UNIT V CRYSTAL PHYSICS Lattice – Unit cell – Bravais lattice – Lattice planes – Miller indices – d spacing incubic lattice – Calculation of number of atoms per unit cell – Atomic radius –Coordination number – Packing factor for SC, BCC, FCC and HCP structures –NaCl, ZnS, diamond and graphite structures – Polymorphism and allotropy – Crystal defects – point, line and surface defects- Burger vector.

INTRODUCTION: A solid consists of atoms or clusters of atoms arranged in close proximity. The physical structure of a solid and its properties are closely related to the scheme of arrangement of atoms within the solid.

a) Amorphous solids – the arrangement of atoms in random.

b) Crystalline solids - regular and periodic arrangement of atoms.

Simple geometrical concepts of a lattice and unit cell are used to describe the atomic arrangement in crystals.

Classification of solids:-

Based on the atomic arrangement solids are classified into

a) single crystalsb) polycrystalline solidsc) amorphous solids.

a) Single crystals (example quartz, diamond, alum, rock salt, etc.,) are polyhedrons that have a distinctive shape fro each material. They have smooth faces and straight edges. When a crystal is broken, it cleaves along certain preferred directions.

The same substances can crystallize under different conditions of crystal growth to form different geometrical shapes but the angles between the faces are constant in different shapes. This is known as law of constancy of angles. A large single crystal of regular shape is called a monocrystals. Monocrystals are anisotropic and exhibit difference in the physical properties in different directions. The arrangement of atoms (mostly in 3-dimension) in specific relation to each other is called order. In crystals the order exists in the immediate neighbourhood of a given atom called short range order or over large distances corresponding to several layers of atoms called long range order.

b) Polycrystalline solids consists of fine grains, having a size of -------------------, separated by well defined boundaries and oriented in different directions.Each such grains are oriented haphazardly, a polycrystalline material is isotropic and exhibit the same properties in all directions. Example- metals.

Some materials maintain a fixed volume and shape and resemble solids in their external features, but internally they do not have the ordered crystalline state. Such materials are known as amorphous solids. Example – Glass, rubber and polymers.

Lattice – Unit cell – Bravais lattice – Lattice planes – Miller indices Space Lattice:

The regular pattern of points which describe the three dimensional arrangement of particles (atoms, molecules or ions) in a crystal structure is called the crystal lattice or space lattice.

Unit cell: The smallest portion of a space lattice which can generate the complete crystal

by repeating its own dimensions in various directions is called unit cell.

• The unit cells shown are cubic

o All sides are equal length

o All angles are 90°

• The unit cell need not be cubic

o The unit cell lengths along the x,y, and z coordinate axes are termed the a, b

and c unit cell dimensions

o The unit cell angles are defined as:

a, the angle formed by the b and c cell edges

b, the angle formed by the a and c cell edges

g, the angle formed by the a and b cell edges

Lattice planes:

A set of parallel & equally spaced planes in a space lattice, which are

formed with respect to the points are called lattice planes.

Bravais lattice:

Bravais space lattices 14 types of space lattice are possible.

The 14 lattices are grouped into seven crystal systems. Seven sets of axes are

sufficient to construct the 14 Bravais lattices. This leads to the classification of all

crystals into seven crystal systems.

1. Cubic

2. Tetragonal

3. Ortho – rhombic

4. Monoclinic

5. Triclinic

6. Trigonal(sometimes called rhombohedral)

7. Hexagonal.

P—simple, C-- base centered, I—body centered, F—face centered

Miller indices: Miller indices are the set of the three integers used to indicate the different

crystal planes in a crystal. Miller indices for a plane are always shown in parenthesis. The

general form for indices of a plane (hkl). These h,k,l are proportional to the reciprocals of

the intercepts of the plane on the three coordinate axes.

Important features of Miller indices: When a plane is parallel to any axis, the intercept of the plane on that

Axis is infinity. Hence the Miller index for that axis is zero.

If a plane cuts an axis on the negative side of the origin,

Corresponding index is negative.

