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Fast method for geometric calibration of detectors and for checking of matching between two detectors
Nuclear Instruments and Methods in Physics Research A 501 (2003) 375–385
Fast method for geometric calibration of detectors and for checking of matching between two detectors Olga Petchenova* Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russia Received 12 June 2002; accepted 8 January 2003
Abstract A fast method of geometric calibration of detectors is described. The main idea of this method is to determine offsets by fitting real data using a set of analytical functions which describe the displacement of one detector relative to another. This method can be applied to determine the offset of one detector relative to another detector or one part of the detector relative to its other part. The form of the analytical functions depends on the geometry of the experiment and on the directions of the coordinate axes. The analytical functions are obtained using the rotation matrices. r 2003 Elsevier Science B.V. All rights reserved. PACS: 02.40.Dr; 02.60.Ed Keywords: Geometric calibration; Matching; Alignment
1. Introduction To match the track segments reconstructed in two detectors, it is necessary to determine the six numbers: the values of detector displacements along X -, Y - and Z-axis and rotation angles around these axes. It is assumed that the tracks are independently reconstructed in two detectors. Tracks can be straight or curved lines. It is necessary to have the parameters of these tracks to define the matched tracks. The main idea of this approach is to determine the offsets by fitting the real data using a set of analytical functions. The method can be applied to determine the offset of one detector relative to another or the offset of one detector’s part relative to its other part. The detectors are assumed to be placed perpendicular to the beam direction. The geometric position of the detector is determined by six parameters: X ; Y ; Z coordinates and rotation matrix around these axes. Z is the axis along the beam.
2. The method description The method is based on the transformation of the coordinate system using translation and rotation. Let us assume that we have two detectors: detectors 1 and 2. The displacement in the space of detector 2 relative *Tel.: +7-09621-63033; fax: +7-09621-63875. E-mail address: [email protected] (O. Petchenova). 0168-9002/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-9002(03)00614-4
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to detector 1 can be described analytically. Let P1 be a vector in the coordinate system X1 Y1 Z1 of the detector 1, P2 be a vector in the coordinate system X2 Y2 Z2 of the detector 2 (Fig. 1). The point ðx1 ; y1 ; z1 Þ is the extrapolated point from detectors 1 to 2. The transformation of coordinates can be obtained by the equation: P2 ¼ P1 A þ B
ð2:1Þ
where A is the rotation matrix, B is the translation vector. The rotation matrix is characterized by the rotation angles a; b; g around X -, Y - and Z-axis, respectively. The matrix A is the result of multiplication of three matrices: A ¼ Mx My Mz where Mx is the rotation matrix around the X -axis, My is the rotation matrix around the Y -axis, and Mz is the rotation matrix around the Z-axis. The translation vector is characterized by linear displacements xoffset ; yoffset ; zoffset along X -, Y - and Zaxis, respectively: B ¼ ðxoffset ; yoffset ; zoffset Þ: If the point ðx1 ; y1 ; z1 Þ is in the coordinate system X1 Y1 Z1 and the point ðx2 ; y2 ; z2 Þ is in the coordinate system X2 Y2 Z2 ; then using Eq. (2.1) the transformed coordinates can be expressed as x2 ¼ a11 x1 þ a21 y1 þ a31 z1 þ xoffset ; y2 ¼ a12 x1 þ a22 y1 þ a32 z1 þ yoffset ; z2 ¼ a13 x1 þ a23 y1 þ a33 z1 þ zoffset
ð2:2Þ
where aij are the elements of the rotation matrix A: If the geometric position of detector 2 was set correctly, the points ðx1 ; y1 ; z1 Þ and ðx2 ; y2 ; z2 Þ should coincide. In the reality, in order to have these points coinciding we need to determine the six values:
Fig. 1. Coordinate systems of detectors 1 and 2.
