Analytical model for event reconstruction in coplanar grid CdZnTe detectors

2.9 MB·English

by  Matthew Fritts

Analytical model for event reconstruction in coplanar grid CdZnTe detectors
MatthewFrittsa,∗,JürgenDurstb,ThomasGöpferta,ThomasWestera,KaiZubera
aInstitutfürKern-undTeilchen-Physik(IKTP),TechnischeUniversitätDresden,D-01062Dresden,Germany
3 bErlangenCentreforAstroparticlePhysics(ECAP),Friedrich-AlexanderUniversitätErlangen-Nürnberg,Erwin-Rommel-Str.1,
1
D-91058Erlangen,Germany
0
2

n
a
J
Abstract

2
Coplanar-grid(CPG) particle detectors were designed for materials such as CdZnTe (CZT) in which charge

] carriers of only one sign have acceptable transport properties. The presence of two independentanode sig-
t
e nals allows for a reconstruction of deposited energy based on the difference between the two signals, and a
d
reconstructionoftheinteractiondepthbasedontheratiooftheamplitudesofthesumanddifferenceofthesig-
-
s nals. Energyresolutionisgreatlyimprovedbymodifyingthedifferencesignalwithanempiricallydetermined
n
weightingfactortocorrectfortheeffectsofelectrontrapping.Inthispaperisintroducedamodifiedinteraction
i
. depthreconstructionformulawhichcorrectsforelectrontrappingutilizingthesameweightingfactorusedfor
s
c energyreconstruction. Theimprovementofthisdepthreconstructionoversimplerformulasisdemonstrated.
si Furthercorrectionsduetothecontributionofholetransporttothesignalsarediscussed.
y
h Keywords: Coplanargrid,CdZnTe,Depth-sensing,γ-rayspectroscopy,Semiconductordetector
p
[

1. Introduction two betas emitted) and to distinguish it from other
2
v backgrounds, including the continuous spectrum
CZT as a γ-ray detector material has a number
4 of the competing process of neutrino-accompanied
0 of attractive properties such as a large band-gap double-beta decay (2νββ). The depth information
6 and good performance at room temperature. The
providedbytheCPG designisalsoveryimportant,
6 COBRA experiment [1] uses CZT detectors in a
for two reasons. First, an important category of
.
1 search for neutrinoless double-beta decay (0νββ) backgroundis α- or β-radiation at the cathode and
1 due to the presence of several 0νββ candidate iso-
anodesurfaces. Theshortpenetrationofsuchback-
2 topesinCdZnTeanditslownaturalbackgroundra-
groundallowsforits efficientdiscrimination, if the
1 dioactivity. CurrentlyCOBRA utilizesa 4×4×2
: interactiondepthiswellknown. Second,forcertain
v array of 1 cm3 CPG detectors operating under low
regionsinthedetectordistortionoftheenergyrecon-
Xi background conditions at the Gran Sasso National structionisanunavoidableresultoftheCPGdesign.
Laboratory(LNGS),withplanstoupgradeto64de-
r Depthinformationcanbeusedtoidentifyeventsin
a tectorsinearly2013.
these regions and remove them from consideration
Good energy resolution is an important consid-
as 0νββ candidates. For both of these phenomena
eration in gamma spectroscopy. It is also key for
analysis cuts are necessary, and an accurate assess-
COBRA in order to enhance the sensitivity to a
mentoftheselectionefficiencyofsuchcutsrequires
0νββ line (producedby the summed energy of the
themostaccuratedepthinformationachievable.
Suchconsiderationsgivestrongmotivationforan
∗Corresponding author. Tel [phone]; fax 49 351 accuratedepthreconstructionforCPGdetectorsfor
46337292. theiruseintheCOBRA experiment,andforallap-
Emailaddresses:
plications in which accurate knowledge of the ac-
[email protected]
(MatthewFritts), tivevolumeisrequired. Inaddition,duetothelarge
[email protected](Jürgen numberofdetectorsusedin COBRA, an automatic
Durst),[email protected](Thomas
methodforgeneratingdepthformulasuniquetoeach
Göpfert),[email protected]
detectorishighlydesirable. Materialpropertiesand
(ThomasWester),[email protected](Kai
Zuber) optimal operational parameters differ significantly
PreprintsubmittedtoElsevier January3,2013
2 CPGPRINCIPLES 2
from detector to detector, necessitating unique for-
mulasforbothenergyanddepthreconstruction.
