• know how atom positions are denoted by fractional coordinates • be able to calculate bond lengths for octahedral and tetrahedral sites in a cube • be able to calculate the size of interstitial sites in a cube
System Cubic Tetragonal Orthorhombic Hexagonal Trigonal (R) Monoclinic Triclinic Essential Symmetry 4 3-fold axes 1 4-fold axis 3 mirrors or 3 2-fold axes 1 6-fold axis 1 3-fold axis 1 2-fold axis no symmetry Symmetry axes along the body diagonals parallel to c, in the centre of ab perpendicular to each other down c down the long diagonal down the ―unique‖ axis
Plane perpendicular to y cuts at , 1, (0 1 0) plane
This diagonal cuts at 1, 1, (1 1 0) plane an index 0 means that the plane is parallel to that axis (1 0 0) ---- (0 0 1) ? (1 1 0) ---- (0 0 1) ?
Tetrahedral sites
Relation of a tetrahedron to a cube:
i.e. a cube with alternate corners missing and the tetrahedral site at the body centre
无机功能纳米材料化学
Chemistry of Functional Inorganic Materials
（ 2）
第三章 无机功能材料的结构
Why study solid structures? 合成 表征 性能 应用
1.组成分析: CHNOS元素分析, 金属元素 (ICP-AES,ICP-MS)
Lattice Planes
Imagine representing a crystal structure on a grid (lattice) which is a 3D array of points (lattice points). Can imagine dividing the grid into sets of “planes” in different orientations
Fractional coordinates
Used to locate atoms within unit cell
1.
2. 3. 4.
0, 0, 0
½, ½, 0 ½, 0, ½ 0, ½, ½
Note 1: atoms are in contact along diagonals (close packed) Note 2: all other positions described by positions above (next unit cell along)
Octahedral Sites
Coordinate ½, ½, ½ Distance = a/2
Coordinate 0, ½, 0 [=1, ½, 0] Distance = a/2
In a face centred cubic anion array, cation octahedral sites at: ½ ½ ½, ½ 0 0, 0 ½ 0, 0 0 ½
Find intercepts on a,b,c: 1/4, 2/3, 1/2 Take reciprocals 4, 3/2, 2 Multiply up to integers: (8 3 4) [if necessary]
General label is (h k l) which intersects at a/h, b/k, c/l (hkl) is the MILLER INDEX of that plane (round brackets, no commas).
NaCl
Fcc: Lattice type F NaF, KBr, MgO...
Side centred unit cell
Counting the number of atoms within the unit cell Many atoms are shared between unit cells Thinking now in 3 dimensions, we can consider the different positions of atoms as follows Atoms Shared Between: corner 8 cells face centre 2 cells body centre 1 cell edge centre 4 cells Each atom counts: 1/8 1/2 1 1/4
Seven unit cell shapes
Symmetry in 3-d
A crystal system is defined in terms of symmetry and not by crystal shape. Thus we need to look at all the symmetry arising from different shapes of unit cell.
2. Primitive and Centred Lattices
Cu
Fcc: Lattice type F Cu, Ag, Au, Al, Ni...
-Irபைடு நூலகம்n
Bcc: Lattice type I -Iron Nb, Ta, Ba, Mo...
CsCl
primitive cubic Lattice type P CsCl, CuZn, CsBr, LiAg...
• All planes in a set are identical • The planes are “imaginary” • The perpendicular distance between pairs of adjacent planes is the d-spacing Need to label planes to be able to identify them
5. d-spacing formula
• For orthorhombic crystals:
1 h2 k 2 l 2 2 2 2 2 d a b c
1 h2 k 2 l 2 2 d a2
•
For cubic crystals a=b=c:
6. Calculations - bond lengths etc. and interstitials
2.元素价态分析: X射线光电子能谱(XPS), UV光电子能谱(UPS)
3.结构分析: X射线衍射(PXRD/SXRD, SAXRD), 红外(IR)/拉曼(Raman)光谱、固体核磁共振(MAS-NMR)
3.形貌表征: 电子显微镜(SEM、TEM、STM、AFM).
Objectives
By the end of this section you should: • understand the concepts in crystals: unit cell, crystal plane, Miller Index, d-spacing • be able to use software to draw a sketch of crystal structure. • understand the concept of diffraction in crystals and be able to use Bragg‟s law • be able to analysis common XRD、HRTEM and SAED patterns.
Unit cell contents are 4(Na+Cl-)
4. Lattice Planes and Miller Indices • understand the concept of planes in crystals • know that planes are identified by their Miller Index and their separation, d • be able to calculate Miller Indices for planes • know and be able to use the d-spacing equation for orthogonal crystals
3. Unit cell contents
Exercise
lattice type P I F C cell contents 1 [=8 x 1/8]
e.g. NaCl Na at corners: (8 1/8) = 1 Na at face centres (6 1/2) = 3 Cl at edge centres (12 1/4) = 3 Cl at body centre = 1
第一节 晶体结构中的基本概念
Early Concepts • Crystals are solid - but solids are not necessarily crystalline • Crystals have symmetry and long range order • Description of solid: Symmetry, Space group