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Chapter 3: Problem 8

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.)

Short Answer

Expert verified
Fractional coordinates after symmetry operations are: (a) For 3-fold: \((x', y', z')\), \((x'', y'', z'')\) (b) For 6-fold: coordinates cycle through all symmetrical positions (c,d) For \(\ref{3}\) and \(\ref{6}\): symmetry operations first invert and then rotate accordance with 3-fold and 6-fold rules respectively

Step by step solution

01

- Understanding Fractional Coordinates and Symmetry Operations

Fractional coordinates are defined within the unit cell. A symmetry operation, such as a rotation, maps a set of coordinates to another set of coordinates within the same unit cell. Understanding how these rotations act on the coordinates is key.
02

- 3-fold Rotation Axis

The 3-fold rotation axis rotates a point by 120° around the z-axis. Let the initial point be \((x, y, z)\). After one rotation, the point becomes: \((x', y', z') = \left(-\frac{1}{2}x - \frac{\sqrt{3}}{2}y, \frac{\sqrt{3}}{2}x - \frac{1}{2}y, z\right)\). Applying another 120° rotation gives \((x'', y'', z'') = \left(-\frac{1}{2}x + \frac{\sqrt{3}}{2}y, -\frac{\sqrt{3}}{2}x - \frac{1}{2}y, z\right)\).
03

- 6-fold Rotation Axis

The 6-fold rotation axis rotates a point by 60° around the z-axis. After one rotation of 60°, the coordinates become: \((x', y', z') = \left(\frac{1}{2}x - \frac{\sqrt{3}}{2}y, \frac{\sqrt{3}}{2}x + \frac{1}{2}y, z\right)\). After two 60° rotations (120°), it becomes as in step 2. Continuing this for 180°, 240°, 300°, and back to 360° shows how the coordinates cycle through all the possible symmetrical positions around this axis.
04

- \(\ref{3}\) Axis

The screw axis combo \(\ref{3}\) means a combined inversion and 3-fold rotation. An inversion at the origin transforms \((x, y, z)\) to \((-x, -y, -z)\). Rotating by 120° around the z-axis changes \((x', y', z')\) according to the same rules as for a simple 3-fold axis but on the inverted coordinates.
05

- \(\ref{6}\) Axis

Similarly, the \(\ref{6}\) axis involves inversion and 6-fold rotation. After inverting coordinates \((x, y, z)\) to \((-x, -y, -z)\), performing 60° rotation gives \((x', y', z') = \left(\frac{1}{2}(-x) - \frac{\sqrt{3}}{2}(-y), \frac{\sqrt{3}}{2}(-x) + \frac{1}{2}(-y), -z\right) = \left(-\frac{1}{2}x + \frac{\sqrt{3}}{2}y, -\frac{\sqrt{3}}{2}x - \frac{1}{2}y, -z\right)\).

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Key Concepts

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

Fractional Atomic Coordinates
Fractional atomic coordinates describe the positions of atoms within a unit cell using fractions of the cell's dimensions.
Unlike Cartesian coordinates, which use absolute distances, fractional coordinates express atom positions relative to the edges of the unit cell (denoted as 0 to 1).
This system simplifies calculations in crystallography and makes it easier to describe periodic structures.
For example, an atom at point \( (x, y, z) \) in fractional coordinates would be positioned at \( x \) times the unit cell length in the x-direction, \( y \) times the unit cell length in the y-direction, and \( z \) times the unit cell length in the z-direction.
Rotation Axes
In crystallography, rotation axes are symmetry elements that involve rotating a point around an axis by a specific angle.
These rotations are based on the unit cell's geometry and dictate how identical points map onto each other.
Common rotation axes include 2-fold, 3-fold, 4-fold, and 6-fold, corresponding to rotations of 180°, 120°, 90°, and 60°, respectively.
The z-axis is often used as the reference for these rotations, which helps visualize how the lattice points rotate around this central line.
Crystallographic Symmetry
Crystallographic symmetry involves operations that map a crystal structure onto itself, ensuring periodicity.
These symmetry operations include rotations, reflections, inversions, and translations.
Each operation leaves the crystal looking indistinguishable from its original form, crucial for classifying crystals into symmetry groups.
Using these symmetry operations, one can predict the arrangement of atoms in a crystal, aiding in the understanding and design of materials.
3-fold Rotation
A 3-fold rotation axis rotates a point by 120° around an axis, often the z-axis.
For a point \( ( x, y, z ) \), the coordinates after a 120° rotation become:
\( ( x', y', z' ) = \left( - \frac{1}{2} x - \frac{\sqrt{3}}{2} y, \frac{\sqrt{3}}{2} x - \frac{1}{2} y, z \right) \).
Another 120° rotation yields: \( ( x'', y'', z'' ) = \left( - \frac{1}{2} x + \frac{\sqrt{3}}{2} y, - \frac{\sqrt{3}}{2} x - \frac{1}{2} y, z \right) \).
These rotations show how the points cycle through symmetrical positions while remaining within the unit cell.
6-fold Rotation
A 6-fold rotation axis involves rotating a point by 60° around the z-axis.
The rotated point after 60° is: \( ( x', y', z' ) = \left( \frac{1}{2} x - \frac{\sqrt{3}}{2} y, \frac{\sqrt{3}}{2} x + \frac{1}{2} y, z \right) \).
This rotation can be repeated to reach 120°, 180°, 240°, 300°, and back to 360°, covering all possible symmetrical positions.
These multiple rotations ensure that the structure maintains its periodic symmetry, a key aspect in crystallographic studies.
Screw Axis
A screw axis combines a rotation with a translation along the axis.
In crystallography, a screw axis \( n \) involves a rotation by \( \frac{360}{n} \) degrees and a simultaneous fractional translation along the axis.
For example, a \( 3_1 \) screw axis rotates a point by 120° and translates it by \ \frac{1}{3} \ of the unit cell length along the axis.
These combined operations are crucial for describing complex periodic structures and enhancing the understanding of crystalline symmetries.

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Most popular questions from this chapter

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.)

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.)

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