材料表征教学资料 diffra.ppt

Dr. Di Wu, Nanjing University,1,衍射,Dr. Di Wu, Nanjing University,2,衍射是什么,衍射技术考察大量散射体对电磁辐射的散射。我们主要考虑晶体材料中原子、原子团和分子对电磁辐射的散射。 各种衍射技术最后都会给出一个衍射谱图，所谓的衍射pattern。这些衍射谱记录了晶体对射线衍射的角度和强度。,Dr. Di Wu, Nanjing University,3,衍射能干什么,从衍射谱我们可以得到如下信息： 晶体中原子层的面间距； 确定单晶或者晶粒的取向； 确定未知材料的晶体结构； 测量晶粒的大小、形状和内应力。,Dr. Di Wu, Nanjing University,4,可见光的衍射,Fraunhofer衍射：远场衍射，即平面波和散射体的相互作用； Fresnel衍射：近场衍射，除了Fraunhofer衍射以外的衍射； 电子衍射在大多数情况下属于Fraunhofer衍射，但是在某些特殊的场合属于Fresnel衍射。,Dr. Di Wu, Nanjing University,5,波前,Huygens: 波前上的每一个点都是一个球面波源，这些球面波相互干射形成新的波前。 Every point on the wave front is a source for spherical waves. These spherical waves interference with each other to form the new wave front.,Dr. Di Wu, Nanjing University,6,双缝衍射,从两个狭缝出射的波之间有光程差，因而有位相差，在距离r, 位相差为 。,Dr. Di Wu, Nanjing University,7,Double slit diffraction,maximum intensity at or,Dr. Di Wu, Nanjing University,8,Multi slit diffraction,Dr. Di Wu, Nanjing University,9,Multi slit diffraction, when,Dr. Di Wu, Nanjing University,10,Multi slit diffraction,m and h are integers.,Dr. Di Wu, Nanjing University,11,Multi slit diffraction,There are N-1 diffraction minimums between two diffraction maximums.,and,correspond to two intensity minimums around a diffraction maximum.,Dr. Di Wu, Nanjing University,12,Multi slit diffraction,The width of the diffraction peak is . Larger N generates sharper peaks. To X-ray diffraction, N is directly associated with the size of crystallites under detection.,Dr. Di Wu, Nanjing University,13,To the slit at the origin O, , while to the slit at y, .,Diffraction from slit with finite width,suppose the slit of in width as a combination of slits of dy in width,Dr. Di Wu, Nanjing University,14,Diffraction from slit with finite width,Dr. Di Wu, Nanjing University,15,Diffraction from slit with finite width,Dr. Di Wu, Nanjing University,16,Diffraction from a small hole (aperture),The peak width is 1.22/D. Any aperture in any microscope would put limitations on the resolution (R).,D is the diameter.,Dr. Di Wu, Nanjing University,17,Geometric of Diffraction,The geometric theory of diffraction is essentially the same for X-rays, electrons and neutrons. Following assumptions has been made,The intensity of the scattered beam is much lower than that of the incident beam. The interaction of the scattered beam and the incident beam is omitted.,The scattering of the scattered beam is not considered.,The source is adequately far from the scattering system so that the incident beam is regarded as a plane wave. The size of the scattering system is sufficiently small.,Dr. Di Wu, Nanjing University,18,点散射体的散射,O,P,2,put the scattering center at the origin O and consider the scattered beam at P,S and S are unit vector in the direction of incident and scattered beam,compared with the phase at O, the phase at P is retarded by,obviously, the beam is scattered in all directions. The scattered beam is spherical.,Dr. Di Wu, Nanjing University,19,点散射体的散射,O,P,2,If the electric component of the incident beam is represented as,the amplitude of the scattered beam at P, E2 should be,we write E2 as,f2 is the scattering coefficient,Dr. Di Wu, Nanjing University,20,点散射体的散射,The beam intensity I in a certain direction is defined as the energy scattered into unit solid angle in this direction in unit time.,O,P,2,The beam intensity in the direction of is,The introduction of beam intensity I2 is much convenient because R, which is related to distance is avoided.,Dr. Di Wu, Nanjing University,21,随机分布的点散射体的散射,O2,O1,O,I0,We first consider a system with two scattering centers at O1 and O2.,Dr. Di Wu, Nanjing University,22,随机分布的点散射体的散射,define the scattering vector,E2 can be written as,Dr. Di Wu, Nanjing University,23,随机分布的点散射体的散射,The scattering coefficient of the system can be written as,The beam intensity can still be expressed as,Dr. Di Wu, Nanjing University,24,随机分布的点散射体的散射,The above results can be generalized to a system of n randomly distributed identical point scattering centers. The scattering coefficient of a system of n randomly distributed point scattering centers identical to each other