Data on the scattering of X-rays from various mirror flats obtained from different vendors. The scattering data discussed in this section is obtained by a one dimensional proportional counter at X-ray energies of 8.06, 6.4 and 2.99 keV. X-rays are generated at one end of a 312.4m long tube which is under vacuum. This long tube provides a parallel beam of X-rays which is then collimated by a slit and allowed to scatter from a pair of mirror flats at small grazing angles of incidence. The slit and mirrors assembly is placed 182.4 meters from the other end of the tube where the proportional counter detector is located. The detector is sitting on an optical table which is pivoted so that it can rotate to intercept X-rays at a range of spatial locations. The optical table pivot is located 2.78 meters in front of the detector, and has an operational angular range of 350 arc minutes. Measured from the position of the mirror, this total angular range of the data is (350) (2.78/182.4) = 5.33 arcminutes

Williams, A. C.

English

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It
t,
TECHNICAL REPORT NO. 1
REDUCTION OF THE ONE-DIMENSIONAL X-RAY SCATTERING DATA ON
MIRROR FLATS AT NASA/MSFC
(NASA-CR-170749) RFECCTICfi CF 7HF
	 Ne_' - 2`r_43
CNE-DIPIENSICNAL X-RAY SCAT7FFIK EA7A CN
HIRFCF FLATS AT NASA/MARSHAII :FACE FLIGHT
CENTER (Alata®a A & C Univ., Ncrwal.)
	 18 p	 Uncld£HC A02 /t1F A01	 CSCI 20F G3/74
	 11(5E
Prepared for
NASA/MSFC
George C. Marshall Space Flight Center
Huntsville, AL 35812
Contract NAS8-34591
.—h_^•
T
MAY 1983By	 RECEIVED
in
Alton C. Williams
	 r' .
fgqLJn,
	
Alabama A. & M. University
	
DEPT.
Normal, Al_
	 35762	 ^^^^,,,^.,,,•^":`
t' r
NOVEMBER 1981
9REDUCTION OF THE ONE-DIMENSIONAL X-RAY
SCATTERING DATA ON MIRROR FLATS AT NASA/MSFC
I. General Discussion of Measurements
The X-ray scattering facility at NASA/MSFC has been utilized to
obtain data on the scattering of X-rays from various mirror flats
obtained from different vendors. The scattering data discussed in this
section is that obtained by a une dimensional proportional counter at
x-ray energies of 8.06, 6.4 and 2.99 keV.
The basic experimental-set up is as follows. X-rays are generated
at one end of a 312.4	 meter long tube which is under vacuum. This
long tube provides a parallel beam of x-rays which is then collimated
by a slit and allowed to scatter from a pair of mirror flats at small
grazing angles of incidence.
	 The pair of mirror flats are in periscope
geometry so that the scattered beam is parallel to the incident beam.
The slit and mirrors assembly is placed 182.4 meters from the other
end of the tube where the proportional counter detector is located.
The detector, itself, is sitting on an optical	 table which is
pivoted so that the detector can rotate to intercept x-rays at a range
of spatial	 locations.	 The optical table pivot is located 2.78 meters
i
in front of the detector (towards the incoming x-rays), and it has an
operational angular rangeof 350 arc minutes which corresponds to an
arc length of 0.283 meters (11.13 in).
	 Measured from the position of
the mirror, this total angular range of the data is (350)
	 (2.78/182.4)
= 5.33 arcminutes.
E
2Because angles can be measured both with respect to the optical
table pivot as well as with respect to the position of the mirrors
assembly, it is necessary to distinguish between these two. The
angles measured with respect to the optical table pivot will be desig-
nated by "Table angles"; whereas, angles measured with respect to the
mirrors will be designated by the angles without the "table" prefix.-
Hence, the range of data that is taken with the one dimensional pro-
portional counter is 350 table arc minutes or 5.33 arcminutes. To
convert from one to the other, one can use the relation
	
L	 -
	
e = L 1eT	 = .0151 eT
2
Where L 1 = distance from the optical table pivot to the detector and
L2 = distance from the mirrors assembly to the detector.	 eT is the
table angle and a is the angle measured with respect to the mirrors
assembly.
In the gathering of data, the following is done. Electronically,
the linear anode wire of the detector is divided into 1,024 equal segments.
X-rays incident within a given segment will then be accumulated in the
corresponding channel of a 1,024 channel multi-channel analyzer. With
the optical table at a given position, data is taken at the grazing
angles of incidence of 0' (,direct slit), 25', 39', and 51'.
	
