1

$3 Crystal Structure Analysis

1. Bragg Equation 2. Single Crystal Diffraction3. X-ray Powder Diffractiony

Crystal Structure Analysis

X-ray diffraction

Electron Diffraction

Neutron Diffraction 

Essence of diffraction:  Bragg Diffraction

Crystals, Powders, and Diffraction

Polycrystalline Specimen

Unit Cell

Crystal (Crystallite) Xray Powder Pattern

Electron Diffraction Powder Pattern

Inte

nsit

y

Bragg Angle0

X-ray Diffraction When an X-ray beam bombards a crystal, the atomic structure of the crystal causes the beam to scatter in a specific pattern. This phenomenon, known 

X-ray Diffraction

p p pas X-ray diffraction, occurs when the wavelength of the X rays and the distances between atoms in the crystal are of similar magnitude.

衍射（diffraction）：波在经过障碍物边缘或孔隙时所发生的扩散或弯曲现象，绕射波随后发生相互干涉，产生加强或减弱相间的许多区域。

复习物理学概念

衍射是晶体的固有特性

衍射是散射波的叠加，是波动的特性

衍射的特点是能量守恒，动量不守恒

衍射的物理意义：

散射（scattering）：描述粒子间的碰撞过程。即，具有足够能量的入射粒子轰击被研究的靶（如原子、原子核等）结果是入射粒子被散射到各个方向。或者说电磁波在其通过的路径上被物质中的粒子所偏转的一种过程。

散射过程可以用粒子的状态是否因碰撞而发生改变而区分为两

复习物理学概念

种类型：1.一种碰撞的结果，粒子间只有动量交换，而粒子的内部状态不变，这类散射称为弹性散射。如Rayleigh scattering（瑞利散射）2. 碰撞使粒子的内部状态发生改变，如粒子被激发、碎裂等，这类散射称为非弹性散射。 如拉曼效应（Raman Effect）和康普顿效应（Compton Effect）

2

复习物理学概念

干涉（interference）：影响介质相同部分的两个或两个以上波运动的互作用，导致的合成波中瞬间干扰为干涉波中瞬间干扰的向量和。当一小光源的光通过屏上的狭缝，利用从狭缝射出的光，照亮第二屏上相邻的两条狭缝，使这两条狭缝射出的光落在第三屏上，就会形成一系列平行干涉条纹。从狭缝射出的两个光波它们的 大值重合时就会产生亮纹（相长干涉），其中一个波的 大值和另一波的 小值重合时就会产生暗纹（相消干涉）。

X ray (Roentgen ray) Roentgen experiment in 1895

Properties?

Cathode anode

+–

U kSo named: X ray!

Electromagnetic radiation of very short wavelength and very high energy produced when high-speedelectrons strike a solid  target.

Unknown!!

Wilhelm Konrad Roentgen,First Nobel Price 1901 for

伦琴拍摄的一张X射线照片，伦琴夫人的手骨与戒指 ←←

• 1895年Wilhem Conrad von RÖntgen发现了X射线。

• 1912年慕尼黑大学结晶学权威Prof. Paul von Groth :晶体是三维周期排列结构，从阿佛加德罗常数知周期在10-8m

光学权威P f A S f ld (和K h)认为 X光是

Laue Equations

• 光学权威Prof.A.Sommerfeld (和Koch)认为： X光是波，且在Walter和Paul的X射线通过不同狭缝的实验上测X光波长，未成；在此基础推测X射线可能有10-9cm数量级的波长。

• P.P. Ewald 博士论文－－晶体光学性质当Ewald 和Laue讲师讨论博士论文时， Laue产生一个想法，X光和晶体作用又如何？

• 在这些基础上，劳埃提出一个设想：在人工做的狭缝光栅上，X光衍射失败是因为狭缝太宽，X的狭缝光栅上，X光衍射失败是因为狭缝太宽，X光波长太短，而三维周期排列的晶体是一个理想的天然立体光栅。

第一次衍射实验

在劳埃的建议下，伦琴实验室的弗里德里赫（W.Friedrich）和尼平（P.Knipping），用硫酸铜晶体作为光栅衍射X射线，得到世界上第一张X射线衍射图。

实验证实--

1. 晶体是分子尺度上的三维周期性排列的；2. X光跟可见光一样是波动的。

这不仅是结晶学的转折点，同样是现代科学的里程碑。从这以后物质的结构可以用X光研究了。

1912年，劳埃发表了《X射线的干涉现象》一文

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Laue--Nobel Prize winner of 1914

德国的Laue第一次成功地进行X射线通过

晶体发生衍射的实验，晶体发生衍射的实验，验证了晶体的点阵结构理论。并确定了著名的晶体衍射劳埃方程式。

crystal ⇔ periodic repetition of identical unit cells ⇔ diffraction grid⇒ constructive interference of scattered waves only in particular directions:

von Laue Diffraction

h)cos(cosa 0

X射线投射到晶体中时，会受到晶体中原子的散射，而散射波就好象是从原子中心发出，每一个原子中心发出的散射波又好比一个源球面波。

von Laue Diffraction

• For different incidence directions, the diffraction patterns are different

k)(b

h)cos(cosa 0

l)cos(cosc

k)cos(cosb

0

0

, , , 0, 0, 0分别为散射光和入射光与三个点阵轴矢的夹角。

不方便！

Singlecrystal

von Laue Diffraction

由于原子在晶体中是周期排列，这些散射球面波之间存在着固定的位相关系，它们之间会在空间产生干涉，结果导致在某些散射方向的球面波相互加强，而在某些方向上相互抵消，从而出现衍射现象，即在偏离原入射线方向上，只有在特定的方向上出现散射线加强而存在衍射斑点，其余方向则无衍射斑点。

Laues ExperimentsEinstein regarded this experiment as the most beautiful

Laue Images of  Zincblende ZnS(a)show its four-fold axis，(b)show its three-fold 

axis

beautiful one in physics.

