Systematic and statistical errors in the Aluminum data runs
Septimiu Balascuta,
James David Bowman,
07/20/2012
1. Introduction
This report is a description of correction methods for the systematic errors due to
transient electronic signals originating in the VME3 module of the NPDGamma Data
Acquisition. The transient is overlapped with the detector signals during the Aluminum
and Chlorine data runs in July 2011. The correction of the systematic errors was done
first by extracting the transient amplitude and decay time from the minimization of a χ2
function. The correction factors in the 48 detector asymmetries are calculated at the end
of each run.
1. The cuts used in the analysis.
The Aluminum data runs were analyzed starting with run 63207 up to run 66200
for 1950 good runs. In the analyzing program the detector background was calculated
for each accepted spin sequence by computing the average signal near the top of the
pulse, from time bin 16 to 32, and near the start of the pulse, from time bin 0 to 8, and
using 48 g-factors that were read from a file. The g-factors were calculated before the
analysis of Aluminum data, for each detector by using the pulse average of the dropped
pulses present in some spin sequences with beam-on.
The cuts were placed on spin-sequences to eliminate any possible systematic
effects. The first cut eliminated the spin sequences with bad header quality bit. The
quality bit is a flag that is zero if turning the Spin Rotator on or off is done in the proper
sequence: the Spin Rotator is one during pulses 0, 3 ,5, 6 and is off during pulses 1,2 4,
7. The neutron spin is rotated with 180 degrees when the Spin Rotator is on. The
second cut eliminated the spin sequences with dropped pulses. The dropped pulse
occurred when the proton beam was not sent into the mercury target. This was releted
with the synchronization of the pulses in the proton beam. To detect a dropped pulse
the average monitor signal in a pulse was calculated. If the average was smaller than
0.12 Volts then the pulse was a dropped pulse and the entire spin sequence was
rejected.
The next three cuts are upper limits placed on monitor 2 and detector signal and
are named Cut1, Cut2 and Cut3. Cut1 is placed on the monitor 2 signal and eliminated
the very slow (wrap around) neutrons. The upper limit for this cut was calculated for
each good spin sequence by summing the monitor 2 signals in the time bin intervals
[30, 35] and [17, 20] for each of the eight pulses in the sequence. The absence of the
wrap around neutrons due to a dropped pulse (two pulse frames ago) can be seen in
the time window [30, 35] closer to the end of the pulse. In contrast the wraps around
neutrons are not present in the time window [17, 20].
35
S (k )   Vm 2 (k , t )
t 30
(1)
For this reason the ratio of the monitor M2 areas integrated in these two windows
is small when the wrap-around neutrons are not present in the pulse. The maximum and
the minimum of the integrated M2 signals in the time bin intervals [30, 35] are calculated
for each spin sequence:
35
S (k )   Vm 2 (k , t )
t 30
(2)
V2max  max{S (k )}k
(3)
V2min  min{S (k )}k
(4)
The average of the eight average M2 signals between time bins 17 and 20 is
calculated for the same spin sequence:
1 7  20 V (t ) 
VS 2     m 2 
8 k 0  t 17 4 
(5)
The upper limit of Cut1 is estimated from the histogram of the ratio between the
maximum and the minimum values V2max and V2min divided with the average Vs2:
V
 V2min
R1  2max
 CUT1
Vs 2
(6)
The limit CUT1 can be between 0.12 and 0.2 as seen in the histogram of this
ratio calculated from all spin sequences with no dropped pulses (figure 1)
CUT1
Figure 1: The histogram of the ratio R1 is calculated from 1950 Aluminum data runs.
The second cut is placed on the spin sequences that passed all the cuts (Cut0 and
Cut1) mentioned above. First the integrated monitor signal is calculated over all 40 time
bins in each pulse (k). The maxim and the minim of the eight pulse areas in M 2 are
calculated. The ratio between the differences S2max – S2min divided with the average of
the eight M2 pulses is calculated and compared with the upper limit CUT2.
39
1 7
A(k )   Vm 2 (k , t )
As   A(k )
8 k 0
t 0
(7)
S2max  max{ A(k ), k  0...7}
(8)
(9)
S2min  min{ A(k ), k  0...7}
(10)
R2  (S2max  S2min ) / As  CUT2
The histogram of the ratio R2 built from 150 Aluminum runs is presented in figure 2.
Figure 2: The histogram of the ratio R2 is calculated over 150 data runs. Cut2 can be
between 0.3 and 0.4.
The third cut is placed on the eight pulses of the detector signals. The maximum and
the minimum of the integrated detector signal are calculated at each of the eight pulses
in the spin sequence.
1 39
Yd (k ) 
(11)
 Yd (k , t )
40 t 0
39
S d (k )  40Yd (k )   Yd (k , t )
(12)
t 0
Dmax  max{Sd (k )}
(13)
Dmin  min{Sd (k )}
(14)
The spin sequence average of the eight integrated detector signals is calculated:
1 7
(15)
 S d (k )
8 k 0
The difference between the maximum and the minimum detector signals divided with
the spin sequence average R3 is calculated for each spin sequence. The histogram of
the ratio R3 is divided with the spin sequence average of the detector signal is
presented in figure 3.
D  Dmin
(16)
R3  max
 CUT3
Dss
Dss 
Figure 3. The histogram of the ratio R3 calculated from the first detector signal is built
from 100 Al data runs of the first detector signal.
2. The counting statistics.
The standard deviation of the pulse average detector signal depends on the
fluctuations in the beam intensity from one pulse to the other, and on random numbers
of detected gamma photons per time bin, when the neutrons are captured in the solid
Aluminum target. The pulse average detector signal is calculated in a time bin interval
[T1, T2] where T1 depends on the switch on time of the Spin Rotator:
T
2
1
Yk (d ) 
Yk (d , t )
T2  T1  1 T1
(17)
The measurement of the beam-off detector signals with a 9 Volts connected to
the 12 inputs of each the sum-difference module in the VME3 [1] showed that there are
two transients correlated with the T0 signal of the Spin Sequencer and the RF input
signal of the Spin Rotator (figure 4). There are four sum-difference modules for each of
the four detector rings. The measurements were done for each of the four sum modules
of the four detector rings. In the presence of the beam the detector signal is a sum of
the beam signal, proportional with the flux of gamma rays, and the two transient signals.
Figure 4: The signal of the first detector, in the first ring, was measured as a function of
time, with a 9 V battery connected to the sum-difference module of the first ring.
In this work it is assumed that the parity term (P) does not depend on time bin.
This approximation is good for the time bin interval [8, 38] inside each pulse. For this
time bin interval the Spin Rotator is completely switched on. The pulse number in the
spin sequence “k” takes values from 0 to 7, with 0 the first pulse in the spin sequence,
with SR on and the neutron spin down.
In figure 4 there is a major transient with amplitude Vd and decay time “tau”,
generated during the 9th beam pulse at the end of each spin sequence of eight pulses.
The DAQ in the VME3 module reads data during eight pulses and transfers the data to
a PC disk during the 9th beam pulse. The transient is due to the discharge of the
capacitor inside the VME3 module, when the ADC inputs in the VME3 module starts
reading again the detector signal. The transient in the above figure was fitted with a
simple exponential decay function with zero offset. The same measurements showed
that there are also minor transients in the seven pulses (k=1…7) of the spin sequence.
These minor transients generated in the short time interval between two consecutive
pulses in a spin sequence, have amplitude Ad smaller than Vd and are generated by the
discharge of the same capacitor in the VME3 module of the Spin Sequencer, during the
time gap between two pulse readings. Therefore the two transients have the same
decay time (tau). If the time for one time bin is t1=0.416E-3 seconds, “k” is the pulse
number and “t” is the time measured relative to the start of the pulse, then the detector
signal at pulse k >0 is equal with:
Yd (k , t )  hd I k (t )(1  Psk )  Vd e
 ( t  kT )/
 Ad e
 t /
k 1
e
 jT /
(18)
j 1
The sum in the last term in the right hand side is the due to the minor exponential
transient generated at the beginning of all k-1 pulses (1, 2… k-1) previous to pulse k in
the same spin sequence. The transients generated in the previous spin sequences are
completely negligible because the transient decay time, is between 15 milliseconds and
20 milliseconds and the time for one pulse is 25 milliseconds. Because the number of
fitting parameters have to be smaller than eight (the number of pulses), for each pulse
starting with the second pulse in the spin sequence (k=1), one has to neglect the tails of
the minor exponential transient generated in previous pulses. This is a rough
approximation because if the asymmetry is calculated for the last pulses, the difference
between the amplitudes of the minor transient introduces a false asymmetry because
the two pulses have different spins and the amplitudes of the minor transient at the start
of the two pulses are always correlated.
The constant sk is +1 for pulses with spin-up (k=1, 2, 4, 7) and -1 for pulses with
spin-down (k=0, 3, 5, 6). By summing the detector signal in equation 4 between time
bins T1 and T2, and dividing with the number of time bins T 2-T1+1, one can calculate
the pulse average detector signal:
Yd (k )  hd I k (1  Psk )  Vd s1 ( ) e  kT /  Ad s1 ( ) f k
(19)
The sum (s1) is given by the relation:
T2
1
s1 ( ) 
 exp(t /  )
T2  T1  1 t T1
T2
1
Ik 
 I k (t )
T2  T1  1 t T1
(20)
(21)
 1 k>0
fk  
0 k=0
q1  exp(1/  )
(22)
The time average detector signal can be fitted with a function that has four fitting
parameters: P (the parity), Vd (the major transient amplitude), q1=exp(-1/tau) (transient
decay rate) and Ad (the minor transient amplitude).
In the following relations the “tau” dependence of the function s1 (tau) will not be
written explicitly. The dependence has to be considered in the calculation of the fitting
errors from the error matrix.
The time average of the gamma ray yield inside a pulse k (equation 21) is a factor
that multiplies (1+P* sk) because the parity has a negligible time dependence. One can
define also the spin sequence-average gamma ray yield the eight pulses:
T2
1
Is 
 I k (t )
T2  T1  1 t T1
(23)
The spin - sequence average of the eight pulse-average detector signals is given by:
1 7
7
Yd ( s )   Yd (k )  Yd (k )  hd I s  Vd s1s8  Ad s1
8 k 0
8
7
s8 
e
(24)
 kT /
k 0
8
(25)
The parity is calculated for each detector “d” by fitting the time averaged detector
signal Yd (k) with a fitting function derived from equation 26:
7
 