The table
is then rotated to get another (but overlapping)field of view. Also,
data is taken at two different table positions on the direct slit image
3so that a correlation between number of channels and table angles can
be made. Background runs (without x-rays on) are made each day data
is taken.
Some special measurements have been made to calibrate the response
of the detector as a function of position on the anode wire. Basically,
the 1,024 different segments of the wire do not all respond to incident
x-rays equally efficiently. The measurements consist of removing the
mirrors and slit assembly and illuminating the entire anode wire.
For a uniform flux of x-rays, any deviations in a constant number of
counts in all of the 1,024 channels indicate a variation in the effi-
ciency of parts of the wire. This data was taken at two different table
positions so that effects due to a non-uniform beam could be removed.
This was done for all x-ray energies of interest.
II. Description of Typical Result
All the data taken for a given mirror at a particular grazing
angle of incidence and x-ray energy was combined to give a single
spectrum corrected for background and for deviations in the relative
efficiencies of different portions of the anode wire of the proportional
counter.
Measuring from the peak of each image, a typical spectrum covers
the angular range of from -50 areseconds to 280 areseconds. Since there
is data on both sides of the peak, a measure of the asymmetry of the
Omage can be given. Typically, the asymmetry is below 20%. There
t
4
r
were some highly asymmetric images which apparently indicates that
something is wrong with the data such as poor zero angle alignment.
Data which extends further on one side of the peak is used for
the other side assuming that the image should be symmetric. This then
gives a spectrum which covers the angular range, typically, of ±280
areseconds measured from the peak. A summary package of each spec-
trum is given by the computer software which includes the following:
1. A printout of total counts, angular range of the data
taken, the full-width-half-maximum, the percent asymmetry,
the reflectivity and a table showing the integrated frac-
tion of counts in the image as a function of the angle ±e
measured from the peak. This fraction of counts is given
for both the image itself and relative to the direct slit
image.
2. A plot showing the entire image is made. See figure 1 for a
typical result. This plot is made on microfiche for con-
venience in filing. A hard copy can be obtained when desired.
3. A second plot is made which shows a blow-up of the image
around the peak. This allows one to see the structure in
the image better. Figure 2 shows a typical plot of this
kind.
4. A third plot is made which shows the integrated fraction of
counts as a function of the angle, ±e, measured from the
peak. This curve always goes to the value of 1.0 at the
maximum angle at which data was taken. The slope of this
curve at the maximum angle gives an indication as to whether
5there is more counts in the image outside the maximum
angle. Zero slope would indicate no more counts. A typical
plot of this kind is shown in figure 3. As is true for this
sample plot, it is generally true that the curve rises very
rapidly reaching a plateau close to its maximum value of
1.0 within a few table arcminutes. This apparently-indicates -
that most of the flux is within the central core of the image
with very little scattered flux.
5. A fourth plot is made which blows up the integrated fraction
curve out to about 5 table arcminutes. This allows one to
look at the details in the rapidly rising part of this curve.
A more slowly rising curve than that for the direct slit
image indicates a spreading out of the peak due to a com-
bination-of figure distortion and scattering due to the
roughness of the mirror surfaces. A typical plot of this
kind is shown in figure l..
The angular resolution of the data is determined by the facts that
-1 channel of the multichannel analyzer covers an angular range of about
8.3 table areseconds and that the inherent resolution of the detector
is about 0.5 mm which corresponds to about 37 table areseconds. In
terms of the angle measured from the mirror assembly, this 37 table
areseconds corresponds to 0.56 areseconds.
Colculation of the statistical error in the data is shown in
Appendix A. As a typical example of the error in the data, the data
on the PE-II-003 (ULEiNi) mirror at the x-ray energy of 2.99 keV is
given below.
Grazing Angle Total
of Incidence Counts FWHM Reflectivity
0'	 (Dir.	 Slit) 7399 ±86 0.15 ±.13 --
25' 6295 ±79 2.01 ± .17 0.92	 ±	 .01
39' 4767 ±69 2.27 ±.19 .80	 ±	 .01
51' 1946 ±54 0.75 t .13 .63	 ±	 .01
6
R
7III. Collected Results To Data
All data collected with the one dimensional detector between
the period November 5, 1980 and May 14, 1981 has been analyzed.
This includes the following thirteen mirrors:
Vendor Substrate Coating X-ray Energist (keV)
Eastman-Kodak Zerodur Ni 2.99
Max Planck Zerodur Ni 2.99
Eastman-Kodak Fused Silica Pt 2.99, 6.4
Eastman-Kodak Zerodur Pt 2.99, 6.4
Perkin-Elmer Zerodur Au 2.99, 6.4, 8.06
Max Planck Zerodur Au 2.99, 6.4, 8.06
Perkin-Elmer Ule Ni 2.99, 6.4, 8.06
Eastman-Kodak Ule Ni 2.99, 6.4, 8.06
Perkin-Elmer Ni-Be Pt 2.99, 6.4, 8.06
(IR & D)
Eastman-Kodak Ule Au 2.99, 6.4, 8.06
(Recoated)
Eastman-Kodak Fused Silica Au 2.99, 6.4, 8.06
-	 Perkin-Elmer Zerodur Pt 2.99, 8.06
Perkin-Elmer Ule Pt 2.99, 6.4, 8.06
t8
Y
The plots mentioned in section II for each scattering spectrum
on these mirrors aregiven in appendix B. In appendix C is given a
summary of the data for each angle, energy, and mirror. The items
in the summary includes 1) the full-width half maximum, 2) the
angle at which 50% of the counts fall within, 3) the fraction of
the counts outside 14 areseconds, 4) the measured reflectivity,
5) the theoretical reflection, and 6) a general comment on the data.
When the general comment mentions "scuzz" it is referring to the fact
that this data had a spurious source of counts in a certain spatial
location which had nothing to do with x-rays from the mirrors or
normal background. Hence, the fraction of counts within a given
R
angle would not be meaningful for these spectra.
9t
IV.	 Summary & Limitations
In general, the mirror surfaces are so smooth that there is very
little scattering due to roughness. In retrospect, there are several
things that should be done differently to obtain better data. Among
these are the zero angle verification procedure, the mirror holder
system, the calibration of the detector efficiency, and the optimi-
zation of the elevation of the detector. For these reasons, the
data presented in this report should be taken as very qualitative.
In particular, the measured reflectivities do not, in general, agree
with the theoretical values because of one or nore of the problems
mentioned above. Other information such as the FWHM and the fraction
of counts outside 14 areseconds is more reliable, however.
ORIGINAL PAG 0
	
