Download Elements of Modern X-ray Physics

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Elements of Modern X-ray Physics
Des McMorrow
London Centre for Nanotechnology
University College London
About this course
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“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
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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
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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
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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
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5 micron disks
red: absorption only
blue: phase only
Image courtesy of Timm Weitkamp
Relationship between scattering and refraction
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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
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⎛ 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
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Absorption contrast imaging
Computer axial tomography (CAT)
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Brain courtesy of Mikael Haggstrom
The Radon transform R(θ,x’)
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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
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Radon transform
FT
IFT
Fourier
slice
theorem
Example of image reconstruction from the Radon transform
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CAT
Three dimensional image reconstruction
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Commercial, fully automated, desktop systems
with resolution < 10 microns available from
various companies
X-ray Microscopy
Mirrors and Refractive Lenses
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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
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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
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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
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Zone radius:
Outermost zone width:
X-ray Fresnel zone plates
Spatial resolution
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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
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Magnification
M=dO-D/dS-O
>1,000
X-ray Absorption Microscopy
Imaging of biological materials
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The water window
X-ray Absorption Microscopy
Cell division in a unicellular yeast
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Image courtesy of Carolyn Larabell
X-ray Magnetic Circular Dichroism (XMCD)
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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
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Image courtesy of Jo Stohr
X-ray Magnetic Circular Dichroism (XMCD)
Sum rules for transitions to 3d states
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Iron thin films, Chen et al. PRL (1995)
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Image courtesy of Jo Stohr
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Image courtesy of Jo Stohr
X-ray magnetic microscopy
STXM
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Image courtesy of Jo Stohr
Alignment of Ferromagnetic by Antiferromagnetic Domains
Nolting et al., Nature 405, 767 (2000)
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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
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n=1-δ+iβ
Phase contrast imaging
While the direction of propagation is:
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The phase of the wave is:
Angular deviation perpendicular to beam:
Phase contrast imaging
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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
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Beam displacement encodes information on the phase gradient
perpendicular to the optical axis
Phase contrast imaging
Example of free-space propagation
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Etched silicon (100) wafer
Images courtesy of Martin Bech and Torben Jensen
Phase contrast imaging
Free space propagation
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Blood cell infected with malaria parasite
Images courtesy of Martin Bech, Martin Dierolf and Torben Jensen
Phase contrast imaging
Grating interferometry
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David et al. (2002), Momose et al. (2003), Weitkamp et al. (2006), Pfeiffer et al. (2006)
Phase contrast imaging
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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
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No sample
With sample
G2
period
p1/2
Resolution determined by pixel size, currently around 10 microns
Phase contrast imaging
Absorption
contrast
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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
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Absorption
contrast
“Dark field”
(scattering)
Differential
phase
contrast
Images courtesy of Franz Pfeiffer and Martin Bech
Phase contrast imaging
Grating interferometry
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“Pictures of the Future”
Siemens Magazine for Research
and innovation
Images courtesy of Franz Pfeiffer
The Phase Problem
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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
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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
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2LL = Nλ
Nλ = (N +1)( λ − Δλ )
λ
⇒N≈
Δλ
Can also define coherence time t 0 = LL /c
Photon Degeneracy Δc
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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
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P. Guertler, HASYLAB
Imaging using coherent X-rays
Speckle patterns
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SAXS thought experiment
single
spherical
particle
Imaging using coherent X-rays
Time average scattering and instantaneous speckle
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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)
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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)
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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)
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Direct measurement of antiferromagnetic domain fluctuations
Shpyrko et al. Nature (2007)
X-ray photon-correlation spectroscopy (XPCS)
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Direct measurement of antiferromagnetic domain fluctuations
Shpyrko et al. Nature (2007)
Temperature dependent domain wall dynamics
Coherent diffraction imaging
Phase retrieval via oversampling
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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
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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
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Coherent diffraction imaging
Reconstruction of three dimensional electron density
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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
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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)
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• 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
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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)
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X-ray holography
Fidelity of reconstructed image
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Where are we heading……
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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
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Chapman et al. Nature Physics (2006)
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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!