O. Petchenova / Nuclear Instruments and Methods in Physics Research A 501 (2003) 375–385
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Fig. 2. The track’s azimuthal ðfÞ and the polar ðYÞ angles in the detectors.
xoffset ; yoffset ; zoffset ; a; b; g for one detector relative to another. These values describe relative misalignment between the two detectors. Fig. 2 shows the track’s azimuthal angle f and the polar angle Y in the detector. This method uses a simultaneous fit of the real data dependence Df versus f and DY versus f using the functions F 1 and F 2: Df ¼ F 1ðf; xoffset ; yoffset ; a; b; gÞ;
ð2:3Þ
DY ¼ F 2ðf; xoffset ; yoffset ; zoffset ; a; b; gÞ;
ð2:4Þ
Df ¼ fdetector 1 fdetector 2 ;
ð2:5Þ
DY ¼ Ydetector 1 Ydetector 2
ð2:6Þ
where f is the track’s azimuthal angle, Y is the track’s polar angle, a is the rotation angle around the X axis, b is the rotation angle around the Y -axis, and g is the rotation angle around the Z-axis. y1 y2 ð2:7Þ Df ¼ arctan arctan ; x1 x2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ y21 x22 þ y22 DY ¼ arctan arctan : ð2:8Þ z1 z2 In this case it is assumed that the detectors have cylindrical shape. To obtain average values Df and DY one should integrate Eq. (2.7) and (2.8) over the radial acceptance of detector 2. In general case one should integrate over the cross-section. It is assumed that tracks are uniformly distributed within the acceptance. R Rmax DfðRÞ dR R Df ¼ min ; ð2:9Þ Rmax Rmin R Rmax DY ¼
Rmin
DYðRÞ dR
Rmax Rmin
:
The initial matrices are the following: 0 1 1 0 0 B C Mx ¼ @ 0 cos a sin a A; 0 sin a cos a
ð2:10Þ
O. Petchenova / Nuclear Instruments and Methods in Physics Research A 501 (2003) 375–385
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0
cos b
B My ¼ @ 0 sin b 0
cos g B Mz ¼ @ sin g
0 sin b 1
0
0
cos b
sin g cos g
0
0
1 C A;
1 0 C 0 A: 1
The resulting rotation matrix is the following: A ¼ Mx My Mz ; 0
cos b cos g B A ¼ @ sin a sin b cos g cos a sin g
cos b sin g sin a sin b sin g þ cos a cos g
1 sin b C sin a cos b A:
cos a sin b cos g þ sin a sin g
cos a sin b sin g sin a cos g
cos a cos b
3. The application of the method for CERES/NA45 experimental setup Fig. 3 shows the CERES experimental setup, which is described in details in Ref. [1]. Z is the axis along the beam. The experimental data without magnetic field were used, therefore all tracks were straight lines. The flow of particles is assumed to be independent of the angle Y: The graph of Df versus f and DY versus f was obtained using the real data, and a simultaneous fit of this distribution by the functions F 1 and F 2 was performed in order to determine the four parameters ðxoffset ; yoffset ; zoffset ; gÞ: The method was used for the geometric calibration of SiDC2 (Silicon Drift Camera) relative to SiDC1 and TPC (Time Projection Chamber) relative to SiDC. The values of a and b were set to 0. It was assumed that Df and DY distributions have Gaussian forms with sigma values comparable to detector resolution in f and Y; respectively. In considered case, the initial matrices are the following: 0 1 0 1 1 0 0 1 0 0 B C B C Mx ¼ @ 0 cos a sin a A ¼ @ 0 1 0 A; sin a
0 0
cos b
B My ¼ @ 0 sin b 0
cos g
B Mz ¼ @ sin g
cos a
0 sin b
1
0
0
0 1
1
0 0
C B 0 A ¼ @0 cos b 0
1 0
sin g cos g
0
0
0
1
C 1 0 A; 0 1
1
C 0 A: 1
For this particular case, the resulting rotation matrix is the following: A ¼ Mx My Mz ;
O. Petchenova / Nuclear Instruments and Methods in Physics Research A 501 (2003) 375–385
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Fig. 3. CERES experimental setup.
