Elements of Modern X-ray Physics

Transcription

1 Des McMorrow London Centre for Nanotechnology University College London

2 About this course 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. X-ray imaging

3 X-ray imaging About this lecture 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

4 Why X-ray imaging? Advantages for real space imaging 1. Imaging contrast: Refractive index n=1-δ+iβ charge, magnetic, orbital. β<δ<<1 2. Penetrative power: 3. Spatial resolution: 4. Temporal resolution: σ abs ( ω) 3 Z 4 λ x rays λ light < 100 XFELs 5. Coherence: Holography, Lensless imaging

5 P and O illuminated by an incident plane wave radiate outgoing circular waves detected at D From the near to the far field Fresnel and Fraunhofer diffraction Expect far-field limit to apply when λ<<δ 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 = O F OF = R Rcosψ a 2 / (2R)

6 The contact, Fresnel and Fraunhofer regions 5 micron disks red: absorption only blue: phase only Image courtesy of Timm Weitkamp

7 Relationship between scattering and refraction Resonant scattering f (Q, ω) = f 0 (Q) + f ( ω) + i f ( ω) Thomson scattering X-rays Rayleigh scattering Visible light Refractive index n =1 δ + iβ δ = ( f 0 (0) + f ) 2πρ r a 0 k 2 β = f 2πρ a r 0 k 2 Scattering and refraction: different ways of understanding the same phenomena

8 Relationship scattering, refraction and absorption n = 1 δ + iβ δ = 2πρ ( f 0 (0) + f )r a 0 k 2 β = 2πρ f r a 0 k 2 Absorption coefficient µ defined by I = I 0 e µz and absorption cross-section σ a = µ / ρ a f k 2 µ = 2πρ a r 0 2k = k 4πr 0 σ a Absorption is proportional to the imaginary part of the forward scattering amplitude (Optical Theorem)

9 X-ray absorption edges

10 Absorption contrast imaging Computer axial tomography (CAT) Brain courtesy of Mikael Haggstrom

11 The Radon transform R(θ,x ) The line integral of the absorption coefficient can be determined from the ratio I 0 /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.

12 Fourier slice theorem Image reconstruction from the Radon transform Radon transform FT Fourier slice theorem IFT

13 Example of image reconstruction from the Radon transform

14 CAT Three dimensional image reconstruction Commercial, fully automated, desktop systems with resolution < 10 microns available from various companies

15 X-ray Microscopy Mirrors and Refractive Lenses 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

16 X-ray lenses The ideal shape Applying Fermat s Theorem to optical path lengths leads to c.f. equation of an ellipse Hence and Therefore

17 X-ray Fresnel zone plates Kinoform 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.

18 X-ray Fresnel zone plates Binary approximation Zone radius: Outermost zone width:

19 X-ray Fresnel zone plates Spatial resolution Rayleigh Criterion Si FZP Δr M =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

20 X-ray Absorption Microscopy FZP based scanning and full-field transmission microscopes Magnification M=d O-D /d S-O >1,000

21 X-ray Absorption Microscopy Imaging of biological materials The water window

22 X-ray Absorption Microscopy Cell division in a unicellular yeast Image courtesy of Carolyn Larabell

23 X-ray Magnetic Circular Dichroism (XMCD) 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

24 Image courtesy of Jo Stohr

25 X-ray Magnetic Circular Dichroism (XMCD) Sum rules for transitions to 3d states Iron thin films, Chen et al. PRL (1995)

26 Image courtesy of Jo Stohr

27 Image courtesy of Jo Stohr

28 X-ray magnetic microscopy STXM Image courtesy of Jo Stohr

29 Alignment of Ferromagnetic by Antiferromagnetic Domains Nolting et al., Nature 405, 767 (2000) a) LaFeO 3 layer b) Co layer Normalized Intensity (a.u.) dark light Normalized Intensity (a.u.) dark 10A Pt 12A Co LaFeO 3 (100) 2 µm medium light Photon Energy (ev) Photon Energy (ev)

30 Phase contrast imaging n=1-δ+iβ

31 The phase of the wave is: Phase contrast imaging Angular deviation perpendicular to beam: While the direction of propagation is:

32 Phase contrast imaging Depends on being able to determine the deflection angle α accurately. Essentially three methods: 1. Free-space propagation 2. Interferometric techniques 3. Analyser crystals

33 Phase contrast imaging Free-space propagation Beam displacement encodes information on the phase gradient perpendicular to the optical axis

34 Phase contrast imaging Example of free-space propagation Etched silicon (100) wafer Images courtesy of Martin Bech and Torben Jensen

35 Phase contrast imaging Free space propagation Blood cell infected with malaria parasite Images courtesy of Martin Bech, Martin Dierolf and Torben Jensen

