Model Answer
0 min readIntroduction
Crystallography, the science dealing with the arrangement of atoms in crystalline solids, relies heavily on the concept of symmetry. Symmetry elements are geometric entities (planes, axes, and centers) about which symmetry operations can be performed, leaving the crystal unchanged. The isometric system, also known as the cubic system, is one of the seven crystal systems and is characterized by high symmetry. Understanding the symmetry elements and their operations within this system is fundamental to mineral identification and understanding material properties. This answer will detail the symmetry elements of the normal class isometric system, its Hermann-Mauguin notation, and demonstrate form generation on a stereogram.
Symmetry Elements in the Normal Class of the Isometric System
The normal class of the isometric system possesses the highest degree of symmetry among all crystal systems. The symmetry elements present are:
- Axes of Symmetry: Three mutually perpendicular four-fold rotation axes (4̄1), denoted as C4.
- Axes of Symmetry: Four three-fold rotation axes (3̄1), denoted as C3. These coincide with the body diagonals of the cube.
- Axes of Symmetry: Six two-fold rotation axes (2̄1), denoted as C2.
- Planes of Symmetry: Nine mirror planes. Three are parallel to the cube faces (cubic {100} forms), six are diagonal planes intersecting the cube edges (cubic {110} forms).
- Centre of Symmetry: A single centre of symmetry, denoted as ī.
Hermann-Mauguin Notation of the Normal Class of Isometric System
The Hermann-Mauguin notation (also known as International Tables notation) for the normal class of the isometric system is m3m. This notation concisely represents all the symmetry elements present:
- 'm' represents mirror planes.
- '3' represents three-fold rotation axes.
- 'm' again represents mirror planes (indicating their presence in multiple sets).
Therefore, m3m signifies the presence of mirror planes and three-fold rotation axes, defining the symmetry of the normal isometric class.
Plotting a Face (hkl) and Form Generation on a Stereogram
Let's consider the face (111) in the isometric system. This face is equivalent to the {111} form. To plot this on a stereogram (specifically, a Wulff net or Schmidt net), we follow these steps:
- Plot the Pole: The pole of the (111) face is located at 19.47° from the centre of the stereogram along the [111] direction.
- Symmetry Operations: Now, we apply the symmetry operations to the (111) face to generate equivalent faces.
Symmetry Operations and Generated Forms
| Symmetry Operation | Generated Face |
|---|---|
| Four-fold rotation about [001] | (11-1), (-111), (-1-11) |
| Three-fold rotation about [111] | (1-11), (-111) |
| Mirror plane perpendicular to [100] | (-111) |
| Centre of Symmetry | (-1-1-1) |
Stereogram Plot (Description):
On a stereogram, the (111) pole is plotted. Applying the four-fold rotation axes generates poles at 90° intervals around the centre. The three-fold rotation axes generate poles at 120° intervals. The mirror planes create reflections of the pole across the respective planes. The centre of symmetry generates a pole diametrically opposite the original pole. The resulting pattern on the stereogram demonstrates the high symmetry of the isometric system and the generation of equivalent faces through symmetry operations. (A visual stereogram would be included here in an actual exam setting).
Conclusion
The isometric system, with its high degree of symmetry, is a cornerstone of crystallography. Understanding its symmetry elements – axes, planes, and the centre of symmetry – is crucial for interpreting crystal structures and predicting their properties. The Hermann-Mauguin notation provides a concise representation of this symmetry. The application of symmetry operations to a single face, as demonstrated with the (111) plane, reveals the generation of equivalent forms, highlighting the inherent symmetry of the isometric system. This knowledge is fundamental for geologists, material scientists, and anyone working with crystalline materials.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.