SPIR 149
Photodisintegration of the Deuteron at
20 MeV using Circularly Polarized Photons
Glen Pridham
April 28th , 2014
Contents
1 Introduction
4
2 Run Summary
4
3 Methods
3.1 Time-of-Flight Cut . . . . . . .
3.2 Pulse-Shape Discrimination Cut
3.3 Light Cuts . . . . . . . . . . . .
3.4 ADC Cut . . . . . . . . . . . .
3.5 Multiplicity Cut . . . . . . . . .
3.6 Background Cut . . . . . . . . .
3.7 Water (1 H2 O) Target Cut . . .
3.8 Cell Exclusions . . . . . . . . .
3.9 Light Scaling Factors . . . . . .
3.10 Computing the Flux . . . . . .
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4 Theory
5 Results
5.1 Neutron Yields . . . . .
5.2 Total Cross Section . . .
5.2.1 Parameterization
5.3 Differential Cross Section
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13
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15
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32
6 Discussion
6.1 Total Cross Section . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Parameterized Differential Cross Section . . . . . . . . . . . .
6.3 Comparison to Kucuker’s Results . . . . . . . . . . . . . . . .
35
35
35
36
7 Sources of Error
41
8 Conclusion
44
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2
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List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Parameterized Theoretical Differential Cross Section . . . .
Long Target Yield (arm 2 up) . . . . . . . . . . . . . . . .
Long Target Yield (arm 4 up) . . . . . . . . . . . . . . . .
Long Target Yield . . . . . . . . . . . . . . . . . . . . . . .
Short Target Yield . . . . . . . . . . . . . . . . . . . . . .
Total Cross Section . . . . . . . . . . . . . . . . . . . . . .
Parameterization Comparison . . . . . . . . . . . . . . . .
Parameter a1 comparison . . . . . . . . . . . . . . . . . . .
Parameter a2 comparison . . . . . . . . . . . . . . . . . . .
Parameter a3 comparison . . . . . . . . . . . . . . . . . . .
Parameter a4 comparison . . . . . . . . . . . . . . . . . . .
Parameter e2 comparison . . . . . . . . . . . . . . . . . . .
Parameter e3 comparison . . . . . . . . . . . . . . . . . . .
Parameter e4 comparison . . . . . . . . . . . . . . . . . . .
Parameter c1 comparison . . . . . . . . . . . . . . . . . . .
Parameter c2 comparison . . . . . . . . . . . . . . . . . . .
Parameter d1 comparison . . . . . . . . . . . . . . . . . . .
Parameter d2 comparison . . . . . . . . . . . . . . . . . . .
Differential Cross Section (long target average) . . . . . . .
Differential Cross Section (short target average) . . . . . .
Comparison of Parameterization to Kucuker (long target) .
Comparison of Parameterization to Kucuker (short target)
Relative Yield of Empty Target . . . . . . . . . . . . . . .
Error Breakdown . . . . . . . . . . . . . . . . . . . . . . .
Simplified Error Breakdown . . . . . . . . . . . . . . . . .
Parameterization Comparison . . . . . . . . . . . . . . . .
Total Cross Section . . . . . . . . . . . . . . . . . . . . . .
3
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14
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37
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48
1
Introduction
An experiment devoted to measuring the photodisintegration of the deuteron
using 20 MeV circularly polarized photons was performed at the High-Intensity
Gamma Source (HIγS) located at Duke University, Durham, North Carolina
in September 2008. The experiment was led by Kucuker, and she initially
analyzed the data, however, her results showed an inconsistency in the padσ
, between the short and long targets
rameterized differential cross section, dΩ
at extreme polar angles (θ ≈ 0◦ and180◦ ) [6]. This discrepancy inspired the
current work: a second, independent, analysis of the data set.
Our results show that the spatial distribution of our yield data agrees
with the theoretical yield, and we see no target length dependence within
the error intervals. There is, however, a hint that there may be a small polar
angle discrepancy between the short and long target yields.
Our total cross section agrees well with theory and previous researchers,
but only if we include a systematic error of approximately 5.5%, due to adc
gain offsets [9].
Once again [4,9] we see that our calibrated adc spectra are systematically
low for the neutrons and use inferred factors to increase the neutron light
output. Interestingly, these factors are larger than the ones used in 2010 and
therefore cannot be solely due to an aging process.
Where desired, it is recommended that readers supplement this document
with the thesis written by Glen Pridham, to be published some time in 2014.
The same methodology was used in both experiments and analyses, except
where explicitly stated otherwise.
2
Run Summary
The data were collected at HIγS on September 25th and 26th , 2008. The
summary of runs used is described in table 1 on the next page.
4
Run
685
700
701
702
703
704
714
715
717
718
720
721
722
723
725
726
728
730
731
732
733
735
3
Table 1: September 2008 Run Summary.
Run Time Target/Source Arm Up
Purpose
th
September 25 , 2008.
9:30-9:31
AmBe Source
2
Gains (runs 700-704).
12:45-13:01 10.7 cm D2 O
2
Align the tdcs; data run.
13:01-14:07 10.7 cm D2 O
2
Data run.
14:07-15:06 10.7 cm D2 O
2
Data run.
15:21-16:17 10.7 cm H2 O
2
Data run.
16:17-17:12 10.7 cm H2 O
2
Data run.
20:16-?
10.7 cm D2 O
4
Data run.
21:09-21:45 10.7 cm D2 O
4
Data run.
21:54-22:05 AmBe Source
4
Gains (runs 714-715) and psd.
th
September 26 , 2008.
8:00-8:21
AmBe Source
4
Gains (runs 720-728).
9:34-?
10.7 cm H2 O
4
Data run.
10:14-10:33 10.7 cm H2 O
4
Data run.
10:39-11:47
2.0 cm D2 O
4
Data run.
11:47-12:35
2.0 cm D2 O
4
Data run.
13:15-13:32
2.0 cm H2 O
4
Align the tdcs.
13:44-15:34
2.0 cm H2 O
4
Data run.
17:13-18:11
2.0 cm D2 O
4
Data run.