2. CPGprinciples
The CPG detector design was introduced by
P. Lukein 1994[2]. A properunderstandingofthe
design principles begins with a treatment based on
the Shockley-Ramo theorem. For a more detailed
descriptionoftheapplicationoftheShockley-Ramo
theoremtoCPGdetectorssee[3].
InaCPGdetectorsuchasthoseusedinCOBRA,
the cathode is patterned on one side as a uniform
rectangle, while on the opposite side two isolated
anodegridsare formedinthe shapeofinterlocking
combs. Figure 1 is a schematic of a COBRA CPG
detector, manufacturedby EI Detection & Imaging
Systems [4]. In operation a large (approximately
1 kV) bias (referred to as HV) is applied between
the cathode and one anode, while a smaller (typi-
cally 50-100 V) bias (GB) is applied between the
anode grids. The anode held at higher potential is
referred to as the collecting anode (CA); the other
anodeisknownasthenon-collectinganode(NCA).
Electronsexcitedintotheconductionbandbyapar-
ticle interaction somewhere in the bulk of the de-
tector drift straight toward the anode plane (in the
negative-z direction) until they get close to the an-
odegridrails,atwhichtimetheyaredivertedbythe
near-anodefieldtobecollectedintheCA.
A first step in the application of the Shockley-
Ramo theorem is to calculate the weighting poten-
tials[5,p.813-8].Figure2showstheweightingpo- Figure 1: Schematic electrode layout for CPG de-
tentials of the CA and NCA along a plane through tectors used by COBRA. Two interleaved anodes
thecenterofthedetector,calculatedusingtheelec- and an outer guard ring (uninstrumented) are pat-
trode geometryof COBRA detectors.1 At most lo- terned on one side (top); the other side (bottom) is
cationsinthedetector,thetwoweightingpotentials a uniform cathode. Shown here is the convention
arenearlyequal,risingwithaslopeof1/2fromthe usedinthispaperforthez axis(interactiondepth).
cathodeplanetotheanodeplane. Atthepositionof Thedashedlinesindicatethesectionoverwhichthe
a CA rail(normallythe finallocationofthemobile calculatedweightingpotentialisshowninFigure2.
electrons created by ionization), the CA and NCA
weighting potentials are 1 and 0, respectively, by
definition.
Thesepropertiesoftheweightingpotentialallow
for a difference signal based on the raw CA and
NCAsignalsthatisproportionaltothechargefrom
ionizationandindependentoftheinteractiondepth.
FromtheShockley-Ramotheoremonecancalculate
1 Incurrentoperationtheguardringelectrode(blackinFig-
ure1)isleftunconnected. Thisinformationisincorporatedinto
theweightingpotentialcalculations.
3 ELECTRONTRAPPING 3
theamplitudesofthetwoanodesignalsproducedby
electrons:
1
∆q = Q (z +1) (1)
CA 0 0
2
1
∆q = Q (z −1) (2)
NCA 0 0
2
Here ∆q represents the change in induced charge,
and thus the amplitude of the signal, for the corre-
ntial 1 spondinganode. Q0 isthemagnitudeofthemobile
ote chargeproducedbytheinteraction.z0isthedistance
p
g betweentheinteractionlocationandtheanodeplane
n
hti expressedasadimensionlessfractionofthedetector
g
A wei0.5 length.z0isreferredtoasthe“interactiondepth”(or
C simply“depth”).
The depth dependencein the two raw anodesig-
nalsisremovedbysubtraction:
0
0 ∆q −∆q =Q (3)
CA NCA 0
0.2
0.4 0.6 0.6 0.8 z1 Notethat∆qNCAisalwaysnegative,sotheequation
0.8 0.4 representsasumoftheabsoluteamplitudes.Bycal-
0.2
x10 ibrating this signal using sources of known spectra
one achieves a well-resolved reconstruction of the
energydepositedbyaninteraction.
Similarly one can remove the charge magnitude
dependencetoreconstructtheinteractiondepth[6]:
al
g potenti 1 ∆∆qqCCAA−+∆∆qqNNCCAA =z0 (4)
n
hti
g The picture of CPG operational principles pre-
ei
A w0.5 sented in this section can be referred to as zeroth-
C
N orderbehavior.First-ordereffects,whichcomplicate
this simple picture, are the subjectof the following
sections.
