Understanding the Crystal Structure
The problem describes a compound with three types of ions: X, Y, and Z.
- The Z ions are arranged in a Face-Centered Cubic (FCC) lattice.
- The X ions occupy a fraction of the tetrahedral voids.
- The Y ions occupy a fraction of the octahedral voids.
Calculating Void Numbers in FCC
In an FCC unit cell, let the number of atoms (Z ions) be denoted by $n$. For FCC, $n=4$.
- Number of octahedral voids = $n = 4$.
- Number of tetrahedral voids = $2n = 2 \times 4 = 8$.
Determining Ion Composition
We can now calculate the number of X and Y ions based on the voids they occupy:
- Number of Z ions = 4 (due to FCC arrangement).
- Number of X ions = $\frac{1}{6}$ of tetrahedral voids = $\frac{1}{6} \times 8 = \frac{8}{6} = \frac{4}{3}$.
- Number of Y ions = $\frac{1}{3}$ of octahedral voids = $\frac{1}{3} \times 4 = \frac{4}{3}$.
Finding the Chemical Formula
The ratio of ions X:Y:Z is $\frac{4}{3} : \frac{4}{3} : 4$. To find the simplest whole number ratio, we multiply the ratio by 3:
Ratio = $(\frac{4}{3} \times 3) : (\frac{4}{3} \times 3) : (4 \times 3)$
Ratio = $4 : 4 : 12$
Now, we simplify this ratio by dividing by the greatest common divisor, which is 4:
Simplified Ratio = $\frac{4}{4} : \frac{4}{4} : \frac{12}{4}$
Simplified Ratio = $1 : 1 : 3$
Therefore, the chemical formula of the compound is $XYZ_3$.
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?
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 ______.
Consider a three-dimensional crystal of $N$ inert gas atoms. The total energy is given by $U(R) = 2N\epsilon \left[p \left(\frac{\sigma}{R}\right)^{12} - q \left(\frac{\sigma}{R}\right)^6\right]$, where $p = 12.13$, $q = 14.45$, and $R$ is the nearest neighbour distance between two atoms. The two constants, $\epsilon$ and $R$, have the dimensions of energy and length, respectively. The equilibrium separation between two nearest neighbour atoms in units of $\sigma$ (rounded off to two decimal places) is ________
