XRD allows Crystal Structure Determination What do we need to know in order to define the crystal structure? - The size of the unit cell and the lattice type The size of the unit cell and the lattice type (this defines the positions of diffraction spots) - The atom type at each point The atom type at each point (these define the intensity of diffraction spots) Conclusion: If we measure positions and intensities of many spots, then we should be able to determine the crystal structure. POSITION OF PEAKS LATTICE TYPE WIDTH OF PEAK PERFECTION OF LATTICE INTENSITY OF PEAKS POSITION OF ATOMS IN BASIS

The scattered x-ray amplitude is proportional to: Can break this sum into a sum over all lattices and a sum over all of the atoms within the basis. Structure Factor S hkl gives intensity of peaks Structure factor To get a diffraction peak, K has to be a reciprocal lattice vector, but even if K is,  f(r)e -ir  K might still be zero!

r ○ K r = n 1 a 1 + n 2 a 2 + n 3 a 3 (real space) Cubic form: Where x i, y i and z i are the lattice positions of the atoms in the basis.

In Class: Simple Cubic Lattice Simplify the structure factor for the simple cubic lattice for a one atom basis. Just let f be a constant. Where x i, y i and z i are the lattice positions of the atoms in the basis.

How Do We Determine The Lattice Constant? For the simple cubic lattice with a one atom basis: Substituting and squaring both sides: Thus, if we know the x-ray wavelength and are given (or can measure) the angles at which each diffraction peak occurs, we can determine a for the lattice! How? So the x-ray intensity is nonzero for all values of (hkl), subject to the Bragg condition, which can be expressed. We know for cubic lattices (a=b=c):

Missing Spots in the Diffraction Pattern In some lattices, the arrangement and spacing of planes produces diffractions from planes that are always exactly 180º out of phase causing a phenomenon called extinction. For the BCC lattice the (100) planes are interweaved with an equivalent set at the halfway position, giving a reflection exactly out of phase, which exactly cancel the signal.

Extinction (out of phase) for 100 family of planes in BCC What about the 101 family of planes? (001) (-101)

Group: The Structure Factor of BCC What values of hkl do not have diffraction peaks? Analysis of more than one lattice point per conventional unit cell E.g: bcc and fcc lattices bcc lattice has two atoms per unit cell located at r 1 = (0,0,0) and r 2 = (1/2,1/2,1/2) r i = x i a 1 + y i a 2 + z i a 3

Group: Find the structure factor for BCC.

Structure Factor body-centered cubic Allowed low order reflections are: – 110, 200, 112, 220, 310, 222, 321, 400, 330, 411, 420 … – Draw lowest on this cube -> Forbidden reflections are: – 100, 111, 210 – Due to identical plane of atoms halfway between causes destructive interference Real bcc lattice has an fcc reciprocal lattice This kind of argument leads to rules for identifying the lattice symmetry from "missing" reflections.

How to determine lattice parameter this time? Just as before, if we are given or can measure the angles at which each diffraction peak occurs, we can graphically determine a for the lattice! For a bcc lattice with a one atom basis, the x-ray intensity is nonzero for all planes (hkl), subject to the Bragg condition, except for the planes where h+k+l is odd. Thus, diffraction peaks will be observed for the following planes: (100)(110)(111)(200)(210)(211)(220)(221) (300) … A similar analysis can be done for a crystal with the fcc lattice with a one atom basis. For materials with more than one atom type per basis in a cubic lattice, a slightly different rule for the values of (hkl) is generated.

Group: Find the structure factor and extinctions for FCC.

Group: Find the structure factor for FCC Allowed low order reflections are: 111, 200, 220, 311, 222, 400, 331, 310 Forbidden reflections: 100, 110, 210, 211

Diamond (Homework due Thursday) Calculate the structure factor and extinctions for the diamond structure. Lattice = FCC. Basis = (000), (¼ ¼ ¼)

Allowed Diffraction Peaks (Trend?) The more atoms in basis, the less peaks

Structure Factor Ni 3 Al structure Simple cubic lattice, with a four atom basis Again, since simple cubic, intensity at all points. But each point is ‘chemically sensitive’. ()()

Common to see an average decrease in intensity of the diffraction peaks despite rules for peak intensities

Atomic Scattering Factor f (key points) (aka Form Factor) Only at 2  =0 does f=Z Also, thermal effects increase the effective size of atom Atoms are of a comparable size to the wavelength of the x-rays and so the scattering is not point like. There is a small path difference between waves scattered at either side of the electron cloud This effect increases with angle For x-rays, scattering strength depends on electron density All electrons in atom (Z of them) participate, core e - density ~spherical

Structure Factor with Different Atoms NaCl (rock salt) structure FCC Reminder:

Extra slides There is a lot of useful information on diffraction. Following are some related slides that I have used or considered using in the past. A whole course could be taught focusing on diffraction so I can’t cover everything here.