Nobel Prize winner of 1914

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直线点阵的衍射

平面点阵的衍射 空间点阵的衍射

Laue实验的意义：

1.证实了Ｘ射线是一种波长很短的电磁波，可以利用晶体来研究Ｘ射线的性质，从而建立了Ｘ射线光谱学；并且对原子结构理论的发展也起了有力的推动作用。

2.证实了几何晶体学提出的空间点阵假说，晶体内部的原子、离子、分子等确实是作规则的周期性排列，使这一假说发展为科学理论。

3.Ｘ射线结构分析（即Ｘ射线晶体学）的诞生：利用Ｘ射线晶体衍射效应来研究晶体的结构，根据衍射方向可确定晶胞的形式和大小，根据衍射强度可确定晶胞的内容（原子、离子、分子的分布位置）。是一种在原子──分子水平上研究化学物质结构的重要实验方法。

the Bragg Condition of Crystal Diffraction

劳厄的文章发表不久，就引起英国布拉格父子的关注。 1913年，亨利·布拉格和他的儿子威廉·劳伦斯·布拉格开始用X射线研究晶体结构。

布拉格在实验中发现 晶体中有一系列原子平面布拉格在实验中发现，晶体中有 系列原子平面反射着白色X光中的某些波长一定的特征X光，

基于这一发现和对它的理论解释，布拉格把劳埃方程变换成布拉格方程。

Strong reflection of the incident wave will occur for the set of incident angles that satisfy the Bragg condition: 2dsin =n

the index n defines the path difference between waves i & ii when the diffraction occurs … for a given value of n this path difference is n.

d

Geometry of Bragg Diffraction

sind2n hkl

Path difference for diffraction of rays from adjacent planes is 2dhklsin, which must correspond to n for constructive interference.

dhklsindhklsin

光程差 = AB + BC = dsin + dsin = 2dsin

满足衍射的条件为： 2dsin = n

d为面间距， 为Bragg角。这即为Bragg方程。

5

n = 2dsinwhere n order of diffraction

X-ray wavelength

d spacing between layers of atom

Bragg's Equation

d spacing between layers of atom

angle of diffraction

Bragg's Law is the fundamental law of X-ray crystallography.

布拉格定律（X射线反射定律）

布拉格方程 + 光学反射定律

布拉格定律 （X射线反射定律）

所有的被照射原子所产生的散射只有满足布拉格方程，散射线满足“光学镜面反射”条件（散射线、入射线与原子面法线共面）时，各原子的散射波将具有相同的位相，干涉结果产生加强，才能产生反射（衍射），或称散射才能发生加强干涉。

Bragg方程反映了X射线在反射方向上产生衍射的

条件，借用了光学中的反射概念来描述衍射现象。与可见光的反射比较，X射线衍射有着根本的区别：

1、单色射线只能在满足Bragg方程的特殊入射角下有衍射。

2、衍射线来自晶体表面以下整个受照区域中所有原子的散射贡献的散射贡献。

3、衍射线强度通常远低于入射线强度。

4、衍射强度与晶体结构有关，有系统消光现象。

在几乎所有的教科书中，常用“反射（reflection）”这个术语描述衍射问题，或者将散射（scattering），“反射(reflection)”和“衍射（diffraction）”作为同义词混合使用。

但应强调指出，X射线从原子面的反射和可见光的镜面反射不同，前者是有选择地反射，其选择条件为布拉格定律；而一束可见光以任意角度投射到镜面上时都可以产生反射，即反射不受条件限制。

因此，将X射线的晶面反射称为选择反射，反射之所以有选择性，是晶体内若干原子面反射线干涉的结果。

Bragg's Law

When X-rays are scattered from a crystal lattice, peaks of scattered intensity are observed which correspond to the following conditions:The angle of incidence = angle of scatteringThe pathlength difference is equal to an integer The pathlength difference is equal to an integer number of wavelengths.The conditions for maximum intensity contained in Bragg's law allow us to calculate details about the crystal structure, or if the crystal structure is known, to determine the wavelength of the X-rays incident upon the crystal.

The Braggs

British physicists William Henry Bragg (1862~1942) and William Lawrence Bragg (1890~1971) won Nobel Physical Prize in 1915 due to their achievements on the Structure Analysis via X-ray.

6

Generation of X-rays

Heated

electrons are produced by thermionic emission from a W filament and are accelerated by a large potential difference the high energy electrons (≃ 50 keV) bombard a metal target (usually Cu, but can also be Mo)X-rays are generated by the interaction between electrons and target

Copper anode

Heated tungsten filament

electrons

X-rays

-

+

cathode

anode

PD ≃50 kV

X-ray Emission Spectrumupon collisions the high energy electrons can knock inner core electrons from the target atoms, leaving vacancies in the innermost shell (K)these vacancies are rapidly filled by electronic transitions from the other  orbitalsnot all transitions are possible

the wavelengths are characteristic of the target element

Copper anode:

Inte

nsit

y

K

L

M

K2K1K1K2

Wavelengthl c

K2K1

K1

K2

Copper anode:K 1.54178 Å  K1 1.540598 Å K2 1.54434 Å

K 1.3922 Å

)3

1

3

2( 2K1KK

K1

K2

K W L1

K1

K2 K1

K2

Spectral Contamination in Diffraction Patterns

K W L1

•The K1 & K2 doublet will almost always be present– Very expensive optics can remove the K2 line– K1 & K2 overlap heavily at low angles and are more 

separated at high angles•W lines form as the tube ages: the W filament contaminates the target anode and becomes a new X-ray source

•W and K lines can be removed with optics

K双线分离现象

CuK1 = 1.5405Å, K2 = 1.544Å, K = 1.5418Å

由Bragg方程：2dsin =

2d cos • =

= tg • /(弧度) = tg • / • 180/(度) = tg • /(弧度) = tg • / • 180/(度)

= 15o = 0.035o

= 35o = 0.09o

= 80o = 0.74o

X-ray Generation Using a Synchrotron

High intensity X-rays can be generated using a particle accelerator such as a synchrotron: charged particles (electron or positrons) are accelerated round a circle and emit radiation tangentially. A particular wavelength can be selected from the continuous spectrum of X-rays generated.Synchrotron radiation: tunable

intenseintense

X-rays

X-rays

beam

Synchrotron RadiationMore intense X-rays at shorter wavelengths mean higher resolution & much quicker data collection.

