1

1 Crystal Basics

2 Symmetry

3 Crystal Structure Analysis

4 Crystal Chemistry

5 Some Important Crystal Structures

Review of Crystallography

1.Crystal

2.Fundamental Characteristics of Crystals

$1 Crystal Basics

Why Solids?

All Elements and compounds are solids under suitable conditions of temperature and pressure. Many exist only as solids.

atoms in ~fixed position

“simple” case crystalline solid Crystal Structure

Why study crystal structures?

description of solid

comparison with other similar materials classification

correlation with physical properties

Crystals are solid but solids are not necessarily crystalline

Crystals have symmetry (Kepler, 1611) and long range order

Spheres and small shapes can be packed to produce regular shapes (Hooke; Hauy,1812)

?

Early Ideas

As a consequence of studies

on cleavage, envisaged calcite

crystals, of whatever habit, as

built up by the packing

together of “constituent

molecules” in the form of

minute rhombohedral units.

Crystals A homogenous solid formed by a repeating, three-dimensional pattern of atoms, ions, or molecules and having fixed distances between constituent parts.

Definition Crystal A crystal may be defined as a collection of

atoms arranged in a pattern that is periodic in 3D.

Crystals are necessarily solids, but not all solids are crystalline (amorphous solids lack long range periodic order).

In a perfect single crystal, all atoms in the crystal are related either through translational symmetry or point symmetry.

Polycrystalline materials are made up of a great number of tiny (m to nm) single crystals

Crystallinity

2

• Single crystal:atoms are in a repeating or periodic array over the entire extent of the material

• Polycrystalline material:comprised of many small crystals or grains. The grains have different crystallographic orientation.There exists atomic mismatch within the regions where grains meet. These regions are called grain boundaries.

Single Crystal and Polycrystalline Materials

repeated arrangement of atoms extends throughout the specimen

all unit cells have the same orientation

exist in nature

can also be grown (eg. Si)

without external constraints, will have flat, regular faces

Single Crystals

Beautiful Crystals

硫酸亚铁

重铬酸钾

Crystals of different

sizes

orientations

shapes

Grain Boundaries

mismatch between two neighboring crystals

Polycrystalline Materials

Most crystalline materials are composed of many small crystals called grains

Crystallographic directions of adjacent grains are usually random

There is usually atomic mismatch where two grains meet this is called a grain boundary

Most powdered materials have many randomly oriented grains

Polycrystalline Materials High-performance bulk thermoelectrics with all-scale hierarchical architectures

Micro and nanostructures in SPS PbTe–SrTe(4 mol%) doped with 2 mol% Na.

All-length-scale hierarchy in thermoelectric materials.

K Biswas et al. Nature 489, 414-418 (2012)

3

Homogeneity Under macroscopic observation, the physics effect and chemical composition of a crystal are the same.

e. g. the crystal has the fixed melting point

Anisotropy Physical properties of a crystal differ according to the direction of measurement.

e. g. self-limitation in the crystal growth

Basic Characteristic of Crystals

Different directions in a crystal have different packing. For instance, atoms along the edge of FCC (face-centered cubic ) unit cell are more separated than along the face diagonal. This causes anisotropy in the properties of crystals, for instance, the deformation depends on the direction in which a stress is applied.

Anisotropy

Trigonal Se

Se Chain

YN Xia*, J. Am. Chem. Soc. 2000, 122, 12582 Adv. Mater. 2002, 14, 279

Nanowires Nanorods Nanotubes

C nanotubes

(a) Structural framework of Cu5V2O10, where polyhedra, large balls, and small balls represent the CuOn, Cu, and O, respectively. Two types of zigzag CuOn chains along the b- and c-axes are seen. The numbers show five different Cu sites. (b) Spin arrangements of Cu2+ ions along the b-axis of Cu5V2O10.

Unusually Large Magnetic Anisotropy in a CuO-Based Semiconductor Cu5V2O10

(a) The temperature dependence of the magnetic susceptibilities measured at H = 0.1 T along different axes. (b) Magnetization (M) as a function of applied field (H) at T = 5 K.

何长振，et al, J. Am. Chem. Soc., 2011, 133, 1298

In some polycrystalline materials, grain orientations are random, so bulk material properties are isotropic.

Some polycrystalline materials have grains with preferred orientations

(texture), so properties are dominated by those relevant to the texture orientation and the material exhibits anisotropic properties.

The interfacial angles are constant for all crystals if a given mineral with identical composition at the same temperature.

Since all crystals of the same substance will have the same spacing between lattice points (they have the same crystal structure), the angles between corresponding faces of the same mineral will be the same.

The symmetry of the lattice will determine the angular relationships between crystal faces.

Law of Constancy of Interfacial Angle

“first law of crystallography” ----Danish physician Nicolas Stenon on quartz crystals, 1869

4

Crystal Shape

The external shape of a crystal is referred to as its Habit

Not all crystals have well defined external faces

Typically see faces on crystals grown from solution

Natural faces always have low indices (orientation can be described by Miller indices that are small integers)

The faces that you see are the lowest energy faces

Surface energy is minimized during growth

Prismatic Pyramidal Tabular Rhombohedra Dodecahedral Acicular Bladed

Crystal Habits

This is a term that refers to the form that a crystal takes as it grows.

Crystal Habits

Law of Symmetry: Only 1,2,3,4,6-fold rotation axis can exist in crystal.

Why snowflakes have 6 corners, never 5 or

7? By considering the packing of polygons in 2 dimensions, demonstrate why pentagons and heptagons shouldn’t occur.