36 Phase contrast imaging Grating interferometry David et al. (2002), Momose et al. (2003), Weitkamp et al. (2006), Pfeiffer et al. (2006)

37 grating period p 1 Phase contrast imaging Grating interferometry Wavevector: For we have Propagated wave Transverse to optical axis, expect periodic wavefield with k x =2π/p Talbot length

38 Phase contrast imaging Grating interferometry No sample With sample G2 period p 1 /2 Resolution determined by pixel size, currently around 10 microns

39 Phase contrast imaging Grating interferometry Sugar Water Absorption contrast Differential phase contrast Dark field (scattering) Fringe Visibility Images courtesy of Franz Pfeiffer and Martin Bech

40 Phase contrast imaging Grating interferometry Absorption contrast Dark field (scattering) Differential phase contrast Images courtesy of Franz Pfeiffer and Martin Bech

41 Phase contrast imaging Grating interferometry Pictures of the Future Siemens Magazine for Research and innovation Images courtesy of Franz Pfeiffer

42 The Phase Problem Measure intensity in a scattering experiment I = F 2, F = f j e iq r j j and it appears that all information on phase is lost.

43 Transverse Coherence Length Sources have a finite size and produce divergent waves Monochromatic waves are in phase at P, and again after a distance 2L T 2L T Δθ = λ with Δθ = D / R

44 Longitudinal Coherence Length Sources are not perfectly monochromatic 2L L = Nλ Nλ = (N +1)(λ Δλ) N λ Δλ Can also define coherence time t 0 = L L /c

45 Photon degeneracy: Photon Degeneracy Δ c 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 Δ c B t 0 ΔΩ tr A S Δν ν B : brilliance; t 0 : coherence time 1/Δν; A S : source area ΔΩ tr : solid angle subtended by tranverse coherence area (L T )2 R 2 1 R 2 λ 2 R 2 D 2 = λ2 Δν ν : bandwidth Δ c Bλ3 λ 3 c A S

46 Photon Degeneracy Δ c Comparison of different sources P. Guertler, HASYLAB

47 Imaging using coherent X-rays Speckle patterns SAXS thought experiment single spherical particle

48 Imaging using coherent X-rays Time average scattering and instantaneous speckle SAXS from dilute suspension of 500 nm silica spheres Coherent beam illumination, average of 200 exposures Data courtesy of Anders Madsen

49 X-ray photon-correlation spectroscopy (XPCS) Direct measurement of antiferromagnetic domain fluctuations Shpyrko et al. Nature (2007) Chromium: SDW below T N =311 K CDW induced at T N

50 X-ray photon-correlation spectroscopy (XPCS) Direct measurement of antiferromagnetic domain fluctuations Shpyrko et al. Nature (2007) (200) (200) static Time dependent speckle reveals information on the domain dynamics

51 X-ray photon-correlation spectroscopy (XPCS) Direct measurement of antiferromagnetic domain fluctuations Shpyrko et al. Nature (2007)

52 X-ray photon-correlation spectroscopy (XPCS) Direct measurement of antiferromagnetic domain fluctuations Shpyrko et al. Nature (2007) Temperature dependent domain wall dynamics

53 Coherent diffraction imaging Phase retrieval via oversampling Three critical factors allow the phase problem to be overcome by illuminating a finite size object of size L 3 with a coherent beam (Sayre, 1980): Diffraction pattern is extended in reciprocal space over volume ~1/L 3 Can sample the diffraction pattern with resolution better than 1/L Phases can be recovered by iterative numerical procedure (Fienup, 1982)

54 Coherent diffraction imaging Phase retrieval via oversampling 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

55 Coherent Diffraction Imaging Phase retrieval via oversampling

56 Coherent diffraction imaging Reconstruction of three dimensional electron density First demonstrated by Miao et al., Nature(1999 ) for a non-crystalline sample

57 Coherent diffraction imaging Reconstruction of three dimensional electron density of a crystal Real space resolution limited by inverse of maximum Q sampled. Currently around 30 nm Data courtesy of Ian Robinson

58 X-ray Holography Eisebitt et al. Nature (2004) Image produced by interference between object and reference beams X-ray resonant magnetic scattering at Co L 3 edge Fourier transform holography record interference pattern in far-field limit Image courtesy of Jo Stohr

59 X-ray holography Fourier transform holography 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 A R (Q) and A O (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 A R (Q), or in other words the real space image of the structure.

60 X-ray holography Eisebitt et al. Nature (2004)

61 X-ray holography Fidelity of reconstructed image

62 Where are we heading 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)

63 .and how far have we come Femtosecond diffractive imaging with a soft XFEL Chapman et al. Nature Physics (2006) FLASH facility at DESY, Hamburg Single pulse 32 nm, 25 fs, 4 x Wcm 2 Coherent diffraction imaging Real space image reconstructed Sample destroyed!