7

X-ray Generators The Synchrotron

Electrons (or positrons) are released from a particle accelerator into a storage ring. The trajectory of the particles is determined by their energy and the local magnetic field. Magnets of various types are used to manipulate the particle trajectory. When the particle

European Synchrotron Radiation FacilityGrenoble, France

beam is “bent” by the magnets, the electrons (or positrons) are accelerated toward the center of the ring. Charged particles moving under the influence of an accelerating field emit electromagnetic radiation, and when they are moving at close to relativistic speeds, the radiation emitted includes high energy X-ray radiation..

Single Crystal X-ray Diffraction

Single crystal X-ray diffraction is a kind of method by putting a crystal in the beam, observing what reflections come out at what angles for what orientations of the crystal with what intensities.

•Advantage: You can learn everything to know about hthe structure.

•Disadvantages: You, however, may not have a single crystal. It is time-consuming and difficult to orient the crystal. If more than one phase is present, you will not necessarily realize that there is more than one set of reflections.

Single Crystal X-ray Diffraction (Cont.)

• Primary application is to determine atomic structure (symmetry, unit cell dimensions, space group, etc.).

• Older methods used a stationary crystal with "white X-ray" beam (X-rays of variable ) such h B ' i ld b i fi d b that Bragg's equation would be satisfied by

numerous atomic planes.

• Modern methods (rotation, Weissenberg, precession, 4-circle) utilize various combination of rotating-crystal and camera setup to overcome limitations of the stationary methods

Laue Method• The Laue method is mainly used to determine the

orientation of large single crystals. White radiation is reflected from, or transmitted through, a fixed crystal.

• The diffracted beams form arrays of spots, that lie on curves on the film. The Bragg angle is fixed for every set of planes in the crystal. Each set of planes picks out and diffracts the particular wavelength from the white

di i h i fi h l f h l f dradiation that satisfies the Bragg law for the values of d and involved. Each curve therefore corresponds to a different wavelength. The spots lying on any one curve are reflections from planes belonging to one zone. Laue reflections from planes of the same zone all lie on the surface of an imaginary cone whose axis is the zone axis.

• Experimental• There are two practical variants of the Laue method, the

back-reflection and the transmission Laue method.

•劳埃照相法原理

a是准直管（保证很细的一束X光照到单晶上）；

b是晶头 晶体就是固定在晶头上；b是晶头，晶体就是固定在晶头上；

c是吸收入射光的小铅片；

d是用来防止可见光使胶片感光的黑纸；

e是照相底片

劳埃照相中所用晶体是固定不动的单晶体，X射线是白色X射线

劳埃照相原理

劳埃照相中底片放置的两种方法

背射透射

在劳埃照相中，衍射斑点在底片上的位置取决于晶胞的形状和大小。调节晶体的某一对称方向与X光平行，衍射斑点在底片上呈现的对称性反映出晶体在这一方向的对称性。

transmission back-reflection

8

•应用

1. 确定晶体宏观对称性

在确定弗里得尔对称性后，可利用倍频效应，旋光性，压电效应，热电效应等判断晶体有无对称中心，进而确定晶体的宏观对称性

劳埃照相

进而确定晶体的宏观对称性。

2. 晶体的定向

在进行X光结构分析时，在工业上使用单晶时，经常

要对晶体进行定向。尤其是熔融法长成的单晶呈圆柱状，不用劳埃照相无法判定其结晶学方向。也可用原理与劳埃照相类似的X光定向仪。

3. 结晶化学分析

劳埃照相比粉末法相比提供了更进一步的结晶学数据。劳埃照相能提供11种弗里得尔对称性，

这就确定了未知晶体的晶系。依据劳埃照相能计算轴率和格子常数 并利用大角度数据对轴

劳埃照相

计算轴率和格子常数，并利用大角度数据对轴率，格子常数算得较精确。劳埃照相的突出优点是衍射点对应于一个晶面。相比之下粉末法对于上述工作是有困难的，尤其对于低级晶系。

取得的数据与手册（Crystal Data,1974)进行全面比较，从而进行鉴定。前提是样品必须长成单晶。

Rotating Crystal MethodIn the rotating crystal method, a

single crystal is mounted with an axis normal to a monochromatic X-ray beam. A cylindrical film is placed around it and the crystal is rotated about the chosen axis. As the crystal rotates, sets

f l tti l ill t of lattice planes will at some point make the correct Bragg angle for the monochromatic incident beam, and at that point a diffracted beam will be formed.

The reflected beams are located on the surface of imaginary cones. When the film is laid out flat, the diffraction spots lie on horizontal lines.

转动照相法也是测量单晶样品。

与劳埃照相法的区别在于单晶必须转动，采用单色X光。

在底片上得到的衍射图形不是连续的线而是

把圆形底片展开，即得旋转图，每一条由若干斑点形成的线而是一

组斑点。条由若干斑点形成的线叫层线，与h=0相对应的叫中央层线，与h=1或h=-1相对应的叫一层线。

转动照相底片层线图

转动照相的三种用途

1）晶体的结构分析

由于转动相的指标化有着根本的缺点，现仅用作的辅助工具。仪器引入了较高的对称性，我们无法从

转动照相辅助 具 仪器引 了较高的对称性 我们无法从

旋转图测定晶体的对称性。

2）纤维周期的测定

由于多数魏森堡照相纤维结构有点像捆好的一捆稻草，在长度方向取向是一致的，在与纤维垂直方向其排列一般是无序的，用转动照相测定其周期是比较合适的。3）单晶的格子常数测定

转动照相的缺点

根本的缺点：指标化时对一些衍射指标无法分辨。

在晶体绕c轴旋转时所有的hk3衍射点都分布在一维

的直线上的每个衍射点都分布在第三层上，要在一维的直线上的每个衍射点解出两个变量h和k，这一般说是困难的，有时是做不到的。

如果晶体具有使hk3 kh3的对称性 这时指标化可如果晶体具有使hk3~kh3的对称性，这时指标化可以进行。

在四方晶系中，hk3和 kh3的衍射将收集在底片的同一点上，在进行强度测量时无法区分强度中hk3和kh3成分。

转动照相中所用晶体是转动的单晶体，X射线是单色X射线

魏森堡照相转动的单晶体，转轴是水平的

9

Chemical CrystallographySingle Crystal Analysis

Stoe IPDS Image Plate Diffraction System

single crystalsize < 0.5 mm

在这种几何安排下，照相底片所记录的hk0这一层衍射点是未受到任何歪曲的倒易点阵的点阵点—

倒易点阵照相，这样的照片指标化就十分简单。

Chemical Crystallography Powder Analysis

powder

STADI-P Stoe Powder diffractometer

powder samplein glass capillary

Schematic Illustration of X-ray Diffraction

To obtain nearly monochromatic X-rays, an X-ray tube is used to produce characteristic X-rays. Matched filters are used in the X-ray beam to optimize the fraction of the energy which is in the K line.

