Space Charge Effects in Noble Liquid Calorimeters and Time Projection Chambers
88 February 2021
Space Charge Effects in Noble Liquid Calorimetersand Time Projection Chambers
Sandro Palestini
CERN, 1211 Geneva 23, Switzerland
Abstract
The subject of space charge in ionization detectors is reviewed, showing how the observations andthe formalism used to describe the effects have evolved, starting with applications to calorimetersand reaching recent, large-size time projection chambers. General scaling laws, and different waysto present and model the effects are presented. The relation between space-charge effects and theboundary conditions imposed on the side faces of the detector are discussed, together with a designsolution that mitigates part of the effects. The implications of the relative size of drift length andtransverse detector size are illustrated. Calibration methods are briefly discussed.
Submitted to Instruments
Special Issue: Liquid Argon Detectors: Instrumentation and Applications a r X i v : . [ phy s i c s . i n s - d e t ] F e b Introduction
The term space charge is referred to the underlying distribution of charge, usually positive ions, reachinga level capable to affect significantly the electric field established by the voltage on the electrodes of anelectronic device. The change in electric field affects the motion of the main charge carriers, usuallyelectrons, and therefore changes the signal being read out. This subject was discussed first in the con-text of gas diodes [1, 2], before becoming relevant for particle detectors, namely in the context of driftchambers with long collection time [3, 4]. In these cases, the positive ions are produced through themechanism of gas multiplication in proximity of the anode. Ionization detectors without gas multiplica-tion can be affected by space charge as well, if the time necessary to dispose of the positive ions becomeslong enough to cause significant accumulation of positive charge density. In particular, this situationoccurs in detectors based on dense media, such as liquid argon or krypton, used in calorimetry for highradiation intensity applications, or for large-size time projection chambers (TPC), where even low levelsof radiation, together with long drift paths, can cause significant space-charge effects.This paper deals with ionization detectors without gas multiplication — with the exception of a shortdiscussion of liquid argon TPCs in which gas multiplication contributes to space charge. Section 2contains first a historical review of relevant observations, sometimes reported together with developmentsof the formalism used to describe the effect. Space-charge effects occurring near the side faces of TPCs,and a design solution for their mitigation are discussed in Section 3, which also contains a comparisonof the predictions of multidimensional models with observations. Section 4 contains the discussion ofspace charge in dual-phase detectors. Fluid motion represents a limit to the validity of electrostaticmodels for space-charge effects, and reinforces the need of calibration procedures. Section 5 provides abrief discussion of calibration methods, in particular for data-driven ones.
This section contains a review of relevant observations of space-charge effects, and of how they led todevelopment in their understanding and description.
Detailed discussion of space charge effects in ionization detectors came first from studies and operationof electromagnetic calorimeters. The NA48 Collaboration provided a report of observation of space-charge effects in a quasi-homogeneous liquid-krypton calorimeter, together with an analytical modeldescribing space and time dependence of the effect [5]. The sensitivity to space charge came from thecombination of several conditions:– The flux to which the detector was exposed was equivalent to a charge density injection of positiveions exceeding 100 pC cm − s − in the central part of the calorimeter.– The nearly longitudinal readout structure made the detector response sensitive to the position ofthe axis of the electromagnetic shower within the readout cell, which had a gap of about 1.0 cm.– The readout cells were operated at a voltage of only 1.5 kV for the first year of data taking.In these conditions, space charge effects were identified and corrected for, including their time depen-dence within the beam spill: the effect increased during the first 1.6 s of exposure to the beam, cor-responding to the time required to reach the equilibrium condition of the density of positive ions, andremained stable during the rest of the spill. After that, the density of positive ions reduced to zero duringthe interval to the next spill. 1 c /E E a /E Fig. 1:
Normalized electric field at the anode E a / E ◦ and cathode E c / E ◦ as a function of the dimensionless param-eter α . Reproduced from reference [5]. The model developed in [5] was based on the use of a continuity equation for the density of positive ions: ∂ ρ + ∂ t + ∂ ( ρ + v + x ) ∂ x = K . (1)The detector geometry allows to neglect the dependence on the coordinates parallel to the electrodes.The parameter K is the injection rate of the charge density of positive ions, which includes the initialrecombination of electrons and ions. Besides this effect, the presence of the electrons is ignored, sincethey are removed from the drift cell in a time about 4 × times shorter than the ions. Under steadyconditions, the space charge density satisfies the equation ρ + ( x ) = K x µ + E ( x ) , (2)where the ion mobility is introduced with v + x = µ + E x . The electric field is determined by the boundaryconditions and by the Gauss equation dE x dx = ρ + ε , (3)which is solved as E x ( x ) = E ◦ (cid:113) ( E a / E ◦ ) + α ( x / L ) , (4)where E a is the value at the anode, determined by the boundary (cid:82) E x dx = V ◦ = E ◦ L , integrated fromanode (at x =
0) to cathode (at x = L ), where V ◦ is the voltage difference between anode and cathode,and E ◦ is the uniform field strength that would be present for K =
0, i.e. for vanishing space charge. Thedimensionless parameter α is defined as α = LE ◦ (cid:115) K ε µ + . (5)Figure 1 shows the dependence of E a and of E c , the electric field at the cathode, on the parameter α . Atthe anode, the field is lower than E ◦ , while the opposite holds at the cathode, to a greater extent. Figure 2shows the values of E x / E ◦ and ρ + / ρ ◦ versus x / L , for values of the parameter α equal to 0.8 and 1.6.The density ρ ◦ , defined as ε E ◦ / L , is the natural scale for the space charge density. It is equal to the ratio2 ig. 2: Electric field (continuous lines) and charge density (dashed line) behavior for α = . α = . x / L ), with x = ( ) at the anode(cathode). The electric field is in units of E ◦ = V ◦ / L , and the charge density in units of ρ ◦ = ε E ◦ / L . Reproducedfrom reference [6]. of the charge density on the surface of the electrodes, when no space charge is present, to the length ofthe drift gap. From Equation 5, α can be described as the charge density injection rate K , multiplied byan effective ion drift-time across the gap L / ( µ + E ◦ ) , and divided by ρ ◦ . Figure 1 shows that the effectsof space charge are relevant for α = . α up to about 1.1, showing good agreement betweenpredictions and observations. Further advancements were driven by interest in the response of liquid argon sampling calorimeters, likethose used in the ATLAS detector, when operated in the high rate environment of the Large HadronCollider, and in particular for the foreseen upgrade to the High Luminosity Large Hadron Collider (HL-LHC). Particular attention was devoted to the study of critical conditions reached at very high intensity.The occurrence of critical conditions had already been remarked in [5], as illustrated in Figure 1 andEquation 4: for α = x as E x = E ◦ ( x / L ) . For larger values of α it had been suggested that the active region contracts to a gap oflength L (cid:48) = L / α , detached from the anode by L − L (cid:48) , with E x = E ◦ ( x (cid:48) / L (cid:48) ) for x (cid:48) = x − ( L − L (cid:48) ) > E x is highly suppressed for x ≤ L − L (cid:48) .These conditions of very large space-charge effects were studied with more detailed analytical descrip-tions [7, 8, 9], including the charge density of electrons and bulk recombination in the continuity andGauss equations. Laboratory measurements were performed on calorimetric cells exposed to β sources.The onset of the critical condition, described by the authors as cell closing down , was observed andfound in substantial agreement with expectations [9]. The transition into critical conditions was used todefine the minimum angle to the collision axis at