0
0
0.2
3. ElectronTrapping
0.4 1
0.6 0.8 z
0.6
0.8 0.4 Electron trapping in CZT detectors is a strong
0.2
x10 enougheffectthat,forCOBRAdetectors,thesignal
amplitudeofaninteractionoccurringatlargedepth
Figure2: Weighting potentialsofCA andNCA as isreducedbyabout10%relativetothatofaninter-
calculated for the COBRA CPG detector geometry action occurringat small depth. This depth depen-
across a plane through the center of the detector. dencewouldsignificantlydegradeenergyresolution
The coordinates are normalized to the detector di- were it notcorrected. Such a correctionis possible
mensions,thusrangingfrom0to1. duetothedepthinformationintrinsicintherawsig-
nals.
P.LukeandE.Eisslerintroducedasimpleweight-
ingfactor(referredtohereas w) tomodifythe dif-
ference signal in order to correct for electron trap-
ping[7]:
∆q −w∆q (5)
CA NCA
3 ELECTRONTRAPPING 4
theAppendixof[8],althoughheretheanodesignals
areseparatelyderived.Firstassumeameantrapping
ht lengthλthatisvalidfortheentireelectrondriftpath.
g
ei1000 This is only true to the extent that both the crystal
h
nal propertiesandtheelectricfieldareuniformthrough-
sig 900 outthedetectorvolume.Inactualitytheelectricfield
A
C isnearlyuniformforalargemajorityoftheelectron
800 pathformostinteractionlocations,andthusformost
ofthedriftdistanceinwhichtrappingoccurs.2 Thus
700 the magnitude of the charge as a function of drift
distancedistakentobe
600 Q=Q e−d/λ (6)
0
500 Aswithz0,λanddareexpressedintermsofdetec-
tor length and are thus dimensionless. λ is consid-
400 eredafreeparameterinthistreatment. Physicallyit
dependsupontheelectronmobility-lifetimeproduct
300 forthecrystalandthevoltageappliedtothecrystal,
both of which vary by detector. Thus λ is detector
200 dependent.
-600 -500 -400 -300 -200 -100 0 100
A second simplification is to assume that the
NCA signal height
weighting potentials of both anodes take their bulk
formalongtheentireelectrondriftpathupuntilthe
Figure3: CAversusNCAsignalheightsfor228Th
anode plane is reached, at which point the CA and
calibration data for one COBRA detector. γ-lines
NCAweightingpotentialsabruptlychangeto1and
from the source appear as upward-sloping lines in
0, respectively. The idealized weighting potentials
this plot. The red line indicates 2.615 MeV events
as functions of depth z are illustrated in Figure 4.
from208Tldecay. Theslopeofthisline,0.80,isthe
Inthismodeltheelectrondriftisreducedtoonedi-
empirically-determined weighting factor w. In the
mension,withthedriftentirelyinthenegativez di-
absenceoftrappingthisslopewouldbe1.
rection. Formostinteractionlocationsthisisarea-
sonable approximation in that, again, most of the
trapping occurs before deviations in the weighting
Theweightingfactorisempiricallydeterminedand
potentialbecomesignificantandbeforetheelectron
isalwayslessthanone. Thecorrectionfortrapping
pathisdivertedbythenear-gridfield. Theexpected
canbeunderstoodqualitatively:asinteractiondepth
increases,−∆q becomessmallerwhilethesig- sizeoftheerrorintroducedbythisapproximationis
NCA
discussedinalatersection.
nalreductionduetotrappingbecomeslarger.Under-
Giventheseassumptionstheexpectedanodesig-
weighting the NCA amplitude thus artificially low-
nalamplitudescanbecalculated. Wedividethecal-
ers the amplitudesof low-depthinteractionsto bet-
culation into two parts. The first part corresponds
termatchthetrapping-degradedamplitudesofhigh-
to the electrons drifting through the region of uni-
depthinteractions.
