Model Answer
0 min readIntroduction
Crystallography, the study of crystal structures, categorizes crystalline solids into seven crystal systems based on their unit cell parameters and symmetry. The orthorhombic system is distinguished by three mutually perpendicular crystallographic axes of unequal lengths (a ≠ b ≠ c) and all interaxial angles equal to 90° (α = β = γ = 90°). The "normal class" or holohedral class (also known as the Rhombic Dipyramidal class, point group mmm) represents the highest symmetry within the orthorhombic system, possessing the maximum number of symmetry elements compatible with its axial parameters. This class is prevalent in many naturally occurring minerals, contributing significantly to their distinct physical properties and appearances.
Symmetry Elements of the Normal Class of the Orthorhombic System (mmm)
The normal class of the orthorhombic system (Rhombic Dipyramidal, mmm) exhibits a high degree of symmetry. Its characteristic symmetry elements include:
- Three mutually perpendicular 2-fold rotation axes (3A2): These axes are coincident with the crystallographic a, b, and c axes. Rotating the crystal by 180° around any of these axes leaves its appearance unchanged.
- Three mutually perpendicular mirror planes (3P): Each mirror plane is perpendicular to one of the 2-fold rotation axes and passes through the other two axes. These planes are designated as {100}, {010}, and {001}.
- A center of inversion (C): This is a point within the crystal such that for every point on the crystal, an identical point exists at an equal distance on the opposite side, passing through the center.
In total, the normal class of the orthorhombic system possesses 8 symmetry operations: identity, three 2-fold axes, three mirror planes, and one center of inversion.
Different Forms for this Class
The forms observed in the normal class of the orthorhombic system are typically closed and include:
- Orthorhombic Dipyramid {hkl}: This is the general form, consisting of eight triangular faces, each intersecting all three unequal axes at different unit lengths (e.g., {123}, {431}). It is a closed form.
- Orthorhombic Prism {hk0}, {h0l}, {0kl}: These are open forms composed of faces parallel to one crystallographic axis and intersecting the other two. There are three types:
- Macroprism {hk0}: Faces parallel to the c-axis, intersecting a and b.
- Brachyprism {h0l}: Faces parallel to the b-axis, intersecting a and c.
- Dipyramid {0kl}: Faces parallel to the a-axis, intersecting b and c.
- Pinacoids {100}, {010}, {001}: These are open forms consisting of a pair of parallel faces, each intersecting only one crystallographic axis and being parallel to the other two.
- Macropinacoid {100}: Faces parallel to the b and c axes, intersecting a.
- Brachypinacoid {010}: Faces parallel to the a and c axes, intersecting b.
- Basal Pinacoid {001}: Faces parallel to the a and b axes, intersecting c.
Sketch Stereogram of the Form {hkl}
A stereogram represents the orientation of crystallographic planes and axes. For the general form {hkl} in the orthorhombic normal class (mmm), the stereogram would show:
[Drawing of a stereogram for the orthorhombic mmm class with a general {hkl} pole and its symmetrically equivalent poles.]
The stereogram for the normal class (mmm) would typically show:
- Three perpendicular 2-fold rotation axes represented by ellipses, often coinciding with the N-S, E-W, and the center (out of plane) axes.
- Three perpendicular mirror planes represented by great circles (one circumferential and two passing through the N-S and E-W points).
- A general pole {hkl} in the upper hemisphere would be represented by a solid circle. Due to the symmetry operations (three 2-fold axes, three mirror planes, and a center of inversion), it would generate 8 equivalent poles in total (4 in the upper hemisphere and 4 in the lower hemisphere, represented by open circles). These would be {hkl}, {hkl̄}, {h̄kl}, {hkl̄}, {h̄kl̄}, {hkl̄}, {h̄kl̄}, {h̄kl̄}.
(Note: As an AI, I cannot directly draw. The description above details what the sketch stereogram should visually represent. A typical sketch would involve a circular projection with symbols for 2-fold axes and mirror planes, and 8 plotted points representing the {hkl} poles and their symmetrical equivalents.)
Examples of Minerals Crystallizing in this System
Two prominent examples of minerals that crystallize in the orthorhombic system, specifically the normal class (mmm), are:
- Barite (BaSO4): A common barium sulfate mineral, often forming tabular or prismatic crystals.
- Topaz (Al2SiO4(F,OH)2): A silicate mineral known for its gemstone quality, typically forming prismatic crystals with a distinct rhombic cross-section.
Other examples include Olivine ((Mg,Fe)₂SiO₄), Aragonite (CaCO₃), and Alpha-sulfur (S₈).
Conclusion
The orthorhombic crystal system, particularly its normal class (mmm), is characterized by three unequal and mutually perpendicular axes, possessing three 2-fold rotation axes, three mirror planes, and a center of inversion. This comprehensive set of symmetry elements gives rise to a variety of crystal forms, including the characteristic orthorhombic dipyramid, prisms, and pinacoids. Minerals like barite and topaz exemplify this system, showcasing the diverse and often aesthetically pleasing crystalline structures found in nature. Understanding these symmetry elements and forms is fundamental for mineral identification and comprehending the physical properties of 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.