第四章电子衍射3_11-10-12讲义
第四章电子衍射2_11-9-30讲义
若样品内某(hkl)晶面满足布拉格条件，则在 与入射束呈2θ角方向上产生衍射。
透射束和衍射束分别与离试样L处的照相底片 相交于O’和P’点。
O’点：透射斑点000 P’点：衍射斑点hkl
若照相底片上中心斑点到衍射斑点的距离为R
R = tan2θ L
电子衍射时，满足布拉格定律的角度θ很小，故
代入布拉格公式2dsinθ=λ可得：
rd = f0λ
由于底片上（或荧光屏上）记录到的衍射花样 是物镜背焦面上第一幅花样的放大像。若中间镜 与投影镜的放大倍率分别为Mi和Mp。则底片上相 应衍射斑点与中心斑点的距离R应为
R = rM iM p
因为
(R / M iM p )d = λfo
(a) 第一幅衍射花样的形成和选 区电子衍射原理
爱瓦尔德球
L、λ、R均已知，故可求 出晶面间距d，晶面夹角。
Rd = Lλ R可在衍射
谱上量出
利用电子衍射谱进行
L
结构分析的依据。
衍射花样
晶体结构、位向
入射 束
O
试样
1 λ 2θ 1 λ
1d
G
倒易 点阵
底板
O' R P'
相机常数为某一定值，所以R反比于d。由此可见，在电子衍射中， 晶体参数d与衍射斑点R之间的关系比Ｘ射线衍射中相应的关系简单。
目前，先进的透射电子显微镜都有自动电子补偿器消除相对磁转 角，为在显微图像上显示出晶体学方向提供了便利。
4.4 倒易点阵平面及其画法
电子衍射花样
倒易点阵与Ewald球面相截的 部分，再在荧光屏上投影。
单晶的电子衍射谱是一个二 维倒易平面的放大。
电子衍射
（ ）表示平面，*表示倒易， 0表示零层倒易面。
这个倒易平面的法线即正空 间晶带轴[uvw]的方向，倒易平 面上各个倒易点分别代表着正空 间的相应晶面。
0
4. 晶带轴的求法
若已知零层倒易面上任意二个倒易矢量的坐标， 即可求出晶带轴指数.由
得
u=k1l2-k2l1
v=l1h2-l2h1
简单易记法 h1 k1 l1 h1 k1 l1
Hot- and Cold-Stage TEM
20oC a 220oC b 25oC
c
AFE-1 AFE FE
d -100oC
a and b: PbZrO3 single crystal C and d: Pb(ZrSnTi)O3 ceramic
AFE-2
七、多晶电子衍射成像原理与衍射花样特征
图8-2 多晶电子衍射成像原理
金的原子力显微镜照片
倒易点阵
正点阵：晶体点阵
倒易点阵：与正点阵存在倒易关系
a*•b=a* • c=b* • a=b* • c=c* • a=c* • b=0
a* • a=b* • b=c* • c=1
写成标量形式
a*=1/a×cosφ b*=1/b×cosψ c*=1/c×cosω
ω
与正点阵的关系
反射式高能电子衍射分析(RHEED):以高能
电子照射较厚样品分析其表面结构，电子 束以掠射方式(与样品表面的夹角小于5o) 照射样品，使衍射发生在样品浅表层。
RHEED用荧光屏作结果显示，在超高真空
环境下工作。
低能电子衍射(LEED)：电子束能量为10~1000eV (一般为10~500) 。由于电子能量低，衍射结果 只能显示样品表面1~5个电子层的结构信息，因 此是分析晶体表面结构的重要方法，广泛用于 表面吸附、腐蚀、催化、外延生长、表面处理 等材料表面科学与工程领域。 低能电子衍射仪器为低能电子衍射仪，也是在 超高真空环境下工作。
实验二 电子衍射实验讲义
2024/10/16
1
0 、历史背景
目录
一、实验目的
二、实验原理
三、实验仪器
四、实验内容及步骤 五、实验数据记录及处理 六、注意事项
0 历史背景
➢ 关于光的“粒子性”和“波动性”的争论，人们最终接 受了光既具有粒子性又具有波动性，即光具有波粒二象 性。
➢ 1924年法国物理学家德布罗意deBeroglie）提出了一 切微观实物粒子都具有波粒二象性的假设。1927年戴 维逊与革末发表了用低速电子轰击镍单晶产生电子衍射 的实验结果，成功地完成了电子衍射实验，验证了电子 的波动性，并测得了电子的波长,与按德布罗意公式计 算出的波长相吻合。
七、思考题
➢ 电子衍射的实验目的是？ ➢ 简述电子衍射管的结构及各部分作用； ➢ 100KV加速电压下电子波波长值为多少？用电子衍射现象 研究晶体结构？对此你能提出一些看法吗？
四、实验内容及步骤
1、定性观察电子衍射图样
调节电子束聚焦，便能得到清晰的电子衍射图样。观察 电子衍射现象，增大或减小电子的加速电压值，观察电子衍 射图样直径变化情况，并分析是否与预期结果相符，用手机 拍摄衍射图样。
2、测量运动电子的波长
对不同的加速电压（10KV、11KV、12KV、13KV）从 荧光屏上直接测量(111), (200), (220), (311) 4个晶面族对电 子的衍射环的直径2r;将测量值分别代入算式，计算实验测 量波长。
➢ 两个月后，英国的汤姆逊和雷德用高速电子穿透金属薄 膜的办法直接获得了电子衍射花纹，进一步证明了德布 罗意波的存在。
一、实验目的
➢ 测量运动电子的波长，验证德布罗意公式 ➢ 理解真空中高速电子穿过晶体薄膜时的衍射现象，
电子衍射
电子衍射与X射线衍射比较相似性：波的叠加导致布拉格公式结构因子消光规律s s v v vK称为电子衍射相机常数λ0S v λS vg hkl vλ0S v λS vg hkl v衍射斑点矢量是产生这一斑点晶面组的倒易矢量的比例放大，K是放大倍数故仅就衍射花样的几何性质而言：单晶花样中的斑点可以直接看成是相应衍射晶面的倒易阵点，各个斑点的就是相应的，之间的夹角就等于产生衍射的两个晶面之间的夹角。
g v R v R v g v R vfr多晶电子衍射花样的标定及其应用二、应用1、已知晶体标定仪器的相机常数KRd =150kv加速电压下拍得多晶金的衍射花样①测量环的半径R i从里向外测得圆的直径：2R 1＝17.6mm 、2R 2＝20.5mm 、2R 3＝28.5mm ，………即R l ＝8.8mm ，R 2＝10.3mm 、R 3＝14.3mm 、……已知金为面心结构，a ＝0.407nm②计算R i 2及R i 2/R 12（R 1—最小半径），根据R i 2/R 12确定衍射环指数8:4:3R :R :R 232221=18:6:4:2 17.9:00.3:98.1:1R :R :R :R 2D2C 2B 2A ==简单立方：1，2，3，4，…体心立方：2，4，6，8，10，12，…h＋k＋l＝2n 面心立方：3，4，8，11，12，16，19，20，…全奇全偶满足体心结构标准花样对照法：由R=Kg可推知：单晶电子衍射花样实质是满足衍射条件的某个零层倒易面的放大像。
∗0]uvw [对于本例，可知，衍射花样是的放大像∗0]110[单晶电子衍射花样分析三、应用1、物相鉴定原理与X射线相同，根据d值和强度查PDF卡片但仅跟据某一晶带的衍射斑点，d值不够8个。
须倾动晶体样品，拍摄不同晶带的衍射花样。
根据化学成分，热处理工艺，可将待测相限制为几种可能，可根据下面三个条件，仅由一张花样鉴别。
<1>点阵类型与PDF卡片相符<2> 衍射斑点必须自洽<3> 底指数晶面间距与卡片的标准相符，允许误差3%左右单晶电子衍射花样分析三、应用2、晶体取向关系的验证和确定<1> 两相取向关系常用两相的一对互相平行的晶面及面上平行的晶向来表示()()[]BA BA w v u //]uvw [l k h //hkl ′′′′′′()()()()()()B 333A 333B 222A 222B 111A 111l k h //l k h l k h //l k h l k h //l k h ′′′′′′′′′表示：面或三对平行的晶向来有时也用三对平行的晶[][][]333A 333B 222A 222B111A 111w v u //]w v u [w v u //]w v u [w v u //]w v u [′′′′′′′′′）），根据()110()011()020()111()111()200(200()202B(h 2k 2l 2)C E F A(h 1k 1l 1)1g v 2g v 'g v D O gv 乘一个系数n，使（hkkl）转化为整数爱瓦尔德球像L 1电子衍射中间镜的物平面与背焦面物镜一次像中间镜投影镜二次像终像。