18:15-19:44
2.0 cm D2 O
4
Data run.
19:44-20:47
2.0 cm D2 O
4
Data run.
20:50-21:12
2.0 cm D2 O
4
Data run.
21:18-21:42
2.0 cm H2 O
4
Data run.
21:55-22:02 AmBe Source
4
Gains (runs 730-733).
Methods
We calibrated the data using standard methods [6,17] and then applied data
analysis cuts to eliminate noise, culminating in a final neutron yield in each
detector.
Each cut had an associated error which was propagated by perturbing
the cut by its error value and measuring the change in the final neutron yield
after all other cuts were applied.
All cuts were applied to the simulated data as well, except for the back5
ground radiation cut and multiplicity cut which are not applicable to simulated data. With the same cuts applied to the simulated data, we then
assumed that the simulated efficiency reproduced the experimental efficiency
in order to compute the total cross section.
3.1
Time-of-Flight Cut
Table 2: Time-of-Flight Cut Summary
Purpose
Eliminate promptly scattered beam photons.
Upper Cut
55 ns
Lower Cut
3 ns
Uncertainty
Assumed to be 0.
The time-of-flight cut’s purpose is to eliminate the on-time beam photons
from the analysis. The cut must therefore be placed after the expected
arrival time of the promptly scattered beam photons (≈ 1.5 ns) and before
the earliest neutron arrival time (≈ 7 ns), while taking into account the
possibility of a poorly aligned tdc (which varies by . 2 ns); we pick 3 ns as
a conservative estimate for the earliest neutron arrival time1 .
The later cut is determined by the length of the neutron window which
cuts off signals arriving after ≈ 55 ns.
Therefore, placing the lower cut at 3 ns and the upper cut at 55 ns
eliminates the promptly scattered beam photons with minimal loss of the
neutron spectrum. We depend on the psd cut to eliminate the out-of-time
beam photons2 and late scattered photons.
The reason for not applying a stronger time-of-flight cut is that there are
a number of features in the tdc spectrum: the 13 C neutron peak, the deuteron
neutron peak, and the 16 O neutron peak; placing a time-of-flight cut near
or inside one of these peaks will inevitably lead to discrepancies due to tdc
drift.
1
Placing the cut at 5-6 ns would put it right in the middle of the 13 C neutron peak
for the upstream and downstream most cells, which would cause any errors in the tdc to
propagate into large uncertainties in the yield.
2
Out-of-time beam photons are those photons which arrive at the target at unexpected
times. They are typically attributed to electrons spreading out of their correct buckets.
6
3.2
Pulse-Shape Discrimination Cut
Purpose
Upper Cut
Lower Cut
Uncertainty
Table 3: PSD Cut Summary
Eliminate photon events from detectors.
None.
Mean - 2 standard deviations.
Estimated using fitting algorithm.
The pulse-shape discrimination (psd) cut was performed to eliminate photons
from the analysis. The cut was placed such that a maximum number of photons were eliminated without significant loss of neutrons. The methodology
performed is outlined in SPIR-148 [7].
The psd was computed using run 717 (an AmBe run) then the parameters
were verified using the other AmBe runs (685, 718, and 735): the psd did
not change significantly during the course of these experimental runs.
3.3
Light Cuts
Purpose
Upper Cut
Lower Cut
Uncertainty
Table 4: Light Cut Summary
Eliminate low energy false positives and adc artifacts.
5500 keVee
2000 keVee
Calculated from uncertainty in adc gain.
The lower light cut is applied to the calibrated adc spectra, it is necessary for
two reasons: to eliminate the region of overlap in the psd spectrum, and to
give all cells the same effective hardware threshold. The lower light cut could
have been placed as low as 1000 keVee, but by placing it at 2000 keVee we also
eliminated most of the neutrons produced from the photodisintegration of
oxygen and, more importantly, a significant portion of the Compton scattered
out-of-time beam photons.
The upper light cut is necessary because the neutron spectrum does not
fit in the adc spectrum for the downstream cells: the gain was set too high.
Since we use the sliding scale technique on the adcs, this destroys the information in the bins past ≈3650, therefore the upper light cut was calculated
7
by considering the cell with the smallest gain and ensuring that the upper
light cut was lower than bin 3650 on that cell. The resulting calculation
yielded 5500 keVee as a conservative upper light cut.
The upper light cut removed the tail of the neutron energy spectrum
for the downstream cells, but not the upstream cells: therefore it had the
potential to cause a spatial discrepancy in the event that it was not properly
reproduced by the simulation.
3.4
ADC Cut
Purpose
Upper Cut
Lower Cut
Uncertainty
Table 5: ADC Cut Summary
Eliminate artifacts from the adc spectrum (redundant).
3650 bin
0 bin
4.3 bins
Using the sliding scale technique on the v792 vme adcs improves linearity, but
causes the later bins to produce inaccurate results. The v792 manual states
that bins ≥ 3840 cannot be used when the sliding scale technique is active [5],
but we observed strange behaviour for all bins & 3700 and therefore placed
the cut at bin 3650 instead: ignoring all adc values > 3650.
This cut is redundant because the upper light cut was selected such that
it was always lower than bin 3650.
3.5
Multiplicity Cut
Purpose
Upper Cut
Lower Cut
Uncertainty
Table 6: Multiplicity Cut Summary
Eliminate and compensate for incomplete signals in detectors.
1
1
Assumed to be 0.
The electronics of Blowfish are configured such that if any single cell records
an event, then all cells have their adcs read. This means that if two different
cells record events in rapid succession then they will both be recorded during
8
the same readout event and the gates generated for the adcs will be inaccurate
for the event which occurred last: causing a partial event to be recorded.
Since the gates are incorrect, the adc values are meaningless (as is the psd).
We excluded all events when more than one cell recorded an event above
threshold (i.e. the discriminator fired) so long as the gates were generated
(i.e. the adc values recorded are > 0). The number of excluded events
for each cell was recorded and the ratio of excluded events to total hits for
each cell was computed to give the multiplicity factor : the factor which
compensates for the multiplicity cut. Finally, the total cell yield was scaled
by its corresponding multiplicity factor.