Empty space not allowed

Law of Symmetry

Quasicrystal: AlFeCu

Allowed rotation axis: 1, 2, 3, 4, 6 NOT 5, > 6

Europhys. Lett., 16 (3), pp. 271-276 (1991)

5

Quasicrystal Structures (First in 1984 Al-Mn)

R0.09Mg0.34Zn0.57

Dodecahedral morphology （十二面体）

R0.1Mg0.4Cd0.5

Rhombic triacontahedral morphology (菱形的三十面体)

ED: 5 fold axis

Face-centred icosahedral R-Mg-Zn 面心二十面体

Primitive icosahedral R-Mg-Cd 简单二十面体

Nonperiodic longrange ordered structures

Rotational symmetry of diffraction patterns (e.g. 5fold, 10fold) impossible for periodic crystals

Quasicrystalline Materials

Quasi-unit cells

Israel's Daniel Shechtman wins Nobel Prize in chemistry (2011)

Excerpts from 2011 Nobel chemistry prize citation: “In all solid matter, atoms were believed to be packed inside crystals in symmetrical patterns that were repeated periodically over and over again. For scientists, this repetition was required in order to obtain a crystal. Shechtman’s image, however, showed that the atoms in his crystal were packed in a pattern that could not be repeated.” “His discovery was extremely controversial. In the course of defending his findings, he was asked to leave his research group. However, his battle eventually forced scientists to reconsider their conception of the very nature of matter.” “Following Shechtman’s discovery, scientists have produced other kinds of quasicrystals in the lab and discovered naturally occurring quasicrystals in mineral samples from a Russian river. A Swedish company has also found quasicrystals in a certain form of steel, where the crystals reinforce the material like armor. Scientists are currently experimenting with using quasicrystals in different products such as frying pans and diesel engines.”

2011.10.05 released

Phys. Rev. Lett. 53, 1951–1953 (1984) Metallic Phase with Long-Range Orientational Order and No Translational Symmetry

Dan Shechtman (Hebrew: דן שכטמן) (born in 1941 in Tel Aviv) is the Philip Tobias Professor of Materials Science at the Technion – Israel Institute of Technology, an Associate of the US Department of Energy's Ames Laboratory, and Professor of Materials Science at Iowa State University. On April 8, 1982, while on sabbatical at the U.S. National Bureau of Standards in Washington, D.C., Shechtman discovered the icosahedral phase, which opened the new field of quasiperiodic crystals.He was awarded the 2011 Nobel Prize in Chemistry for "the discovery of quasicrystals". After receiving his doctorate, Prof. Shechtman was an NRC fellow at the Aerospace Research Laboratories at Wright Patterson AFB, Ohio, where he studied for three years the microstructure and physical metallurgy of titanium aluminides. In 1975 he joined the department of materials engineering at Technion. In 1981-1983 he was on Sabbatical at Johns Hopkins University, where he studied rapidly solidified aluminum transition metal alloys (joint program with NBS). During this study he discovered the Icosahedral Phase which opened the new field of quasiperiodic crystals. In 1992-1994 he was on Sabbatical at NIST, where he studied the effect of the defect structure of CVD diamond on its growth and properties. Prof. Shechtman's Technion research is conducted in the Louis Edelstein Center, and in the Wolfson Centre which is headed by him. He served on several Technion Senate Committees and headed one of them.

http://en.wikipedia.org/wiki/Dan_Shechtman

1982年，两位主要从事航空用高强度铝合金研究的以色列科学家Shechtman和Blech

，无意中在急冷Al6Mn合金中发现五次对称衍射图，由于两人的晶体学基础一般，就到处

请教晶体学专家，专家们认为那不过是晶体学中常见的五次孪晶，抱着试试看的态度，他

们还是决定把文章寄到美国《应用物理杂志》，不幸被不识货的杂志编辑直接退稿，成名

后的Shechtman对此事仍耿耿于怀，他作学术报告时总喜欢把那封退稿信作为第一张透明

片，来讽刺那位有眼无珠的编辑。

后来他们又去请教法国CNRS冶金化学研究所的D. Gratias，由于实验结果与传统晶体

学的周期性相矛盾，Gratias认为很难被主流接受发表。1984年秋，Gratias在加州大学的

一次理论物理讨论会中听了Steinhardt的报告，发现他们关于二十面体理论模型的衍射花

样与Shechtman等人的实验结果完全一致，两人会后这么一碰，火花就出来了，他们决定

把理论和实验结果同时寄到物理学最权威的 Physical Review Letters，独具慧眼的编辑让

两篇文章以最快的速度先后发表，从此准晶（Quasicrystal）这个新名称诞生了。

准晶的发现引发了上世纪八十年代全球性的准晶热，中日美成为引领准晶研究的三驾

马车，各种准晶材料和结构被发现，当然，也有不少研究者“顿足捶胸”，这不是自己N

年前就发现的东西吗？准晶的发现也刺激了某些权威的神经，以双料诺贝尔奖获得者鲍林

（Pauling：1954年诺贝尔化学奖，1962年和平奖，1995年去世）为代表的保守势力，要

誓死捍卫传统晶体理论的“纯洁性”，他们认为所谓准晶就是众人皆知的孪晶，在Nature

发文用“Nonsense”这个词形容准晶的发现，并利用自己的特殊身份在美国科学院院报

上连发檄文，歇斯底里地反对准晶。

准晶的发现

http://blog.sciencenet.cn/home.php?mod=space&uid=480705

&do=blog&id=383415

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Ideal solid crystals exhibit structural long range

order (LRO)

Real crystals contain imperfections, i.e., defects and

impurities, which spoil the LRO

Amorphous solids lack any long range order

(LRO) ,though may exhibit short range order (SRO)

Crystal Glass (amorphous)

Gas

Amorphous Solids

Quartz Glass Quartz Crystal

Quartz Crystal and Quartz Glass

The Fixed melting point

non-crystal ：Some substances, such as wax, pitch

and glass, which posses the outward appearance of

being in the solid state, yield and flow under

pressure, and they are sometimes regarded as

highly viscous liquid.

