Asymmetries in Silicon Microstrip Response Function and Lorentz Angle
aa r X i v : . [ phy s i c s . i n s - d e t ] M a r Preprint typeset in JINST style - HYPER VERSION
Asymmetries in Silicon Microstrip ResponseFunction and Lorentz Angle
Gregorio Landi a ∗ , and Giovanni E. Landi b a Dipartimento di Fisica e Astronomia, Universita’ di Firenze,Largo E. Fermi 2 50125 Firenze Italyand INFN, Sezione di Firenze,Firenze,ItalyE-mail: [email protected] b UBICA s.r.l.,Via S. Siro 6/1,Genova, Italy. A BSTRACT : An experimental set up, dedicated to isolate an error present in the h -algorithm, gavean unexpected result. The average of a center of gravity algorithm at orthogonal particle incidenceturns out to be non zero. This non zero average signals an asymmetry in the response function of thestrips, and introduces a further parameter in the corrections: the shift of the strip response centerof gravity respect its geometrical position. A strategy to extract this parameter from a standarddata set is discussed. Some simulations with various asymmetric response functions are exploredfor this test. The method is able to detect easily the asymmetry parameters introduced in thesimulations. Its robustness is tested against angular rotations, and we see an almost linear variationwith the angle. This simple property is used to simulate a determination of a Lorentz angle withand without the asymmetry of the response function.K EYWORDS : Particle tracking detectors; Si microstrip and pad detectors; Data processingmethods; Pattern recognition, cluster finding, calibration and fitting methods. ∗ Corresponding author. ontents
1. Introduction 12. Correction of the systematic errors 3 h -Algorithms 52.3 Correction of the h Algorithms 6
3. Determination of d g e k ( x gk ) x g ( e )
4. Non orthogonal incidence and Lorentz angle 16
5. Conclusions 21
1. Introduction
In many high-energy physics experiments, arrays of silicon microstrip detectors are fundamen-tal tools to track charged particles. The excellent position resolution of these detectors is essen-tial in the event reconstruction. To obtain the best performance, the role played by the position-reconstruction algorithms becomes crucial. For example, the final alignments are corrected withtrack reconstructions; any inaccuracy in the position reconstruction algorithms is systematicallydiffused to all the data. The use of reconstruction algorithms in the detector alignment and in thedata creates correlations that renders almost impossible to verify their consistency. Thus, an apriori exploration of their systematic errors is essential.In a previous article [3] we applied to silicon microstrip detectors the general equations wedeveloped in [1, 2] for the center of gravity (COG) algorithm. Among the many properties demon-strated for the COG, we underlined the presence of a systematic error in the so called h -algorithm [4],when used outside the symmetry conditions. The authors in [4] recommended the limitation to asymmetric configuration without demonstration. Thus, in the last years, the recommendation has– 1 –een neglected, and the h -algorithm has been used well outside its range of validity. It is easy toguess the production of many incorrect position reconstructions.The h -algorithm improves the COG-algorithm with a global analysis of a set of equivalentdata. Our procedure to define the h − algorithm is substantially different from that used in [4].We deduce it from the solution of a first order differential equation that has an easy solution fora uniform distribution of impact points. But, any first order differential equation always requiresan initial constant, in this case an exact impact point corresponding to a COG value. This type ofdatum is never available excluding some special cases. The initial constant is easily selected forsymmetrical configurations, and is zero with the definitions of [4]. For unsymmetrical configura-tion, for example at non-orthogonal incidence angles, an angle dependent shift is produced by theuse of the zero constant of the symmetric case. The shift depends on the form of the signal distri-bution. Thus, detectors aligned with minimum ionizing particles (MIP) could show non alignmentswith heavy ions (in reality there are non alignments in both cases). Similar apparent shift of a de-tector could be induced by the modification of the depleting tension or any other deformation of thesignal distribution. Simulations show shifts greater than the root mean square (RMS) error in somedirections, and always larger than the full width half maximum (FWHM) of the error distributions.In ref. [3], we demonstrate a method to correct it.We have to underline the importance of the h -algorithm in improving the position reconstruc-tions. The comparison of the RMS-error of the COG and h -algorithm does not show dramaticdifferences in favor of the latter, as the comparison of the FWHM. The reason of the small sen-sitivity of the RMS-error to the improvement of the h -algorithm is connected to the non-lineardependence of the two algorithms from their component stochastic variables. As it is well known,non-linearities introduce drastic deviation from the gaussian distributions toward slow decreasingprobability distributions. The Cauchy distribution is a typical member of this class. These non-gaussian distributions tend to have infinite variances as the Cauchy distribution. In this case, theRMS-error is essentially limited the selection strategy of the finite sample and it is insensible to thequality of the reconstruction algorithms. On the contrary the FWHM saves its sensitivity.In a test beam with a set of sensors of the PAMELA tracker [5], a special set up was exposedto the beam with the aim to measure the systematic error of the h -algorithm. The analysis of thecollected data [6] clearly confirms the presence of an angle-dependent shift, and the correctionproposed in ref. [3] is able to cancel the shift at any measured angle.In this work we concentrate the attention