电子衍射讲义
Aims previous | next On completion of this TLP you should:∙Understand why the spots on an electron diffraction pattern appear where they do.∙Know how to index a diffraction pattern from a sample with a known lattice.Introduction:Electrons can act as waves as well as particles; this is a consequence of quantum mechanics. A series of electrons hitting an object is exactly equivalent to a beam of electron waves hitting the object and it produces a diffraction pattern in the same way as a beam of X-rays does.The two important differences between electron and X-ray diffraction are that (1) electrons have a much smaller wavelength than X-rays, and (2) the sample is very thin in the direction of the electron beam (of the order of 100 nm or less) - it has to be thin so that enough electrons can get through to form a diffraction pattern without being absorbed. These factors conspire to have a fortunate effect on the Ewald sphere construction (see The Ewald sphere in the X-ray Diffraction TLP) and diffraction pattern:1. The thin sample makes the reciprocal lattice points longer in thereciprocal direction corresponding to the real-space dimension in which the sample is thin:It should be noted that there is not necessarily always a particular plane oriented like this. However, it is usual for identification of crystallinephases in a sample to orient the sample so that the electron beam isparallel to a low index lattice direction, as this makes the electrondiffraction pattern easier to interpret.2. The small electron wavelength makes the radius of the Ewald spherevery large (recall its radius is 1/λ). The small electron wavelength alsomakes the diffraction angles θ small (1-2°); this can be seen bysubstituting a wavelength of 2.51 x 10-12 m into the Bragg equation (see The Bragg law in the X-ray Diffraction TLP).These make the Ewald sphere diagram look like this so that whole layers of the reciprocal lattice end up projected onto the film or screen:Note that the large (strong) spot in the middle is the straight-through beam (the beam which has passed through the sample without diffracting). This always has the index 000.Caution 1: systematic (kinetic) absences appear in electron diffraction patterns just as in X-ray diffraction patterns, for the same reason: the various features of the lattice or motif diffract electrons in the same direction but the phase factors from the various features cancel, leaving an absence.Caution 2: sometimes where there should be a systematic absence, the spot appears to be still there. This is because of the strong interaction between electrons and atoms: there is a small but significant probability that an electron will be diffracted twice, from two planes one after another - i.e. in two different reciprocal lattice directions one after another. These two directions can add up so that the twice-diffracted electron may arrive at a position in reciprocal space where there is a systematic absence. As an example, the diagram below is a schematic of the [011] electron