Animated Solution for Chemistry - States of Matter: Which primitive unit cell has unequal edge lengths ($a\ne b\ne c$) and all axial angles different from $90^{\circ }$?

Imagine you are given a set of magical building blocks to construct a crystal lattice. Depending on how you stretch, compress, or tilt these blocks, you can create seven fundamental geometric shapes. These are known in solid-state chemistry as the seven primitive crystal systems.

We define the geometry of these unit cells using six parameters: three edge lengths ($a$, $b$, and $c$) and the three axial angles between them ($\alpha$, $\beta$, and $\gamma$).

Symmetry is a beautiful and defining characteristic of crystalline solids. At the absolute peak of symmetry sits the Cubic system. In a cubic unit cell, everything is perfectly balanced: all sides are equal ($a=b=c$), and all angles are exactly right angles ($\alpha =\beta =\gamma =90^{\circ }$).

But what happens if we take that perfect cube and completely crush it? What if we stretch every single side to a different length and tilt every single angle so that nothing is straight anymore?

Look closely at the parameters given in our question. We are told that the edge lengths are completely unequal:

This means our building block is not a cube, nor is it a regular rectangular box. It is stretched unevenly in all three spatial dimensions.

Furthermore, the question states that all axial angles are different from $90^{\circ }$:

There are absolutely no right angles anywhere in this structure. It is completely skewed and tilted in every possible direction. This total lack of equality in both edges and angles makes it the most unsymmetrical crystal system possible.

The system that fits this chaotic, completely asymmetrical description is the Triclinic system. The prefix 'tri' means three, and 'clinic' refers to an incline or tilt—perfectly describing a unit cell that is tilted in all three directions.

For a quick revision of the other options: Hexagonal: $a = b eq c$, with $\alpha =\beta =90^{\circ }$ and $\gamma =120^{\circ }$. Monoclinic: $a eq b eq c$, but it retains some symmetry with $\alpha =\gamma =90^{\circ }$ and $\beta eq 90^\circ$. Tetragonal:* $a = b eq c$, with all angles perfectly straight at $\alpha =\beta =\gamma =90^{\circ }$.

Therefore, the correct answer is definitively the Triclinic system.