which liquid argon detectors of a given design could beoperated at HL-LHC. Naturally, such limits could be overcome operating the calorimetric cells at highervoltage or, more effectively, with smaller gaps, as the value of the α is proportional to L / V ◦ , as shownin Equation 5. Besides, for a given ratio L / V ◦ , the value of α is affected by the uncertainty in µ + , forwhich values in the range of 0.8 to 2.0 × − m s − V − have been reported (references in [6]).3 .2 Liquid Argon Time Projection Chambers Large liquid argon TPCs were proposed as neutrino detectors [10] as well as for searches of nucleondecays [11]. Such detectors combine the function of tracking devices, with resolution at the scale of1 mm, determined by diffusion of the drifting electrons, together with calorimetric measurement fromthe collected charge. Detectors of this kind have been operated underground, in condition of very lowintensity, or near surface. When operated near surface, the ionization due to cosmic rays corresponds toa charge density injection of approximately 2 × − C m − s − , and may cause visible effects of spacecharge. In fact, the six orders of magnitudes between this value and the one quoted in Section 2.1.1 arebalanced by the increase of the drift length from 1 cm to a few meters, together with a reduction of theaverage electric field from 1.5 kV cm − to typical values of 0.5 kV cm − .Compared to the calorimetric cells discussed above, the liquid argon TPCs add additional complexity tothe effects of space charge, because:– These detectors may have lateral dimensions comparable to the drift length, so that border effectsare relevant and the description cannot be reduced to one dimension.– Fluid motion may alter the drift of positive ions, moving them away from electrostatic equilibrium,and therefore changing the way they affect the electric field.Large liquid argon TPCs have been built and operated near surface in ICARUS, MicroBooNE, and, morerecently, ProtoDUNE. A brief review of their observations is given in the next three sections. The ICARUS detector is formed by two modules with a double drift-gap of 1.5 m and transverse activesize of 3 . × . . The detector collected data at the Gran Sasso underground laboratory, and onsurface, where space-charge effects were studied [12]. The observations were analyzed in terms of thedelay in the collected charge, due to the distortion to the electric field. The apparent value of the driftcoordinate is shifted by δ x ( x ) = v e ◦ × δ t ( x ) = (cid:90) x (cid:18) v e ◦ v e ( x (cid:48) ) − (cid:19) dx (cid:48) = (cid:90) x (cid:18) v e ◦ µ + E x ( x (cid:48) ) − (cid:19) dx (cid:48) , (6)where v e ◦ is the electron drift velocity at the average electric field E ◦ , and v e ( x (cid:48) ) is the drift velocity atthe field strength E ( x (cid:48) ) present at the drift coordinate x (cid:48) , as in Equation 4. The authors refer to δ t and δ x as bending parameters . Figure 3 illustrates the observable, and shows the expected values of δ t ( x ) with E ◦ =
500 V cm − , for K = . × − C m − s − , µ + = . × − m kV − s − . These valuescorrespond to α = .
4. The prediction for the bending parameter includes corrections from modeling ofelectrodes and field-cage, and the small contribution to space charge from negative ions, due to electroncapture from impurities. The result of the measurement is shown in Figure 4, which is in good agreementwith the expectation, for the quoted values of K / µ + . The MicroBooNE liquid argon TPC has collected data from neutrino interactions at the BNB facility ofFermilab. The detector has a drift gap L = .
56 m (along the x axis, horizontal), transverse size of 2.32 malong the vertical direction ( y axis) and 10.37 m along the beam direction ( z axis). It has been operatedat a nominal electric field of 274 V/cm. Under these conditions, and using µ + = . × − m s − V − ,the α parameter is approximately equal to 0.81, and space charge effects were clearly observed andextensively reported by the MicroBooNE Collaboration [13]: uncorrected crossing tracks appear bentin the xy and xz projections more than in ICARUS, with sagittas on the centimeter scale. Besides, the4 Fig. 3:
Left: schematic view of the drift time delay δ t . Right: drift time delay dependence on the drift coordinate x . The anode is at x = x =
150 cm. Reproduced from reference [12].