diffraction pattern of silicon: the 200 type reflections are systematically absent. The intensity at the 200 reflections is caused by double diffraction (arising from the addition of the two reciprocal lattice vectors shown). Thus, in words, intensity can occur in the 200 reflectionfrom, firstly, diffraction from the 11planes, followed by, secondly, difraction by the 1 1 planes as the electron wave passes throught the specimen.Mathematics relating the real space to the electron diffraction patternThe distance, r hkl , on the pattern between the spot hkl and the spot 000 is related to the interplanar spacing between the hkl planes of atoms, d hkl , bythe following equation:(Derivation )where L is the distance between the sample and the film/screen.We can therefore say that the diffraction pattern is a projection of thereciprocal lattice with projection factor L , because reciprocal lattice vectors have length 1/d hkl .Relation 2Since the diffraction pattern is a projection of the reciprocal lattice, the angle between the lines joining spots h 1k 1l 1 and h 2k 2l 2 to spot 000 is the same as the angle between the reciprocal lattice vectors [h 1k 1l 1]* and [h 2k 2l 2]*. This is also equal to the angle between the (h 1k 1l 1) and (h 2k 2l 2) planes, orequivalently the angle between the normals to the (h 1k 1l 1) and (h 2k 2l 2)planes. This angle is θin the diagram below.Using these two relations between the diffraction pattern and the reciprocal lattice, we are now able to index the electron diffraction pattern from a specimen of a known crystal structure.The two pages linked to here refer only to indexing the central region of the diffraction pattern - the rest will be dealt with later.back Indexing with the orientation of the electron beamknownFrom the Ewald sphere diagram, we know that the zero order Laue zone (ZOLZ) contains reflections hkl where hu + kv + lw= 0 (the Weiss zone law). This ZOLZ can be identified by finding two reciprocal lattice vectors in theZOLZ. Suppose these two reciprocal lattice vectors are h1a* + k1b* + l1c* and h2a* + k2b* + l2c*. Then we knowh1u + k1v + l1w= 0andh2u + k2v + l2w= 0and that the angle between these reciprocal lattice vectors is the angle between the h1k1l1 and h2k2l2 planes.Other reflections in the electron diffraction pattern can then be deduced from simple vector addition, with the proviso that the indices of the reciprocal lattice vectors are integers and that they are not forbidden by the lattice. The pattern can then be built up manually or by computer.Example of indexing with a known electron beam orientationbackSuppose the material under examination is copper, and suppose the electron beam direction is [211]. Copper has a cubic close packed structure with a lattice parameter, a, of 0.361 nm. Allowed reflections must have h,k,l either all even or all odd. Thus the planes with the highest interplanar spacings (and hence those that give rise to reflections with the smallest r hkl values) are {111}, {200}, {220}, {311}, {222}, etc.Looking at the {111} planes, it is apparent that the Weiss zone law is obeyedfor (11) when [uvw] =[211]. Hence 11 is a possible reciprocal latticevector.No {200} plane will obey the Weiss zone law for [uvw] = [211], but of the{220} planes it is apparent