# Application of Single and Dual Multilayer Optics for Powder X-Ray Diffraction

Authors: Scott T. Misture
Journal: X-Ray Optics and Instrumentation
 (2008)

---

## Abstract

The performance of parallel beam multilayer optics, including a parabolic multilayer Osmic MaxFlux GO-13N and a flat custom multilayer, was evaluated experimentally and compared to Bragg-Brentano and traditional
‘‘parallel beam’’ or ‘‘thin film’’ optical geometries. A novel arrangement of a parabolic multilayer in the incident beam with a flat multilayer in the diffracted beam functioning as an analyzer crystal was proven effective for powder diffraction applications. The dual-optic configuration improves resolution while eliminating sample displacement and transparency errors as expected for a configuration with equatorial divergence below 100 arcseconds. Fundamental parameters fitting showed that the parabolic multilayer can be accurately modeled using a constant Gaussian function, while a long parallel-plate soller collimator requires a constant hat function. No additional convolutions are necessary for the diffracted-beam flat multilayer because of the lower acceptance angle.


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## Body

## 1. Introduction

Powder diffraction broadly encompasses
many measurements; powder diffraction, thin film diffraction, diffuse and
specular reflectivity, residual stress, and texture analysis among them. Of these measurements, powder diffraction,
thin film diffraction, and reflectivity are routinely performed usingΘ–2Θ diffractometers with appropriate
attachments and optics. The
incorporation of parabolic multilayer optics for powder diffraction has been
studied by several groups, with varied results that depend upon the particular
application [1–9].
Research to date has focused mainly on the use of a parabolic multilayer
in the incident beam to collimate the beam, with diffracted-side optics that
include either a long soller parallel-plate collimator or collimator with
crystal combination [4, 6, 8, 10, 11]. Additional studies focus on the use of parabolic
multilayers to couple with 2- or 4-bounce channel cut crystal monochromators [7, 12], and dual mirror combinations
where a secondary parabolic multilayer is used as an analyzer crystal [13].Ray tracing calculations for prediction
of intensities and diffraction geometries by Toraya and Hibino [9] as well as by Gutman and Verman [3] demonstrate that one should
expect a shift in the α
1 and α
2 peak separations as a result of
slightly different diffraction angles from the multilayer. Of course, the different diffraction angles
for the α
1 and α
2 radiation are not an issue if
one employs a channel-cut monochromator to further condition the beam.Applications of parallel beam systems
for powder and thin film diffraction, either using parallel-plate collimators
or multilayers, are reasonably well established. A recent paper [8] compared the use of a single
parabolic multilayer with cross-coupled (Kirkpatrick-Baez) multilayers, and
clearly demonstrated that the incident beam parabolic multilayer is well-suited
for the study of very thin films. The
present paper focuses on the use of a dual-multilayer configuration for powder diffraction,
however, with a novel arrangement that includes a parabolic multilayer on the
incident side and a flat multilayer as an analyzer crystal.


## 2. Experimental Procedures

A Siemens
D-500 diffractometer, fitted with a Cu line-focus tube, scintillation counter,
and 200 mm radius, was used in all of the measurements. Five diffraction geometries were used for the
comparisons. All experiments were run as
step scans, using0.02
° step size and
2-second count times with the X-ray tube running at 1200 W. Each powder specimen was mounted in a 15
×
15
mm side-drifted sample holder.  In all
cases, axial collimation was provided by a 3
° collimator in the incident beam side.The instrumental arrangements included the following.Bragg-Brentano (BB) geometry with 1
° divergence slits,
0.05 mm receiving slit, 6
° tube takeoff
angle, diffracted-beam graphite monochromator, and 13 mm diameter detector
aperture. All measurements were made as
coupled Θ–2Θ step scans.Parallel-Beam (PB) geometry using 2
Θ scans with Θ fixed at 2.0
°.
The (P-B) arrangement included divergence slits set to 0.14
° to illuminate
15 mm of the specimen at Θ
=
2
°. A thin film
attachment (long Soller collimator) with divergence of 0.15
° was installed
on the diffracted beam side of the diffractometer, with a detector aperture 13 mm in diameter. A Ni beta filter 0.03 mm
thick was installed on the diffracted beam side of the diffractometer as the
only energy discrimination.Parabolic Multilayer (PM). An Osmic MaxFlux optic (GO-13N) was
installed in a modified Huber monochromator housing (see Figure 1), while the
diffracted beam side included the 0.15
° thin film attachment and a detector aperture 13 mm in
diameter. The multilayer and sample were
arranged in either ++ or +− configurations.
The GO-13 is characterized by an average reflectivity of 66% for Cu
radiation with physical dimensions of 40 by 20 mm. No beta filter was used. 2
Θ scans were made with Θ fixed at 3.6
° or as coupled measurements, as noted
below. Because the optic provides a beam
which is ~0.9 mm in height, the Θ angle was chosen to illuminate 14 mm of the 15 mm powder samples when illuminating the sample at 3.6
°
Θ.Schematic of the beam
configurations tested, including (a) a single parabolic multilayer in ++ arrangement,
and (b) the novel dual multilayer configuration in the +++ configuration.
(a)
(b)Parabolic incident multilayer and diffracted-beam flat
multilayer (PM
+
FM). This arrangement is shown
in Figure 1 and includes a flat multilayer in the diffracted beam side. The flat multilayer was produced by Osmic,
Inc., Mich, USA, with physical dimensions of 32
by 65 mm and average reflectivity for Cu radiation of 72%. The rocking curve width on average is 96 arcseconds. Measurements were
made using either coupled Θ−2Θ scans or with Θ fixed at 3.6
°, such that the entire beam (~
0.8
×
12 mm) illuminated the 15 mm sample. The
multilayers and sample were arranged in either +++ or +−+ configurations.The samples
are listed in Table1 and include a wide range of absorption coefficients for
evaluating sample transparency errors.
Each powder sample was loaded into a side-drifted powder holder with
dimensions 15 mm by 15 mm by 4 mm deep.  Data analysis was completed using HighScore [14] and Topas [15].Table 1
Samples used in the investigation.
Sample
Form
Silver
powder
Linde
C (α-Al
2
O
3)
powder
Urea
[CO
(
N
H
2
)
2]
powder
SRM
640b & c Si
powder
SRM
660LaB
6
powder
Quartz
50
×
50 mm plate