The indices (hkl ) do not define a particular plane but a set of parallel

planes. Thus the planes whose intercepts are 1,1,1; 2,2; -3,-3,-3 etc.,

are all represented by the same set of Miller indices.

d’ spacing in cubic lattice :

The separation between lattice planes in a cubic

Crystal d = ___a_______

√(h2 +k2 +l2.)

Calculation of number of atoms per unit cell – Atomic radius –Coordination number – Packing factor for SC, BCC, FCC and HCP structures –

CALCULATION OF NUMBER OF ATOMS PER UNIT CELL, ATOMIC RADIUS,

COORDINATION NUMBER, PACKING FACTOR

ATOMIC RADIUS :

Atomic radius is defined as half the distance between two nearest

neighbouring atoms in a crystal.

CO- ORDINATION NUMBER :

Coordination number is the number of nearest neighbour atoms

that an atom has in the given crystal structure.

PACKING FACTOR :

Atomic packing factor is the ratio of the volume of atoms in the unit cell to

the volume of the unit cell.

SIMPLE CUBIC STRUCTURE (SC)

(i) NUMBER OF ATOMS PER UNIT CELL :

There are 8 atoms at the 8 corners of the cell.

The share of each corner atom to a unit cell is 1/8 of an atom.

Total number of atoms per unit cell is 8* 1/8 = 1.

The Simple Cubic Lattice

(ii) ATOMIC RADIUS : In SC cell the atoms are in contact along the edges of the cube. 2r = a r = a/2 .(iii) COORDINATION NUMBER, CN CN = 6

(iv) PACKING FACTOR: Packing factor = 52 % Simple cubic structure is a loosely packed structure. Polonium is the only element which exhibits this structure

BODY CENTRED STRUCTURE (BCC)

i) NUMBER OF ATOMS PER UNIT CELL : There are 8 atoms at the 8 corners, each shared by 8 unit cells and one atoms at the body centre. The atom at the centre of the body of the cell wholly belongs to the unit cell.

(ii) ATOMIC RADIUS : In this case, atoms touch each other along the diagonal of the cube. Atomic radius r = √3 a/4

iii) COORDINATION NUMBER, CN: CN = 8

(iv) PACKING FACTOR : Packing factor = 68% Thus the packing factor is equal to 68%. So it is not a closely packed structure. Tungsten, Sodium, iron and chromium have this type of structure.

FACE CENTRED CUBIC (FCC) STRUCTURE:

(i) NUMBER OF ATOMS PER UNIT CELL : There are 8 atoms, each shared by 8 unit cells at 8 corners and 6 face centered atoms, each shared by 2 cells.The total number of atoms per unit cell = contribution due to 8 corner atoms + contribution due to 6 face centered atoms. = 1 /8 *8 + 1 /2 *6 = 1+3 = 4

Thus the total number of atoms per unit cell is 4.

ATOMIC RADIUS:

In a FCC cell, atoms are in contact along the face diagonal of the cube

Atomic radius, r = a /2 √2

(iii) COORDINATION NUMBER, CN:

CN = 12

(iv) PACKING FACTOR:

Packing factor = 74 %

When compared with SC and BCC, FCC has the highest packing factor.

HEXAGONAL CLOSE PACKED (HCP) STRUCTURE

There are three layers of atoms in the unit cell.

(i) At the bottom layer, central atom has six nearest neighbor atoms in the same plane.

(ii) Top layer has the same atomic arrangement as the bottom layer.

(iii) At c/2 distance from the bottom layer there is a middle layer containing three atoms.

NUMBER OF ATOMS PER UNIT CELL:

The top layer contains 6 atoms at the corners and one atom at the centre.

The corner atoms are shared between 6 unit cells. Hence 6 atoms contribute only 1 atom

to the unit cell. The central atom is shared between 2 unit cells and therefore contributes

1/2 atom to unit cell.

Thus, the total number of atoms/ unit cell.

= 3 + 3 + 3 = 6

2 2

(ii) ATOMIC RADIUS :

The atoms are in contact along the edges of the hexagon

2r = a ∴ r = a /2

(iii) COORDINATION NUMBER, CN

CN = 12.

