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Chapter 3: Problem 2

Show that if there is a centre of symmetry at \(0,0,0\) in an orthorhombic crystal there will also be centres of symmetry at seven other positions in the unit cell. (Hint: You will need to recognise the role of planes of reflection.)

Short Answer

Expert verified
The centers of symmetry in an orthorhombic crystal are at \((0,0,0)\), \((a,0,0)\), \((0,b,0)\), \((0,0,c)\), \((a,b,0)\), \((a,0,c)\), \((0,b,c)\), and \((a,b,c)\).

Step by step solution

01

Understand the Concept of a Center of Symmetry

A center of symmetry in a crystal is a point where any vector drawn from this point has an equivalent vector in the opposite direction at an equal distance. When a center of symmetry exists at the origin \((0,0,0)\), the crystal is invariant under inversion through this point.
02

Identify Reflection Planes

In an orthorhombic crystal, planes of reflection are key to finding additional centers of symmetry. Identify the reflection planes along the x, y, and z axes.
03

Reflect Through Each Plane Individually

Reflect the center of symmetry at \((0,0,0)\) through the plane perpendicular to the x-axis at zero, which gives another center at \((a,0,0)\). Similarly, reflect through the yz-plane and xz-plane to get centers at \((0,b,0)\) and \((0,0,c)\).
04

Reflect Through Combinations of Planes

Next, reflect through combinations of planes, such as the ones perpendicular to both x and y axes, to get centers at \((a,b,0)\), after reflecting \(a,0,0\) through the yz-plane, and so on.
05

Confirm the Total Number of Symmetry Centers

Finally, combine all reflections to identify all the unique centers. The eight positions are: \((0,0,0)\), \((a,0,0)\), \((0,b,0)\), \((0,0,c)\), \((a,b,0)\), \((a,0,c)\), \((0,b,c)\), and \((a,b,c)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Crystal Symmetry
Crystal symmetry refers to the repetitive and regular arrangement of atoms in a crystal that leads to certain symmetrical properties. In a symmetrical crystal, identical patterns can be found in different orientations. The highest level of symmetry includes operations that leave the crystal looking indistinguishable from its original form.
Reflection Planes
Reflection planes are imaginary planes within a crystal that divide it into symmetrical halves. When you reflect the crystal across these planes, the resulting image mirrors the original. Reflection planes are central to identifying additional symmetry centers. In the process of finding all centers of symmetry in an orthorhombic crystal, these planes help generate new positions by reflecting existing ones.
Orthorhombic Crystal Structure
The orthorhombic crystal structure has three distinct axes that differ in length but intersect at 90-degree angles. This structure forms the basis of our given problem. Within the orthorhombic unit cell, there are multiple symmetrical operations possible, thanks to its defined reflection planes along the x, y, and z axes.
Symmetry Operations in Crystals
Symmetry operations in crystals are mathematical actions that move each atom in a crystal structure to another position in such a way that the final structure is indistinguishable from the original. These can include reflections, rotations, inversions, and translations. In our specific case with an orthorhombic crystal, we performed a series of reflections through the crystal's planes to identify additional symmetry centers.

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Most popular questions from this chapter

Show that the point group \(\mathrm{mmm}\) contains three orthogonal 2-fold axes.

List the set of fractional atomic coordinates related by the following symmetry operations, assuming each passes through the origin and is oriented parallel to \(z\) : (a) 3 -fold rotation axis; (b) 6 -fold rotation axis; (c) \(\overline{3}\) axis; (d) \(\overline{6}\) axis. (Note: in each case use the axes of the lattice, for which the angle between \(x\) and \(y\) is \(120^{\circ}\), rather than orthogonal axes.)

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.)
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