## 3. Results and Discussion

All of the
diffraction geometries provided high-quality data within expectations of the
optical configurations, as shown for example in Figure2. The novel arrangement of PM
+
FM provided
data of outstanding quality without any obvious complications. Visual inspection of the range of data
naturally demonstrates large differences in resolution and intensity among the
various configurations.Figure 2
Qualitative comparison of data from
four optical configurations for a thick powder sample of urea. Data are normalized to a common background
intensity and offset vertically for clarity.A point of
concern with multilayer optics is the distortion of theK
α
1 and K
α
2 beam directions, which has been
established both experimentally and by ray tracing calculations [3, 8, 9]. As shown by Toraya and Hibino [9], the spacing between the K
α
1 and K
α
2 lines is a function of several
parameters, most importantly the spectral resolution of the multilayers. These
authors used an optical configuration nearly identical to our PM configuration,
but with an older-generation longer (150 mm) multilayer. Their work incorporated a high-resolution
encoded goniometer; however, the α
1
/
α
2 separation
remained unchanged within the experimental error. Likewise, we were unable to detect abnormal α
1
/
α
2 separation in
any of our measurements, either PM or PM
+
FM, in either ++ or +− arrangement for
the former or +−+ or +++ for the latter.
Both profile fitting and Rietveld analyses were successful without
altering the α
1
/
α
2 separation or intensity ratios,
as described in more detail below.


### 3.1. Intensity

The absolute
intensities were compared on a peak-by-peak basis for Ag, alumina, urea, and
quartz. The overall average absolute
peak intensities were used for the comparisons.
Figure3 shows the absolute intensities relative to the intensity
measured in the standard Bragg-Brentano configuration and highlights the
improved intensity, of approximately 4 times, in the configuration with the
parabolic multilayer and fixed incident angle of 3.6
°.Figure 3
Ratio of absolute intensities
obtained for each optical configuration.While several
authors have used ray tracing calculations to estimate intensities and beam
paths with multilayer optics [3, 8, 9], the differences in the measured
intensities shown in Figure 3 can be explained qualitatively by considering the
beam paths. For example, in the BB
geometry, ~
1
° of divergence
is used in the incident and diffracted beams.
In stark contrast, the traditional PB measurement includes very fine
slits to irradiate a 15 mm sample at an incident angle of 2
°
Θ, or ~
0.14
° of divergence. Therefore, one would expect substantially fewer photons incident upon
the sample, by a factor of ~7, in the PB configuration. Further, to obtain
similar measured intensities as shown in Figure 3, the diffracted-beam side
optics must pass ~7 times more photons than the diffracted-beam graphite
monochromator in the BB configuration.Extending
similar arguments to the multilayers, one finds that the parabolic multilayer
captures~
0.5
° of
divergence, diffracts ~65% of the incident radiation, and is equipped with the
same diffracted-beam side optics as the PB configuration. Therefore, the total intensity should be on
the order of ~2.5 times larger than that measured in the BB configuration, a
prediction in reasonable agreement with the measured value of 3.8 times,
considering minor differences in alignment.Adding the
second multilayer in the diffracted beam side substantially reduces the
intensity, but, as shown later, also improves resolution. The second multilayer, with reflectivity of
~70%, offers approximately the same throughput as a graphite diffracted beam
monochromator. However, the multilayer
also offers beam collimation that is much tighter than that offered by the long
Soller collimator, by a factor of ~6 (recall that the rocking curve width of
the multilayer is 96 arcseconds, and the long Soller collimator has divergence of~
0.15
° or 540 arcseconds). Therefore, one can employ the multilayer
without a long Soller collimator to achieve parallel beam conditions. The efficiency of the multilayer is close to
that of the Soller collimator used here, such that the overall intensity of the
PM
+
FM design is approximately the same as for the PB and BB configurations.


### 3.2. Peak-to-Background Ratios

The
peak-to-background ratios were calculated on a peak-by-peak basis exactly as
were the intensity ratios. The results
are shown in Figure4. The BB
configuration provides a peak-to-background ratio that is substantially higher
than for the PB configuration, about the same as the PM configuration and worse
by a factor of ~0.5 than the PM
+
FM configuration. One would expect the PB configuration to provide
lower peak-to-background ratios because the only energy discrimination in the
system is the Ni foil K
β filter. In
contrast, the PM configuration yields peak-to-background ratios that are more
than double those for the PB configuration.
Here, one must consider the incident beam spectrum after diffraction
from the incident beam parabolic multilayer.
Because the multilayer passes only a small window around the K
α
1 and K
α
2 lines, scattering and
fluorescence are reduced before the sample, thus improving the
peak-to-background ratio. Adding the
second multilayer again provides additional energy discrimination and tight
angular acceptance, improving the peak-to-background ratio further.Figure 4
Peak-to-background ratios for each
configuration relative to the BB configuration.


### 3.3. Resolution

The
instrumental resolution was determined using the traditional approach by
measuring the “5 fingers” of quartz, as recommended by Jenkins and Snyder [16]. The divergence of the PB, PM, and PM
+
FM
configurations, although “parallel” in comparison to BB optics, is a function
of the optics. For the ‘‘parallel’’
configurations, the incident/diffracted beam side divergence is on the order of
540/540 for PB, 96/540 for PM, and 96/96 arcseconds for the PM
+
FM configurations. Naturally, one expects improved instrumental
resolution with lower divergence optics.Figure5 shows
the diffraction patterns for quartz in the various diffractometer
configurations. Calculated resolution
values between 1.5 and 2.0, based upon peak-to-valley ratios in the quartz
peaks, are the accepted range for “high-resolution” Bragg-Brentano measurements
[16].  As shown in Figure 5, the BB configuration
resolution value is 1.7, while those for PB, PM, and PM
+
FM are 1.5, 2.0, and
4.0, respectively. Note that the
resolution when using the PM optics is some 20% better than a Bragg-Brentano
measurement and ~40% better than the traditional PB optics. Incorporating the diffracted beam multilayer
optic again improves the resolution to a value of 4, as would be expected from
the smaller acceptance angle of the diffracted-beam multilayer compared to the
Soller collimator.Figure 5
Quartz diffraction data for each
optical configuration.