2
1
2
1
Y (k )  U d (1  Pd sk  vd s1 ( ) e
2  d

k 0
d ,k
 kT /
 wd f k 
(26)
The relative transient amplitudes are given by the equation:
vd 
Vd
hd I k
ad 
wd  ad f k
Ad
hd I k
(27)
(28)
The relative transients amplitude depend on the detector gains and the multiplicative
noise in the detectors preamplifiers.
For simplicity of the fit, it is convenient to substitute v d∙ s 1 with the fitting parameter
α d and ad∙ s 1 with the fitting constant w d:
 d  vd s1 ( )
wd  ad s1 ( )
(29)
Due to the beam intensity fluctuations from one spin sequence to the other, it is
expected that the standard deviation in the pulse-average detector signal Yk(d) over all
good Aluminum runs is bigger than the standard deviation in the ratio Yd(k)/Yd(s)
calculated at each accepted spin sequence (figure 5). As expected the contribution of
the beam intensity fluctuations cancel in the ratio Yd(k)/Yd(s) where Yd(k) is the timeaverage of detector signal at pulse k and Yd(s) the average of all 8 detector timeaverage signals Yd(k) (k=0 to 7) in a spin sequence.
Figure 5: The standard deviation (RMS) of the time average detector signal for a
pulse “k” , Yd(k), and the standard deviation of the ratio Yd(k)/Yd(s) calculated at each
spin sequence, where Yd(s) is the spin sequence average, and for each detector.
The spin sequence average detector signal (background subtracted) Yd(s) is
calculated over all Aluminum runs in figure 6.
Figure 6: The pulse average detector voltage, background subtracted, is calculated
for each detector, for all 1950 Aluminum runs.
The ratio of the pulse average signals of two opposite detectors (d1, d2) is written in
equation 30. The gains of the two detectors (hd1, hd2) do not cancel out in the ratio. This
ratio, calculated over all good spin sequences in a run, contains not only the statistical
errors due to the counting statistics but also the contribution of the electronic transients
integrated over time bins [T1, T2]. The contribution of the transient increases the root
mean square of the histogram of this ratio and introduces a systematic error in the parity
calculated from the difference over the sum of two ratios calculated (equation 32).
This ratio can be calculated for each pulse in the spin sequence and for each
detector “d1” in the first half (beam-right) of the 12 detectors in each ring.
The parities for the two opposite detectors in a ring are Pd1=P and Pd2 = -P. In
equation 30 the average beam intensity cancels in the ratio. In reality due to the
multiplicative noise at the detector output and to the transient signal the time-average of
the product hd*Ik(t) has a contribution from integration over multiplicative noise
correlated with the noise in the transient signal. The product of the time average beam
intensity and the detector gains is present in the relative transient amplitudes. The
transients do not cancel in the ratio of the average signals of opposite detectors and the
root mean square of a histogram of this ratio has contributions from both the counting
statistics and the transient signals.
 kT /1
 wd1 f k )
Y (d ) hd I k (1  Ps
1 k   d1 e
k
R1,2
 k 1  1
(29)
 kT / 2
Yk (d 2 ) hd2 I k (1  Ps
 wd2 f k )
1 k   d2 e
Let F1(k) and F2(k) be the sum of the two transients integrated over time bins [T1,T2]
in pulse k such that:
F1 (k )  d1 ekT /1  wd1 f k
(31)
F2 (k )   d2 e
 kT / 2
 wd2 f k
Neglecting the terms in the second error, equation (31) becomes:
k
R1,2
 R1,2j
R R
k
1,2
j
1,2
1
1
2 P1,2k , j  ( F1 (k )  F2 (k ))  ( F1 ( j )  F2 ( j ))
2
2
(32)
The terms F1 (k) - F2 (k) and F1 (j) –F2 (j) do not cancel in equation 32 (even if the
relative transient amplitudes for the two detectors are close) because the time average
factors (hd1*Ik) and (hd2*Ik) are different. Equation (32) is valid for each pairs of pulses
with opposite spin: k (spin up) and j (spin down). The average parity is calculated at
each spin sequence over all 16 pairs of opposite pulses and contains a factor
proportional with the transient in addition to the average parity:
Without correction, the parity for two pairs of pulses with opposite neutron spin (k,
spin up and j spin down) is given in equation 34, and the average parity over all 16 pairs
of pulses is given by equation 35.
k, j
1,2
P

k
R1,2
 R1,2j
k
2( R1,2
 R1,2j )
(33)
 P
k, j
1,2
P
k
j
16
(34)
In equation (35) the false asymmetry due to the transients increases the 24 parities
and the two asymmetries calculated from the minimum of the chi2 function for 24 pairs.
If the parity is calculated at each time bin and averaged over 16 pairs of pulses, the
transient contribution to the asymmetry is reduced. The transient terms we introduce the
ratio of the signals of two opposite detectors at the same time bin “t”:
 ( t  kT )/1
 ad1 e  t /1 f k )
Yk (t , d1 ) hd1 I k (t )(1  Ps
1 k  vd1 e
k
R1,2 (t ) 

 ( t  kT )/ 2
Yk (t , d 2 ) hd2 I k (t )(1  Ps
 ad2 e  t / 2 f k )
1 k  vd 2 e
(35)
This time the two time functions I k (t) exactly cancel in the ratio. To cancel also the
detector gains, the parity is calculated at time bin “t” from the difference divided with the
sum of the opposite detectors ratio measured at pulses (k, j) with opposite spins:
q1k  q1j
q2k  q2j (ad1 s1  ad2 s2 )( f k  f j )
 2Pk , j (t )  vd1 s1
 vd2 s2