Fig. 4:
Bending parameter δ x = v e ◦ × δ t ( x ) as a function of the drift coordinate x , for the two drift volumes of anICARUS detector module. Reproduced from reference [12]. most apparent effect of space charge occurs on the side faces: the transverse coordinates of entry or exitpoints of ionizing particles crossing the field-cage are shifted towards the center of the detector. Theeffect is illustrated in Figure 5 for tracks crossing the top or bottom side faces of the detector, far fromthe upstream and downstream side faces, for which the apparent y coordinate is plotted against the driftcoordinate x . The distortion vanishes at the anode ( x =
0) and it is maximum for electrons that havedrifted all the distance from the cathode. The distortion of the x coordinate of the end-point is expectedto be very small, since near the field-cage the field is constrained to E x = E ◦ , apart from minor effectsdue granularity of the field-cage electrodes. The transverse distortions due to space charge are discussedfurther in Section 3.2.MicroBooNE uses calibration procedures based on an ultraviolet laser system and on a data-driven ap-proach, which exploits cosmic muons and the limited distortion of track end-points, as discussed below5 ig. 5: Entry/exit points of reconstructed cosmic muon tracks in the MicroBooNe detector, projected on the xy plane. The anode is located at x = x =
256 cm. The data points accumulate on the cathodeplane and on the profile of the top and bottom side faces, which appear distorted towards the center of the detectorbecause of a transverse component in the electric field due to space-charge. Reproduced from reference [13]. in Section 5.
The ProtoDUNE Single-Phase LAr TPC has collected data at the CERN Neutrino Platform. The detectorhas an active volume of 7 . × . × . , with a cathode plane separating two regions with drift gaps of3.6 m. Operated at 500 V cm − , the parameter α is about 20% smaller than in MicroBooNE, but spacecharge effects are expected to be larger, because spatial distortion scales approximately as α L . Figure6 [14] shows the transverse distortions to the tracks end-points, when they cross the top or bottom faceof the active volume. The drift occurs horizontally, along x , with the cathode at x = y . The top panels show the results of a numerical model thatprovides an approximate description of the effects of space charge and of the boundary condition imposedby the field cage. The bottom ones show the corresponding observation. Some features of the model,such as larger effects near the cathode and no effects on the boundaries with the anodes and the lateralfaces, are visible in data, but they come together with large changes in the amplitude of the distortionand with large asymmetries. A similar situation occurs near the lateral side faces of the detector, wherethe end-points are shifted along the z direction.Fluid motion, induced by thermal gradients and by recirculation of liquid argon, is likely to be a maincontributor to the differences between the model and the observation, since its range in velocity, fromfraction of mm to several cm per second, exceeds the drift velocity of positive ions, which is equal to a fewmm per second. The large asymmetries between the two sides of the cathode observed in ProtoDUNEmay be related to the construction of the electrode as a closed surface, and to the asymmetric location of6 ig. 6: Predicted and observed transverse distortion on the top and bottom side faces, in the ProtoDUNE Single-Phase detector. The top panels show the distortion as predicted by an approximate numerical model. The amplitudeof the distortion is maximal near the cathode ( x = x = ±
360 cm) and with the other side faces. The observed distortion, bottom panels, is qualita-tively comparable, but reaches larger values and shows large asymmetries between the top and bottom faces, andbetween the two sides of the cathode surface. Reproduced from reference [14]. the inlet/outlet ports for liquid argon circulation.