that (02) will. Hence 02is a second possible reciprocal lattice vector.The angle between the 11 and 02reciprocal lattice vectors is 90° - the dot product of these two reciprocal lattice vectors is zero. The ratio of thelengths of these two reciprocal lattice vectors is . These are the two shortest reciprocal lattice vectors in the [211] electron diffraction pattern. Thus the pattern looks like:back Indexing with the orientation of the electron beamunknownIf we do not know the beam orientation, it is rather more difficult to find which reciprocal plane is the one projected down onto the film.One approach is to consult tables of angles and distance ratios for the low index reflections for the structure of the crystal we are imaging. Again, we will use copper as an example.Table of anglesThe angles in this table are the angles between the reciprocal lattice vectors given at the sides of the table in the appropriate row and column. Such aTable of distance ratiosThe dimensionless numbers in the central portion are the ratios of the 1/d hklexcluded. Thus for copper, which has an F lattice, the reflections are those with h ,k ,l all even or all odd.Now we pick two spots on the diffraction pattern and measure the angle between them and the ratio of their distances from the 000 spot - and see if they correspond to any of the values in the tables.View an exampleOnce we know for sure what two of the non-collinear dots are, we can index the rest of the pattern by vector addition.Example of indexing with an unknown electron beamorientationbackIf were 72.5° and we were to measure the ratio x/y and find it to benumerically equal to 1.66 then we could be reasonably convinced that the dot at Y could be labelled as the 200 spot, and the dot at X could be labelled as 113. In this case our predicted electron beam direction is in thedirection common to the 200 and 113 planes, i.e. [03]. This particular electron diffraction pattern has a central rectangular repeat. If this is correct, further spots on this diffraction pattern can be indexed in aself-consistent manner by vector addition.Laue zoneSo far we have been looking at the central region of the diffraction pattern. This is only a part of the total diffraction pattern. If we look again at the Ewald sphere construction, we have:We have been indexing the portion in the middle with the 000 spot in it. However, there are also areas of diffraction spots at the edges of the film, caused by the Ewald sphere intersecting points in an adjacent parallel plane containing reciprocal lattice points. (If the film was small or the camera length large it is possible that it did not catch these spots at the side, so that we sometimes only have the middle part.)These outlying parts of the diffraction pattern are called Higher Order Laue Zones (HOLZs). Each of the HOLZs can be described by an equation of the general formhu + kv + hw Nwhere:∙N is always an integer, and is called the order of the Laue zone.∙[uvw] is the direction of the incident electron beam.