Single Crystal Diffraction Arthur J. Schultz Argonne National - PowerPoint PPT Presentation

Single Crystal Diffraction Arthur J. Schultz Argonne National Laboratory National School on Neutron and X-Ray Scattering August, 2013

What is a crystal? Atoms (molecules) pack • together in a regular pattern to form a crystal. Periodicity: we superimpose • (mentally) on the crystal structure a repeating lattice or unit cell. A lattice is a regular array of • geometrical points each of which has the same Unit cells of oxalic acid dihydrate environment. Quartz crystals 2

Why don’t the X -rays scatter in all directions? X-ray precession photograph (Georgia Tech, 1978). • X-rays (and neutrons) have wave properties. • A crystal acts as a diffraction grating producing constructive and destructive interference. 3

Bragg’s Law Jointly awarded the 1915 Nobel Prize in Physics William Henry Bragg William Lawrence Bragg 4

Crystallographic Planes and Miller Indices c b a (221) d -spacing = spacing between origin and first plane or between neighboring planes in the family of planes. 5

Laue Equations Scattering from points Max von Laue 1914 Nobel Prize for Physics S s a a • S s a • ( - S i ) S i a • S s + a • ( - S i ) = a • ( S s – S i ) = h λ a • ( S s – S i ) = h λ In three dimensions → b • ( S s – S i ) = k λ c • ( S s – S i ) = l λ 6

Real and reciprocal Space a * • a = b * • b = c * • c = 1 a * • b = … = 0 Laue equations: S s a • ( S s – S i ) = h λ , or a • s = h | S | = 1/  b • ( S s – S i ) = k λ , or b • s = k s c • ( S s – S i ) = l λ , or c • s = l | s | = 1/d where S i s = ( S s – S i )/ λ = h a * + k b * + l c * 7

The Ewald Sphere 1/d 1/(2d) θ θ b* O θ 1/ λ a* 8

The Ewald sphere: the movie Courtesy of the CSIC (Spanish National Research Council). http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html 9

Bragg Peak Intensity 0 b a Relative phase shifts related to molecular structure. b i is the neutron scattering length. It is replaced by f i , the x-ray form factor. 10

The phase problem 𝐽 ℎ𝑙𝑚 = 𝐺 ℎ𝑙𝑚 𝐺 ℎ𝑙𝑚 = 𝐺 ℎ𝑙𝑚 𝑓 𝑗𝜚 𝐺 ℎ𝑙𝑚 𝑓 −𝑗𝜚 = 𝐺 ℎ𝑙𝑚 2 Euler’s formula: 𝐺 ℎ𝑙𝑚 = 𝐺 ℎ𝑙𝑚 𝑓 𝑗𝜚 = 𝐺 ℎ𝑙𝑚 cos 𝜚 + 𝑗 sin 𝜚 = 𝐵 + 𝑗𝐶 𝐵 − 𝑗𝐶 = 𝐵 2 + 𝐶 2 𝐽 ℎ𝑙𝑚 = 𝐵 + 𝑗𝐶 𝐺 ℎ𝑙𝑚 𝑗 sin 𝜚 𝐺 ℎ𝑙𝑚 𝜚 𝐺 ℎ𝑙𝑚 cos 𝜚 11

θ -2 θ Step Scan Counts Two-theta 12

Omega Step Scan Mosaic spread Omega 1. Detector stationary at 2 θ angle. 2. Crystal is rotated about θ by +/- ω . 3. FWHM is the mosaic spread. 13

Something completely different - polycrystallography What is a powder? - polycrystalline mass All orientations of crystallites possible Sample: 1 m l powder of 1 m m crystallites - ~10 9 particles Packing efficiency – typically 50% Spaces – air, solvent, etc. Single crystal reciprocal lattice - smeared into spherical shells Courtesy of R. Von Dreele

Powder Diffraction    *   2 sin Bragg’s Law: d Counts 2  • Usually do not attempt to integrate individual peaks. • Instead, fit the spectrum using Rietveld profile analysis. Requires functions that describe the peak shape and background. 15

Why do single crystal diffraction (vs. powder diffraction)?  Smaller samples – neutrons: 1-10 mg vs 500-5000 mg – x-rays: μ g vs mg  Larger molecules and unit cells  Neutrons: hydrogen is ok for single crystals, powders generally need to be deuterated  Less absorption  Fourier coefficients are more accurate – based on integrating well- resolved peaks  Uniquely characterize non-standard scattering – superlattice and satellite peaks (commensurate and incommensurate), diffuse scattering (rods, planes, etc.) But:  Need to grow a single crystal  Data collection can be more time consuming 16

Some history of single crystal neutron diffraction • 1951 – Peterson and Levy demonstrate the feasibility of single crystal neutron diffraction using the Graphite Reactor at ORNL. • 1950s and 1960s – Bill Busing, Henri Levy, Carroll Johnson and others wrote a suite of programs for singe crystal diffraction including ORFLS and ORTEP. • 1979 – Peterson and coworkers demonstrate the single crystal neutron time- of- flight Laue technique at Argonne’s ZING - P’ spallation neutron source. 17

The Orientation Matrix U is a rotation matrix relating the unit cell to the instrument coordinate system. The matrix product UB is called the orientation matrix . 18

