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An analytical approach to space charge distortions for time projection chambers
ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 617 (2010) 193–195
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
An analytical approach to space charge distortions for time projection chambers S. Rossegger a,b,, B. Schnizer b, W. Riegler a a b
CERN PH, CH-1211 Geneve 23, Switzerland Institut f¨ ur Theoretische Physik—Computational Physics, Technische Unisersit¨ at, A-8010 Graz, Austria
a r t i c l e in fo
abstract
Available online 1 July 2009
In a time projection chamber (TPC), the possible ion feedback and also the primary ionization of high multiplicity events result in accumulation of ionic charges inside the gas volume (space charge). This charge introduces electrical field distortions and modifies the cluster trajectory along the drift path, affecting the tracking performance of the detector. In order to calculate the track distortions due to an arbitrary space charge distribution in the TPC, novel representations of the Green’s function for a TPC geometry were worked out. This analytical approach finally permits accurate predictions of track distortions due to an arbitrary space charge distribution by solving the Langevin equation. & 2009 Elsevier B.V. All rights reserved.
Keywords: Time projection chamber Calibration Green’s function Space charge
1. Introduction The Langevin equation can be used to model track and space point distortions in TPCs. In order to do so, a detailed knowledge of all field imperfections is required. Static E and B fields inhomogeneities due to magnet and field cage imperfections can be quantified directly by means of laser measurements. Modeling dynamic distortions due to space charges calls for a solution of the three-dimensional Laplace equation and as a result the electrical field distortions for a geometry close to a typical time projection chamber field-cage, namely a coaxial cavity.
2. Analytical solution of the Laplace equation for a coaxial cavity Following the methods as described in Ref. [1], three series representations of the Green’s function were derived [2]. They have in common that each term of the sum comprises two eigenfunctions with eigenvalues chosen such that they fulfill the Dirichlet boundary conditions. The third factor of each representation consists of non-oscillating functions introduced by the method of particular integrals. The potential of a charge cloud is obtained by integrating the Green’s function times the charge distribution over the volume. Each representation proves its right to exist by providing a fast converging expression (e 107 for r30 terms) for one specific electric field component in cylindrical coordinates ðEr ; Ef ; Ez Þ. Thus, they finally permit the treatment of Corresponding author at: CERN PH, CH-1211 Geneve 23, Switzerland.
E-mail address: [email protected] (S. Rossegger). 0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.06.056
any 3D space charge distribution rðr; f; zÞ within a TPC field cage, and not only radially symmetric ones. Example plots demonstrating the smooth properties of all three E field components are given in Fig. 1.
3. Simulated space charge distributions for the Alice TPC TPC clusters from Pb–Pb collisions were simulated (HiJing and AliRoot) in order to estimate the charge distributions from single events and after a pile up. Within the Alice TPC [3] the expected event rate is 3000 Hz, thus about 480 min-bias events contribute to the space charge distribution within the gas volume (considering an ion clearing time of 0:156 s). Every event can contribute by means of the following aspects: 1. PI: Primary ionization from the tracks within the gas volume. 2. ROC-IFB: Ion feedback from tracks within the read out chamber (from the prev. event). 3. ILK: Ion leakage—typically suppressed by the gating grid. An expected scenario within the Alice TPC (PI plus ROC-IFB) would lead to a radially symmetric charge distribution of rðr; zÞ ð3 0:9 zÞ=r 2 1010 C=m3 . Magnitudes of the slope and the zero crossing in z heavily depend on gain settings and the readout geometry (by means of ROC-IFB) as well as on the multiplicity of the events. Unexpected asymmetries or problems with the gating grid could destroy the radial symmetry. However, the analytical solution discussed in the previous section can be used to handle even such scenarios.
ARTICLE IN PRESS 194
S. Rossegger et al. / Nuclear Instruments and Methods in Physics Research A 617 (2010) 193–195
1 2 0.9 1.5 0.8 1 0.7 0.5 0.6 0 0.5 0.4 -0.5 -1 0.3 0.2 -1.5 0.1 -2 0
ρ(r, φ) [a.u.]
Er [a.u.]
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Eφ [a.u.]
2
Ez [a.u.]
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
-2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Fig. 1. E field components due to a constant space charge distribution within the region ð1:5rrr1:8; 4:88rfr5:23; 1:0rzr1:5Þ in the ðr; fÞ-plane at z ¼ 1:7. General geometry of the cavity: inner radius a ¼ 0:8, outer radius b ¼ 2:5, and height h ¼ 2:5.