∙hkl are the co-ordinates of an allowed reflection in the N th order Laue zone.The middle part of the diffraction pattern, with 000 in it, is the zero order Laue zone (ZOLZ), because it comes from the plane for which N =0: an allowed reflection hkl in the ZOLZ is joined to the origin 000 by a reciprocal lattice vector that lies in the ZOLZ. For the ZOLZ the electron beam [uvw] and the allowed reflection hkl satisfy the Weiss zone law hu + kv + lw= 0. The next layer up has a value N =1, then N =2, and so on, as shown.From the geometry of the way in which the Ewald sphere intersects the HOLZs, the radius of the N th HOLZ ring, R n, in reciprocal space, is given to a very good approximation by the formulaassuming that the wavelength of the electrons is much less than the modulus |uvw| of the direction [uvw] in the crystal parallel to the electron beam direction. Thus, HOLZs are seen more easily at lower voltages (e.g. 100 kV rather than 300 kV) and when the electron beam is parallel to a relatively high index direction in a crystal.It is possible to index the reflections in the HOLZs on a diffraction pattern. Examples of such indexing are given in the book Transmission Electron Microscopy of Materials by D B Williams and C B Carter.Kikuchi lines previous | next Kikuchi lines often appear on electron diffraction patterns:An example of a "two-beam" electron diffraction pattern with a number of Kikuchi lines.A pair of Kikuchi lines is arrowed.[The term "two-beam" denotes the fact that the straight-through beam, 000, and one diffraction spot are both diffracting very strongly. The intensity of all spots in this electron diffraction pattern are significantly weaker by comparison with these two beams.](Click on image to view larger version)We will not learn to index the Kikuchi lines in this TLP. Instead, we will explain their origin and behaviour with the help of the following animation.Kikuchi lines are interesting because of what they do when the crystal is moved in the beam. Diffraction spots fade or become brighter when the crystal is rotated or tilted, but stay in the same places; the Kikuchi lines move across the screen.The difference in behaviour can be explained by the position of the effective source of the electrons that are Bragg-scattered to produce the two phenomena. The diffraction spots are produced directly from the electron beam, which either hits or misses the Bragg angle for each plane; so the spot is either present or absent depending on the orientation of the crystal. The source of the electrons that are Bragg-scattered to give Kikuchi lines is the set of inelastic scattering sites within the crystal. When the crystal is tilted the effective source of these inelastically scattered electrons is moved, but there are always still some electrons hitting a plane at the Bragg angle - they merely emerge at an angle different to the one that they did before the crystal was tilted.Using polycrystalline materials in the TEM previous | next Just as with X-rays, a completely isotropic fine-grained polycrystalline sample will give a diffraction pattern of concentric rings in the zero order Laue zone (ZOLZ), as the many small crystals at random orientations produce a continuous angular distribution of hkl spots at distance 1/d hkl from the 000 spot - a ring of radius 1/d hkl around the 000 spot for each allowed reflection. The rings are then indexed according to the order of allowed reflections within the ZOLZ.As the grain size increases, the rings within the diffraction