UNIT 1.2 THE SOLID STATE: CRYSTAL LATTICE & UNIT CELL

LEC 2            CRYSTAL LATTICE & UNIT CELL

all crystal have a regular repetition of some constituent particles (atoms, molecules or ions). to represent the relative arrangement of these particles in a crystal, each particles (atom, molecule or ion) is considered as a point, position taken by these particles in three dimension crystal are called as lattice points or lattice sites.

the representation of crystal in which the location of the ions or atoms are shown by lattice points is called crystal lattice or space lattice. space lattice or crystal lattice can be define as an array of lattice points showing the arrangement of constituent particles in different positions in three dimensional space. There are only 14 possible three dimensional lattices. These are called Bravais Lattices. The following are the characteristics of a crystal lattice:

(a) Each point in a lattice is called lattice point or lattice site.

(b) Each point in a crystal lattice represents one constituent particle which may be an atom, a molecule (group of atoms) or an ion.

(c) Lattice points are joined by straight lines to bring out the geometry of the lattice.

in space lattice, a certain group of lattice points set the pattern of whole lattice. the smallest group of lattice points is called unit cell. so the Unit Cell can be defined as the smallest portion of the crystal lattice which define completely the repeating pattern in the crystal in all directions.

A unit cell is characterised by:

(i) its dimensions along the three edges, a, b and c. These edges may or may not be mutually perpendicular.

(ii) angles between the edges, α (between b and c) β (between a and c) and γ (between a and b). 

Thus, a unit cell is characterised by six parameters, a, b, c, α, β and γ.

Unit cells can be broadly divided into two categories:

1) Primitive / simple Unit Cells:  When constituent particles are present only on the corner positions of a unit cell, it is called as primitive unit cell.

2) Centred Unit Cells: When a unit cell contains one or more constituent particles present at positions other than corners in addition to those at corners, it is called a centred unit cell. 

Centred unit cells are of three types:

(i) Body-Centred Unit Cells: Such a unit cell contains one constituent particle (atom, molecule or ion) at its body-centre besides the ones that are at its corners.

(ii) Face-Centred Unit Cells: Such a unit cell contains one constituent particle present at the centre of each face, besides the ones that are at its corners.

(iii) End-Centred Unit Cells: In such a unit cell, one constituent particle is present at the centre of any two opposite faces besides the ones present at its corners.

On the basis of geometrical consideration edge length and axial angle (a, b, c, ∝, β, γ), main seven crystal system system are present also known as seven primitive Unit Cell and on the basis of arrangement of particles in primitive or centred unit cell.

Number of Atom in a Unit Cell & Packaging Efficiency

we know that any crystal lattice is made up of a very large number of unit cell and every lattice point is occupied by one constituent particles (atom, molecule or ion).
In whatever way the constituent particles are packed there is always some free space in the form of voids. Packaging Efficiency is the percentage of total space filled by the Particles in a unit cell.
 now we are going to study how many atoms are present in different type of unit cell and what is the Packaging Efficiency of the different Structure.

1) Primitive Cubic Unit Cell:

In primitive Unit cell has atoms only at its corner. each atom at a corner is shared by eight adjacent unit cell and a unit cell has 8 corners so

Now for packing efficiency lets consider atom is a spherical type of radius "r" and unit cell is a cube of edge length "a" then

2) Body Centred Unit Cell

A body-centred cubic (BCC) unit cell has an atom at each of its corners and also one atom at its body centre, It can be seen that the atom at the body centre wholly belongs to the unit cell in which it is present.

The atom at the centre will be in touch with the other two atoms diagonally arranged as in fig shows

3) Face Centred Cubic Unit Cell (FCC)

A face centred cubic (fcc) unit cell contains atoms at all the corners and at the centre of all the faces of the cube and each atom located at the face centre is shared between two adjacent unit cells.
No of atoms present in one unit cell is given as

now packing Efficiency is given as

it means that FCC structure have maximum Efficiency of packaging.

Packing of Constituent in Metallic Crystal

In the formation of Crystals, the constituent particles may be atoms, ions or molecules. these constituent particles may be of different size and so it may give different mode of particles in the crystal.
The concept of packing of constituent in metallic crystal is different from Ionic crystal but in this class we only study about metallic crystal according to syllabus.
The number of nearest neighbours of a particle is called its coordination number. Let us consider the constituent particles as identical hard spheres and build up the three dimensional structure in three steps.

(a) Close Packing in One Dimension

In this arrangement, each sphere is in contact with two of its neighbours. The number of nearest neighbours of a particle is called its coordination number. Thus, in one dimensional close packed arrangement, the coordination number is 2.

(b) Close Packing in Two Dimensions

Two dimensional close packed structure can be generated by stacking (placing) the rows of close packed spheres. This can be done in two different ways.

(i) The second row may be placed in contact with the first one such that the spheres of the second row are exactly above those of the first row. The spheres of the two rows are aligned horizontally as well as vertically. If we call the first row as ‘A’ type row, the second row being exactly the same as the first one, is also of ‘A’ type. Similarly, we may place more rows to obtain AAA type of arrangement.

