Calculate the lengths of intercepts made by planes of Miller indices (111) and (123) in \(\mathrm{BaCl}_{2}\) crystal with primitive translation vectors \(a=6.69 \AA, b=10.86 \AA\) and \(c=7.15 \AA\), if both the planes cut an intercept \(7.15 \AA\) along \(z\) -axis. $$ \begin{aligned} &\text { Ans. (i) } l_{1}=6.69 \AA, l_{2}=10.86 \AA, l_{3}=7.15 \AA \text { and } \\ &\text { (ii) } l_{1}=20.07 \AA, l_{2}=16.29 \AA, l_{3}=7.15 \AA \end{aligned} $$

Crystallographic planes are critical when studying the atomic structure of crystals. These planes are represented by Miller indices, such as (111) or (123). The Miller indices (hkl) are derived from the reciprocals of the intercepts that the plane makes with the crystallographic axes.
For example, the Miller indices (111) indicate a plane that intercepts each of the x, y, and z axes equally. In contrast, the (123) plane intercepts the x-axis at 1 unit, the y-axis at 2 units, and the z-axis at 3 units.
Understanding these intercepts and Miller indices is crucial for calculating distances, angles, and positions within the crystal lattice. It also helps in determining the orientation of different planes in the crystal structure.

Primitive translation vectors are fundamental building blocks in a crystal's lattice structure. Commonly denoted as **a, b,** and **c**, they define the smallest repeating unit cell in the crystal.
For a given crystal, such as \(\text{BaCl}_2\), the primitive translation vectors might be defined as:
* a = 6.69 Å
* b = 10.86 Å
* c = 7.15 Å
These vectors help in calculating the intercepts of crystallographic planes. By expressing the length of each intercept using the primitive translation vectors, one can understand the spatial arrangement of atoms in the crystal better.
In the context of Miller indices, these vectors provide a basis for translating abstract numerical indices into physical planes within the crystal.

The concept of reciprocals of intercepts is foundational for understanding Miller indices in crystallography. To find the Miller indices (hkl) for any plane, you follow these steps:
1. Determine where the plane intersects the crystallographic axes (x, y, z). The intercepts can be expressed as multiples of the primitive translation vectors **a, b,** and **c**.
2. Take the reciprocals of these intercepts (if an intercept is at infinity, the reciprocal is zero).
3. Simplify to the smallest set of integers to get the Miller indices.
For instance, if we have intercepts of 1a, 2b, and 3c, the Miller indices would be (1,2,3).
This method helps in converting a plane's position in space into a straightforward numerical form, making it easier to work with and analyze crystal structures.