Space charge affects the electric field and may causes significant deviations from the behavior of anideal TPC. As seen in the examples discussed above, the distortions caused by space charge may exceedby two order of magnitude the resolution limit of about 1 mm imposed by electron diffusion. Besides,particle identification by dQ / dX is affected by local variations of the ionization yield, which depends onthe electric field through initial recombination, and by changes in the absolute length scales over whichthe charge yield is measured, which varies with the gradients of the spatial distortion.A discussion of numerical results and analytic approximations has been recently presented in refer-ence [6]. We refer to that analysis for the discussion of local variation in the ionization yield and in thelength scale, and for the effect of a negative ions, formed in electron capture by impurities. The followingsections deal with effects related to the side faces and to the detector aspect ratio, i.e. the relative size ofthe drift gap and of the detector lateral extension. The term longitudinal is used for the effects that occuralong the nominal drift direction, denoted as x , while the term transverse is used for the effects along the7 and z directions. The anode is at x =
0, and one detector side face, namely a face of the field cage, is at y = While calorimetric cells might be described with the one-dimensional model of Section 2.1.1, TPC detec-tors needs a treatment in more dimensions, which cannot be described with analytical equation as simpleas equations 2 and 4. As the number of dimensions increases, additional care must be taken in dealingwith boundary conditions. For example, when making the approximate assumption on the space-chargedensity ρ + (cid:39) K x / ( µ + E ◦ ) and using it as source for a perturbative treatment E = E ◦ + E (cid:48) , the term E (cid:48) cannot be solved using ∇ · E (cid:48) (cid:39) ρ + / ε together with empty-space boundary conditions. The problemof such approach is that electrodes and field cage react to the presence of ρ + , effectively producing adistribution of image charge of comparable density. For this reason, describing the inward distortion of sections 2.2.2 and 2.2.3 just as the result of theattraction from positive space charge in the detector volume, is a simplification which may preclude awider comprehension of the situation. To this purpose, consider Figure 7, where a TPC with L = ρ + and of the components of E . A cross section in the xy plane is shown, near the side face and far from the boundary in z , drawingequipotential contours and drift paths. The anode and cathode are at x = y =
0. The space charge density corresponds to α = .
15. On the plot on the left, the boundarycondition is a linear voltage distribution along the field cage: V fc = − V ◦ x / L . Far from the field cage,for y (cid:38) L , the equipotential surfaces are parallel, at a distance to each other that depends on x followingthe pattern of equation 4. As the boundary y = x = y = V fc ( x ) is instead modified to follow the behavior of V ( x )= − (cid:82) E x dx at y ≥ L , the equipoten-tial surfaces remain parallel all the way to y =
0, without any transverse component of the electric field.Following reference [6], the plot on the right in Figure 7 shows the result of a very simple modificationto the field cage: the slope of V fc changes at x = . V fc is given the value V ( x , y ≥ L ) corre-sponding at the same x but far inside the detector. V fc ( x ) still follows a linear behavior, with differentslopes, between this point and the electrodes. As a result, the electric field E x ( x , y = ) imposed along thefield cage is much closer to E x ( x , y (cid:38) L ) , the transverse component E y is suppressed for x (cid:39) . L are reduced by a factor 2. The correction to thefield cage could be tuned to larger values, producing an outward distortion of the drift paths, despite theforce due to the space-charge density ρ + remains attractive.Indeed, the irreducible effect of space charge is the distortion on the longitudinal electric field E x , whichat x = L / α E ◦ L /
16, at lowestorder in α . The drift delay δ t and the corresponding longitudinal distortion δ x receive contributions ofopposite sign from the regions close to the cathode (where E x > E ◦ ) and to the anode ( E x < E ◦ ), and theeffects are further reduced by the partial saturation of the electron drift velocity for typical values of E ◦ .On the other hand, the transverse distortion δ y ( x , y , z ) = (cid:90) x = ( x , y , z ) E y E x dx (7) As an illustration, consider for simplicity the one-dimensional model at lowest level in α : for a space charge of integrateddensity Σ + = (cid:82) ρ + dx , the cathode reacts with an increase in the absolute value of its surface density by ( / ) Σ + , and the anodewith a reduction by ( / ) Σ + . (m) x (m)0 61 2 3 4 50123456 0 61 2 3 4 5 y ( m ) -
300 0 c m w/ correctionw/o correction
73 cm - - - - - - - - - - - - - - Fig. 7:
