, below). Each face of the 2nd order prism, {h00}/{100}, is perpendicular to and intercepts one a-axis and is parallel to the other. Faces of the 3rd order form, {hk0}/{120} intercept both a-axes, but are neither perpendicular nor parallel to either one. In all cases, the prism faces are parallel to and surround the c-axis. Pyramid faces of the same order are located above, or below, the corresponding faces of prisms. Pyramids & Dipyramids: Tetragonal Pyramids: 1st, 2nd and 3rd order: 4 faces. Positive, the faces converge on and meet at the upper end of the c- axis; and negative, the faces converge and meet on the lower end of the c-axis. A 1st order pyramidal face, {hhl}/{111}, intercepts both a-axes at equal distances and the c-axis. A 2nd order face, {h0l}/{101} intercepts one a-axis, the c- axis, and is parallel to the remaining a- axis. A 3rd order pyramidal face, {hkl}/{121}, intercepts all three axes, each at a different distance. Faces, ideally, are isosceles triangles. 3rd order faces are small and rarely seen, . Ditetragonal Pyramids: 4 faces. There are two of them. Four faces of the positive form converge and meet on the positive end of the c-axis. Those of the negative form converge on the negative end of the c-axis. . Tetragonal dipyramids: 8 faces. 1st, 2nd, and 3rd order. The dipyramids have the same general symbols as the pyramids. . Ditetragonal dipyamids: 16 faces. Eight faces on the upper half and eight on the lower half. Ideally, isosceles triangles. The symbol is {hkl}/{211}.

Prisms & Diprisms: Tetragonal Prisms: 4 faces; parallel to and enclosing the c-axis. An open form (requires other forms to enclose space); 1st order prism {hh0}/{110} intercepts both a-axes equally. The 2nd order prism {h00}/{100) intercepts and is perpendicular to one a- axis and is parallel to the other. The 3rd order prism {hk0}/{120} intercepts both a-axes and is neither parallel to nor perpendicular to either of them (See figure). . Ditetragonal Prism: 8 Faces, surrounding and parallel to the c-axis; each face of a pair intercepts the two a-axes at unequal distances.

Trapezohedrons: Tetragonal Trapezohedron: 8 faces, closed form (the form can enclose space); Left and Right forms result in enantiomorphism (left and right handed crystals).. The trapezohedron has eight 4 sided faces, each with edges not parallel to the others. Disphenoid & Scalenohedron . Tetragonal Disphenoid: 4 faces {hhh}/{111}. Resembles a tetrahedron, but has one longer axis. There is a positive and a negative form. When positive and negative forms appear on the same crystal, they generally are different in size. . Tetragonal Scalenohedron: 8 Faces, each is a scalene triangle. A closed form, but not found itself as a crystal. It is found modifying chalcopyrite crystals. The symbol is {hkl}/{322}. There is a positive and a negative form. Pinacoid & Pedion: . Pinacoid: 2 faces. {001}, perpendicular to the c-axis, one at the positive end and one at the negative end. . Pedion: 1 face, basal, intercepts the c- axis, usually at its lower end.

General Morphology: Of the 300 or so tetragonal minerals, somewhat more than half are prismatic/pyramidal (33%), equant (16%), or tabular (28%). In these cases, when viewing along the c-axis one usually sees an essentially square cross- section. There are relatively few that show an acicular habit, although roughly 10% form radiating groups. According to the International Union for Crystallography:

Apophyllite: Ditetragonal Dipyramidal Class.

Chalcopyrite: Tetragonal Scalenohedral Class.

Wulfenite: Tetragonal Pyramidal Class.

Diaboleite: Tetragonal Dipyramidal Class.

Lemanskiite: Tetragonal Trapezohedral Class. Model: Chalcopyrite

Chalcopyrite: Tetragonal Scalenohedral Class:

Tetragonal Dipyramidal: Model: Wulfenite

Model: Scheelite

THE END.