Magboltz vs. Langevin calc. for E = 135 V/cm, B = 0.5 T
9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4
Lorenzangle
φBtrans φE×B
10 20 30 40 50 60 70 80 Angle between E and B [degrees]
Angle residuals [degrees]
Angle residuals [degrees]
Magboltz vs. Langevin calc. for E = 400 V/cm, B = 0.5 T 18 16 14 12 10 8 6 4 2 0 -2 -4 -6 -8 -10
φBtrans
Lorenzangle φE×B
10 20 30 40 50 60 70 80 Angle between E and B [degrees]
Fig. 2. Differences of the velocity-component-angles between the Magboltz and the Langevin method; left: NeCO2 N2 (90/10/5); right: ArCH3 (90/10).
4. Lorenz angle calculations The Lorenz angle depends on the gas composition and the angle between the E and B fields. It can be calculated by means of MC techniques (see Ref. [4]) or by solving the Langevin equation for a const. mobility m. Sizable differences between the Magboltz and Langevin velocity components are possible, whereas the Magboltz method, being based on the atomic cross-sections, is more accurate. Angles between the resultant velocity components are compared for two different gas compositions at different operating conditions (drift field). NeCO2 N2 (90/10/5) is used for the Alice TPC. The Langevin method displays an excellent agreement in
comparison to the angles calculated via Magboltz ðDfmax 0:13 Þ. Another commonly used drift gas is P10, namely ArCH3 ð90=10Þ. In this case, the Langevin method reveals a good agreement at small angles between E and B, but considerably larger differences at larger angles (e.g. Df13 at +ðE; BÞ103 ). The differences between the Langevin and the Magboltz method are plotted in Fig. 2.
5. Space point distortions due to space charges By solving the Langevin equation e.g. within Garfield [5] we can include almost every effect which disturbs the drifting electron. These effects are the inhomogeneous B field, the E field
ARTICLE IN PRESS
0.4
0.4
0.3
0.3
0.2 0.1 0 -0.1
r = 90cm φ = 0°
-0.2
r = 240cm φ = 0°
r = 90cm φ = 180°
radial distortion Δ r [cm]
radial distortion Δ r [cm]
S. Rossegger et al. / Nuclear Instruments and Methods in Physics Research A 617 (2010) 193–195
0.2 0.1 0 -0.1
r = 90cm φ = 0°
-0.2
r = 240cm φ = 0°
r = 240cm φ = 180°
-0.3
195
r = 90cm φ = 180° r = 240cm φ = 180°
-0.3 -200 -100 0 100 200 initial position z [cm]
-200 -100 0 100 200 initial position z [cm]
Fig. 3. Space point distortions due to field inhomogeneities within the Alice TPC; left: due to B field imperfections; right: due to B field and (expected) space charge.
distortions due to space charges or field cage imperfections (by means of an 3D finite element model of the field cage) as well as possible gas density deviations (drift velocity variations due to P=T differences). For example, Fig. 3 contains simulated radial distortions due to B field imperfections and expected space charges within the Alice TPC.
6. Summary and outlook We present a fast converging analytical solution of the Laplace equation for a typical TPC geometry which can be used to calculate the electric field inhomogeneities due to arbitrary space charge configurations. Furthermore, we show that the Langevin approximation of the Lorenz angle is sufficient for a gas composition of NeCO2 N2 (90/10/5) which is used in the Alice TPC. Together, this approach yields to fast and accurate predictions of space point distortions due to any space charges in
combination with B field inhomogeneities and gas density variations by means of solving the Langevin equation. This analytical approach is the basis of an inverse model which will permit the identification of space charge clouds within the gas volume by means of laser measurements. It is planned to be used in an event-by-event space charge correction within the Alice TPC. References [1] W.R. Smythe, Static and Dynamic Electricity, third ed., McGraw-Hill, New York, NY, 1968. [2] S. Rossegger, Static Green’s functions for a coaxial cavity including an innovative representation, Technical Report CERN-OPEN-2009-003, CERN, Geneva, February 2009. [3] ALICE TPC Collaboration, TPC technical design report, Technical Report CERNLHCC-2000-001, CERN, Geneva, 2000. [4] S. Biagi, Magboltz version 7.07, April 2008. [5] R. Veenhof, Garfield version 9, February 2009.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
An analytical approach to space charge distortions for time projection chambers S. Rossegger a,b,, B. Schnizer b, W. Riegler a a b
CERN PH, CH-1211 Geneve 23, Switzerland Institut f¨ ur Theoretische Physik—Computational Physics, Technische Unisersit¨ at, A-8010 Graz, Austria
a r t i c l e in fo
abstract
Available online 1 July 2009
In a time projection chamber (TPC), the possible ion feedback and also the primary ionization of high multiplicity events result in accumulation of ionic charges inside the gas volume (space charge). This charge introduces electrical field distortions and modifies the cluster trajectory along the drift path, affecting the tracking performance of the detector. In order to calculate the track distortions due to an arbitrary space charge distribution in the TPC, novel representations of the Green’s function for a TPC geometry were worked out. This analytical approach finally permits accurate predictions of track distortions due to an arbitrary space charge distribution by solving the Langevin equation. & 2009 Elsevier B.V. All rights reserved.