We attempted to improve the precision of the multiplicity factor by processing the first event normally when several events were recorded in rapid
succession.
3.6
Background Cut
Table 7: Background Cut Summary
Purpose
Subtract background radiation from detectors.
Background Interval
[-40√ns, -30 ns]
Uncertainty
counts
The background cut is an estimate of the number of non-neutron events
(i.e. false positives) which have made it past the other cuts: this estimate is
subtracted from the final neutron yield.
The number of false positives is estimated by looking at the interval [-40
ns, -30 ns] on the time-of-flight spectrum for each cell: the number of photons
in the interval are counted, then divided by the interval length in order to
get an estimated background rate. The background rate is then scaled to the
time-of-flight window and subtracted from the neutron yield (equation 1a).
I
B =N −I ·
N =N−
15ns
0
9
R −15ns
T (t)dt
15ns
−30ns
(1a)
Background Corrected Yield. Where: N 0 is the background corrected
neutron yield, N is the neutron yield before correction, B is the background
yield, T (t) is the time-of-flight spectrum (in hits/ns), and I is the time-offlight cut interval.
We propagate the error using the normal technique but neglecting the error
√
in the time interval, then the error in the background counts is given by B:
dN 0 = dN + (
3.7
I
dB)
15ns
(1b)
Water (1 H2 O) Target Cut
Purpose
Cut
Uncertainty
Table 8: Water Target Cut Summary
Compensate for false-positives and neutrons from other sources.
1
H2 O yield was fully analyzed then subtracted from other runs.
Full estimate: treated like D2 O data.
After all other cuts have been applied, it is unlikely that there are any photons
remaining in the yield spectrum for a run. It is probable, however, that
there are neutrons generated via reactions other than 2 H(γ, n)H in the yield.
Kucuker assumed the number of neutrons due to other reactions was trivial
after her cuts were applied [1]: this is not true for the cuts we applied during
this analysis. The H2 O target runs were fully analyzed and then the yield
was subtracted from the D2 O yield, where the nearest in time H2 O target
run was subtracted.
This subtraction removes neutrons from: the target holder, the target
cell, oxygen in the target, and any other background radiation which made
it through the cuts (and remained constant in time).
3.8
Cell Exclusions
Cells which showed pathological behaviour were excluded from analysis, this
included: empty cells (cell 56), cells with time-of-flight spectra which could
10
not be aligned to within 1 ns of the expected time-of-flight, cells which gave
no distinct separation in the psd spectrum, and cells which demonstrated
the presence of out-of-time beam photons after the psd cut has been applied.
The presence of out-of-time beam photons in the final yield causes a timedependent photon background which cannot be reliably subtracted via the
H2 O yield subtraction, and therefore these cells had to be excluded.
3.9
Light Scaling Factors
In these data we observed that the photon energy spectra agree very well
with the simulated data, but the neutron spectra do not: the neutron adc
outputs are too small. This observation is not new [4, 9], and so we used the
standard methodology: the neutron adc gains were scaled by inferred factors
in order to show good agreement between the simulated and experimental
adc spectra; we call these the light scaling factors.
These factors were extracted by fitting a simple edge function (the same
used to extract the Compton edge) to the calibrated adc spectra for both the
simulation and experimental runs (after all cuts were applied). The location
of the inflection point was recorded and the simulated result was divided by
the average experimental result to give the light scaling factors for each cell.
The light scaling factors were observed to be stable during the course of
the experiment: they did not change significantly from run to run.
It is unknown what is causing the simulated and experimental adc spectra
to disagree only for neutrons [9], and there is no good leading theory.
The light scaling factors used are reported in table 9 on the next page.
11
Cell:
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
22:
Factor
1.210
1.157
1.156
1.155
1.172
1.196
1.128
1.181
1.161
1.142
1.161
1.158
1.166
1.168
1.218
1.176
1.136
1.158
1.128
1.159
1.137
1.119
Cell:
23:
24:
25:
26:
27:
28:
29:
30:
31:
32:
33:
34:
35:
36:
37:
38:
39:
40:
41:
42:
43:
44:
Factor
1.150
1.108
1.117
1.098
1.150
1.123
1.135
1.127
1.140
1.143
1.140
1.145
1.140
1.107
1.123
1.128
1.116
1.095
1.097
1.094
1.114
1.104
Cell:
45:
46:
47:
48:
49:
50:
51:
52:
53:
54:
55:
56:
57:
58:
59:
60:
61:
62:
63:
64:
65:
66:
Factor
1.130
1.128
1.110
1.095
1.099
1.099
1.081
1.111
1.096
1.105
1.139
n/a
1.110
1.090
1.134
1.144
1.128
1.099
1.128
1.130
1.122
1.083
Cell:
67:
68:
69:
70:
71:
72:
73:
74:
75:
76:
77:
78:
79:
80:
81:
82:
83:
84:
85:
86:
87:
88:
Factor
1.124
1.086
1.092
1.085
1.102
1.089
1.087
1.082
1.130
1.090
1.119
1.003
1.121
1.086
1.076
1.045
1.106
1.090
1.073
1.128
1.099
1.098
Table 9: Light Scaling Factors.
3.10
Computing the Flux
The number of photons incident on the target was calculated using the
methodology delineated in SPIR-140 [8] and the linear mass attenuation
coefficient for light water [3]. This required inverting equation 2.
Nγ
Nm = Be−v (1.−m ) B (1. − e−m
12
Nγ
B
)
(2)
Where: Nm is the number of events measured by the flux monitor, B is the
number of bunches, v is the veto paddle efficiency, m is the flux monitor
efficiency, and Nγ is the number of photons leaving the target.
Equation 2 on page 12 cannot be robustly inverted using an series expansion, however, taking the derivative one finds that the slope is always
positive and therefore the inversion has exactly one finite solution.
Since we know that the number of photons per bunch is usually 1-2, we
can put an upper limit on the value of Nm by setting NBγ > 2 and then use the
bisection method to find the only solution to an arbitrary level of precision.
