The correct relationship among the following, for a tetragonal (a = b ≠ c ; α = β = γ = 90°) crystal system, is
Tetragonal Crystal System Properties
A tetragonal crystal system is characterized by specific lattice parameters and angles. The unit cell has sides of length a, b, and c, and interaxial angles α, β, and γ. For a tetragonal system, the relationships are:
- Lattice parameters: a = b ≠ c
- Interaxial angles: α = β = γ = 90°
This means the base of the unit cell is a square, and the height is different. All angles between the axes are right angles.
Bragg's Law and Interplanar Spacing
Bragg's Law relates the angle of diffraction (θ), the wavelength of the incident X-rays (λ), and the distance between crystallographic planes (interplanar spacing, \(d_{hkl}\)) for a given set of Miller indices (h, k, l). The general form is:
\[n\lambda = 2d_{hkl} \sin\theta\]For first-order diffraction (n = 1), this simplifies to:
\[\lambda = 2d_{hkl} \sin\theta\]We can rearrange this equation to express \(\sin^2\theta\):
\[\sin^2\theta = \frac{\lambda^2}{4d_{hkl}^2}\]To find the relationship specific to a tetragonal system, we need the formula for \(d_{hkl}^2\) in this system.
Interplanar Spacing in a Tetragonal System
The formula for the reciprocal of the square of the interplanar spacing, \(1/d_{hkl}^2\), for an orthogonal crystal system (which includes cubic, tetragonal, and orthorhombic systems, as all angles are 90°) is:
\[\frac{1}{d_{hkl}^2} = \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2}\]where h, k, and l are the Miller indices of the plane.
For a tetragonal system, we know that a = b. Substituting a for b in the formula:
\[\frac{1}{d_{hkl}^2} = \frac{h^2}{a^2} + \frac{k^2}{a^2} + \frac{l^2}{c^2}\]Combine the terms with the same denominator:
\[\frac{1}{d_{hkl}^2} = \frac{h^2+k^2}{a^2} + \frac{l^2}{c^2}\]To combine the terms on the right side, find a common denominator, which is \(a^2c^2\):
\[\frac{1}{d_{hkl}^2} = \frac{c^2(h^2+k^2)}{a^2c^2} + \frac{a^2l^2}{a^2c^2}\] \[\frac{1}{d_{hkl}^2} = \frac{c^2(h^2+k^2) + a^2l^2}{a^2c^2}\]Relationship among θ, λ, and Lattice Parameters
Now substitute this expression for \(1/d_{hkl}^2\) back into the rearranged Bragg's Law formula \(\sin^2\theta = \frac{\lambda^2}{4d_{hkl}^2}\):
\[\sin^2\theta = \frac{\lambda^2}{4} \times \frac{1}{d_{hkl}^2}\] \[\sin^2\theta = \frac{\lambda^2}{4} \times \left( \frac{c^2(h^2+k^2) + a^2l^2}{a^2c^2} \right)\]Rearranging the terms gives the final relationship for a tetragonal crystal system:
\[\sin^2\theta = \frac{\lambda^2}{4 a^2 c^2} \left[ c^2(h^2+k^2) + a^2l^2 \right]\]Comparing with Options
Let's compare our derived formula with the given options:
- sin2θ = \(\frac{\lambda^2}{4 a^2}\left[c^2\left(h^2+k^2\right)+a^2 \ell^2\right]\) - Incorrect denominator.
- sin2 θ = \(\frac{\lambda^2}{4 a^2 c^2}\left[c^2\left(h^2+k^2\right)+a^2 l^2\right]\) - Matches the derived formula.
- sin2 θ = \(\frac{\lambda^2}{4 c^2}\left[a^2\left(h^2+k^2\right)+c^2 l^2\right]\) - Incorrect structure inside the bracket.
- sin2 θ = \(\frac{\lambda^2}{4 a^2}\left[h^2+k^2+\ell^2\right]\) - This formula is for a cubic system (where a=b=c).
The derived relationship matches Option 2.
Important Questions from Solid State
Which of the following is molecular solid?
Which of the following equations represents Bragg’s law?
The number of cation vacancies per mole, when NaCl is doped with 10−3 mol% of BCl3, is
Separation of {123} and {246} planes of an orthorhombic unit cell with a = 0.82 nm, b = 0.94 nm and c = 0.75 nm are __________ and _________.
Materials possessing both amorphous and polycrystalline behaviour with abrupt changes in crystal orientation across individual grain boundaries in the extended lattice are ________.
