A synchronization method for the multi-channel silicon telescope
P. Žugec, M. Barbagallo, J. Andrzejewski, J. Perkowski, N. Colonna, D. Bosnar, A. Gawlik M. Sabaté, -Gilarte M. Bacak F. Mingrone, E. Chiaveri
AA synchronization method for the multi-channel silicon telescope
P. ˇZugec a, ∗ , M. Barbagallo b,c , J. Andrzejewski d , J. Perkowski d , N. Colonna b , D. Bosnar a , A. Gawlik d , M. Sabat´e-Gilarte c,e ,M. Bacak c,f , F. Mingrone c , E. Chiaveri c ,The n TOF Collaboration a Department of Physics, Faculty of Science, University of Zagreb, Croatia b Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy c European Organization for Nuclear Research (CERN), Geneva, Switzerland d Uniwersytet Ł´odzki, Lodz, Poland e Universidad de Sevilla, Spain f Technische Universit¨at Wien, Vienna, Austria
Abstract
A simple method is presented for the simultaneous o ff -line synchronization of the digitally recorded data-streams from a multi-channel silicon telescope. The method is based both on the synchronization between the separate pairs of silicon strips and onthe synchronization relative to an external timing device. Though only a reduced subset of these constraints is necessary in idealcircumstances, it is shown that this minimal set of conditions may not be su ffi cient for adequate synchronization in all cases. Allavailable sources of information are therefore considered, in order to constrain the final synchronization as well as possible. Keywords:
Silicon telescope, Multi-channel synchronization, Neutron time of flight, n TOF facility
1. Introduction
The synchronization between multiple sampling channels isa common enough challenge in experimental nuclear physics,as well as other areas of research and technology. To thisend, many di ff erent solutions were developed (see, for exam-ple, Refs. [1, 2, 3, 4, 5]). Older data acquisition systems, relyingon the analogue electronic units such as the time-to-digital con-verters (TDC) and signal discriminators, have to be synchro-nized in advance, by a careful adjustment of the delay linesand the signal intake settings. The more recent types of dig-ital electronics, such as the fast signal digitizers, profit fromthe possibility of implementing the complex on-the-fly or post-processing synchronization algorithms (to be applied during orafter the signal acquisition). Naturally, an absolute time cali-bration requires a timing reference, typically an external timingdevice (see Ref. [3] for a concise and succinct description). Im-plementation of such on-the-fly algorithms is, of course, morechallenging than of the post-processing ones, as the additionalhardware and signal interlacing requirements need to be met.We provide here a simple, purely post-processing method thatcan be applied after the pulse-processing stage of extracting thephysical data from the registered signals. The obvious practicaladvantage of such a posteriori method is that it can be utilized atthe very late stage in the data analysis, without having to repro-cess the signals in case the time o ff sets between multiple chan-nels were belatedly identified. In that, the method simultane-ously takes into account both the absolute timing constraints – ∗ Corresponding author. Tel.: +
385 1 4605552
Email address: [email protected] (P. ˇZugec) / ntof in respect to the external timing device – and the relative timingconstraints between all admissible pairs of channels. This featis based on observing the statistical properties of already iden-tified pulses, which could hardly be achieved by other means.Section 2 describes the details of the experimental setup andthe context of the synchronization issues. Section 3 presents theproposed synchronization method: all the necessary considera-tions to be taken into account, as well as the necessary imple-mentation details. Section 4 summarizes the main conclusionsof this work.
2. Experimental setup
The neutron time of flight facility n TOF at CERN is thehighly luminous white neutron source spanning 12 orders ofmagnitude in neutron energy – from 10 meV to 10 GeV. Its op-eration is based on the 20 GeV proton beam from the CERNProton Synchrotron irradiating a massive lead spallation target,serving both as the neutron source and the primary moderatorof the initially fast neutrons. The second stage of moderationtakes place in the borated or demineralized layer of water fromthe cooling system surrounding the spallation target. The gen-eral features of the n TOF facilty are well documented and maybe found in Ref. [6].Today the n TOF facility accommodates two experimentalareas: Experimental Area 1 (EAR1) located at the horizontaldistance of 185 m from the spallation target [6] and the Exper-imental Area 2 (EAR2) situated 20 m above the same target[7, 8, 9]. Each experimental area is specially suited to the par-ticular set of challenges in measuring the di ff erent types of neu-tron induced reactions: from the neutron capture and the neu- Preprint submitted to Nuclear Instruments and Methods A September 9, 2020 a r X i v : . [ phy s i c s . i n s - d e t ] S e p eutrons
50 mm
ΔE (20 μm) E (300 μm) ΔE (20 μm)50 mm E (ΔE (20 μm) E (300 μm)