Since the uncertainty in the flux monitor is on the order of ≈2%, we see no
loss of precision by inverting equation 2 on page 12 to < .2%.
The actual inversion algorithm used was not the bisection method, but
instead the function was simply graphed using the ROOT class TF1; it was
then verified to be accurate to within 4 photons in all cases (typically within
1): a completely trivial inaccuracy relative to the uncertainty (which is on
the order of hundreds of millions of photons).
Finally, the number of photons incident on the target was estimated using
the linear mass attenuation coefficient for 20 MeV photons [3] and N0 =
N e−µl .
4
Theory
The full theoretical calculation by Schwamb and Arenhövel [12–14] was parameterized using the Legendre polynomial expansion to fourth order in the
‘a’ parameters: the rest were set to 0 (unpolarized beam). The result of the
fit to Schwamb and Arenhövel’s data is given in figure 1 on the next page.
13
Figure 1: Parameterized Theoretical Differential Cross Section.
The fit (figure 1) has a precision better than 0.27% at all points.
The values of the parameters are compared to Kucuker’s results in table 10. Kucuker also fit to the e parameters and so we expect that our
results are more accurate since they had fewer degrees of freedom.
Parameter
a1
a2
a3
a4
Current Work
-0.165378
-0.872064
0.151289
-0.0179751
Kucuker [6]
-0.1652
-0.8750
0.1504
-0.0181
% Error
0.1%
0.3%
0.6%
0.7%
Table 10: Legendre Expansion Parameters of Theoretical Calculation.
14
5
Results
5.1
Neutron Yields
Once all of the cuts were applied, we were left with a neutron yield for each
included cell in Blowfish. These cells were parameterized using the Neutron
Yield Expansion: equation 3.
Nd =A[(1 −
... +
4
X
3
3
sim
ak − 3e2 − 6e3 − 10e4 − c1 − c2 − d1 − d2 )Nd,00
2
2
k=1
4
X
sim
sim
sim
sim
ak Nd,0k
+ 3e2 Nd,22
+ 6e3 Nd,23
+ 10e4 Nd,24
k=1
3
3
sim
sim
sim
sim
+ d1 Nd,11
d2 Nd,12
... + c1 Nd,11
+ c2 Nd,12
0 +
0]
2
2
(3)
Neutron Yield Expansion. Where: Nd is the neutron yield in a detector,
sim
d, and Nd,lk
is the simulated neutron yield for the probability density function
ρlk .
Equation 3 is believed to be a homeomorphism of the Legendre expansion of the differential cross section (equation 5 on page 32), therefore the
parameters extracted from equation 3 can be directly used to calculate the
dσ
(so long as the total cross section, σ, is known).
differential cross section, dΩ
Several tests have been applied to verify that equation 3 is a homeomorphism,
it is3 : so long as the simulation accurately reproduces the experimental efficiency after all cuts have been applied (i.e. sim. = exp. ).
The neutron yield for the long target average with arm 2 up (runs 700,
701 and 702) are given in figure 2 on the next page along with their parameterization and the theoretical calculation mapped into a neutron yield4 .
3
Equation 3 has been empirically shown to be a homeomorphism by simulating arbitrarily parameterized data then parameterizing the resulting yield in order to verify the
initial parameter values are recovered.
4
In all cases the theoretical neutron yield has been scaled to the experimental yield.
15
Figure 2: Long Target Yield (arm 2 up). We see excellent agreement
between the yield parameterization (χ2fit = 0.63) and theory. Notice that
there is a dramatic asymmetry in the yields within the same ring for cells
near cell 40: this may be due to bad gains (the Gain Monitoring System
couldn’t be used for these runs), or a change in beam polarity.
Similarly, the yield for the long target average with arm 4 up (runs 714,
715 and 720) are given in figure 3 on the next page.
16
Figure 3: Long Target Yield (arm 4 up). We see excellent agreement
between the yield parameterization (χ2fit = 0.98) and theory.
The total average of all of the long target runs is given in figure 4 on the
next page.
17
Figure 4: Long Target Yield. As expected, we see excellent agreement
between the yield parameterization (χ2fit = 0.96) and theory.
Finally, the short target yield is given in figure 5 on the next page.
18
Figure 5: Short Target Yield. We see fair agreement between the yield
parameterization (χ2fit = 1.60) and theory with the possibility for a systematic
disagreement in the last ring of cells (81-88).
We see good agreement in all cases. Although the short target yield hints
that the parameterization may be systematically higher than the theoretically
predicted yield for the last couple of rings.
It is worth emphasizing that the yield plots are the best way to compare
our results to theory because there may be some correlation between param19
eters which allows them to fit to the yield plot even if their parameter values
are, in fact, incorrect. Correlation matrices are reported in section 5.2.1 on
page 22, along with the values of the parameters with error.
5.2
Total Cross Section
The total cross section was calculated by assuming that the simulation was
able to replicate the experimental efficiency. Knowing the photon flux (section 3.10 on page 12), and neutron yield (section 5.1 on page 15) we calculated
the total cross section using equation 4.
σ=
Nd µ
N sim
sim
Nd ρnN0 (1 − e−µl )
(4)
Where: σ is the total cross section, N sim is the number of neutrons simulated,
and Nd is the number of neutrons measured by detector d, µ is the linear
mass attenuation coefficient, and l is the target length.
Run
Target
Cross Section (µbarns)
700
701
702
714
715
720
722
723
728
730
731
732
10.7 cm
10.7 cm
10.7 cm
10.7 cm
10.7 cm
10.7 cm
2.0 cm
2.0 cm
2.0 cm
2.0 cm
2.0 cm
2.0 cm
520+80
−51
521+64
−36
531+68
−38
564+74
−43
554+76
−45
512+76
−47
535+82
−52
534+83
−53
540+89
−59
519+82
−53
508+97
−69
515+97
−69
Table 11: Run-by-Run Total Cross Section Data. Note: the systematic
shifts between runs 702 to 714, 715 to 720, 728 to 730 were all accompanied
by both a change in the background run used and a change in the gain run
used.