on an anomaly observed on the data of ref. [6] wherethe average of the COG distribution is appreciably different from zero for orthogonal particle inci-dence. In the absence of magnetic field, the maximal symmetry is expected for this configurationwith the COG probability distribution symmetric respect to the origin and zero average. The nonzero average could be originated by an asymmetry in the charge drift to the collection pads or someother (linear) distortion in the read-out chain.Our correction to the h -algorithm works identically for asymmetrical response functions, but,a further detector parameter must be known: the COG position of the strip response function. Infact, the COG algorithm assumes that the strip signals are concentrated in the COG position of thestrip response function. The asymmetry moves the COG response function from the strip axis, andthis shift must be accounted for in any reconstruction at any angle, not only in the h -algorithm.We have no control on the physics of the showering particle, but, we suppose to know all the– 2 –etector parameters, being the detector production under our control. In practice the situation isnot so simple. Various types of material depositions are performed in specialized places and slightasymmetries could be easily introduced during these operations, no visual or electronic inspectioncan isolate these defects. In addition to this, subtle asymmetries could be introduced in the path ofthe data from the detector to final user.Direct measurements could be performed, but they require auxiliary detectors with resolutionsmuch better than the tested detectors. It is evident the complexity of this task. We will tray to esti-mate the asymmetry from the charge collected by the strips for MIP at orthogonal incidence angle.In this way, a good angular measurement can replace a high resolution position measurements.In section 2 we give a direct demonstration of the h -algorithm correction in general cases toisolate the effects of the asymmetry. Section 3 is devoted to define our strategy to estimate theasymmetry parameter of the response functions and to test it on simulated data with two differenttype of asymmetry. Our simulations are tuned on the double sided silicon microstrip detectors ofthe type introduced by ref. [7, 8], and used in the PAMELA detector. In one side a strip each twois left unconnected, and it distributes the charge in a peculiar mode. We call this side floating stripside. The other side is normal (in the sense that it has no floating strips).Section 4 deals with the non orthogonal particle incidence and its relation with the asymmetry.The angular rotation introduces a simple and almost linear effect that allows a better determinationof the asymmetry. It gives even an indication of the angular precision to obtain significative results.This sensitivity to the angular rotation suggests a method to measure the Lorentz angle when amagnetic field is present. Here, the effect of a magnetic field an a silicon microstrip detector issimulated as an effective rotation of the incoming particle direction. A proper angular rotationis able to restore the maximal symmetry to the signal distribution. Our method easily find thiscondition even in presence of an asymmetry of the response function. The simulations of this caseshow an excellent sensitivity of the method.We are aware that these developments are very formal and complex, but the asymmetry cor-rection and the Lorentz angle are deeply buried in the properties of the COG algorithm. It is inter-esting that analytical developments are able to isolate them and reach the consistency displayed bythe simulations.
2. Correction of the systematic errors
In ref. [1, 2, 3] we extensively utilized the Fourier Transform (FT) and Poisson identity [9, 10](or the Shannon sampling theorem). Now, we will proceed in a different way that avoids sometechnical complications and underlines its generality.Let us derive the COG average. With the notation of [1, 2] and considering all the strips witha non zero energy, we have the following definition for the COG ( t is the strip dimension): x g ( e ) = (cid:229) n ∈ Z n t f ( n t − e ) (cid:229) n ∈ Z f ( n t − e ) (2.1)where f ( n t − e ) is the energy collected by a strip centered in n t for a signal distribution with itsCOG in e (for any e ∈ R ). We use an infinite sum, but the function f ( n t − e ) is expected to go– 3 –o zero for a fixed range of its argument (finite support function). An identical transformation onequation 2.1 gives: x g ( e ) − e = (cid:229) n ∈ Z ( n t − e ) f ( n t − e ) (cid:229) n ∈ Z f ( n t − e ) . (2.2)Equation 2.2 explicitly shows the t -periodicity of x g ( e ) − e and justifies the use of Fourier Series(FS). The assumption of absence of signal loss gives a flat efficiency surface and f ( n t − e ) has thesum rule: (cid:229) n ∈ Z f ( n t − e ) = ∀ e ∈ R , (2.3)allowing the suppression of the denominator in equation 2.2. The energy f ( n t − e ) is defined as theconvolution of the strip response function g ( x ) with the signal distribution j ( x − e ) . The responsefunction g ( x ) is centered on the fiducial strip position and j ( x ) has its COG in e :f ( n t − e ) = Z + ¥ − ¥ g ( n t − x ′ ) j ( x ′ − e ) d x ′ (2.4)With equation 2.3, the e -average on a period t of equation 2.2 acquires an easy aspect. The intro-duction of the integration variables x n = n t − e gives:1 t Z + t / − t / ( x g ( e ) − e ) d e = t (cid:229) n ∈ Z Z n t + t / n t − t / x n f ( x n ) d x n , the sum on n can be absorbed in the definition of the integration limits:1 t Z + t / − t / ( x g ( e ) − e ) d e = t Z + ¥ − ¥ x f ( x ) d x (2.5)Equation 2.5 is the first momentum of f ( x ) , and the convolution theorem for the first momenta [9]gives: Z + ¥ − ¥ x f ( x ) d x = d g t + d j Where d g and d j are defined as: d g = t Z + ¥ − ¥ x g ( x ) d x d j = Z + ¥ − ¥ xj ( x ) d x For their normalizations ( R + ¥ − ¥ j ( x ) d x = R + ¥ − ¥ g ( x ) d x = t ), d g is the COG position of theresponse function and d j is the COG position of the signal distribution. The COG d j is zero forour definition of e , and the average of equation 2.2 remains:1 t Z + t / − t / ( x g ( e ) − e ) d e = d g . (2.6)Equation 2.6 shows that the COG algorithm is a biased estimator of the impact point. To eliminatethis bias, equation 2.6 imposes that the fiducial strip position must be coincident with the COGof its response function g ( x ) , in this case d g =