Picker 4-Circle Diffractometer 19

Kappa Diffractometer Brucker AXS: KAPPA APEX II • Full 360 ° rotations about ω and φ axes. • Rotation about κ axis reproduces quarter circle about χ axis. 20

Monochromatic diffractometer • Rotating crystal • Vary sin  in the Bragg equation: 2 d sin  = n  Reactor HFIR 4-Circle Diffractometer 21

Laue diffraction I (  )  Polychromatic “white” spectrum 22

Laue photo from white radiation X-ray Laue photos taken by Linus Pauling 23

Time-resolved X-ray Laue diffraction of photoactive yellow protein at BioCARS using pink radiation Coumaric acid cis-trans isomerization 24

Quasi-Laue Neutron Image Plate Diffractometer Select D/ of 10-20% 2012 at HFIR: IMAGINE 25

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Pulsed Neutron Incident Spectrum 𝜇 = ℎ ℎ 𝑢 𝑛𝑤 = 𝑀 𝑛 L = 10 m  = wavelength COUNTS h = Planck’s constant m = neutron mass v = velocity t = time-of-flight L = path length 1.25 msec 12.5 msec 0.5 Å 5.0 Å SOURCE COUNTS PULSED AT 30 HZ t 0 t 0 t 0 33 1/3 msec 27

Time-of-Flight Laue Technique 28

SCD Instrument Parameters Moderator liq. methane at 105 Detector distances on locus of constant Source frequency 30 Hz solid angle in reciprocal space. Sample-to-moderator dist. 940 cm Number of detectors 2 Detector active area 155 x 155 mm 2 Scintillator GS20 6 Li glass Closed-cycle Closed-cycle Scintillator thickness 2 mm 105 K liquid 105 K liquid 105 K liquid He refrigerator He refrigerator Efficiency @ 1 Å 0.86 methane moderator, methane moderator, methane moderator, Typical detector channels 100 x 100 9.5 m upstream 9.5 m upstream 9.5 m upstream 15 x 15 cm 2 15 x 15 cm 2 Resolution 1.75 mm detectors detectors Detector 1: Incident Incident angle 75 ° neutron neutron sample-to-detector dist. 23 cm beam beam Detector 2: angle 120 ° Sample Sample sample-to-detector dist. 18 cm vacuum vacuum Typical TOF range 1 – 25 ms chamber chamber wavelength range 0.4 – 10 Å d -spacing range ~0.3 – 8 Å TOF resolution, Δ t / t 0.01 Sample Environments Now operating in Los Alamos. Hot-Stage Displex: 4-900 K Displex Closed Cycle Helium Refrigerator: 12 – 473 K Furnaces: 300 – 1000 K Helium Pressure Cell Mounted on Displex: 0 – 5 kbar @ 4 – 300 K 29

ISAW hkl plot 30

ISAW 3D Reciprocal Space Viewer Diffuse Magnetic Scattering Analysis of ZnMn 2 O 4 by William Ratcliff II (NIST). 31

Topaz  Project Execution Plan requires a minimum of 2 steradian (approx. 23 detectors) coverage.  Each detector active area is 150 mm x 150 mm.  Secondary flight path varies from 400 mm to 450 mm radius and thus cover from 0.148 to 0.111 steradian each. Danny Williams, Matt Frost, Xiaoping Wang, Christina Hoffmann, Jack Thomison 32

Natrolite structure from TOPAZ data 33

Outline of single crystal structure analysis  Collect some initial data to determine the unit cell and the space group. – Auto-index peaks to determine unit cell and orientation – Examine symmetry of intensities and systematic absences  Measure a full data set of observed intensities.  Reduce the raw integrated intensities, I hkl , to structure factor amplitudes, | F obs | 2 .  Solve the structure.  Refine the structure. 34

Data reduction – single crystal TOF Laue Data reduction: convert raw integrated intensities, I hkl , into relative structure factor amplitudes, | F hkl | 2 . 𝐺 ℎ𝑙𝑚 2 𝜇 4 sin 2 𝜄 𝑑 2 𝐽 ℎ𝑙𝑚 = 𝑙 𝜚 𝜇 𝜁 𝜇 𝐵 𝜇 𝑧 𝜇 𝑊 𝑡 𝑊 k = scale factor f = incident flux spectrum e = detector efficiency as a function of wavelength  A (  ) = sample absorption y (  ) = secondary extinction correction V s = sample volume V c = unit cell volume 35

Intensity vs. sample volume and unit cell volume 𝐺 ℎ𝑙𝑚 2 𝜇 4 sin 2 𝜄 𝑑2 𝐽 ℎ𝑙𝑚 = 𝑙 𝜚 𝜇 𝜁 𝜇 𝐵 𝜇 𝑧 𝜇 𝑊 𝑡 𝑊 𝐺 ℎ𝑙𝑚 2 𝑊 𝜇 4 sin 2 𝜄 𝐽 ℎ𝑙𝑚 = 𝑙 𝜚 𝜇 𝜁 𝜇 𝐵 𝜇 𝑧 𝜇 𝑊 𝑡 𝑊 𝑑 𝑑 𝐽 ℎ𝑙𝑚 = 𝑙 𝜚 𝜇 𝜁 𝜇 𝐵 𝜇 𝑧 𝜇 𝑂 𝑡 𝐺 ℎ𝑙𝑚 2 𝑊 𝜇 4 sin 2 𝜄 𝑑 Scattering per Number of unit volume unit cells in approximately the sample constant 36