can be expressed by the scattering coefficient of the scattering center at the origin multiplying the sum of their phase factors relative to the origin.,Dr. Di Wu, Nanjing University,25,随机分布的点散射体的散射,The results can also be easily generalized to a system of n randomly distributed different point scattering centers. Here, the same s for all these different scattering centers is assumed. This is common for X-ray diffraction.,Dr. Di Wu, Nanjing University,26,随机分布的点散射体的散射,Radiation beam diffracted by crystals is a special case of scattering. The scattering centers can be electrons, Coulomb potential, and nucleus, depending on the type of incident beam. The atoms in a crystal are periodically distributed. So are the scattering centers. Therefore, diffraction beam is formed by strengthening the scattered beam in certain directions due to interference.,Dr. Di Wu, Nanjing University,27,劳厄方程,The intensity is expressed similar to that of the multi silt Fraunhofer diffraction of visible light.,Consider an array of identical atoms separated equally,scattering coefficient,intensity of scattered beam,Dr. Di Wu, Nanjing University,28,劳厄方程,The diffraction maximum appears when and the peak width ,In 3-dimensional situation, Laues equation should be,h, k, and l are integers.,Dr. Di Wu, Nanjing University,29,Fortunately, we know by definition that the reciprocal lattice vector is the vector that satisfies the Laues equation.,劳厄方程,In practical problems, is known. If we could find , we will determine the direction of diffraction .,Dr. Di Wu, Nanjing University,30,衍射矢量,H is called diffraction vector. It can also be expressed as,Reciprocal lattice vectors compensate wave vectors so that wave vectors are kept constant during diffraction.,Dr. Di Wu, Nanjing University,31,Bragg定律,L,Dr. Di Wu, Nanjing University,32,Bragg定律,Braggs law can also be deduced in the following way.,X-ray photons reflected from crystalline planes with a spacing of d have a path difference BA+AC.,The reflected beam will be strengthened when,Dr. Di Wu, Nanjing University,33,Bragg定律,If the lattice spacing of (h, k, l) planes is dhkl, the spacing of (nh, nk, nl) planes is dhkl/n.,Dr. Di Wu, Nanjing University,34,Ewald球,Consider a wave incident on a crystal. The crystal is represented by its reciprocal lattice, with origin O.,The incident wave is represented by a wave vector k0.,Draw the incident wave vector, k0, ending at O.,Construct a sphere with radius 1/ (i.e. |k0|), which passes through O.,H,Dr. Di Wu, Nanjing University,35,Ewald球,Wherever a reciprocal lattice point touches the sphere, Braggs Law is obeyed and a diffracted beam will occur. LO represents the incident beam and LH is a diffracted beam. The angle between them must be 2. The diffraction of different planes can be achieved either by rotating the crystal or by varying the wavelength to have reciprocal lattice points touching the sphere.,animation,Dr. Di Wu, Nanjing University,36,极限球,Draw a large sphere having O as the center and 2/ as the radius. It is called limit sphere. When the crystal rotates around O, the points within the limit sphere have the possibility to touch the Ewald sphere and to give rise to diffraction. However, the points out of the limit sphere will never touch the Ewald sphere. This is due to the small d value for the points out of the limit sphere.,2/,Dr. Di Wu, Nanjing University,37,Ewald球,Since very few reciprocal lattice points are intersected by the Ewald sphere, very few sets of planes give rise to diffracted beams. In general, from a single crystal, only a few diffraction spots are recorded. Put it in another way: in a single crystal only a few sets of planes are orientated at their Bragg angle at any one time.,Dr. Di Wu, Nanjing University,38,X-ray diffraction method,Diffraction can occur whenever Braggs 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.,Dr. Di Wu, Nanjing University,39,X-ray diffraction method,This problem can be overcome by continuously varying or over a range of values, to satisfy Braggs law. Practically this is done : by using a range of x-ray wavelengths (i.e. white radiation), or by rotating the crystal or, using a powder or polycrystalline specimen.,Dr. Di Wu, Nanjing University,40,The 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.,Dr. Di Wu, Nanjing University,41,The Laue method,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.,Dr. Di Wu, Nanjing University,42,The Laue method,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.,Dr. Di Wu, Nanjing University,43,Back-reflection Laue method,In the back-reflection method, the film is placed between the x-ray source and the crystal. The beams which are diffracted in a backward direction are recorded.,One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an hyperbola.,Dr. Di Wu, Nanjing University,44,Transmission Laue method,In the transmission Laue method, the film is placed behind the crystal to record beams which are transmitted through the crystal.,One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an ellipse.,Dr. Di Wu, Nanjing University,45,The Laue method,Crystal orientation is determined from the position of the spots. Each spot can be indexed, i.e. attributed to a particular plane. The Laue technique can also be used to assess crystal perfection from the size and shape of the spots. If the crystal has been bent or twisted in anyway, the spots become distorted and smeared out.,Dr. Di Wu, Nanjing University,46,The powder method,The powder method is used to determine the value of the lattice parameters accurately. It uses monochromatic x-ray beams.,As mentioned before, if a monochromatic x-ray beam is directed at a single crystal, then only one or two diffracted beams may result.,Dr. Di Wu, Nanjing University,47,The powder method,If the sample consists of some tens of randomly oriented single crystals, the diffracted beams are seen to lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards.,A sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones.,Dr. Di Wu, Nanjing University,48,The powder method,A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film.,Dr. Di Wu, Nanjing University,49,Indexing a pattern,We remove the film strip from the Debye camera after exposure, then develop and fix it. From the strip of film we make measurements of the position of each diffraction line.,From the results it is possible to associate the sample with a particular type of structure and also to determine a value for its lattice parameter.,Dr. Di Wu, Nanjing University,50,Indexing a pattern,When the film is laid flat, S1 can be measured. This is the distance along the film, from a diffraction line, to the center of the hole for the transmitted direct beam.,For back reflections, i.e. where 2 900, you can measure S2 as the distance from the beam entry point.,Dr. Di Wu, Nanjing University,51,Indexing a pattern,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:,or,Dr. Di Wu, Nanjing University,52,If we know that the sample is of certain symmetry, cubic for example, we know that And, we have Thus,Indexing a pattern,Dr. Di Wu, Nanjing University,53,Indexing a pattern,From the measurements of each arc we can now generate a table of S1, and sin2 .,We now multiply the values of sin2 by some constant value to give nearly integer values for all the h2+k2+l2 values. Integer values are then assigned.,Dr. Di Wu, Nanjing University,54,Indexing a pattern,The integer values of h2+ k2+ l2 are then equated with their hkl values to index each arc, using the table.,For some structures e.g. bcc, fcc, not all planes reflect, so some of the arcs may be missing.,Dr. Di Wu, Nanjing University,55,Indexing a pattern,It is then possible to identify certain structures, in this case fcc ( the planes have hkl values: all even, or all odd). For each line we can also calculate a value for a, the lattice parameter. For greater accuracy the value is averaged over all the lines.,Dr. Di Wu, Nanjing University,56,Powder X-ray Diffractrometer,Sample is mounted at the center of the rotation stage.,X-ray source,detector,amplifier,computer,sample,source,sample,detector,X-ray source and detector are located at the same circle. When the sample