k
R1,2
(t )  R1,2j (t )
2
2
2
k
R1,2
(t )  R1,2j (t )
(36)
The last term always cancel when the pulses k and j are both bigger than zero. Because
the relative transient amplitude (vd1, vd2 and q1, q2) and decay times are about the same
(between -4E-5 and -1E-5) the difference between the second and the third term are
even smaller. For each time bin the average P(t) is calculated over all pulses with
opposite spin:
k
vd s1
R1,2
(t )  R1,2j (t )
v s
1
2 P (t )   k
  1 (q1k  q1j )   d 2 2 (q2k  q2j )
j
16 k , j , R1,2 (t )  R1,2 (t ) k , j 2
2
k, j
(37)
Eventually the time average parity is calculated:
T2
P
 P (t )
t T1
T2  T1  1
(38)
In the average P (t) over 16 pairs of pulses with opposite spins (equation 37) the
two differences cancel out because the relative amplitudes of the transient fluctuates
from one spin sequence to the next with a Gaussian distribution with almost zero
average. This cancelation does not happen when the ratio R1, 2 is calculated from the
time average of opposite detector signals. This is why the standard deviation of the
parity calculated from the ratio of time-average detector signals is bigger than the
standard deviation of the parity from the ratio of the opposite detector signals at each
time bin (figure 7).
Figure 7: The root mean square (RMS) of the histograms build from the ratios
calculated from equation 1.
This result indicates that the multiplicative noise cancels in the ratio of the two
detectors signals measured at the same time bin, but does not cancel in the ratio of the
two time-average detector signals calculated over the same pulse (k).
The background voltage (V ped) for Aluminum runs is calculated for each detector in
figure 8. The standard error in the pedestal voltage is between 7E-6 and 1.4E-5, after
1950 Al runs.
Figure 8: The background is calculated for each detector from the dropped pulses,
and averaged over all data runs. The error of mean Vped is between 7E-6 and 1.4E-5
Volts.
The pulse average signals of the two opposite detectors in pair are Yk(d1) and Yk(d2)
and the ratio of the pulse-average beam intensity is R1,2k =Yk(d1)/Yk(d2). In the absence
of the transients, both the detector gains and the beam intensity cancel in the ratio
(R1,2k-R1,2j) divided with the sum (R1,2k+R1,2j) where “i” and “j” are two pulses with
opposite spins (equation 38). However the transient cannot be neglected in this ratio
because the parity in Aluminum is of the order 1E-7 and the relative transient amplitude
is about 2E-5. Taking in account the transient this ratio is approximately equal with twice
the parity (2*P*sk) plus the transient terms.
The contribution of the transient to the false asymmetry can be seen from the width
of the histogram (Rk - Rj )/ (Rk + Rj) calculated from the ratio of pulse-average signals of
opposite detectors (figure 9).
Figure 9: The histogram of the ratio (R1-R2)/(R1+R2) for the first pair of opposite
detectors (0 and 6) and averaged over all opposite-spin pulses is calculated over
1.08E6 Aluminum data sequences. The RMS of the histogram is 0.001446.
For each good spin sequence, the time average of the parity for the first pair of
detectors, calculated from equation 37 over the time bin interval [8 , 38], is input in the
histogram (figure 10). The root mean square of this histogram is 4E-4, about 4 times
smaller than the root mean square of the root mean square calculated from the
histogram of the ratio in equation 38.
Figure 10: The histogram of the time averaged parity for the pair of opposite
detectors 0 and 6 is calculated over all good spin sequence in Aluminum data runs. The
parity at each time bin (from 8 to 38) is calculated for this pair of detectors and for all 16
pairs of opposite spin pulses in a spin sequence and then averaged over the time bin
interval. The RMS of the parity histogram is 0.000398.
The standard deviation of pulse-average detector signal can be calculated with and
without background subtraction. The calculation of the standard deviation due to
counting statistics starts from the ratio R of pulse-average signals of two opposite
detectors. The standard deviation of the detector voltage σY calculated at the end of
each run is between 10-3 and 10-2. The standard deviation due only to the counting
statistics can be calculated from the ratio of pulse average detector signals. The square
of the relative standard deviation of the ratio of detector signals is equal with:
 Y2,k ,d
1
Yk ,d1
2

 k2,d
Yk ,d2
2
2

 R2
k
2
k
R
(39)
The two terms in the sum are about equal because the ratio does not depend on
detector gains. Therefore the counting statistics contribution to the pulse average can
be calculated at the end of the run, from the average of the ratio (σR /R) over all spin
sequences:
 k ,d 
1
Yk (d1 )  Rk

2 Rk
(40)
The beam intensity fluctuations contribute to the standard deviation sig (Yk) and is
about 8 times bigger than the standard deviation calculated from the ratio Rk .
The standard deviation of the pulse-average detector signals, due only to counting
statistics, is calculated for each detector from the ratio of pulse-average signals of a pair
of opposite detectors. At the end of the run the average of the standard deviation is
calculated over all accepted (Ns) spin sequences in the run:
 Y2( k ,r ,d )
1
Yk2,r ,d1
1 N s  Rk ,s
1

 2 
N s s 1 2 Rs 2 N s
1
 X2 ( k ,d )
2
Nr

r 1
NR
X k ,d  
r 1
Ns
7

s 1 k  0
( Rk , s  Rs ) 2
7 Rs2
(41)
1
 Y2( k ,r ,d )
Yk ,d ,r
 Y2( k ,d ,r )
(42)
NR

r 1
1
2
Y ( k ,d ,r )
Sig(X,d) is calculated for each detector and for two pulses (0 and 1) in figure 11.
Figure 11. The standard error sigma(X, d) of the weighted pulse average detector
signal X(d, k) is calculated for each detector.
If the transient terms would be zero and for the same pulse average of the number
of gamma rays in the opposite detectors, the sum over all 16 pairs of pulses (k, j) with
opposite spins would be zero:
Rk  R j
0
(43)

k , j Rk  R j
The contribution of the transient terms and the counting statistics makes the average
of the above sum not zero due to the relative transient terms.
If R1 and R2 are the ratio of opposite detectors at pulses 1 and 2, then the ratio (R1R2)/(R1+R2) calculated for all 4*4 pairs of opposite pulses is presented in figure 12.
Figure 12: The ratio of the pulse average voltages of a pair of opposite detectors
(R1) is calculated for all 16 pairs of pulses with opposite neutron spin in the spin
sequence, and over all 24 pairs of conjugate detectors.
The standard deviation of the ratio (R1-R2)/(R1+R2) calculated for each accepted
spin sequence is presented in figure 13.
Figure 13: The standard deviation of the ratio (R1-R2)/(R1+R2) is calculated for
each pair of opposite detectors, for two pulses with spin up and spin down respectively.
Figure 14: The standard deviation of the ratio R of two pulse-average signal of
opposite detectors is plotted against the run number for the first three pairs of opposite
detectors in the first ring.
The pulse average detector signal is calculated with and without pedestal
subtraction in figure 15. The detector pedestal is calculated for each detector.
Figure 15: The pulse average detector voltage with and without background
subtraction is calculated for each detector.
The error bars in this figure are the standard deviation in the pulse-average detector
signal for Aluminum runs for 1472 runs (from 63200 to 65000) and are between 1.2E-2
and 1E-3 (Volts). The error of the mean for the pulse average detector signal and is
between 10-4 and 2*10-4 (Volts).
4. The calculation of the asymmetry for each detector.
The evidence of the systematic error due to the electronic transients, ring offsets and
beam fluctuations is seen clearly when the asymmetry is calculated for each detector.
Ignoring the electronic transients the detector signal at each time bin “t” and pulse “k” is
equal with:
Yk ,d (t )  hd I k (t )(1  Psk )
(44)
At each spin sequence, the asymmetry at a time bin “t” is calculated for each of the 16
pairs of pulses with opposite spins (“k” spin up and “j” spin down), from the ratio of the
difference and the sum of the detector signals:
Yk ,d (t )  Y j ,d (t )
Yk ,d (t )  Y j ,d (t )
 Pdk , j (t )
(45)
The parity at time bin “t” is equal with the average of the 16 parities calculated at time
bin “t”:
Pdk , j (t )
16
Pd (t )  
k , j ,
(46)
The time average parity is calculated over the time bin interval [T1, T2] where T1, and
T2 depends on the switch-on time of the Spin Rotator and closing time of the chopper
respectively.
T2
Pd 
 P (t )
t T1
d
(47)
T2  T1  1
The asymmetry is calculated according to the equations (45, 46, and 47) for each of the
48 detectors. The raw asymmetries (A1, B1) and the four ring offsets (C0, C1, C2, C3)
are calculated in the end of the loop over the Aluminum data runs, from the minimization
of the chi2 function with 48 -6 degrees of liberty. In this equation the detector number is
d=r*12+j, the ring number is “r” and the detector inside the ring is “j”:
3
11
 482  
r 0 j 0
1