### 3.4. Profile Shapes

Diffraction
peak shapes influence all aspects of data analysis, and for low absorption
materials measured with Bragg-Brentano geometry, substantial systematic error
should be expected. The peak shapes/widths
were evaluated as a function of angle for the PM and the new PM + FM
configurations by fitting data from NIST SRM640c Si powder.  Peak shapes were easily modeled using pseudo-Voigt
profiles and the peak widths follow a typical behavior with diffraction angle [17].  Figure 6 shows the FWHM as a function of angle
fitted with a parabolic function, which suggests that the dual multilayer
optics result in well-behaved line shapes and widths.Figure 6
Peak FWHM (K
α
1) for SRM640c Si powder using the
PM and PM
+
FM configurations. Error
bars are ±1 estimated standard deviation
from profile fitting.


### 3.5.2
Θ Accuracy and Precision

#### 3.5.1. Traditional Lattice Parameter Refinement

Refined
lattice parameters, estimated standard deviations of the lattice parameters,
and figures of merit are all good indicators for accuracy and precision when
coupled with NIST certified standards [16]. To compare the diffractometer
configurations, each powder was measured with an internal 2
Θ standard, NIST 640b silicon
powder. Lattice parameters were then
refined from the data without calibrating the peak positions and then again after calibrating the peak positions (also called “internal
standard 2
Θ calibration”). In addition, the NIST standards 640c and 660
were measured independently. The results
are summarized in Tables 2 and 3, with estimates of error that are the
estimated standard deviation values from the multivariable refinement.Table 2
Refined lattice parameters using uncalibrated peak positions.
Material/configuration
a axis (A)
c axis (A)
No. of Peaks indexed
Smith/SnyderF
N
No. of ESD from calibrated
BB (a axis/c axis)
Ag
B-B
4.0878(1)
9
f
(
8
)
=
168 (0.0060, 8)
10.3
Ag PB
4.0859(4)
9
f
(
8
)
=
31 (0.0326, 8)
1.7
Ag PM
4.0862(3)
9
f
(
8
)
=
49 (0.0204, 8)
1.2
Linde C BB
4.7616(2)
12.9993(9)
32
f
(
30
)
=
33 (0.0204, 44)
11.9
7.6
Linde C PB
4.7591(1)
12.9887(9)
30
f
(
30
)
=
38 (0.0174, 45)
0.80
4.19
Linde C PM
4.7587(1)
12.9915(7)
30
f
(
30
)
=
46 (0.0147, 45)
3.47
1.37
Urea
BB
5.660(1)
4.712(1)
22
f
(
21
)
=
16 (0.0261, 50)
13.4
8.90
Urea
PB
5.6470(6)
4.706(1)
19
f
(
19
)
=
21 (0.0281, 32)
0.50
2.9
Urea
PM
5.6464(4)
4.701(1)
18
f
(
18
)
=
99 (0.0057, 32)
0.25
2.10
UreaPM
+
FM
5.6481(3)
4.702(2)
19
f
(
19
)
=
65 (0.0051, 58)
1.8
1.5Table 3
Refined lattice parameters using internal standard calibrated data.
Material/configuration
a axis (A)
c axis (A)
No. of peaks indexed
Smith/SnyderF
N
No. of ESD from calibrated
BB (a axis/c axis)
Ag BB
4.08657(12)
8
f
(
8
)
=
128 (0.0078, 8)
0.0
Ag PB
4.08618(41)
8
f
(
8
)
=
32 (0.0309, 8)
0.95
Ag PM
4.08632(14)
8
f
(
8
)
=
71 (0.0141, 8)
1.79
Linde C BB
4.75922(15)
12.9925(7)
31
f
(
30
)
=
53 (0.0125, 45)
0.00
0.00
Linde C PB
4.75810(2)
12.9893(9)
29
f
(
29
)
=
55 (0.0116, 45)
1.67
3.52
Linde C PM
4.75854(13)
12.9931(6)
30
f
(
30
)
=
45 (0.0148, 45)
4.53
0.89
Urea BB
5.6466(8)
4.7031(8)
21
f
(
21
)
=
14 (0.0294, 48)
0.00
0.00
Urea PB
5.6473(7)
4.705(1)
19
f
(
19
)
=
24 (0.0248, 32)
0.875
1.9
Urea PM
5.6486(5)
4.7025(5)
18
f
(
18
)
=
74 (0.0076, 32)
2.5
0.75The principal
source of error in peak positions from Bragg-Brentano diffractometers is the
sample, and it is well known that the Bragg-Brentano parafocusing geometry is
highly prone to sample displacement and sample transparency errors [16, 17].
The former results in systematic peak shifts while the latter results in
asymmetric peak shapes. The use of a
parallel beam on both the incident and diffracted beam sides of the sample
eliminates, in principle, sample displacement and transparency errors.  Likewise, using grazing angles (Θ
<
3
°) reduces transparency errors by
minimizing beam penetration into the sample.
The lattice parameters obtained using the “parallel” optics, PB, PM, and
PM
+
FM configurations, are therefore expected to yield the highest accuracy
lattice parameters.Assuming that
the ‘‘correct’’ value for any particular lattice constant is the value determined
from internal standard calibrated BB data, one can compare results for the
various samples with different absorption coefficients in Tables2 and 3. Examination of Tables 2 and 3 shows that, with one exception (the a axis of  Linde C alumina), all of the lattice constants from uncalibrated data agree within error with the “ideal” value from
calibrated Bragg-Brentano data. In
contrast, all of the BB data and most of the PB data are significantly
different from calibrated BB data. After
calibration, the results are generally in agreement with the calibrated BB
data, as expected.One point of
note relates to the low absorption specimen, urea. Urea manifests massive sample transparency
error, with highly anisotropic line shapes even at intermediate diffraction angles
in parafocusing geometry. Indeed, as
shown in Figure7, the transparency error is so large that the BB measurement
provides substantially incorrect lattice constants but the ‘‘parallel’’
measurements are largely immune to sample transparency error. This point is visually obvious from the data
in Figure 2.Figure 7
Refined lattice constants for urea for each
instrumental configuration. Error bars
are ±3 estimated
standard deviations from the refinements.