(iv) PACKING FACTOR :

Packing factor = 74% Since the density of packing is 74%, it is a close

packed structure. Magnesium, zinc and cadmium crystallize in this structure.

Metallic Crystal Structures

Important properties of the unit cells are

The type of atoms and their radii R.

• cell dimensions (side a in cubic cells, side of base a and height c in HCP) in terms of R.

• n, number of atoms per unit cell. For an atom that is shared with m adjacent unit cells, we only count a fraction of the atom, 1/m.

• CN, the coordination number, which is the number of closest neighbors to which an atom is bonded.

• APF, the atomic packing factor, which is the fraction of the volume of the cell actually occupied by the hard spheres. APF = Sum of atomic volumes/Volume of cell.

Unit Cell n CN a/R APF

SC 1 6 2 0.52

BCC 2 8 4√ 3 0.68

FCC 4 12 2√ 2 0.74

HCP 6 12 0.74

The closest packed direction in a BCC cell is along the diagonal of the cube; in a FCC cell is along the diagonal of a face of the cube.

The crystal structure of sodium chloride

The unit cell of sodium chloride is cubic, and this is reflected in the shape of NaCl crystals

The unit cell can be drawn with either the Na+ ions at the corners, or with the Cl- ions at the corners.

• If the unit cell is drawn with the Na+ ions at the corners, then Na+ ions are are also present in the center of each face of the unit cell

• If the unit cell is drawn with the Cl- ions at the corners, then Cl- ions are are also present in the center of each face of the unit cell

Within the unit cell there must be an equal number of Na+ and Cl- ions.

For example, for the unit cell with the Cl- ions at the center of the faces

• The top layer has (1/8+1/8+1/8+1/8+1/2)=1 Cl- ion, and (1/4+1/4+1/4+1/4)=1 Na+ ion • The middle layer has (1/2+1/2+1/2+1/2)=2 Cl- ions and (1/4+1/4+1/4+1/4+1)=2 Na+

ions • The bottom layer will contain the same as the top or 1 each Cl- and Na+ ions • The unit cell has a total of 4 Cl- and 4 Na+ ions in it. This equals the empirical

formula NaCl.

The crystal structure of diamond is equivalent to a face-centred cubic (FCC) lattice, with a basis of two identical carbon atoms: one at (0, 0, 0) and the other at (1/4, 1/4, 1/4), where the

coordinates are given as fractions along the cube sides. This is the same as two interpenetrating FCC lattices, offset from one another along a body diagonal by one-quarter of its length.

The conventional unit cell is cubic (see diagram), with a side length a0 approximately equal to 3.567 Å (0.3567 nm) at room temperature. From this we can derive a few other quantities.

The C–C bond length d is equal to 1/4 of the cubic body diagonal, so that d = (√3)a0/4 ≈ 1.54 Å. The conventional cell contains the equivalent of 8 whole C atoms. The atomic density is therefore 8/(a0

3) ≈ 8/((3.567×10-10 m)3) ≈ 1.76×1029 atoms/m3

(1.76×1023 atoms/cm3).

Each atom can be thought of as a sphere with a radius of 1/8 of the cubic body diagonal. The packing fraction is therefore [8×(4/3)π((√3)a0/8)3]/[a0

3], which simplifies to (√3)π/16 ≈ 0.34.

Polymorphism and allotropy:

Polymorphism is a physical phenomenon where a material may have more than one crystal structure. A material that shows polymorphism exists in more than one type of space lattice in the solid state. If the change in structure is reversible, then the polymorphic change is known as allotropy. The prevailing crystal structure depends on both the temperature and the external pressure.

One familiar example is found in carbon: graphite is the stable polymorph at ambient conditions, whereas diamond is formed at extremely high pressures.

The best known example for allotropy is iron. When iron crystallizes at 2800 oF it is B.C.C. (δ -iron), at 2554 oF the structure changes to F.C.C. (γ -iron or austenite), and at 1670 oF it again becomes B.C.C. (α -iron or ferrite).

Figure 1. Cooling curve for pure iron. (Allotropic behavior of pure iron)

α -iron (alpha) :

Figure 2. Alpha iron (B.C.C) unit cell

The other name for α -iron is ferrite. This crystal has body centered cubic structure. The unit cell and the micrograph of the crystal are shown in Figures (2) and (3).