Crystal non-crystal

• Transparent, amorphous solid

–Composition almost all silicon dioxide (SiO2 – Quartz sand)

Ordinary glass 75% SiO2

Pyrex glass SiO2 with B2O3

Lead glass SiO2 + PbO, and K2O

Green glass (cheap bottles) FeO + SiO2

Blue glass Cobalt oxide + SiO2

Violet glass Manganese + SiO2

Yellow glass Uranium oxide + SiO2

Red glass Gold and copper + SiO2

Glass

Long-Range Topological Order in Metallic Glass

Jianzhong Jiang & Ho-Kwang Mao, Science, 2011, 332, 1404

Computational simulation of Ce75Al25 MG structure and XRD patterns at 300 K and ambient pressure.

In-situ high-pressure XRD of Ce75Al25 MG in a DAC. (A) Integrated XRD patterns, (B) two-dimensional (2D) XRD image below 24.4 GPa showing typical glass pattern, and (C) 2D XRD image at 25.0 GPa showing typical single-crystal zone-axis pattern. A focused (15 m by 15 m) monochromatic x-ray (wavelength, 0.36806 Å) through the DAC axis without rotation was used for (B) and (C). Red spots are masks of diamond single-crystal XRD spots.

Liquid Crystal

Liquid crystals are a phase of matter whose order is intermediate between that of a liquid and that of a crystal. The molecules are typically rod-shaped organic moieties about 25 Angstroms in length and their ordering is a function of temperature.

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(a) crystal，(b)、(c) anisotropic liquids，(d) isotropic liquid

This is the structure change process of some molecules with long chains when increasing temperatures

From Crystal to Liquid Crystal to Liquid

Temperature Increasing

Entropy driven formation of liquid crystals of rodlike colloids

Nematic (向列相) Isotropic

Crystal Smectic（层列相）

= Direction of increasing density

Principles of Liquid Crystal Displays

No voltage voltage

Smaller, lighter, with no radiation problems. Found on portables and notebooks, and starting to appear more and more on desktops.

Less tiring than c.r.t. (Cathoderay tube) displays, and reduce eyestrain, due to reflected nature of light rather than emitted. Use of supertwisted crystals have improved the viewing angle, and response rates are improving all the time (necessary for tracking cursor accurately).

Liquid Crystal Displays

Free-standing mesoporous silica films with tunable chiral nematic structures

Mark J. MacLachlan*， Nature, 2010,468, 422–425

a, Schematic of the chiral nematic ordering present in nanocrystalline cellulose

(NCC), along with an illustration of the half-helical pitch P/2 (~150–650 nm). b, POM image of a TEOS/NCC suspension observed during

slow evaporation at room temperature (22C) clearly shows a fingerprint texture characteristic of chiral

nematic ordering. c, POM image of an NCC/silica composite film. Strong birefringence and domains with different orientations are

present. d, POM image of the mesoporous silica film

obtained from the calcination of the film in c. A shift in color

from red to blue was observed, while the overall texture remained essentially

unchanged. All micrographs were taken with crossed polarizers (scale bar, 100 μm).

a photonic mesoporous inorganic solid that is a

cast of a chiral nematic liquid crystal formed from

nanocrystalline cellulose.

Optical characterization of NCC/silica composite films and the corresponding mesoporous silica films

a, Transmission spectra of four NCC/silica

composite films with reflectance peaks in the near-infrared part of the spectrum. The

proportion of TMOS:NCC was increased from samples S1 to S4, resulting in a

redshift in the reflectance peaks of the films. b, Transmission spectra of the

mesoporous silica films obtained from the calcination of composite films S1 to S4. The reflectance peaks were all blueshifted

by approximately 300 nm, resulting in films that reflect light across the entire

visible spectrum. c, Photograph showing the different colours of mesoporous silica

films S1 to S4. The colours in these silica films arise only from the chiral nematic pore structure present in the materials.

The dime is included for scale (diameter, 18 mm). d, Photograph of a yellow

mesoporous silica film (S3) taken at normal incidence. e, Photograph of the

same film taken at oblique incidence appears blue owing to the sinθ dependence of the reflected wavelength.

8

SEM images of chiral nematic mesoporous silica films and comparison of fingerprint textures in the solid state and liquid crystal phase.

a, Top view of a cracked film shows the relatively smooth top surface and a layered structure looking down the edge (scale bar, 10m). b, Side view of a cracked film shows the stacked layers that result from the helical pitch of the chiral nematic phase (scale bar, 3m). c, Higher magnification reveals the helical pitch distance to be of the order of several hundred nanometres (scale bar, 2m). d, Very high magnification shows a rod-like morphology with the rods twisting in a left-handed orientation (scale bar, 200 nm). e, Fingerprint defect in a solid mesoporous silica film (scale bar, 10

m). f, Fingerprint defect observed by POM in the liquid crystal phase of an NCC/TEOS mixture (scale bar, 30 m).