2
Ar , j
P
r, j
r, j
r, j
 AG
1 ud  B1Glr  Cr 
2
(48)
The detector asymmetries calculated with this procedure have a standard deviation
between 6.3E-4 and 7.6E-4. The up-down and left-right asymmetries are calculated
from the raw asymmetries by dividing with the Spin Rotator efficiency (eSR=0.97 +/-0.01)
neutron beam Polarization (Pn=0.94+/-0.02) and the coefficient of neutron spin
depolarization (Ds=0.99 +/-0.01) in the Aluminum target.
A1
 SR Pn  s
B1
ALR 
 SR Pn  s
AUD 
(49)
(50)
Figure 16: The asymmetry is calculated from equation (47) for each detector, from
all 1950 Aluminum data runs. For each detector asymmetry the error bars are the root
mean squares of the asymmetry histogram. The fitting parameters from the chi2 fit are
A1=(-0.80 +/- 0.65)E-8, B1=(-2.13 +/-0.65)E-7, C0=(1.12+/-0.08)E-6, C1=(9.97 +/-0.73)
E-7, C2=(1.18 +/- 0.07)E-6 and C3=(1.03 +/-0.08)E-6, the physical asymmetries are
Au,d= (- 0.87 +/-0.71)E-8 and Al, r=(-2.33+/-0.71)E-7
All the four positive ring offsets are due to the presence of the transient signals. In
the limits of their standard errors, the up-down and left –right asymmetries are about
equal with the two physical asymmetries calculated from the ratio of opposite detector
signals.
4. The fit of the time-average detector signal with one transient signal.
The average of the detector signal over the time bin interval [8, 38] was calculated
for each pulse “k” in all accepted spin sequences in a run [2]. At the end of each run, the
average detector signal was fitted with a simple function that has only one transient
term:
U  hd I s
(51)
The sum “s1” is given by the relation:
T2
1
s1 ( ) 
 exp(t /  )
T2  T1  1 t T1
1 T2
Is   Ik
8 t T1
(52)
(53)
  vd s1 ( )
(54)
q  exp(40 /  )
(55)
The parameter α in equation 46 was introduced to simplify the fitting function. The
relative transient amplitude vd is about two times bigger than α.
The average detector voltage was calculated from the weighted average X (d, k) of
the average detector voltage Y (d, k) at the end of each run according to equations 41
and 42. The weighted average of the time average detector signals X (d, k) and its
standard error σ X (d, k) enter in the chi2 function:
7
 
2
1
k 0
1

2
X ( d ,k )
( X d ,k  U (1  Psk   q k )) 2
(56)
The fitting parameters are U, P, α and q=exp (-40/tau) with the transient decay time
“tau” in units of time bins. The 48 parities were calculated from the minimum of eightstep chi2 function (equation 49) at the end of 1950 Aluminum data runs. To take into
account the four offsets for each ring, the 48 detector parities were fitted with a second
chi2 function that depends on 6 fitting parameters A1, B1, C0, C1, C2 and C3:
3
11
  
2
2
k 0 j 0
1

2
P(k , j )
( Pk , j  A1Gudk , j  B1Glrk , j  Ck )2
(57)
5.1 The results for Aluminum data runs
The parity (P), transient amplitude (α) and decay parameter (q) were calculated from a
fit to equation (23), for each of the 48 detectors. The time-averaged detector signals in
each pulse X (d, k) and the standard deviation due to counting statistics σ (d, k) were
calculated as described above.
The calculations were done using the code prae2.cc located in the folder
/home/balascuta/analysis/prae2.cc. The fitting parameters parity (P), 1st transient
amplitude (V), 1st transient decay time (tau) and 2nd transient amplitude (W) are
calculated from the chi2 fit at the end of the runs. The 48 parities calculated from the fit,
presented in figure 16, are shifted to negative values due to the partial correction of the
ring offsets and of the transient signal contribution to the asymmetry.
Figure 16: The Parity calculated from the fit of pulse-average detector signal
normalized to the spin-sequence average detector signal, is calculated for each
detector, using 6.77E6 accepted spin sequences (runs from 63207 to 66200).
The relative transient amplitude (v) and the product v*s1=alpha are presented in
figure 17. Because s1 is an average of the numbers exp (-t /τ) smaller than 1, α is
smaller than the relative transient amplitude “v” as expected.
Figure 17: The relative amplitude of the first transient signal (v ) and alpha =
v*s1(tau) is calculated for each detector and for 6.77E6 accepted spin sequences (1950
data runs).
The transient decay time is calculated from the fitting parameter q=exp (-1/tau) for
each detector, at the end of all the runs.
Figure 18: The decay time of the major transient is calculated for each detector
number.
Figure 19: The relative amplitude of the linear transient signal w=W(d) /Ys(d) is
calculated for 6.77E6 accepted spin sequences for Aluminum runs 63207 to 66200.
The errors in the fitting parameters were calculated from the error matrix. The
measured values are the weighted over runs of the time-average detector signal Xk (d).
The weights are the inverse variances 1/σ(X, d)2 calculated over all accepted spinsequences at the end of the runs. The 48 parities are input in the second chi2 function
to get six fitting parameters: A1, B1 proportional with the up-down and left-right
asymmetries and the four rings offsets Ck, k=0, 1, 2, 3 , determined from equation (24).
The plot of the 48 parities and the fitting function of the 48 geometry factors are
presented in figure 20.
Figure 20: The Aluminum parities calculated from the 8-pulses fitting function are used
in the second chi2 function to extract the two asymmetries A1, B1 and the four detector
rings offsets.
The parities are calculated from 1950 Al data with the of three cuts (cut1 =0.12, cut2 =
0.34 and cut3 = 0.4) named “cuts A“, in table 1.
The same data runs were analyzed with the second sets of cuts “B” (cut1=0.2, cut2=0.4
and cut3 = 0.4).
The reduced chi2 is about 3.83 when the fit of the eight time averaged detector signal is
done with one transient and with the variances calculated from counting statistics.
Table 1.
Method
Fit with one
transient(cut I)
Fit with one
transient(cut II)
Ratio opposite
detector signals
Number A1
parities E-7
B1
E-7
C0
E-7
C1
E-8
C2
E-8
C3
E-7
chi2(r)
48
-2.85
0.53
-2.35
±0.53
-2.11
±0.33
-4.73
±0.73
-4.59
±0.73
0.06
±4.53
-7.73
±5.78
-4.75
±0.58
0.07
±3.58
-6.75
±5.77
-2.24
±0.57
0.06
±3.56
-1.98
±0.72
-5.78
±0.71
3.82
±4.52
3.83
48
48
-1.71
±0.53
-0.59
±0.53
-0.69
±0.33
3.82
1.63
The significant change in the fitting parameters when the cuts are slightly
changed (rows 1 and 2) indicate that the 8 pulses chi2 fit is not a reliable way to
calculate the 48 parities. The fact that χ2 is bigger than one is due to calculation of the
variances in the average detector signals over the runs, from the ratio of average
detector signals in which the beam intensity fluctuations cancel out. The expected
reduced chi2 is one if the weights are calculated with the total standard deviation that
includes all the contributions to the parities. Considering σ the uncertainty in the final
asymmetry, one can calculate the systematic errors in the final asymmetry due to the
incomplete fitting model used and to the beam intensity fluctuations:
2
2
 tot2   syst
  Cs