0. Any deviation from this condition introduces aconstant shift in the reconstructed position. – 4 –n principle, the extraction of d g from the data is easy, one has to take a set of (uniform)events, where the values of { e j } are known, and to average the differences x g ( e j ) − e j . In practice,the value of e j is very difficult (or impossible) to measure with the due precision. Thus, we have tofind another strategy to obtain a reasonable estimation of d g from the data of a standard test beamexperiment.Equation 2.6 is evidently valid for a noiseless case. The data are surely noisy. Assuming asymmetric additive noise, it is easy to figure out how it will modify the COG. At fixed impactpoint the noise will spread the data around the noiseless COG value. The symmetry of the noisedistribution induces a symmetric distribution of COG values around the noiseless one and theaverages of the noisy data will converge to the noiseless ones. So, for a large data sample, ournoiseless equations will work identically even in presence of noise. h -Algorithms Let us see how d g modifies the correction of the h algorithms. As we proved in refs. [1, 3], h -algorithms may be extended beyond the two strip case used in ref. [4], and identified as a generalproperty of any COG algorithms. Due to their strict similarity, we will continue to call h -algorithmsall these extensions.The COG algorithms with different numbers of signal strips have very different properties andsystematic errors, and a great care must be devoted to avoid to mix them. For example, the cuts onsmall or negative values of signal strips may produce the mixing. In ref. [3], 2-strips, 3-strips and4-strips algorithms exhausted our needs, there we limited to consider incidence angles up to 20 ◦ .Above 20 ◦ , 5 or more strips are relevant, and other strategies can be used to reduce these cases tothe present developments.In the simulations, the set of events has e -values with a uniform distribution on a strip. Thisassumption supports our averages over e . As in ref. [3] we calculate the COG in a reference systembound to the event, we choose the maximum signal strip. The experimental events are spread over alarge number of strips. To be consistent with our simulations, we will assume that the set of events { e ( j ) } produces the uniform distribution of points { (cid:229) K ∈ Z e ( j ) + K t } . Thus, on a given strip, onehas the uniform distribution of points { e ( j ) + K j t } , where K j t is the distance of the strip with theimpact point e ( j ) from the given strip. This will be the definition of uniformity of events on a strip.Let us recall some aspects of the h algorithm [3] to define the notation. Assuming the existenceof a single valued function x gk ( e ) which is randomly sampled by our COG algorithm with k − strips(in the following the index k will indicate the number of strips used in the algorithm), the probabilityto have x gk is: P ( e ) (cid:12)(cid:12)(cid:12) d e d x gk (cid:12)(cid:12)(cid:12) = G ( x gk ) , where P ( e ) is the probability to have a value e and G ( x gk ) is the corresponding probability for x gk .The positivity of the derivative is reported in ref. [1] and it turns out that any incoming signal,with average positive signal distribution, has positive derivative. Assuming a constant probability P ( e ) = / t , one arrives to the first order differential equation:1 t d e d x gk = G ( x gk ) . (2.7)– 5 –he integration of equation 2.7 requires an initial constant (i.e., an exact value of the impact point e ( x gk ) ). For symmetric signal distribution and symmetric response function, the initial constant isthe center of the strip or one of its border. These special points have x gk = e . For the asymmetriccase, the initial constant must be determined resorting to other properties of the COG algorithms.The presence of noise modifies this picture introducing an average over the noise realization.To render the approach less heavy we will neglect this average, but now equation 2.7 becomes thedefinition of the function e k ( x gk ) .The uniform distribution of events e j on the array of periodic detector generates a periodicprobability distribution G p ( x gk ) (normalized on a period), and the solution of equation 2.7, for thesymmetric configuration, is given by: e k ( x gk ) = − t + t Z x gk − t / G p ( x ) d x (2.8)The initial constant used in [4] is e k ( x gk = ) =
0, but, as we discussed above, e k ( x gk = − t / ) = − t / e k ( x gk = ) = j ( x ) and for symmetric response function. Inall the other cases, the required correction will be indicated with D k .It is easy to show the periodicity of e k ( x gk ) − x gk , in fact, due to the periodicity and the nor-malization of G p we may rewrite the equation 2.8 as: e k ( x gk ) = x gk + Z x gk − t / ( t G p ( x ) − ) d x . (2.9)The integral is a periodic function of x gk , and we express it as a FS: e k ( x gk ) = x gk + + ¥ (cid:229) n = − ¥ a n e ( i2 p nx gk / t ) a n = t Z + t / − t / [ e k ( x gk ) − x gk ] e ( − i2 p nx gk / t ) d x gk , (2.10)and with the correction D k : e k ( x gk ) = x gk + + ¥ (cid:229) n = − ¥ a n exp ( i2 p nx gk / t ) + D k . (2.11)In the definition of the a n , the k -index, the number of strips used in the algorithm, is not explicitlyreported, but it is evident that a n depends from k .With low noise, the function e k ( x gk ) is a good approximation of noiseless form, and it sitson the most probable values of e for any x gk . This property is crucial for any best fit in a trackreconstruction. The absence of the correction D k introduces an average systematic shift of e k respect to the true e , quite evident in the simulations. h Algorithms