32 Point Groups of Crystals

Unit Cell, 7 Crystal Systems, Lattice Planes, Miller indices

Lattices and 14 Bravias Types of Lattices

230 Space Groups

Symmetry: Point Symmetry

Space Symmetry

$2 Symmetry

• Mathematics of Symmetry

• Crystal’s Symmetry

• Physical Properties Caused by Symmetry

Crystal Symmetry Symmetry is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection. Hermann Weyl

Eiffel tower in Paris, France is a wonderful example of symmetry

Symmetry in Nature, Art and Math

Point symmetry elements operate to change the orientation of structural motifs

A point symmetry operation does not alter at least one point that it operates on

Symmetry Elements and Symmetry Operations:

1. Mirror Planes ——Reflection or Mirror

2. Center of Symmetry ——Inverse

3. Rotation Axis ——Rotate

4. Rotoinversion Axis ——Rotate and inverse

Macroscopic Symmetry Elements (Point Symmetry Elements)

Mirror plane symmetry arises when one half of an object is the mirror image of the other half

•Can be folded in half •Seen externally with animals

s

s

Mirror Plane Symmetry

9

This molecule has two mirror planes: One is horizontal, in the plane of the

paper and bisects the H-C-H bonds Other is vertical, perpendicular to the

plane of the paper and bisects the Cl-C-Cl bonds

A crystal has reflectional symmetry if an imaginary plane can divide the crystal into halves, each of which is the mirror image of the other.

Mirror Plane Symmetry

Symmetry Operation Reflection flips all points in the

asymmetric unit over a line, which is called the mirror and thereby changes the handedness of any figures in the asymmetric unit. The points along the mirror line are all invariant points under a reflection.

Mirror planes in Cube

Rotated about a point

Allows chirality

In crystals limited to 1,2,3,4, and 6 rotations

Rotational Symmetry

Symmetry Operation

Rotation

turns all the points in the asymmetric unit around one point, the center of rotation. A rotation does not change the handedness of figures in the plane. The center of rotation is the only invariant point.

Symmetry Axis of Rotation

We say a crystal has a symmetry axis of rotation when we can turn it by some degree about a point and the pattern looks exactly the same. Think of the center of a pizza. If it is made so that all the pieces are the same size and have the same ingredients in the same places, then the pizza could be turned and you couldn't tell the difference. This means the pizza has rotational symmetry. The pizza below has rotational symmetry of 60 degrees.

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Rotational Symmetry

coincidence upon rotation about the axis of 360/n n-fold axis of rotational symmetry

graphite

O

H

Symbol for a symmetry element for which the operation is a rotation of 360/n C2 = 180, C3=120, C4 = 90, C5 = 72, C6 = 60, etc.

Can rotate by 120 about the C-Cl bond and the molecule looks identical the H atoms are indistinguishable. This is called a rotation axis

in particular, a three fold rotation axis, as rotate by 120 (= 360/3) to reach an identical configuration

Rotation Axis (Cn)

In general: n-fold rotation axis = rotation by (360/n)

Rotation Axis in Cube

“present if you can draw a straight line from any point, through the center, to an equal distance the other side, and arrive at an identical point”.

Center of Symmetry (Inversion symmetry)

Center of symmetry at S

No center of symmetry

(x,y,z)

(-x,-y,-z)

i

Symmetry Operation

Inversion

every point on one side of a center of symmetry has a similar point at an equal distance on the opposite side of the center of symmetry.

Center of symmetry in Cube

11

Rotoinversion Axis

Symmetry Axis of Rotary Inversion

Rotoinversion Axis (Sn or ) : n-fold rotation combined with an inversion.

n

i1 m2 3 = 3fold rotation + inversion

4

6

=3fold rotation with perpendicular mirror plane 4 axis in CH4 molecule

Macroscopic Symmetry Elements: Point Groups

Electrical resistance Thermal expansion Magnetic susceptibility Elastic constants

Macroscopically measured properties

Macroscopic symmetry

X

Translation symmetry

Combination of mirror, center of symmetry, rotational symmetry, center of inversion point groups

Point Groups

Point groups have symmetry about a single point at the center of mass of the system. Symmetry elements are geometric entities about which a symmetry operation can be performed. In a point group, all symmetry elements must pass through the center of mass (the point). A symmetry operation is the action that produces an object identical to the initial object. Group theory is a very powerful mathematical tool that allows us to rationalize and simplify many problems in chemistry. A group consists of a set of symmetry elements (and associated symmetry operations) that completely describe the symmetry of an object.

Point Group

Point group (point symmetry)All crystalline solids can be characterized by 32 different combinations of symmetry elements. There are two naming systems commonly used in describing symmetry elements 1. The Schoenflies notation used extensively by spectroscopists 2. The Hermann-Mauguin or international notation preferred by crystallographers

Schoenflies Symbols

Cn: cyclic, the point group which only one rotation axis, n is the order of the rotation axis.

Dn: dihedral, the group point which generated from the combination of 2-fold axis, n is the order of the main rotation axis.

