Model Answer
0 min readIntroduction
Crystallography, the science dealing with the arrangement of atoms in crystalline solids, is fundamental to understanding the physical properties of minerals. Crystal systems are classified based on their symmetry elements – planes, axes, and centers – which dictate the external form of the crystal. The orthorhombic system is one of the seven crystal systems, characterized by three mutually perpendicular axes of unequal length. Understanding the symmetry elements and their combinations within the orthorhombic system is crucial for mineral identification and geological interpretation. This answer will detail the symmetry elements of the normal class of orthorhombic, illustrate a stereographic projection, and provide the Hermann-Mauguin notations for all its classes.
Symmetry Elements in the Normal Class of Orthorhombic System
Symmetry elements are geometric entities that, when applied to a crystal, leave it unchanged. These include:
- Axes of Symmetry: Lines around which the crystal can be rotated by a specific angle (e.g., 2-fold, 3-fold, 4-fold, 6-fold) and appear identical.
- Planes of Symmetry: Imaginary planes that bisect the crystal, creating mirror images.
- Center of Symmetry: A point within the crystal such that any line drawn through the point intersects the crystal surface at two points equidistant from the center.
The ‘normal class’ of orthorhombic (also denoted as 2/m 2/m 2/m or simply mmm) possesses the highest symmetry within the orthorhombic system. It contains the following symmetry elements:
- Three mutually perpendicular 2-fold rotation axes: These are aligned with the crystallographic axes (a, b, and c).
- Three mutually perpendicular mirror planes: These are perpendicular bisectors of the crystallographic axes (perpendicular to a, b, and c).
- A center of symmetry: Located at the intersection of the crystallographic axes.
Stereographic Projection of a Crystal Face (hkl) for Normal Class Orthorhombic System
A stereographic projection is a method of representing a three-dimensional crystal face as a two-dimensional plane. For the normal class orthorhombic system, the projection involves plotting the poles of the crystal faces on a Wulff net.
Consider a face (110) in an orthorhombic crystal. The normal to this face is [1 1 0]. To project this pole, imagine a sphere surrounding the crystal with its center at the origin. The pole [1 1 0] is the intersection of the normal with the sphere. This point is then projected onto a tangent plane. The resulting point on the Wulff net represents the (110) face. Due to the orthorhombic system's lack of rotational symmetry, the projection will not exhibit the same rotational symmetry as in cubic systems. The projection will show a single point representing the (110) face, and other faces will be projected accordingly, reflecting the orthorhombic symmetry.
(Note: A visual diagram of the stereographic projection would be ideal here, but is not possible in this text-based format. The description above aims to convey the process and expected outcome.)
Hermann-Mauguin Notations of all Classes of Orthorhombic System
The Hermann-Mauguin notation is a shorthand system for representing the symmetry of a crystal class. The orthorhombic system has three main classes:
| Crystal Class | Hermann-Mauguin Notation | Symmetry Elements |
|---|---|---|
| Pmm2 | 2/m 2/m | Two 2-fold axes, two mirror planes perpendicular to each axis. |
| Cmm2 | 2/m 2/m | One 2-fold axis, two mirror planes perpendicular to each axis, and a center of symmetry. |
| mmm (Normal Class) | 2/m 2/m 2/m | Three 2-fold axes, three mirror planes perpendicular to each axis, and a center of symmetry. |
The notation uses the following symbols:
- n: n-fold rotation axis
- m: Mirror plane
- /: Indicates a glide plane
- - : Indicates a center of symmetry
Conclusion
In conclusion, the orthorhombic crystal system is characterized by three unequal axes and a unique set of symmetry elements. The normal class, with its three 2-fold axes, three mirror planes, and a center of symmetry, represents the highest symmetry within this system. Understanding these symmetry elements and their representation through stereographic projections and Hermann-Mauguin notations is crucial for accurate mineral identification and geological analysis. The orthorhombic system, while less symmetrical than cubic or hexagonal systems, plays a significant role in the formation of many important minerals.
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.