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材料科学基础英文版课件.ppt

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材料科学 基础 英文 课件
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Chapter 1 ?Fundamental Concepts of Crystal,1,,,1) Definition of Crystal,1-1. Basic Concepts of Crystal,What is Crystal?Crystal and Amorphous,2,People realized that there are two kinds of mineral in nature: Crystal and Amorphous. Are there some commonness for crystals? Regular external shapes, flat-geometric faceThe crystal firstly found was quartz with regular shape.Quartz with geometrical polyhedron shape was called as crystal.,Quartz,3,Later, all solids with geometrical polyhedron shape were defined crystal, such as salt, calcite, magnetite.,Salt,Calcite,magnetite,4,5,The building blocks of these two are identical, but different crystal faces are developed,Crystal Structure,Cleaving a crystal of rocksalt,6,The external appearance of crystals are all characterized by flat bounding planes which intersect at characteristic angles.,Why many rounded stones and man-made cast solid objects with no external evidence are observed?,Question:,7,8,Definition of crystal,Crystals are built up of regular arrangements of atoms, molecules or ions in three dimensions.,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。,8,2) Space Lattice and Equivalent Points,Equivalent Point: Identical Surroundingsthe grouping of lattice points about any given point is identical to the grouping about any other point in the lattice.,9,Equivalent Points, Space Lattice,,periodic array of points in three dimension ? ? ? ? ? ? ? ? ? ? ? ?—— Space Lattice,,Lattice Points,Equivalent Point: Identical position and identical surroundingi.e. Cl- center、Na+ center、between Cl- & Na+ Geometric point,No weight and sizeSpace lattice with 3D period repeating pattern is built by abstraction equivalent point in crystal structure, which shows .,10,Lattice,Lattice = An infinite array of points in space, in which each point has identical surroundings to all others. It can be described by associating with each lattice point a group of atoms called the Motif (Basis) Crystal Structure = The periodic arrangement of atoms in the crystal.,11,Lattice: Periodic repeating array,,,?=,?,=,12,Lattice, Basis, Crystal Structure,Don't mix up atoms with lattice points Lattice points are infinitesimal points in space Atoms are physical objects Lattice Points do not necessarily lie at the centre of atoms,13,Differences of Lattice and Crystal Structure,LatticeAn array of points in space,Crystal StructureThe arrangement of atoms or molecules which actually exists in a crystal,Contrived, abstract geometrical array Geometrical point,Objective Physical object atoms or ions,A space lattice is a geometrical abstraction which is a reference in describing and correlating symmetry of actual crystals.,14,3) Lattice Elements ?空间点阵的要素,Lattice point (dot) ?结点 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。),15,(3)Nets 面网,空间点阵中不在同一行列上的任意3个结点就可联成一个平面 任意2个相交的行列就可决定一个面网,面网密度:dot number in unit area面网间距:distance between two nets.,相互平行的面网的面网密度和面网间距相等 面网密度大的面网其面网间距也大,The intersections of ?one dimension array build a space lattice.,16,(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,空间格子由一系列平行重叠的平行六面体构成,17,4) Properties of Crystals,Crystallizing Uniformity ?AnisotropyConfining GeometrySymmetryLowest Energy,18,1-2.Symmetry of Crystals,1) Symmetrythe regular repeating of equivalence, or identity among constituents of an entity.,R & L Hand:Minor Electric Fan: 120°,Symmetric condition ?equivalent or identical parts of an entity ?recurrence through a specific action,Rotor,,19,2) ?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.,20,Symmetry,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.,Snowflake ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?flowers,21,3) Symmetry Operations & Symmetry Elements,Symmetry operations,22,(2)Symmetry elements,Symmetry elements— Point, line, plane,Mirror plane,Rotation axis,Symmetry elements ———————— Symmetry operations,Corresponding,23,4) Symmetry Elements of Crystals,A point, an identical arrangement found on the other side. Any part of a structure can be reflected through this centre of symmetry.,,Centre of Symmetry,24,25,Mirror Plane ?(P/m) —— Reflect,(mirror plane),Chirality Object,,Two halves of a molecule can be interconverted by carrying out the imaginary process of reflection across the mirror plane.,25,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.,? = 360/n, ?n: fold of axis,26,Representation of Rotation Axis,n fold Rotation Axis: 1次、2次、3次、4次、 6次Symbols: ? ? ? ? ? ? ? L1 ? ? ? ? L2 ? ? ?L3 ? ? L4 ? ? ? L6 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 ? ? ? ?2 ? ? ? 3 ? ? ? 