rotate , the detector should rotate 2.,2,Dr. Di Wu, Nanjing University,57,2 XRD pattern,By comparison of the pattern, including calculated d values with the standard pattern, the diffraction peaks can be indexed and the structure of the materials can be identified.,Dr. Di Wu, Nanjing University,58,Diffraction and Fourier Transform,Real space and reciprocal space are fourier transform of each other. Reciprocal space is also the space of diffraction intensity since there are diffraction intensity values only on reciprocal lattice sites.,Dr. Di Wu, Nanjing University,59,Fourier Transform,The fourier transform of a function f(x) is defined as,And f(x) is the fourier transform of F(),Dr. Di Wu, Nanjing University,60,Basic characteristics,Dr. Di Wu, Nanjing University,61,Convolution,The convolution of function f(x) and g(x) is defined as If,and, then,and,Dr. Di Wu, Nanjing University,62,Convolution,Dr. Di Wu, Nanjing University,63, function,Definition: function is used to describe discrete characteristics, lattice sites for example.,Dr. Di Wu, Nanjing University,64,许多函数都可以表现出 函数的特性。比如，高斯函数, function,Dr. Di Wu, Nanjing University,65, function, 函数可以展开成傅立叶积分:,Dr. Di Wu, Nanjing University,66, function,If describes the property on the origin of the lattice site, describes the property of the whole lattice, where,Dr. Di Wu, Nanjing University,67, function,The convolution integral of f(x) and (x-a) is,The result is that the function is horizontally shifted to x=a.,Dr. Di Wu, Nanjing University,68, function,The convolution integral of and the lattice function is,That is to say if describes the character of one cell, represents the character of the whole crystal. If a function is convoluted with the function representing the lattice sites, the character of this function is transferred to the lattice.,Dr. Di Wu, Nanjing University,69,Diffraction and Fourier Transform,点散射体,一维晶格,点散射体,一维晶格,Dr. Di Wu, Nanjing University,70,Diffraction and Fourier Transform,h is an integer.,Dr. Di Wu, Nanjing University,71,Diffraction and Fourier Transform,晶格的傅立叶变换得到另一套晶格，倒异晶格。 衍射强度在倒空间的分布是不连续的，只有在 ，即倒格点上才有强度。 衍射图谱就是倒异晶格的记录。,Dr. Di Wu, Nanjing University,72,If is the distribution function of the scattering centers, for example, electron density for X-ray diffraction, the scattering centers in volume at is . The phase of this volume of scattering centers is . Thus, the diffraction amplitude should be,Diffraction and Fourier Transform,unit: the scattering amplitude of a single scattering center,Dr. Di Wu, Nanjing University,73,Theoretical diffraction intensity,Why are there variations in the intensity of diffraction spots from different planes ? Why are certain diffraction forbidden for certain structures?,scattering center distribution function in a single cell, stands for the cell in an ideal infinite crystal,lattice function of a infinite crystal,for the lattice of finite crystal,inside in the crystal,outside in the crystal,Dr. Di Wu, Nanjing University,74,Theoretical diffraction intensity,the scattering center distribution in a finite crystal can be expressed as,the relative diffraction amplitude is therefore,Dr. Di Wu, Nanjing University,75,Theoretical diffraction intensity,is the diffraction amplitude of a single cell, called the structure factor,is the fourier transform of the size factor,Dr. Di Wu, Nanjing University,76,Structure factor,For an non-primitive cell, the symmetry of the crystal may probably make the structure factor of certain zero. The diffraction is thus forbidden.,Dr. Di Wu, Nanjing University,77,Structure factor,bcc structure: there are two units in every cell, one at (0,0,0), another at (1/2,1/2,1/2),even,odd,(110) (200) (220) (121) ,(100) (111) (102) (300) ,Dr. Di Wu, Nanjing University,78,Structure factor,NiAl structure: two simple cubic lattice, with Ni at (0,0,0) and Al at (1/2,1/2,1/2).,even,odd,Dr. Di Wu, Nanjing University,79,Structure factor,fcc structure: there are four units in every cell, at (0,0,0), (1/2,1/2,0), (1/2,0,1/2) and (0,1/2,1/2), respectively.,h, k, l: all even or all odd,h, k, l: mixed even and odd,diffr