T: tetrahedral

O: octahedral

The combination of rotation axis with higher order

12

宏观对称元素

对称元素

旋转轴

对称中心

反映面

反轴

1 2 3 4 6

1

2

3 4 6

惯用符号

L1 L2 L3 L4 L6

C

P

L3 L4 L6

圣佛里斯符号

C1 C2 C3 C4 C6 i(Ci)

Cs

C3i S4 C3h

国际符号

1 2 3 4 6

1

m

3 4 6

图 示

双线或粗线

i i i

The 32 Point Groups

Add mirror plane to the above 11 basic point groups, the adding mirror plane intersect at one point with other symmetry elements, and in addition, no new symmetry types are formed, thus there are three ways: 1)Mirror plane is horizontal with the main rotation axis, Ph 2)Mirror plane is vertical to the main rotation axis, Pv 3)Mirror plane is vertical to the main rotation axis,and is diagonal to the neighboring 2-fold axis, Pd

C1 C2 C3 C4 C6 D2 D3 D4 D6 T O 11

+Ph Cs C2h C3h C4h C6h D2h D3h D4h D6h Th Oh 22

+Pv --- C2v C3v C4v C6v --- --- --- --- --- --- 26

+Pd --- --- --- --- --- D2d D3d --- --- Td --- 29

＋C Ci --- C3i --- --- --- --- --- --- --- --- 31

n S4 32

晶体的32点群

L 2L22P / [重复]

[L4PC] / [重复] L (S4)

3L44L36L29PC/ [重复]

3L44L36L29PC (Oh) 3L44L36L2 (O)

3L 4L36P (Td) 3L24L33PC (Th) 3L24L3 (T)

/ [重复] L66L27PC (D6h) L66L2 (D6)

/ [重复] L44L25PC (D4h) L44L2 (D4)

L33L23PC (D3d) L 3L23P (L33L24P) (D3h)

L33L2 (D3)

L 2L22P (D2d) 3L23PC (D2h) L22L2 (D2)

L66P (C6v) L6PC (C6h) L6 (C6)

L44P (C4v) L4PC (C4h) L4 (C4)

L (L3C) (C3i)

L33P (C3v) L (L3P) (C3h) L3 (C3)

L22P (C2v) L2PC (C2h) L2 (C2)

C (Ci) [ P ] / [重复] P (Cs) L1 (C1)

加C 加 Pv（Pd) 加Ph 轴型

4

6

6

4

4

4

3

12 11 7 2 32 Crystallographic Point Groups

Crystal System Number of

Point Groups

Herman-Mauguin

Point Group

Schoenflies

Point Group

Triclinic 2 1,1 C1, Ci

Monoclinic 3 2, m, 2/m C2, Cs, C2h

Orthorhombic 3 222, mm2, mmm D2, C2v, D2h

Trigonal 5 3,3, 32,

3m,3m

C3, C3i, D3,

C3v, D3d

Hexagonal 7 6,6, 6/m, 622,

6mm,62m, 6mm

C6, C3h, C6h, D6,

C6v, D3h, D6h

Tetragonal 7 4,4, 4/m, 422,

4mm,42m, 4/mmm

C4, S4, C4h, D4,

C4v, D2d, D4h

Cubic 5 23, m3, 432,

432, m3m

T, Th, O,

Td, Oh

Crystal Systems

There are seven crystal systems which can be defined either on the basis of symmetry, or, upon the basic building block of the crystal. The seven main symmetry groups into which all crystals, whether natural or artificial, can be classified. All crystals grow in one of following seven shapes on the microscopic level.

1 Cubic or Isometric (3 axes of equal length intersect at 90˚) 2 Tetragonal (2 axes of same length, all at 90˚) 3 Orthorhombic (3 axes of different length at 90˚) 4 Hexagonal (3 horizontal axis at 60˚. vertical axis at 90˚) 5 Monoclinic (3 axes of different length, 2 intersect at 90˚, the

other is oblique to the others) 6 Triclinic (3 axes of different length are all oblique to one

another) 7 Trigonal (Rhombohedral) (3 axes of equal length)

Seven Unit Cell Shapes Seven Crystal Systems

Cubic a=b=c ===90

Tetragonal a=bc ===90

Orthorhombic abc ===90

Monoclinic abc ==90, 90

Triclinic abc 90

Hexagonal a=bc ==90, =120

Trigonal (Rhombohedral ) a=b=c ==90

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a+b为x轴和y轴的平分线方向；

a+b+c为体对角线方向；

晶系 对 称 方 向

第一 第二 第三

三斜 无

单斜 b [010]

正交 a [100] b [010] c [001]

四方 c [001] a [100]/[010] a+b [110]

六方 c [001] a [100]/[010] 2a+b [120]

三方 (R) c’ [001] (a+b+c [111])

a’ [100]

(a-b [ 011

])

立方 a [100]/[010]/[001] a+b+c [111] a+b [110]

各晶系的晶体学方向

We can use a flow chart such as this one to determine the point group of any object. The steps in this process are: 1. Determine the symmetry is special. 2. Determine if there is a principal rotation axis. 3. Determine if there are rotation axes perpendicular to the principal axis. 4. Determine if there are mirror planes. 5. Assign point group.

Identifying Point Groups

Definition of Crystal Structures

Crystal Structure: The spatial order of the atoms is called the crystal structure, or the periodic arrangement of atoms in the crystal.

It can be described by associating with each lattice point a group of atoms called the Motif (Basis).

Structure=Lattice+Motif

Lattice = An infinite array of points in space, in which each point has identical surroundings to all others. Crystal Structure = The periodic arrangement of atoms in the crystal. It can be described by associating with each lattice point a group of atoms called the Motif (Basis)

Lattice: Periodic arrangement of points in space.

Motif: Collection of atoms to be placed equivalently

about each lattice point. Consists of atomic identities

and fractional coordinates.