4 ? ? ? ?6,Figure Symbols:,Crystals may display rotational symmetries 2, 3, 4 and 6. Others, such as n=5, 7, are never observed.,27,That is not to say that molecules which have pentagonal symmetry, n=5, cannot exist in the crystalline state. They can, but their fivefold symmetry cannot be exhibited by the crystal as ?a whole.As shown in the figure, where a fruitless attempt has been made to pack pentagons to form a complete layer; thus, individual pentagons have fivefold symmetry but the array of pentagons does not.,The impossibility of forming a complete layer of pentagons,A complete layer of hexagons,28,,,Symmetry Elements of Cube,Symmetry Elements: (A) center of symmetry; (B) mirror plane; (C) twofold rotation axis; (D) fourfold rotation axis,29,Inversion Axis,,A combined symmetry operation involving rotation (according to n) and inversion through the center.,Possible inversion axes in crystals are limited to ?and ?for the same reason that only certain pure rotation axes are allowed.,30,The onefold inversion axis is not a separate symmetry element, but is simply equivalent to the center of symmetry; also, the twofold inversion axis is the same as a mirror plane perpendicular to that axis.,Combined symmetry operation,C/i ? ? ?P/m ? ? ?3+i ? ? ? ? ? ? ? ? ? ?3+m,8 symmetry elements:1,2,3,4,6,i,m,,,,,,Separate symmetry element,,31,把一个结晶多面体所具有的全部宏观对称要素以一定的顺序组合起来,就构成了该结晶多面体的对称型。排列顺序为:对称轴Ln(由高到低)、对称面P、对称心C.,5) ?Combination of Symmetry Elements,宏观晶体中对称要素的集合,包含了宏观晶体中全部对称要素的总和以及它们相互之间的组合关系,对称变换的集合 —— 对称变换群 对称要素的集合 —— 对称要素群,对称群,,32,,,,,6) Point Group,A crystallographic point group is a set of symmetry operations, which is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. An infinite crystal would look exactly the same before and after any of the operations in its point group. In the classification of crystals, each point group corresponds to a crystal class. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.,33,1-3. Symmetry and Choice of Unit Cell,34,,高级晶族:高次轴 n>2 的多于1个 ?中级晶族:高次轴 n>2 的只有1个 ?低级晶族:无高次轴,高级晶系/1个:等轴/立方晶系/Cubic,低级晶系/3个:正交/斜方晶系/Orthorhombic ? ? ? ? ? ? ? ? ? ? ? ? ?单斜晶系/ Monoclinic ? ? ? ? ? ? ? ? ? ? ? ? ?三斜晶系/ Triclinic,Symmetry Classification of Crystals,,,中级晶族,低级晶族,高级晶族,,中级晶系/3个:六方晶系/Hexagonal ? ? ? ? ? ? ? 四方晶系/Tetragonal ? ? ? ? ? ? ? 三方晶系/Trigonal,35,2) ?Abstraction of Parallelepiped / Prism,A geometrical abstraction which is useful as a reference in describing and correlating symmetry of actual crystals. There are several different prisms (parallelepiped) Shown below:,36,Principle of Abstracting Parallelepiped,在空间点阵中,选取出符合以下四条原则的平行六面体称为单位平行六面体。,首要条件是要求所选择的平行六面体能反映整个空间点阵的宏观对称特征。 在不违反对称的条件下,应选择棱与棱之间的直角关系为最多的平行六面体。,在遵循前2条的前提下,所选的平行六面体的体积应为最小。 当对称性规定棱间交角不为直角时,在遵循前3条的前提下,应选择结点间距小的行列作为平行六面体的棱,且棱间交角接近于直角的平行六面体。,37,Example: Choice of quadrangle in a 2D lattice,L44P,L22P,38,3) ?Unit Cells ?晶胞,如果晶体结构中划分晶胞的平行六面体单位是对应的空间格子中的单位平行六面体时,这样的晶胞称为单位晶胞。,,是指能够充分反映整个晶体结构特征的最小结构单位,其形状大小与对应的单位平行六面体完全一致,并可用晶胞参数来表征,其数值等同于对应的单位平行六面体参数。,The smallest repeating unit which shows the full symmetry of the crystal structure.,39,Unit Cells vs Primitive Cells,The primitive cell may be defined as a geometrical shape which, when repeated indefinitely in three dimensions, will fill all space and is the equivalent of one atom.Primitive cells are drawn with lattice points at all corners, and each primitive cell contains the equivalent of one atom. The unit cell differs from the primitive cell in that it is not restricted to being the equivalent of one atom. In some cases the two coincide.,平行六面体,晶 ?胞,由实在的具体质点所组成,由不具有任何物理、化学特性的几何点构成,40,Graphite Lattice,4) ?The Bravais Lattice,To specify a given arrangement of points in a space lattice, it is customary to identify a unit cell with a set of coordinate axes, chosen to have an origin at one of the lattice points.,41,Specification of Unit Cell parameters,Each space lattice has some convenient set of axes, but they are not necessarily equal in length or orthogonal. Seven different systems of axes are used in crystallography, each possessing certain characteristics as to equality of angles and equality of lengths.,Unit Cell Parameters:a, b, c, ?,?,?,42,Bravais Lattice,Primitive(P):,A cell has lattice points only at the corners is Primitive, P.A primitive cell contains one lattice point.,Body Center(I):,A cell has lattice points at each corner and an extra ?lattice point at the body centre.Body centre lattice contains two lattice points.,,43,Bravais Lattice,Face Center(F):,A cell has a lattice points at each corner and additional lattice points in the centre of each face.A face centre lattice contains four lattice points.,Side Center (A,B,C):,A side centre lattice has lattice points at each corner, and one in the centres of one pair of opposite