Divergence slit

Attenuator

Detector

G hit

Anti-scatter slit

Beam knife

Receiving slit

Typical Setup for Reflectivity Measurements

Sample

Graphite monochromator

-2Reflectometer

θ-2θ入射光不动

衍射仪构造示意图

G – 测角仪圆

S – X射线源

、2连动

D – 样品

H – 样品台

F – 接收狭缝

E – 支架

C – 计数管

K – 刻度尺

10

样品竖直测角仪 Common Mechanical Movement in Powder Diffractometers

Type Tube Specimen Receiving Slit

r1 r2

Brag-Brentano :2

Fixed Varies as Varies as 2 Fixed =r1

Brag-Brentano :

Varies as Fixed Varies as Fixed =r1

Seeman-Bohlin Fixed Fixed Varies as 2 Fixed variableTexture

Sensitive (Ladel)

Fixed Varies as processes about

Varies as 2 Fixed variable

* Generally fixed, but can rotate about or rock about goniometer axis.

- Reflectometer

PC and software

x-ray tube(source) detector

1 21 2

1- angle between incoming beam,and the sample plane 2- angle between detectorand the sample plane

Sample

θ-θ样品不动，立式（样品平躺）

、连动

样品水平型测角仪 Bragg-Brentano diffractometer

The Bragg-Brentano diffractometer is the dominant geometry found in most laboratories. In this system, if the tube is fixed, this is called -2 geometry. If the tube moves (and the specimen is fixed), this is called - geometry. The essential characteristics are:(1) The relationship between (the angle between the specimen surface and the incident X-ray beam) and 2 (the angle between the incident beam and the receiving slit detector) is maintained throughout the analysis.(2) r1 and r2 are fixed and equal and define a diffractometer circle in which the specimen is always at the center.

11

Radiation Method

White Laue: stationary single crystal Monochromatic Powder: specimen is polycrystalline, and

therefore all orientations are simultaneously presented to the beam

Rotation Weissenberg: oscillation Rotation, Weissenberg: oscillation De Jong-Bouman: single crystal rotates

or oscillates about chosen axis in path of beam

Precession: chosen axis of single crystal precesses about beam direction

• Diffraction can occur whenever Bragg's law is satisfied. With monochromatic radiation, an arbitrary setting of a single crystal in an X-ray beam will not generally produce any diffracted beams. There would therefore be very little information in a single crystal diffraction pattern from using monochromatic radiation.Thi bl b b i l • This problem can be overcome by continuously varying or over a range of values, to satisfy Bragg's law. Practically this is done by:

(1)using a range of X-ray wavelengths (i.e. white radiation), or

(2)by rotating the crystal or, using a powder or polycrystalline specimen.

A single crystal specimen in a Bragg-Brentano diffractometer would produce only one family of peaks in the diffraction pattern.

At 2=20.6, Bragg’s law fulfilled for the (100) planes, producing a diffraction peak.

The (110) planes would diffract at 2=29.3; however, they are not properly aligned to produce a diffraction peak (the perpendicular to those planes does not bisect the incident and diffracted beams). Only background is observed.

The (200) planes are parallel to the (100) planes. Therefore, they also diffract for this crystal. Since d200 is 1/2 d100, they appear at 2=42.

2

A polycrystalline sample should contain thousands of crystallites. Therefore, all possible diffraction peaks should be observed.

2 2 2

For every set of planes, there will be a small percentage of crystallites that are properly oriented to diffract (the plane perpendicular bisects the incident and diffracted beams). Basic assumptions of powder diffraction are that for every set of planes there is an equal number of crystallites that will diffract and that there is a statistically relevant number of crystallites, not just one or two.

①参加衍射的衍射面（hkl）与样品面平行。不平行的晶面即使发生衍射也不能被检测器检测到。

②多晶每一个衍射峰都不是一批晶粒衍射，如参加(100)衍射就不参加(111)衍射。检测器检测到的是一个统计结果，所以要把样品颗粒磨的尽可能均匀，这样使得平行于样品面的衍射面尽量一样多。衍射强度大小与衍射面面积有关，如(111)面大于(220)面，则其衍射强度也大。

布拉格公式再理解

①不能决定是三维晶体，只能证明一个方向有无周期性。XRD研究的是低维体系，且这里的维度指

的是周期性而不是形貌的。如：三维：一般晶体；二维：石墨烯；一维：分子链；2+1维：石墨。②沿面方向平移不破坏布拉格条件②沿面方向平移不破坏布拉格条件。

③只要“高度”不变，原子替换不破坏布拉格条件。④面内点阵排列还是无序排列，不破坏布拉格条件。

⑤θ角实际不能测，2θ角实际能测。实际测的是入射光和衍射光的夹角。

12

“R k” S l

k Sample normalk

kfki

XRD: “Rocking” Curve Scan

Vary orientation of k relative to sample normal while maintaining its magnitude.How? “Rock” sample over a very small angular range. Resulting data of Intensity vs. theta (, sample angle) shows detailed structure of diffraction peak being investigated.

“Rock” Sample

The detector is fixed at 2 positionThe sample is scanned around The defects in the sample will cause the width of

the peak broadenRocking curve is usually used to indicate the

quality of the thin film

Rocking Curve

依据d2θ,将探测“停”在2θ位，扫样品（对卧

式探测仪）立式探测仪相应实现此目的。

XRD: Rocking Curve Example

16000

GaN Thin Film002 Reflection

y  /s)

Rocking curve of single crystal GaN around 002 diffraction peak showing its detailed structure.