0
cos g B A ¼ @ sin g 0
sin g cos g 0
0
1
C 0 A: 1
Then using Eqs. (2.7), (2.8), (2.9) and (2.10) one obtains R Rmax ðarctan ðy1 =x1 Þ arctan ðy2 =x2 ÞÞ dR R Df ¼ min ; Rmax Rmin
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R Rmax
x21 þ y21 =z1 arctan x22 þ y22 =z2 dR Rmin arctan DY ¼ : Rmax Rmin
ð3:1Þ
ð3:2Þ
Using the resulting matrix A we can convert Eqs. (2.2) to obtain x2 ¼ cos g x1 sin g y1 þ xoffset ; y2 ¼ sin g x1 þ cos g y1 þ yoffset ; z2 ¼ z1 þ zoffset :
ð3:3Þ
Then using the equations x1 ¼ R cos fdetector 1 ;
y1 ¼ R sin fdetector 1 ;
x2 ¼ R cos fdetector 2 ;
y2 ¼ R sin fdetector 2
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we transform Eq. (3.3) to obtain x2 ¼ cos g R cos f1 sin g R sin f1 þ xoffset ; y2 ¼ sin g R cos f1 þ cos g R sin f1 þ yoffset ; z2 ¼ z1 þ zoffset :
ð3:4Þ
We know the trigonometric relation arctan A arctan B ¼ arctan
AB : 1þAB
Using this Eq. (3.5), Eq. (3.1) is transformed to obtain R Rmax arctan ððy1 x2 y2 x1 Þ=ðx1 x2 þ y1 y2 ÞÞ dR R Df ¼ min : Rmax Rmin
ð3:5Þ
ð3:6Þ
Then using Eqs. (3.4), the arctan expression in Eq. (3.6) is transformed to arctan y1 ðcos gR cos f1 sin gR sin f1 þ xoffset Þ ðsin gR cos f1 þ cos gR sin f1 þ yoffset Þx1 x1 ðcos gR cos f1 sin gR sin f1 þ xoffset Þ þ y1 ðsin gR cos f1 þ cos gR sin f1 þ yoffset Þ
Fig. 4. The Df and DY distributions before SiDC2 geometric calibration. Matching between the SiDC1 and SiDC2 detectors.
O. Petchenova / Nuclear Instruments and Methods in Physics Research A 501 (2003) 375–385
Fig. 5. The Df and DY distributions after SiDC2 geometric calibration. Matching between the SiDC1 and SiDC2 detectors.
and using Eqs. (3.4), the arctan expression in Eq. (3.2) is transformed to R arctan arctan z1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcos gR cos f1 sin gR sin f1 þ xoffset Þ2 þ ðsin gR cos f1 þ cos gR sin f1 þ yoffset Þ2 : z1 þ zoffset Then using the trigonometric relations one obtains R Rmax arctanððM1 R þ M2 Þ=ðL1 R þ L2ÞÞ dR R Df ¼ min ; Rmax Rmin R Rmax DY ¼
Rmin
where Z ¼ z1 ; f ¼ f1 ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½arctan ZR arctanðð N1 R2 þ N2 R þ N3Þ=ðT1 R þ T2ÞÞ dR Rmax Rmin
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382
Fig. 6. The Df and DY distributions before any geometric calibration. Matching between the SiDC and TPC detectors.
R¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ y21 ;
M1 ¼ sinðf gÞE cosðf gÞH; M2 ¼ Z F sinðf gÞ þ Z sin a cosðf gÞ Bp ; L1 ¼ cosðf gÞE þ sinðf gÞH; L2 ¼ Z F cosðf gÞ Z sin a sinðf gÞ Ap ; N1 ¼ E 2 þ H 2 ; N2 ¼ 2 EðZ F Ag Þ 2 HðZ sin a Bg Þ; N3 ¼ ðZ F Ag Þ2 þ ðZ sin a Bg Þ2 ; T1 ¼ cos f sin b þ sin f sin a cos b; T2 ¼ Z cos a cos b zoffset ;
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Fig. 7. The Df and DY distributions after SiDC2 geometric calibration. Matching between the SiDC and TPC detectors.
Ap ¼ xoffset cos f þ yoffset sin f; Bp ¼ xoffset sin f yoffset cos f; E ¼ cos f cos b þ sin f sin a sin b; H ¼ cos a sin f; F ¼ cos a sin b; Ag ¼ xoffset cos g þ yoffset sin g; Bg ¼ xoffset sin g yoffset cos g: Rmax and Rmin are the maximum and minimum radii of detector 2, respectively. The fit is performed for Df and DY simultaneously. The form of the functions F 1 and F 2 depends on the geometry of the experiment and on the direction of the coordinate axes. The calibration was performed in two steps: (1) SiDC1+SiDC2 (offset determination for SiDC2 relative to SiDC1) and (2) SiDC+TPC (offset determination for TPC relative to SiDC).
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Fig. 8. The Df and DY distributions after SiDC2 and TPC geometric calibration. Matching between the SiDC and TPC detectors.