The two dimensional coordination number is 4.if the centres of these 4 immediate neighbouring spheres are joined, a square is formed. Hence this packing is called square close packing in two dimensions.

(ii) The second row may be placed above the first one in a staggered manner such that its spheres fit in the depressions of the first row. If the arrangement of spheres in the first row is called ‘A’ type, the one in the second row is different and may be called ‘B’ type. When the third row is placed adjacent to the second in staggered manner, its spheres are aligned with those of the first layer. Hence this arrangement is of ABAB type. In this arrangement there is less free

space and this packing is more efficient than the square close packing. Each sphere is in contact with six of its neighbours and the two dimensional coordination number is 6. The centres of these six spheres are at the corners of a regular hexagon hence this packing is called two dimensional hexagonal close packing.

The triangular voids are of two different types. In one row, the apex of the triangles are pointing upwards and in the next layer downwards.

c) Close Packing in Three Dimensions

All real structures are three dimensional structures. They can be obtained by stacking two dimensional layers one above the other. Let us see what types of three dimensional close packing can be obtained from these.

(i) Three dimensional close packing from two dimensional square close-packed layers: While placing the second square close-packed layer above the first we follow the same rule that was followed when one row was placed adjacent to the other. The second layer is placed over the first layer such that the spheres of the upper layer are exactly above those of the first layer. this lattice has AAA.... type pattern. The lattice thus generated is the simple cubic lattice, and its unit cell is the primitive cubic unit cell.

(ii) Three dimensional close packing from two dimensional hexagonal close packed layers: Three dimensional close packed structure can be generated by placing layers one over the other.

(a) Placing second layer over the first layer: Let us take a two dimensional hexagonal close packed layer ‘A’ and place a similar layer above it such that the spheres of the second layer are placed in the depressions of the first layer. Since the spheres of the two layers are aligned differently, let us call the second layer as B. It can be observed that not all the triangular voids of the first layer are covered by the spheres of the second layer. This gives rise to different arrangements. Wherever a sphere of the second layer is above the void of the first layer (or vice versa) a tetrahedral void is formed. These voids are called tetrahedral voids because a tetrahedron is formed when the centres of these four spheres are joined. They have been marked as ‘T’.

At other places, the triangular voids in the second layer are above the triangular voids in the first layer, and the triangular shapes of these do not overlap. One of them has the apex of the triangle pointing upwards and the other downwards. These voids have been marked as ‘O’ Such voids are surrounded by six spheres and are called octahedral voids. The number of these two types of voids depend upon the number of close packed spheres.

Let the number of close packed spheres be N, then:

The number of octahedral voids generated = N

The number of tetrahedral voids generated = 2N

(b) Placing third layer over the second layer When third layer is placed over the second: there are two possibilities.

(i) Covering Tetrahedral Voids: Tetrahedral voids of the second layer may be covered by the spheres of the third layer. In this case, the spheres of the third layer are exactly aligned with those of the first layer. Thus, the pattern of spheres is repeated in alternate layers. This pattern is often written as ABAB ....... pattern. This structure is called hexagonal close packed (hcp) structure. This sort of arrangement of atoms is found in many metals like magnesium and zinc.

(ii) Covering Octahedral Voids: The third layer may be placed above the second layer in a manner such that its spheres cover the octahedral voids. When placed in this manner, the spheres of the third layer are not aligned with those of either the first or the second layer. This arrangement is called “C’ type. Only when fourth layer is placed, its spheres are aligned with those of the first layer as shown in Figs. 1.18 and 1.19. This pattern of layers is often written as ABCABC ........... This structure is called cubic close packed (ccp) or face-centred cubic (fcc) structure. Metals such as copper and silver crystallise in this structure. 

Both these types of close packing are highly efficient and 74% space in the crystal is filled. In either of them, each sphere is in contact with twelve spheres. Thus, the coordination number is 12 in either of these two structures.

Formula of a Compound and Number of Voids Filled

In ionic solids, the bigger ions (usually anions) form the close packed structure and the smaller ions (usually cations) occupy the voids. If the latter ion is small enough then tetrahedral voids are occupied, if bigger, then octahedral voids. Not all octahedral or tetrahedral voids are occupied. In a given compound, the fraction of octahedral or tetrahedral voids that are occupied, depends upon the chemical formula of the compound. for example

Problem: Atoms of element B form hcp lattice and those of the element A occupy 2/3rd of tetrahedral voids. What is the formula of the compound formed by the elements A and B?

Solution: The number of tetrahedral voids formed is equal to twice the number of atoms of element B and only 2/3rd of these are occupied by the atoms of element A.
No of atom of B at HCP lattice = N
No of Tetrahedral Voids = 2N
only 2/3 Tetrahedral voids are accupied by A then No of atom A = 2N x 2/3 = 4N/3
Ratio of A and B is  4:3 hence the formula of the compound is A4B3.