faces.A side centre contains two lattice points.,44,The seven primitive cell shapes,45,The combination of crystal system and lattice type gives the Bravais lattice of a structure. There are fourteen possible Bravais lattices.,Bravais Lattices,46,Notice:,There are no more than fourteen ways can be found in which points can be arranged in space so that each point has identical surroundings. Of course, there are many more than fourteen ways in which atoms can be arranged in actual crystals; thus there are a great number of crystal structures.,Too often the term “lattice” is loosely used as a synonym for “structure”, an incorrect practice which is frequently confusing.,The distinction can be clearly seen that a space lattice is an array of points in space. It is a geometrical abstraction in describing and correlating symmetry of actual crystals. However, a crystal structure is the arrangement of atoms or molecules which actually exists in a crystal.,47,1-4. Representation of Lattice Geometric ? ?Elements,结晶轴的选取原则:1. 符合晶体内部的空间格子规律2. 考虑晶体本身的对称特点,结晶轴可能的位置: 适当的对称轴和对称面的法线方向无对称轴或对称面,则选择三个适宜的晶棱方向原点一般放在晶体的几何中心,1) Principle for Selection of coordinate axes,48,2) ?Orientation of Crystal System,Three AxesCubic: ? ? ? ? ? ? ? ? ? ? a=b=c, ??=?=?= 90°Tetragonal: ? ? ? ? ? ? a=b≠c, ??=?=? =90°Orthorhombic: ? ? ?a≠b≠c, ??=?=?= 90°Monoclinic: ? ? ? ? ? a≠b≠c, ??=?=90°?≠90°Triclinic: ? ? ? ? ? ? ? ?a≠b≠c, ??≠?≠?≠ 90°Four AxesHexagonal: ? ? ? ? ?a=b≠c, ??=?= 90° ?=120°Trigonal: ? ? ? ? ? ? ?a=b≠c, ??=?= 90° ?=120°,49,3) Miller Indices,,To determine Miller indices (hkl) of a plane, we take the following steps: Find the intercepts on the three axes in multiples or fractions of the edge lengths along each axis. Determine the reciprocals of these numbers. Reduce the reciprocals to the three smallest integers having the same ratio as the reciprocals. Enclose these three integral numbers in parentheses, e.g., (hkl).,50,The plane in figure cuts the x axis at a/2, the y axis at b and the z axis at c/3; the fractional intersections are therefore, ?, 1, 1/3.Take reciprocals of these fractions; this gives (213).,51,Miller indices,,ABC plane ?(332) ADC plane,,,(010),52,4) Indices of directions,Directions in crystals and lattices are labelled by first drawing a line that passes through the origin and parallel to the direction in question. Let the line pass through a point with general fractional coordinates x, y, z; the line also passes through 2x, 2y, 2z; 3x, 3y, 3z, etc. These coordinates, written in square brackets [x, y, z], are the indices of the direction, by division or multiplication throughout by a common factor.,53,B:111 OB:[111] A:1 2/3 1 OA:[323],54,1-5. Space Symmetry and Space Group,晶体具有宏观对称性,,来源于晶体内部的点阵构造,,,,点阵构造的对称性,宏观多面体是有限的图形 宏观性质表现出来是连续的 只有方向的特点 没有位置的概念,55,1) Space Symmetry,Translation, Glide Plane, Screw Axis,Translation,Translation repeats objects by movement along a line at specific distances and angles.,56,57,Glide Plane,a combination of mirror operations and translation,A glide plane is a reflection in a plane, followed by a translation parallel with that plane.,57,This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face.The d glide, which is a fourth of the way along either a face or space diagonal of the unit cell, which is often called the diamond glide plane as it features in the diamond structure.,58,59,Screw Axis (ns),a combination of rotation axes and translation,A screw axis is a rotation about an axis, followed by a translation along the direction of the axis.,59,These are noted by a number, n, to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation (e.g., 3 would mean a rotation one third of the way around the axis each time). The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So, 21 is a twofold rotation followed by a translation of 1/2 of the lattice vector.,60,61,2) Space Group and ?Notation,The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices which belong to one of 7 crystal systems. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.,62,(1) Sch?nflies notation,即在其点群符号右上方加上一个指数,表示属于这个点群中的不同空间群 如单斜晶系的C2h点群,,这种表示法不能立即告诉人们某一种空间群的特征对称要素,In Sch?nflies notation, point groups are denoted by a letter symbol with a subscript.,63,(2) Hermann-Mauguin / International notation,The Hermann-Mauguin (or international) notation consists of a set of four symbols The first describes the centering of the Bravais lattice (P, A, B, C, I, R or F). The next three describe the most prominent symmetry operation visible when projected along one of the high symmetry directions of the crystal. These symbols are the same as used in point groups, with the addition of glide planes and screw axis, described above.,

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