xBSM Beam Size Calibration
Dan Peterson
CesrTA general meeting 2012-12-07
introduction to the optics elements
introduction to the x-ray energy distribution
program for determining the x-ray energy distribution
beam size range of usefulness of the optics
the problem with taking more data
20121207 Peterson xBSM Optics, Beam Size Calibration
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There are two optics elements:
“Pinhole”, “Slit”, “Gap”
a single slit, or one dimensional pinhole
image is characteristic of a single slit diffraction pattern
the opening is optimized to provide the minimum image size for a point source
larger opening broadens the image because it becomes a projection
smaller opening broadens the image because of diffraction.
Coded Aperture array of slits, size varies
light passing through different slits interferes,
the mask has transparency >0, with phase shift, modifying the interference pattern
and providing sharp features in the image
With either optics element,
The calibration of the image to measure the beam size requires
a determination of the image for a point source;
the image for any other beam shape is a convolution with this point source shape (or size).
The determination of the point source image is dependent
on a determination of the x-ray energy distribution.
20121207 Peterson xBSM Optics, Beam Size Calibration
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So, we must determine the x-ray energy distribution
this depends on the generated distribution
and the effects of the various filters placed in the beam line:
the silicon substrate of the coded aperture,
(known)
any filters used, e.g. 4 μm diamond,
(known)
absorbing materials in the detector,
(sort of known )
and the energy dependent efficiency of the detector.
(not well known)
I take generated energy dependent intensity distribution from Jackson:
δI(ω)/δΩ|θ=0 = 3/(2π) e2/c γ2 ω/ωc exp( -2ω/ωc ) (Jackson 14.88)
This is the
and is
energy distribution in the synchrotron plane,
valid if the light cone is wide
compared to the optics element.
The critical angle is
θc = 1/γ (ω/ω c ) -1/3
(Jackson 14.89)
for the highest beam energy used with the low-energy coded aperture, 2.3 GeV, γ = 4501,
for a high x-ray energy at this beam energy, 7.5 keV,
ω/ω c = 4.307
thus, the smallest critical angle to be considered is
θc = 1.4 x 10-4 radians .
The full height at the optic corresponding to this angle is , θc * 2 * ObjectDistance = 1200 μm.
The height of the Coded Aperture is 300 μm (600 μm including extra interfering sidebands),
so the equation for energy spectrum on the orbit plane is valid
20121207 Peterson xBSM Optics, Beam Size Calibration
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While the Jackson formula is valid for the production distribution,
we do not know much about the effect of the detector.
We can place various filters in the beam line to
probe the relative x-ray intensity in various energy ranges.
The diamond filter low energy cut-off (50% of high energy
transmission) is at about 1.6 keV.
The aluminum filter has a peak at 1.5 keV,
but also passes the high energy x-rays.
The silicon filter is the substrate of the
coded aperture
The molybdenum filter has a peak at 2.5 keV,
but also passes the highest energy x-rays.
20121207 Peterson xBSM Optics, Beam Size Calibration
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April 2012
Total intensity measurements were taken with 3 beam energies and 3 filter conditions.
(Not all combinations were recorded.)
A constant size pinhole was used.
E0
pinhole image relative integrated area
(normalized to constant pinhole slit width)
(normalized to particle beam current)
{model} in red Jackson formula, without any modification
data in black
1.800 GeV
2.085 GeV
2.300 GeV
no filter
4 μm diamond filter
0.304 { 0.478 }
1
{1}
--{ 1.500 }
0.079 { 0.046 }
0.520 { 0.195 }
1.917 { 0.418 }
2 μm molybdenum filter
----- { 0.0039 }
0.050 { 0.021 }
0.299 { 0.051 }
The table shows the comparison between the data and model for the relative total intensity.
The unknown properties of the detector have a large effect.
20121207 Peterson xBSM Optics, Beam Size Calibration
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We have been using an exponent value of 2.31 to model detector effects.
- Relative integrated image area values for data and model are compared below.
- The beam energies and filters provide comparison over a range of average x-ray energy.
- The model has a finite number of x-ray energy bins but is being improved.
E2.31
pinhole image relative integrated area (normalized to constant pinhole slit width, particle beam current)
average x-ray energy in blue
{model} in red Jackson formula, with applied transmission: (Ex-ray)2.31
data in black
no filter
4 μm diamond
1.800 GeV
1.96 keV
0.304 { 0.284 }
2.42 keV
0.079 { 0.076 }
2.25 keV
--- { 0.030 }
2.085 GeV
3.09 keV
1
{ 1
3.50 keV
0.520 { 0.521 }
3.95 keV
--- { 0.175 }
4.47 keV
0.050 { 0.066 }
4.23 keV
1.917 { 1.595 }
4.77 keV
--- { 0.613 }
5.27 keV
0.299 { 0.273 }
2.300 GeV
3.95 keV
--- { 2.356 }
}
7.2 μm aluminum
20121207 Peterson xBSM Optics, Beam Size Calibration
2 μm molybdenum
2.70 keV
--- { 0.0081}
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A minimization of the
differences between model and data
yield a slight different exponent:
E(x-ray)2.25
The RMS difference between
data and model is 15%.
pinhole image relative integrated area
(normalized to constant pinhole slit width)
(normalized to particle beam current)
E2.25
{model} in red Jackson formula, with applied transmission: (Ex-ray)2.25
data in black
normalized
1.800 GeV
2.085 GeV
2.300 GeV
no filter
0.304 { 0.290 }
1
{1}
--{ 2.319 }
4 μm diamond filter
0.079 { 0.075 }
0.520 { 0.511 }
1.917 { 1.547 }
2 μm molybdenum filter
----- { 0.0081 }
0.050 { 0.065 }
0.299 { 0.261 }
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The EDGE provides more information about the
x-ray photon energy distribution.