Figure 3. Ferrite crystals.

γ -iron (Gamma):

Figure 4. Face centered cubic crystal unit cell.

The other name for γ -iron is austenite. This crystal has face centered cubic (F.C.C) structure. The unit cell and the micrograph of the crystal are shown in Figures (4) and (5).

Figure 5. Austenite crystals.

Crystal defects – point, line and surface defects- Burger vector.

Crystal Defects:

A perfect crystal, with every atom of the same type in the correct position, does not exist. All crystals have some defects. Defects contribute to the mechanical properties of metals. In fact, using the term “defect” is sort of a misnomer since these features are commonly intentionally used to manipulate the mechanical properties of a material. Adding alloying elements to a metal is one way of introducing a crystal defect. Nevertheless, the term “defect” will be used, just keep in mind that crystalline defects are not always bad. There are basic classes of crystal defects:

• point defects, which are places where an atom is missing or irregularly placed in the lattice structure. Point defects include lattice vacancies, self-interstitial atoms, substitution impurity atoms, and interstitial impurity atoms

• linear defects, which are groups of atoms in irregular positions. Linear defects are commonly called dislocations.

• planar defects, which are interfaces between homogeneous regions of the material. Planar defects include grain boundaries, stacking faults and external surfaces.

It is important to note at this point that plastic deformation in a material occurs due to the movement of dislocations (linear defects). Millions of dislocations result for plastic forming operations such as rolling and extruding. It is also important to note that any defect in the regular lattice structure disrupts the motion of dislocation, which makes slip or plastic deformation more difficult. These defects not only include the point and planer defects mentioned above, and also other dislocations. Dislocation movement produces additional dislocations, and when dislocations run into each other it often impedes movement of the dislocations. This drives up the force needed to move the dislocation or, in other words, strengthens the material. Each of the crystal defects will be discussed in more detail .

Point Defects:

Point defects are where an atom is missing or is in an irregular place in the lattice structure. Point defects include self interstitial atoms, interstitial impurity atoms, substitutional atoms and vacancies. A self interstitial atom is an extra atom that has crowded its way into an interstitial void in the crystal structure. Self interstitial atoms occur only in low concentrations in metals because they distort and highly stress the tightly packed lattice structure.

A substitutional impurity atom is an atom of a different type than the bulk atoms, which has replaced one of the bulk atoms in the lattice. Substitutional impurity atoms are usually close in size (within approximately 15%) to the bulk atom. An example of substitutional impurity atoms is the zinc atoms in brass. In brass, zinc atoms with a radius of 0.133 nm have replaced some of the copper atoms, which have a radius of 0.128 nm.

Interstitial impurity atoms are much smaller than the atoms in the bulk matrix. Interstitial impurity atoms fit into the open space between the bulk atoms of the lattice structure. An example of interstitial impurity atoms is the carbon atoms that are added to iron to make steel. Carbon atoms, with a radius of 0.071 nm, fit nicely in the open spaces between the larger (0.124 nm) iron atoms.

Vacancies are empty spaces where an atom should be, but is missing. They are common, especially at high temperatures when atoms are frequently and randomly change their positions leaving behind empty lattice sites. In most cases diffusion (mass transport by atomic motion) can only occur because of vacancies.

Linear Defects – Dislocations:

Dislocations are another type of defect in crystals. Dislocations are areas were the atoms are out of position in the crystal structure. Dislocations are generated and move when a stress is applied. The motion of dislocations allows slip – plastic deformation to occur.

Before the discovery of the dislocation by Taylor, Orowan and Polyani in 1934, no one could figure out how the plastic deformation properties of a metal could be greatly changed by solely by forming (without changing the chemical composition). This became even bigger mystery when in the early 1900’s scientists estimated that metals undergo plastic deformation at forces much smaller than the theoretical strength of the forces that are holding the metal atoms together. Many metallurgists remained skeptical of the dislocation theory until the development of the transmission electron microscope in the late 1950’s. The TEM allowed experimental evidence to be collected that showed that the strength and ductility of metals are controlled by dislocations.

