3.012 Structure – An Introduction to X-ray Diffraction This handout summarizes some topics that are important for understanding x-ray diffraction. The following references provide a thorough explanation of x-ray diffraction in materials: http://capsicum.me.utexas.edu/ChE386K/ B.J. Cullity Elements of X-ray Diffraction C. Hammond The Basics of Crystallography and Diffraction (QD905.2.H355) X-ray diffraction is a tool to identify the phases and crystal structure of a material. To understand how this happens, several concepts come together. Part I of this handout presents reciprocal space and how to visualize it. Part II summarizes Huygens construction, how scattering waves interfere, and why they must be detected from particular directions. Part III talks about the Laue conditions and constructive interference of x-rays in a crystal. Part IV brings together Parts I, II, and III to explain how to identify the key element of the crystal structure that leads to constructively-interfering x-rays. PART I: Direct (or real) and reciprocal space Let's imagine a regular array in space (we’ll call this usual space the “real space”), and for simplicity, let's imagine the most simple case of a simple-cubic lattice where we put one atom at every lattice point. We could have put any other motif at every lattice point – it could have been 2 atoms, 3 atoms, a pair of skis, or just one atom but displaced a bit from the lattice point, but for now, we’ll stick with one atom on a lattice point. Now imagine you have a vector a1 that points from the atom at the origin toward one of its nearest neighbors (see figure below). A property of every vector is that it has both magnitude and direction. The direction of a1 in this particular simple-cubic case is along X and the magnitude we'll call a. We can define other vectors a2 and a3 that lie along Y and Z, respectively, and they each have magnitude a. a1, a2, and a3 are the “basis” for our regular array. We can define the position of any point in the array (i.e. of any atom in the space) by taking integer combinations of a1, a2, and a3. (Formally a1, a2, and a3 are called the principal lattice vectors of the Bravais lattice.) The position of point P that has crystallographic coordinates (2,2,1) is R(P) = 2a1 + 2 a2 + 1a3. (Note the difference between crystallographic coordinates and Cartesian coordinates – Cartesian coordinates are given with respect to the usual three perpendicular axes X, Y, Z, while crystallographic coordinates are given with respect to the axes on which a1, a2, and a3 lie and in units of a1, a2, and a3) 3.012, A. Predith, N. Marzari, 5 Dec 05 1Now let's look at the reciprocal lattice. The formal definition is that the reciprocal lattice vector G belongs to the reciprocal lattice of a Bravais lattice if the plane wave exp(i G • r) has the same amplitude at every point of the direct Bravais lattice of which G is a reciprocal lattice vector. In practice, this is how you can think at it: we define a basis in the reciprocal space in terms of three principal lattice vectors b1, b2, and b3 : b1 = 2π * (a2 x a3) / V, where V = a1 • (a2 x a3) b2 = 2π * (a3 x a1) / V b3 = 2π * (a1 x a2) / V Note that the scalar product between ai and bj will be 2π if i=j, or 0 otherwise. The units of reciprocal space are 1/distance (1/Å, for example). As any direct Bravais lattice vector can be defined by integer combinations of a1, a2, and a3, similarly, any reciprocal lattice vector can be defined by integer combinations of b1, b2, and b3. Combinations of b1, b2, and b3 create a regular array of points, and the points form a lattice in reciprocal space. b1, b2, and b3 are called the reciprocal lattice basis vectors. If you change the direct lattice (e.g. making the basis vectors longer), the reciprocal lattice vectors will change in an inverse way (e.g. becoming shorter). On the other hand, the scalar product between a direct lattice vector that has crystallographic coordinates (p,q,r) and a reciprocal lattice vector that has coordinates (s,t,u) will not change (think about this, and use the scalar product above). PART II: Huygen’s construction Now let's talk about diffraction. This topic is nicely covered in the handwritten notes "Description and Determination of Atomic Positions in Crystalline Solids." Please read those notes to fully understand the material. Part II and Part III gives a brief summary. When x-rays (λ ~1 Å) impinge on a materials, the atoms scatter the x-rays because the distance between atoms is about the same size as the x-ray wavelength. Huygens construction shows what happens when a plane wave (an x-ray, for example) hits a regular array of scatterers (a row of atoms, for example). In the 'Normal Incidence' figure on page 6 of the notes and in the figure below, consider an arc with the smallest radius next to a single scatterer. The arc represents where the wave has equal amplitude or is at the same phase. The next largest arc shows the same amplitude when the wave is one wavelength farther away from the scatterer. 3.012, A. Predith, N. Marzari, 5 Dec 05 2a3a1a2PXZYFigure by MIT OCW.When viewed from very far away and from a few very specific directions, however, it is possible to see x-rays from different scatterers all with the same amplitude; the x-rays are in phase. In fact, if you look at the array of scatterers from far away (using your x-ray vision) from a direction perpendicular to the scatterers (you're looking at them head-on), the x-rays from all the scatterers will have the same amplitude. This is the zero-order direction. If you look at the scatterers from a direction perpendicular to the 'first-order' tangent, you will see when the scattered waves on that tangent are in phase. The same holds true for 'second-order', 'third-order' and so on. The purpose of Huygens construction is to show that it is possible to see waves in phase coming from different scatteres only when you look at the scatterers from very specific directions. It is also important to notice that when viewed at a great distance, the individual waves in phase with each other appear like plane waves. The wavefront becomes flat at large distances. Notice that different combinations of scatterers contribute to each order of in-phase wavefronts. PART III: Laue conditions Laue took a step further to quantify what is necessary for constructive interference of x-rays
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