3.014 Materials Laboratory Oct. 13th – Oct. 20th , 2006 Lab Week 2 – Module γ1 Derivative Structures Instructor: Meri Treska OBJECTIVES 9 Review principles of x-ray scattering from crystalline materials 9 Learn how to conduct x-ray powder diffraction experiments and use PDFs 9 Study the inter-relationship of different crystal structures SUMMARY OF TASKS 1) Calculate structure factor for materials to be investigated 2) Prepare samples for x-ray powder diffraction 2) Obtain x-ray scattering patterns for all materials 3) Compare obtained patterns with calculations and powder diffraction files (PDFs) 4) Perform peak fitting to determine percent crystallinity and crystallite size 1BACKGROUND X-ray Diffraction from Crystalline Materials As discussed in 3.012, a periodic arrangement of atoms will give rise to constructive interference of scattered radiation having a wavelength λ comparable to the periodicity d when Bragg’s law is satisfied: n = 2sin λ d θ where n is an integer and θ is the angle of incidence. Bragg’s law tells us necessary conditions for diffraction, but provides no information regarding peak intensities. To use x-ray diffraction as a tool for materials identification, we must understand the relationship between structure/chemistry and the intensity of diffracted x-rays. Recall from 3.012 class that for a 1d array of atoms, the condition for constructive interference can be determined as follows: unit vector of ur diffracted beam S a ν µ y x 0S ur x= a cos ν unit vector of yacos µ incident beam= −=cos ν− a cos µλ The total path difference: xya = h uruurr SS ah⋅= λ(− )0uruur −r (SS0 )Defining s = , the condition for 1d constructive interference becomes: λ rr sah⋅= 2rr =For 3 dimensions, we have: s⋅ ah rr s⋅bk= rr s⋅ cl= where h, k and l are the Miller indices of the scattering plane. For a single unit cell having M atoms, the scattered amplitude is proportional to the structure factor, defined as: M rur () f exp 2 πis ⋅ rFs=∑ n  n n=1 ur where rnis the atomic position vector for the nth atom in the unit cell: r r r r rn= xn + nbznay+ c where (xn, yn, zn) are the atomic position coordinates. Example: for a BCC structure, there are 2 atoms/cell at (0,0,0) and (1/2,1/2,1/2). The parmater fn is the atomic scattering factor, proportional to the atomic number Z of the nth atom. Hence, atoms of high Z scatter more strongly than light elements. The atomic scattering factor is a function of θ and λ. f sinθ/λ Z 3rSubstituting rn into the structure factor: M r r r r () = f exp 2 is ⋅(xa + yb + zc )Fs ∑ n  π n n n  n=1 M ∑ n [ ihx n )]Fhkl = f exp 2 π ( n + ky + lzn n=1 For a BCC crystal:  hk l F = f exp 2 πi(0) + f exp 2 πi + + = f + f exp πih ( + k + l)hkl [ ]  222  [ ] Fhkl= 2 f h+k+l = even Fhkl = 0 h+k+l = odd The scattered intensity is related to the structure factor: 22 Icoh ∝ FF* = = 4 f h+k+l = evenFhkl Icoh = 0 h+k+l = odd Note that the total coherent intensity will be a sum of the contributions of all unit cells in the crystal. For a BCC crystal, reflections from planes with Miller indices where h+k+l is an odd integer will be absent from the diffraction pattern, while reflections from (110), (200), (211), etc. will be present with reduced intensity as h+k+l increases. I 110 200 211 220 2θ 4In our hypothetical case above, constructive interference occurs only at the exact Bragg angle and the I vs. 2θ curve exhibits sharp lines of intensity. In reality, diffraction peaks exhibit finite breadth, due both to instrumental and material effects. An important source of line broadening in polycrystalline materials is finite crystal size. In crystals of finite dimensions, there is incomplete destructive interference of waves scattered from angles slightly deviating from the Bragg angle. If we define the angular width of a peak as: B = 12 (2θ1 − 2θ2 ) then the average crystal size can be estimated from the Scherrer formula as: 0.9λt = B cos θB Interplanar spacings can be calculated for different hkl planes from geometric relationships for a given crystal system: −2 h2 + k 2 + l2 Cubic: d = 2a h2 k 2 l2 Orthorhombic: d −2 = 2 + 2 + 2a b c Tetragonal: d −2 = h2 + 2 k 2 + l22a c −24  h2 + hk + k 2  l2 Hexagonal: d = 3  a2 + c2  −21  h2 k 2 sin 2 β l22hl cos β Monoclinic: d = sin2 β  a2 + b2 + c2 − ac  5Bonding-Structure Relationships Covalent bonding Inorganic materials Materials that exhibit covalent bonding pack in arrangements that reflect the directional nature of their bonds. For example, sp3 hybridization in diamond and silicon mandates that atoms pack in these materials with tetrahedral coordination. These materials adopt the diamond cubic structure, a derivative of the FCC structure in which ½ of the tetrahedral interstitial sites are filled. diamond cubic structure in 2 orientations Images from http://www.uncp.edu/home/mcclurem/lattice/ 2 tetrahedral sites in a FCC lattice Along with silicon, numerous semiconductor alloys exhibit significant covalent character in their bonding, often adopting the zinc blende structure, a derivative structure of the diamond cubic structure. Examples include the III-V compounds GaAs, GaP, GaSb, AlP, AlAs, InSb, InP, InAs, and the II-VI compounds ZnS, ZnSe, ZnTe, CdTe. Zinc blende structure of ZnS: a FCC arrangement of sulfur atoms with zinc atoms filling ½ of the total tetrahedral sites 6 Courtesy of Dr. Mark McClure. Used with permission.Chalcopyrite, CuFeS2, a mineral, has a crystal structure that can be viewed as derivative of the zinc blende structure. Structure of CuFeS2: Cu (solid), Fe (shaded), S (unfilled). Polymers Polymers are covalently bonded long chain molecules composed of repeating units made of carbon and hydrogen, and sometimes oxygen, nitrogen, sulfur, silicon and/or fluorine. As with inorganic materials, the covalent bonding in polymers imposes directionality on their spatial arrangement into periodic structures. Polymer chains exhibit weak intermolecular forces due to van der Waals attractions. The ability of polymer chains to pack into an ordered array depends strongly on the stereoregularity of their pendant groups. For example, depending on the method of polymerization, polystyrene may exhibit isotactic, syndiotactic or atactic structure. Atactic polystyrene, is entirely amorphous due to