XRD: “Rocking” Curve Scan Vary ORIENTATION of K relative to sample normal while maintaining its magnitude. How? “Rock” sample over a very small angular range. Resulting data of Intensity vs. Omega (  sample angle) shows detailed structure of diffraction peak being investigated. Can inform about quality of sample. “Rock” Sample Sample normal K K

XRD: Rocking Curve Example Rocking curve of single crystal GaN around (002) diffraction peak showing its detailed structure GaN Thin Film (002) Reflection Intensity (Counts/s) Omega (deg) How do you know if this is good? Compare to literature to see how good (some materials naturally easier than others) Generally limited by quality of substrate

X-ray reflectivity (XRR) measurement Si Mo  t [Å]  [Å] Calculation of the density, composition, thickness and interface roughness for each particular layer W The surface must be smooth (mirror-like) Kiessig oscillations (fringes) A glancing, but varying, incident angle, combined with a matching detector angle collects the X rays reflected from the samples surface

The X-ray Shutter is the most important safety device on a diffractometer X-rays exit the tube through X-ray transparent Be windows. X-Ray safety shutters contain the beam so that you may work in the diffractometer without being exposed to the X-rays. Being aware of the status of the shutters is the most important factor in working safely with X rays.

XRD: Reciprocal-Space Map Vary Orientation and Magnitude of  k. Diffraction-Space map of GaN film on AlN buffer shows peaks of each film.  /2   GaN(002) AlN

If the wavelength of the incident x-rays and the scattering angle are known, then one can deduce the distance (already done) between the planes, d hkl, responsible for each scattering peak. The following twelve lines were obtained from a crystalline powder, known to belong to a cubic system. Line d( Å ) relative intensity Index the lines in terms of their Miller indices (hkl) and calculate the lattice constant of the cubic lattice. Establish the type of cubic lattice. Conditions for Peak SCAllpoints BCCSum =even FCCAllEven/odd DiamAllOdd or sum4n NaClAllEven/odd FCC Ni 3 AlAllPoints (SC) Similar to HW, but turned theta dependence to d. How?

Neutron λ = 1A° E ~ 0.08 eV interact with nuclei Highly Penetrating Electron λ = 2A° E ~ 150 eV interact with electron Less Penetrating Non-xray Diffraction Methods (more in later chapters) Any particle will scatter and create diffraction pattern Beams are selected by experimentalists depending on sensitivity – X-rays not sensitive to low Z elements, but neutrons are – Electrons sensitive to surface structure if energy is low – Atoms (e.g., helium) sensitive to surface only For inelastic scattering, momentum conservation is important X-Ray λ = 1A° E ~ 10 4 eV interact with electron Penetrating

Group: Consider Neutron Diffraction Qualitatively discuss the atomic scattering factor (e.g., as a function of scattering angle) for neutron diffraction (compared to x-ray) by a crystalline solid. For x-rays, we saw that f is related to Z and has a strong angular component. For neutrons? The same equation applies, but since the neutron scatters off a tiny nucleus, scattering is more point-like, and f is ~ independent of .

Systematic Extinction Systematic extinction is a consequence of lattice type At right is table of systematic extinctions for symmetry elements Other extinctions can occur as a consequence of screw axis and glide plane translations (Dove, Ch.6 Structure and Dynamics) Accidental Extinctions may occur resulting from mutual interference of other scattering vectors SymmetryExtinction Conditions simplenone Chkl; h + k = odd Bhkl; h + l = odd Ahkl; k + l = odd bodyhkl; h + k + l = odd All faceshkl; h, k, l mixed even and odd Key: C, B, A = side-centered on c-, b-, a-face; I = body centered; F = face centered (001)

Preferred Orientation (texture) Preferred orientation of crystallites can create a systematic variation in diffraction peak intensities – can qualitatively analyze using a 1D diffraction pattern – a pole figure maps the intensity of a single peak as a function of tilt and rotation of the sample this can be used to quantify the texture (111) (311) (200) (220) (222) (400) Two-Theta (deg) x Intensity(Counts) > Gold - Au Diffracting crystallites