Figure 1: Top: detector configuration from the joint measurement of the C( n , p ) and C( n , d ) reactions, comprising two identical silicon telescopes(all widths are exaggerated). At the center is the bearing structure for the LiFcalibration sample (we use these calibration data in this work). Bottom: SITEclose-up. The striped structure of ∆ E -layer (shown here) is clearly visible. tron induced fission to the reactions with the light charged par-ticles in the exit channel [10]. EAR1 o ff ers an excellent neutronenergy resolution and allows for the high neutron energy mea-surements due to the increased neutron flight path. Thanks tothe extremely high instantaneous neutron flux EAR2 providesthe unprecedented capabilities for measuring very low neutronreaction cross sections, including the measurements with verysmall and / or highly radioactive samples.Many di ff erent types of detectors are employed at n TOF,each specially suited to the measurements of the particular typeof reaction. A general review of these detectors and the asso-ciated signal analysis procedures may be found in Ref. [11].One of these detectors – relevant to this work – is the multi-channel silicon telescope (SITE), recently introduced at n TOFfor measurements of the neutron induced reactions with thelight charged particles in the exit channel, such as the ( n , p ),( n , d ), ( n , t ), ( n , α ) reactions [12]. This detector was first usedin the highly challenging measurement of the Be( n , p ) reaction[13], which was also accompanied by the measurement of the Be( n , α ) reaction [14], relying on the similar type of the siliconsandwich detector [15]. Both of these measurements, of centralimportance for the famous and as yet unresolved Cosmologi-cal Lithium Problem, became feasible only with the successfulconstruction of EAR2.Two such multi-channel telescopes were recently used in s] m Time [0 1 2 3 4 5 6 7 8 S i gna l [ a . u .] - - - - - Signal PS fit
Figure 2: Pulse from one of the SITE strips and the optimally adjusted numer-ical pulse shape. The amplitude and the timing properties are determined fromthe fit. the joint energy-di ff erential measurement of the C( n , p ) and C( n , d ) reactions [16], performed at EAR1 of the n TOF fa-cility. This measurement was motivated by the unexpected re-sults from an earlier integral measurement of the C( n , p ) re-action [17, 18], yielding an integral cross section higher thanindicated by any past dataset. The analysis of the data fromthe latest C( n , p ) and C( n , d ) measurements is under way,and the special analysis procedure has already been developedin order to properly take into account the challenging nature ofthese reactions [19]. The geometric configuration of the useddetector setup is shown in Fig. 1. We refer to the top telescopeas the front SITE and the bottom one as the rear
SITE. In short,each SITE consists of two segmented layers of silicon strips,each layer comprising 16 strips, all of them oriented in the samedirection (rather than perpendicular between the layers). Bothlayers are 5 cm × × ∆ E -layer and the second, E -layer – are distanced by 7 mm.Their respective thicknesses are 20 µ m and 300 µ m. Further de-tails about the telescope construction and readout may be foundin Ref. [12].The signals from two SITE were recorded and digitized at125 MS / s sampling rate with a 14-bit resolution. They wereanalyzed by the pulse shape fitting procedure described inRef. [11] – specifically, by adjusting a numerical pulse shapeto the baseline-corrected pulses and extracting the pulse ampli-tude and timing properties from such fit. An example of theSITE pulse, together with the adjusted pulse shape is shown inFig. 2.Another detector of importance to this work is the Wall Cur-rent Monitor (WCM) [20] – an induction device specificallydesigned for registering the proton pulses delivered by the Pro-ton Synchrotron. WCM o ff ers a reliable response to a protonpulse, registering with high fidelity the intensity of the beam, aswell as the arrival time of the pulse.The main purpose of this work is to provide a simple methodfor a simultaneous synchronization of the digitally recordeddata-streams from all involved detector channels, i.e. from allsilicon strips. The proper temporal synchronization of all chan-nels is crucial for the proper identification of the coincidentalpulses between the two ( ∆ E and E ) layers, signaling the detec-tion of a charged particle. The basic idea has already been used2o determine the synchronization between the entire ∆ E -layerand entire E -layer in measurement with the Be sample [12],and to determine the appropriate coincidence window width forthe Be( n , p ) data. However, no notable time o ff sets withina given layer were observed, so it was su ffi cient to consideronly a single spectrum of time di ff erences between the pulsesfrom any silicon strip in ∆ E -layer and any strip in E -layer(the overall distribution of di ff erences), immediately yieldingthe data-recording time o ff set between the two layers as themean value of this distribution. During the joint C( n , p ) and C( n , d ) measurement the time o ff sets between several stripsfrom within the same layer were observed. The reason may beas simple as using transmission lines of mismatched length intransmitting the signals from the particular strips to the digitaldata recording system. There may also be other, unaccountedsources of time o ff set within the entire data acquisition chain.In regard to the Be measurement, it must be taken into accountthat a di ff erent acquisition chain was used – the one dedicatedto EAR2, as opposed to the one from EAR1 in case of the Cmeasurement – thus justifying the di ff erence in the synchro-nization issues between the two experiments. Therefore, weexpand here the basic idea from Ref. [12], providing the gen-eral procedure for the simultaneous synchronization of all datachannels, taking into account any and all timing informationavailable in order to constrain the time o ff sets between the par-ticular strips and some external clock as well as possible.For purposes of energy calibration of the silicon strips, ameasurement of the Li( n , t ) reaction was performed during the C campaign, using