16.995 17.195 17.395 17.595 17.7950

8000

Inte

nsit

y(C

ount

s/

theta (deg)

10000

100000ZnSe/GaAs

004 rocking curve

FWHM = 112 arcsec

ZnSe Layer

FWHM = 21 arcsec

GaAs

Substrate

ensi

ty (

cps)

21 arcsec112 

arcsec

Rocking Curve to Evaluate the Crystal Quality

32,6 32,8 33,0 33,2 33,4100

1000

Theta (deg)

Int

Lattice parameter fluctuations

Mosaic structure ?

金刚石

2Ɵ=43.95

13

2Ɵ=75.30

Rocking Curves assessing crystal quality

To estimate the crystal quality, a crystal is rotated through with the counter set at a known Bragg angle, 2. The resulting intensity versus curve is known as a rocking curve. The width of the rocking curve is a direct measure of the range of orientation curve is a direct measure of the range of orientation on mosaic spread present in the irradiated area of the crystal, as each sub-grain of the crystal will come into orientation as the crystal is rotated. For a film that isn't truly epitaxial, the width of a rocking curve of the layer peak will be a measurement of the quality of the layer.

①看多晶取向。片材、丝材方向不同，取向不同（注意装样方向）。②看大晶粒分布。③找某一相的晶粒。④看单晶质量：微观应力，开裂。

Rocking Curves

⑤找单晶衍射面，进而作θ-θ联动。（补差角）⑤找单晶衍射面 进而作 联动 补差角

The Debye-Sherrer Camera

多晶样品中的小晶粒是随机取向的,当某个晶面满足布拉格方程时 衍射就

平行单色光入射，粉末或多晶块状样品。

格方程时，衍射就会发生。由于小晶粒的取向是随机的，所以会有许许多多的小晶粒满足布拉格方程，这些衍射点就连成一个圆锥。粉末法成像原理

粉末照相法样品的制备

•将粉末塞在毛细管内（这时粉末压缩）

•将粉末用透明胶涂在细的（直径要小于百分之一毫米）玻璃丝上．

•将粉末塞在透明胶或任何其他快干有机漆所制的小管内．•用液体透明胶将粉末调成糊状，滚成圆柱体．

•纤维定向样品的制样可以在纤维方向用胶磋成柱状，再

通常将多晶粉末制成细棒状样品。

纤维定向样品的制样可以在纤维方向用胶磋成柱状，再粘于玻璃丝上。

粉末量很少时，则只在玻璃丝顶部涂上一点点胶粘剂。

为防止样品氧化，需在毛细管内加入玻璃丝后将管口封起来。

注意

粉末法的 大优越性在于它不要求一个大晶体，而任何物质均可磨成粉末，这样扩大了X射线的应用范围，如金属、合金、无数的微晶质、化学沉淀物、矿物晶体以及岩石等。

14

Powder Diffraction

x-rays

Bragg’s Law : sind2nλ

衍射线束与X射线的夹角为2，由于小晶粒是无

数的，某一晶面的反射实际上是以4为顶角的

德拜-谢乐法的衍射几何图

R是相机半径2L是一对相应衍射弧线间的中线距离

圆锥面

π

180

R4

L2θ

L2180

πRθ4

The Debye-Sherrer Camera

L2R4

22

Front reflections

L2180

R4

180

R4

L2

is diffraction angle, R is radii of camera, 2L is the distance of every pair of arcs in the image

22

Back reflections

Debye-Scherrer Camera

• A very small amount of powdered material is sealed into a fine capillary tube made from glass that does not diffract X-rays. The specimen is placed in the Debye-Scherrer camera and is accurately aligned to be in the center of the camera. X-rays enter the camera through a collimator.

• The powder diffracts the X-rays in accordance with Bragg’s law to produce cones of diffracted beams. These cones intersect a strip of photographic film located in the cylindrical camera to produce a characteristic set of arcs on the film.

Debye-Scherrer CameraCan record sections on these cones on film or some other X-ray detector

– Simplest way of doing this is to surround a capillary sample with a strip of film

– Can convert line positions on film to angles and intensities by electronically scanning film or measuring positions using a ruler and or measuring positions using a ruler and guessing the relative intensities using a “by eye” comparison

1916

X-ray powders diffraction

Powder Diffraction Film• When the film is removed from the camera,

flattened and processed, it shows the diffraction lines and the holes for the incident and transmitted beams.

• There are always two arcs in the x-ray beams K1and K2, this causes the highest angle back-reflected arcs to be doubled. From noting this, it is always clear which hole is for the transmitted beam and which is for the incident beam in the film.

Dutch post stamp, 1936, memorizing Peter Josephus Wilhelmus Debye and his Nobel prize.

15

Powder X-ray Diffraction (Powder diffraction film)

The distance S1 corresponds to a diffraction angle of 2. The angle between the diffracted and the transmitted beams is always 2. We know that the distance between the holes in the film, W, corresponds to a , , pdiffraction angle of = . So we can find from:

W2

S1 )

W

S1(

22

or在进行粉末照相时，相机为圆柱形，样

品位于相机中心，每一衍射圆锥为圆柱形底片所截，得到一对弧线，将底片张开，得到长条状粉末照相底片。

从实验量得W，根据布拉格公式可以计算出（以弧度为单位）

The scheme shows the Debye cones that intersect the film in the camera, and how diffractions are measured on the film to determine the d-spacings for the reflections measured.

•In a linear diffraction pattern, the detector scans through an arc that intersects each Debye cone at a single point; thus giving the appearance of a discrete diffraction peak.

•If the crystallites are randomly oriented, and there are enough of them, then they will produce a continuous Debye cone.

Measurement of Debye-Scherrer Photographs

The pattern of lines on a photograph (left figure) represents possible values of the Bragg angles which

Film from powdercamera laid flat.

satisfy Bragg’s equation:

hklhkl sind2n

We know Bragg's Law: n = 2dsin and the equation for inter-planar spacing, d, for cubic crystals is given by:

where a is the lattice parameter this gives:

ad

From the measurements of each arc we can now generate a table of S1, and sin2.

222hkllkh

d

)lkh(a4

sin 2222

2

Indexing a diffraction pattern means assigning Miller indices hkl to each value of d.If we know the unit cell, we can assign hkl values to each d value using:

222 lkh1

Indexing a Diffraction Pattern

Similarly, if we know hkl values, we can calculate the unit cell.