Keywords: Time projection chamber Calibration Green’s function Space charge
1. Introduction The Langevin equation can be used to model track and space point distortions in TPCs. In order to do so, a detailed knowledge of all field imperfections is required. Static E and B fields inhomogeneities due to magnet and field cage imperfections can be quantified directly by means of laser measurements. Modeling dynamic distortions due to space charges calls for a solution of the three-dimensional Laplace equation and as a result the electrical field distortions for a geometry close to a typical time projection chamber field-cage, namely a coaxial cavity.
2. Analytical solution of the Laplace equation for a coaxial cavity Following the methods as described in Ref. [1], three series representations of the Green’s function were derived [2]. They have in common that each term of the sum comprises two eigenfunctions with eigenvalues chosen such that they fulfill the Dirichlet boundary conditions. The third factor of each representation consists of non-oscillating functions introduced by the method of particular integrals. The potential of a charge cloud is obtained by integrating the Green’s function times the charge distribution over the volume. Each representation proves its right to exist by providing a fast converging expression (e 107 for r30 terms) for one specific electric field component in cylindrical coordinates ðEr ; Ef ; Ez Þ. Thus, they finally permit the treatment of Corresponding author at: CERN PH, CH-1211 Geneve 23, Switzerland.
E-mail address: [email protected] (S. Rossegger). 0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.06.056
any 3D space charge distribution rðr; f; zÞ within a TPC field cage, and not only radially symmetric ones. Example plots demonstrating the smooth properties of all three E field components are given in Fig. 1.
3. Simulated space charge distributions for the Alice TPC TPC clusters from Pb–Pb collisions were simulated (HiJing and AliRoot) in order to estimate the charge distributions from single events and after a pile up. Within the Alice TPC [3] the expected event rate is 3000 Hz, thus about 480 min-bias events contribute to the space charge distribution within the gas volume (considering an ion clearing time of 0:156 s). Every event can contribute by means of the following aspects: 1. PI: Primary ionization from the tracks within the gas volume. 2. ROC-IFB: Ion feedback from tracks within the read out chamber (from the prev. event). 3. ILK: Ion leakage—typically suppressed by the gating grid. An expected scenario within the Alice TPC (PI plus ROC-IFB) would lead to a radially symmetric charge distribution of rðr; zÞ ð3 0:9 zÞ=r 2 1010 C=m3 . Magnitudes of the slope and the zero crossing in z heavily depend on gain settings and the readout geometry (by means of ROC-IFB) as well as on the multiplicity of the events. Unexpected asymmetries or problems with the gating grid could destroy the radial symmetry. However, the analytical solution discussed in the previous section can be used to handle even such scenarios.
ARTICLE IN PRESS 194
S. Rossegger et al. / Nuclear Instruments and Methods in Physics Research A 617 (2010) 193–195
1 2 0.9 1.5 0.8 1 0.7 0.5 0.6 0 0.5 0.4 -0.5 -1 0.3 0.2 -1.5 0.1 -2 0
ρ(r, φ) [a.u.]
Er [a.u.]
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Eφ [a.u.]
2
Ez [a.u.]
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
-2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Fig. 1. E field components due to a constant space charge distribution within the region ð1:5rrr1:8; 4:88rfr5:23; 1:0rzr1:5Þ in the ðr; fÞ-plane at z ¼ 1:7. General geometry of the cavity: inner radius a ¼ 0:8, outer radius b ¼ 2:5, and height h ¼ 2:5.