By centering the x-ray beam on the “edge”
of a rectangular opening in the gold mask on the optics chip,
we measure the power through
a silicon filter
and a combination of gold and silicon filters.
The transmission through the gold leads to the
low (but not zero) pulse height region.
The transmission is energy dependent.
( I do not compare these rates directly to the
pinhole rates because there are different
horizontal widths. )
But the ratio, PHlow/PHhigh, is a value that
can be compared to the model.
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2012-08-24
review of LOW/HIGH values for EDGE
key: { DATA } { model with (Ex-ray)2.31 scaling, 4.0 μm diamond}
R[Low/High]
no filter
DATA {model}
R[Low/High]
diamond filter
RDATA
/Rmodel
DATA {model}
R[Low/High]
molybdenum filter
RDATA
/Rmodel
DATA {model}
C-line 1.800 GeV
0.167 { 0.113 } 1.48
0.177 { 0.162 }
1.10
{ 0.213 }
C-line 2.085 GeV
0.191 { 0.174 } 1.10
0.205 { 0.214 }
0.96
{ 0.325 }
C-line 2.300 GeV
0.268 { 0.240 } 1.12
0.276 { 0.276}
1.00
0.324 { 0.406 }
0.147 { 0.144 }
1.02
D-line 2.085 GeV
E2.31
RDATA
/Rmodel
0.80
Differences between model and data
are minimized with an exponent:
E(x-ray)2.5 ,
in some agreement with the
results of the filter measurements.
The RMS difference between
data and model is again ~15%.
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Another test that can be performed is comparing the resultant image shapes with the data.
Lines show image templates for various exponents,
all with 2.085 GeV,
4 micron diamond,
(illustrated at 7 micron beam height).
These are fit to the data.
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Three runs are used for the fit.
Each has a corresponding pinhole run taken under the same bean conditions.
The pinhole run is used to fix the beam size in the fit.
The image shape suggests an exponent: ~ 2.8, but is “consistent” with 2.25.
(This the chisquared/dof, while the other results are RMS.)
fits with Hybrid Template fitter
Brian Heltsley
20121207 Peterson xBSM Optics, Beam Size Calibration
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Is the image template improved with the particular energy distribution?
Or, is the image template sensitive only to the average energy?
Lines show image templates for mono-energetic x-rays
covering the range of the possible average energies.
20121207 Peterson xBSM Optics, Beam Size Calibration
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The fits to image template for various exponents, plotted as a function of the average energy.
Also shown are the fits to mono-energetic x-rays.
- The distribution of energies improves the fits.
- Something interesting at 2.3 keV is due to rapidly changing phase shifts.
fits with Hybrid Template fitter
Brian Heltsley
20121207 Peterson xBSM Optics, Beam Size Calibration
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In summary, the model of an x-ray distribution:
(Jackson 14.88) x En
is compared to the data in 3 ways:
- transmission through filters affecting the pinhole image area
- the relative transmission through the gold mask in the CA,
- and the shape of the observed CA image.
The three comparisons indicate that
an exponent in the range { 2.2 : 2.8 }
reasonably models the x-ray energy distribution.
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The modeled x-ray energy distribution
results in an image template that can be
used to fir CA data.
Measurements agree with
corresponding pinhole measurements.
C: RD_04029
0.53 μm Au standard CA
root=xr2m
σimage=0.5751 pixel
σbeam=9.85 μm
20121207 Peterson xBSM Optics, Beam Size Calibration
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December 2012, D-line
preliminary
E2.31
pinhole image relative integrated area (normalized to constant pinhole slit width, particle beam current)
average x-ray energy in blue
{model} in red Jackson formula, with applied transmission: (Ex-ray)2.31
data in black
no filter
4 μm diamond
7.2 μm aluminum
2 μm molybdenum
particle energy
1.800 GeV
1.96 keV
--- { 0.284 }
2.42 keV
--- { 0.076 }
2.25 keV
--- { 0.030 }
2.70 keV
--- { 0.0081}
2.085 GeV
3.09 keV
1
{ 1
}
3.50 keV
0.51 { 0.521 }
3.95 keV
0.14 { 0.175 }
4.47 keV
0.066 { 0.066 }
2.300 GeV
data/0.73
3.95 keV
1.82 { 2.356 }
2.49
4.23 keV
1.11 { 1.595 }
1.52
4.77 keV
0.31 { 0.613 }
0.42
5.27 keV
0.17 { 0.273 }
0.24
20121207 Peterson xBSM Optics, Beam Size Calibration
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The model can be used to predict the resolution power of the optics element.
Shown:
a coded aperture calculated image,
smeared to beam sizes
5, 7 microns
16, 18 microns.
Calculate the RMS difference
of the smeared images.
20121207 Peterson xBSM Optics, Beam Size Calibration
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20121207 Peterson xBSM Optics, Beam Size Calibration
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