2
tot
2
Cs
(58)
 12  3.83
 (3.83 1)  
2
Cs
 syst  1.68 Cs
(59)
2
syst
(60)
(61)
The calculations were done for different number of pulses in each spin sequence: all
pulses from 0 to 7 (N=8), pulses from 1 to 7 (N=7) and the last two pulses 6 and 7
(N=2). To calculate the parity for pulses 6 and 7 for each pair of detectors the ratio of
the differences and the sums are calculated at each time bin, and the average over the
time bins is saved in the 24 histograms for detector pairs.
Rd71 ,d2 (t )  Rd61 ,d2 (t )
Pd1 ,d2 (t )  7
Rd1 ,d2 (t )  Rd61 ,d2 (t )
(62)
Table2: The fitting parameters in the chi2 function with 48 geometry factors.
Number of
pulses
N
Number A1
parities E-7
B1
E-7
C0
E-7
C1
E-7
C2
E-7
C3
E-7
chi2(r)
8
48
-0.59
±0.53
-2.35
±0.53
-4.59
±0.73
-4.76
±0.58
-2.24
±0.57
-5.78
±0.71
3.82
7
48
-0.67
±0.55
-3.33
±0.55
-9.82
±0.76
-6.96
±0.59
-6.15
±0.59
-11.98
±0.74
4.66
2
48
-3.33
±0.09
-6.24
±0.91
-2.87
±0.13
-1.92
±0.10
2.38
±0.10
-3.49
±0.12
4.36
Table 3: The physical asymmetries (Aud and Alr) corrected for the average neutron
polarization for spin rotator efficiency, and neutron depolarization in the target.
Number of AUD
ALR
chi2r
pulses N E-7
E-7
8
-0.65
-2.58
3.82
±0.58
±0.58
7
-0.74
-3.65
4.66
±0.60
±0.61
2
-1.69
-9.10
4.36
±0.66
±0.66
The increase in the left right asymmetry when the number of pulses are decreased
with one show that the correlation between parity and the other fitting parameters (U, V,
q) makes this procedure unreliable for the accurate calculation of the parity. The
average of the detector signal is calculated over many spin sequences. The beam
fluctuations increase the uncertainty in the calculation of the four fitting parameters.
For comparison the same data was used for the calculation of the asymmetry from sum
and differences of detector ratio. In this case the fitting parameters and corrected
asymmetry are almost independent on the change in the number of pulses.
Table 4: The fitting parameters in the chi2 function with 24 and 48 geometry factors.
Number Numb A1
B1
C0
C1
C2
C3
chi2(r)
of
er
E-7
E-7
E-7
E-7
E-7
E-7
pulses
paritie
N
s
8
24
-0.68
-1.61 0.01
-0.45 0.72
1.44
1.303
± 0.47 ±1.16 ±0.87 ±0.85 ±0.83 ±0.85
7
48
-0.69
-2.11 0.006 0.008 0.006 0.004 1.63
± 0.33 ±0.33 ±0.453 0.358 0.356 0.452
Table 5: The physical asymmetries (Aud and Alr) corrected for the average neutron
polarization for spin rotator efficiency, and neutron depolarization in the target.
Number of
pulses N
AUD
E-7
ALR
E-7
chi2r
8
Number of
parities in the
chi2 function
24
-0.75 ±0.51
-1.76 ±1.27
1.303
7
48
-0.74 ±0.60
-2.31 ±
0.37
4.66
Because two equivalent detectors in a pair have the same up-down geometry factors
and opposite left-right geometry factors it is expected that the up-down asymmetry for
pairs of equivalent detectors is much smaller than their left-right asymmetry. The
precision of this method of calculation of the 48 parities can be seen by comparing
asymmetries calculated for pairs of equivalent and opposite detectors in figures 19 and
20. As expected the up-down asymmetry for equivalent detectors is zero Ae (u, d) = (2.44 +/-2.93) E-8 but not the left-right asymmetry Ae (l, r) = (-1.93 +/-0.71)E-8.
Figure 21: The parity for pairs of equivalent detectors and the fitting parameters: A1=
(-2.23+/-2.67) E-8 and B1 = (-1.76 +/-0.65) E-7, C=(-4.63 +/- 7.74) E-8.
Figure 22: The parity for pairs of conjugate detectors. The fitting parameters are
A1=(-8.45 +/-2.67)E-8 B1=(-1.76 +/- 0.65)E-7, C=(-4.71 +/- 7.74)E-8 and the up-down
and left right asymmetries are A(u,d) = (-9.27 +/- 2.95)E-8 and A(l,r)=(-1.93 +/-0.71)E-7
5.2. The results for the chlorine data runs
For Chlorine the 48 detector parities were calculated for 86 good runs (between runs
63098 and 63189). The parities were fitted with the same chi2 function in equation 25.
The fitting parameters and the physical asymmetries are written in the first row of the
table 6 and 7 respectively. The physical asymmetries are calculated with an average
neutron polarization 0.94+/-0.02, spin flipper efficiency 0.98 +/-0.02 and neutron
depolarization due to inelastic scattering in the target 0.99 +/-0.01.
These results can be compared with the fitting parameters and the physical
asymmetries calculated from the ratio of the difference and the sum of opposite detector
signals, presented in the second rows of the tables 2 and 3.
Table 6. The six fitting parameters are calculated from the minimization of the chi2
function with 48 detector parities from 86 Chlorine data runs. For each detector the
parity is calculated from the minimization of eight pulses chi2 function with one transient
and from of a chi2 function (without transient terms) and with 48 parities calculated from
the ratio of the difference and the sum of conjugate detector ratios.
Method
Number A1
B1
C0
C1
C2
C3
chi2
parities
E-5
E-7
E-7
E-6
E-7
E-6
(r)
Fit with one
48
-2.05
-3.09
-1.57
-8.62
-1.46
-1.64 1.56
transient
±0.03
2.90
±0.42 ±2.95 ±0.32 ±0.37
Ratio of opp. 48
-1.99
-2.99
0.49
1.72
det. Signals
±0.02
±1.71 ±1.04
Table 7. The physical asymmetries corrected for neutron beam polarization.
Method
Number Aud
Alr
parities
E-5
E-7
Fit with one
48
-2.25
-3.38
transient
±0.07
±3.18
Ratio of opp. 48
-2.19
-3.25
Det. Signals
±0.07
±1.86
The two reduced chi2 values suggest a small contribution of the systematic errors
due to the transient signal. The absolute value of the systematic errors can be
calculated from the difference Aud(1)-Aud(2) and Alr(1)-Alr(2).
The up-down and the left-right asymmetries calculated from the eight pulses chi2 fit
(figure 23) is Aud = (-2.39 +/- 0.08) E-5 Alr = (-6.502 +/- 4.514)E-7.
In figure 24, the parities for opposite detectors pairs are fitted with a chi2 function
that depends on 24 geometry factors. The three fitting parameters (A1, B1, C0)
calculated from the minimization of the transient chi2 function and the two physical
asymmetries (Aud, Alr) are written in the first row of table 7. The results can be
compared with the three fitting parameters and the two asymmetries calculated from the
geometry mean of the product of opposite detectors signals. In this case the reduced
chi2=1.5. The differences in the up-down and left-right asymmetries calculated with and
without correction are 0.013E-5 and 0.198E-6 respectively.
Figure 23: The 48 detector parities for the Chlorine target are calculated from the fit
of the eight-pulses chi2 function, from 82 Chlorine data runs. The fit in this figure is
calculated from the second chi2 function with 48 geometry factors.
Figure 24: The 24 corrected parities for Chlorine are calculated from the 8-pulses
chi2 fit. The 3 parameters fit to the 24 parities are calculated from the minimization of
the chi2 function (equation 24). The three fitting parameters are A1 = (-1.978 +/-0.075)
E-5 and B1= (-1.52 +/-1.82) E-6, C = (-1.114 +/-1.047) E-6, reduced chi2=1.045.
The big offset in the fit of the 24 pairs (figure 24) proves that the systematic error due to
the transient was not completely removed by the 8-pulses chi2 fit.
Table 8: The parities for the 24 detector pairs are calculated from the 86 Chlorine
runs using the geometry mean of conjugate detector ratio and the minimization of the
chi2 function with transients.
Method
Fit with one
transient
Ratio diff/sum
Number
parities
24
24
A1
E-5
-2.03
±0.03
-2.02
±0.01
B1
E-6
-1.87
±0.69
-1.69
±0.30
C0
E-6
-1.97
±0.83
-0.18
±0.04
Aud
E-6
-2.22
±0.19
-.2.21
±0.09
Alr
E-6
-2.05
±0.77
1.85
±0.34
chi2
reduced
1.06
1.5
The calculation of the parity from the fit with one transient does not remove the
systematic errors due to the transient because the offset constant C0 increases to 1.9E-6. In the next section, the correction is done with a fitting function that is a sum of
the major and minor transient signals.
6. The fit of the time-average detector signal with two transients.
In this section, both the major and the minor transients are considered in the 8-pulses
chi2 function:
7
1
k 0
 X2 ( d ,k )
 