formweightingpotentials,where dVw =−1. Since
The weightingfactorw canbe determinedby an dz 2
the charge magnitude changes along the drift path
optimizationprocedureseekingthebestenergyres-
wecalculatethiswithanintegralformoftheRamo
olution. Alternativelyifinformationfrombothsig-
equation,yielding
nalsisrecordedseparatelytheweightingfactorcan
bedeterminedfromasinglecalibrationrunbyanal- ∆q = z0 1Q e−(z0−z)/λdz
1 0
ysisoftheCAandNCAsignalamplitudesforacal- Z 2
0 (7)
ibration line. Figure 3 illustrates this procedure as 1
= Q λ 1−e−z0/λ
performedinCOBRA. 2 0
(cid:16) (cid:17)
In the following paragraphs an analytical treat-
ment of the electron trapping effect in CPG detec- 2This is true considering the electrode geometry if one as-
sumes that space charge has a negligible effect. The shape of
tors is presented, using a few simplifying idealiza-
chargepulsesinCOBRAdetectorsshownoevidenceofsignifi-
tions. This treatment is similar to the derivation in cantfielddistortionsduetospacecharge.
4 EVALUATIONOFTHETRAPPINGMODEL 5
thetwosignalstoreconstructthechargeexcitedinto
theconductionbandbytheinteraction:
al 1
g potenti Collecting Anode (CA) ∆qCA− λλ−+11∆qNCA = λ+λ 1Q0 (12)
htin0.8 Non-collecting Anode (NCA)
eig Itisnoteworthythatthetrappingeffects,themselves
W
ofexponentialform,arecanceledexactlywithalin-
0.6 ear combination. For this it is necessary only that
the z -dependent terms in equations 10 and 11 are
0
identical in form. On the left-hand side of equa-
0.4
tion12isaweighteddifferencesignalequivalentto
theoneintroducedbyLuke. Ontheright-handside
istheexcitedchargereducedbyaconstantfactor,as
0.2
expected from the qualitative understanding of the
weighted difference. The constant is of course ul-
timately absorbed into the energy calibration. The
0
0 0.2 0.4 0.6 0.8 1
z relationshipbetweentheparameterλandtheempir-
icalweightingfactorwis
Figure 4: Idealized weighting potentials of CA
1+w
and NCA correspondingto path of chargesdrifting λ= (13)
1−w
throughaCPGdetector.
Anequivalentrelationshipisderivedin[8].
It is also possible to construct an analytical ex-
This result is valid for both the CA and NCA sig- pression for the interaction depth z from the raw
0
nals. It is closely related to the Hecht equation [5, signal formulas by eliminating Q from equations
0
p. 489], differing only by the factor one-half. In a 10and11:
signal formedby the sum of the CA and NCA sig-
1∆q +∆q
nals(sometimesreferredtoasthecathodesignal)the λln 1+ CA NCA =z (14)
exactHechtequationisrecovered.Thiscorresponds (cid:18) λ∆qCA−∆qNCA(cid:19) 0
to the weightingpotentialofa simple parallel-plate
Note that as expectedthe zeroth-orderdepth for-
electrodegeometry.
mula is recovered in the limit of large λ. This
The second part of the calculation of the anode
trapping-corrected depth formula is easily imple-
signalamplitudescorrespondstotheabruptchanges
mented because the parameter λ can be calculated
inweightingpotentialatz=0: +1/2forCAand-1/2
from w which is alreadydeterminedduringthe en-
for NCA. Inserting the final charge into the Ramo
ergycorrectionprocedure.
equationonefinds
∆q = 1Q e−z0/λ (8) 4. Evaluationofthetrappingmodel
2,CA 0
2
In the framework of the simple model for trap-
1
∆q =− Q e−z0/λ (9) pingpresentedherewecanpredicttheperformance
2,NCA 0
2 of the zeroth-order depth formula, now seen as an
Thefullexpressionsfortheanodesignalsare approximation.Itisclearthatitisnon-linear;more-
over,thezeroth-orderformulawilloverestimatethe
1
∆qCA = Q0 λ 1−e−z0/λ +e−z0/λ (10) interactiondepthinadetector-dependentway. From
2
h (cid:16) (cid:17) i the empirical weighting factors w determined for
∆q = 1Q λ 1−e−z0/λ −e−z0/λ (11) COBRA detectors the overestimation can be ex-
NCA 0
2 h (cid:16) (cid:17) i pected to reach a maximum of around 3-10% for
By inspection we see that, in the limit of large z =1.