However, often we don’t know hkl or the unit cell….

2222 c

l

b

k

a

h

d

1

16

Debye-Scherrer powder camera photographs of gold (Au), a Face centered cubic structure that exhibits a fairly simple diffraction

Debye-Scherrer powder camera photographs of Zircon (ZrSiO4). Zircon is a fairly complex tetragonal structure and this complexity is reflected in the diffraction pattern.

2dsin = n

The split of the XRD lines:The symmetry of structure decreased, the lines increased.

Interpretation of Powder PhotographsFirst task is to familiarize ourselves with these patterns. The three most common structures are called face-centered cubic (FCC), body-centered cubic (BCC) and hexagonal close-packed (HCP).

P d tt f th

222 lkhN

Powder patterns of three common types of simple crystal structures.

(a) Face-centered cubic

(b) Body-centered cubic

(c) Hexagonal close-packed

h k l h2 + k2 + l2 h k l h2 + k2 + l2 1 0 0 1 2 2 1, 3 0 0 9 1 1 0 2 3 1 0 10 1 1 1 3 3 1 1 11

In a cubic material, the largest d-spacing that can be observed is 100=010=001. For a primitive cell, we count according to h2+k2+l2

How Many Lines Are Possible?

1 1 1 3 3 1 1 112 0 0 4 2 2 2 12 2 1 0 5 3 2 0 13 2 1 1 6 3 2 1 14 2 2 0 8 4 0 0 16

Note: 7 and 15 impossibleNote: we start with the largest d-spacing and work downNote: not all lines are present in every case

Cubic Structures•For a cubic structure，only one quantity is involved, the cell edge or the lattice parameter, we have

)lkh(a4

sin 2222

2

N

222hkllkh

ad

nsind2

Not all values of h2 + k2 + l2, which we shall call N, are possible. Numbers such as 7, 15, 23, 28, 60 are said to be forbidden.

For small values of N, the values of h, k and l are easily deduced.

N nsind2

Observable diffraction peaks

222 lkh

Primiti e

PC: 1,2,3,4,5,6,8,9,10,11,12..

Cubic System

RatioN

Primitivecubic BCC: 2,4,6,8,10, 12….

FCC: 3,4,8,11,12,16,24….

222hkllkh

ad

nsind2

17

Ex: An element, BCC or FCC, shows diffraction peaks at 2: 40, 58, 73, 86.8,100.4 and 114.7. Determine: (a) Crystal structure?

(b) Lattice constant?(c) What is the element?

2theta theta (hkl)

40 20 0 117 1 (110)

2sin 222 lkh

40 20 0.117 1  (110)58 29 0.235 2 (200)73 36.5 0.3538 3 (211)

86.8 43.4 0.4721 4 (220)100.4 50.2 0.5903 5 (310)114.7 57.35 0.7090 6 (222)

a =3.18 Å , BCC,   W

Systematic Absences and Centering

The presence of a centered lattice leads to the systematic absence of certain types of peak in the diffraction pattern

For P lattice, no systematic absence（衍射全部存在,没有衍射消光现象）

For I centered lattice:h + k + l = 2n for a line to be presentp

（ h+k+l=偶数衍射存在, h+k+l=奇数衍射消光）

For F centered lattice:h + k =2n, k + l = 2n and h + l = 2n for a line to be present

（h, k, l全奇或全偶衍射存在， h, k, l奇偶混杂时，衍射消光）

For C centered lattice:h + k =2n for a line to be present

（h+k=偶数衍射存在，h+k=奇数衍射消光

立方晶系三种格子的系统消光

h, k, l h2+k2+l2 (sin2)比

简单P 都可能1:2:3:4:5:6:8:9:10:11:12:13(缺7,15,23,28,31等比例项)

6 8 6 8体心I

面心F

h+k+l=偶数衍射存在h+k+l=奇数衍射消光

2:4:6:8:10:12:14:16:18:20=1:2:3:4:5:6:7:8:9:10(不缺)

h,k,l全奇全偶衍射存在h,k,l奇偶混杂消光

3:4:8:11:12:16:19:20:24:27:32衍射线呈单线双线交替排列

Tetragonal and Hexagonal Structures

For tetragonal structures, we have

)lc

akh(

a4sin 2

2

222

2hkl2

For hexagonal structures, or trigonal structures referred to hexagonal axes

)lc

akhkh(

a4sin 2

2

222

2hkl2

Powder X-ray Diffraction

Measuring samples consisting of a collection of many small crystallites with random orientations.

Powder XRD is used routinely to assess the purity and crystallinity of materials

Each crystalline phase has a unique powder diffraction patterndiffraction pattern

Measured powder patterns can be compared to a database for identification

•Advantages over Single Crystal Diffraction

It is usually much easier to prepare a powder sample. You are guaranteed to see all reflections.

JCPDS – Joint Committee on Powder Diffraction Standards

ICDD – International Centre for Diffraction Data

一、PDF – Powder Diffraction File

索引 Alphabetical 从物质名称检索索引：Alphabetical – 从物质名称检索。

Hanawalt – 从三条 强衍射线检索。

Fink – 按照d值大小排序检索。

二、PCPDFWIN（电子版）

18

卡片序号三条 强线及第一条线d值和强

度

化学式及名称

数据的可信度：星号，i，O，空白，C，R

靶材及波长

单色器类型：石墨单色器或滤波

片

相机直径

实验方法能测到的 大

d值

衍射强度的检测方

式

样品 强线与刚玉 强线强度比（50/50）

参考文献

晶系 空间群, Pna21

晶胞参数

a/b和c/b值单胞化学

式量数

理论密度

光学数据

Information from Powder XRD

Phase purity

– both qualitative and quantitative

Crystallinity

amo pho s content pa ticle si e and – amorphous content, particle size and strain

Unit cell size and shape

– from peak positions

Amorphous Polycrystalline Crystalline

20 30 40 50 60 70 800

10000

20000

30000

40000

50000

20 30 40 50 60 70 800

50

100

150

200

20 30 40 50 60 70 80200

400

600

800

1000

1200

1400

1600

1800

2000

X-ray Diffraction

For perfect crystals, I(2) consists of functions (perfectly sharp scattering).

For imperfect crystals, the peaks are broadened. p

For liquids and glasses, it is a continuous, slowly varying function.