#### 3.5.2. Evaluation of Sensitivity to Specimen Displacement

To evaluate
the tolerable amount of displacement for the parallel beam configurations,
samples of SRM 640b silicon were intentionally displaced from the
diffractometer center by up to 0.5 mm.
Although 0.5 mm is an extreme displacement, it is not uncommon for high
temperature systems to introduce sample shifts [2, 13, 18] or for curved or rough surfaces
to present large sample shifts, for example in residual stress analysis.Figure8 shows
the lattice constants obtained for samples displaced from the goniometer
center, and clearly demonstrates that the PM (and PM
+
FM, not shown) configuration
is essentially immune to systematic sample displacement errors. The PB configuration has beam divergence 540 arcseconds on both sides of the sample, while the PM system has divergence of ~100 arcseconds and 540 arcseconds on the incident and diffracted sides. Therefore, it appears that the largest
divergence that can be tolerated while effectively eliminating sample
displacement errors is somewhere between ~100 and 500 arcseconds, which is unlikelyy to be achieved without a
diffractive optical element—either a crystal or multilayer.Figure 8
Lattice constants for SR640b silicon powder
as a function of intentional specimen displacement for two configurations.A final set of experiments focused on
absolute accuracy using NIST Standard Reference Materials and the novel optical
configuration of combined parabolic and flat multilayers. NIST SRM640c Si powder and the SRM1976
alumina plate were measured, both of which have certified lattice
constants. Lattice parameter refinements
for both materials resulted in unprecedented internal consistency, based upon
the Smith-Snyder figure of merit (F
N). For SRM640c, the lattice constant refined to
within 0.0006
Å of the certified value, with F
N
=
521 (0.0019,
8). Likewise, the data for SRM1976 refined to produce a very high F
N
=
213 (0.0026, 33), with the lattice constants within 0.0007
Å of the certified value. These F
N values are enormous for any optical geometry, and
very unusual for a laboratory instrument.


#### 3.5.3. Rietveld Refinement

Rietveld
analysis of powder data from each configuration was performed using traditional
pseudo-Voigt profiles and fundamental parameters peak shapes.  Refinement of data from the PM and PM+FM
configurations did not demonstrate any measurable abnormalities in theα
1 and α
2 peak positions. Refined models for alumina, LaB6, and Si were
all reasonable, including isotropic atomic displacement parameters. For example, the displacement parameters for
Si powder were 0.37(1) for the PM configuration and 0.42(1) for the PM
+
FM
configuration. Figure 9 shows the
refined data for SRM640c in the PM
+
FM configuration.Figure 9
Rietveld plot for SRM640c Si on a square root scale. Inset shows high angle region on a linear
scale.The
set of refinements using fundamental parameters line shapes also provided reasonable
structure models, and the refined lattice constants from the PM and PM + FM
configurations for the NIST standards were within0.0006
Å of the certified values. To optimize the fundamental parameters models
for the PM and PM
+
FM configurations, the data were modeled without full
structures using Pawley or individual profile fitting. The refinements included models for the axial
and equatorial divergences and then additional convolutions to accommodate the
multilayers [19]. Additional convolutions with refinable parameters included hat (square
wave) and constant Gaussian functions.For
thePM
+
FM configuration, adding a constant Gaussian convolution notably
improved the fit from Rwp
=
19.5 to 16%, a difference that is visible by eye as
shown in Figure 10. In the case of the
PM optical configuration, addition of both a constant hat function and a
constant Gaussian function improved the model fits to the line profiles
significantly, reducing Rwp from 13.3 to 7%.
The Gaussian convolutions added to both models refined to nearly the
same values, 0.0440(5) for the system with two multilayers and 0.041(1) for the
single multilayer, suggesting that the incident beam parabolic multilayer
contributes the Gaussian component to the line shapes. The hat function is a result of the long
soller collimator and is absent in the case of two multilayers, presumably
because the angular acceptance for the flat multilayer is some 5.5 times
smaller than that of the parallel slit collimator. We are currently studying the effects of
axial divergence and beam dimensions in the axial and equatorial directions to
better define the convolutions.Profiles fit to SRM660a LaB6 using
fundamental parameters: (a) is without
an additional constant Gaussian convolution; (b) includes the constant
Gaussian.
(a)
(b)After
removing the instrumental effects from the peak locations using the fundamental
parameters approach, the peak positions can be compared directly. Figure11 shows the difference in peak
location after deconvolution of the instrumental effects and shows a shift in zero
error between the PM and PM
+
FM measurements of ~
0.005
°
2
Θ.
Beyond the zero shift, however, the peak locations from the two configurations
are within about ±
0.004
°
2
Θ, which is on the order of the reproducibility
of the standard commercial goniometer used in the study (i.e., ±
0.003
°
2
Θ, without encoders [20]). This result demonstrates that the two
configurations are modeled correctly by incorporating the additional
convolutions during fundamental parameters fitting.Figure 11
Difference in peak locations
after deconvolution of the instrumental effects for SRM660a between the PM andPM
+
FM configurations after modeling with fundamental parameters.


## 3.1. Intensity

The absolute
intensities were compared on a peak-by-peak basis for Ag, alumina, urea, and
quartz. The overall average absolute
peak intensities were used for the comparisons.
Figure3 shows the absolute intensities relative to the intensity
measured in the standard Bragg-Brentano configuration and highlights the
improved intensity, of approximately 4 times, in the configuration with the
parabolic multilayer and fixed incident angle of 3.6
°.Figure 3
Ratio of absolute intensities
obtained for each optical configuration.While several
authors have used ray tracing calculations to estimate intensities and beam
paths with multilayer optics [3, 8, 9], the differences in the measured
intensities shown in Figure 3 can be explained qualitatively by considering the
beam paths. For example, in the BB
geometry, ~
1
° of divergence
is used in the incident and diffracted beams.
In stark contrast, the traditional PB measurement includes very fine
slits to irradiate a 15 mm sample at an incident angle of 2
°
Θ, or ~
0.14
° of divergence. Therefore, one would expect substantially fewer photons incident upon
the sample, by a factor of ~7, in the PB configuration. Further, to obtain
similar measured intensities as shown in Figure 3, the diffracted-beam side
optics must pass ~7 times more photons than the diffracted-beam graphite
monochromator in the BB configuration.Extending
similar arguments to the multilayers, one finds that the parabolic multilayer
captures~
0.5
° of
divergence, diffracts ~65% of the incident radiation, and is equipped with the
same diffracted-beam side optics as the PB configuration. Therefore, the total intensity should be on
the order of ~2.5 times larger than that measured in the BB configuration, a
prediction in reasonable agreement with the measured value of 3.8 times,
considering minor differences in alignment.Adding the
second multilayer in the diffracted beam side substantially reduces the
intensity, but, as shown later, also improves resolution. The second multilayer, with reflectivity of
~70%, offers approximately the same throughput as a graphite diffracted beam
monochromator. However, the multilayer
also offers beam collimation that is much tighter than that offered by the long
Soller collimator, by a factor of ~6 (recall that the rocking curve width of
the multilayer is 96 arcseconds, and the long Soller collimator has divergence of~
0.15
° or 540 arcseconds). Therefore, one can employ the multilayer
without a long Soller collimator to achieve parallel beam conditions. The efficiency of the multilayer is close to
that of the Soller collimator used here, such that the overall intensity of the
PM
+
FM design is approximately the same as for the PB and BB configurations.