0
λ,thecorrespondingno-trappingforms(Equations1 Figure 5 illustrates the advantages of the new
and2)arerestored. depth formula. Sources of alpha radiation at the
As in the zeroth-order case the position depen- cathode surfaces for the detectors currently in op-
dence can be removed by a linear combination of eration at LNGS produce an excess of events near
5 ADDITIONALFIRST-ORDEREFFECTS 6
s s
of event 200 ZTrearpopthin-og-rcdoerrr efocrtemdu floarmula detector 78
er of
mb er 6
Nu 150 umb 5
N
4
100
3
2
50
1
0 0
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0 0.01 0.02 0.03
z µτ (cm2/V)
Figure5:Interactiondepthzofnear-cathodeevents Figure 6: Electron mobility-lifetime products of
intheCOBRAdatacollectedatLNGSin31detec- 30 COBRA detectors, estimated by the methodde-
torswithdepositedenergiesgreaterthan2MeV.An scribed in the text. The mean value is 1.1×10−2
excess of interactionsoccurringvery nearthe cath- cm2/V.
odeisknowntoexist.Uncorrecteddepthreconstruc-
tionresultsindepthsoverestimatedbyafewpercent
andawiderdistributionduetodetectordifferences. 5. Additionalfirst-ordereffects
5.1. Holedrift
z = 1. Thetrueinteractiondepthfortheseeventsis In CZT holes have much poorer transport prop-
expectedtobeverycloseto1. Thefigurecompares erties than electrons; nevertheless they contribute
the depth distribution using both the zeroth-order significantly to the raw signalamplitudes. Because
andtrapping-correctedformulas. Thepredictedad- holesdriftmuchmoreslowlythanelectrons,itisfirst
vantagesareevident: thecorrecteddepthsaremuch necessarytoconsiderwhethertheircontributionsto
closer to one, and they are much more tightly dis- signalamplitudesarefullyrecordedinthedatapro-
tributed. cessing procedure. In standard COBRA pulse pro-
The empirically determined parameter λ can be cessingtheeffectivetimewindowwithinwhichthe
usedtoestimatetheelectronmobility-lifetimeprod- amplitude is measured is greater than 3 µs. The
uct(µτ)foreachdetectorasfollows: lifetime of holes in CZT is estimated at 1 µs, so
that more than 95% of the hole charge is trapped
L2λ within the processingtime window—assumingthat
µτ = (15)
V theholesdonotreachanelectrodewithinthistime.
bulk
Thustoareasonableapproximationwecanassume
where L is the detector length (1 cm) and Vbulk holescontributefullytotheprocessedsignalampli-
is calculated to be the potential difference between tudes.
the cathodeand the averageof the anodepotentials Inanalogyto theparameterλinthetreatmentof
(HV - 1GB). Figure 6 shows the resulting µτ esti- electrontrapping,wetakeρtobethemeantrapping
2
matesfor30COBRAdetectors. Themeanvalueof length for holes, normalized to the detector length.
1.1×10−2cm2/VisclosetothosequotedforCZT Becausetypicalλvaluesarearound10,andbecause
inseveralpublications(seeTable7in[9]).However themobility-lifetimeproductforholesinCZTises-
the wide distribution of values suggests significant timated to be about 1% that of electrons, ρ is esti-
variation across detectors, and possibly additional mated to be around0.1. Thusfor manyinteraction
effectsunaccountedforbythissimplifiedmodel.3 depthswedonotexpectholestodriftfarenoughto
3 One potential effect is an imbalance in the CA and NCA Howeverforthisstudysuchimbalancesweremeasuredandtheλ
signalamplificationfactorsarisingfromthespecificpropertiesof andµτ valuescorrectedaccordingly. Neithercantemperaturebe
theelectronics.Ifsuchanimbalanceexistsitwillcontributetothe afactorsincealldetectorsareoperatedunderessentiallyidentical
weightingfactorwindependentlyofelectrontransportproperties. conditionsatroomtemperature.