Fig. 4 shows the Df and DY distributions before using this procedure of geometric calibration. The positions of the detectors were taken from prior measurements. The same distributions after the calibration are shown in Fig. 5. One can see that after the offsets for SiDC2 had been determined relative to SiDC1, the matching between SiDC1 and SiDC2 changed slightly, within statistical errors, but matching between SiDC and TPC changed significantly (Figs. 6 and 7). After the second step of the calibration procedure, the matching between SiDC and TPC gets further improvement (Fig. 8). Fig. 9 shows the plot of Df versus f (an interval of 0–6:28 rad) and DY versus f (6.28–12:56 rad). The real data distribution is denoted by black stars and the results of the fit—by rhombuses. After the calibration some small misalignment still remains, which can be due to non-geometric factors (drift velocity, etc). The units in the histograms are radians.
4. The results and discussion The results are sumarized in Table 1. This method allows to determine parameters of relative misalignment between two detectors. Having the real data plot of Df versus f and DY versus f and performing a simultaneous fit of these distributions by the functions F 1 and F 2; one can obtain the six parameters: xoffset ; yoffset ; zoffset ; a; b; g: It is necessary that the systematic errors in the mean values of experimental data were negligible compared to the statistical ones.
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Fig. 9. Fit of real data distribution by the functions F 1 and F 2: The plot of Df versus f (an interval of 0–6:28 rad) and DY versus f (6.28–12:56 rad). Black stars represent the real data and rhombuses correspond to the fitting functions.
Table 1 The SiDC–TPC matching
Before any correction After correction of SiDC2 and target disks positions After correction of TPC position
Mean of Df
Mean of DY
Sigma of Df
Sigma of DY
3.3 2.5 0.0
0.0 1.5 0.0
14.1 12.9 4.2
2.7 2.5 1.3
The units in the table are mrad.
5. Conclusion The application of this method is promising. This method can be applied to two detectors or to two parts of the detector and can be used to check the geometric matching between two detectors. Acknowledgements The author is grateful to Yu.A. Panebratsev, Joint Institute for Nuclear Research, to Johanna Stachel, Heidelberg University, to Peter Braun-Munzinger, GSI, for their permanent interest and support in this investigation and to members of the CERES collaboration for all the useful discussions. The successful test of the method could not be achieved without the collected experimental data of the CERES collaboration. References [1] G. Agakichiev, et al., (CERES Collaboration), Nucl. Phys. A 661 (1999) 23c.
Fast method for geometric calibration of detectors and for checking of matching between two detectors Olga Petchenova* Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russia Received 12 June 2002; accepted 8 January 2003
Abstract A fast method of geometric calibration of detectors is described. The main idea of this method is to determine offsets by fitting real data using a set of analytical functions which describe the displacement of one detector relative to another. This method can be applied to determine the offset of one detector relative to another detector or one part of the detector relative to its other part. The form of the analytical functions depends on the geometry of the experiment and on the directions of the coordinate axes. The analytical functions are obtained using the rotation matrices. r 2003 Elsevier Science B.V. All rights reserved. PACS: 02.40.Dr; 02.60.Ed Keywords: Geometric calibration; Matching; Alignment
1. Introduction To match the track segments reconstructed in two detectors, it is necessary to determine the six numbers: the values of detector displacements along X -, Y - and Z-axis and rotation angles around these axes. It is assumed that the tracks are independently reconstructed in two detectors. Tracks can be straight or curved lines. It is necessary to have the parameters of these tracks to define the matched tracks. The main idea of this approach is to determine the offsets by fitting the real data using a set of analytical functions. The method can be applied to determine the offset of one detector relative to another or the offset of one detector’s part relative to its other part. The detectors are assumed to be placed perpendicular to the beam direction. The geometric position of the detector is determined by six parameters: X ; Y ; Z coordinates and rotation matrix around these axes. Z is the axis along the beam.