1
APPENDIX A	 OF POOR QUALITY	 Y
GENERAL FORMALISM FOR STATISTICAL ERRORS
I. Error in the Number of Counts
The usual assumption of using the square root of the number of
counts as the uncertainty, o, in these counts will be used:
o f	 y N	 (Al)
where, N i is either the number of counts in channel i, or the total
number of counts in all channels of a given run i .
II. Error in Corrected Counts for Each Channel
Since the counts in each channel are corrected for background
counts,as well as, for variations in the response of the detector as
a function of the channel number, the error, a i ', may be above and
beyond that calculated by equation (A1). This will be examined now.
-The corrected number of counts N i ' is calculated from the
uncorrected number of counts N i
 in each channel i by the relation:
Ni ' _ (N i - NB ) RCE i .	 (A2)
NB
 is the number of background counts in each channel which is assumed
to be a constant. RCE i
 is the correction factor which corrects for the
efficiency of channel number i . The equation for RCE i is given in a
separate article. 'he important equation for error analysis, however,
is the following:
M
RCE	
_ ( RCE i ) 2	
Ei[(_Nj_jT71 1 	 + j j1	 11	 N 1
1 i	 i	 i
V-1 o
(A3)
!'r
	
ORIGINAL I^r --'	 2
OF POOR 4_'JAL rkly
Where Nij (1) and Nij (2) are in reference to the fact that to calcu-
late RCEi a data run had to be done with the detector located in two
different spatial positions. 	 N1 1 (1) is the number of counts in
channel number U when the detector is aL location number 1, and
ij - i + j - k where k is the number cf channels between location 1
and location 2.
	 1  is an arbitrary reference channel number normally
taken as channel number 512. The summation contains M  terms where
M 
	 -
	
(A4)
As long as the channel numbers i are limited to channels which
have reasonably good efficiencies, it is true that
	