We calculate D k exploring the mean value of the differences e k ( j ) − e ( j ) in a case of a largenumber N of events and uniform distribution on a given strip as defined. The mean value must be– 6 –ero in the absence of systematic errors:1 N N (cid:229) j = [ e k ( j ) − e ( j )] = N N (cid:229) j = [ x gk ( j ) − e ( j )]+ N N (cid:229) j = [ + ¥ (cid:229) n = − ¥ a n e ( i2 p nx gk ( j ) / t ) ] + D k (2.12)The mean value of the FS, weighted with the probability G p ( x gk ) , gives a [3]. Adding and sub-tracting the position of the strip with the maximum signal m j , equation 2.12 becomes:1 N N (cid:229) j = [ e k ( j ) − e ( j )] = N N (cid:229) j = [ x gk ( j ) − m j ] − N N (cid:229) j = [ e ( j ) − m j ] + a + D k (2.13)The mean of ( e ( j ) − m j ) is independent from the COG algorithm, and we calculate it in the easiercondition. We use equation 2.6 with all the signal strips in the COG algorithm (four or five at most),to near the condition of equation 2.3. We will call this x g ¥ , and equation 2.6 becomes:1 N N (cid:229) j = ( x g ¥ ( j ) − m j ) = N N (cid:229) j = ( e ( j ) − m j ) + d g . (2.14)The average of e ( j ) − m j of the unknown exact impact points is reduced to known quantities.Substituting in equation 2.13, and imposing the zero average of ( e k ( j ) − e ( j )) the equation for D k becomes: 1 N N (cid:229) j = ( x gk ( j ) − m j ) − N N (cid:229) j = ( x g ¥ ( j ) − m j ) + d g + a + D k = D k = N N (cid:229) j = ( x g ¥ ( j ) − m j ) − N N (cid:229) j = ( x gk ( j ) − m j ) − a − d g We have to recall that the two expressions (cid:229) Nj = ( x g ¥ ( j ) − m j ) / N and (cid:229) Nj = ( x gk ( j ) − m j ) / N are theaverages of the COGs calculated in our reference system of the maximum signal strip. The constant a embodies the initial conditions (not limited to 0 or − t /
2) and the correction D k eliminates anyreference to the initial integration contant.To be complete, the correction D k + a eliminates even the systematic error of the COGalgorithm (with k -strips) due to the non zero d g and to the eventual loss given by the limitation inthe strip number. The loss has rarely a significative effect, but we are able to consider it. For theCOG indicated with x g ¥ the correction is simply − d g . The residual non zero average is the meanvalue of ( e ( j ) − m j ) that the asymmetry modifies respect to its zero value in the symmetric case.In the simulations of ref. [3], the COG algorithm with four strips was a good approximationfor x g ¥ , here we will use even the five strip algorithms. Due the physical meaning of d g , onceits value is obtained, the correction of any position algorithm and for any incidence angle can beimplemented. In the following we will need the correction D ¥ to the h -algorithm obtained startingfrom x g ¥ , this correction is given by − a − d g .An indication of d g = ( x g ¥ ( j ) − m j ) for theorthogonal incidence. This averages can not be zero for equation 2.14. It is a sum of two unknownquantities, and another equation is necessary to extract their values. To estimate d g , we need toreconstruct j ( e ) and explore its asymmetry. – 7 – . Determination of d g In refs. [1, 3] we demonstrated an equation to obtain the signal distribution from the COG algo-rithm. The j ( e ) is given by: d x g ( e ) d e = t (cid:229) n ∈ Z j ( n t − e − t / ) . (3.1)In the derivation of equation 3.1 the response function is assumed to be the lossless interval func-tion, and j ( x ) is the signal distribution. The expression 3.1 is the sum of copies of j ( t / − e ) shifted of n t with n ∈ Z . This gives a periodic function with overlaps of the tails (aliasing) if therange of j ( x ) is greater than t . For ranges less than t the reconstruction is faithful. If the range of j ( x ) is greater than t , the assembly of a set of contiguous interval functions avoids this limitation.For fluctuating signal distributions, as in the case of a MIP, equation 3.1 defines an averagesignal distribution. The reconstruction of j ( x ) of equation 3.1 requires the response function as a pure interval functionof size t , rarely this condition is verified, and a generic response function produces a redefinitionof j ( x ) . If the lossless condition equation 2.3 is maintained, we proved in ref. [1] that the responsefunction must be a convolution of an interval function with another (arbitrary and eventually asym-metric) function g ( x ) . In this case, the function of Eq. 3.1 is the convolution of the true signaldistribution with g ( x ) . We will continue to call j ( x ) any result of d x g ( e ) / d e even if it deviatesfrom the true signal distribution.In ref. [3], we explored a possible form of the response function for microstrip detector withfloating strips, and the following form reproduces the main aspects of the data: p ( x ) = Z + ¥ − ¥ P ( x − x ′ ) (cid:8) . [ d ( x ′ − / ) + d ( x ′ + / )]+ . [ d ( x ′ − / ) + d ( x ′ + / )] (cid:9) . (3.2)(This form is surprisingly similar to that measured in ref. [11].) Here, the reconstruction of equa-tion 3.1 generates the convolution of the signal distribution with the four Dirac d -functions ofequation 3.2, the low intensity Dirac d -functions have a negligible effect, but the effects of the twomain d ’s are clearly seen in figure 1 as two copies of the signal distribution.If the response function is asymmetric, the asymmetry is contained in x g ( e ) and transferred tothe reconstructed function. The asymmetry transfer to j ( x ) does not allow a direct extraction of d g . We have to resort to an indirect procedure. e k ( x gk ) The form of equation 2.10 for e k ( x gk ) is not well suited for our needs. Its inverse function x gk ( e k ) is of better use, and it is expressed by: x gk ( e k ) = e k + + L (cid:229) n = − L b n exp ( i 2 p n t e k ) (3.3)– 8 – a) b)c) d) Figure 1.
