Crystal Structure Analysis
This solution calculates the ratio of interplanar spacings for the (101) plane when a simple cubic structure transforms into a tetragonal structure.
Interplanar Spacing Formulas
The formula for interplanar spacing ($d_{hkl}$) varies by crystal system:
- Cubic: $d_{hkl} = \frac{a_c}{\sqrt{h^2 + k^2 + l^2}}$
- Tetragonal: $d_{hkl} = \frac{1}{\sqrt{\frac{h^2}{a_t^2} + \frac{k^2}{b_t^2} + \frac{l^2}{c_t^2}}}$ (where $a_t = b_t$)
Cubic (101) Plane Spacing
Calculate the spacing for the $(101)$ plane in the cubic phase.
- Given: Lattice parameter $a_c$. Plane $(101)$.
- $d_{101, cubic} = \frac{a_c}{\sqrt{1^2 + 0^2 + 1^2}} = \frac{a_c}{\sqrt{2}}$
Tetragonal (101) Plane Spacing
Calculate the spacing for the $(101)$ plane in the tetragonal phase.
- Given: Lattice parameters $a_t = b_t = \sqrt{2} a_c$ and $c_t = 2 a_c$. Plane $(101)$.
- $d_{101, tetragonal} = \frac{1}{\sqrt{\frac{1^2}{a_t^2} + \frac{0^2}{b_t^2} + \frac{1^2}{c_t^2}}} = \frac{1}{\sqrt{\frac{1}{a_t^2} + \frac{1}{c_t^2}}}$
- Substitute parameters: $d_{101, tetragonal} = \frac{1}{\sqrt{\frac{1}{(\sqrt{2} a_c)^2} + \frac{1}{(2 a_c)^2}}} = \frac{1}{\sqrt{\frac{1}{2 a_c^2} + \frac{1}{4 a_c^2}}}$
- Simplify: $d_{101, tetragonal} = \frac{1}{\sqrt{\frac{2+1}{4 a_c^2}}} = \frac{1}{\sqrt{\frac{3}{4 a_c^2}}} = \frac{1}{\frac{\sqrt{3}}{2 a_c}} = \frac{2 a_c}{\sqrt{3}}$
Ratio Calculation
Determine the ratio $\frac{d_{101, cubic}}{d_{101, tetragonal}}$.
- Ratio = $\frac{a_c / \sqrt{2}}{2 a_c / \sqrt{3}}$
- Ratio = $\frac{a_c}{\sqrt{2}} \times \frac{\sqrt{3}}{2 a_c}$
- Cancel $a_c$: Ratio = $\frac{\sqrt{3}}{2 \sqrt{2}} = \sqrt{\frac{3}{2^2 \times 2}} = \sqrt{\frac{3}{8}}$
The calculated ratio of the interplanar spacings for the (101) planes is $\sqrt{\frac{3}{8}}$.
Important Questions from Crystal Structure Bravais Lattices Unit Cell
- For a two-dimensional hexagonal lattice with lattice constant $ a $, the atomic density is
- Consider a crystal that has a basis of one atom. Its primitive vectors are $ \vec{a_1} = a\hat{i} $, $ \vec{a_2} = a\hat{j} $, $ \vec{a_3} = \frac{a}{2}(\hat{i} + \hat{j} + \hat{k}) $, where $ \hat{i}, \hat{j}, \hat{k} $ are the unit vectors in the $ x, y $ and $ z $ directions of the Cartesian coordinate system and $ a $ is a positive constant. Which one of the following is the correct option regarding the type of the Bravais lattice?
- A compound consists of three ions X, Y and Z. The Z ions are arranged in an FCC arrangement. The X ions occupy $\frac{1}{6}$ of the tetrahedral voids and the Y ions occupy $\frac{1}{3}$ of the octahedral voids. Which one of the following is the CORRECT chemical formula of the compound?
For the given unit cells of a two dimensional square lattice, which option lists all the primitive cells?
The number of distinct ways the primitive unit cell can be constructed for the two dimensional lattice as shown in the figure is ______.
