Quantitative extraction of spectral line intensities and widths from x-ray spectra recorded with gated microchannel plate detectors.

Abstract

Plasma spectroscopy requires determination of spectral line intensities and widths. At Sandia National Laboratories Z facility we use elliptical crystal spectrometers equipped with gated microchannel plate detectors to record time and space resolved spectra. We collect a large volume of data typically consisting of five to six snapshots in time and five to ten spectral lines with 30 spatial elements per frame, totaling to more than 900 measurements per experiment. This large volume of data requires efficiency in processing. We have addressed this challenge by using a line fitting routine to automatically fit each spectrum using assumed line profiles and taking into account photoelectron statistics to efficiently extract line intensities and widths with uncertainties. We verified that the random data noise obeys Poisson statistics. Rescale factors for converting film exposure to effective counts required for understanding the photoelectron statistics are presented. An example of the application of these results to the analysis of spectra recorded in Z experiments is presented.

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Quantitative extraction of spectral line intensities and widths from x-ray
spectra recorded with gated microchannel plate detectors
Greg Dunham
Ktech Corporation, 1300 Eubank Boulevard, SE Albuquerque, New Mexico 87123
J. E. Bailey, G. A. Rochau, and P. W. Lake
Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87185-1196
L. B. Nielsen-Weber
Ktech Corporation, 1300 Eubank Boulevard, SE Albuquerque, New Mexico 87123
?Received 5 February 2007; accepted 21 May 2007; published online 27 June 2007?
Plasma spectroscopy requires determination of spectral line intensities and widths. At Sandia
National Laboratories Z facility we use elliptical crystal spectrometers equipped with gated
microchannel plate detectors to record time and space resolved spectra. We collect a large volume
of data typically consisting of five to six snapshots in time and five to ten spectral lines with 30
spatial elements per frame, totaling to more than 900 measurements per experiment. This large
volume of data requires efficiency in processing. We have addressed this challenge by using a line
fitting routine to automatically fit each spectrum using assumed line profiles and taking into account
photoelectron statistics to efficiently extract line intensities and widths with uncertainties. We
verified that the random data noise obeys Poisson statistics. Rescale factors for converting film
exposure to effective counts required for understanding the photoelectron statistics are presented.An
example of the application of these results to the analysis of spectra recorded in Z experiments is
presented. © 2007 American Institute of Physics. ?DOI: 10.1063/1.2748674?
I. INTRODUCTION
X-ray emission spectroscopy is a powerful diagnostic
techniquefornumeroushigh
plasmas.1–5Applications include laser produced plasmas,
z-pinch radiation sources, and inertial confinement fusion
?ICF? capsule implosions. Information about the plasma
electron temperature is commonly obtained by measuring
spectral line intensity ratios, while the electron density is
inferred from the spectral line shapes. These measurements
involve several steps: ?1? data acquisition; ?2? quantitative
extraction of desired line intensities and widths; ?3? applying
instrument calibrations to the measured intensities and
widths; and ?4? inference of plasma conditions using an ap-
propriate plasma spectroscopy model.
Each of these steps is complex and must be carefully
tailored to the particular experiment. In the first step it is
frequently desirable to resolve the temporal and spatial gra-
dients. Hence spectra are often recorded with time-gated mi-
crochannel plate ?MCP? detectors,6,7since this permits the
acquisition of a sequence of snapshots that resolve the spec-
tral wavelengths and simultaneously provide one dimen-
sional spatial resolution. The second step is the topic of this
paper. The measured spectral lines are fitted with model
spectral line profiles, accounting as rigorously as possible for
the statistical data fluctuations ?noise?. The product of this
step is a collection of spectral intensities and widths along
with their attendant uncertainties. The third step consists of
applying instrument calibrations to the measurements in or-
der to determine the properties of the plasma source emission
from the signals measured at the detector. It is critical that
energydensity
?HED?
this step be performed after the fitting performed in step 2,
since otherwise the intensity dependence of the statistical
fluctuations will be altered and incorrect uncertainties will
result. The fourth step includes decisions about the model to
be used, approximations appropriate to the application, and
which measured intensities and widths offer the greatest sen-
sitivity to the relevant plasma parameters.