20
Source
Cross Section (µbarns)
Long Target Average
Long Target Average*
Short Target Average
Total Average
Theory
Skopik et al [15]
Ahrens et al [2]
534+73
−43
545+75
−53
527+88
−59
531+81
−51
596.49
604 ± 29
585 ± 14
Table 12: Averaged Total Cross Section Data. *Arm 4 up only.
Figure 6: Total Cross Section. Our experimental results compared to the
theoretical prediction, and the results of previous experiments. Our results
agree with theory and previous experiments: within error.
The errors had to be propagated such that the systematic errors were
preserved (see section 7 on page 41), this prevented an increase in precision
for the overall average compared to the individual runs.
An additional error was added to the propagated error in all cases: this
was the estimated effect of a gain offset in the v792 adc gain which propagates
21
into a ≈5.5% uncertainty in the final cross section when the gain offset is
175 keVee5 .
5.2.1
Parameterization
Although our neutron yield plots already demonstrate that our parameterization and yield are consistent with theoretical predictions, we would like
to verify that our parameters agree with the theoretical ones. This tests
the consistency of both the parameterization of the neutron yield and the
Legendre parameterization of the theoretical differential cross section.
We included 11 Legendre parameters and one overall scaling factor in the
parameterization: only 4 of these (ai ) are predicted by theory (the ei vanish
when averaged over the polarization), the ei were included to test for linear
polarization in the beam, and the ci and di were included to compensate for
a poorly aligned target.
Parameter
a1
a2
a3
a4
e2
e3
e4
c1
c2
d1
d2
Arm 2 Up
Arm 4 Up
-0.1615 ± 0.0099 -0.1611 ± 0.0079
-0.878 ± 0.016
-0.851 ± 0.011
0.106 ± 0.019
0.117 ± 0.016
-0.006 ± 0.021
-0.030 ± 0.019
-0.0238 ± 0.0043 -0.0084 ± 0.0042
0.0014 ± 0.0020 0.0036 ± 0.0019
-0.0012 ± 0.0018 -0.0033 ± 0.0018
0.0204 ± 0.0010 0.0150 ± 0.0099
-0.0014 ± 0.0065 -0.0131 ± 0.0064
0.016 ± 0.013
0.016 ± 0.011
-0.0060 ± 0.0082 0.0041 ± 0.0070
Average
-0.1617 ± 0.0059
-0.8624 ± 0.0087
0.119 ± 0.012
-0.018 ± 0.014
-0.0170 ± 0.0029
0.0021 ± 0.0014
-0.0020 ± 0.0013
0.0209 ± 0.0069
-0.0056 ± 0.0045
0.0147 ± 0.0080
-0.0001 ± 0.0052
Table 13: Legendre Expansion Parameters (long target).
Table 13 gives the values of the parameters for the long target averages.
Since arm 2 was performed without the gain monitoring system, a drift in the
gain of a cell could cause a systematic shift: for this reason it is recommended
that the average of runs taken with arm 4 up be reported as the best long
target values. We compare the results graphically later.
5
175 keVee is the average cell gain offset for the latest gain linearity test in July 2008 [9].
22
Parameter
a1
a2
a3
a4
e2
e3
e4
c1
c2
d1
d2
Long Target
Short Target
Theory
-0.1617 ± 0.0059 -0.1440 ± 0.0059 -0.165378
-0.8624 ± 0.0088 -0.8352 ± 0.0081 -0.872064
0.119 ± 0.012
0.147 ± 0.011
0.151289
-0.018 ± 0.014
-0.019 ± 0.014 -0.0179751
-0.0170 ± 0.0029 -0.0101 ± 0.0033
0
0.0021 ± 0.0014 0.0020 ± 0.0014
0
-0.0020 ± 0.0013 0.0002 ± 0.0013
0
0.0209 ± 0.0069 0.0148 ± 0.0077
0
-0.0056 ± 0.0045 0.0052 ± 0.0048
0
0.0147 ± 0.0080 -0.0040 ± 0.0077
0
-0.0001 ± 0.0052 0.0018 ± 0.0046
0
Table 14: Legendre Expansion Parameters.
Table 14 gives the final parameterizations for the long target and short
target averages, as well as the theoretical parameters for an unpolarized
beam.
It is important to note how the parameterizations as a whole compare for
the short target, long target, and theory; this comparison is given in figure 7
on the next page.
23
Figure 7: Parameterization Comparison. The differential cross section
has been evaluated at each cell position for the long target (red), short target
(blue) and theoretical calculation (yellow). We see good agreement between
all three: all data points agree within their error bars. The last ring of cells
(81-88) hint that the theoretical calculation may be systematically lower than
the short target parameterization.
Figure 7 demonstrates that the parameterizations all agree with each
other and theory at the data points. Next we compare the validity of extrap24
olating and interpolating with the parameters by comparing them one-by-one
starting with figure 8 (parameter a1 ) and ending with figure 18 on page 30
(parameter d2 ).
Figure 8: Parameter a1 comparison.
25
Figure 9: Parameter a2 comparison.
Figure 10: Parameter a3 comparison.
26
Figure 11: Parameter a4 comparison.
Figure 12: Parameter e2 comparison.
27
Figure 13: Parameter e3 comparison.
Figure 14: Parameter e4 comparison.
28
Figure 15: Parameter c1 comparison.
Figure 16: Parameter c2 comparison.
29
Figure 17: Parameter d1 comparison.
Figure 18: Parameter d2 comparison.
30
All of the ei , ci , and di parameters are within two error bars of zero,
implying that they may be zero (as they should be); except for e2 for the
long target average with arm 2 up. The large value of e2 relative to its error
implies that there was some linear polarization which changed when Blowfish
was rotated (which is possible since they killed the beam after run 702) , or,
that the e2 parameter was compensating for some other problem in the data.
We see that the ai parameters could agree between the short and long
targets, but no single target length agrees with all of the theoretical parameters. This may be a manifestation of our lack of cells at extreme polar
angles: because the parameterization agrees very well with theory at the cell
locations (figure 7 on page 24), or it could be due to nonphysical correlations
between parameters.