Reconstruction of j ( x k ) (with noise) using different numbers of strips: a) with 2 strips, b) with 3strips, c) with 4 strips, d) with 5 strips. The asymmetry parameter is 0.07 and 45000 events. b n = t Z t / − t / (cid:2) x gk − e k ( x gk ) (cid:3) exp (cid:0) − i 2 p n t e k ( x gk ) (cid:1) G p ( x gk ) d x gk where e k ( x gk ) is the result of equation 2.8 and L the maximum wave-number used, L around 45 isa reasonable cut off even if the limits ± ¥ will be often used. All the forms of j ( x ) are obtaineddifferentiating equation 3.3 respect to e k .As discussed, deformations are introduced in j ( x ) by the differences of p ( x ) respect to aninterval function. Another set of important deformations are produced by the loss. Two type ofloss are encountered: the intrinsic loss of the strip and the loss given by the suppression of non-zero signal strips. The first type of loss operates as an additional smooth deviation of the responsefunction from the pure interval function, it has a negligible effect on our procedures. The secondtype of loss introduces a strong deformations in j ( x ) . The presence of any type of loss is explicitlyexcluded by the form of equation 3.1, but we can, in any case, differentiate equation 3.3 and exploreits results. In the absence of noise, the deformations given by the second type of loss assume the– 9 –orm of Dirac d -functions. The limitation in the number of terms in equation 3.3 gives finitepeaks. The exclusion of signal strips in the COG algorithm generates forbidden x g -values. Herethe probability G p is zero and e ( x g ) has an interval of constant value. If we insist to invert thefunction e ( x g ) this constant horizontal segment becomes a vertical segment, and the differentiationgenerates a Dirac d -function. In general, if the strip number is even, one aspects peaks around e k ≈
0, for odd strip numbers the peaks are for e k ≈ ± t /
2. The amplitudes of the peaks areproportional to amplitude of the signal function acquired by the excluded strips [1]. For k = , d g = k =
2, the peakis not in zero due to the − / e k = − / x gk = − /
2. In fact, the peaks of x g are at e = ± /
2. With the lower integrationlimit to zero, e k = x gk = k = e =
0. It is evidentthat with an asymmetry in the response function nor x gk = − / x gk = D k fixes the correct e k and the correct positions of the peaks. The almost total suppression of the losseliminates the peaks for k = k = G p ) are easily encountered in noiseless case. The noise helpsto avoid G p =
0, but it easily adds other unwanted artifacts. The Cesaro’s method of arithmeticmeans [10] attenuates some numerical instabilities. x g ( e ) The exploration of the analytical form of x g ( e ) clarifies our path toward d g . Here all the propertiesof the detector and the signal distribution are explicitly underlined. In the case of orthogonalincidence, the incoming signal distribution is symmetric and its FT F ( w ) is real and symmetric.The response function p ( x ) has asymmetries and its FT P ( w ) is complex with P ( − w ) = P ∗ ( w ) .The form of x g ( e ) , with d g the first momentum of p ( x ) , is [3] (with t = x g ( e ) = e + d g + i + ¥ (cid:229) k = , k = − ¥ F ( − k p ) P ′ ( − k p ) exp ( i k pe ) , (3.4)where P ′ ( w ) is the first derivative of P ( w ) respect to w , and e is the impact point. We know that,even in the best condition, the equation 2.8 for e k has an incorrect initial constant. To handle theasymmetric case, we have to generalized a new e k ( x gk ) defined for any initial condition x gk beyondthe x gk = − / e k ( x gk ) = x gk + Z x gk x gk G ( x ) p d x The correction procedure must work for any x gk . It is evident that x gk = − / x gk = e ( x gk ) − x gk is now thedifference of e k from e , given the initial constant x gk . Substituting e with e k in equation 3.4, wehave: e = e k + ( e ( x gk ) − x gk ) x g ( e k ) = e k + ( e ( x gk ) − x gk ) + d g + + ¥ (cid:229) n = , n = − ¥ e b n exp [ i n p ( e k + e ( x gk ) − x gk )] . (3.5)– 10 –emembering equation 3.3, the comparison with equation 3.4 gives: b n = e b n exp [ i n p ( e ( x gk ) − x gk )] n = b = ( e ( x gk ) − x gk ) + d g e b n = i F ( − n p ) P ′ ( − n p ) (3.6)as expected b is the sum of the two unknown d g and the shift ( e ( x gk ) − x gk ) . To extract d g we needanother equation. The derivative d x g ( e k ) / d e k does not contain b , it has a shift of ( e ( x gk ) − x gk ) respect to the differentiation in the exact e . This shift is present as a phase factor in equation 3.6,and it goes to increase the asymmetry of j ( e k ) . In the symmetric configuration the phase relationsare easy: all the F ( − n p ) P ′ ( − n p ) are real and e b n imaginary.Due to the special form of the of the relation of b n and e b n , we can add a fictitious phaseparameter 2 p n x to b n to modify the asymmetry of j ( e k ) . For small asymmetry, we expect that thisasymmetry variation of j ( e k ) reaches its minimum when all the b n coincide with e b n . The phasefactors of the e b n are given by the intrinsic asymmetry of the response function, and are non trivialfunctions of n . These phase relations are not eliminated by the trivial transform implied by a globalshift of j and the intrinsic asymmetry cannot be reduced. An asymmetry parameter with a smallsensitivity to the noise is: W ( x ) = Z / − / (cid:2) j ( x + e k ) − j ( x − e k ) (cid:3) d e k . (3.7)Here j ( e k ) is d x g ( e k ) / d e k . The minimum of W ( x ) is obtained for a x m given by: x m = − ( e ( x gk ) − x gk ) , this is the second equation that allows the use of b in equation 3.6 to extract d g : d g = b + x m . (3.8) W ( x ) is expressed with b n as: W ( x ) = (cid:229) n ∈ Z (cid:2) | b n | + b n exp ( i4 p n x ) (cid:3) ( p n ) . (3.9)The first term is a constant and the x -dependence is a periodic function of period 1 /
2. When d g = x gk = − / b n = e b n it is easy to verify the minimum for x = e b n = −| b n | .In general, the minima of equation 3.9 produce the corrections of the h -algorithm for all the initial x gk . To see the effectiveness of the minimization of equation 3.9, we calculated W ( x ) with x = x gk from − W ( ) has minima where b n = e b n or,more precisely, in the points where e k − x gk =
0. In figure 2 we report W ( ) in function of theinitial x gk and effectively it has evident minima when e k − x gk =
0. In the simulations we use alow asymmetry z = .
02, and a noiseless simulation and x g algorithm for a floating strip sensor.Figure 2 gives an empirical support to our research of the minima for W ( x ) .– 11 – Figure 2.
