11.4
結晶学的点群
結晶学的点群は、結晶内で起こりうるさまざまな対称性操作を表します。これらの行動中に少なくとも一つの点が常に変わらないという点で、彼らは独特です。例えば、三斜晶系を考えてみましょう。この系は軸や対称面を持たず、C1点群とC i 点群に整列します。ここでCiは中心反転のみで特徴付けられます。
対照的に、単…
結晶学的点群は結晶の対称操作を記述し、少なくとも一つの点が固定されたままである。
例えば、三斜晶系は対称性の平面や軸を持たず、 Ci 点群と C1 点群を含みます。
単斜晶系は、1つの平面と1つの対称軸を持ち、3つの点群( C2h、 C2、 Cs)を持ちます。
直交方晶系は、3つの対称面と3つの軸を持ち、D2h、C2v、D2 点群を含みます。5つの平面と5つの対称軸を持つ正方形系は7つの点群を持ちます:C4h、C 4、S4、D4h、C4v、D4、D 2d。
同様に、7つの平面と7つの対称軸を持つ六角形系でも7点群を含みます。
興味深いことに、三方系は3つの平面と4つの対称軸を持ち、点群は5つだけで構成されます:C3、C3i、D3、C3v、D3dです。
最後に、最大対称要素を持つ立方系には、Th、T、Oh、O、T d 群が含まれます。
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Q1: What defines a crystallographic point group?
A crystallographic point group describes the symmetry operations that occur within crystals, with at least one point remaining fixed during these operations. Point groups classify crystals based on their symmetry elements, such as rotation axes, mirror planes, and centers of inversion. Each crystal system contains specific point groups that characterize its geometric organization.
Q2: How many point groups does the triclinic crystal system have?
The triclinic system has two point groups: C1 and Ci. This system lacks any plane or axis of symmetry, making it the simplest crystal system. The Ci point group is characterized solely by a center of inversion, while C1 has no symmetry elements at all.
Q3: What symmetry elements distinguish the monoclinic system?
The monoclinic system contains one mirror plane and one two-fold rotation axis, giving it three point groups: C2h, C2, and Cs. The C2h group combines both elements with a center of inversion at their intersection. The C2 and Cs groups are non-centrosymmetric, while C2h is centrosymmetric.
Q4: How does the orthorhombic system compare to monoclinic in symmetry?
The orthorhombic system has greater complexity than monoclinic, with three perpendicular symmetry axes and three mirror planes. It includes three point groups: D2h, C2v, and D2, all belonging to the dihedral family. These groups feature multiple C2 axes arranged perpendicular to each other, often accompanied by mirror planes and a center of inversion.
Q5: Why do tetragonal and hexagonal systems contain seven point groups each?
The tetragonal and hexagonal systems have higher symmetry than orthorhombic, with five and seven symmetry planes and axes respectively. This increased symmetry allows for more distinct point group combinations. The tetragonal system includes C4h, C4, S4, D4h, C4v, D4, and D2d groups, while hexagonal includes C6h, C6, C3h, D6h, C6v, D6, and D3h.
Q6: What makes the trigonal system unique among crystal systems?
The trigonal system has an unusual symmetry arrangement with three mirror planes and four rotation axes, yet contains only five point groups: C3, C3i, D3, C3v, and D3d. This asymmetry between the number of symmetry elements and point groups distinguishes trigonal from other systems with comparable symmetry.
Q7: Which crystal system exhibits maximum symmetry?
The cubic system exhibits maximum symmetry with nine mirror planes and thirteen rotation axes. It contains five point groups: Th, T, Oh, O, and Td. This highest degree of symmetry reflects the cubic system's highly ordered geometric structure and represents the most symmetric crystal arrangement possible.