The quality of the fit obtained in step 2 is described
using the normalized ?2, defined as
i=1?
?i
?2=
1
N − m?
1
2?yi− y?xi??2?,
?1?
where N−m is the number of degreees of freedom for fitting
N data points with m parameters, yiis the data, y?xi? is the fit,
and ?i
The signal in adjacent detector channels must be uncor-
related in order for Eq. ?1? to be valid. For MCP detectors the
minimum channel size is determined by the 30–50 ?m di-
ameter intensity spot that corresponds to the bundle of pho-
toelectrons emerging from the rear of a single MCP channel.
The channel size used in the work reported here was 66
?66 mm2or larger, ensuring that the data statistics are not
affected by adjacent channel correlations. In addition, a pre-
scription must be available for the uncertainty ??i? of the
signal intensity measured in each detector channel. It is com-
monly assumed that the data obey Poisson statistics and the
uncertainty in each channel is equal to the square root of the
intensity. The measurements in this article were designed to
test whether that assumption is valid for the MCP detectors
used in our research.6
2is the variance.8
REVIEW OF SCIENTIFIC INSTRUMENTS 78, 063106 ?2007?
0034-6748/2007/78?6?/063106/6/$23.00 © 2007 American Institute of Physics
78, 063106-1
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Page 2

The Poisson statistics assumption was previously vali-
dated for streak camera and x-ray film detectors.9,10The
strategy followed here was similar. Data are recorded by ex-
posing the detector to a source that illuminates the detector
with an intensity that is spatially uniform. If Poisson statis-
tics are a reasonable approximation for the actual detector
response, then the fluctuations in the intensity about the
mean value will be well represented by a fit with a Poisson
distribution. A second test is performed by recording data
with a constant intensity source and varying exposure dura-
tion to produce a collection of exposures with varying mean
signal strengths. The data are converted from film exposure
units into counts in each channel using a process described
below. If Poisson statistics are valid, then a log plot of the
signal-to-noise ratio ?SN=yi/?i=yi
should be a straight line of slope of 1/2.
The data obtained in these experiments confirm that
Poisson statistics are a good approximation for the actual
fluctuations in spectra recorded with our MCP detectors. In
addition, data were recorded using MCP bias voltages of
500, 600, and 700 V. This enables a determination of
whether the mechanism that dominates the noise is the initial
conversion of incident x-ray photons into photoelectrons or
if the amplification within the MCP pores adds additional
noise. We find that most of the noise appears to originate in
the initial photoelectron production.
The methods described here also provide techniques for
converting experimental spectra from exposure units into
counts. This is an essential part of any fitting method that
seeks to quantitatively determine the uncertainties in spectral
line intensities and widths. An illustration of the extraction of
quantitative information from ICF capsule implosions con-
ducted at the Sandia National Laboratories Z facility is de-
scribed in Sec. IV.
1/2? as a function of signal
II. EXPERIMENT
The experimental setup was similar to Fig. 1 of Ref. 9. A
Manson source using a silver anode was used to uniformly
expose a six strip MCP to x rays. The spectrum from this
source consists primarily of characteristic Ag L lines.11The
distance between the x-ray source and detector was approxi-
mately 2 m. A 12.7 ?m beryllium filter was placed in front
of the MCP detector to block visible light. The MCP detector
was the type used in recent Z experiments2–5,12with sensi-
tivity calibrations described in Ref. 6. Each strip line was
40 mm across and 4 mm tall, with a 0.75 mm gap between
each strip line. The MCP used was manufactured by Burle
Inc., had a pore length/diameter ratio ?L/D? of 46, and had
an open area that was 65% of the total surface area. The
MCP front surface was coated in four separate layers with
the following materials and thicknesses: 75 Å of chromium,
5000 Å of copper, 75 Å of chromium, and 1000 Å of gold.
The phosphor type was P43, the phosphor to MCP gap was
0.7 mm, and the phosphor bias was 3000 V. Kodak TMAX
400 film was used to record the signal exiting the MCP de-
tector. Eight to ten exposure times ranging from 30 s to 2 h
were chosen so that the full range of optical densities from
film fog to near saturation were obtained. MCP bias voltages
of 500, 600, and 700 V were used.