## 3.2. Peak-to-Background Ratios

The
peak-to-background ratios were calculated on a peak-by-peak basis exactly as
were the intensity ratios. The results
are shown in Figure4. The BB
configuration provides a peak-to-background ratio that is substantially higher
than for the PB configuration, about the same as the PM configuration and worse
by a factor of ~0.5 than the PM
+
FM configuration. One would expect the PB configuration to provide
lower peak-to-background ratios because the only energy discrimination in the
system is the Ni foil K
β filter. In
contrast, the PM configuration yields peak-to-background ratios that are more
than double those for the PB configuration.
Here, one must consider the incident beam spectrum after diffraction
from the incident beam parabolic multilayer.
Because the multilayer passes only a small window around the K
α
1 and K
α
2 lines, scattering and
fluorescence are reduced before the sample, thus improving the
peak-to-background ratio. Adding the
second multilayer again provides additional energy discrimination and tight
angular acceptance, improving the peak-to-background ratio further.Figure 4
Peak-to-background ratios for each
configuration relative to the BB configuration.


## 3.3. Resolution

The
instrumental resolution was determined using the traditional approach by
measuring the “5 fingers” of quartz, as recommended by Jenkins and Snyder [16]. The divergence of the PB, PM, and PM
+
FM
configurations, although “parallel” in comparison to BB optics, is a function
of the optics. For the ‘‘parallel’’
configurations, the incident/diffracted beam side divergence is on the order of
540/540 for PB, 96/540 for PM, and 96/96 arcseconds for the PM
+
FM configurations. Naturally, one expects improved instrumental
resolution with lower divergence optics.Figure5 shows
the diffraction patterns for quartz in the various diffractometer
configurations. Calculated resolution
values between 1.5 and 2.0, based upon peak-to-valley ratios in the quartz
peaks, are the accepted range for “high-resolution” Bragg-Brentano measurements
[16].  As shown in Figure 5, the BB configuration
resolution value is 1.7, while those for PB, PM, and PM
+
FM are 1.5, 2.0, and
4.0, respectively. Note that the
resolution when using the PM optics is some 20% better than a Bragg-Brentano
measurement and ~40% better than the traditional PB optics. Incorporating the diffracted beam multilayer
optic again improves the resolution to a value of 4, as would be expected from
the smaller acceptance angle of the diffracted-beam multilayer compared to the
Soller collimator.Figure 5
Quartz diffraction data for each
optical configuration.


## 3.4. Profile Shapes

Diffraction
peak shapes influence all aspects of data analysis, and for low absorption
materials measured with Bragg-Brentano geometry, substantial systematic error
should be expected. The peak shapes/widths
were evaluated as a function of angle for the PM and the new PM + FM
configurations by fitting data from NIST SRM640c Si powder.  Peak shapes were easily modeled using pseudo-Voigt
profiles and the peak widths follow a typical behavior with diffraction angle [17].  Figure 6 shows the FWHM as a function of angle
fitted with a parabolic function, which suggests that the dual multilayer
optics result in well-behaved line shapes and widths.Figure 6
Peak FWHM (K
α
1) for SRM640c Si powder using the
PM and PM
+
FM configurations. Error
bars are ±1 estimated standard deviation
from profile fitting.


## 3.5.2
Θ Accuracy and Precision

### 3.5.1. Traditional Lattice Parameter Refinement

Refined
lattice parameters, estimated standard deviations of the lattice parameters,
and figures of merit are all good indicators for accuracy and precision when
coupled with NIST certified standards [16]. To compare the diffractometer
configurations, each powder was measured with an internal 2
Θ standard, NIST 640b silicon
powder. Lattice parameters were then
refined from the data without calibrating the peak positions and then again after calibrating the peak positions (also called “internal
standard 2
Θ calibration”). In addition, the NIST standards 640c and 660
were measured independently. The results
are summarized in Tables 2 and 3, with estimates of error that are the
estimated standard deviation values from the multivariable refinement.Table 2
Refined lattice parameters using uncalibrated peak positions.
Material/configuration
a axis (A)
c axis (A)
No. of Peaks indexed
Smith/SnyderF
N
No. of ESD from calibrated
BB (a axis/c axis)
Ag
B-B
4.0878(1)
9
f
(
8
)
=
168 (0.0060, 8)
10.3
Ag PB
4.0859(4)
9
f
(
8
)
=
31 (0.0326, 8)
1.7
Ag PM
4.0862(3)
9
f
(
8
)
=
49 (0.0204, 8)
1.2
Linde C BB
4.7616(2)
12.9993(9)
32
f
(
30
)
=
33 (0.0204, 44)
11.9
7.6
Linde C PB
4.7591(1)
12.9887(9)
30
f
(
30
)
=
38 (0.0174, 45)
0.80
4.19
Linde C PM
4.7587(1)
12.9915(7)
30
f
(
30
)
=
46 (0.0147, 45)
3.47
1.37
Urea
BB
5.660(1)
4.712(1)
22
f
(
21
)
=
16 (0.0261, 50)
13.4
8.90
Urea
PB
5.6470(6)
4.706(1)
19
f
(
19
)
=
21 (0.0281, 32)
0.50
2.9
Urea
PM
5.6464(4)
4.701(1)
18
f
(
18
)
=
99 (0.0057, 32)
0.25
2.10
UreaPM
+
FM
5.6481(3)
4.702(2)
19
f
(
19
)
=
65 (0.0051, 58)
1.8
1.5Table 3
Refined lattice parameters using internal standard calibrated data.
Material/configuration
a axis (A)
c axis (A)
No. of peaks indexed
Smith/SnyderF
N
No. of ESD from calibrated
BB (a axis/c axis)
Ag BB
4.08657(12)
8
f
(
8
)
=
128 (0.0078, 8)
0.0
Ag PB
4.08618(41)
8
f
(
8
)
=
32 (0.0309, 8)
0.95
Ag PM
4.08632(14)
8
f
(
8
)
=
71 (0.0141, 8)
1.79
Linde C BB
4.75922(15)
12.9925(7)
31
f
(
30
)
=
53 (0.0125, 45)
0.00
0.00
Linde C PB
4.75810(2)
12.9893(9)
29
f
(
29
)
=
55 (0.0116, 45)
1.67
3.52
Linde C PM
4.75854(13)
12.9931(6)
30
f
(
30
)
=
45 (0.0148, 45)
4.53
0.89
Urea BB
5.6466(8)
4.7031(8)
21
f
(
21
)
=
14 (0.0294, 48)
0.00
0.00
Urea PB
5.6473(7)
4.705(1)
19
f
(
19
)
=
24 (0.0248, 32)
0.875
1.9
Urea PM
5.6486(5)
4.7025(5)
18
f
(
18
)
=
74 (0.0076, 32)
2.5
0.75The principal
source of error in peak positions from Bragg-Brentano diffractometers is the
sample, and it is well known that the Bragg-Brentano parafocusing geometry is
highly prone to sample displacement and sample transparency errors [16, 17].
The former results in systematic peak shifts while the latter results in
asymmetric peak shapes. The use of a
parallel beam on both the incident and diffracted beam sides of the sample
eliminates, in principle, sample displacement and transparency errors.  Likewise, using grazing angles (Θ
<
3
°) reduces transparency errors by
minimizing beam penetration into the sample.
The lattice parameters obtained using the “parallel” optics, PB, PM, and
PM
+
FM configurations, are therefore expected to yield the highest accuracy
lattice parameters.Assuming that
the ‘‘correct’’ value for any particular lattice constant is the value determined
from internal standard calibrated BB data, one can compare results for the
various samples with different absorption coefficients in Tables2 and 3. Examination of Tables 2 and 3 shows that, with one exception (the a axis of  Linde C alumina), all of the lattice constants from uncalibrated data agree within error with the “ideal” value from
calibrated Bragg-Brentano data. In
contrast, all of the BB data and most of the PB data are significantly
different from calibrated BB data. After
calibration, the results are generally in agreement with the calibrated BB
data, as expected.One point of
note relates to the low absorption specimen, urea. Urea manifests massive sample transparency
error, with highly anisotropic line shapes even at intermediate diffraction angles
in parafocusing geometry. Indeed, as
shown in Figure7, the transparency error is so large that the BB measurement
provides substantially incorrect lattice constants but the ‘‘parallel’’
measurements are largely immune to sample transparency error. This point is visually obvious from the data
in Figure 2.Figure 7
Refined lattice constants for urea for each
instrumental configuration. Error bars
are ±3 estimated
standard deviations from the refinements.