X-ray diffraction works on the principle that x-rays form predictable diffraction patterns when interacting with a crystalline matrix of atoms.

The width of the peaks in a powder pattern contains information about the crystallite size in the sample (and also the presence of microstrain)

Scherrer equation

Crystallite Size

K Scherrer equation

t  mean size of crystallitesK  constant, roughly 1: depends on shape of crystallites (0.89)B  width of reflection in radian

cosB

Kt

弧度

19

Effect of Particle Size in X-ray Diffraction

Smaller Crystallite

nten

sity

Larger Crystallite

nten

sity

Scherrer Equation

B

9.0t

2

B

2 22

Rel

ativ

e In

2

B

2 22

Rel

ativ

e In

BcosB

Bcost

9.0B

or

Nanocrystal X-ray Diffraction

Types of Diffraction ExperimentX-rayRoutinely used to provide structural information on compounds and to identify samplesUsed with both powder and single crystal samplesX-rays produced in the home lab or using synchrotronsCan also be used to examine liquids and glasses

Electron diffractionElectron diffractionprimarily used for phase identification, and unit cell determination on small crystallites in the electron microscope also used for gas phase samples

Neutronsuseful source of structural information on crystalline materials, but expensiveAlso useful for spectroscopy and structure of liquids/glassesgood for looking at light atoms

iti t ti t

Evidence of superdense Aluminum synthesized by ultrafast microexplosion

Nature Communications, 2011, 2, 445

micro-X-ray diffraction (-XRD) image acquired (a) in the centre of the shockwave compressed area, and (b) outside the shockwave compressed area. (c) comparison between a radially integrated experimental (-XRD profile obtained from the experimental data shown in (a), and theoretical (-XRD profile expected for bcc-al refined using materials studio package, assuming 18 nm crystalline size. the crystalline (hkl) plane indices are shown next to the peaks. for comparison, simulated profiles of host sapphire (vertically offseted grey) and native fcc-Al (purple) are also shown.

(a) Micro-X-ray diffraction (-XRD) patterns from the laser-structured region (the pattern of Figure 2) where bcc-Al signatures or fcc-Al are dominant. (b) Comparison of the two integrated -XRD patterns shown in (a). Numerical fit of the experimentally observed peaks by the fcc-Al structure with a cell size of a=3.584 Å is achieved; bottom purple marks show the expected position of the fcc-Al peaks at room conditions with a=4.049 Å. The retrieved fcc-Al is subjected to pressures ~60–100 GPa.

$4 Crystal Chemistry

close packed structuresoctahedral and tetrahedral holesbasic structuresbasic structures

Inorganic Crystal Structures

All crystal structures may be described in terms of the unit cell and atomic coordinates of the contents

M i i bMany inorganic structures may be described as arrays of space filling polyhedra  tetrahedra, octahedra, etc.

Many structures  ionic, metallic, covalent  may be described as close packed structures. 

20

Close Packing Concept

2D close-packed layer  Most efficient way to fill space

Random arrangement of atoms (hard, neutral spheres)

Two-Dimensional Packing

• What is the most efficient way to arrange circles on a plane surface?

Y YY

Square packingcoordination number = 4

Close packingcoordination number = 6

XY

Y

Y XY Y

YY

Close Packed Structures - Metals

Most efficient way of packing equal sized spheres. In 2D, have close packed layers

Coordination number (CN) = 6. This is the maximum possible for 2D packing.

Can stack close packed (CP) to give 3D structures.

Three Dimensional Packing

The simplest arrangement is to place a second layer of spheres on top of the first layer.

Two Main Stacking Sequences

If we start with one close packed layer, two possible ways of adding a second layer (can have one or other, but not a mixture) :

If we start with one close packed layer, two possible ways of adding a second layer (can have one or other, but not a mixture) :

Two Main Stacking Sequences

21

3 layers

ABA Hexagonal close-packing (HCP)

把第三层放在与第一层一样的位置

ABC Cubic close-packing (CCP)

把第三层放在堵住头二层漏光的三角形空隙上

Three-dimensional Packing of Spheres (Atoms)

hexagonal close cubic close hexagonal close packing (HCP)

cubic close packing (CCP)

hexagonal close packing (HCP)(Be, Mg, Zn, Cd, Ti, Zr, Ru ...)

Common Unit Cells for HCP and CCP

cubic close packing (CCP or FCC)(Cu, Ag, Au, Al, Ni, Pd, Pt ...)

Cubic Close Packing (CCP)

Hexagonal Close Packing(HCP)

Coordination number = 12packing efficiency 74%

No matter what type of packing, the coordination number of each equal size sphere is always 12

22

Packing Fraction

The fraction of space which is occupied by atoms is called the “packing fraction”, , for the structure.

space available

atoms by occupied spaceη

For cubic close packing:

74.023r216

r34

4η3

3

For cubic close packing:

The spheres have been packed together as closely as possible, resulting in a packing fraction of 0.74.

One layer of atoms placed directly over another layer of atoms will give rise to the simple cubic unit cell.

Here 52% of the unit cell is filled with the atoms

d 8% i th

Simple Cubic Packing(Primitive Cubic Packing)

Coordination number = 6packing efficiency 52%

and 48% is the spaces between atoms.

a = 2r

a3 = 8r3

No. of atoms = (8 x 1/8) = 1

52.06r8

r34

3

3

Body centred cubic unit cell (BCC) and its lattice point representation

A layer of atoms placed in the spaces between the first layer of atoms gives rise to the body-centered cubic unit cell. The first and third layer li H 68% f th

fractional counting of atoms with respect to the content of a unit cell !

(Fe, Cr, Mo, W, Ta, Ba ...)

line up. Here 68% of the unit cell is filled and 32% is the spaces between spheres. Examples are iron, chromium, and all group 1A elements.

6802.0

Unit Cell of Body-centered Close Packing

Tungsten

Body-centered packing

Metals usually have one of three structure types:FCC, HCP or body centered cubic (BCC). The reasons why a particular metal prefers a particular structure are still not well understood.

