1 $3 Crystal Structure Analysis 1. Bragg Equation 2. Single Crystal Diffraction 3. X-ray Powder Diffraction Crystal Structure Analysis X-ray diffraction Electron Diffraction Neutron Diffraction Essence of diffraction: Bragg Diffraction Crystals, Powders, and Diffraction Unit Cell Crystal (Crystallite) Xray Powder Pattern Electron Diffraction Powder Pattern Intensity Bragg Angle 0 Polycrystalline Specimen 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 as X-ray diffraction, occurs when the wavelength of the X rays and the distances between atoms in the crystal are of similar magnitude. X-ray Diffraction 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(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。 衍射的物理意义: 复习物理学概念 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 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) 复习物理学概念
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1
$3 Crystal Structure Analysis
1. Bragg Equation
2. Single Crystal Diffraction
3. X-ray Powder Diffraction
Crystal Structure Analysis
X-ray diffraction
Electron Diffraction
Neutron Diffraction
Essence of diffraction: Bragg Diffraction
Crystals, Powders, and Diffraction
Unit Cell
Crystal (Crystallite) Xray Powder Pattern
Electron Diffraction Powder Pattern
Inte
nsi
ty
Bragg Angle 0
Polycrystalline Specimen
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 as X-ray diffraction, occurs when the wavelength of the X rays and the distances between atoms in the crystal are of similar magnitude.
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 scattering 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
Copper anode
Heated tungsten filament
electrons
X-rays
-
+
cathode
anode
PD ≃ 50 kV
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
X-ray Emission Spectrum
upon 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 orbitals not all transitions are possible
the wavelengths are characteristic of the target element
Inte
nsi
ty
K
L
M
K2 K1 K1 K2
Wavelength l c
K2
K1
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
•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
Spectral Contamination in Diffraction Patterns
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)
= 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
intense
X-rays
X-rays
beam
Synchrotron Radiation
More intense X-rays at shorter wavelengths mean higher resolution & much quicker data collection.
7
X-ray Generators The Synchrotron
European Synchrotron Radiation Facility Grenoble, France
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 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 the 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 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 radiation 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.
In 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 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.
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.
2
Divergence slit
Attenuator
Sample
Detector
Graphite
monochromator
Anti-scatter slit
Beam knife
Receiving slit
Typical Setup for Reflectivity Measurements
-2 Reflectometer
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
D – 样品
H – 样品台
F – 接收狭缝
E – 支架
C – 计数管
K – 刻度尺
、2连动
10
样品竖直测角仪
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 variable
Texture
Sensitive
(Ladel)
Fixed Varies as
processes about
Varies as 2 Fixed variable
* Generally fixed, but can rotate about or rock about goniometer axis.
Common Mechanical Movement in Powder Diffractometers
- Reflectometer
PC and software
x-ray tube (source) detector
1 2
1- angle between incoming beam,
and the sample plane
2- angle between detector
and 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 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.
• 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
2 2 2
A polycrystalline sample should contain thousands of crystallites. Therefore, all possible diffraction peaks should be observed.
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.
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
kSample normal
k
kf ki
XRD: “Rocking” Curve Scan
The detector is fixed at 2 position The sample is scanned around The defects in the sample will cause the width of
the peak broaden Rocking 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
Rocking curve of single crystal GaN around 002 diffraction peak showing its detailed structure.
16.995 17.195 17.395 17.595 17.795
0
8000
16000
GaN Thin Film
002 Reflection
Inte
nsi
ty
(Co
un
ts/s
)
theta (deg)
32,6 32,8 33,0 33,2 33,4
100
1000
10000
100000
ZnSe/GaAs
004 rocking curve
FWHM = 112 arcsec
ZnSe Layer
FWHM = 21 arcsec
GaAs
Substrate
Theta (deg)
Inte
nsi
ty (
cps)
Lattice parameter fluctuations
Mosaic structure ?
21 arcsec 112
arcsec
Rocking Curve to Evaluate the Crystal Quality
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 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.
is diffraction angle, R is radii of camera, 2L is the distance of every pair of arcs in the image
22
Front reflections
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 Camera
Can 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 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 K1 and 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.
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 diffraction angle of = . So we can find from:
W2
S1 )
W
S1(
2
2
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
Film from powder camera laid flat.
The pattern of lines on a photograph (left figure) represents possible values of the Bragg angles which 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: From the measurements of each arc we can now generate a table of S1, and sin2.
222hkl
lkh
ad
)lkh(a4
sin222
2
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: Similarly, if we know hkl values, we can calculate the unit cell.
However, often we don’t know hkl or the unit cell….
2
2
2
2
2
2
2c
l
b
k
a
h
d
1
Indexing a Diffraction Pattern
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 Photographs
First 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).
Powder patterns of three common types of simple crystal structures.
