The arrangement of atoms or molecules which actually exists in a crystal • Objective
• Contrived, abstract geometrical array • Geometrical point
• Physical object
= 360/n, n: fold of axis
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n=2, twofold rotation axis
n=4, fourfold rotation axis
Centre of Symmetry
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② Mirror Plane (P/m) —— Reflect
Two halves of a molecule can be interconverted by carrying out the imaginary process of reflection across the mirror plane.
2) Point Symmetry
Character:
• The external appearance of crystals is the confining geometry with flat bounding planes which intersect at characteristic angles. • Macroscopic symmetry of crystals reflects the symmetry of space lattice.
• Crystal Structure = The periodic arrangement of atoms in the crystal.
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=
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Lattice: Periodic repeating array
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Lattice, Basis, Crystal Structure
♪ Don't mix up atoms with lattice points ♪ Lattice points are infinitesimal points in space ♪ Atoms are physical objects
Chapter 1
Fundamental Concepts of Crystal
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1-1. Basic Concepts of Crystal
1) Definition of Crystal
• What is Crystal？ • Crystal and Amorphous
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• People realized that there are two kinds of mineral in nature: Crystal and Amorphous. • Are there some commonness for crystals? • Regular external shapes, flatgeometric face
1) Symmetry
the regular repeating of equivalence, or identity among constituents of an entity.
R & L Hand： Minor Electric Fan: 120°
Rotor
Symmetric condition ① equivalent or identical parts of an entity ② recurrence through a specific action 19
晶体是内部质点在三 维空间呈周期性重复 排列的固体；或者说， 晶体是具有格子构造 的固体。
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2) Space Lattice and Equivalent Points
Equivalent Point: Identical Surroundings the grouping of lattice points about any given point is identical to the grouping about any other point in the lattice.
•面网密度：dot number in unit area •面网间距：distance between two nets.
• 面网密度大的面网其面 网间距也大
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（4）Space Lattice 空间格子
联结分布在三维空间内的结点—— Space Lattice 空间格子
• 由三个不共面的行列就可决定一个空间格子
(mirror plane)
Chirality Object
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③ Rotation Axis (Ln / n) —— Rotate
Rotation about the axis by 360/n degrees give an identical orientation and the operation is repeated n times before the original configuration is regained.
♪ Lattice Points do not necessarily lie at the centre of atoms
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Differences of Lattice and Crystal Structure
Lattice
An array of points in space
Crystal Structure
• Space lattice with 3D period repeating pattern
is built by abstraction equivalent point in crystal structure, which shows .
periodic array of points in three dimension —— Space Lattice
(1) Lattice point (dot) 结点 (2) One dimension array 行列 在空间点阵中，分布在同一直线上的结点构成一个行列。 空间点阵中，任意两个结点连接起来就是一条行列方向。 行列中两个相邻结点间的距离 ——distance between lattice points 结点间距 • 在同一行列中结点间距是相等的； • 在平行的行列上结点间距是相等的； • 不同的行列，其结点间距一般是不等的（某些方向的行 列结点分布较密；另一些方向行列结点的分布较疏。）
空间格子由一系列 平行重叠的平行六 面体构成
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4) Properties of Crystals
Crystallizing Uniformity Anisotropy Confining Geometry Symmetry Lowest Energy
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1-2.Symmetry of Crystals
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3) Symmetry Operations & Symmetry Elements
(1) Symmetry operations
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（2）Symmetry elements
Symmetry elements— Point, line, plane
Rotation axis Mirror plane Corresponding Symmetry elements ———————— Symmetry operations
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Symmetry
Snowflake
flowers
Imagine an axis passing through the center of the snowflake, if the snowflake is rotated through 1/6 of a revolution, then the new position is indistinguishable from the old.
Question: • Why many rounded stones and man-made cast solid objects with no external evidence are observed?
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Definition of crystal
Crystals are built up of regular arrangements of atoms, molecules or ions in three dimensions.
Cleaving a crystal of rocksalt
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• The external appearance of crystals are all characterized by flat bounding planes which intersect at characteristic angles.
• atoms or ions
A space lattice is a geometrical abstraction which is a reference in describing and correlating symmetry of actual crystals.
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3) Lattice Elements 空间点阵的要素
Quartz
• The crystal firstly found was quartz with regular shape.
• Quartz with geometrical polyhedron shape was called as crystal.
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• Later, all solids with geometrical polyhedron shape were defined crystal, such as salt, calcite, magnetite.