## Solid State

# Calculations involving unit cell parameters and Imperfection in solids

- Cell parameters: A unit cell can be characterised by two types of parameters

(i) It's dimension along three edges a, b and c

(ii) angles between the edges α (between b & c)

β (between a & c)

γ (between a & b) - It can be represented as follows.
- Calculations involving unit cell parameters:

Let a = length of edge of a unit cell

r = radius of atom (point)

rank of unit cell ← z = Number of atoms in unit cell

d = density of unit cell

N_{A}= Avogadro's number

M = Molecular mass of unit cell - Contribution of an atom to unit cell from corner of a cubic crystal lattice is "\frac{1}{8}" since atom at corner of cubic unit cell shared by 8 neighboring unit cells.
- Contribution of an atom to unit cell from face centre is \frac{1}{2} since atom at face centre is shared by two neighboring unit cells.
- Contribution of an atom to unit cell from body centre is 1 since one full atom at centre of unit cell.
- Contribution of an atom to unit cell from edge centre is \frac{1}{4} since atom at edge centre is shared by four neighboring unit cells.
- Number of atoms in primitive cubic unit cell is 1

Number of atoms = 8 \times\frac{1}{8}=1 - Number of atoms in one F.C.C unit cell is Z = 4

Z_{(effective)}= From all the corners = 8 \times\frac{1}{8}=1

From all the face centres = 6 \times\frac{1}{2}=\frac{3}{4}

Rank (or) Z_{effective}(or) Z = 4- Number of atoms in one B.C.C unit cell Z
_{eff}= 2

Z_{(effective)}= From all the corners = 8 \times\frac{1}{8}=1

From body centres = 1 \times 1=\frac{1}{2}- Trick to calculate Z
_{effective}for cubic unit cell.

Z_{eff}=\frac{C}{8}+\frac{B}{1}+\frac{F}{2}+\frac{E}{4}

C = Number of atoms from all corner

B = Number of atoms from body centre

F = Number of atoms from all available faces

E = Number of atoms from all available edges - Density of a unit cell can be calculated by using following formula d=\frac{ZM}{N_{A}a^{3}}
- Relations between "a" and "r" of unit cells

(1) For F.C.C atoms touch along face diagonal - \therefore 4r = \sqrt{2}a

a = 2\sqrt{2}r - For B.C.C atoms touch along body diagonal
- \therefore 4r = \sqrt{3}a

a = \frac{4}{\sqrt{3}}r - For simple cubic atoms touch along edge length
- ∴ a = 2r

### Part1: View the Topic in this Video from 0:07 to 12:08

### Part2: View the Topic in this Video from 0:08 to 10:57

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Bragg's equation, nλ = 2d sinθ

where, n = 1, 2, 3......(diffraction order)

λ = wavelength of X - rays incident on crystal and

d = distance between atomic planes

θ = angle at which interference occurs.