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s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Elements of Modern X-ray Physics Des McMorrow London Centre for Nanotechnology University College London About this course s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je “To explain the physics underlying the production and exploitation of X-rays with emphasis on application in condensed matter and materials physics” 1. Sources of X-rays 2. X-rays and their interaction with matter: scattering 3. Case studies 4. 4. X-ray imaging X-ray imaging About this lecture s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je 1. Introduction: contact region, the near and far fields 2. Absorption contrast imaging - Radiography and tomography - Microscopy 3. Phase contrast imaging - Free-space propagation - Grating interferometry 4. Coherent diffraction imaging - Coherent beams and speckle patterns - Phase retrieval via oversampling 5. Holography Why X-ray imaging? Advantages for real space imaging s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je 1. Imaging contrast: Refractive index n=1-δ+iβ β<δ<<1 charge, magnetic, orbital…. 2. Penetrative power: σ abs ∝ (ω )−3 Z 4 3. Spatial resolution: λ x − rays λlight 4. Temporal resolution: < 100 fs @ XFELs 5. Coherence: Holography, Lensless imaging From the near to the far field Fresnel and Fraunhofer diffraction s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je P and O illuminated by an incident plane wave radiate outgoing circular waves detected at D In the far-field limit (Fraunhofer diffraction) the phase difference is Q ⋅ r = −k ′ ⋅ r In the near field (Fresnel diffraction) must allow for shortening of path length difference by Δ = OF ′ − OF = R − Rcos ψ ≈ a 2 / (2R) Expect far-field limit to apply when λ <<Δ The contact, Fresnel and Fraunhofer regions s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je 5 micron disks red: absorption only blue: phase only Image courtesy of Timm Weitkamp Relationship between scattering and refraction s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Resonant scattering f (Q,ω ) = f (Q) + f ′(ω ) + if ′′(ω ) 0 Thomson scattering X-rays Rayleigh scattering Visible light Refractive index n = 1 − δ + iβ 2πρa r0 δ = ( f 0 (0) + f ′) k2 ⎛ 2πρa r0 ⎞ β = − f ′′⎜ ⎟ ⎝ k2 ⎠ Scattering and refraction: different ways of understanding the same phenomena Relationship scattering, refraction and absorption s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je ⎛ 2πρa ( f 0 (0) + f ′ )r0 ⎞ ⎛ 2πρa f ′′ r0 ⎞ δ =⎜ n = 1 − δ + iβ ⎟ β = −⎜ 2 ⎟ k k2 ⎝ ⎠ ⎝ ⎠ Absorption coefficient µ defined by I = I 0 e− µ z and absorption cross-section σ a = µ / ρa ⎛ k2 ⎞ f ′′ = − ⎜ ⎟ ⎝ 2πρa r0 ⎠ ⎛ k ⎞ µ = −⎜ ⎟ σa 2k 4 π r ⎝ 0⎠ Absorption is proportional to the imaginary part of the forward scattering amplitude (Optical Theorem) X-ray absorption edges s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Absorption contrast imaging Computer axial tomography (CAT) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Brain courtesy of Mikael Haggstrom The Radon transform R(θ,x’) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je The line integral of the absorption coefficient can be determined from the ratio I0/I. The Radon transform is the line integral evaluated at a specific viewing angle θ as a function of the coordinate x’ perpendicular to the direction of viewing. Fourier slice theorem Image reconstruction from the Radon transform s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Radon transform FT IFT Fourier slice theorem Example of image reconstruction from the Radon transform s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je CAT Three dimensional image reconstruction s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Commercial, fully automated, desktop systems with resolution < 10 microns available from various companies X-ray Microscopy Mirrors and Refractive Lenses s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je For X-rays, n<1 which implies total external reflection for angles below a critical angle of order 0.1 degrees Due to absorption, refractive lenses operate efficiently above 10 keV X-ray lenses The ideal shape s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Applying Fermat’s Theorem to optical path lengths leads to c.f. equation of an ellipse Hence and Therefore X-ray Fresnel zone plates Kinoform s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Performance of lens is insensitive to relative change in the optical path length by an integer number of wavelengths For X-rays Λ~10-100µm. X-ray Fresnel zone plates Binary approximation s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Zone radius: Outermost zone width: X-ray Fresnel zone plates Spatial resolution s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Rayleigh Criterion Si FZP ΔrM=30 nm The resolution of a FZP depends on the width of the outermost zone State-of-the-art is around 15 nm Image courtesy of Christian David X-ray Absorption Microscopy FZP based scanning