Contours of equal voltage V(x,y) and drift paths, with a usual voltage gradient at the field cage (left), andwith a third voltage connection at x = 3.5 m, as discussed in the text (right). The anode is at x =
0, the cathode at x = y =
0. Reproduced from reference [6]. receives in general coherent contributions along the entire drift path. With the usual, linear field-cagevoltage profile, the maximum magnitude of δ y near the side face is significantly larger than the max-imum longitudinal distortion in the central region of the detector. Moving away from the field cage,the amplitude of the transverse component of the electric field and of the transverse distortion decreasesapproximately as exp [ − y / ( . L )] [6], and the main manifestation of space charge is the longitudinaldistortion. The Gauss equation can be written as ∂ E x ∂ x = ρ + ε − ∂ E y ∂ y − ∂ E z ∂ z , (8)where the derivative of the main component E x is related to the density of the space charge and tothe derivatives of the transverse components E y , E z . As discussed in Section 3.1, in usual detectorconfigurations the transverse components and their derivatives are due to space charge, and since theyare directed outward and their magnitude decreases moving inside the detector, they reduce the effect of ρ + on E x .In detectors with lateral size at least as large as twice the drift length, in both transverse directions, thederivatives of the transverse components of the electric field can be neglected in the central region of thedetector, defined here as the region far from the side faces by a distance greater than L , but still extendingfrom the anode to the cathode. In this region the one-dimensional description of space charge is still9pplicable. This is close to the situation reported by ICARUS [12], where only small corrections areneeded because of the limited lateral extension of the detector.For narrower detectors, E y and E z still vanish in the center of the detector because of contributions ofopposite sign from opposite faces, but their derivatives collect same-sign contributions from all sides.In this condition, the voltage gradient established by the field cage is able to reduce the effects of spacecharge across the entire detector volume. Naturally, narrow detectors are affected by transverse distortionover a large fraction of their volume, in a way similar to pincushion aberration, maximal near the centerof the side faces, and vanishing at the corners and at the center of the detector.In such condition, the lateral distortion may be reduced with a modification of the voltage profile at thefield cage, as discussed above in Section 3.1, but the advantage of such option should be weighed againsta corresponding increase in the longitudinal distortion. Numerical examples can be found in reference[6]. A discussed above in section 2.2.1, the observation of space-charge effects made by the ICARUS Col-laboration [12] has been compared by the authors to the prediction of the one-dimensional model. Thetransverse distortions due to the limited vertical extension of the drift volume are of low relevance be-cause the tracks crossing the region near the top and bottom field cage are excluded from the test sample,unless they are also close to the anode. The prediction for the amplitude of the observed longitudinal ( δ x )distortion is corrected by about −
6% for the detector geometry and −
12% for the presence of negativeions from capture of the drifting electrons. The procedure is consistent with the considerations and theexamples provided in reference [6]. In that study, for a detector with lateral extension twice as large asthe drift gap along one direction, and much larger along the other, the field cage changes the amplitudeof the longitudinal distortion near the center by about − −
15% when the electron lifetime measured in ICARUS is used.The amplitude of the longitudinal distortion observed by ICARUS is used to obtain the ratio of chargeinjection rate K to the ion mobility µ , with the latter assumed equal for positive and negative ions. With K set at the value of 1 . × − C m − s − , as usually done for detectors on the Earth surface and operatedat an electric field of 500 V cm − , the ion mobility is µ (cid:39) . × − m kV − s − , a value on the low sideof the range of the available measurements.The MicroBooNE Collaboration has provided an extensive report on transverse and longitudinal distor-tions caused by space charge [13]. As mentioned above in section 2.2.2, the vertical extension of thedetector (along y ) is similar to the drift gap (along x ), while the extension along the third coordinate ( z )is much larger. Far from the detector boundaries in z , the effects of space charge can be predicted withthe two-dimensional model described in [6]. Considering first the transverse distortion, its maximumvalue is given by | δ y max | = . α L for a detector with transverse size much larger than the drift gap L . For the aspect ratio of MicroBooNE, namely with L = .
