• The index of a section (i) and a coordinate in the section (s) are calculated according to thevposition whereγ-ray is reconstructed.
• The vectorshγ anddγ are calculated by interpolatingi-th andi+1-thhanddaccording to sfor the case where γ-ray is in section 1-23. In section 0 or 24, 0-th or 24-th vectors are used.
• The point P is calculated by interpolating points ofi-th andi+1-th hole on the US-side arch. The ratio of the interpolation is calculated froms.
• The point Q is calculated by adding a vector to the point P, the vector is parallel tohγ and the length of the vector is calculated from theuposition of theγ-ray.
• The point Q is projected onto the surface of the inner window with respect to the axis of the fitted cylinder face; the point is named point R.
• The vectorw·dγis added to the point R. The point is the corrected reconstructed position of theγ-ray.
In the procedure of position correction, thermal expansions of the detector parts are con-sidered. The main structure of the detector cryostat is made of stainless steel, SUS316L (α∼16×10−6/K), and PMT holders are made of PEEK4 resin (α ∼50×10−6/K).
effect of the correction Average shift of γ-ray position over acceptance is (-0.6, -1.9, 2.1) (mm) in (x,y,z). The shift is LXe detector shift with respect to the COBRA frame. Additional zshift to connect the COBRA frame to drift chamber frame is determined from measurements which use both LXe detector and drift chamber, alignment run by cosmic ray and AIF, result is +2.0±0.4 mm for shift inzcoordinate.
Figure 4.15 shows the effect by the correction. Shift toward negative y is consistent with thermal shrink, but also consistent with the zero shift with other measurement within uncertainty.
The effect of thermal shrink of PMT holder was not considered in previous analysis but is included in the correction, although the thermal effect cannot be confirmed in any calibration. Average difference in stereo angle is∼4 mrad, it is not negligible compared with the angular resolution ofγ-ray.
u [cm]
−20 −10 0 10 20
v [cm]
−80
−60
−40
−20 0 20 40 60 80
(a) Shift inuandvcoordinate,wis fixed at 1 cm. The arrow shows the direction and distance of correction at the position. The length is magnified by factor 10.
stereo angle shift [mrad]
0 1 2 3 4 5 6 7 8 9 10
0 20 40 60 80 100 120 140 160 180
(b) Difference of reconstructed points with previous and current methods in stereo angle.
Figure 4.15: Result of the LXe detector alignment
4.2.2 Time calibration
The timing offset between channels is caused by the difference of the amplifier characteristics, cable length and electronics at down-stream. The time offset information is extracted from the Michael and cosmic ray data. The leading edge timings after z coordinate correction and the track timing fitting are plotted, and the timing for the channel is found by fitting the leading edge timing distribution.
4.2.3 Alignment
The relative position of each DCH module is quite important as a high precision tracker. We have two methods to measure the alignment of the drift chamber modules, an optical survey with laser-tracker, and a method by reconstruction of the tracks.
4.2.3.1 Optical method
For the optical survey, a drift chamber module is equipped with cross-hair markers and corner cubes at both ends of the module, as shown in Fig. 4.17. The position of the wire and cathode against the DCH module is measured when it was produced. The position is related to a
[cm]
anode
z
-40 -30 -20 -10 0 10 20 30 40
hood A
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Cell #57
Figure 4.16: zanodeversus asymmetry found in cathode. The wave length of the sinusoidal curve is 5 cm of the cathode pattern.
alignment pin, which is also measured in survey. In each year before data taking, measurement with theodolite5is performed for cross-hairs and pins, target markers are also surveyed (Sec. 4.4).
The accuracy is expected to be 0.2-0.3 mm in x and y and 1.5/2.5 mm in z for DS/US side of the modules. The difference in US and DS is because the theodolite is located in DS side of the detectors. From year 2011, the optical survey is updated with corner cube and laser tracker6.
The accuracy of the method is determined from the uncertainty of the alignment of the corner cube and expected to be 0.15-0.25 mm in x, y and z, although the spacial resolution of the tracker is 0.015 mm for each axis.
ĐŽƌŶĞƌĐƵďĞ
ĐƌŽƐƐŵĂƌŬĞƌ
Figure 4.17: View from down-stream side of drift chambers. Corner cubes and cross-hair markers are seen.
4.2.3.2 Software method
The relative positions of the modules were also measured with two independent methods of cosmic ray and Michel positrons.
The cosmic ray counter (CRC) was installed for this calibration, to trigger events in which the cosmic rays pass through the DCH modules. The CRC consists of 10 plastic scintillator bars with PMT read out, and are mounted on the outer surface of COBRA magnet as illustrated in Fig. 4.18. The magnet is turned off during the cosmic ray dedicated runs. We processed the cosmic ray data by the algorithm called "Millipede" [85]. It is a kind of linear fitting with a lot of free parameters in a matrix of a large number of laws and columns. The optimal displacement vector for each module is calculated with the algorithm, by minimizing the residuals with respect to straight track of cosmic rays. The parameters are determined within 150µm accuracy by the method.
Z
Z ,
ŵŽĚƵůĞƐ ĐŽƐŵŝĐƌĂLJ
LJ dž nj
Figure 4.18: CRC mounted on the COBRA magnet. The figure is shown in cross-sectional view of a plane perpendicular to the beam axis.
The other method is an iterative algorithm to optimize radial and longitudinal displacement of each module using Michel positron tracks. This method is conducted by minimizing the residual of radial and longitudinal directions between the reconstructed track and hits in each module.
The process repeated iteratively, updating positron tracks with corrected modules position. This method is confirmed by checking consistency with Mott calibration data (Sec. 4.2.4).
4.2.4 Mott calibration
Mott scattering Mott calibration [86] takes a use of Mott scattering, which is elastic scattering of a relativistic Dirac particle by point like nucleus. We make a use of Mott data to evaluate the positron track resolution, and alignment of chamber.
When the nucleus satisfyZ 137, the scattering cross section is approximated as [87], dσ
dΩ = Z e2 2E
!2cos2(θ/2)
sin4(θ/2) (4.10)
whereEis the initial energy of the particle (positron). The energy of the positron after scattering is given as,
E0= E 1− E Mc2
1−cosθ
1+(E/Mc2)(1−cosθ)
!
. (4.11)
operation In operation, we select positron beam of momentum 53 MeV/c (rejecting pulse-correlated particles with RF signal from accelerator). The positrons with uniform energy hits target and undergo Mott scatting, a noticeable point is that in our setting, energy of scattered positron can be regarded approximately monochromatic (Fig. 4.19). The energy spread of measured positrons are 620 keV inσ.
E(MeV)
40 42 44 46 48 50 52 54 56 58 60
Counts/200keV
0 100 200 300 400 500 600
Mott events - RF selection
Figure 4.19: Spectrum of Mott scattered positron