### 3.5.2. Evaluation of Sensitivity to Specimen Displacement

To evaluate
the tolerable amount of displacement for the parallel beam configurations,
samples of SRM 640b silicon were intentionally displaced from the
diffractometer center by up to 0.5 mm.
Although 0.5 mm is an extreme displacement, it is not uncommon for high
temperature systems to introduce sample shifts [2, 13, 18] or for curved or rough surfaces
to present large sample shifts, for example in residual stress analysis.Figure8 shows
the lattice constants obtained for samples displaced from the goniometer
center, and clearly demonstrates that the PM (and PM
+
FM, not shown) configuration
is essentially immune to systematic sample displacement errors. The PB configuration has beam divergence 540 arcseconds on both sides of the sample, while the PM system has divergence of ~100 arcseconds and 540 arcseconds on the incident and diffracted sides. Therefore, it appears that the largest
divergence that can be tolerated while effectively eliminating sample
displacement errors is somewhere between ~100 and 500 arcseconds, which is unlikelyy to be achieved without a
diffractive optical element—either a crystal or multilayer.Figure 8
Lattice constants for SR640b silicon powder
as a function of intentional specimen displacement for two configurations.A final set of experiments focused on
absolute accuracy using NIST Standard Reference Materials and the novel optical
configuration of combined parabolic and flat multilayers. NIST SRM640c Si powder and the SRM1976
alumina plate were measured, both of which have certified lattice
constants. Lattice parameter refinements
for both materials resulted in unprecedented internal consistency, based upon
the Smith-Snyder figure of merit (F
N). For SRM640c, the lattice constant refined to
within 0.0006
Å of the certified value, with F
N
=
521 (0.0019,
8). Likewise, the data for SRM1976 refined to produce a very high F
N
=
213 (0.0026, 33), with the lattice constants within 0.0007
Å of the certified value. These F
N values are enormous for any optical geometry, and
very unusual for a laboratory instrument.


### 3.5.3. Rietveld Refinement

Rietveld
analysis of powder data from each configuration was performed using traditional
pseudo-Voigt profiles and fundamental parameters peak shapes.  Refinement of data from the PM and PM+FM
configurations did not demonstrate any measurable abnormalities in theα
1 and α
2 peak positions. Refined models for alumina, LaB6, and Si were
all reasonable, including isotropic atomic displacement parameters. For example, the displacement parameters for
Si powder were 0.37(1) for the PM configuration and 0.42(1) for the PM
+
FM
configuration. Figure 9 shows the
refined data for SRM640c in the PM
+
FM configuration.Figure 9
Rietveld plot for SRM640c Si on a square root scale. Inset shows high angle region on a linear
scale.The
set of refinements using fundamental parameters line shapes also provided reasonable
structure models, and the refined lattice constants from the PM and PM + FM
configurations for the NIST standards were within0.0006
Å of the certified values. To optimize the fundamental parameters models
for the PM and PM
+
FM configurations, the data were modeled without full
structures using Pawley or individual profile fitting. The refinements included models for the axial
and equatorial divergences and then additional convolutions to accommodate the
multilayers [19]. Additional convolutions with refinable parameters included hat (square
wave) and constant Gaussian functions.For
thePM
+
FM configuration, adding a constant Gaussian convolution notably
improved the fit from Rwp
=
19.5 to 16%, a difference that is visible by eye as
shown in Figure 10. In the case of the
PM optical configuration, addition of both a constant hat function and a
constant Gaussian function improved the model fits to the line profiles
significantly, reducing Rwp from 13.3 to 7%.
The Gaussian convolutions added to both models refined to nearly the
same values, 0.0440(5) for the system with two multilayers and 0.041(1) for the
single multilayer, suggesting that the incident beam parabolic multilayer
contributes the Gaussian component to the line shapes. The hat function is a result of the long
soller collimator and is absent in the case of two multilayers, presumably
because the angular acceptance for the flat multilayer is some 5.5 times
smaller than that of the parallel slit collimator. We are currently studying the effects of
axial divergence and beam dimensions in the axial and equatorial directions to
better define the convolutions.Profiles fit to SRM660a LaB6 using
fundamental parameters: (a) is without
an additional constant Gaussian convolution; (b) includes the constant
Gaussian.
(a)
(b)After
removing the instrumental effects from the peak locations using the fundamental
parameters approach, the peak positions can be compared directly. Figure11 shows the difference in peak
location after deconvolution of the instrumental effects and shows a shift in zero
error between the PM and PM
+
FM measurements of ~
0.005
°
2
Θ.
Beyond the zero shift, however, the peak locations from the two configurations
are within about ±
0.004
°
2
Θ, which is on the order of the reproducibility
of the standard commercial goniometer used in the study (i.e., ±
0.003
°
2
Θ, without encoders [20]). This result demonstrates that the two
configurations are modeled correctly by incorporating the additional
convolutions during fundamental parameters fitting.Figure 11
Difference in peak locations
after deconvolution of the instrumental effects for SRM660a between the PM andPM
+
FM configurations after modeling with fundamental parameters.


