Inter-Planar Spacing Basics

The inter-planar spacing, denoted by $d$, refers to the distance between adjacent parallel planes of atoms within a crystal lattice.

Relationship Between Indices and Spacing

The spacing $d$ for a set of planes $\{hkl\}$ is related to the Miller indices $(h, k, l)$. Consider a specific set of planes $\{hkl\}$ with inter-planar spacing $d$.

Now, consider a new set of planes denoted by $\{nh, nk, nl\}$, where $n$ is an integer. These planes represent points or lattice planes that are further apart or closer together, depending on the value of $n$ and the original indices.

Deriving the Spacing for {nh nk nl} Planes

In crystallography, the inter-planar spacing $d_{hkl}$ is inversely proportional to the magnitude of the reciprocal lattice vector $\mathbf{G}_{hkl}$. Mathematically, $d_{hkl} \propto \frac{1}{|\mathbf{G}_{hkl}|}$.

• The reciprocal lattice vector for planes $\{hkl\}$ is $\mathbf{G}_{hkl}$.
• The reciprocal lattice vector for planes $\{nh, nk, nl\}$ is $\mathbf{G}_{nh, nk, nl}$.
• It's established that $\mathbf{G}_{nh, nk, nl} = n \mathbf{G}_{hkl}$.
• Therefore, the magnitude relationship is $|\mathbf{G}_{nh, nk, nl}| = |n \mathbf{G}_{hkl}| = n |\mathbf{G}_{hkl}|$.

Final Calculation

Let $d'$ be the inter-planar spacing for the planes $\{nh, nk, nl\}$. Since spacing is inversely proportional to the magnitude of the reciprocal lattice vector:

$ d' \propto \frac{1}{|\mathbf{G}_{nh, nk, nl}|} = \frac{1}{n |\mathbf{G}_{hkl}|} $

Comparing this to the original spacing $d \propto \frac{1}{|\mathbf{G}_{hkl}|}$, we find:

$ d' = \frac{1}{n} \left( \frac{1}{|\mathbf{G}_{hkl}|} \right) \propto \frac{d}{n} $

Thus, the inter-planar spacing for the planes $\{nh, nk, nl\}$ is $d/n$. For example, if $n=2$, the spacing is halved.