5 ADDITIONALFIRST-ORDEREFFECTS 7
be collected at an electrode; rather they will be al-
mtheosdtecteocmtoprleletenlgythtr.appedafter drifting a fraction of z1.1 25 vents
e
Qualitatively the hole effect can be pictured as 20 of
follows: holes will drift a short distance toward 1 ber
m
the cathode, along a path in the bulk region of the 0.9 15 Nu
weightingpotential.Theythuscontributeequalpos-
itive contributions to the CA and NCA signal am- 0.8
10
plitudes.Sincetheenergyreconstructionisbasedon
0.7
thedifferenceofthesesignals,theholecontributions
willcanceleachother. Ontheotherhandthedepth 5
0.6
reconstructionisbasedonthesumoftheanodesig-
nals, andso the holeeffectcanlead toa significant 0.5 0
500 550 600 650 700 750 800
overestimationofinteractiondepth. Energy (keV)
Quantitativelywecanconsidertheholeeffectus-
Figure7:Distributionofeventsindepthandenergy
ing the same framework as was used for electron
inaCOBRAdetectorirradiatedwitha137Cssource.
trapping.Assumingthattheinteractiondepthoccurs
The reconstructed energy of 662 keV γ events is
inthebulkregionoftheweightingpotential,wefind
somewhatunderestimatedatdepthsnear1duetothe
1 holeeffectleadingtoadeformationofthelinenear
∆qholes = Q0ρ 1−e−(1−z0)/ρ (16) thetopoftheplot.
2
(cid:16) (cid:17)
This contribution applies to both anode signals.
Considering the small value of ρ we see by in- depthwillbeoverestimatedbyapproximatelyρ.For
spectionthattheholecontributionisapproximately near-cathodeinteractionsthereisnooverestimation,
1Q ρ for interaction depths far from the cathode as is clear from Figure 5. Figure 8 illustrates the
2 0
(where 1 − z is several times ρ). In contrast the relationshipbetweentrueandmeasureddepth. The
0
expressionfallstozeroasz approaches1,sincefor clearestevidenceofthiseffectfromdataisthesmall
0
near-cathodeinteractionstheholeshavenodistance numberofeventsbelowacertainmeasureddepthfor
to drift. Thereforethe hole contributionis zero for populations of interactions known to have uniform
z =1. depthdistributions,asexemplifiedinFigure9.
0
The contribution of holes to the signals has Inprincipleofcourseonecanalsocorrectforthe
a second-order effect on the energy reconstruc- hole effect in the depth reconstruction. Howeverit
tion, producing a small reduction for near-cathode is more difficult to calculate the parameter ρ in the
events[7, 8]. Forinteractionsfar from the cathode calibration process. Another difficulty is that the
the weighted difference signal (Equation 12) is en- analytic expressionfor hole-correcteddepth has no
hanced by approximatelyρ/λ, on the orderof 1%. solution if the zeroth-orderdepth is greater than 1.
Thisconstantfactorwillbeabsorbedintotheenergy Sincethefiniteresolutionofmeasureddepthsome-
calibration. However the hole enhancement disap- timesresultsinvaluesgreaterthan1,thisisproblem-
pearsforinteractionsveryclosetothecathode.Thus aticforanevent-by-eventholecorrectionalgorithm.
theenergydepositonofnear-cathodeeventswillbe
underestimatedbyabout1%. Thisiscomparableto 5.2. Near-anodedistortions
theenergyresolutionofCOBRAdetectors. Theef- CPG event reconstruction depends on the nearly
fectisnoticeableonlyinalayerwhosethicknessis equalCAandNCAweightingpotentialsattheinter-
afewpercentofthedetectorlength. Figure7shows actionpoint. Howeverthetwoweightingpotentials
this small effect on a calibration line with roughly differsignificantlyatlocationsneartheanodeplane,
uniformdepthdistribution. asillustratedinFigure 2.
The hole effecton depthreconstructionis larger. The correspondingeffectsare somewhatcompli-
Itcanbeadequatelydescribedbyconsideringthead- cated, and depend not only on depth z but also
0
dition of the hole contribution to the zeroth-order on the x and y coordinatesof the interactionpoint.
equations (Equation 1 and 2) and the correspond- The energycan beeitherunderestimatedoroveres-
ingeffectonthezeroth-orderdepthformula(Equa- timated dependingon whetherthe interaction point
tion 4). For interactions far from the cathode, the liesaboveaNCAoraCArail. Thethicknessofthe
5 ADDITIONALFIRST-ORDEREFFECTS 8
z 1
d
e
ur
s
a
e
M0.8
0.6
0.4
0.2
s
nt
ve 400
e
of
0 er
0 0.2 0.4 0.6 0.8 1 b
z um 300
N
Figure8:Relationshipbetweenmeasureddepthand
200
truedepth(red)basedonan analyticaltreatmentof
the hole effect, assuming ρ = 0.1. The measured
depthis correctedforelectrontrappingbutnotcor- 100
rectedfortheholeeffect.