Continuous line (blue): Asymmetry W ( ) calculated for all the initial conditions x g from -1 to 0.The dash-dotted line (red) is the correction e − x g , the asymmetry has minima when e − x g = . Noiselesssimulation with z = . and floating strip sensor In general, any initial condition can be used, but, the x m are widely different with an inefficientminimum search. The special form of ( b ) and the following substitution simplifies the search andeliminates the explicit dependence from the initial conditions: b ′ n = b n exp ( − i 2 p n b ) b ′ n = e b n exp ( − i 2 p n d g ) , (3.10)and, for any x gk , equation 3.8 is reduced to: d g = x m . A presence of a small loss has a negligible effect on this approach. Large loss, signaled by thepresence of peaks around zero or ± /
2, can strongly modify the minimum search.For example, inour first set of simulations, x g and x g have minima very different from that of x g , x g . The simulated data are generated as discussed in ref. [3]. For the floating strip case, we modifythe response function breaking the symmetry of the two most important Dirac- d functions of equa-tion 3.2. We add to the first Dirac- d function a constant z and a z is subtracted to the other oneto save the normalization, figure 1 has z = .
07 . This type of asymmetry looks similar to theone observed in the test-beam data, but we amplify the effect. In any case, this is only a numeri-cal experiment to see the efficiency of the d g determination. We will compare with the noiselesssimulation to see the effect of the noise.Figure 3 shows the determination of d g with the procedure illustrated above. The simulateddata are noiseless, but even here we see fluctuations of d g from equation 2.14. The fluctuations– 12 – z d g ( m m ) −0.1 −0.05 0 0.05 0.1−2−1.5−1−0.500.511.52 z d g ( m m ) a) b) Figure 3.
Noiseless case, the blue dots are the results of equation 2.14, the red asterisks are the d g obtainedby the minima of equation 3.9. The plot a is given by x g with four strips and b with five strips. originate from the reconstruction that requires the extraction of j ( e ) from histograms and, due toa finite set of data (45000 events), the procedure adds an effective noise that is lower in the caseof d g calculated with five strips. Here the attenuation of the fluctuations could be due a reductionof the slight loss, that is present in the four strip simulation due to the suppression of the signal(convolution of gaussians [3]) collected by the fifth strip. This loss is too low to produce a peak,but it contributes to the effective noise of G p .The realistic case (with noise) fluctuates more than the noiseless case. Even here the d g calcu-lated with five strips has less fluctuations than that calculated with four strips. The RMS error is 0.1 m m for the five strips case and 0.2 m m for the four strips case. In figure 4 we reported the averages (cid:229) Nj = ( x gk ( j ) − m j ) / N that is the signal of a non zero d g . The form of asymmetry generation, weused, gives an amplification of the COG averages by (relatively) small d g . For the floating strip– 13 – z d g ( m m ) −0.08 −0.04 0 0.04 0.08−4−2024 z d g ( m m ) a) b) Figure 4.
Noisy case. The dots are the results of equation 2.14.The asterisks are the d g given by the minimaof equation 3.9,and the squares are the COG averages. Plot a ) is for the four strip x g and b ) is for the fivestrip x g . side, the introduction of d g could be a minor correction around half micron, with all the sensorsoriented identically a parallel shift of the track is implied. If some sensor has a reverse orientation d g change sign and gaps of a micron are present in the tracks. Even if these constant shifts couldbe corrected by the alignment procedures, it is a good practice to have estimators free of bias whenpossible. – 14 – .6 Normal strips We explore the strategy of the extraction of d g for the case of "normal" strips. Even now the impactdirection is orthogonal to the detector plane. At this angle, the detector resolution is low due to theconcentration of a large parte of the signal in a single strip. To be consistent with the real detector,the simulated noise is doubled respect to the case of the floating strip sensor, and its effect stronglydeteriorates the extraction of d g . Here we have no indications of the type of asymmetry, the meanvalue of x g is different from zero, but the reconstruction does not show evident asymmetry. Weproduce the asymmetry with an additional Dirac- d function zd ( x − t / ) convoluted with the usualinterval function to have an everywhere flat efficiency. The values of z are all positive, we mustavoid negative values of the response function and of d x g ( e ) / d e .For the noiseless case, the reproduction of d g is reasonable for all the z values even if thefluctuations introduced by the finite number of events is higher than the corresponding case ofthe floating strips. The addition of the noise changes drastically the results, the determinationof d g degrades rapidly at increasing z , now large values of d g are connected to lower values of (cid:229) Nj = ( x gk ( j ) − m j ) / N . Here we report even the results of the two-strip algorithms, and they arebetter than the four strip case. In general, the loss of the two strip COG could give incorrect results,the peak around zero can be very high and it drastically deforms j . In this case, the noise washesaway the peaks, and the noise reduction of the two strip algorithm gives a j ( e ) more sensible tothe asymmetry parameter d g than j ( e ) . The use of the two strip algorithms could be interesting inpresence of high noise. z d g ( m m ) Figure 5.
Noiseless case. The blue dots indicate the results of equation 2.14.The red asterisks are the d g obtained by the minima of equation 3.9 with four-strip algorithm. The blue squares are the COG averages – 15 – z d g ( m m ) Figure 6.