Ni^(1) s Ni^(2) - N i (1) 'g N i	 -	 N R	 (A5)
o
That is in the data taken to calibrate the efficiency of the detector,
the number of counts in each channel are approximately equal to some
average number NR . For the data analyzed in this report:
500	 Fe Line (1.94 A)
NR	 -	 450	 Ag Line (4.16 ^)	 (A6)
100	 Cu Line (1.54 ^1)
Hence, the variance in RCE i
 can be written as
RCE	 ;RCEi )2 (2 Mi + 2)	 (A1)i	 N
R
In the counter efficienLy runs used to calibrate the counter reponse
efficiencies, the value of k in equation (A4) was about 244. Hence,
	
i - 5121
	 (P.8)
Mi244
PRFCEDiNO PATIP HANK NOT MMry)
M
Substituting for aRCE i 2 from eq. (A7) this becomes
oN I _ ( RCEi)2	
ai l + 
2N12 (Mi 
+ 1)
	 I (A16)i	 R
For the data analyzed in this report, the worst case is in the peak
channel where typically these are 2300 counts. If it is a l so assumed
that RCE i and Mi are the worst case, then RCE i = 2, Mi = 2, Nit = 450
which givesaN 	535 or a 23% error.
i
In channels which are not close to the peak, the second term in
egUation (A16) becomes negligible and the error reduces to the sq mare
root of the nLmber of corrected counts.
III. Error in total Counts for Corrected Data
The variance in the total number of counts for the corrected
data is given by	 2
0 2	 E ON
N	 i	 i
	 (A17)
Since the dominant channel variance comes from the peak channel, this
becomes
a 	
s 
apk .
	 (A18)
As a typical example, considar the data taken on the PE-II-003 (ULE/Ni)
mirror with Ag line x-rays. Near the peak RCE s 1.33. The total n umber
of counts in the direct slit imayz, is 7400; at 25' there are 6300 counts;
«t 39' there are 4800 counts; and at 51' there are 3000 counts. This
gives the following errorst
ORIGINAL PACC 3
OF POOR QUALITY
ORIGINAL PACE fS
OF POOR QUALITY	 0
Angle of Incidence	 Total Counts	 Error
	0' (D.S.)	 7400	 + 194 (± 2.6%)
	
25'	 6300	 ± 145 (± 2.3%)
	
39'	 4800	 ± 115 (± 2.4%)
	
51'	 3000_	 ± 80 (± 2.7%)
IV. Error in Reflectivities
The "reflectivities" as quoted in this report are calculated with
the equation.
	
R =	
'Total (Refl)	 'TR	
2	 (A19)
NTotal (D.S.)	 NTS
	
The variance in
	
R is given by
Cr 2	 a R 2	
°N 2
	 + /a R	 Q	 2
R	 a NT	 TR	 a "1T 
2
	 NTS
R2
	
+	
2	 2
4 N2	 ^NTR	
o _
	
ANTS	 (A20)
TR	 TS
R2	 °2NTR	 +	 ^2NTS
4	 NTR2	 NTS2
For the typucal mirror PE-II-003 ( ULE/Ni) at the Ag line, the
reflectivities and errors are given below:
tORIGINAL PACL r^$	 6
OF POOR QUALITY
	 r
Angle of Incidence	 Reflectivity	 Error
25'	 0.92	 + .02
39'	 0.80	 + .014
51'	 0.63	 + .01
V. Error in the FWHM
Let M be the number of channels one has to go from the-peak
of the image before the number counts falls below one half of the peak
value, and let	 D be the number of channels/table arc second ( -.12).
Then,
FWHM =
	
	 2M	 (A21)
D•
This gives
6	
=	
FWHM	 Cr _	
(A22)
FWHM	 D -	
D
The uncertainty in	 D is estimated from the data to be
op.. = .01
	
with	 D = 0.12. Hence,
Q	 =(.083) FWHM.	 (A23)
FWHM
This number, known, may turn
resolution of the dector whi
(.125 areseconds). Hence,
v	 __
FWHM
out to be less than the physical spatial
ch is one channel or 8.:3 table areseconds
c'FWHM is given by
Larger of	
(.083) FWHM	 (A24)
0.13 areseconds
HLv
m
1
Z
O
2
Lr
1
W
CL
11
Ix
0
ORIGINAL
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ORIGINAL PAGE 19
OF POOR QUALITY.• c ain6LI
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Reduction of the one-dimensional X-ray scattering data on mirror flats at NASA/Marshall Space Flight Center

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