pattern break up into discontinuous rings containing discrete reflections. If there is any texture (preferred orientation) within the specimen, arcs may be seen instead of complete rings.Convergent beam electron diffraction (CBED)previous | next When a convergent beam is used instead of a parallel beam of electrons, the rays converge to a point within the specimen and come out the other side inverted like a camera. However, we do not look at the inverted image; we look at the diffraction pattern, with the spots magnified:Depending on the camera length chosen, either the zero order Laue zone can be examined or the zero order Laue zone and higher order Laue zones. Two examples of CBED images are shown below. The symmetry seen is such patterns can be related to the space group symmetry of the specimen.Examples of CBED images:Diffraction pattern showing a zero order Laue zone with a mirror in the pattern as shown(Click on image to view larger version)Diffraction pattern showing a first order Lauezone(Click on image to view larger version)Using other methods in conjunction with electrondiffraction previous | next Electron diffraction is a powerful technique - but other techniques must be used with it to put the results in context. This is a brief synopsis of how other methods can be used to help.Optical imagingThis is a very important way of analysing a specimen. Using the naked eye and optical microscopes we can determine down to a point-to-point resolution limited by the wavelength of light how many phases there are and how they relate to one another. We can also infer what type of material they are likely to be and how they may have been processed.Chemical analysisA wide range of chemical techniques can be used to find out what components are present in the different phases and in what proportions. This will narrow the field of possible elements that we need to consider when analysing ourdiffraction results. These techniques range from simple chemical tests, through infrared spectroscopy of organic samples, to a wide variety of chemical characterisation techniques that can often be performed within thetransmission electron microscope.TEM imagingUsing the TEM to image the same area of sample that is being used toproduce the diffraction pattern is an invaluable technique:Nitrided surface layer of austenitic stainlesssteel(Click on image to view larger version)Diffraction pattern from nitrided surface layer of austenitic stainless steel (Click on image to view larger version) Using the image to verify that the double dots in the diffraction pattern arebeing caused by the two crystal structures either side of the twin boundary, we can index the pattern and determine the twin plane and the crystal structures either side of it.Summary previous | next In this teaching and learning package we have considered how electron diffraction patterns are formed in the transmission electron microscope. The principles of how to index spot electron diffraction patterns have been discussed in some detail. Although we have considered how to index electron diffraction patterns from relatively simple crystal structures toillustrate the basic principles, these principles are generic and can therefore be applied to any crystal structure. We have also considered other features of electron diffraction patterns such as the formation of Kikuchi lines, the formation of convergent beam electron diffraction patterns and theformation of higher index Laue zones.。