Structure=Lattice+Motif Square a=b =90

Rectangular ab =90

Centered Rectangular ab =90

Hexagonal a=b =120

Oblique ab 90

5 Bravais Lattice in 2D

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Orthorhombic: P, I, F, C

Symmetry in Crystals

C F

Primitive (P)

Body-centered (I)

Side-centered (C)

Face-centered (F)

Hexagonal

Monoclinic Triclinic

=

Side-centered tetragonal Primitive tetragonal

=

Face-centered tetragonal Body-centered tetragonal

a

b

c

a b

g

1. Primitive Triclinic

a

b

c

a b

g

2.Primitive Monoclinic

a

b

c

a b

g

3. Side(or C)-centered Monoclinic

a

b

c

4.Primitive orthorhombic

5.C-centered orthorhombic

a

b

c

a

b

c

6. Body-centered orthorhombic

a

b

c

7. Face-centered orthorhombic

Unit Cells of the 14 Bravais Lattices

a

c

a

8. Primitive tetragonal

a

c

a

9. Body-centered tetragonal

a

c

a

120

10. Primitive hexagonal

11. Primitive rhombohedral (trigonal)

a a a

12.Primitive cubic

a

a

a

13. Body-centered cubic

a

a

a

a

a

a

14.Face-centered cubic

Bravais Lattice: an infinite array of discrete points with an

arrangement and orientation that appears exactly the same from whichever of the points the array is viewed.

3D: 14 Bravais Lattice, 7 Crystal System

15

The 14 possible Bravais Lattices

14 Bravais Lattices connect the macroscopic morphology of the crystals and their inner periodic structure

• Lattice: Periodic arrangement of points in space. Must be one of the 14 Bravais lattices.

Trigonal P : 3-fold rotation

Trigonal P

a=b=c ==90

Simple Cubic Lattice

Cesium Chloride (CsCl) is primitive cubic

Different atoms at corners and body center. NOT body centered, therefore.

Lattice type P

Also CuZn, CsBr, LiAg

BCC Lattice (Body Centered Cubic)

-Iron is body-centered cubic Identical atoms at corners and body center (nothing at face centers) Lattice type I Also Nb, Ta, Ba, Mo...

FCC Lattice (Face Centered Cubic)

Copper metal is face-centered cubic Identical atoms at corners and at face centers Lattice type F also Ag, Au, Al, Ni...

16

Structures for Simple Elemental Metals at Room Temp

• Metallic elements with more complicated structures are left blank

Sodium Chloride (NaCl) Na is much smaller than Cs Face Centered Cubic Rocksalt structure Lattice type F Also NaF, KBr, MgO….

FCC Lattice (Face Centered Cubic)

Unit Cell Contents Counting the number of atoms within the unit cell

Many atoms are shared between unit cells

Atoms Shared Between: Each atom counts: corner 8 cells 1/8 face center 2 cells 1/2 body center 1 cell 1 edge center 4 cells 1/4

lattice type cell contents P 1 [=8 x 1/8] I 2 [=(8 x 1/8) + (1 x 1)] F 4 [=(8 x 1/8) + (6 x 1/2)] C 2 [=(8 x 1/8) + (2 x 1/2)]

Density Calculation

n: number of atoms/unit cell A: atomic mass VC: volume of the unit cell NA: Avogadro’s number (6.023x1023 atoms/mole)

Calculate the density of copper.

RCu =0.128nm, Crystal structure: FCC, ACu= 63.5 g/mole

n = 4 atoms/cell

8.94 g/cm3 in the literature

17

Microscopic Symmetry Elements

(Space Symmetry Elements)

1. Lattice —— its corresponding operation is translational symmetry.

2. screw axis —— combination of a rotation and a translational symmetry.

3. glide plane —— combination of a reflection and a translational symmetry.

All these actions are space symmetry. Every point in the space is changed, but the space do not change after the action. So, their symmetry is called space groups.

Symmetry Elements

translational symmetry

Lattice

Symmetry Operation

Symmetry Operation:

Translation

moves all the points in the asymmetric unit the same distance in the same direction. This has no effect on the handedness of figures in the plane. There are no invariant points under a translation.

A C

a

ab

A screw axis with symbol nm is a combination of an nfold rotation followed by a translation of m/n of the unit cell repeat parallel to the axis.

Screw Axis Symmetry Element

the operation of 31 axes

e.g. a 41 axis parallel to z axis involves rotation of 90 followed by translation of 1/4 c.

front view top view

41 axis in Diamond Structure

Symmetry Operation rotation + translation

the operation of 61 axes

21，

31，32，

41，42，43，

61，62，63，64，65

Various Kinds of Screw Axes

18

Glide Reflection reflects the asymmetric unit across a mirror line and then translates parallel to the mirror. A glide reflection changes the handedness of figures in the asymmetric unit. There are no invariant points under a glide reflection.

Glide Reflection: A glide reflection combines a reflection with a translation along the direction of the mirror line.

Symmetry Operation

A glide plane is a combination of a reflection and a translation. The orientation of the plane and its symbol determine what sort of translation is involved.

b glide parallel to b axis

Glide Planes Symmetry Element

Diagonal Glide

the diagonal glide (nglide) have a displacement vector of ½(a+b).

