Flash Spectroscopy using Meridionally- or Sagittally-bent Laue Crystals: Three Options
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Flash Spectroscopy using Meridionally- or Sagittally-bent Laue Crystals: Three Options
Zhong Zhong National Synchrotron Light Source, Brookhaven National
Laboratory
Collaborators: Peter Siddons, NSLS, BNL
Jerome Hastings, SSRL, SLAC
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Agenda• The problem we assume• X-ray diffraction by bent crystals
– Meridional– Sagittal
• Sagittally bent Laue crystal– Focusing mechanism, focal length– Condition for no focusing
• Three Laue approaches– Meridionally bent, whole beam – Meridionally bent, pencil beam– Sagittally bent, whole beam
• Some experimental verification • Conclusions
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The problem we “assume”• Would like to measure, in one single pulse, the
spectrum of spontaneous x-ray radiation of LCLS
• Energy bandwidth: 24 eV at 8 keV, or 3X10-3 E/E
• Resolution of dE/E of 10-5, dE= 100 meV
• 5 micro-radians divergence, or 1/2 mm @ 100 m
• Source size: 82 microns
• N (1010 assumed) ph/pulse
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The general idea
• Use bent Laue crystals to disperse x-rays of different E to different angle.
• Go far away enough to allow spatial separation.
• Use a linear or 2-D intensity detector to record the spectrum.
• Un-diffracted x-rays travel through and can be used for “real” experiments.
y
E1
E2
R
O
d2
T
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Laue vs. Bragg, perfect vs. bent
Bragg
Laue
Symmetric Asymmetric
B B
BB
Angular acceptance Energy bandwidth (micro-radians) (E/E)Perfect Crystal a few-10’s 10-4- 10-5
Meri. Bent Laue xtal 100’s-1000’s 10-3 - 10-2
Sag. Bent Laue xtal 100’s 10-3
Order-of-Magnitude
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Diffraction of 8-keV X-rays by Si CrystalReflection
Bragg
Angle (deg)
Darwin
Width (micro-radians)
Extinction length (microns)
dE/E
111 14.3 34 3.0 1310-5
220 23.8 25 2.6 5.7 10-5
311 28.3 14 4.1 2.710-5
400 34.8 17 3.2 2.410-5
511 47.9 9.1 5.4 0.8310-5
440 53.8 13 4.1 0.9110-5
533 69.4 10.7 6.7 0.4010-5
• 511 or 440 can be used to provide 10-5 energy resolution
• Absorption length ~ 68 microns
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• What bending does? – A controlled change in angle of lattice planes and d-spacing of
lamellae through the crystal• Lattice-angle change- determines dispersion• D-spacing change – Does not affect the energy resolution,
as it is coupled to lattice-angle change …diffraction by lamellae of different d-spacing ends up at different spot on the detector.
• Both combine to increase rocking-curve width - energy bandwidth
• Each lamella behave like perfect crystal –resolution• Reflectivity: a few to tens of percent depends on diffraction
dynamics and absorption– Small bending radius: kinematic – low reflectivity– Large bending radius: dynamic – high reflectivity
• A lamellar model for sagittally bent Laue crystals, taking into account elastic anisotropy of silicon crystal has recently been developed. (Z. Zhong, et. al., Acta. Cryst. A 59 (2003) 1-6)D
Diffraction of X-rays by Bent Laue Crystal
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Sagittally-bent Laue crystal
• : asymmetry angle • Rs: sagittal bending radius • B: Bragg angle• Small footprint for high-E x-rays• Rectangular rocking curve• Wide Choice of , and crystal thickness, to control the energy-resolution•Anticlastic-bending can be used to enable inverse-Cauchois geometry
Side View
Top view
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Anisotropic elastic bending of silicon crystal
}2{
}2{
}{
3343532
4323632
332
43232
132
xzSxySSxEu
xzSxySSxEu
SzzySSySxEu
z
y
x
'23
33
23
232
2
332
2
ratioPoisson
2
curvature bending cAnticlasti
2
curvature bending Sagittal
SS
S
ESy
u
ESz
u
s
m
ym
xs
Displacement due to bending
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For Sagittally-bent crystals
Lattice-angle change
]cos)tan(cossin)[(
/)cossin(
2'63
'23
'13
'23
//////
ssssR
T
luu
Bs
yxangle
)]coscossinsin([tan
/)cossin(tan
2'23
'63
2'13
sssR
T
luul
l
Bs
yxBspaced
d-spacing change
Rocking-curve width
)]coscossinsin(tan
cos)tan(cossin)([
2'23
'63
2'13
2'63
'23
'13
'230
sss
ssssR
T
B
Bs
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For Meridionally-bent Crystals
Lattice-angle change ])tan(cossin)1[( '
53'13 ss
R
TB
sangle
)]coscossinsin([tan 2'53
2'13 ss
R
TB
sspaced
d-spacing change
Rocking-curve width
]sintancossin)tan1(
costan)tan([
2'13
'53
'13
'53
20
sss
sR
T
BB
BBs
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Three Laue Options
0.5 mm
E1
E2
Meridionally bent, “whole” beam
Meridionally bent, pencil beam
Sagittally bent, whole beam
E1
E2
0.5 mm
E1
E2
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Meridionally bent, “whole” beam• How it works
– Using very thin (a few microns) perfect Silicon crystal wafer.