## 3.5.1. Traditional Lattice Parameter Refinement

Refined
lattice parameters, estimated standard deviations of the lattice parameters,
and figures of merit are all good indicators for accuracy and precision when
coupled with NIST certified standards [16]. To compare the diffractometer
configurations, each powder was measured with an internal 2
Θ standard, NIST 640b silicon
powder. Lattice parameters were then
refined from the data without calibrating the peak positions and then again after calibrating the peak positions (also called “internal
standard 2
Θ calibration”). In addition, the NIST standards 640c and 660
were measured independently. The results
are summarized in Tables 2 and 3, with estimates of error that are the
estimated standard deviation values from the multivariable refinement.Table 2
Refined lattice parameters using uncalibrated peak positions.
Material/configuration
a axis (A)
c axis (A)
No. of Peaks indexed
Smith/SnyderF
N
No. of ESD from calibrated
BB (a axis/c axis)
Ag
B-B
4.0878(1)
9
f
(
8
)
=
168 (0.0060, 8)
10.3
Ag PB
4.0859(4)
9
f
(
8
)
=
31 (0.0326, 8)
1.7
Ag PM
4.0862(3)
9
f
(
8
)
=
49 (0.0204, 8)
1.2
Linde C BB
4.7616(2)
12.9993(9)
32
f
(
30
)
=
33 (0.0204, 44)
11.9
7.6
Linde C PB
4.7591(1)
12.9887(9)
30
f
(
30
)
=
38 (0.0174, 45)
0.80
4.19
Linde C PM
4.7587(1)
12.9915(7)
30
f
(
30
)
=
46 (0.0147, 45)
3.47
1.37
Urea
BB
5.660(1)
4.712(1)
22
f
(
21
)
=
16 (0.0261, 50)
13.4
8.90
Urea
PB
5.6470(6)
4.706(1)
19
f
(
19
)
=
21 (0.0281, 32)
0.50
2.9
Urea
PM
5.6464(4)
4.701(1)
18
f
(
18
)
=
99 (0.0057, 32)
0.25
2.10
UreaPM
+
FM
5.6481(3)
4.702(2)
19
f
(
19
)
=
65 (0.0051, 58)
1.8
1.5Table 3
Refined lattice parameters using internal standard calibrated data.
Material/configuration
a axis (A)
c axis (A)
No. of peaks indexed
Smith/SnyderF
N
No. of ESD from calibrated
BB (a axis/c axis)
Ag BB
4.08657(12)
8
f
(
8
)
=
128 (0.0078, 8)
0.0
Ag PB
4.08618(41)
8
f
(
8
)
=
32 (0.0309, 8)
0.95
Ag PM
4.08632(14)
8
f
(
8
)
=
71 (0.0141, 8)
1.79
Linde C BB
4.75922(15)
12.9925(7)
31
f
(
30
)
=
53 (0.0125, 45)
0.00
0.00
Linde C PB
4.75810(2)
12.9893(9)
29
f
(
29
)
=
55 (0.0116, 45)
1.67
3.52
Linde C PM
4.75854(13)
12.9931(6)
30
f
(
30
)
=
45 (0.0148, 45)
4.53
0.89
Urea BB
5.6466(8)
4.7031(8)
21
f
(
21
)
=
14 (0.0294, 48)
0.00
0.00
Urea PB
5.6473(7)
4.705(1)
19
f
(
19
)
=
24 (0.0248, 32)
0.875
1.9
Urea PM
5.6486(5)
4.7025(5)
18
f
(
18
)
=
74 (0.0076, 32)
2.5
0.75The principal
source of error in peak positions from Bragg-Brentano diffractometers is the
sample, and it is well known that the Bragg-Brentano parafocusing geometry is
highly prone to sample displacement and sample transparency errors [16, 17].
The former results in systematic peak shifts while the latter results in
asymmetric peak shapes. The use of a
parallel beam on both the incident and diffracted beam sides of the sample
eliminates, in principle, sample displacement and transparency errors.  Likewise, using grazing angles (Θ
<
3
°) reduces transparency errors by
minimizing beam penetration into the sample.
The lattice parameters obtained using the “parallel” optics, PB, PM, and
PM
+
FM configurations, are therefore expected to yield the highest accuracy
lattice parameters.Assuming that
the ‘‘correct’’ value for any particular lattice constant is the value determined
from internal standard calibrated BB data, one can compare results for the
various samples with different absorption coefficients in Tables2 and 3. Examination of Tables 2 and 3 shows that, with one exception (the a axis of  Linde C alumina), all of the lattice constants from uncalibrated data agree within error with the “ideal” value from
calibrated Bragg-Brentano data. In
contrast, all of the BB data and most of the PB data are significantly
different from calibrated BB data. After
calibration, the results are generally in agreement with the calibrated BB
data, as expected.One point of
note relates to the low absorption specimen, urea. Urea manifests massive sample transparency
error, with highly anisotropic line shapes even at intermediate diffraction angles
in parafocusing geometry. Indeed, as
shown in Figure7, the transparency error is so large that the BB measurement
provides substantially incorrect lattice constants but the ‘‘parallel’’
measurements are largely immune to sample transparency error. This point is visually obvious from the data
in Figure 2.Figure 7
Refined lattice constants for urea for each
instrumental configuration. Error bars
are ±3 estimated
standard deviations from the refinements.