The robust correlation matrix [10] for the long target average is given in
table 15.
a1
Parameter
a1
1
0.42
a2
-0.10
a3
a4
-0.11
a2
a3
0.42 -0.10
1
0.23
0.23
1
-0.10 0.47
a4
-0.11
-0.10
0.47
1
Table 15: Correlation of Key Fit Parameters (long target average). All
of the ‘a’ parameters have significant correlations, with strong correlations
between a2 and a1 as well as a3 and a4 .
Similarly, a robust correlation matrix for the short target average is given
in table 16.
a1
Parameter
a1
1
a2
0.00
a3
-0.31
a4
-0.06
a2
a3
a4
0.00 -0.31 -0.06
1
-0.03 -0.39
-0.03
1
0.24
-0.39 0.24
1
Table 16: Correlation of Key Fit Parameters (short target average).
The ‘a’ parameters have much weaker correlations than in the long target fit,
though there are still strong correlations between the a1 and a3 parameters
as well as the a2 and a4 parameters.
31
Parameters a1 and a3 are physically correlated such that their ratio should
be ≈ −1. The correlation in between a2 and a1 in table 15 on page 31 may
indicate why the parameters do not agree very well between the short and
long targets (since that correlation is not present in table 16 on page 31).
a1
a3
5.3
Differential Cross Section
The differential cross section was calculated from the parameterization using
equation 5.
"
4
4
X
X
dσ
σ
0
ek Pk2 (cos θ) cos 2φ
ak Pk (cos θ) +
=
1+
dΩ 4π
k=2
k=1
... +
2
X
ck Pk1 (cos θ) cos φ +
2
X
#
dk Pk1 (cos θ) sin φ
(5)
k=1
k=1
Legendre Expansion of Differential Cross Section. Where: Pki are the
associated Legendre polynomials, and σ is the total cross section.
The differential cross section for the long target average is shown in figure 19 on the next page.
32
Figure 19: Differential Cross Section (long target average). We see
good agreement except at extreme angles.
Similarly, the differential cross section for the short target average is
shown in figure 20 on the next page.
33
Figure 20: Differential Cross Section (short target average). We see
good agreement except at small polar angles (θ . 50◦ ) where the theory
underestimates the cross section.
The discrepancies at extreme polar angles observed in both figure 19 on
page 33 and figure 20 are manifestations of the discrepancy between the
parameter values in section 5.2.1 on page 22. The data which do not agree
are extrapolations and are therefore of little impact: the parameterizations
agree well for angles that Blowfish covers.
34
6
Discussion
6.1
Total Cross Section
Our reported cross section agrees with the theoretical calculation [12–14]
and previous experimental results [2, 15] but only with the inclusion of a
5.5% systematic error due to gain offsets in the adcs.
We can suspect Skopik et al’s results may not be accurate because they
used an untagged bremsstrahlung beam and therefore their results were dependent on a model for the photon flux. Ahrens et al also used untagged
bremsstrahlung but they calculated the cross section using the total photon
attenuation by comparing the results of a D2 O target and an H2 O target,
and therefore their results have no photon model dependence. Since Skopik
et al and Ahrens et al produced consistent results we must conclude that our
total cross section is too low, probably because of the adc gains.
6.2
Parameterized Differential Cross Section
Our measured neutron yields agree very well with the theoretically predicted
yields, although the short target hints that theory may be slightly underestimating the yield at forward angles (though the last ring still agrees within
error). Assuming our simulated Legendre probability density functions6 and
parameterization of the theoretical cross section, are correct: our results are
consistent with the theoretical prediction.
If there truly is a discrepancy between the short target yields and the theoretically predicted yields, this may be attributed to spin-dependent scattering (which is not considered in the simulation); the theory would then have
to be too low, which may be a manifestation of the lack of a full Lorentz
transformation7 .
It is also possible that our multiplicity assumption is incorrect which
would lead to a discrepancy between the short and long target yields in the
downstream-most rings: we assumed that the ratio of the hits in all cells was
much greater than the ratio of hits in any single cell, allowing us to truncate
cell hits 2
) = 0).
the multiplicity series (i.e. we assumed ( any single
total hits
6
It is important to note that the simulated data did not include a Lucite target holder,
thus it is possible that the simulated neutron yields are incorrect due to this omission.
7
Schwamb and Arenhövel [12–14] used a non-relativistic deuteron wavefunction then
applied relativistic corrections later.
35
The parameterizations of the long target (with arm 4 up) and short target
agree reasonably well: all of the important parameters (a1 , a2 , and a3 ) agree
within 2 error bars and nearly agree within 1. Plotting the parameterizations
to account for parameter correlations showed excellent agreement between
the short and long targets except at the downstream-most ring, but which
still agreed within error.
Comparison of the long and short target parameters to the theoretical
parameters tells a similar story: the parameterizations agree quite well with
theory, but the individual values of the parameters do not. This is likely a
manifestation of parameter correlations and a ‘bumpy’ χ2 space: different
parameter combinations can produce very similar spectra. The individual
values of the parameters probably aren’t accurate, but the resulting parameterization agrees well with theory (and between target lengths).
The short target does suggest a larger yield in the downstream cells than
theory predicts (though it still agrees within error), this is consistent with
the yield observation mentioned earlier.
Finally, the differential cross section shows good agreement with theory
for all target lengths except at the extreme polar angles (θ ≈ 0 − 20◦ and
θ ≈ 160 − 180◦ ). The theoretical cross section is slightly lower than the
experimental for the long target at upstream angles greater than 160◦ and
barely agrees with the downstream theoretical prediction near 0◦ (theory is
too high). Although the short target differential cross section also predicts a
higher expected cross section upstream (θ & 160◦ ), it predicts a higher cross
section downstream (θ . 50◦ ) than theory (whereas the long target predicts
a lower cross section downstream). These discrepancies are of little weight
since they are extrapolations based on the parameterization (Blowfish only
covers polar angles θ ∈ [22.5◦ ,157.5◦ ]).
6.3
Comparison to Kucuker’s Results
Kucuker [6] did not report a total cross section, and so we can only compare
the normalized, parameterized neutron yields.