– Use symmetric Laue diffraction, with S53
’=0, to achieve perfect crystal resolution
• Bandwidth:
– Easily adjustable by bending radius R, R~ 100 mm to achieve E/E~3x10-3.
• Resolution
dE/E~10-5 for thin crystals, T~ extinction length, or a few microns
BB R
yEE
sintan/
22
22
)tan/()/(
tan/
BD
B
D
RT
dEdE
y
E1
E2
R
O
d2
T
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Meridionally bent, “whole” beam• Advantages
– Wide range of bandwidth, 10 –4 - 10-2 achievable.
– High reflectivity ~ 1.– Very thin crystal (on the
order of extinction length, a few microns) is used, resulting in small loss in transmitted beam intensity.
y
E1
E2
R
O
d2
T
•Disadvantages–Different beam locations contribute to different energies in the spectrum
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Meridionally bent, “whole” beam• Our choice
– Assuming y=0.5 mm– Si (001) wafer– 440 symmetric Laue
reflection– T=5 microns– R=200 mm
y
E1
E2
R
O
d2
T
• Yields (theoretically) – 310-3 bandwidth– 2.6 10-5 dE/E, dominated by xtal thickness
contribution– Dispersion at 10 m is 80 mm– 107 ph/pulse on detector, or 104 ph/pulse/pixel
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Meridionally bent, Pencil BeamE1
E2
• How it works– Bending of asymmetric crystal
causes a progressive tilting of asymmetric lattice planes through beam path.
• Bandwidth: – Adjustable by bending radius
R, thickness, and asymmetry angle , possible to achieve E/E~3x10-3 with large .
• Resolution• dE/E is dominated by
beam size y, dE/E ~ y/(RtanB)
• Y must be microns to allow 10-5 resolution
])tan(cossin)1[( '53
'13 ss
R
TB
sangle
y
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Meridionally bent, Pencil BeamE1
E2
• Advantages– Can perform spectroscopy
using a small part of the beam
• Disadvantages: – Less intensity due to cut in
beam size, and typically 10% reflectivity due to absorption by thick xtal.
• Our pick (out of many winners)
– Si (001) wafer
– 333 reflection, =35.3
– T=50 microns
– R=125 mm
• Yields– 310-3 bandwidth– 0.8 10-5 dE/E, – Dispersion at 10 m is 71 mm– 10% reflectivity– 106 ph/pulse on detector, or
103 ph/pulse/pixel
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Sagittally bent, whole beam
0.5 mm
E1
E2
• How it works– Sagittal bending causes a
tilting of lattice planes– The crystal is constrained
in the diffraction plane, resulting in symmetry across the beam.
– Symmetric reflection used to avoid Sagittal focusing, which extends the beam out-of-plane.
• Bandwidth: – Adjustable by bending
radius R, thickness, and crystal orientation.
– E/E~1x10-3.
• Resolution• dE/E probably will be
dominated by the variation in lattice angle across the beam, must be less than Darwin width over a distance of .5 mm.
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Sagittally bent, whole beam
0.5 mm
E1
E2
• Advantages– Uses most of the photons
• Disadvantages: – Limited bandwidth due to the
crystal breaking limit.
• Our choice – Si (111) wafer– 4-2-2 symmetric
Laue reflection– T=20 microns– R=10 mm
• Yields– 0.610-3 bandwidth– 1 10-5 dE/E– Dispersion at 10 m is 21
mm– 70% reflectivity– 109 ph/pulse on detector
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Testing with White Beam
• Four-bar bender • Collimated fan of white incident beam • Observe quickly sagittal focusing and dispersion • Evaluate bending methods: Distortion of the diffracted beam variation in the angle of lattice planes
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Observation of previous data
1 cmh=15 mm
h=0
h=–12
h=–12
h=15 mm
On the wall at 2.8 meters from crystal
Behind the crystal
• 0.67 mm thick, 001 crystal (surface perpendicular to [001]), Rs=760 mm•111 reflection, 18 keV• Focusing effects: Fs=5.7 m agrees with theory of 6 m• “Uniform” region, a few mm high, across middle of crystal• Dispersion is obvious at 2.8 meters from crystal.