The effect of the noise. The meanings of the symbols are that of figure 5. The magenta crossesindicate the d g obtained with the two-strip algorithm.
4. Non orthogonal incidence and Lorentz angle
In the previous calculations, the orthogonality of the incoming particles was often recalled as afundamental condition to access to the asymmetry of the response function. But, the effects of thedeviations from the orthogonality must be explored to test the robustness of the algorithm. The b n of equation 3.9 have terms F ( − p n ) (FT of the true incoming j ( x ) ) in their definition 3.6, forthe orthogonal incidence any F ( − p n ) is real (and symmetric). An angular deviation q ( q = F ( − p n ) and it introduces a largeasymmetry in equation 3.9. Some plots of j ( x ) with q = o are reported in ref. [3]. For example, avalue of q = . o gives x m = . m m for a symmetric response function (floating strip case). Thus,the asymmetry of the signal distribution, can easily mask the asymmetry of the response function.The q data must have sufficient accuracy to detect small effects. In any case, the asymmetryinduced by q = q and d g remains constant. So, for sufficientlysmall angles, where the total asymmetry is almost linear, the collection of data at various anglesaround q = x m with a low degree polynomial function can give abetter value of d g .To explore the variation of x m from q , we process the convolution ( j ∗ g ) of our model [3]of j ( x ) , with the machinery of equation 3.7. This is a very easy operation due the explicit FTexpression of ref. [3]. The results are illustrated in figure 7 for the g of floating strip sensors.A good linear relation is obtained for small q values, this linearity is driven by two effects: aphase factor proportional to the angle in the model function, and the two copies of j given by thetwo Dirac-delta of g . For comparison, in the normal strip case the absence of the two delta adds– 16 –on linear distortions. This linearity of x m is saved (with a small reduction of the slope) in ourreconstructed j ( e ) and it is almost insensible to the noise.The direct application of equation 3.7 on j ∗ g gives x m -values that depend very weakly onthe asymmetry z due an almost complete cancelation of the first order terms. In any case, j ∗ g isaccessible only in the simulations and this cancelation is irrelevant in the data. On the contrary, the h -algorithms introduce phase factors proportional to d g in the FS-amplitudes of j ( e ) as a globalshift of the function. Thus, W ( x ) and equation 3.10 allow the extraction of d g from the data. Afterthe proper d g -correction of the h -algorithm, W ( x ) gives a minimum for x m ≈
0. The simulations −2 −1 0 1 2−3−2−10123 q o x m m m Figure 7.
Asymmetry z = , floating strip sensors. Dotted line (blue) x m on the convolution ( g ∗ j ) ofthe model signal distribution, asterisk line (red) x m of j ( e ) and crosses line (magenta) is the x m for thenoiseless case. with an asymmetry z = .
04 are reported in figure 8 at different angles q (step 0 . o ) of a floatingstrip sensor. As in figure 7, the x m -values have a linear relation with the angles as in the symmetriccase, but the line is shifted by − . m m that is its crossing with the q = e ( x g ) corrected with D for a symmetric response function, and, as expected,is constant in q and equal to d g . The addition of the correction − d g to D completely eliminatesthe systematic error in e ( x g ) . The average of x g is different from zero at q =
0, signaling theasymmetry of the response function.Similar results can be obtained for the normal strip case. The absence of the floating strip andthe high noise make the plots of x m to deviate from the good linearity of the floating case. Or better,the linear approximation has a restricted range of validity. As in figure 6, the asymmetry obtainedfrom x m is less than the right one and part of the systematic effect remains uncorrected. The x m of j ( e ) gives a better estimation of d g than that of j ( e ) , but it has a strong deviation from linearity.A fit with a low degree (3,4) polynomial function could be used. In any case, it is under studya more refined extraction of j ( e ) from the data with a strong suppression of the noise distortion.Preliminary results [12] support a drastic improvement of the method– 17 – Figure 8.
Asymmetry z = . , floating strip sensor. Crosses line (blue) x m of j ( e ) , dotted line (red)systematic error of e without the correction d g and asterisks line (magenta) is the average of x g . −3 −2 −1 0 1 2 3−1.5−1−0.500.511.522.53 Figure 9.
Asymmetry z = . , normal strip sensor. Dot-blue line x m of j ( e ) , cross-green line x m for j ( e ) . Dot-Red line, systematic error of e without the correction d g , and asterisks-magenta line is theaverage of x g . The effect of the magnetic field on the particle-holes drift in a silicon detector is usually parame-terized as a rotation of the particle path of an angle q L . The rotation is around an axis parallel tofield containing the impact point. The effective COG of the track is shifted from the true one if thestrip direction is non orthogonal to the field. The strips of the floating strip side of the PAMELA– 18 –etector are parallel to the magnetic field, and the assumed value of q L is 0 . o . If the magnetic fieldhas an effect similar to a rotation on the signal distribution, the present approach naturally measure q L . Usually, this measure is performed on the average length of the clusters produced by the MIPat various incidence angles. The minimum of the cluster size is at an incidence angle of − q L (inthe geometry of ref. [3] where the impact point is always in the collection plane). At this angle, theapparent signal distribution is probably similar to that of an orthogonal incidence, or in any case itobtains its maximal symmetry.The method to measure q L with the average cluster size has a low sensitivity just around theLorentz angle. The data reported in ref. [13] shows clearly this limitation. It would be better to havea method with an high sensitivity just around q L . Our averages of x g , and x m have the propertyto go to zero at q = q = − q L , the averages of x g , and x m are able to measure q L . With the definition of q e f f : tan ( q e f f ) = tan ( q ) + tan ( q L ) , the COG algorithm sees a particle track with q e f f bending angle, and its reconstruction has aneffective shift of the true COG of: D L = d ( q L ) , with d the depletion length of the detector (300 m m in our case of completely depleted sensors).The correction D L must be subtracted by any reconstruction algorithm. −3 −2 −1 0 1 2 3−4−20246 q o x m m m Figure 10.