2
1
X
d ,k
 U (1  Psk  vs1q140 k  ws1 f k ) 
2
(63)
In the above equation qk =q140k and the factors s1(tau), sk , fk were explained in
equations 4, 5, 6 and 7. The fitting parameters are U, P, v, w and q1=exp (-1/tau). The
average detector voltage X (d, k) was calculated for each of the eight pulses in spinsequence from time bin 8 to 38. All four fitting parameters depend on the detector
number “d” that is not written explicitly.
To calculate the up-down and left-right asymmetries the 48 parities (Pd) are using in the
second chi2 function (equation 24) that has six fitting parameters. The fitting
parameters and the two asymmetries calculated from 48 parities of the eight-pulses chi2
fit are presented in table 9 and 10.
Table 9. The six fitting parameters calculated by minimization of the chi2 function with
48 parities with and without transient correction.
Method
Number A1
B1
C0
C1
C2
C3
parities E-5
E-7
E-6
E-6
E-6
E-7
Fit with two
48
-1.98
-6.27 -1.99 -1.58 -1.77 -2.35
transients
±0.04 ±3.76 ±0.35 ±0.46 ±0.51 ±0.45
Ratio of diff.
48
-2.00
-2.97 0.054 0.065 0.33
0.47
and sum
±0.02
±1.71 ±0.25 ±0.19 ±1.87 ±2.20
Table 10. The up-down and left-right Chlorine asymmetries corrected for the neutron
spin polarization and SR efficiency are calculated from A1 and B1 respectively.
Method
Number Aud
Alr
chi2
q
parities E-5
E-7
Fit with two
48
-2.17
-6.88 1.11
0.287
transients
±0.07
±4.13
Ratio of diff.
48
-2.19
-3.25 0.93
0.607
and sum
±0.07
±2.65
The fitting parameters and the two asymmetries calculated from an eight steps chi2
function with three fitting parameters are presented in tables 11 and 12.
Table 11: The fitting parameters for Chlorine calculated by minimization of the
chi2 function with 48 parities and three fitting parameters (A1, B1, C0) are calculated
from the eight steps chi2 fit and from the ratio of opposite detectors signals.
Method
Number A1
B1
C0
parities E-5
E-7
E-6
Fit with two
48
-1.98
-6.27 -1.99
transients
±0.04 ±3.76 ±0.35
Ratio of diff.
48
-2.00
-2.97 0.054
and sum
±0.02
±1.71 ±0.25
Table 12: The up-down and left-right physical asymmetries for Chlorine are calculated
by dividing the fitting parameters A1 and B1 (in table 11) with the neutron polarization
and spin rotator efficiency.
Method
Number Aud
Alr
chi2
q
parities E-5
E-7
Fit with two
48
-2.17
-6.88 1.11
0.287
transients
±0.07
±4.13
Ratio of diff.
48
-2.19
-3.25 0.93
0.607
and sum
±0.07
±2.65
The four ring offsets are all negative and between -2E-6 and -3E-7 when the 48 parities
are calculated from the 8-pulses chi2 fit. The offsets are all zero in the limits of the
uncertainty when the 48 parities are calculated from ratio of opposite detector signals.
The two asymmetries calculated from geometry mean of the ratio of conjugate detectors
and from the 8-step chi2 fit are compared in table 13 and 14.
Table 13
Method
Fit with two
transients
Geometry
mean of ratios
Table 14
Method
Fit with two
transients
Geometry
mean of ratios
Number A1
parities E-5
24
-1.98
±0.04
24
-2.00
±0.02
B1
E-6
-1.52
±0.91
-1.64
±0.58
C
E-7
-5.57
±5.23
-8.94
±3.55
Number Aud
parities E-5
24
-2.17
±0.09
24
-2.19
±0.09
Alr
E-6
-1.67
±1.00
-1.80
±0.65
chi2
q
1.05
0.403
1.54
0.053
The differences between the two asymmetries calculated with the two methods are
0.02E-5 and 0.13E-6 respectively. In both cases the left-right asymmetry calculated for
pairs of conjugate detectors is 8 times bigger than the left-right asymmetries calculated
for 48 detector parities.
7. The fit from the sum of time-average signals of opposite detectors.
In this section the fit is done for each time bin of the sum of detector signals.
Because of the ring offset, the transient amplitudes and decay times are about the same
for the detectors in a pair. The transient amplitude and decay time is extracted from the
sum opposite detector signals at each time. This allow for a precise determination of the
parity. The calculations were done first for the sum of the time-average detector
signals. In this section the time dependence of the detector signals is given by the
equations:
Y0,k (t )
 1  P0 sk 
h0 I k (t )
Y6,k (t )
 1  P6 sk 
h6 I k (t )
V0
W0
e (t  kT )/ 0 
f k (t  20)
h0 I k (t )
h0 I k (t )
(64)
V6
W6
e (t  kT )/ 6 
f k (t  20)
h6 I k (t )
h6 I k (t )
(65)
To get a fitting model for the eight pulse average signals, one has to substitute h0*Ik (t)
with U0 (t) and h6*Ik (t) with U6 (t) in the fitting function. In the first approximation U0 and
U6 are the spin-sequence average of the two opposite detectors signals. Because of
their opposite geometry factors the parities are equal and opposite:
P6 = - P0.
To obtain a fitting function that depends only on the relative transient amplitudes
and the decay time of the major transient, the above equations are added at each time
bin and the detector yield for each time bin Ik(t) is replaced with the spin sequence
averaged detector yield at time bin t:
7
I
I (t ) 
k 0
k
(t )
8
(66)
The average detector yield at a time bin can be related with the average of the eight
detector signals at the same time bin, in each spin sequence.
7
e
Y0 (t )  h0 I (t )  V0e t / 0  k 0
8
7
e
Y6 (t )  h6 I (t )  V6e t / 6  k 0
7
s0 
e
k 0
8
s8 
 W0 (t  20)
7
8
(67)
 W6 (t  20)
7
8
(68)
 kT / 6
8
7
 kT / 0
 kT / 0
e
 kT / 8
k 0
8
(69)
The spin sequence averaged detector signals over the eight pulses are equal with:
39
Y0, s 
 Y0 (t )
t 0
40
39
Y6, s 
 Y (t )
t 0
6
40
(70)
With the approximation v0=v6=v, w0=w6=w and tau0=tau6=tau, the above two equations
are added at each time bin, and the fitting parameters are the transient relative
amplitude v, w, and decay time “tau”.
With the above notations for the sums, and considering the relative transient amplitudes
v0 = V0 / Y0(s) and v6= V6/ Y6(s) the two average detector yields at time “t” are given by:
Y
Y

7
h0 I (t )  Y0 (t ) 1  v0 0, s et / 0  s0  w0 0, s (t  20) 
Y0 (t )
Y0 (t )
8

(71)
Y
Y

7
h6 I (t )  Y6 (t ) 1  v6 6, s et / 6 s6  w6 6, s (t  20) 
Y6 (t )
Y6 (t )
8

(72)
In the first approximation, the quantities h0I(t) and h6I(t) in the above equation are
replaced with Y0(t) and Y6(t).The mixed products (v0,6*p0,6, w0,6*p0,6) are negligible
because v0 and v6 are about 3* 10-5 while p0 and p6 are smaller than 2*10-7:
Y0,k (t )
Y0 (t )
Y6,k (t )
Y6 (t )
 1  P0 sk  v0
 1  P6 sk  v6
Y0, s
Y0 (t )
Y6, s
Y6 (t )
e (t  kT )/ 0  w0
e (t  kT )/ 6  w6
Y0, s
Y0 (t )
Y6, s
Y6 (t )
f k (t  20)
(73)
f k (t  20)
(74)
With the approximation v0=v6=v, w0=w6=w and tau0=tau6=tau, the above two equations
are added at the same time bin, and the fitting parameters are the transient relative
amplitude v, w, and decay time “tau”.
The parity cancels in the sum of the above equations:
Y0,k (t ) Y6,k (t )
Y 
Y 
 Y
 Y

 1  v  0, s  6, s  e(t kT )/  w  0, s  6, s  f k (t  20)
2Y0 (t ) 2Y6 (t )
 2Y0 (t ) 2Y6 (t ) 
 2Y0 (t ) 2Y6 (t ) 
(75)
The average of the eight detector signals at the time bin “t” in the 8 pulses Y0(t) and
Y6(t) and the global spin sequence average detector signals Y0s, Y6s are calculated for
each spin sequence. Let’s introduce the notation for the quantity:
G0.6 (t ) 
Y0, s
2Y0 (t )

Y6, s
2Y6 (t )
(76)
This sum of ratios can be calculated for each spin sequence. There are (T2-T1+1) such
constants in the chi2 function that is a sum over time bins from T1 to T2 and pulse
numbers (k), equation 66. The measured quantity is:
R0,6 (k , t ) 
Y0,k (t )
2Y0 (t )

Y6,k (t )
2Y6 (t )
1
(77)
T2
12    R0,6 (k , t )  vG0,6 (t )e  (t  kT )/  wG0,6 (t ) f k (t  20) 
7
2
k  0 t T1
(78)
The fitting parameters v, w and tau are calculated from the minimum of the chi2 function
at each accepted spin sequence. After this fit, the parity is calculated separately for the
two detectors, at each time bin inside the interval [T1, T2] according to equations:
Y0,k (t )
Y0 (t )
Y6,k (t )
Y6 (t )
1  v
1  v
Y0, s
Y0 (t )
Y6, s
Y6 (t )
e (t  kT )/  w
e (t  kT )/  w
Y0, s
Y0 (t )
Y6, s
Y6 (t )
f k (t  20)  P0 (t ) sk
(79)
f k (t  20)  P6 (t ) sk
(80)
An average parity is calculated at each time bin and over pulses with spin up and spins
down. For each detector the time average parity is calculated over the time bins [T1, T2]
and written in histograms.
Y
Y
 Y0,k (t )