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Experimental test: sagittally bent, whole beam
• 4-2-2 reflection, (111) crystal, 0.35 mm thick, bent to 500 mm radius, 9 keV• Exposures with different film-to-crystal distance.• No sagittal focusing due to zero asymmetry.• The height at 0.75 m is larger than just behind the crystal, demonstrating dispersion. • Distortion is noticeable at 1 m, could be a real problem at 10 meters.
4-2-2
0.11 m 0.37 m 0.75 m
0.11 m
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Measuring the Rocking-curves
•NSLS’s X15A. 111 or 333 perfect-crystal Si monochromator provides 0.1(v) X 100 mm (h) beam, 12-55 keV• (001) crystal, 0.67 mm thick, 100 mm X 40 mm, bent to Rs=760 mm, active width=50 mm• Rm=18.8 m (from rocking-curve position at different heights)• Rocking curves measured with 1 mm wide slit at different locations on crystal (h and x)
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Rocking-curve Measurement
• 111 reflection on the (001) crystal, =35.3 degrees• FWHM~ 0.0057 degrees (100 micro-radians)• Reflectivities, after correction by absorption, are close to unity (80-90%) dynamical limit• Model yields good agreement.
-200 -100 0 100 200Rocking Angle (microradians)
0.0
0.2
0.4
0.6
0.8
Re
flect
ivity
20 keV
25 keV
30 keV
40 keV
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Depth-resolved Rocking-curve Measurement
)]coscossinsin(tan
cos)tan(cossin)([
2'23
'63
2'13
2'63
'23
'13
'230
sss
ssssR
T
B
Bs
Rocking-curve width
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Two crystals, many reflections tested
18 keV incident beam, 20 micron slit size0.67 mm thick crystal, bent to Rs=760 mm
Rocking-curve width
)]coscossinsin(tan
cos)tan(cossin)([
2'23
'63
2'13
2'63
'23
'13
'230
sss
ssssR
T
B
Bs
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Comparison: 001 crystal and 111 crystal
100 xtal, 111 reflection =35 degS31
'=-0.36, S32'=-0.06, S36
'=0Upper-case:0=92-16=76 radLower-case: 0=-73-16=-89 rad
111 xtal, 131 reflection =32 degS31
'=-0.16, S32'=-0.26, S36
'=0Upper-case:0=-73-35=-108 radLower-case: 0=177-35= 141 rad
Upper-case
Lower-case
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Future Directions
• Other crystals?– Diamond? for less absorption– Harder-to-break xtals? To increase energy
bandwidth of sagittally-bent Laue
• Experimental testing– 10 m crystal-to-detector distance is hard to come by– 3-5 m may allow us to convince you
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Summary• 3 possible solutions for the “assumed” problem.
• Option 3, sagittally-bent Laue crystal, is our brain child.
• Option 1 has better chance.
• They all require – distance of ~ 10 m
– 2theta of ~ 90 degrees -> horizontal diffraction and square building
– linear or 2-D integrating detector
• With infrastructure in place, it is easy to pursue all options to see which, if any, works.
• Typical of bent Laue, unlimited knobs to turn for the true experimentalists … asymmetry angle, thickness, bending radius, reflection, crystal orientation …
• We have more questions than answers …
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Focal Length
Real Space
Reciprocal Space
• Diffraction vector, H, precesses around the bending axis change in direction of the diffracted beam
sinsin2
sin
)2/sin(sin2
B
ss
s
RF
Rx
HH
• Fs is positive (focusing) if H is on the concave side• No focusing for symmetric Laue: At =0 Fs is infinity - H points along the bending axis
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Inverse-Cauchois in the meridional plane
• At =0, E/E is the smallest inverse-Cauchois geometry• E/E determined by diffraction angular-width 0~ a few 100’s micro-radians• Source and virtual image are on the Rowland circle. • No energy variation across the beam height
])cos(
1[
tan
)/(/
1
21
20
2
Bmv
B
s
R
F
FEE
Meridional plane
)cos()cos(1 Bs
Bm C
RRF
Condition for Inverse-Cauchois