There are two basic types of dislocations, the edge dislocation and the screw dislocation. Actually, edge and screw dislocations are just extreme forms of the possible dislocation structures that can occur. Most dislocations are probably a hybrid of the edge and screw forms but this discussion will be limited to these two types.

EdgeDislocations:The edge defect can be easily visualized as an extra half-plane of atoms in a lattice. The dislocation is called a line defect because the locus of defective points produced in the lattice by the dislocation lie along a line. This line runs along the top of the extra half-plane. The

inter-atomic bonds are significantly distorted only in the immediate vicinity of the dislocation

line.

Screw DislocationsThere is a second basic type of dislocation, called screw dislocation. The screw dislocation is slightly more difficult to visualize. The motion of a screw dislocation is also a result of shear stress, but the defect line movement is perpendicular to direction of the stress and the atom displacement, rather than parallel. To visualize a screw dislocation, imagine a block of metal with a shear stress applied across one end so that the metal begins to rip. This is shown in the upper right image. The lower right image shows the plane of atoms just above the rip. The atoms represented by the blue circles have not yet moved from their original position. The atoms represented by the red circles have moved to their new position in the lattice and have reestablished metallic bonds. The atoms represented by the green circles are in the process of moving. It can be seen that only a portion of the bonds are broke at any given time. As was the case with the edge dislocation, movement in this manner requires a much smaller force than breaking all the bonds across the middle plane simultaneously.

]

CRYSTAL PHYSICS1. Define unit cell? Unit cell is defined as the fundamental elementary pattern or the smallest volume from which the entire crystal is built up by translational repetition in three dimensions. The unit cell is formed by the primitives a,b and c and it reflects the structural properties of the given crystal lattice.

2. Explain Bravais lattices. Bravais lattices are the 14 types of space lattices in the seven systems of rystals.One

triclinic, two monoclinic, four orthorhombic two tetragonal, one hexagonal, one rhombohedral and three cubic lattices are the fourteen Bravais lattices.

3. Define space lattice. An infinite three dimensional array of points showing how atoms or molecules

are arranged in a crystal is known as space lattice array.

4. Define lattice planes? A set of parallel & equally spaced planes in a space lattice, which are formed with respect to the lattice points are called lattice planes

5. Define primitive cell The unit cell formed by the primitives a,b and c having only one lattice point

is called the primitive cell. Example: simple cubic unit cell.

6. What are Miller indices? Miller indices are the set of the three integers used to indicate the different crystal planes in a crystal. Miller indices for a plane are always shown in parenthesis. The general form for indices of a plane is (hkl). These h,k,l are proportional to the reciprocals of the intercepts of the plane on the three coordinate axes.

7. Write down the expression for interplanar spacing for a cubic system in terms of lattice constant?

Interplanar spacing d = a √h2+k2+l2

where a = lattice constant and h,k,l = Miller indices.

8. What is expression for density of the crystal in terms of lattice constant?

Density ‘ρ’ = mass = (nA/N) Volume a3

Where n = number of atoms in the unit cell. A = atomic weight or molecular weight of that crystal and N = Avagadro number.

9. Define coordination number. What are the values of coordination number in S.C. B.C.C.and F.C.C.?Coordination number is the number of nearest neighbor atoms that an atom has in the given crystal structure at a distance 2r where r is the atomic radius. The coordination number for S.C, B.C.C and F.C.C are 6,8 and 12 respectively.

10. Define packing factor? Packing factor is the ratio between the volume occupied by the atoms in the unit cell ‘V’ and total volume of the unit cell’V’. Thus packing factor, P.F = v/V11. Derive the packing factor value for a crystal belonging to simple cubic system? Since there is only one atom in the simple cubic unit cell, P.F = v = 4/3 πr3*1 V a3

But, r a/2

P.F = 4/3π{a3 /8 } a3

= π 6 = 0.52

12. Write down the packing factor of FCC crystal?

Packing factor = Volume of atoms in the unit cell Unit cell volume = 4*(4/3)* πr3

a3

But a = 2√2r P.F. = 16π a 3 = π = 0.74