## 3.5.2. Evaluation of Sensitivity to Specimen Displacement

To evaluate
the tolerable amount of displacement for the parallel beam configurations,
samples of SRM 640b silicon were intentionally displaced from the
diffractometer center by up to 0.5 mm.
Although 0.5 mm is an extreme displacement, it is not uncommon for high
temperature systems to introduce sample shifts [2, 13, 18] or for curved or rough surfaces
to present large sample shifts, for example in residual stress analysis.Figure8 shows
the lattice constants obtained for samples displaced from the goniometer
center, and clearly demonstrates that the PM (and PM
+
FM, not shown) configuration
is essentially immune to systematic sample displacement errors. The PB configuration has beam divergence 540 arcseconds on both sides of the sample, while the PM system has divergence of ~100 arcseconds and 540 arcseconds on the incident and diffracted sides. Therefore, it appears that the largest
divergence that can be tolerated while effectively eliminating sample
displacement errors is somewhere between ~100 and 500 arcseconds, which is unlikelyy to be achieved without a
diffractive optical element—either a crystal or multilayer.Figure 8
Lattice constants for SR640b silicon powder
as a function of intentional specimen displacement for two configurations.A final set of experiments focused on
absolute accuracy using NIST Standard Reference Materials and the novel optical
configuration of combined parabolic and flat multilayers. NIST SRM640c Si powder and the SRM1976
alumina plate were measured, both of which have certified lattice
constants. Lattice parameter refinements
for both materials resulted in unprecedented internal consistency, based upon
the Smith-Snyder figure of merit (F
N). For SRM640c, the lattice constant refined to
within 0.0006
Å of the certified value, with F
N
=
521 (0.0019,
8). Likewise, the data for SRM1976 refined to produce a very high F
N
=
213 (0.0026, 33), with the lattice constants within 0.0007
Å of the certified value. These F
N values are enormous for any optical geometry, and
very unusual for a laboratory instrument.


## 3.5.3. Rietveld Refinement

Rietveld
analysis of powder data from each configuration was performed using traditional
pseudo-Voigt profiles and fundamental parameters peak shapes.  Refinement of data from the PM and PM+FM
configurations did not demonstrate any measurable abnormalities in theα
1 and α
2 peak positions. Refined models for alumina, LaB6, and Si were
all reasonable, including isotropic atomic displacement parameters. For example, the displacement parameters for
Si powder were 0.37(1) for the PM configuration and 0.42(1) for the PM
+
FM
configuration. Figure 9 shows the
refined data for SRM640c in the PM
+
FM configuration.Figure 9
Rietveld plot for SRM640c Si on a square root scale. Inset shows high angle region on a linear
scale.The
set of refinements using fundamental parameters line shapes also provided reasonable
structure models, and the refined lattice constants from the PM and PM + FM
configurations for the NIST standards were within0.0006
Å of the certified values. To optimize the fundamental parameters models
for the PM and PM
+
FM configurations, the data were modeled without full
structures using Pawley or individual profile fitting. The refinements included models for the axial
and equatorial divergences and then additional convolutions to accommodate the
multilayers [19]. Additional convolutions with refinable parameters included hat (square
wave) and constant Gaussian functions.For
thePM
+
FM configuration, adding a constant Gaussian convolution notably
improved the fit from Rwp
=
19.5 to 16%, a difference that is visible by eye as
shown in Figure 10. In the case of the
PM optical configuration, addition of both a constant hat function and a
constant Gaussian function improved the model fits to the line profiles
significantly, reducing Rwp from 13.3 to 7%.
The Gaussian convolutions added to both models refined to nearly the
same values, 0.0440(5) for the system with two multilayers and 0.041(1) for the
single multilayer, suggesting that the incident beam parabolic multilayer
contributes the Gaussian component to the line shapes. The hat function is a result of the long
soller collimator and is absent in the case of two multilayers, presumably
because the angular acceptance for the flat multilayer is some 5.5 times
smaller than that of the parallel slit collimator. We are currently studying the effects of
axial divergence and beam dimensions in the axial and equatorial directions to
better define the convolutions.Profiles fit to SRM660a LaB6 using
fundamental parameters: (a) is without
an additional constant Gaussian convolution; (b) includes the constant
Gaussian.
(a)
(b)After
removing the instrumental effects from the peak locations using the fundamental
parameters approach, the peak positions can be compared directly. Figure11 shows the difference in peak
location after deconvolution of the instrumental effects and shows a shift in zero
error between the PM and PM
+
FM measurements of ~
0.005
°
2
Θ.
Beyond the zero shift, however, the peak locations from the two configurations
are within about ±
0.004
°
2
Θ, which is on the order of the reproducibility
of the standard commercial goniometer used in the study (i.e., ±
0.003
°
2
Θ, without encoders [20]). This result demonstrates that the two
configurations are modeled correctly by incorporating the additional
convolutions during fundamental parameters fitting.Figure 11
Difference in peak locations
after deconvolution of the instrumental effects for SRM660a between the PM andPM
+
FM configurations after modeling with fundamental parameters.


## 4. Summary

The novel
configuration incorporating parabolic and flat multilayers is effective for
powder diffraction measurements. Using
only a parabolic multilayer on the incident beam side resulted in all of the
advantages expected from a parallel-beam system, and improved the measured
intensity while maintaining the same peak-to-background ratio as the
Bragg-Brentano configuration. Adding the
flat multilayer in the diffracted beam side reduced the overall intensity but
improved the resolution appreciably, as might be expected from the lower
overall equatorial beam divergence.
Systematic shifts in the theα
1 and α
2 peak separations were not
observed in the configurations tested, and the refined lattice constants for a
range of materials were reasonable.
Measurements of NIST SRM660, 660a, SRM1976, and SRM640c showed that the
multilayers do not adversely affect the peak width versus angle, and provide
refined lattice constants within 0.0006
Å of the
certified values in the worst case.Fundamental
parameters fitting of the line shapes demonstrated that the incident parabolic
multilayer adds a Gaussian convolution (constant versus angle) to the line shape, which is the same for both the PM and PM+FM configurations. The
long soller collimator on the diffracted beam side is best described using a
constant hat function. The secondary
flat mirror as an analyzer crystal does not alter the peak shapes appreciably,
so that the line shapes from the system with two multilayers can be modeled
with only the constant Gaussian function.



---

*Source: 408702-2008-07-16.xml*