Fundamentals of X-ray diffraction and scattering

Transcription

1 Fundamentals of X-ray diffraction and scattering Don Savage 1231 Engineering Research Building (608)

2 X-ray diffraction and X-ray scattering Involves the elastic scattering of X-rays Diffraction is primarily used for structure determination. How are atoms or molecules arranged? What is the crystal structure? Scattering uses differences in electron density and looks at larger structures. X-rays are part of the electromagnetic spectrum

3 Laboratory X-ray sources Copper is a common anode choice ev = hn =hc/l, V (volts) =1239.8/l(nm) Electrons bombard target, give off X-rays Water cooling can be used to increase the power to the target Optics can be used to filter and focus the X- rays produced

4 Lab sources Point source Useful with area detector Line source Useful when you have a large uniform sample (e.g., for a perfect crystal or uniform smooth film) Bruker d-8 source has crossed multilayer mirrors to make a parallel point beam Panalytical Source has a multilayer mirror and a channel cut crystal to make a monochromatic, parallel line source Mirror only, used for reflectivity Slits only for Bragg-Brentano method

5 X-ray interactions with matter

6 X-ray interactions with matter

7 X-ray scattering by an atom X-rays are scattered by electrons in an atom into (approximately) all directions, though peaked in the forward direction. Wave picture of light is useful here: Strength of the scattering depends on the number of electrons ~ Z 2 (Z is the atomic number)

8 X-ray scattering by two (or several) atoms Two atoms: Several atoms: Constructive interference in some places. Destructive interference in others. From: C. Barret and T. B. Massalski, Structure of Metals, (1980).

9 X-ray diffraction from periodic arrangements of atoms Important Concept :Xrays reflect from crystal planes (only those that scatter in-phase from multiple planes yield peaks) All Peaks in Diffraction Satisfy Bragg s Law: nl=2 d sin( ) l=2 d hkl sin( ) d sin(θ)

10 What does a lab diffractometer measure? Angles and X-ray intensities (counts) w theta-two theta diffraction geometry additional degrees of rotational freedom

11 X-ray detectors Panalytical Empyrean 255 x 255 diode array Would like to count single x-ray photons with high dynamic range as quickly as possible 0-d Traditional: Scintillation counter - serial detector (slow) - x-ray photon generates electron pulse 1-d linear photo diode array can now count in parallel Bruker d8 Vantec detector 2048 x 2048 pixel 14cm active area 2-d photo plate (first x-ray detectors) - not quantitative wire array charged coupled device (CCD) array 2-d photo diode array

12 Powder diffraction Widely used Phase identification Amorphous to crystalline ratio Common industrial use: Quality control (do I have the same mix) Other uses: Grain size Film texture Stress measurement

13 Intensity (counts) 2 (degrees) Example of powder diffraction data Bruker d8 using 0.5 mm collimator 3 minute acquisition time Corundum

14 Phase identification The diffraction pattern for a particular phase is unique Phases with the same composition can have drastically different diffraction patterns The peak positions and relative intensities are compared with reference patterns in a database

15 Example: Mixture of SiO 2 phases The scattering from a mixture is a simple sum the scattering from each component phase (reference to a standard, as different compositions scatter more or less strongly) Note: The amorphous to crystalline ratio is determined from relative intensities (each phase is SiO2)

16 Quantification: Phases with different compositions RIR calcite [CaCO 3 ] = 3.45 RIR dolomite [CaMg(CO 3 ) 2 ] = 2.51

17 Crystallite size determination Crystallites smaller than ~100nm broaden diffraction peaks Analyze peak width with the Scherrer equation Must include instrument broadening Microstrain may also broaden peaks but can be separated out by measuring peak width over a wide 2 range B(2 ) = K l/[t cos( )], B is the peak full width at half maximum (radians), K is a shape factor ( ), t is the crystallite size and l the wavelength

18 Texture: Best observed with an area detector Thousands of crystalline grains are sampled Intensity in preferred directions shows the orientations are not random (from the deposition process or cold working) 2d detector with point source shows texture directly

