Electronic Materials Research Lab in Physics,Ch2. Wave Diffraction and Reciprocal Lattice Prof. J. Joo Department of Physics, Korea UniversitySolid State Physics Electronic Materials Research Lab in Physics, 2-1 Diffraction of Waves by Crystals Crystal structure studied by the diffraction of photons, neutrons, and electrons (keV) (0.01eV) (100eV) Diffraction depends on the crystal structure and on the wavelength in crystal, the of radiation lattice constant (a) diffracted beam (different from the incident, refracted direction) Consider.. reflection beam , beam : 2dsin for constructive interference : 2dsin=n (n : ) sin 1 2d Bragg refraction can 2d : can not use visible light ( ) Bragg law : consequence of the periodicity of the lattice a d Electronic Materials Research Lab in Physics, : ( ) mm ( ) . : (Reference) ( ) Electronic Materials Research Lab in Physics,(Reference) X-ray , , , For Pb (lead, 82) =7.4eV ( energy ) e - =88eV (K-shell energy ) Incident photon ejected K-shell el. if L x-ray K x-ray N M L K e.g.) for Cu, (z=29) K-shell I.E. : 8.99 keV K x-ray : 8.06 keV (transition from L to K shell) Electronic Materials Research Lab in Physics,(Reference) Experimental Techniques Used in Collecting XRD data (1) The scattering angle 2between the diffracted and incident beams. By substituting sin into Braggs law, one determines the interplanar spacing(d) as well as the orientation of the plane responsible for the diffraction. The intensity I of the diffraction beam. This quantity determines the cell-structure factor, F hkl, and hence gives information concerning the arrangement of atoms in the unit cell. 1.The rotating-crystal method For analysis of the structure of a single crystal A monochromatic incident beam of wavelength The specimen is then rotated until a diffraction condition obtains (and satisfy Braggs law) Electronic Materials Research Lab in Physics,(Reference) Experimental Techniques Used in Collecting XRD data (2) For a rapid determination of the symmetry and orientation of a single crystal A white x-ray beam (spectrum of continuous wavelength) is made to fall on the crystal, which has a fixed orientation relative to the incident beam Cannot determine the actual values of the interplanar spacings only their ratios Can determine the shape but not the absolute size of the unit cell 2. The Laue method 3. The powder method For the specimen which is not a single crystal (polycrystalline) A monochromatic beam impinges on the specimen, and the diffracted beams are recorded on a cylindrical film surrounding it Electronic Materials Research Lab in Physics,2-2 Scattered Wave Amplitude Bragg law : condition for the constructive interference ( ) of diffracted beam Diffracted beam scattering intensity or crystal translation ( ) invariant ( , crystal (local) - , moment - invariant ) Fourier Analysis : sine cosine e ikx idea : crystal Electronic Materials Research Lab in Physics,2-3 Fourier Analysis (1) Consider electron number density, n : invariant under T , n (r + T)=n (r ) : , 1 Fourier n(x) : 1 el. # density can be expanded in a Fourier series of sines and cosines a a a x n(x) The factor 2/a : ensures that n(x) has the period a p: positive integers C p, S p : (Fourier ) Electronic Materials Research Lab in Physics,2-3 Fourier Analysis (2) Compact and generalized form of n(x) 3 , Inversion of Fourier Series 3 (reciprocal lattice) vector 2p / a (1 ) Invariant under T n G : determines the x-ray scattering intensity Electronic Materials Research Lab in Physics,2-4 Reciprocal Lattice Vectors (1) 3-D Fourier series we must know G Construct b 1, b 2, b 3 (the axis vectors of the reciprocal lattice) If are primitive vectors, then are primitive vectors in reciprocal lattice and so forth i.e., Crystal structure crystal lattice ( , real space) reciprocal lattice ( , reciprocal space) Diffraction pattern of crystal : a map of the reciprocal lattice ( ) (where : integers) (reciprocal lattice) Electronic Materials Research Lab in Physics,2-4 Reciprocal Lattice Vectors (2) Dimension of direct lattice (or crystal lattice) : [ length ] Dimension of reciprocal lattice : [ 1/ length ] Reciprocal lattice : a lattice in the Fourier space associated with the crystal , k (wave vector) dimension : (2/) : 1/ length drown in Fourier space Fourier space wave . Integer! This result proves that the Fourier representation of a function n(r) periodic in the crystal lattice can contain components only at the reciprocal lattice vector Electronic Materials Research Lab in Physics,2.5 Diffraction Condition (1) Theorem : the set of reciprocal lattice vectors G determines the possible x-ray reflection Consider.. 