Diamond Glides

the diamond glide (dglide) have a displacement vector of ¼(a+b).

d glide in diamond structure

Translation component

Glide plane element Direction Magnitude Symbol

Axial glide | | to a axis 1/2 a a

Axial glide | | to b axis 1/2 b b

Axial glide | | to c axis 1/2 c c

Diagonal glide | | to face diagonal 1/2 a + 1/2 b, 1/2 b + 1/2 c,

1/2 c + 1/2 a

n

Diamond glide | | to face diagonal

for a face centred cell

1/4 a + 1/4 b, 1/4 b + 1/4 c,

1/4 c + 1/4 a

d

Diamond glide | | to body diagonal

for a body centred cell

1/4 a + 1/4 b + 1/4 c d

Various Kinds of Glide Operations

A FloorTiling Problem

19

Seven Types of Symmetry

Points symmetry

mirror

inversion

rotation

rotoinversion

translation

screw axis

glide plane

Space symmetry

Can not restore the left-handed and the right-handed, only return the equivalent figures

Can restore the left-handed and right-handed

Space Group

Space group ( point & translational symmetry) There are 230 possible arrangements of symmetry elements in the solid state. Any crystal must belong to one (and only one) space group.

Space Groups in 3 Dimension

14 Bravais lattices + 32 point groups

+ screw axes

+ glide planes

230 space groups

Space group symbol Bravais lattice + basis symmetry

Ex) Fm3m Cubic face-centered lattice + m3m (point group)

F (face-centered)

I (body-centered)

C (side-centered)

P (primitive)

了解Herman-Mauguin空间群符号

空间群是经常用简略Herman-Mauguin符号(即

Pnma、I4/mmm等)来指定。 在简略符号中包含能

产生所有其余对称元素所必需的最少对称元素。

从简略H-M符号，我们可以确定晶系、Bravais点阵、

点群和某些对称元素的存在和取向（反之亦然）。

空间群符号LS1S2S3

运用以下规则，可以从对称元素获得H-M空间群符号。

1.第一字母(L)是点阵描述符号，指明点阵带心类型： P， I， F，

C， A， B。

2.其于三个符号(S1S2S3)表示在特定方向（对每种晶系分别规定）

上的对称元素。

3.如果没有二义性可能，常用符号的省略形式 (如Pm，而不用写成

P1m1)。

* 由于不同的晶轴选择和标记，同一个空间群可能有几种不同的符号。

如P21/c,如滑移面选为在a方向，符号为P21/a;如滑移面选为对角

滑移，符号为P21/n。

① 从首位符号知，属于体心格子； ② 从后面的符号知，属于四方晶系4／mmm 对称型； ③ 由对称要素知，平行c轴方向为螺旋轴41 ，垂直c轴有滑移 面a，垂直a轴为对称面m，垂直a轴与b轴的角平分线为滑 移面d 。

对 称 方 向 晶系

第一 第二 第三

三斜 无

单斜 b [010]

正交 a [100] b [010] c [001]

四方 c [001] a [100]/[010] a+b [110]

六方 c [001] a [100]/[010] 2a+b [120]

三方 (R) a+b+c [111] a-b [ 011

]

立方 a [100]/[010]/[001] a+b+c [111] a+b [110]

举例说明：I41／amd 空间群

20

Symmetry combinations

Three Translational vectors

Five rotation axes Four lattice types

32 symmetry point groups

11 basic symmetry elements

Seven crystal systems

14 Bravais lattices

translational screw axes glide planes

230 space groups

Three simple

symmetry elements

230 Space Groups 1-2 : Triclinic, classes 1 and –1 3-15 : Monoclinic, classes 2, m and 2/m 16-24 : Orthorhombic, class 222 25-46 : Orthorhombic, class mm2 47-74 : Orthorhombic, class mmm 75-82 : Tetragonal, classes 4 and -4 83-88 : Tetragonal, class 4/m 89-98 : Tetragonal, class 422 99-110 : Tetragonal, class 4mm 111-122 : Tetragonal, class -42m 123-142 : Tetragonal, class 4/mmm 143-148 : Trigonal, classes 3 and -3 149-155 : Trigonal, class 32 156-161 : Trigonal, class 3m 162-167 : Trigonal, class -3m 168-176 : Hexagonal, classes 6, -6 and 6/m 177-186 : Hexagonal, classes 622 and 6mm 187-194 : Hexagonal, classes -6m2 and 6/mmm 195-206 : Cubic, classes 23 and m-3 206-230 : Cubic, classes 432, -43m and m-3m

2 13 59 68 25 27 36

立方Cu2O沿c方向投影 六方Mg沿c方向投影

O 0

O c/2

Cu 3c/4

Cu c/4

c/2

0

空间群：Pn3m

六组等同点

两套等效点

空间群：P63/mmc

两组等同点

一套等效点

• 晶向指数

点阵中穿过若干阵点的直线方向称为晶向，其指数为[uvw]。晶向指数代表的是一族平行的直线。

晶向指数可如下求得：

1、通过原点作一平行于该晶向的直线；

2、求出该直线上任一点的坐标(u’,v’,w’,);

3、 u’,v’,w’的互质整数为u,v,w, 则[uvw]为晶向指数。

OA [110] OA’ [110]

A crystal structure can be regarded as a grid (lattice) which is a 3D array of points (lattice points).

The grid can be divided into sets of “planes” in different orientations.

Lattice Planes and Miller Indices

It is possible to describe certain directions and planes with respect to the crystal lattice using a set of three integers referred to as Miller Indices.