56 m and the opposite faces of the field cageseparated by 2.32 m, the effect is reduced by 7%. From the observed value | δ y max | (cid:39)
23 cm, the value ofthe space charge dimensionless parameter is determined as α (cid:39) .
0. Near the Earth surface, and with thevalue of the charge injection rate expected for an electric field of 274 V/cm, the corresponding mobilityis µ + (cid:39) . × − m kV − s − .For the longitudinal distortion, obtained after the calibration procedure described in section 5, the Mi-croBooNE collaboration has observed in the xy plane, far from the detector boundaries on the z axis, amaximum value δ x max (cid:39) . δ x max = . α γ L , for a detector withtransverse size much larger than the drift gap L , with γ = ( dv e / dE )( E / v e ) describing the dependence ofthe electron drift velocity on the electric field. The distortion is reduced by about 50% for a detector with10he aspect ratio of MicroBooNE. The value of the space charge parameter corresponding to the observed δ x max is α (cid:39) .
8, and the corresponding ion mobility is µ + (cid:39) . × − m kV − s − .It is possible that the different values of the ion mobilities reported here reflect an incomplete descriptionof the physical effects. The value obtained by ICARUS is in reasonable agreement with the lower valuederived from MicroBooNE, but they refer to different observables. MicroBooNE reports asymmetriesin the pattern of the transverse distortion between the top and bottom side faces, and between the op-posite faces at the far sides of the detector. The vertical distortion on the top face shows an unexpecteddependence on the z coordinate. These observations suggest that the distribution of space charge is af-fected by fluid convective motion in a significant way — although not at the level apparently found withthe ProtoDUNE Single-Phase detector. In order to explain the difference in the mobility values derivedfrom MicroBooNE, fluid motion should effectively enhance the effect of space charge near the field cageand/or reduce it in the central region of the detector. In dual-phase TPCs, at the end of the drift regions the electrons cross an extraction grid before enteringthe vapor phase, where they undergo gas multiplication. Part of the positive ions produced in the mul-tiplication process are driven back to the vapor-liquid interface, and a fraction of them may cross theextraction grid and contribute to the space charge density together with the ions from primary ionization.Large dual-phase liquid argon TPCs have been constructed [15], and the subject of positive ions transportat the vapor-liquid interface has been considered, with different conclusions, in [11] and [16]. While sofar positive ions from gas multiplication have not been directly observed, their contribution to space-charge effects in the drift volume has been discussed in [6]. In this section we illustrate the extension ofthe one-dimensional analysis that includes positive ions from primary ionization and feedback ions frommultiplication in gas.The extraction grid is designed to be transparent to drifting electrons and to facilitate their transport intothe vapor phase. Ignoring electron capture, the flux of electrons at the extraction grid is J e = K L , i.e. itis equal to the rate of charge density injection integrated over the drift length. The corresponding fluxof positive ions entering the drift region is J + = β K L , where the parameter β is the product of differentfactors: (a) the fraction of electrons driven to the amplification region; (b) the gas multiplication gain; (c)the efficiency in driving the positive ions to the vapor-liquid interface; (d) the transport into the liquid;and (e) the limited transparency of the extraction grid for positive ions. The value of J + at x = ρ + is ρ + ( x ) = K ( x + β L ) µ + E x ( x ) . (9)The Gauss equation (3) can be directly integrated, obtaining: E x ( x ) = E ◦ (cid:113) ( E a / E ◦ ) + α [( x / L ) + β ( x / L )] . (10)In the comparison with equation 4, we see that the solution is now determined by two dimensionlessparameters: the space charge parameter α defined in equation 5 and the ion feedback parameter β . Thevalues of the electric field at the anode E a is an integration constant determined as usual by (cid:82) E x dx = V ◦ .The reduction in E a as a function of α is significantly enhanced by the presence of ion feedback, asshown in Figure 8. For β ≥
1, the critical condition of vanishing E a is reached for values α < = 20, 10, 5, 2, 1, 0 left to right E a /E o Fig. 8:
