Show that the presence of a plane of reflection at \(x=0\) in a crystal of orthorhombic symmetry will be accompanied by a mirror plane at \(x=1 / 2\). Similarly show that if there is a plane of reflection at \(x=1 / 4\) there will also be a plane of reflection at \(x=3 / 4 .\) (Hint: You will need to recognise the role of translational symmetry.)

Orthorhombic symmetry is a fundamental concept in crystallography. It refers to a crystal system where the crystal shows three mutually perpendicular axes with different lengths. Imagine a box where the edges are all at right angles, but each of the three dimensions—the length, width, and height—are distinct.
In simpler terms, think of it as a slightly stretched or compressed cube. Orthorhombic crystals have specific symmetry properties. These include a variety of reflective and rotational symmetries.
One key feature is the presence of translational symmetry along these three perpendicular axes. This symmetry helps predict the presence of additional symmetry elements, like reflection planes.
Understanding orthorhombic symmetry aids in visualizing how certain symmetry elements appear repeatedly within the crystal structure. This repetitive nature allows us to identify and confirm the presence of reflection planes in specific positions like at \(x=1/2\) if there is one at \(x=0\).

Translational symmetry is the concept that you can shift the entire crystal lattice by a specific distance in any direction, and it will still look the same. This symmetry is crucial in understanding how reflection planes can appear at predictable intervals.
For example, in the exercise, if there is a reflection plane at \(x=0\), translational symmetry tells us that every unit cell will be a repeat of the first. By shifting the crystal by half a unit cell (\(1/2\) along the x-axis), we identify another reflection plane at \(x=1/2\).
Similarly, if there is a reflection plane at \(x=1/4\), shifting by \(1/2\) will result in another reflection plane at \(x=3/4\). This behavior is a direct consequence of the crystal lattice repeating itself.
In simpler terms, translational symmetry helps us deduce that if a crystal has a repeating structure, it also has repeating symmetry elements. This reduces the complexity of understanding crystal structures, making it easier to predict the occurrence of reflective planes across the lattice.

Mirror plane reflection is when a plane divides the crystal such that one half is the mirror image of the other. In other words, for any point \((x, y, z)\) on one side, there is a corresponding point \((-x, y, z)\) on the other side.
For instance, if there’s a reflection plane at \(x=0\), any point on one side of this plane has an identical point mirrored across to the other side. If we establish that there is such a plane at \(x=0\) in a crystal with orthorhombic symmetry, we can use translational symmetry to identify other mirror planes.
This means reflecting the entire structure by a translational amount, like \(1/2\), will place a corresponding mirror plane at \(x=1/2\). Similarly, a reflection plane at \(x=1/4\) implies a plane at \(x=3/4\) due to the translation along the x-axis by \(1/2\).
Mirror plane reflections are straightforward yet powerful tools to deduce symmetry properties within crystal structures, making it easier to predict and understand complex lattice behavior.