Lorentz angle q L = . o symmetric floating strip sensor. Triangles-blue line x m of j ( e ) .Crossed-green line: linear interpolation of x m . Dot-red line systematic error of e without the correction D L , and asterisks-magenta line is the average of x g . Figure 10 illustrates the sensitivity of x m and the of average of x g to q L =
0, each one crossesthe q = − . o ( x m at − . o and x g at − . o ). Here the detector is perfectly– 19 –ymmetric, thus, the average of x g is an easy and sensible tool to extract q L . The asymmetryparameter x m is equally sensible, but more complex to calculate. It is clear that the symmetrycondition can be verified in the detector without magnetic field, and the average of x g must bezero for q =
0. In the case of an asymmetric response function one has to resort to x m . Forits structure x m is produced by two independent effects: the asymmetry of the response functionand the effective deviation from orthogonality. The asymmetry of the response function d g mustbe measured without the magnetic field and the x g , corrected accordingly. With the corrected x g , , x m goes again to zero for q =
0. The correction d g is constant with q , thus, the addition of themagnetic field gives x m = q = − q L . The presence of the asymmetry d g = x g or x g , and they never go to zero for q =
0, or q = − q L with the magnetic fieldand are not usable to measure q L .The combined effect of the Lorentz angle and the asymmetry d g is illustrated in figure 11.Here, a simulation of the floating strip sensor with the asymmetry of figure 8 and a q L rotation, iselaborate as in figure 10. −3 −2 −1 0 1 2 3−4−20246 q o x m m m Figure 11.
Lorentz angle q L = . o asymmetric floating strip sensor z = . . Triangles-blue line x m of j ( e ) . Cross-green line: linear interpolation of x m . Dot-red line systematic error of e without thecorrection D L , and asterisks-magenta line is the average of x g − d g . Now, figure 11 shows that x m , corrected with d g continues to cross the zero line at q L = − . o .The corrected average of x g does not cross the zero line at q L = − . o and its use as an estimator of q L is destroyed by d g =
0. An interesting property of x m is its sensitivity to two types of asymmetrythat combine in a non interfering way. The correction of d g = x g atthe beginning of the calculation of x m or implemented at the end (subtracting its value from x m ).These two different procedures give identical results. This property resembles a linear combinationof effects.Similar analysis performed on the normal strip sensors gives analog results. The quality of the– 20 – L determination has a similar precision, its resolution is better than that of d g .
5. Conclusions
The properties of the COG algorithms are able to access at a very detailed aspects of the detec-tor: the COG position of the response function and the Lorentz angle. The direct measurement issufficiently complex and could be unnecessary in many typical case. This extraction can estimatethese parameters from the data acquired in standard test beam (or in a running experiment), witha simpler requirement of precise angular positioning of the detector. The noise introduces pertur-bation, but a strong relation to the asymmetry is saved even in the worst case. The Lorentz angledetermination shows a modest sensitivity to the noise.The present procedure is able to separate the intrinsic asymmetry of the j ( e ) and the inducedasymmetry due to incorrect initial conditions. The minimal asymmetry should be the intrinsic one,but it is conceptually difficult to separate the two. In spite of this, the simulations show an excellentability to detect d g in the noiseless cases, giving to the phase shifts of equation 3.10 a robustmeaning in the explored range of asymmetry. The noise modifies this picture adding a blurringin the reconstructions that perturbs the efficacy of equation 3.10. But, the moderate noise of thefloating strip side has a negligible effect on d g . In the simulations, the x g has a RMS of 4 . m m ona strip pitch of 51 m m . For the normal strips, the noise is drastically higher (a RMS of 10 . m m on astrip pitch of 63 m m ) and d g estimation appreciably degrades. The noise tends to mask the effect ofthe asymmetry adding deformations that round j ( x ) with a decreasing of the resulting d g .The indicator of a non zero d g is the average of x gk at orthogonal incidence. In the two caseswe explored, this average has a quite different relation to the asymmetry. In the first case, to alarge x gk -averages corresponds a small asymmetry, the reverse in the second case. Equation 3.1, asin the example of figure 1, allows a visual inspection of j ( x ) . For normal strips, the noise in the x g -algorithm masks almost completely the asymmetry, but the two strip algorithm is able to giveinteresting results.The asymmetries we consider have their principal effect on the central strip. The capacitivecoupling introduces long range interactions in the nearby strips, and these interactions can be asym-metric. The x gk -averages are sensible to very small effects and they may signal even long rangeasymmetries. Equation 3.1 is not fit to handle these effects, it overlaps the tails of the j ( x ) outsidea strip range creating fake distortions. Assembly of strips must be explored if an indication of theselong range effects is acquired.The robustness of the approach is tested at non orthogonal incidence angle. The parameter x m shows a surprising strict linear behavior, in the case of floating strip sensor, that allows an increaseof precision with a linear interpolation of the data. For the normal strip case, appreciable deviationfrom linearity are observed, but, even in this case an interpolation with a low degree polynomialhas beneficial effects on the d g determination.The simulations at non orthogonal incidence suggest that the approach can be used for theLorentz angle determination. The approximation of the magnetic field effect as an effective rota-tion of the reference system is probably very rough, in any case x m is able to detect the angle ofmaximal symmetry with an excellent precision. In the case of d g = x g and of x g cross the q =
0– 21 –ine at the maximal symmetry. These indicators become useless in presence of small d g =
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