19 Stress can be inferred by measuring strain

20 Macrostrain determination in a polycrystalline sample y Look a at a high 2 angle hkl peak position at different angles y with respect to the surface normal

21 Residual stress using the sin2y method

22 Single-crystal diffraction: requires high-resolution Requires accurate control of the sample orientation. To satisfy Braggs law, the incident beam and the detector have to be located precisely. Obtain crystal structure and orientation Measure crystal symmetry, lattice constants and defects In epitaxial film growth Determine strain (film relaxation), crystal mosaic, and film thickness

23 Panalytical Empyrean for high-resolution measurements Hybrid monochromator: curved multilayer mirror coupled with 4-bounce Ge(220) crystal Pixcel detector for fast mapping Channel-cut analyzer crystal with 12 Arc-second acceptance angle Sample stage moves in x, f, and chi

24 High-resolution X-ray analysis Si(004) SiGe (004) thickness SiGe deposited on Si(001) Thickness 79 nm Alloy composition Si 80.5 Ge period InGaAs/InAlAs deposited on InP (001) 4.47 nm In 79 Ga 19 As 3.91 nm In 24.3 Al 75.7 As SL period Fits assume 100% coherent growth

25 Introduction to reciprocal space and the Ewald construction Reciprocal lattice vectors perpendicular to crystal planes spaced = 2pn/d hkl Ewald construction links the experiment to the lattice with q (the scattering vector) When q (the scattering vector) is centered on a reciprocal lattice point, Braggs law is satisfied

26 Possible ways to navigate in reciprocal space Q =k f - k i

27 Why use reciprocal space mapping? The relative positions of Bragg peaks allow one to determine the degree of relaxation (coherency) Maps can take a long time to acquire

28 Reciprocal space maps of epitaxial SiGe (-2-2 4) (-2-2 4)

29 Ultra-fast reciprocal space mapping (-2-2 4) reciprocal space map of SiGe on Si Acquired in 3 minutes Uses 255 lines of diodes at different 2 values In parallel during an w-2 scan

30 X-ray reflectivity Near surface and interface information Density Porosity Film thickness Surface and interface roughness Works for amorphous films as well as crystalline

31 Log intensity X-ray reflectivity Contrast mechanism is differing refractive indices (electron densities) Film thickness measurements from 2nm - 300nm Simulation and fitting: Determine interface roughness and film porosity

32 X-ray reflectivity information content

33 X-ray reflectivity from a thin layer

34 X-ray reflectivity data fitting

35 SAX (small angle x-ray scattering) To look at larger periodic structures or particle sizes, look close to the incident beam. Use transmission Cu radiation Need a vacuum to reduce air scatter

36 Rigaku SAX system Sample to detector distance 2 meters Cu Ka micro source Fixed area detector 10 cm with 1024 pixel diameter PIN diode on beam stop measures beam transmission Sample heating to 350 C

37 Bruker d8 in SAX mode Use when higher angles are needed Sample to detector distance from 15 to 33.6 cm Beam stop to block direct transmitted x-ray beam Sample heating to 350 C

38 Log Intensity (cps) Log Intensity (cps) SAX measurements from silver behenate Rigaku Sax q ~ 0.08 to 1.2 nm -1 d ~ 80 nm to 5 nm q (inverse Angstroms) Bruker d8 q ~ 0.4 to 7.2 nm -1 d ~ 16 nm to 0.9 nm q (inverse Angstroms) Smaller d possible by moving the detector closer

39 Some SAX applications Block copolymer ordering Nanoparticle size and distribution DNA in solution

40 X-ray diffraction summary Diffraction is ideally suited for looking at order in materials Polycrystalline samples: Phase determination, stress, grain size, and texture Single-crystal diffraction: Epitaxial coherency, mosaic spread, film thickness, and strain Bruker d8

41 Crystallinity not needed X-ray reflection and SAX XRR of thin films: Thickness, density, and interface roughness SAX: Particle size (average) and long-range domain ordering Panalytical Empyrean