0
0 0.2 0.4 0.6 0.8 1
z
layer nearthe anode in which these effects are sig-
nificantis greaternearthelateralsurface ofthe de- Figure 9: Depth distribution of events from a
tector4 thanitisnearthe centeroftheanodeplane, COBRA detectorfordata taken at LNGS in an en-
whereitissmallerthan0.5mm.Whereitisthinnest, ergy range of 180-300 keV. The events consist al-
holes can actually cancel the distorting effect, by most entirely of intrinsic β-decay of 113Cd, and
drifting toward the cathode into the bulk region of thus are known to be evenly distributed throughout
theweightingpotential. the detector (although some events near the detec-
Where the effective near-anode layer is thickest, torsurfaceswillbelostduetoescapingβ particles).
holeswillnotdriftfarenoughtocancelthedistorting The lowcountofmeasureddepthsbelowabout0.1
effects. Aswiththereconstructedenergy,therecon- is evidence of the hole effect. The hole effect is
structeddepthisalsodistorted,butitwillgenerally also partiallyresponsibleforthe nonuniformdistri-
besmallerthanρ,sothatajudiciousdepthselection butionatgreaterdepths. Distortionsin energy-and
can remove such events from consideration in data depth-reconstructionat lower depths due to the de-
analysis. tailsoftheweightingpotentialalsocontributetothis
Very near the anode the electric field in the de- nonuniformity.
tector is dominated by the grid bias, which pro-
ducesadifferenteffect.Theregionofeffectdepends
strongly on the applied bias, but is estimated to be
typicallyaround0.1mm. Forinteractionsinthisre-
giontheholeswilldrifttowardtheNCA insteadof
the cathode. The corresponding hole contributions
tothesignalswillbeoppositeinsign,thusstrongly
4Thedistortionsintheweightingpotentialcanbereducedif
abiasisappliedtotheguardring[10]. Howeverforsimplicity
COBRAdetectorsarecurrentlyoperatedwiththeguardringun-
connected.
6 SUMMARY 9
enhancing the difference signal. Due to the short
hole drift distance, most holes are collected before
theyaretrapped. Theshapeoftheweightingpoten-
tialscorrespondingtothepathsofbothelectronsand
holes differ significantly from those corresponding
toeventsinthebulk.Theneteffectisthattherecon-
structed energy will be approximately doubled [6].
Thereconstructeddepthwillbeveryclosetozero.
Signals from interactions far from the anode are
also affected by the detailed form of the weighting
potentialsneartheanode.Ifsucheffectsareconsid-
ered Equations 10 and 11 must be corrected. This
correctionwillvarywiththexandypositionsofthe
interaction. However we can approximate the size
of the correctionby consideringa specific case: an
interactionoccurringona line leadingto thecenter
ofaCArailinthecentralregionoftheanodeplane.
Inthiscasethe weightingpotentialasafunctionof
zfitswelltoanexponentialformwithcharacteristic
lengthαofapproximately0.015(0.15mm).Wefind
thenthatEquations10and11arecorrectedbyterms
oforderα/λ. Thisisathird-ordereffect,muchless
than1%,andthuswhollynegligibleconsideringthe
energyanddepthresolutionsachievablewiththede-
tectors.
6. Summary
It has been shown that a simple model for elec-
trontrappingwithanidealizedformfortheweight-
ingpotentialprovidesamathematicalframeworkto
explaintheuseofaweighteddifferencesignaltore-
constructeventenergiesinacoplanargriddetector.
Using the same modela new formulafor the inter-
actiondepthwhichcorrectsforelectrontrappingef-
fectshasbeenintroduced. Thisformulademonstra-
blyimprovesthedepthreconstruction,especiallyfor
eventsnearthe cathode. Other effectsthat can dis-
torttheenergyanddepthcalculationshavebeendis-
cussed in an effort to understand event reconstruc-
tioninCPGdetectorsasaccuratelyaspossible.
Acknowledgments
TheauthorswouldliketothankTobiasKöttigfor
usefuldiscussions,theCOBRAcollaborationforuse
of data, and the Deutsche Forschungsgemeinschaft
foritssupport.
REFERENCES 10
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