• 晶面指数

如某一不通过原点的点阵平面在三个轴矢方向上的截距为m（以a为单位），n（以b为单位）和p（以c为单位）。令

1/m : 1/n : 1/p = h : k : l

h : k : l为互质整数比，称为米勒指数（miller indices)，记为(hkl)。它代表一族相互平行的点阵平面，该指数用于表征相应的晶面，也称为晶面指数。

21

Miller Index

x

y

z

O A

B

C

a

b c

x

y

z

a b

c

-y

Miller indices describe the orientation and spacing of a family of planes

(100) (111)

(200) (110)

Examples of Miller Indices

Miller indices describe the orientation and spacing of a family of planes.

the spacing between adjacent planes in a family is referred to as a “d-spacing”

Three different families of planes

d-spacing between (300) planes is one third of the (100) spacing (100) (200) (300)

Families of Planes All planes in a set are identical The planes are “imaginary” The perpendicular distance between pairs

of adjacent planes is the d-spacing

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]

Exercise

What is the Miller index of the plane below?

Find intercepts on a,b,c: 1/2, 1, 1/2 Take reciprocals 2, 1, 2 Multiply up to integers: (2, 1, 2)

Plane perpendicular to y cuts at , 1, (0 1 0) plane

General label is (h k l) which intersects at a/h, b/k, c/l (hkl) is the Miller Index of that plane.

This diagonal cuts at 1, 1, (1 1 0) plane

(0 means that the plane is parallel to that axis)

22

晶系的划分和选晶轴的方法

晶系 特征对称元素 晶胞类型 选晶轴的方法

立方 4个按立方体的对角线取向的三重旋转轴

a=b=c

===90

4个三重轴和立方体的4个对角线平行，立方体的3个互相垂直的边即为a、b、c的方向，a、b、c与三重轴的夹角为54 44

四方 四重对称轴 a=b≠c

===90

c四重对称轴

a，b二重轴或对称面或a，b选c的晶棱

正交 2个互相垂直的对称面或3个互相垂直的二重对称轴

a≠b≠c

===90

a，b，c二重轴或对称面

三方 三重对称轴

菱面体晶胞

a=b=c

==<120≠90

a，b，c选3个与三重轴交成等角的晶棱

六方晶胞

a=b≠c

==90,=120

c三重轴

a，b二重轴或对称面或a，b选c的晶棱

六方 六重对称轴 a=b≠c；==90

=120 c六重对称轴；a，b二重轴或对称面或选a，bc的恰当晶棱

单斜 二重对称轴或对称面 a≠b≠c

==90≠

b二重轴或对称面

a，c选b的晶棱

三斜 无 a≠b≠c

≠≠≠90

a，b，c选3个不共面的晶棱

三轴定向 四轴定向Miller-Bravais指数：

(hkil) i = - ( h + k )

Indexing in the Hexagonal System

In hexagonal unit cells it

is common to refer the

orientation of planes and

lines to four coordinate

axes

The fourth axis a3 is just

= -a2-a1 . This approach

reflects the three fold

symmetry associated with

the unit cell

Indices are expressed as (hkil) h + k = i All cyclic permutations of h, k and i are symmetry equivalent

So are equivalent

- - -

(10 1 0), (1100), (0 110)

Properties of Hexagonal Indices

d-spacing Formula

For orthogonal crystal systems (i.e. ===90): For cubic crystals (special case of orthogonal) a=b=c: e.g. for (100) d = a (200) d = a/2 (110) d = a/ etc.

2

2

2

2

2

2

2c

l

b

k

a

h

d

1

2

222

2a

lkh

d

1

2

A tetragonal crystal has a=4.7 Å, c=3.4 Å. Calculate the separation of the:

(100) 4.7 Å

(001) 3.4 Å

(111) Planes 2.4 Å

A cubic crystal has a=5.2 Å (=0.52nm). Calculate the d-spacing of the (110) plane

o2

22

222

2

A7.32

2.5d

2.5

11

a

lkh

d

1

]ba[c

l

a

kh

d

12

2

2

22

2

23

• Directions:

specific directions in brackets: [uvw]

negative directions “one bar-two one”

equivalent directions: written <111>

• Planes:

specific planes in parentheses: (hkl)

negative indices “bar-three two bar-one”

equivalent planes: written {110}

Indexing of Planes and Directions

[ ]

121

[ ], [ ] [ ]and

111 111 111

( )

3 21

( ), ( ) ( )and

110 011 1 0 1

• 晶带

晶体中若干个晶面平行于某个轴线方向，这些晶面称为晶带，轴线方向为该晶带的晶带轴。用该轴线的晶向指数[uvw]作为带轴符号。

在立方晶体中，属于[001]晶带的晶面有：(100), (010), (100),

(010), (110), (110), (110),

(110), (210), (120)等等。

a b

c

晶带方程：hu + kv + lw = 0

即： 晶面(hkl)属于带轴[uvw]的条件。

u:v:w = (k1l2-k2l1) : (l1h2-l2h1) : (h1k2-h2k1)

h:k:l = (v1w2-v2w1) : (w1u2-w2u1) : (u1v2-u2v1)

总结

阵点指数 (m, n, p)

晶向指数 [uvw]

等效晶向族 <uvw>

晶面指数 (hkl)

等效晶面族{hkl}

晶带[uvw]

晶面指数（hkl）与晶向指数[hkl]的关系

o若a, b无特殊关系，晶面法线不能用晶向指数表示

o并不是所有的晶面指数的法线方向的晶向指数都和晶面指数相同

立方：a=b=c α=β=γ=900 (hkl) ⊥ [hkl]

四方：a=b≠c α=β=γ=900

(hk0) ⊥ [hk0], (00l) ⊥[00l]

正交：a ≠b ≠c α=β=γ=900

(h00) ⊥ [h00] ,(0k0) ⊥ [0k0], (00l) ⊥ [00l]

单斜：a ≠b ≠c α=γ=900≠β

(0k0) ⊥ [0k0]