(a) Face-centered cubic
(b) Body-centered cubic
(c) Hexagonal close-packed
222lkhN
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
2 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
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
Note: 7 and 15 impossible
Note: we start with the largest d-spacing and work down
Note: not all lines are present in every case
How Many Lines Are Possible?
Cubic Structures
•For a cubic structure,only one quantity is involved, the cell edge or the lattice parameter, we have
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.
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)
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)
2sin 222
lkh
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 present
( 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:
For hexagonal structures, or trigonal structures referred to hexagonal axes
)lc
akh(
a4sin
2
2
222
2hkl
2
)lc
akhkh(
a4sin
2
2
222
2hkl
2
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 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。
Hanawalt – 从三条最强衍射线检索。
Fink – 按照d值大小排序检索。
二、PCPDFWIN(电子版)
18
卡片序号 三条最强线及第一条线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
实验方法能测到的最大
d值
衍射强度的检测方
式
样品最强线与刚玉最强线强度比(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
– amorphous content, particle size and strain
Unit cell size and shape
– from peak positions
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
Amorphous Polycrystalline Crystalline
X-ray Diffraction
For perfect crystals, I(2) consists of functions (perfectly sharp scattering).
For imperfect crystals, the peaks are broadened.
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
t mean size of crystallites
K constant, roughly 1: depends on shape of crystallites (0.89)
B width of reflection in radian
Crystallite Size
cosB
Kt
弧度
19
Effect of Particle Size in X-ray Diffraction
2
B
22 21
2B
Smaller Crystallite
Rela
tive In
ten
sity
2
B
22 21
2B
Larger Crystallite
Rela
tive In
ten
sity
Scherrer Equation
BcosB
9.0t
Bcost
9.0B
or
Nanocrystal X-ray Diffraction
Types of Diffraction Experiment
X-ray Routinely used to provide structural information on compounds and to identify samples Used with both powder and single crystal samples X-rays produced in the home lab or using synchrotrons Can also be used to examine liquids and glasses
Electron diffraction primarily used for phase identification, and unit cell determination on small crystallites in the electron microscope also used for gas phase samples
Neutrons useful source of structural information on crystalline materials, but expensive Also useful for spectroscopy and structure of liquids/glasses good for looking at light atoms sensitive to magnetic moments
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 structures
octahedral and tetrahedral holes
basic structures
Inorganic Crystal Structures
All crystal structures may be described in terms of the unit cell and atomic coordinates of the contents
Many 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?
Square packing coordination number = 4
Close packing coordination number = 6
X Y
Y
Y
Y X Y
Y Y
Y
Y Y
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) :
cubic close packing (CCP or FCC) (Cu, Ag, Au, Al, Ni, Pd, Pt ...)
Common Unit Cells for HCP and CCP Cubic Close Packing (CCP)
Hexagonal Close Packing (HCP)
Coordination number = 12 packing 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
74.023r216
r3
4
4η3
3
For cubic close packing:
The spheres have been packed together as closely as possible, resulting in a packing fraction of 0.74.
Coordination number = 6 packing efficiency 52%
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 and 48% is the spaces between atoms.
Simple Cubic Packing (Primitive Cubic Packing)
a = 2r
a3 = 8r3
No. of atoms = (8 x 1/8) = 1
52.06r8
r3
4
3
3
Body centred cubic unit cell (BCC) and its lattice point representation
fractional counting of atoms with respect to the content of a unit cell !
(Fe, Cr, Mo, W, Ta, Ba ...)
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 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
ABABAC
2m6PD1
h3
ABAC
mcm
6PD 34
h6
(金属La,Ce,Pr,Nd等)
23
Other kinds of Closest Packing
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 interstitial holes within this anion array.
Two main types of interstitial site:
Tetrahedral : CN = 4
Octahedral: CN = 6 Tetrahedral T+ Tetrahedral T
Octahedral O
Two Kinds of Interstitials Holes in Close Packing Structures
Two kinds of Holes
Octahedral Hole Tetrahedral Hole
Different Kinds of Holes
C.N. = 3
C.N. = 4
C.N. = 6
cubic hole
cuboctahedral hole
24
Fractional Coordinates Used to locate atoms within unit cell
1. 0, 0, 0
2. ½, ½, 0
3. ½, 0, ½
4. 0, ½, ½
Note: atoms are in contact along face diagonals (close packed)
Two main types of interstitial site in CCP Octahedral Sites
Coordinate ½, ½, ½ Distance = 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) 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
A variety of different structures form by occupying T+ 、T and O sites to differing amounts: they can be empty, part full or full.
)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:
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
2 anions per unit cell at: )2
1,
3
1,
3
2(),0,0,0(
a b
b3
1
a3
2
Two main types of interstitial site in CCP Octahedral Holes in HCP Unit Cell