 1  v 0, s e (t  kT )/  w 0, s f k (t  20)   P0

Y0 (t )
Y0 (t )
k
t
 Y0 (t )

(81)
Y
Y
 Y (t )

sk  6,k  1  v 6, s e (t  kT )/  w 6, s f k (t  20)   P6

Y6 (t )
Y6 (t )
k
t
 Y6 (t )

 s
k
(82)
The average parity and its standard deviations are read from the 48 histograms and
saved in two arrays at the end of the 1950 runs. A second χ2 function with 48 geometry
factors is used to fit the data:
3
11
  
2
2
k 0 j 0
1

2
P(k , j )
( Pk , j  A1Gudk , j  B1Glrk , j  Ck )2
(83)
The 48 parities calculated from equations 77 and 78 for all 24 pairs of detectors are
presented in figure 25. The cuts used in this calculations are CUT1=0.12, CUT2=0.34
and CUT3=0.4.
Figure 25: The 48 detector parities P corrected for the transient at each time bin value
are read from the 48 histograms at the end of the runs. The parities are calculated at
each accepted spin sequence.
The error bars for each of the 48 parities are between 2E-7 and 3E-7, about two times
bigger than the error bars of the 24 uncorrected parities calculated from the ratio of
detector signals.
Two of the fitting parameters (A1, B1) are related with the two physical asymmetries are
calculated for different sets of cuts in table 17. The change in the two corrected
asymmetries with the change in the cuts is in the limit of their errors.
Table 17: The six fitting parameters A1,B1,C0,C1,C2,C3 and the corrected asymmetries
are calculated from the same 1950 Al data runs, but with different cuts.
Cuts
A1
B1
C0
C1
C2
C3
Aud
Alr
chi2
1E-7
1E-7
1E-7
1E-7
1E-7
1E-7
1E-7
1E-7
cut1=0.11 -0.82
-2.03 3.61
3.93
5.0
2.94
-0.90 -2.23
0.75
cut2=0.32 ±0.64
±0.64 ±0.81 ±0.72 ±0.71 ±0.81 ±0.69 ±0.69
cut3=0.38
cut1=0.12 -0.83
-2.09 2.61
3.04
4.58
1.73
-0.91 -2.29
0.795
cut2=0.2 ±0.64
±0.64 ±0.81 ±0.72 ±0.71 ±0.81 ±0.70 ±0.70
cut3=0.4
cut1=0.2 -0.76
-2.02 -2.05 -1.42 0.06
-2.75 -0.84 -2.2
0.81
cut2=0.4 ±0.64
±0.64 0.81
±0.71 ±0.71 ±0.81 ±0.70 ±0.70
cut3=0.4
The corrected parities for 24 detector pairs are calculated from the corrected 48
average parities according to the formula:
P0,6  ( P0  P6 ) / 2
The 24 parities were calculated from the chi2 function with 24 geometry factors:

23
2
2, p

k 0
1

2
P,k
( Pk  A1Gudk  B1Glrk  C )2
(84)
The chi2 function in the above table is smaller than one because the variance in
equation (80) is a sum of the variances due to both the transient and beam fluctuations.
The up-down and left –right asymmetries calculated from the 24 parities for pairs of
opposite detectors are presented in table 18. The uncertainty in the left-right asymmetry
increases because there are only 24 points in this case.
Table 18: The fitting parameters and the two corrected asymmetries are
calculated from the minimization of the chi2 function with 24 pairs of geometry factors.
Cuts
A1
B1
C
Aud
Alr
chi2
1E-7
1E-7
1E-7
1E-7
1E-7
cut1=0.12 -0.82
-1.44 -0.43 -0.90 -1.57
0.81
cut2=0.2 ±0.64
±1.56 ±0.93 ±0.70 ±1.72
cut3=0.4
cut1=0.11 -0.81
-1.39 0.41
-0.90 -1.53
0.79
cut2=0.32 ±0.64
±1.56 ±0.93 ±0.70 ±1.72
cut3=0.38
The results can be compared with the asymmetries calculated from the ratio of
detector signals without transient correction. The asymmetries are equal in the limit of
the experimental errors. The left-right asymmetry calculated from 24 detector pairs is
smaller than that calculated from 48 detector parities. The errors in the calculation of the
24 parities from the ratio of the detector signals are smaller than the errors of the 48
parities calculated with correction. The agreements between the two results indicate
that the systematic errors in the asymmetries calculated without correction are small.
Table 19
Cuts
A1
1E-7
cut1=0.12 -0.75
cut2=0.2 0.47
cut3=0.4
cut1=0.11 -0.75
cut2=0.32 0.47
cut3=0.38
B1
1E-7
-1.38
1.14
C0
1E-7
-0.48
0.68
Aud
1E-7
-0.82
0.52
Alr
1E-7
-1.51
1.26
chi2
-1.42
1.14
0.49
0.68
-0.82
0.51
-1.56
1.25
1.45
1.45
To best test for this correction method is provided by the asymmetry calculated
for the last two pulses (6, 7) in the spin sequence. Without transient correction this
asymmetry is about 4 times bigger than the asymmetry calculated from the average
over all 16 pulses with opposite spin. The increase in the parity from the first (ring 0) to
the last one (ring 4) could be a systematic effect due to the transient. In this figure the
parity P(t) was calculated first at each time bin for the pair of two pulses 7 (up) and 6
(down) according to the equation:
Yd1 ,7 (t )
Pd1 ,d2 (t ) 

Yd1 ,6 (t )
1 Yd2 ,7 (t ) Yd2 ,6 (t )
2 Yd1 ,6 (t ) Yd1, j (t )

Yd2 ,6 (t ) Yd2 , j (t )
(85)
Next the parity was averaged over all time bins between 8 and 38. From the last chi2 fit
with 48 geometry factors the fitting parameters are A1=(-1.69+/-0.66)E-7 ;
B1= (-9.10 +/- 0.66)E-7 ; C0=(2.36 +/-90.72)E-9 ;C1=(2.89 +/-71.68)E-9
C2=(2.50 +/-71.34)E-9 ; C3=(1.38 +/-90.55)E-9. Taking into account the average beam
polarization and the spin flipper efficiency, the two physical asymmetries are Aud=(-1.85
+/- 0.73)E-7; Alr=(-9.98 +/- 0.78)E-7. The corrected parity for the 48 detectors is
presented in figure 27.
Figure 26. The parity, not corrected for the transient, is calculated from the last pulses
(6, 7) in the spin sequence, from the ratio of detector signals, at each time bins in the
interval [8, 38].
Figure 27: The parity is calculated for each detector from the minimization of a chi2
function for each time bin (7< t< 39) and each pulse number (6, 7). The six fitting
parameters are: A1=(-8.33 +/6.39)E-8; B1=(-2.09+/-0.64)E-7, C0=(2.61 +/-0.81)E-7,
C1=(3.04+/-0.71)E-7, C2=(4.59+/-0.71)E-7, C3=(1.63 +/-0.81)E-7. The up-down and leftright are Aud = (-9.13 +/-7.02)E-8 and Alr = (-2.29 +/-0.70)E-7.
This method of correction of false asymmetry is not sensitive to the cuts and can be
used to correct for the false asymmetry in each of the 16 pairs of pulses with opposite
spins.
8. Asymmetry calculated from the ratio of opposite detector signals, at each time bin.
This section presents a method of parity calculation for each of the 24 pairs of
opposite detectors The parity for each of the 24 pairs of opposite detectors is calculated
from the ratio of the detector signals at each time bin in [T1, T2] and for 16 pairs of
opposite pulses and averaged over all 16 pairs. For the same spin sequence, the time
average parity is calculated over the time bin intervals [T 1, T2].The contributions of the
transients and of the beam fluctuations cancel in the time-averaged parity over all eight
pulses in a spin sequence. For a pulse “k” with spin up, at time bin “t” relative to the start
of the pulse one can write the ratio of the signals of the two opposite detectors: 0 and 6.
h0 I k (t )(1  P0 sk )  V0e  (t  kT )/ 0  W0 f k (t  20)