The film was developed using a Wing-Lynch Model 5
automatic film processor using the following processing
times: water presoak—1 min; T-Max developer—15 min;
stop—1 min; fixer—10 min; wash—6 min. The film was
scanned onaPerkin-Elmer
matched NA=0.1 optics and a scan box of 22?22 ?m2.
MCP detectors have typical spatial resolution of approxi-
mately 50 ?m, resulting from the lateral spread in electrons
as they are accelerated from the MCP rear surface to the
phosphor. Therefore, statistically meaningful fits require that
the digitization of the MCP data be performed with channel
size of approximately 50 ?m or greater. In our work we
typically digitize the data with a 22?22 ?m2scanning ap-
erture, then rebin the data by a factor of 3. This process is
designed to avoid errors that result when low intensity film
data are scanned with an aperture that is larger than the MCP
spatial resolution element ?see Ref. 10?. For each batch of
film a calibrated step wedge was exposed and then developed
and scanned with the film to minimize any errors caused by
variations in development or light intensity during the devel-
opment and scanning processes. The step wedge was pro-
cessed to generate a density versus exposure correction curve
which was then applied to all of the image data from a single
batch to convert it from optical density to film exposure.
microdensitometerusing
III. DATA ANALYSIS AND RESULTS
The first approach is to examine whether the MCP fluc-
tuations are well fitted by a Poisson distribution. In that case
the uncertainty in the data is equal to the square root of the
number of counts.8
To construct a probability distribution from the data, lin-
eouts were taken across the length of the strip line. The lin-
eout averages over a 100 ?m tall region in the data, the same
height used in many capsule implosion experiments.2The
lineout height is important since the average flux is indepen-
dent of the height, but the number of counts contributing to
the average increases linearly with the height. Thus, the
signal-to-noise ratio improves for taller lineouts ?see below?.
Each point in a lineout may be considered an independent
measurement of the flux hitting the detector. The distribution
of flux values in individual channels about the mean expo-
sure is illustrated in Fig. 1. Figure 1?a? shows a sample im-
age from one exposure time with the black box indicating
where the Fig. 1?b? lineout was taken. In Fig. 1?c? a subset of
the data points from the lineout are shown with the mean and
standard deviation of the distribution marked by the solid
and dashed lines, respectively. For each lineout the data are
first rebinned by a factor of 3 to ensure that each channel is
uncorrelated with its neighbors. The strip line flux measure-
ments are not perfectly spatially uniform, as illustrated in
Fig. 1?b?. Flux variations over broad spatial scales result
from source and detector nonuniformities. In order to ac-
count for these broad spatial scale nonuniformities the data
are fitted with a second order polynomial, which is sub-
tracted from the data, then the mean is added back in to
063106-2 Dunham et al.Rev. Sci. Instrum. 78, 063106 ?2007?
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Page 3

retain the original mean flux. The distribution of flux values
represented by points were binned into a histogram using 50
uniformly spaced bins spanning the range of flux values to
create a plot of frequency of occurrence as a function of flux.
The frequency of occurrence for the ith histogram bin is
defined as the number of measurements falling within the
boundary of that bin. A fit to the histogram is performed
using either a Poisson or a Gaussian distribution. Prior to the
fit, the data are converted from the original film exposure
flux units of ergs/cm2into counts, using the rescale factor
defined by RSF=I/E, where I is the flux expressed in units
of counts and E is the flux in units of ergs/cm2. For data
expressed in counts, the Poisson distribution is
P?x? =?xe−x
x!
,
where x is the number of recorded counts in a given mea-
surement and ?x? is the average number of counts recorded
over all measurements. For large numbers of counts simpli-
fications lead to the Gaussian distribution, which is
exp?−?x − x ¯?2
For Poisson statistics, the standard deviation ?=??, where ?
is the mean. The full width at half maximum for a Gaussian
is ?Gauss=2.354?. Thus, there is a well-defined relationship
between ? and ?. Note that a priori the correct value of RSF
is unknown. However, the correct relationship between the
distribution width and the mean will only exist if the correct
value of RSF is used. We perform a sequence of fits for each
lineout using a range of RSF values. A good quality fit ?as
determined by the fit ?2? can only be obtained if the correct
value of RSF was used and if the data obey Poisson statis-
tics. The minimum of the fit ?2as a function of RSF deter-
mines the optimum value of the RSF and the data are judged
to obey Poisson statistics if the minimum ?2approaches one.
