Chapter 9: Problem 11
The unit cell of highest symmetry is (a) cubic (b) triclinic (c) hexagonal (d) monoclinic
Short Answer
Expert verified
The unit cell with the highest symmetry is (a) cubic.
Step by step solution
01
Understanding the Concept of Crystal Symmetry
Crystal systems are categorized by their symmetry properties, which describe how a unit cell's shape can be preserved under different operations such as rotations and reflections. The more symmetry operations a unit cell admits, the higher its symmetry.
02
Reviewing the Symmetry of Given Options
Each option represents a type of crystal system. The cubic system has the highest number of symmetry operations including rotations by 90 degrees about multiple axes, reflections across various planes, and inversions through a center point. The triclinic system has the lowest symmetry, with no axes of symmetry other than the identity. Hexagonal and monoclinic systems have intermediate levels of symmetry with fewer symmetry operations than cubic, but more than triclinic.
03
Determining the Highest Symmetry
Since cubic crystals have symmetry operations along three axes and can undergo four 3-fold rotations, three 4-fold rotations, and six 2-fold rotations in addition to several reflections and an inversion, it has the highest symmetry among the options provided.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Unit Cell Symmetry
Understanding the concept of unit cell symmetry is integral to the study of crystalline materials. A unit cell is the smallest repeating structural unit of a crystal, and its symmetrical properties dictate the overall symmetry of the crystal lattice. Symmetry in this context refers to the invariance of the unit cell's shape when subjected to various spatial transformations such as rotations, reflections, inversions, and translational operations.
Symmetry elements, which include points, lines, and planes, are imaginary geometric entities around which these symmetry operations occur. For instance, a rotation axis is a line about which the unit cell can be rotated by certain angles to coincide with itself. A plane of symmetry allows a reflection that maps the unit cell onto itself. The presence and arrangement of these symmetry elements in a unit cell classify it into different crystal systems, each exhibiting a unique level of symmetry.
For students striving to comprehend unit cell symmetry, visualizing these operations on a three-dimensional model or using graphical software can be incredibly helpful. Think of a square tile pattern on a wall; the design repeats itself in all directions, and flipping or rotating a single tile doesn't change the overall pattern. This is an everyday example of symmetry in two dimensions, while unit cells extend this principle into the third dimension.
Symmetry elements, which include points, lines, and planes, are imaginary geometric entities around which these symmetry operations occur. For instance, a rotation axis is a line about which the unit cell can be rotated by certain angles to coincide with itself. A plane of symmetry allows a reflection that maps the unit cell onto itself. The presence and arrangement of these symmetry elements in a unit cell classify it into different crystal systems, each exhibiting a unique level of symmetry.
For students striving to comprehend unit cell symmetry, visualizing these operations on a three-dimensional model or using graphical software can be incredibly helpful. Think of a square tile pattern on a wall; the design repeats itself in all directions, and flipping or rotating a single tile doesn't change the overall pattern. This is an everyday example of symmetry in two dimensions, while unit cells extend this principle into the third dimension.
Cubic Crystal System
Among the seven crystal systems, the cubic or isometric system showcases the highest degree of symmetry. Characterized by three equal axes that are perpendicular to each other, cubic crystals appear the same from all directions, a consequence of their high symmetry. Cubic unit cells are the building blocks of a cubic crystal system and include variations such as simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC) structures.
Consider a dice, a perfect example of a real-world object with cubic symmetry; no matter how you turn it, its shape remains consistent — it's isotropic. The strength of the cubic system lies in its multiple 4-fold rotation axes (aligning with the cube's edges), 3-fold rotation axes (diagonals of the cube), and 2-fold rotation axes (intersecting at the cube's face centers), alongside planes of symmetry cutting through the cube's faces, edges, and vertices.
In practical applications, materials with cubic symmetry such as salts, metals, and some oxides, present uniform properties regardless of direction, which can be crucial in engineering and materials science. For students, grasping the concept of a cube, easily found in daily life, aids in understanding the complexities of the cubic crystal system.
Consider a dice, a perfect example of a real-world object with cubic symmetry; no matter how you turn it, its shape remains consistent — it's isotropic. The strength of the cubic system lies in its multiple 4-fold rotation axes (aligning with the cube's edges), 3-fold rotation axes (diagonals of the cube), and 2-fold rotation axes (intersecting at the cube's face centers), alongside planes of symmetry cutting through the cube's faces, edges, and vertices.
In practical applications, materials with cubic symmetry such as salts, metals, and some oxides, present uniform properties regardless of direction, which can be crucial in engineering and materials science. For students, grasping the concept of a cube, easily found in daily life, aids in understanding the complexities of the cubic crystal system.
Crystal Systems and Symmetry Operations
Crystal systems are the categories under which all crystals can be classified based on their symmetry content. There are seven unique crystal systems: cubic, hexagonal, tetragonal, orthorhombic, monoclinic, triclinic, and trigonal, each with a distinct combination of symmetry operations that govern the spatial arrangement of unit cells.
Symmetry operations are the movements or transformations that can be applied to a crystal without altering its appearance. They include rotations, where a crystal can be turned about an axis; reflections, involving a mirror-like inversion across a plane; inversions, where points are mapped through a center point to the opposite end; and translational symmetry, a shift of the entire lattice by a vector that coincides with the lattice points.
Understanding these operations and their corresponding symmetry elements, like axes of rotation (2-fold, 3-fold, 4-fold, etc.), mirrors (planes of symmetry), and centers of inversion, helps students visualize and differentiate between each crystal system. As a mental exercise, picturing a kaleidoscope's repeating, mirrored patterns may assist learners in conceptualizing how symmetry operations generate the diverse and beautiful array of crystal structures found in nature.
Symmetry operations are the movements or transformations that can be applied to a crystal without altering its appearance. They include rotations, where a crystal can be turned about an axis; reflections, involving a mirror-like inversion across a plane; inversions, where points are mapped through a center point to the opposite end; and translational symmetry, a shift of the entire lattice by a vector that coincides with the lattice points.
Understanding these operations and their corresponding symmetry elements, like axes of rotation (2-fold, 3-fold, 4-fold, etc.), mirrors (planes of symmetry), and centers of inversion, helps students visualize and differentiate between each crystal system. As a mental exercise, picturing a kaleidoscope's repeating, mirrored patterns may assist learners in conceptualizing how symmetry operations generate the diverse and beautiful array of crystal structures found in nature.
