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
          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

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.