Y6,k (t ) h6 I k (t )(1  P6 sk )  V6e  (t  kT )/ 6  W6 f k (t  20)
Y0,k (t )
(86)
The factor f k is zero for the first pulse in the sequence (k=0) and is one for k>0. The
integrated gamma-ray flux per time bin I k (t) is a common factor in the above equation.
To cancel the detector gains h0 and h6 one needs to multiply the above ratio with the
inverse ratio of the detector signals at time bin “t” in pulse j with spin down.
Y6, j (t )
Y0, j (t )

h6 I j (t )(1  P6 s j )  V6e (t  jT )/ 6  W6 f j (t  20)
h0 I j (t )(1  P0 s j )  V0e (t  jT )/ 0  W0 f j (t  20)
(87)
To simplify the notations, let’s define the time dependent functions:
V
V
W (t  20)
W (t  20)
; w0 (t )  0
v6 (t )  6 ; v0 (t )  0 ; w6 (t )  6
Y6 (t )
Y0 (t )
Y6 (t )
Y0 (t )
(88)
In the above equations the average detector signal at a time bin, over the eight
pulses is given by:
1 7
Yd (t )   Yd (k , t ) ; d=0 or 6
(89)
8 k 0
By separating the beam related terms on the right side of the equations 86 and 87,
one gets a simple relation for the ratio of opposite detector signals at pulse k and time
bin “t”:
h0 I k (t )(1  P0 sk ) (1  v6 (t )e  (t  kT )/ 6  w6 (t )e  t / 6 )


Y6,k (t ) h6 I k (t )(1  P6 sk ) (1  v0 (t )e  (t  kT )/ 0  w0 (t )e  t / 0 )
Y0,k (t )
(90)
Let’s introduce the notations for the product of the ratios of opposite detector ratio:
Y0,k (t ) Y6, j (t )

Y6,k (t ) Y0, j (t )
k, j
R0,6
(t ) 
1  v6 (t )e  (t  kT )/ 6  w6e  t / 6 1  v0 (t )e  (t  kT )/ 0  w0e  t / 0
Q (t ) 

1  v0 (t )e  (t  kT )/ 0  w0e  t / 0 1  v6 (t )e  (t  kT )/ 6  w6e  t / 6
(91)
k, j
0,6
k, j
R0,6
(t ) 2 
(1  p0 sk ) (1  p6 s j ) k , j 2

Q0,6 (t )
(1  p6 sk ) (1  p0 s j )
(92)
(93)
The two detectors are opposite so their parities are p6= - p0. If pulses k and j have
neutron spin up and down, then sj = - sk = -1 and sk = 1. With these notations the
equation for parity becomes:
k, j
R0,6
(t ) 
(1  p0 (t ) sk ) k , j
 Q0,6 (t )
(1  p0 (t ) sk )
R (t )  Q (t )
k, j
0,6
k, j
0,6
k, j
0,6
k, j
0,6
R (t )  Q (t )
(94)
k, j
 p0,6
(t )
(95)
There are 16 pairs of pulses with spin up and down. The parity at time bin “t” is
calculated from the average over the 16 pairs of pulses:
k, j
k, j
R0,6
(t )  Q0,6
(t )
1
  k , j (t )  Qk , j (t )  pc (t )
16 k 1,2,4,7 j 0,3,5,6 R0,6
0,6
(96)
The average parity over the time bins interval [T1, T2] can be calculated at each
accepted spin sequence using the formula:
T2
1
pc 
 pc (t )
T2  T1  1 t T1
(97)
There are 24 corrected parities Pc calculated according to the above equation. The
transient correction of the parity can be done at the end of the loop over data runs,
using the relative transients amplitudes v(0,6), w(0,6) and decay time tau(0,6) calculated
from the minimization of the chi2 function.
Equation 91 can be written as a sum of a parity term uncorrected for the transient
and a correction because the factor (1-Q (t)) is smaller than 10-4.
k, j
k, j
R0,6
(t )  1  (1  Q0,6
(t ))
1
  k , j (t )  1  (1  Qk , j (t ))  pc (t )
16 k 1,2,4,7 j 0,3,5,6 R0,6
0,6
(98)
The parity not corrected for the transient is given by:
p (t ) 
k, j
k, j
R0,6
(t )  1
k, j
R0,6
(t )  1
(99)
The corrected parity is equal with the sum of uncorrected parity and the correction
terms up to the second order:
p (t )  p (t )  (1  Q (t ))
k, j
c
k, j
k, j
0,6
k, j
2 R0,6
k, j
( R0,6
 1)2
 (1  Q (t ))
k, j
0,6
2
k, j
2 R0,6
k, j
( R0,6
 1)3
(100)
The average over pairs of opposite detector pulses is given by:
pc (t )  p(t )   (1  Q (t ))
k, j
0,6
k, j
k, j
2 R0,6
k, j
( R0,6
 1)2
(101)
The correction factor of the time averaged parity is calculated at the end of the runs
for all 24 pairs of opposite detectors:
k, j
2 R0,6
1
k, j
pc  p 
 (1  Q0,6 (t )) ( Rk , j  1)2
T2  T1  1 t k , j
0,6
(102)
The standard deviation of the parity calculated from the histogram of a pair of
detectors is between 3 * 10-4 and 4.2*10-4. The corner detectors (ex: 1, 4) have
bigger standard deviation than the other detectors.
Figure 28: The standard Deviation of the parity is calculated from the product of
ratios of opposite detector signals for time bins from 8 to 38.
From the Aluminum data runs from 63200 to 66200, the raw asymmetry is presented
in the next figure. The three fitting parameters are A1 = (-7.12+/-3.32) E-8, B1= (-2.14+/0.33)E-7 and offset constant consistent with zero C=(0.0634 +/- 1.98)E-8.
With a Spin Rotator efficiency 98%, average neutron polarization 94% and spin flip
depolarization in the target 99%, the physical asymmetries are AUD=(-7.81+/-3.65)E-8
and ALR=(-2.33 +/-0.37)E-8.
Figure 29: The asymmetry for Aluminum calculated for each detector number, from
the ratio of opposite detector signals.
One can notice that the up-down asymmetry seems to be bigger for the 12 detectors
the last ring. The parity calculated from the model fit of single detector pulse-average
signal is shifted to a negative asymmetry (about -4E-7) while the raw parity calculated
from the detector ratio is not shifted. The difference between the two sets of pair
asymmetries is due to the systematic error of the transient signal and the asymmetry of
the beam intensity fluctuations that are not removed in the fit of single detector signals.
The two asymmetries calculated in figure 29 are not corrected for the transient but
are about the same with the asymmetry corrected for the transient.
Instead of taking the square root, one can calculated the parity at each time bin
according to the following equation and for all 16 pairs of pulses with opposite neutron
spin. The time average parity (between time bins 8 and 38) is calculated for each
accepted spin sequence and written to a histogram.
Y0,k (t ) Y0, j (t )

Y
(
t
)
Y6, j (t )
1
6, k
2 P0,6 (t )  
16 k  j  Y0,k (t ) Y0, j (t )

Y6,k (t ) Y6, j (t )
P
(103)
T2
1
 P0,6 (t )
T2  T1  1 t T1
(104)
There are 48 time average parities (equation 40) for each of the 48 detectors in the
experiment.
The asymmetry calculated from the ratio of opposite pairs of detector signals at each
time bin from 8 to 38, is calculated in the end of 1950 runs. The fitting parameters are
A1=(-7.52 +/-3.32)E-8 , B1=(-2.17 +/-0.33)E-7 and C=(6.51 +/-198.82)E-10. The
reduced chi2 is 1.473, and goodness of the fit is 0.0266. The up-down and left-right
asymmetries are (-8.24 +/- 3.65) E-8 and (-2.36+/-0.37) E-7.
9. Conclusion.
The calculation of the relative transient amplitude and decay rate can be done with
good precision by fitting each detector signal with a sum of three terms: the gamma ray
detector yield, the exponential transient and the linear transient. The parity calculated
from the fit contains the contribution of the transient. This means that the beam intensity
fluctuations can mix the false asymmetry with the real asymmetry.
The most precise method for the calculation of the asymmetry is the product of
opposite detector ratios. The contribution of the transient in the parity computed from
the product of opposite detector ratio, partially cancel. A complete correction of the false
asymmetry can be done using the fitting parameters calculated using the single detector
fit.
10. Reference
[1] Mark McCrea: “A Study of the Transients in Detector Sum/Difference Signals”,
March 11, 2012.
[2] /home/balascuta/ basestar/ prae2.cc (fit with one transient)
[3] /home/balascuta/basestar/prae3.cc (fit with two transients)
[4] /home/balascuta/basestar/prae5.cc (sum of opposite detector signals)