Figure 2 shows sample histograms with fits overlaid for
low, medium, and high numbers of counts. The minimum ?2
value of the fits is shown on each plot. The average chi-
squared values for these fits was 1.06±0.2. The quality of the
fits confirms that the data fluctuations obey Poisson statistics.
An alternate method of determining whether the data
obey Poisson statistics exploits the fact that if Poisson statis-
tics are valid, then a log plot of the signal-to-noise ratio
?SN=I/?I=I1/2? as a function of signal I should be a straight
line of slope of 1/2. To implement this method, a sequence
of eight to ten measurements at varying exposure levels were
made. Each measurement provides data recorded on six
MCP strip lines. From each strip line a set of 100 lineouts
were taken, each 4000 ?m long by 100 ?m wide, to make a
grid of lineouts 5 columns wide and 20 rows high. Individual
lineouts were taken over 1/10 the total strip line length in
order to minimize contributions to the fluctuations that result
from broad scale nonuniformities. This was devised as an
alternative to the polynomial fitting method described above.
We obtain six collections of 100 lineouts from each exposure
measurement, for a total of 600 lineouts. The data from the
lineouts are rebinned by a factor of 3. In order to evaluate
whether the fluctuations in the data scale according to Pois-
son statistics, we must convert the exposure E into counts I.
This could, in principle, be done using the fitting approach
described above, but the volume of data makes this imprac-
tical. Instead, we determine the RSF from each lineout by
measuring the mean exposure ?E? and the standard deviation
in the exposure ?E. Then, if we assume that the data obey
Poisson statistics, ?I
the standard deviation we also have ?I
fore RSF=?E?/?E
P?x? =
1
?2?x ¯
2x ¯?.
2=?I?=RSF?E. From the definition of
2=RSF2?E
2. We can test the assumption of Poisson
2and there-
FIG. 1. ?a? Sample MCP image with lineout area shown by black line. ?b?
Points—data from a 100 ?m wide lineout taken from the area shown in ?a?.
Black line—second order fit to data. ?c? Points—individual flux values to a
short 100 ?m lineout taken from the region between the dashed lines in part
?b?. Solid line—mean flux value. Dashed lines—1? error bars.
063106-3 Quantitative extraction of spectral line intensities and widths Rev. Sci. Instrum. 78, 063106 ?2007?
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Page 4

statistics by determining the RSF individually from a se-
quence of measurements at different exposure values and
averaging the result. The mean exposure ?E? from each mea-
surement is converted into counts using the average RSF. We
can then construct a log plot of SN=I/?Ias a function of I.
If a linear fit to the collection of data from the different
exposures has ?2close to 1 and a slope of 1/2, the assump-
tion of Poisson statistics is valid. This process was repeated
for the three bias voltages used in the experiment. The slope
of the lines fitted to the data were 0.57±0.12, 0.51±0.17, and
0.48±0.03 for the 500, 600, and 700 V bias voltages, respec-
tively ?Fig. 3?. These results provide an additional confirma-
tion that the MCP detector fluctuations are well represented
by Poisson statistics.
The lineout height is selected during the analysis of any
HED experiment depending on a balance between competing
FIG. 2. ?Color? Data in red, fit in black dash. Black arrow indicates mean.
?a? Low number of counts with a Poisson fit. ?b? Medium number of counts
with a Gaussian fit. ?c?. High number of counts with a Gaussian fit.
FIG. 3. Data points are average values from 600 measurements at each
exposure. Black line is fit to the data. Slopes of the fit lines are 0.57±0.12,
0.51±0.17, and 0.48±0.03, for the 500, 600, and 700 V MCP bias voltages,
images ?a?, ?b?, and ?c?, respectively.
063106-4Dunham et al.Rev. Sci. Instrum. 78, 063106 ?2007?