第四章 电子衍射讲解
多晶电子衍射谱的形成
X射线花样形成示意图
电子衍射花样形成示意图
电子衍射基本公式
R Rd L
Ld
通常将K=λL=Rd称为 相机常数，而L被称 为相机长度。
3、现在的电镜极靴缝都非常小，放入样品台以后很 难再放得下一个光阑；
现在电镜的选区光阑可以做到非常小，如JEOL 2010 的选区光阑孔径分别为：5μm,20μm,60μm,120μm
衍射与选区的对应
A 磁转角
一束平行于主轴的入射电子束通过电磁 透镜将被聚焦在轴线上一点，即焦点F
类比光学玻璃凸透镜
Mi M p
mmM, 误i 差
Mp 3.3%
仪器误差——电子波长的不稳定性
内标像机常数
随物镜电流变化的校正曲线
电子衍射花样的标定与分析
电子衍射谱的标定就是确定电子衍射图谱中的诸 衍射斑点（或者衍射环）所对应的晶面的指数和 对应的晶带轴（多晶不需要）。
电子衍射谱主要有多晶电子衍射谱和单晶电子衍 射谱。电子衍射谱的标定主要有以下几种情况：
简立方：N=1, 2, 3, 4, 5, 6, 8, 9, 10, …
体 心：N=2, 4, 6, 8, 10, 12,…
=1:2:3:4:5:6:7:8
面 心：N=3,4,,,,8,,,11,12,,,,16,,,19,20,,,,24,,,27,28,…
金刚石：N=3, ,,,,8,,,11, ,,,,16,,,19, ,,,,24,,,27, ,…
在 透 射 电 子 显 微 分 析 中 ， 即 有 Fresnel （ 菲 涅 尔 ） 衍 射 （近场衍射）现象，同时也有Fraunhofer（夫朗和费） 衍射（远场衍射）。 Fresnel（菲涅尔）衍射（近场衍 射）现象主要在图像模式下出现，而Fraunhofer（夫朗 和费）衍射（远场衍射）主要是在衍射情况下出现。
电子衍射和中子衍射110315(2024版)
单晶电子衍射谱，可以视为由某一特征平行四边形 (斑点为平行四边形的四个顶点)，按一定周期扩展而成。 可以找出许多平行四边形，作为一个衍射谱的基本单元， 我们选择与中心斑点最邻近的几个斑点为顶点构成的四 边形为基础，按下列定义的平行四边形为基本特征平行 四边形——约化平行四边形。
约化平行四边形
在底片透射斑点附近，取距透射斑点O最近的两个不共 线的班点A、B。由此构成的四边形, 如满足下列约化条件： 1) 如R1、R2夹角为锐角
L=f0MiMp f0为物镜的焦距，Mi中间镜放大倍数，Mp投影镜的放大倍数，在透射电 镜的工作中，有效的相机长度L，一般在照相底板中直接标出，各种类 型的透射电镜标注方法不同，λ为电子波长，由工作电压决定，工作电 压一般可由底板标注确定，对没有标注的早期透射电镜在拍摄电子衍射 花样时，记录工作时的加速电压，由电压与波长对应表中查出λ。
此时要求相对误差为
i
dEi dTi dTi
<3%～5%。
例一
• 试标定γ-Fe电子衍射图(图6-10a) • 1、选约化四边形OADB(图6-10b)，
测得 • R1=9.3mm,R2=21.0mm,R3=21.0mm,Ф
=75°，计算边长比得 • R2/R1=21.0/9.3=2.258 • R3/R1=21.0/9.3=2.258 • 2、已知γ-Fe是面心立方点阵，故
衍射花样
NiFe多晶纳米薄膜的电子衍射
La3Cu2VO9晶体的电子衍射图
•
非晶态材料电子衍射图的特征
由于电子束穿过结晶物质时表现得和X射线相似，所以 衍射最强点的位置由布拉格定律所确定:
2d sin n
式中是波长，d 是晶体的晶面间距， 是入射电子束和晶 面间的夹角，n 是整数，称为衍射级次。
第4章 电子衍射PPT课件
f {1 exp[i(h k l)]}
当 h+k +l = 偶数时, F = 2f ， I 4 f 2
当 h+k+l = 奇数时, F = 0， I = 0
体心晶胞当 h+k +l = 偶数时，衍射强度不为零 当h+k+l = 奇数时消光。
(4) 面心晶胞
一个晶胞内有四个同种原子，分别位于000, 1 1 0, 1 0 1 ,0 1 1
第4章 电子衍射
透射电镜的最大特点是既可 以得到电子显微像又可以得到电 子衍射花样。晶体样品的微观组 织特征和微区晶体学性质可以在 同一台仪器中得到反映。
电子束 试样
物镜 物镜后焦面
微区晶体学性质 电子衍射花样
物镜像平面
微观组织
电子衍射实验得出：
多晶体
单晶体
非晶体 菊 池 线
问题的提出
这些点、环、线对携带着晶 体结构信息，对这些点、环、线 对等怎样进行分析，需要对电子 衍射基本知识有所了解。
底心晶胞h, k为全偶、全奇时衍射强度不为零。 h, k为奇偶混合时消光。
(3)体心晶胞
一个晶胞内有两个同种原子，分别位于 000 和 1 1 1
222
½½ ½
000
体心晶胞 F(hkl) 的计算
一个晶胞内有两个同种原子，分别位于 000 和 1 1 1
则
222
F (hkl) f exp[2i(o)] f exp[2i( h k l )]
可以用倒易矢量g来表示。
g
ha
*
kb *
lc
*
a*, b*, c*为倒空间的基矢量，hkl为倒易点 的坐标，即相应的衍射晶面指数。
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同理可知，简单立方点阵的可能N值为：1, 2, 3, 4, 5, 6, 8, 9, 10, …(注意N不 为7)。金刚石立方点阵可能的N值为：3, 8, 11, 16, 19, 24, …
R = λL d
由此可见，晶面间距不 同的晶面族产生衍射，会得 到以中心斑点为圆心的不同 半径的圆心环。
具有大d值的低指数晶面 族的衍射环在内，小d值的高 指数晶面族的衍射环在外。
电子衍射的基本公式
Rd = Lλ
Lλ:衍射常数或仪器常数
多晶环半径Ｒ正比于相 应的晶面间距d的倒数
R = Lλ / d
对于同一个衍射花样，Lλ是个定值，所以，
对于四方晶系，晶面间距与晶面指数的关系为
a
d=
1
令N = h2 + k 2
h2 + k2 + l2
a2
c2
d= 1 N + l2 a2 c2
a 简单四方 a=b
α= β=γ =90o 根据消光条件，对应四方晶体l=0的衍射环半径Ｒ满足比值
R12 : R22 : R32 : ... = N1 : N2 : N3 : ...
2
尽管以上给出的规律是对多晶衍射而言的，事实上该规律对单晶衍 射也适用，但这时的Ｒ是从倒空间原点到衍射点hkl的距离。
入射 束
厄瓦尔德球o′试样来自1 λ 2q 1 λ1d
G
倒易点阵
L
O ''
底板
R G ′′
4.6.2 多晶电子衍射花样的标定
∑ N = hi2
简单立方
12 34 5 6
8 9 10 11 12 13 14 16 17 18 19 20 21 22 24 25 26 27
4.6 多晶体电子衍射花样及其应用
4.6.1 多晶衍射花样的产生及几何特征 与X射线衍射法所得花样的几何特征相似，多晶电子衍射花样是
由一系列不同半径的同心圆环组成。
多晶电子衍射花样为何呈现一系列不同半径的同心圆环？
多晶指的是多种晶形共存，单晶指只 有一种晶形。 单晶内部的晶格方位 完全一致，多晶体则由许多晶粒组成。
R1
:
R2
: ...:
Rj
: ...