36
Figure 21: Comparison of Parameterization to Kucuker (long target).
Figure 21 illustrates that our results show good agreement with Kucuker’s
parameterization for the long target runs with arm 4 up. The short target
data, figure 22 on the next page, shows a similar agreement, but there appears
to be a systematic disagreement in the upstream-most ring’s yield (though
it is within error).
37
Figure 22: Comparison of Parameterization to Kucuker (short target)
Figure 22 shows no significant deviation between our results and those of
Kucuker: the two agree within error. There may be a systematic disagreement in ring 1 (cells 1-8): our results gave a larger yield than Kucuker’s. This
explains why our yield data (section 5.1 on page 15) hinted at a disagreement
between short and long targets in the downstream cells, but not the upstream
cells; whereas Kucuker reported a disagreement in both the upstream and
38
downstream cells. We can explain the possible disagreement between our parameterization and Kucuker’s by considering the role of neutrons produced
by reactions other than 2 H(γ,n)1 H.
Analysis of the empty target runs has revealed the presence of a high
energy neutron source which is consistent with the photodisintegration of 13 C
in the Lucite target and target holder. At 20 MeV, the neutrons produced
from 13 C(γ,n)12 C are higher energy than those produced via 2 H(γ,n)1 H and
therefore a light cut will be unable remove them from the data8 . Since the
psd cut cannot differentiate from neutrons with the same energy produced
from different sources, this leaves only the time-of-flight cut to remove the
13
C neutrons.
The time-of-flight cut cannot remove the 13 C neutrons either because the
13
C neutrons arriving in the midstream cells (θ ≈ 90◦ ) are approximately
concurrent with the neutrons from 2 H. This is because the dominant sources
of 13 C neutrons are the ends of the Lucite target holder which have to travel
farther to reach the middle cells than the 2 H neutrons (recall the 13 C neutrons
are higher energy).
Therefore: the time-of-flight cut, the psd cut, and the light cut, are unable
to remove the neutrons from the Lucite target holder and target cell. This
leaves only one option: the neutron yield from the empty target must be
scaled by the photon flux and subtracted from the final neutron yield for all
of the D2 O target runs.
Kucuker reportedly did not perform this subtraction [1], and consequently
must have had false positive neutrons in her final yield plots. Since the
neutron yield is proportional to the target length, the relative contribution
of the false positive neutrons was larger in the short target than in the long
target: resulting in a discrepancy in the final parameterizations of the short
and long targets. The polar angle dependence of the discrepancy reported by
Kucuker was due to the polar angle dependence of the empty target neutrons:
the empty target yield is highest (by a factor of ≈2) in the last two rings of
cells.
Analysis of the empty target runs shows that the background neutron
yield due to the Lucite holder and target cell in the D2 O target were trivial
for the long target runs, but significant for the short target runs (figure 23).
8
An upper light cut wouldn’t remove the 13 C neutrons because the neutrons can scatter
and they usually only deposit a fraction of their energies in the detectors, therefore there
are neutrons from 13 C with the same energy as those produced from 2 H.
39
Figure 23: Relative Yield of Empty Target. Left: long target; right:
short target. The contribution of the empty target neutrons to the final yield
is trivial in the long target, but significant for the short target: especially
in the last ring. Notice that the empty target yield has a θ dependence: as
the cell number moves from 1 to 88 (i.e. θ = 0◦ → 180◦ ) the yield increases.
Note: some cells have photons in the yield e.g. cell 1.
Figure 23 predicts that our results will weakly disagree with Kucuker’s at
the extreme angles due to the θ dependence of the empty target yield. This
explains why our results hinted at a disagreement with Kucuker’s in the
upstream-most ring: because she did not perform a background subtraction.
This introduces the question: why did the downstream yield results
agree? Evidently there was some additional difference in the data processing methodologies employed by Kucuker and our current work. This is may
be due to the multiplicity correction. The Blowfish-ROOT (BFROOT) [16]
package was used as the basis for both the current analysis and Kucuker’s
analysis, it uses a zeroth-order multiplicity correction which predicts no spatial bias in the multiplicity scaling factor (i.e. the total number of multiplicity > 0 events are excluded, then all cell yields are scaled by the same
factor to compensate). The first-order (cell by cell) multiplicity correction
(which was used for this analysis) shows a significant polar angle dependence
1
) and therefore failure to perform it correctly would cause
(θ ∝ multiplicity
factor
the downstream cells to have underestimated yields. This explains why our
results agree with Kucuker’s in the downstream yields: her failure to perform
40
a background H2 O target subtraction may have been compensated for by an
approximate multiplicity correction. This would require that the multiplicity
compensates almost perfectly for the H2O neutron yield for the long target
(because the long target parameterizations agree between our results and
Kucuker’s).
Interestingly, our yield results (section 5.1 on page 15) hint that there
may be a short and long target discrepancy in our data as well which may
be due to our failure to include a second-order multiplicity correction9 .
We conclude that our yield parameterizations agree with Kucucker’s within
error. We see the possibility for a downstream yield discrepancy between the
short and long targets, but not an upstream yield discrepancy. This discrepancy may be due to the multiplicity correction, or some physical process not
accounted for in the simulation e.g. spin-dependent scattering, or the Lucite
target holder.
7
Sources of Error
The sources of error for an arbitrary run (715) are analyzed in this section.
9
This is because the multiplicity factors are larger for the long target than the short
target due to increased neutron and photon yields.
41
Figure 24: Error Breakdown.
Figure 24 illustrates the relative contributions of all of the accounted
sources of error, a simplified view is also provided in figure 25 on the next
page.
42
Figure 25: Simplified Error Breakdown.
As figure 25 illustrates: the uncertainties in this experiment were primarily systematic: the psd10 , the target, and detector alignment, together
contribute ≈76% to the total yield uncertainty. This means that even though
we took 12 useful experimental runs, our final average results are negligibly
more precise than any one of the individual experiments. On the other, this
means that our random errors are low, implying that we’re getting more than
enough statistics in each run, thanks to the high flux provided by HIγS.
