11.2
結晶中の対称要素
結晶対称性操作は、距離、角度、体積を保持しつつ、物体を区別できないコピーに写す等角変換です。最も単純な対称性演算は平行移動であり、これは無限結晶格子全体を平行に移動させます。
結晶学的回転は、軸上の点の位置を変えずに軸を中心に2π/n の角度で回転させるものです。これは対称性の回転軸と呼ばれ、 nで表…
結晶が対称性を示すのは、その原子が三次元で規則的かつ繰り返しのパターンで配置されているからです。
対称操作は結晶の全体的な外観を変えずに部分を動かします。
最も単純な対称演算の一つが平行移動です。平行移動では、構造内の各点が同じ方向に同じ距離で移動し、繰り返す格子を再現します。
その他の一般的な対称演算には回転、反射、反転があります。回転は構造物を特定の角度で軸の周りに回転させ、外観を変えません。反射は平面全体に鏡像を形成します。反演は各点を中心点を通って、対側の同等位置へ移動させます。
対称性演算が行われる幾何学的特徴は対称要素と呼ばれます。例えば、直線は回転軸を表し、平面は鏡面を表し、点は反転中心を表します。
結晶は複合対称素子を含むこともあります。例えば、ねじ軸は同じ軸に沿って回転と平行移動を組み合わせています。
同様に、滑空面は平面に平行な反射と平行移動を組み合わせています。
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Q1: What is a symmetry operation in a crystal?
A symmetry operation moves parts of a crystal without changing its overall appearance. These isometric transformations preserve distances, angles, and volumes while mapping the structure onto indistinguishable copies. Common symmetry operations include translation, rotation, reflection, and inversion, each fundamental to understanding crystal geometry.
Q2: How does translation work as a symmetry operation?
Translation shifts every point in a crystal structure by the same distance in the same direction, reproducing the repeating lattice pattern. This operation is one of the simplest symmetry operations and is essential for describing how atoms arrange periodically in three dimensions throughout the infinite crystal.
Q3: What are symmetry elements and how do they relate to symmetry operations?
Symmetry elements are geometric features about which symmetry operations are performed. A line represents a rotation axis, a plane represents a mirror plane, and a point represents a center of inversion. These elements define the axes, planes, and points through which specific symmetry operations occur in crystals.
Q4: What is a screw axis and how does it combine symmetry operations?
A screw axis combines rotation with translation along the same axis, creating a combined symmetry element. Similarly, a glide plane combines reflection with translation parallel to the plane. These composite operations demonstrate how crystals can exhibit multiple symmetry transformations simultaneously.
Q5: What is inversion and how does it function as a symmetry operation?
Inversion moves each point through a central point to an equivalent position on the opposite side, creating a center of symmetry or inversion center. This operation preserves the crystal's appearance while reflecting all atomic positions through a single point, a key symmetry element in many crystal structures.
Q6: How many symmetry elements can a crystal have?
A crystal can have multiple planes or axes of symmetry, but only one center of symmetry. The total number of symmetry elements varies by crystal type; for example, a cubic crystal like NaCl has 23 symmetry elements. This collection of symmetries defines the crystal's overall geometric properties.
Q7: What is the difference between rotation and roto-inversion in crystals?
Crystallographic rotation turns the structure around an axis by a specific angle without changing its appearance. Roto-inversion, denoted as n̅, involves a 2π/n rotation coupled with inversion at a point on the same axis. Both are distinct symmetry operations that contribute to the crystal's overall symmetry classification.