=
1 d1
:
1 d2
: ...:
1 dj
: ...
下面我们将此式用于讨论立方晶系、四方晶系和六方晶系的多晶 衍射花样。
（１）立方晶系
立方晶系结构是材料科学研究中最经常碰到，也是最简单的。
立方晶体的晶面间距：
d=
a
=a
h2 + k2 + l2 N
式中a是点阵常数， N = h2 + k 2 + l 2
平行的入射电子束照射到辐照 区内大量取向杂乱无章的细小晶体 颗粒上，使各个晶粒中d值相同的 {hkl}晶面族内符合衍射条件的晶面 组所产生的衍射束，构成以入射束 为轴，2θ为半顶角的圆锥面。
多晶体样品电子衍射花样的产生
事实上，属于同一晶面族{hkl}， 但取向杂乱的那些晶面族的倒易阵 点，在空间构成以O﹡为中心，g(＝1/d) 为半径的球面，它与爱瓦尔德球面的 交线是一个圆，记录到的衍射环就是 这一交线的投影放大像。 其半径为
立方
d=
a
h2 + k2 + l2
N = h2 + k2 + l2
四方
d=
1
h2 + k2 + l2
a2
c2
N = h2 + k2
六方
d=
1
4 h2 + hk + k 2 + l 2
3
a2
c2
N = h2 + hk + k 2
不同晶系可能含有不同类型的结构。根据结构消光原理，不同结构有 各自不同的消光条件，因而显示出自己固有的特征衍射环，这是鉴别不同 结构类型晶体的依据。
根据消光条件，对应六方晶体l=0的衍射环半径Ｒ满足比值
R12 : R22 : R32 : ... = 1: 3 : 4 : 7 : 9 :12 :13 :16 :19 : 21: ...
可见，不同晶系的衍射环半径平方的比值满足不同的规律。根据这个规律， 可以帮助我们判断所鉴定材料的对称性。
R12 : R22 : R32 : ... = N1 : N2 : N3 : ...
对于面心立方(fcc)点阵，只有hkl全为奇数或全为偶数，结构因子F才不为 零，可以产生衍射，则下列(hkl)晶面： (111), (200), (220), (311), (222), (400), (331), (422), …可产生衍射。这些晶 面对应的N值为：3，4，8，11，12，16，19，20，…
R = Lλ / d
R1 : R2 : R3 : ... = N1 : N2 : N3 : ... 或
R12 : R22 : R32 : ... = N1 : N2 : N3 : ... 上式表明，在立方晶体多晶电子衍射花样中，各个环的半径平方 一定满足整数的比例关系。
1
在立方晶体中，各种不同的点阵（简单立方、面心立方、体心立方）由于 受到消光条件（F=0）的限制，N可能有各种取值。
∑ N = hi2
简单立方
12 34 5 6
8 9 10 11 12 13 14 16 17 18 19 20 21 22 24 25 26 27
体心立方 面心立方
金刚石立方
立方晶体中各种点阵的可能的N值
这样，由多晶体衍射花样的N值，很容易确定立方晶系的点阵结构。
由于结构振幅的原因，立方晶系中不同结构类型对应于衍射可能出现 的N值：
简单四方：1，2，4，5，8， … 体心四方：2，4，8，10，16，18，20， …
c
a a
体心四方
（３）六方晶系 对于六方晶系，晶面间距与晶面指数的关系为
1 = 4 h2 + hk + k 2 + l 2 = 4N + l 2
d2 3
a2
c2 3a2 c2
有一个6次对称轴（c轴）。 α=β=90°， γ=120°，a=b≠c。
简单立方结构：1，2，3，4，5，6，8，10，…
体心立方结构：2，4，6，8，10，12，14，16， …
面心立方结构：3，4，8，11，12，16，19，20， …
金刚石结构：3，8，11，16，19，24， …
金刚石立方
a
a
a a 简单立方
a
a
a a 体心立方
a
a
a a 面心立方
（２）四方晶系 c