We can reduce the psd uncertainty by taking longer AmBe runs or by
reworking the psd algorithm so that it is more robust; but it doesn’t appear
that this will reduce the uncertainty by much11 . A number of cells are outputting essentially no separation between the neutron and photons lines (i.e.
10
The psd uncertainty is systematic because it is only calculated once for all of the runs,
calculating it multiple times would make it more random.
11
An alternative would be to compute the psd parameters several times then average
the results together.
43
there are two distinct lines but they overlap) in the raw psd plot, indicating that the psd properties of our BC-505 detectors are failing for some of
the cells, possibly due to an aging effect (e.g. the presence of oxygen in the
cells). Based on the 2008 data, there are at least 5 cells which cannot give
psd separation without overlap (cells 1, 6, 14, and 78).
Improving the detector and target alignment precision would also significantly reduce the yield uncertainty, but most of these sources of error
would require major overhauls. The uncertainty in the radius of Blowfish
is currently estimated at 3 mm based on the rotating metal arm procedure:
perhaps it is time to find a more reliable way to set the Blowfish radius.
Finally, it must be mentioned that the gain offsets of the v792 adcs led to
an estimated uncertainty of 5.5% in the total neutron cross section, and could
have caused a spatial bias (we assume it didn’t). We can compensate for gain
offsets12 , but only if we take multiple radioactive source runs (preferably 3):
in this experiment we only took AmBe runs. Ideally, we’d like to settle why
the v792 adcs are not acting linearly (with zero offset) and fix them; or give
up on them and commit to the flash adcs.
8
Conclusion
The long target data taken with arm 2 up did not have the Gain Monitoring
System enabled and so two sets of results for the long target have been
reported: the average of all six runs, and the average from only when arm
4 was up. It is recommended that only the arm 4 data be used because the
arm 2 data indicates an azimuthal angle, φ, dependence (e.g. by having a
large e2 parameter value relative to its error) which, at best, indicates some
remnant polarization in the beam or a bad parameterization, and at worse
indicates a few cells had systematic shifts in their yields (e.g. due to gain
drift).
Our neutron yields agree with the theoretical prediction for both short
and long targets. Comparison of the parameterizations (figure 26 on page 46)
shows that the theoretical, short and long target parameterizations all predict the same cell yields and interpolated differential cross section, but the
extrapolated differential cross section at extreme polar angles (low and high)
shows some discrepancy. The actual parameter values agree reasonably well
between the short target and the long target (with arm 4 up), but do not
12
It is unclear how we could use the Gain Monitoring System with gain offsets.
44
agree with the theoretical parameters. This indicates that our yields agree
with the theoretical predictions, but our extrapolated parameterization is
poor: our parameterization must, therefore, be incorrect (at extreme polar
angles).
45
Figure 26: Parameterization Comparison. The differential cross section
has been evaluated at each cell position for the long target (red), short target
(blue) and theoretical calculation (yellow). We see good agreement between
all three: all data points agree within their error bars. The last ring of cells
(81-88) hint that the theoretical calculation may be systematically lower than
the short target parameterization.
The parameterizations we extracted from the yields agree with Kucuker’s
results both for short and long targets, except that in the case of the short
46
target our parameterizations reflect larger yields in the upstream cells. We
see the same potential disagreement in the short and long target downstream
yields that Kucuker reported, however, we see good agreement between the
short and long targets in the upstream cells.
This disagreement could be physical: the residual polarization in the
neutrons may be leading to preferential scattering towards the downstream
cells; the disagreement would have to be masked in the long target runs by
multiple scattering events.
This disagreement could also be an artifact of the analysis: the multiplicity correction used in this work is a first-order approximation, which may lead
to a discrepancy which is strongest at small polar angles (i.e. downstream).
We also neglected to include the Lucite target holder in the simulation which
has been shown to cause a change in the extreme polar angle yields of a few
% for the short target13 .
It is also possible that the theory is simply incorrect: several Blowfish
researchers have shown that the downstream yield predicted by Schwamb
and Arenhövel appears to be too small: Sawatzky at 3.5, 4 and 6 MeV (but
importantly not at 10 MeV) [11] Blackston at 14 and 16 MeV [4], and the
18 MeV 2010 data. Most of the time the disagreement with theory is small
and well within errors, though. If the theory is incorrect, it may be because
Schwamb and Arenhövel use a non-relativistic deuteron then add corrections
later (rather than a fully relativistic deuteron).
13
These simulations were performed based on estimated dimensions for the Lucite target
holder using a photograph in Kucuker 2010 [6].
47
Figure 27: Total Cross Section. Our experimental results compared to the
theoretical prediction, and the results of previous experiments. Our results
agree with theory and previous experiments: within error.
Finally, the total cross section we report is unexpectedly low: it is lower
than theory and two previous direct measurements (figure 27). Since the
two direct measurements agree (they also agree with theory), we must conclude that our results are systematically low. This can be explained by the
existence of a gain offset in the v792 adcs [9]. Unfortunately, we could not
determine the gain offsets for these runs, so we estimated them based on
those determined earlier in 200814 (175 keVee) [9]. This offset propagates
into a 5.5% error in the total cross section (≈ 40 µbarns) and explains the
observed discrepancy. With the systematic error our results agree with the
previous experiments and the theoretical prediction.
14
The values of the gain offsets were found to be smaller in October 2008 (after these
runs) and negligible in 2010, which may indicate that they disappeared due to the electronics rebuild between July and September 2008, or it may be due to a correlation between
the gain and the gain offset (the gains were much lower in October 2008 and 2010). We
assumed the worst case scenario (that the offsets are correlated to the gain) and estimated
the offsets based on the July 2008 data.
48
References
[1] Private Communications with Serpil Kucuker; 2014.
[2] J. Ahrens, H. Eppler, H. Gimm, M. Krning, P. Riehn, H. Wffler,
A. Zieger, and B. Ziegler. Photodisintegration of the deuteron at 1525
mev photon energy. Physics Letters B, 52(1):49 – 50, 1974.
[3] S. S. C. J. C. J. S. R. Z. D. Berger MJ, Hubbell JH and O. K. Xcom:
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