Behavior of E a / E ◦ vs. α , for different values of the feedback parameter β . Reproduced from reference [6]. Calibration procedures are necessary because of the large uncertainties in the value of µ + , and for uncer-tainties and possible space dependence in the value of K . Both parameters affect the value of α . Besides,calibration may be necessary because of uncertainties in the way fluid motion affects the distribution ofspace charge.Laser beams for calibration purposes have been used in MicroBooNe [17] and design is underway forDUNE [18]. Two methods are usually considered. In the first, the laser is used to provide well definedsources of photoelectrons at predetermined locations. For example, photo targets placed on the cathoderespond to laser pulses providing a precise measurement of the total drift time and of the transversedistortion for drift paths across the full gap. In the second method, UV lasers are used to generatedionization tracks in liquid argon TPCs via multi-photon ionization [19]. Movable mirrors can be used tosteer laser beams across different regions of the drift volume, suitable for comparison with uncalibratedtracks from reconstruction. In MicroBooNE, an iterative correction procedure is used to determine thespatial displacement between true and apparent coordinates, using laser systems placed on opposite sidefaces of the detector. Alternatively, a calibration procedure can be applied to the crossing points oftracks originating from the two laser beams: the expected (true) coordinates of the crossing point can becompared to the corresponding values from uncalibrated tracks, providing directly a local measurementof all components of the spatial distortion.Besides the laser system, MicroBooNE has extended the method of crossing tracks, replacing laser beamswith crossing cosmic muons [13]. The method relies on the position of the end-points of tracks crossingthe detector boundaries. As discussed in Section 2.2.2, on the side faces of the detector the spatialdistortion is normal to the surface and it can be directly measured on observed tracks, before calibration,while no distortion is present for the end-points at the anode. Once all true end-points are determined,pairs of nearly crossing muons can be used to compare the expected coordinate of the (near) interceptpoint with the corresponding one from uncalibrated tracks. The calibration of the transverse distortionsfor electrons drifting from the cathode is not directly available, and MicroBooNE uses an approximationof it, interpolating the ratio of observed to expected distortion measured on the edges of cathode and sidefaces. In principle, a detector with a double-drift, anode–cathode–anode configuration could avoid suchapproximation using anode–to–anode crossing muons.Deep underground detectors would not benefit from calibration based on cosmic muons, but they should12lso be free from space-charge effects, and calibration methods are foreseen to reduce instrumental un-certainties of different origin. In recent years, the subject of space charge in ionization detectors has seen an increase of interest, drivenby the foreseen operation of calorimeters under unprecedented level of radiation, and by the develop-ment of large size liquid argon TPCs for neutrino experiment. In this paper, the experience gained withtwo calorimeter technologies and with three TPCs has been reviewed. The formalism used to describethe main features and the scaling laws of the space charge phenomenology has been discussed along itsmain lines. A new way to consider the transverse effect near the side faces has been presented. Themultidimensional model has been compared to the observations made with the ICARUS and the Micro-BooNE detectors. A brief description of calibration methods has been provided, discussing in particulara data-driven method, that may have further application in the SBN neutrino physics programme whichis about to start. Even with detailed modeling, calibration may remain fundamental for dealing with theeffects of fluid motion, which can alter significantly the equilibrium configuration of space charge andelectric field. It is possible that modeling of fluid motion will be proven sufficiently accurate, and maybea space-charge aware detector design, in a wider sense than done so far, will improve the reliability ofthe predictions. The SBN programme, and further developments on DUNE detector prototypes operatedon surface will provide the ground and the motivation for advancement during the next few years.
Acknowledgements
The author thanks Filippo Resnati, Flavio Cavanna, Kirk McDonald, Michael Mooney, Francesco Pietropaolo,Stephen Pordes, Tingjun Yang and Bo Yu for useful discussions on the subject of this paper.
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