 # 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
NA = 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) Zeffective (or) Z = 4
• Number of atoms in one B.C.C unit cell Zeff = 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 Zeffective 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

### 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.