FCC HCP BCC

Other kinds of Closest Packing

Packing Fraction=0.74

ABABAC2m6PD1

h3

ABAC

mcm

6PD 34

h6

(金属La，Ce，Pr，Nd等)

23

Other kinds of Closest Packing

(金属Tb）

Packing Fraction=0.74

ABCAB

mPD d

—

313

ABCACB

mcm

6PD 34

h6

(金属Tb）

Close Packed Ionic Structures

Ionic structures cations (+ve) and anions (-ve)

In many ionic structures, the anions, which are larger than the cations, form a close packed array and the cations occupy close packed array and the cations occupy interstitial holes within this anion array.

Two main types of interstitial site:

Tetrahedral : CN = 4

Octahedral: CN = 6

Two Kinds of Interstitials Holes in Close Packing Structures

Tetrahedral T+ Tetrahedral T

Octahedral O

Two kinds of Holes

Octahedral Hole Tetrahedral Hole

Different Kinds of Holes

C.N. = 3

C N 4

cubic hole

C.N. = 4

C.N. = 6

cuboctahedral hole

24

Fractional CoordinatesUsed to locate atoms within unit cell

1. 0, 0, 02. ½, ½, 03 ½ 0 ½

Two main types of interstitial site in CCP

3. ½, 0, ½4. 0, ½, ½

Note: atoms are in contact along face diagonals (close packed)

Octahedral Sites

Coordinate ½, ½, ½Distance = a/2

Coordinate 0, ½, 0 [=1, ½, 0 ]Distance = a/2

In a face centered cubic anion array, cation octahedral sites at:

)0,2

1,0(),0,0,

2

1(),

2

1,0,0(),

2

1,

2

1,

2

1(

Location and Number of Octahedral Holes in a FCC (CCP) Unit Cell

Z = 4(number of atoms in the unit cell)unit cell)

N = 4 (number of octahedral holes in the unit cell)

Can divide the FCC unit cell into 8 ‘minicubes’ by bisecting each edge; in the center of each minicube is a tetrahedral site

So 8 Tetrahedral Sites in a FCC

)2

1,0,

2

1(),

2

1,

2

1,0(),0,

2

1,

2

1(),0,0,0(

)0,2

1,0(),0,0,

2

1(),

2

1,0,0(),

2

1,

2

1,

2

1(

4 anions per unit cell at:

4 octahedral sites at:)

4

3,

4

3,

4

1(),

4

3,

4

1,

4

3(),

4

1,

4

3,

4

3(),

4

1,

4

1,

4

1(4 tetrahedral T+ sites at:

)4

3,

4

3,

4

3(),

4

1,

4

3,

4

1(),

4

3,

4

1,

4

1(),

4

1,

4

1,

4

3(4 tetrahedral T sites at:

A variety of different structures form by occupying T+ 、T and O sites to differing amounts: they can be empty, part full or full. 

444444444444

Sizes of Interstitials

FCC 

Spheres are in contact along face diagonals

octahedral site, bond distance = a/2

radius of octahedral site = (a/2)  r

tetrahedral site, bond distance =

radius of tetrahedral site =         r

a4

3

a4

3

25

Octahedral and Tetrahedral Sites in a FCC

)2

1,0,

2

1(),

2

1,

2

1,0(),0,

2

1,

2

1(),0,0,0(

)0,2

1,0(),0,0,

2

1(),

2

1,0,0(),

2

1,

2

1,

2

1(

4 anions per unit cell at:

4 octahedral sites at:)

4

3,

4

3,

4

1(),

4

3,

4

1,

4

3(),

4

1,

4

3,

4

3(),

4

1,

4

1,

4

1(4 tetrahedral T+ sites at:

)4

3,

4

3,

4

3(),

4

1,

4

3,

4

1(),

4

3,

4

1,

4

1(),

4

1,

4

1,

4

3(4 tetrahedral T sites at:

Fractional Coordinates in HCP Unit Cell

Two main types of interstitial site in CCP

2 anions per unit cell at: )2

1,

3

1,

3

2(),0,0,0(

a b

b3

1a

3

2

Octahedral Holes in HCP Unit Cell

)0,2

1,0(),0,0,

2

1(),

2

1,0,0(),

2

1,

2

1,

2

1( )

4

3,

3

2,

3

1(),

4

1,

3

2,

3

1(

4 anions per unit cell 2 anions per unit cell

Tetrahedral Holes in HCP Unit Cell

T+

T-

4 anions per unit cell

)4

3,

4

3,

4

1(),

4

3,

4

1,

4

3(),

4

1,

4

3,

4

3(),

4

1,

4

1,

4

1(

)4

3,

4

3,

4

3(),

4

1,

4

3,

4

1(),

4

3,

4

1,

4

1(),

4

1,

4

1,

4

3(

)8

7,

3

1,

3

2(),

8

1,

3

1,

3

2(),

8

5,0,0(),

8

3,0,0(

2 anions per unit cell

Tetrahedral and Octahedral Holes in HCP Unit Cell

)8

7,

3

1,

3

2(),

8

1,

3

1,

3

2(),

8

5,0,0(),

8

3,0,0( )

4

3,

3

2,

3

1(),

4

1,

3

2,

3

1(

26

八面体空隙中心坐标: (1/3,2/3,1/4), (1/3,2/3,3/4)

四面体空隙中心坐标: (0,0,3/8), (0,0,5/8), (2/3,1/3,1/8), (2/3,1/3,7/8)

Octahedral Holes and Tetrahedral Holes in HCP Unit Cell

Each face of this octahedron is shared with a tetrahedral hole. The fourth atom of the tetrahedron is at a corner of the unit cell.

2 anions per unit cell at:

2 octahedral sites at:

4 tetrahedral sites at: )8

7,

3

1,

3

2(),

8

1,

3

1,

3

2(),

8

5,0,0(),

8

3,0,0(

)4

3,

3

2,

3

1(),

4

1,

3

2,

3

1(

)2

1,

3

1,

3

2(),0,0,0(

2 anions per unit cell at:

2 octahedral sites at:

4 tetrahedral sites at: )8

7,

3

1,

3

2(),

8

1,

3

1,

3

2(),

8

5,0,0(),

8

3,0,0(

)4

3,

3

2,

3

1(),

4

1,

3

2,

3

1(

)2

1,

3

1,

3

2(),0,0,0(

Octahedral Holes and Tetrahedral Holes in HCP Unit Cell