Magboltz vs. Langevin calc. for E = 135 V/cm, B = 0.5 T
9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4
Lorenzangle
φBtrans φE×B
10 20 30 40 50 60 70 80 Angle between E and B [degrees]
Angle residuals [degrees]
Angle residuals [degrees]
Magboltz vs. Langevin calc. for E = 400 V/cm, B = 0.5 T 18 16 14 12 10 8 6 4 2 0 -2 -4 -6 -8 -10
φBtrans
Lorenzangle φE×B
10 20 30 40 50 60 70 80 Angle between E and B [degrees]
Fig. 2. Differences of the velocity-component-angles between the Magboltz and the Langevin method; left: NeCO2 N2 (90/10/5); right: ArCH3 (90/10).
4. Lorenz angle calculations The Lorenz angle depends on the gas composition and the angle between the E and B fields. It can be calculated by means of MC techniques (see Ref. [4]) or by solving the Langevin equation for a const. mobility m. Sizable differences between the Magboltz and Langevin velocity components are possible, whereas the Magboltz method, being based on the atomic cross-sections, is more accurate. Angles between the resultant velocity components are compared for two different gas compositions at different operating conditions (drift field). NeCO2 N2 (90/10/5) is used for the Alice TPC. The Langevin method displays an excellent agreement in
comparison to the angles calculated via Magboltz ðDfmax 0:13 Þ. Another commonly used drift gas is P10, namely ArCH3 ð90=10Þ. In this case, the Langevin method reveals a good agreement at small angles between E and B, but considerably larger differences at larger angles (e.g. Df13 at +ðE; BÞ103 ). The differences between the Langevin and the Magboltz method are plotted in Fig. 2.
5. Space point distortions due to space charges By solving the Langevin equation e.g. within Garfield [5] we can include almost every effect which disturbs the drifting electron. These effects are the inhomogeneous B field, the E field
ARTICLE IN PRESS
0.4
0.4
0.3
0.3
0.2 0.1 0 -0.1
r = 90cm φ = 0°
-0.2
r = 240cm φ = 0°
r = 90cm φ = 180°
radial distortion Δ r [cm]
radial distortion Δ r [cm]
S. Rossegger et al. / Nuclear Instruments and Methods in Physics Research A 617 (2010) 193–195
0.2 0.1 0 -0.1
r = 90cm φ = 0°
-0.2
r = 240cm φ = 0°
r = 240cm φ = 180°
-0.3
195
r = 90cm φ = 180° r = 240cm φ = 180°
-0.3 -200 -100 0 100 200 initial position z [cm]
-200 -100 0 100 200 initial position z [cm]
Fig. 3. Space point distortions due to field inhomogeneities within the Alice TPC; left: due to B field imperfections; right: due to B field and (expected) space charge.
distortions due to space charges or field cage imperfections (by means of an 3D finite element model of the field cage) as well as possible gas density deviations (drift velocity variations due to P=T differences). For example, Fig. 3 contains simulated radial distortions due to B field imperfections and expected space charges within the Alice TPC.
6. Summary and outlook We present a fast converging analytical solution of the Laplace equation for a typical TPC geometry which can be used to calculate the electric field inhomogeneities due to arbitrary space charge configurations. Furthermore, we show that the Langevin approximation of the Lorenz angle is sufficient for a gas composition of NeCO2 N2 (90/10/5) which is used in the Alice TPC. Together, this approach yields to fast and accurate predictions of space point distortions due to any space charges in
combination with B field inhomogeneities and gas density variations by means of solving the Langevin equation. This analytical approach is the basis of an inverse model which will permit the identification of space charge clouds within the gas volume by means of laser measurements. It is planned to be used in an event-by-event space charge correction within the Alice TPC. References [1] W.R. Smythe, Static and Dynamic Electricity, third ed., McGraw-Hill, New York, NY, 1968. [2] S. Rossegger, Static Green’s functions for a coaxial cavity including an innovative representation, Technical Report CERN-OPEN-2009-003, CERN, Geneva, February 2009. [3] ALICE TPC Collaboration, TPC technical design report, Technical Report CERNLHCC-2000-001, CERN, Geneva, 2000. [4] S. Biagi, Magboltz version 7.07, April 2008. [5] R. Veenhof, Garfield version 9, February 2009.