2. The method description The method is based on the transformation of the coordinate system using translation and rotation. Let us assume that we have two detectors: detectors 1 and 2. The displacement in the space of detector 2 relative *Tel.: +7-09621-63033; fax: +7-09621-63875. E-mail address: [email protected] (O. Petchenova). 0168-9002/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-9002(03)00614-4
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O. Petchenova / Nuclear Instruments and Methods in Physics Research A 501 (2003) 375–385
to detector 1 can be described analytically. Let P1 be a vector in the coordinate system X1 Y1 Z1 of the detector 1, P2 be a vector in the coordinate system X2 Y2 Z2 of the detector 2 (Fig. 1). The point ðx1 ; y1 ; z1 Þ is the extrapolated point from detectors 1 to 2. The transformation of coordinates can be obtained by the equation: P2 ¼ P1 A þ B
ð2:1Þ
where A is the rotation matrix, B is the translation vector. The rotation matrix is characterized by the rotation angles a; b; g around X -, Y - and Z-axis, respectively. The matrix A is the result of multiplication of three matrices: A ¼ Mx My Mz where Mx is the rotation matrix around the X -axis, My is the rotation matrix around the Y -axis, and Mz is the rotation matrix around the Z-axis. The translation vector is characterized by linear displacements xoffset ; yoffset ; zoffset along X -, Y - and Zaxis, respectively: B ¼ ðxoffset ; yoffset ; zoffset Þ: If the point ðx1 ; y1 ; z1 Þ is in the coordinate system X1 Y1 Z1 and the point ðx2 ; y2 ; z2 Þ is in the coordinate system X2 Y2 Z2 ; then using Eq. (2.1) the transformed coordinates can be expressed as x2 ¼ a11 x1 þ a21 y1 þ a31 z1 þ xoffset ; y2 ¼ a12 x1 þ a22 y1 þ a32 z1 þ yoffset ; z2 ¼ a13 x1 þ a23 y1 þ a33 z1 þ zoffset
ð2:2Þ
where aij are the elements of the rotation matrix A: If the geometric position of detector 2 was set correctly, the points ðx1 ; y1 ; z1 Þ and ðx2 ; y2 ; z2 Þ should coincide. In the reality, in order to have these points coinciding we need to determine the six values:
Fig. 1. Coordinate systems of detectors 1 and 2.
O. Petchenova / Nuclear Instruments and Methods in Physics Research A 501 (2003) 375–385
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Fig. 2. The track’s azimuthal ðfÞ and the polar ðYÞ angles in the detectors.
xoffset ; yoffset ; zoffset ; a; b; g for one detector relative to another. These values describe relative misalignment between the two detectors. Fig. 2 shows the track’s azimuthal angle f and the polar angle Y in the detector. This method uses a simultaneous fit of the real data dependence Df versus f and DY versus f using the functions F 1 and F 2: Df ¼ F 1ðf; xoffset ; yoffset ; a; b; gÞ;
ð2:3Þ
DY ¼ F 2ðf; xoffset ; yoffset ; zoffset ; a; b; gÞ;
ð2:4Þ
Df ¼ fdetector 1 fdetector 2 ;
ð2:5Þ
DY ¼ Ydetector 1 Ydetector 2
ð2:6Þ
where f is the track’s azimuthal angle, Y is the track’s polar angle, a is the rotation angle around the X axis, b is the rotation angle around the Y -axis, and g is the rotation angle around the Z-axis. y1 y2 ð2:7Þ Df ¼ arctan arctan ; x1 x2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ y21 x22 þ y22 DY ¼ arctan arctan : ð2:8Þ z1 z2 In this case it is assumed that the detectors have cylindrical shape. To obtain average values Df and DY one should integrate Eq. (2.7) and (2.8) over the radial acceptance of detector 2. In general case one should integrate over the cross-section. It is assumed that tracks are uniformly distributed within the acceptance. R Rmax DfðRÞ dR R Df ¼ min ; ð2:9Þ Rmax Rmin R Rmax DY ¼
Rmin
DYðRÞ dR
Rmax Rmin
:
The initial matrices are the following: 0 1 1 0 0 B C Mx ¼ @ 0 cos a sin a A; 0 sin a cos a
ð2:10Þ
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0
cos b
B My ¼ @ 0 sin b 0
cos g B Mz ¼ @ sin g
0 sin b 1
0
0
cos b
sin g cos g
0
0
1 C A;
1 0 C 0 A: 1
The resulting rotation matrix is the following: A ¼ Mx My Mz ; 0
cos b cos g B A ¼ @ sin a sin b cos g cos a sin g
cos b sin g sin a sin b sin g þ cos a cos g
1 sin b C sin a cos b A:
cos a sin b cos g þ sin a sin g
cos a sin b sin g sin a cos g
cos a cos b
3. The application of the method for CERES/NA45 experimental setup Fig. 3 shows the CERES experimental setup, which is described in details in Ref. [1]. Z is the axis along the beam. The experimental data without magnetic field were used, therefore all tracks were straight lines. The flow of particles is assumed to be independent of the angle Y: The graph of Df versus f and DY versus f was obtained using the real data, and a simultaneous fit of this distribution by the functions F 1 and F 2 was performed in order to determine the four parameters ðxoffset ; yoffset ; zoffset ; gÞ: The method was used for the geometric calibration of SiDC2 (Silicon Drift Camera) relative to SiDC1 and TPC (Time Projection Chamber) relative to SiDC. The values of a and b were set to 0. It was assumed that Df and DY distributions have Gaussian forms with sigma values comparable to detector resolution in f and Y; respectively. In considered case, the initial matrices are the following: 0 1 0 1 1 0 0 1 0 0 B C B C Mx ¼ @ 0 cos a sin a A ¼ @ 0 1 0 A; sin a
0 0
cos b
B My ¼ @ 0 sin b 0
cos g
B Mz ¼ @ sin g
cos a
0 sin b
1
0
0
0 1
1
0 0
C B 0 A ¼ @0 cos b 0
1 0
sin g cos g
0
0
0
1
C 1 0 A; 0 1
1
C 0 A: 1
For this particular case, the resulting rotation matrix is the following: A ¼ Mx My Mz ;
O. Petchenova / Nuclear Instruments and Methods in Physics Research A 501 (2003) 375–385
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Fig. 3. CERES experimental setup.