and full-field transmission microscopes s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Magnification M=dO-D/dS-O >1,000 X-ray Absorption Microscopy Imaging of biological materials s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je The water window X-ray Absorption Microscopy Cell division in a unicellular yeast s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Image courtesy of Carolyn Larabell X-ray Magnetic Circular Dichroism (XMCD) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Photon polarization dependent absorption Dipole selection rules Δl=±1 ΔS=0 Conservation of angular momentum and Pauli exclusion principle produce different absorptions of left and right circular polarized light s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Image courtesy of Jo Stohr X-ray Magnetic Circular Dichroism (XMCD) Sum rules for transitions to 3d states s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Iron thin films, Chen et al. PRL (1995) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Image courtesy of Jo Stohr s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Image courtesy of Jo Stohr X-ray magnetic microscopy STXM s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Image courtesy of Jo Stohr Alignment of Ferromagnetic by Antiferromagnetic Domains Nolting et al., Nature 405, 767 (2000) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je 10A Pt 12A Co LaFeO3(100) a) LaFeO3 layer b) Co layer 705 Normalized Intensity (a.u.) Normalized Intensity (a.u.) 2 µm dark light 710 715 720 725 Photon Energy (eV) 730 770 dark light medium 775 780 785 790 795 Photon Energy (eV) 800 Phase contrast imaging s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je n=1-δ+iβ Phase contrast imaging While the direction of propagation is: s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je The phase of the wave is: Angular deviation perpendicular to beam: Phase contrast imaging s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Depends on being able to determine the deflection angle α accurately. Essentially three methods: 1. Free-space propagation 2. Interferometric techniques 3. Analyser crystals Phase contrast imaging Free-space propagation s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Beam displacement encodes information on the phase gradient perpendicular to the optical axis Phase contrast imaging Example of free-space propagation s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Etched silicon (100) wafer Images courtesy of Martin Bech and Torben Jensen Phase contrast imaging Free space propagation s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Blood cell infected with malaria parasite Images courtesy of Martin Bech, Martin Dierolf and Torben Jensen Phase contrast imaging Grating interferometry s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je David et al. (2002), Momose et al. (2003), Weitkamp et al. (2006), Pfeiffer et al. (2006) Phase contrast imaging s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je grating period p1 Grating interferometry Wavevector: For we have Propagated wave Transverse to optical axis, expect periodic wavefield with kx=2π/p Talbot length Phase contrast imaging Grating interferometry s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je No sample With sample G2 period p1/2 Resolution determined by pixel size, currently around 10 microns Phase contrast imaging Absorption contrast s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Water Grating interferometry Sugar Fringe Visibility Images courtesy of Franz Pfeiffer and Martin Bech Differential phase contrast “Dark field” (scattering) Phase contrast imaging Grating interferometry s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Absorption contrast “Dark field” (scattering) Differential phase contrast Images courtesy of Franz Pfeiffer and Martin Bech Phase contrast imaging Grating interferometry s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je “Pictures of the Future” Siemens Magazine for Research and innovation Images courtesy of Franz Pfeiffer The Phase Problem s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Measure intensity in a scattering experiment 2 I= F , F = ∑ f je j and it appears that all information on phase is lost. iQ⋅r j Transverse Coherence Length Sources have a finite size and produce divergent waves s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Monochromatic waves are in phase at P, and again after a distance 2LT ∴ 2LT Δθ = λ with Δθ = D / R Longitudinal Coherence Length Sources are not perfectly monochromatic s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je 2LL = Nλ Nλ = (N +1)( λ − Δλ ) λ ⇒N≈ Δλ Can also define coherence time t 0 = LL /c Photon Degeneracy Δc s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Photon degeneracy: The number of photons per eigenstate of the photon field. This is equivalent to the number of photons in a coherence volume. The photon degeneracy can be calculated from the brilliance: Thus we have Δν ν AS : source area Δ c ∝ B × t 0 × ΔΩtr × AS × B : brilliance; t 0 : coherence time ≈ 1/Δν; (LT ) 2 1 λ2 R 2 λ2 ΔΩtr : solid angle subtended by tranverse coherence area ≈ ≈ 2 = R D2 R2 AS Δν : bandwidth ν Bλ3 Δc ∝ c Photon Degeneracy Δc Comparison of different sources s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je P. Guertler, HASYLAB Imaging using coherent X-rays Speckle patterns s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je SAXS thought experiment single spherical particle Imaging using coherent X-rays Time average scattering and instantaneous speckle s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je SAXS from dilute suspension of 500 nm silica spheres Coherent beam illumination, average of 200 exposures Data courtesy of Anders Madsen X-ray photon-correlation spectroscopy (XPCS) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Direct measurement of antiferromagnetic domain fluctuations Shpyrko et al. Nature (2007) Chromium: SDW below TN=311 K CDW induced at TN X-ray photon-correlation spectroscopy (XPCS) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Direct measurement of antiferromagnetic domain fluctuations Shpyrko et al. Nature (2007) (200) static (200) Time dependent speckle reveals information on the domain dynamics X-ray photon-correlation spectroscopy (XPCS) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Direct measurement of antiferromagnetic domain fluctuations Shpyrko et al. Nature (2007) X-ray photon-correlation spectroscopy (XPCS) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Direct measurement of antiferromagnetic domain fluctuations Shpyrko et al. Nature (2007) Temperature dependent domain wall dynamics Coherent diffraction imaging Phase retrieval via oversampling s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Three critical factors allow the phase problem to be overcome by illuminating a finite size object of size L3 with a coherent beam (Sayre, 1980): • Diffraction pattern is extended in reciprocal space over volume ~1/L3 • Can sample the diffraction pattern with resolution better than 1/L • Phases can be recovered by iterative numerical procedure (Fienup, 1982) Coherent diffraction imaging Phase retrieval via oversampling s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Example of real space constraint: auto-correlation function is a measure of the sample size and can be obtained by the FT of the intensity Coherent Diffraction Imaging Phase retrieval via oversampling s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Coherent diffraction imaging Reconstruction of three dimensional electron density s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je First demonstrated by Miao et al., Nature(1999 ) for a non-crystalline sample Coherent diffraction imaging Reconstruction of three dimensional electron density of a crystal s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Real space resolution limited by inverse of maximum Q sampled. Currently around 30 nm Data courtesy of Ian Robinson X-ray Holography Eisebitt et al. Nature (2004) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je • Image produced by interference between object and reference beams • X-ray resonant magnetic scattering at Co L3 edge • Fourier transform holography – record interference pattern in far-field limit Image courtesy of Jo Stohr X-ray holography Fourier transform holography s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je In the far-field limit the intensity can be simply calculated as: auto-correlations cross-correlations If reference and object beams are spatially offset, then cross-correlation and auto-correlations terms are offset when the diffraction pattern is Fourier transformed. From the convolution theorem, the FT of the cross-correlation term is the convolution of the FT of AR(Q) and AO(Q). In the case that the reference hole is small it may be represented as a delta function, and the convolution integral is then simply the FT of AR(Q), or in other words the real space image of the structure. X-ray holography Eisebitt et al. Nature (2004) s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je X-ray holography Fidelity of reconstructed image s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Where are we heading…… s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Utilizing the highly brilliant coherent beams from XFELs it is envisaged (hoped!) that in the near future we will be able to: • Obtain atomic-scale, real-space images of single molecules (particles) • Have a time resolution of < 100 fs • Study chemical/biological processes as they happen Potential for biomolecular imaging with femtosecond X-ray pulses, Neutze et al Nature (2000) ….and how far have we come Femtosecond diffractive imaging with a soft XFEL s ic ow ys orr Ph M c y ra M X- es n D er d od an M n of lse ts ie en ls-N em A El ns Je Chapman et al. Nature Physics (2006) • • • • • FLASH facility at DESY, Hamburg Single pulse 32 nm, 25 fs, 4 x 1013 Wcm−2 Coherent diffraction imaging Real space image reconstructed Sample destroyed!