1.The phase difference between incident beam and scattered beam from volume elements (dV), r apart : 2.The amplitude of the scattered wave 3.Total amplitude of scattered EM wave 4. : rsin phase difference : 2 (rsin/ ) Incident beam : Scattered beam : dV 0 Electronic Materials Research Lab in Physics,2.5 Diffraction Condition (2) Define scattering amplitude F : In elastic scattering of photon, h is conserved. ; Constructive interference condition (Bragg law) The diffracted condition for a constructive interference is If G is a reciprocal lattice vector, so is G : another statement of the Bragg condition..eq ..eq [H.W] problem 1 in Ch2 : d (h k l) = 2/ G (for elastic scattering) Electronic Materials Research Lab in Physics,2.5 Diffraction Condition (3) Using eq ; Generalize, 2dsin = n Various statements of the Bragg condition : 2dsin = n Laue Equations ; ; geometrical representation Consider.. : k lies on a certain cone k : satisfy eqs. , , and (h k l) planes Electronic Materials Research Lab in Physics,2.6 Brillouin Zone Significant for diffraction condition : Wigner-Seitz primitive cell in the reciprocal lattice Gives a vivid geometrical interpretation of the diffraction condition : ) Any k1 vector reached to plane1, satisfy the diffraction condition : Construction of B.Z. ( : 2D square lattice) 1 st, 2 nd,3 rd B.Z : same area Can be translated into 1 st B.Z. by G [ ] , B.Z, is not the language of x-ray, but is an essential part of energy band structure (p.40) Plane 1 Plane 2 2dsin = n ( Fig. 9(a) ) Electronic Materials Research Lab in Physics,2.7 Reciprocal Lattice to SC Lattice Electronic Materials Research Lab in Physics,2.8 Reciprocal Lattice to BCC Lattice Primitive vectors of a fcc lattice in reciprocal lattice Primitive translational vector of bcc lattice Electronic Materials Research Lab in Physics,2.9 Reciprocal Lattice to FCC Lattice Primitive vectors of a bcc lattice in reciprocal lattice 4/a a Electronic Materials Research Lab in Physics, Fourier Analysis of the Basis (1) , basis r 0 Electronic Materials Research Lab in Physics, Fourier Analysis of the Basis (2) The structure factor Define atomic form factor And Electronic Materials Research Lab in Physics, Structure Factor of the BCC Lattice E.g.) Metallic sodium : bcc no diffraction pattern : (1 0 0), (3 0 0), (1 1 1) exist diffraction pattern : (2 0 0), (1 1 0), (2 2 2) For bcc, identical atoms at (0,0,0) and ( , , ) and f is atomic form factor Electronic Materials Research Lab in Physics, Structure Factor of the FCC Lattice Assumed that the basis of the fcc structure referred to the cubic cell has identical atoms at ; 0 ; 0 ; 0 f is the same Electronic Materials Research Lab in Physics, Atomic Form Factor(1) : f A measure of the scattering power of the jth atom in the unit cell Depends on the number and distribution of atomic electrons, and the and of the radiation Consider that the integral extended over the el. concentration associated with a single atom and the el. distribution is spherically symmetric about the origin Electronic Materials Research Lab in Physics, Atomic Form Factor(2) If the same total el. density were concentrated at r=0 f : the ratio of the radiation amplitude scattered by the actual el. distribution in an atom to that scattered by one el. localized at a point overall el. distribution in a solid as seen in x-ray diffraction is fairly closed to that of the appropriate free atoms ; 1.Thermal motion does not broaden a diffraction line, but reduces the intensity 2.Broadening of diffraction line due to the lose of crystallization Electronic Materials Research Lab in Physics,Reference vector (h k l) plane in real space interplanar distance, (h k l) plane H.W. (#1) 1 Can show the relations (#2) 5