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Page 5

desires for good spatial resolution and signal to noise. The
rescale factor RSF should be linearly dependent on the lin-
eout height, as described above. To evaluate this dependence,
the above analysis was repeated using 400 and 1000 ?m tall
lineouts. The resulting RSF values are shown in Table I, in
comparison with the original 100 ?m tall lineout result. The
RSF values increase with lineout height, but the increase is
73%–82% of the linear dependence that is expected. This
discrepancy is not fully understood but may be due to sys-
tematic nonrandom fluctuations in the data that grow in im-
portance as the lineout area increases.
The confirmation that Poisson statistics are a valid ap-
proximation for MCP data analysis does not rely on under-
standing the exact origin of data fluctuations. Nevertheless, it
is an interesting question whether the bulk of the statistical
fluctuations arise in the initial conversion of x-ray photons
into photoelectrons or if additional noise is added during the
acceleration of the photoelectron cascade through the MCP
pores that provide the gain. The answer to this question helps
determine the optimum gain setting for a particular experi-
ment. In order to begin answering this question we can ex-
amine whether SN=?RSF?E is independent of the MCP
gain that is controlled by the applied bias voltage. If so, then
the noise must originate in the initial conversion process
?Fig. 4?. Each group of points represents the average signal
to noise compiled from 600 measurements at a given expo-
sure time. Data for five exposure times of 1, 2, 4, 8, and
16 min are shown. There is a significant amount of scatter in
these data but the data are consistent with the claim that no
significant degradation in SN occurs as the bias voltage is
increased. This result is similar to the findings in Ref. 10 and
implies that the SN is dominated by the initial photoelectrons
produced at the MCP pore entrance, prior to any gain. More
definitive information would require additional experiments.
IV. DISCUSSION
The results and methods developed here are illustrated
by application to x-ray spectra from an ICF capsule implo-
sion experiment described in Ref. 2. A sequence of space-
resolved 50 ?m tall lineouts were taken on the data shown in
Fig. 8 of Ref. 2, at the time snapshot designated t=−0.1 ns.
The 50 ?m lineout height corresponds to the size at the
source plasma and is equal to 100 ?m at the detector, the
same size as in the calibrations described above. In order to
fit the lineouts we must first determine the RSF that converts
the data from flux in erg/cm2into counts. Ideally, this could
be done using calibrations that uniformly expose the detector
to x rays, with the detector operated in the same pulsed mode
that is employed in the HED experiment. In practice, re-
source limitations prevent such calibrations. Instead, we de-
termine the RSF using the methods described above to ana-
lyze the fluctuations in a portion of the lineout that consists
of continuum with no spectral lines. In this experiment we
used the bound-free recombination between approximately
3.03–3.12 Å. This has limited accuracy since the number of
channels is small compared to the number used in the cali-
brations described here. We overcome this problem by aver-
aging the RSF determined from multiple MCP frames and
from as many multiple HED experiments as possible, subject
to the constraints that every MCP detector must be analyzed
separately and that changes between experiment series make
it impractical to compare measurements that are not from the
same experimental campaign. In this case measurements
from seven frames were averaged to yield a rescale factor of
1084±103.
After converting the spectrum to counts we fit the lin-
eouts using the line fitting code ROBFIT.13An example for
high-intensity data near the center of the spatial distribution
is shown in Fig. 5. ROBFIT determines the intensity, width,
and wavelength for each spectral feature self-consistently
with the underlying continuum. Various line shape standards
TABLE I. MCP voltage vs lineout width scaling data.
MCP voltage 1000 ?m/100 ?m400 ?m/100?m
700 V
600 V
500 V
Average
7.201±0.282
8.247±0.684
6.724±1.726
7.34±0.26
3.123±0.183
3.658±0.301
3.256±0.365
3.27±0.14
FIG. 4. Average signal to noise for a given exposure time at the three bias
voltages of 500, 600, and 700 V.
FIG. 5. ?Color? Fit to spectra from ICF capsule experiment. Black line—
data. Blue line—base line. Green line—fit to individual lines. Red line—
composite fit to spectrum.
063106-5Quantitative extraction of spectral line intensities and widths Rev. Sci. Instrum. 78, 063106 ?2007?
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