0
cos g B A ¼ @ sin g 0
sin g cos g 0
0
1
C 0 A: 1
Then using Eqs. (2.7), (2.8), (2.9) and (2.10) one obtains R Rmax ðarctan ðy1 =x1 Þ arctan ðy2 =x2 ÞÞ dR R Df ¼ min ; Rmax Rmin
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R Rmax
x21 þ y21 =z1 arctan x22 þ y22 =z2 dR Rmin arctan DY ¼ : Rmax Rmin
ð3:1Þ
ð3:2Þ
Using the resulting matrix A we can convert Eqs. (2.2) to obtain x2 ¼ cos g x1 sin g y1 þ xoffset ; y2 ¼ sin g x1 þ cos g y1 þ yoffset ; z2 ¼ z1 þ zoffset :
ð3:3Þ
Then using the equations x1 ¼ R cos fdetector 1 ;
y1 ¼ R sin fdetector 1 ;
x2 ¼ R cos fdetector 2 ;
y2 ¼ R sin fdetector 2
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O. Petchenova / Nuclear Instruments and Methods in Physics Research A 501 (2003) 375–385
we transform Eq. (3.3) to obtain x2 ¼ cos g R cos f1 sin g R sin f1 þ xoffset ; y2 ¼ sin g R cos f1 þ cos g R sin f1 þ yoffset ; z2 ¼ z1 þ zoffset :
ð3:4Þ
We know the trigonometric relation arctan A arctan B ¼ arctan
AB : 1þAB
Using this Eq. (3.5), Eq. (3.1) is transformed to obtain R Rmax arctan ððy1 x2 y2 x1 Þ=ðx1 x2 þ y1 y2 ÞÞ dR R Df ¼ min : Rmax Rmin
ð3:5Þ
ð3:6Þ
Then using Eqs. (3.4), the arctan expression in Eq. (3.6) is transformed to arctan y1 ðcos gR cos f1 sin gR sin f1 þ xoffset Þ ðsin gR cos f1 þ cos gR sin f1 þ yoffset Þx1 x1 ðcos gR cos f1 sin gR sin f1 þ xoffset Þ þ y1 ðsin gR cos f1 þ cos gR sin f1 þ yoffset Þ
Fig. 4. The Df and DY distributions before SiDC2 geometric calibration. Matching between the SiDC1 and SiDC2 detectors.
O. Petchenova / Nuclear Instruments and Methods in Physics Research A 501 (2003) 375–385
Fig. 5. The Df and DY distributions after SiDC2 geometric calibration. Matching between the SiDC1 and SiDC2 detectors.
and using Eqs. (3.4), the arctan expression in Eq. (3.2) is transformed to R arctan arctan z1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcos gR cos f1 sin gR sin f1 þ xoffset Þ2 þ ðsin gR cos f1 þ cos gR sin f1 þ yoffset Þ2 : z1 þ zoffset Then using the trigonometric relations one obtains R Rmax arctanððM1 R þ M2 Þ=ðL1 R þ L2ÞÞ dR R Df ¼ min ; Rmax Rmin R Rmax DY ¼
Rmin
where Z ¼ z1 ; f ¼ f1 ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½arctan ZR arctanðð N1 R2 þ N2 R þ N3Þ=ðT1 R þ T2ÞÞ dR Rmax Rmin
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382
Fig. 6. The Df and DY distributions before any geometric calibration. Matching between the SiDC and TPC detectors.
R¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ y21 ;
M1 ¼ sinðf gÞE cosðf gÞH; M2 ¼ Z F sinðf gÞ þ Z sin a cosðf gÞ Bp ; L1 ¼ cosðf gÞE þ sinðf gÞH; L2 ¼ Z F cosðf gÞ Z sin a sinðf gÞ Ap ; N1 ¼ E 2 þ H 2 ; N2 ¼ 2 EðZ F Ag Þ 2 HðZ sin a Bg Þ; N3 ¼ ðZ F Ag Þ2 þ ðZ sin a Bg Þ2 ; T1 ¼ cos f sin b þ sin f sin a cos b; T2 ¼ Z cos a cos b zoffset ;
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Fig. 7. The Df and DY distributions after SiDC2 geometric calibration. Matching between the SiDC and TPC detectors.
Ap ¼ xoffset cos f þ yoffset sin f; Bp ¼ xoffset sin f yoffset cos f; E ¼ cos f cos b þ sin f sin a sin b; H ¼ cos a sin f; F ¼ cos a sin b; Ag ¼ xoffset cos g þ yoffset sin g; Bg ¼ xoffset sin g yoffset cos g: Rmax and Rmin are the maximum and minimum radii of detector 2, respectively. The fit is performed for Df and DY simultaneously. The form of the functions F 1 and F 2 depends on the geometry of the experiment and on the direction of the coordinate axes. The calibration was performed in two steps: (1) SiDC1+SiDC2 (offset determination for SiDC2 relative to SiDC1) and (2) SiDC+TPC (offset determination for TPC relative to SiDC).
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Fig. 8. The Df and DY distributions after SiDC2 and TPC geometric calibration. Matching between the SiDC and TPC detectors.
Fig. 4 shows the Df and DY distributions before using this procedure of geometric calibration. The positions of the detectors were taken from prior measurements. The same distributions after the calibration are shown in Fig. 5. One can see that after the offsets for SiDC2 had been determined relative to SiDC1, the matching between SiDC1 and SiDC2 changed slightly, within statistical errors, but matching between SiDC and TPC changed significantly (Figs. 6 and 7). After the second step of the calibration procedure, the matching between SiDC and TPC gets further improvement (Fig. 8). Fig. 9 shows the plot of Df versus f (an interval of 0–6:28 rad) and DY versus f (6.28–12:56 rad). The real data distribution is denoted by black stars and the results of the fit—by rhombuses. After the calibration some small misalignment still remains, which can be due to non-geometric factors (drift velocity, etc). The units in the histograms are radians.
4. The results and discussion The results are sumarized in Table 1. This method allows to determine parameters of relative misalignment between two detectors. Having the real data plot of Df versus f and DY versus f and performing a simultaneous fit of these distributions by the functions F 1 and F 2; one can obtain the six parameters: xoffset ; yoffset ; zoffset ; a; b; g: It is necessary that the systematic errors in the mean values of experimental data were negligible compared to the statistical ones.
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Fig. 9. Fit of real data distribution by the functions F 1 and F 2: The plot of Df versus f (an interval of 0–6:28 rad) and DY versus f (6.28–12:56 rad). Black stars represent the real data and rhombuses correspond to the fitting functions.
Table 1 The SiDC–TPC matching
Before any correction After correction of SiDC2 and target disks positions After correction of TPC position
Mean of Df
Mean of DY
Sigma of Df
Sigma of DY
3.3 2.5 0.0
0.0 1.5 0.0
14.1 12.9 4.2
2.7 2.5 1.3
The units in the table are mrad.
5. Conclusion The application of this method is promising. This method can be applied to two detectors or to two parts of the detector and can be used to check the geometric matching between two detectors. Acknowledgements The author is grateful to Yu.A. Panebratsev, Joint Institute for Nuclear Research, to Johanna Stachel, Heidelberg University, to Peter Braun-Munzinger, GSI, for their permanent interest and support in this investigation and to members of the CERES collaboration for all the useful discussions. The successful test of the method could not be achieved without the collected experimental data of the CERES collaboration. References [1] G. Agakichiev, et al., (CERES Collaboration), Nucl. Phys. A 661 (1999) 23c.