the Li-enriched LiF calibration sample.The synchronization issues, i.e. the time o ff sets between mul-tiple channels are, of course, independent of the used sampleand / or the measured reaction. In this work we show the datafrom the Li( n , t ) measurement for a simple reason: they yieldslightly more presentable plots. In addition, if the synchroniza-tion procedure needs to be repeated in a course of some othercampaign, the main reaction of interest will change. On theother hand, the Li( n , t ) measurement is a standard calibrationprocedure employed at n TOF, so that by presenting these par-ticular data we provide the possibility for the fully consistentcheck against the measurement that is repeated between di ff er-ent campaigns.Incidentally, the time o ff sets – which, in our case, are mostlycontained below 100 ns – would not, if left uncorrected, notablya ff ect the time of flight spectra of the C( n , p ) and C( n , d ) re-actions, as their relevant energy range around 20 MeV (aimedby the measurement from n TOF) corresponds to EAR1 timeof flight to of approximately 3 µ s. However, the identificationof coincidences between the strips does critically depend on theadequate synchronization, as the o ff sets may become compara-ble to or larger than the appropriate coincidence windows. Inaddition, if the reactions of interest were taking place at higherneutron energies, or even if the intermediate-energy measure-ments were performed at shorter flight paths – such as EAR2 ofthe n TOF facility – the measured time of flight spectra mightbecome severely a ff ected even by the time o ff sets in excess oftens of nanoseconds.
3. Synchronization method
Let n enumerate the relevant events, be it the instance ofsome reaction of interest or some separate event. Let s enu-merate the particular silicon strips (of which there are 32 in asingle SITE from n TOF). Then the time instant t ns of the n -thevent, as registered by the s -th strip may be expressed as: t ns (cid:39) T n + τ s , (1)with T n as some reference time of that event, and τ s as thisparticular strip’s o ff set relative to the timing device yielding T n .For simplicity of notation we use throughout this paper the sym-bol (cid:39) (to be read as ”is expected to be”) to indicate the statisti-cally expected values, in a sense that (cid:104) t ns (cid:105) = (cid:104) T n + τ s (cid:105) . It is ourgoal to determine the o ff sets τ s , taking into account the max-imum amount of information available from the experimentaldata, thus constraining the set of τ s as well as possible.There are two types of pulses regularly recorded by any de-tection system adopted at n TOF. One is the so-called γ -flashpulse, caused by an intense burst of γ -rays and ultrarelativis-tic particles from the spallation process producing the neutronbeam. The other type consists of the those pulses followingthe γ -flash, related to the detection of the neutron-induced re-actions of interest, and to the background processes caused bythe competing neutron reactions and the environmental radioac-tivity. We will use the first type – the γ -flash pulses – for theabsolute timing calibration, together with the second type – thecoincidental pulses from the detection of the neutron-inducedreactions – for the relative calibration of the strip o ff sets τ s . For the reference time instants T n of the particular γ -flashesone could consider the γ -flash pulses from one selected siliconstrip. However, due to the insensitivity-by-design (in order for adetector not to be blinded by the γ -flash) individual strips havea low e ffi ciency for detecting the γ -flash. This is also the rea-son why, following each γ -flash, no strip can be consistentlycalibrated relative to its own γ -flash pulse, thus necessitatingthe external timing information. As an external timing devicewe use the WCM, whose reliable response to each and everyproton pulse is immediately followed by the release of the γ -flash. Thus, we take the instant T n of the proton beam delivery,as registered by WCM, as the reference point for the absolutetime calibration of silicon strips.Let us define the time o ff set: ∆ ns ≡ t ns − T n (2)for the n -th γ -flash pulse registered at t ns by the s -th strip. Wewill take into account the fidelity of the γ -flash detection bythe particular strip (or its proper recognition during the pulseshape analysis of electronic signals) by weighting its contribu-tion by the γ -flash pulse amplitude A ns , as registered by thatstrip. The choice of the weighting factors – these particularones having been selected for the conceptual simplicity – is es-sentially arbitrary, as the final results are rather insensitive to3 wide class of alternative selections (e.g. A / ns ), provided thatthe selected function of A ns (here linear) is neither pathologi-cal nor inordinately selective of any particular range of ampli-tudes. As opposed to the later unweighted procedure – relatedto Eq. (8) and motivated therein by the amplitude variations re-flecting an intrinsically meaningful spectrum – weighting thestrips’ response to a γ -flash is justified by the registered γ -flashamplitudes indeed being a measure of the reliability of this re-sponse. The reason in twofold: (1) the intensity of the protonbeam from the Proton Synchrotron – as the primary cause ofthe γ -flash – is well defined and usually limited to one of thetwo particular values (the lower and higher intensity, known in-ternally as the parasitic and dedicated mode), with very littlevariations, while there are broad amplitude variations in eachstrip’s response; (2) there are broad variations in di ff erent strips’response to the same γ -flash.For those strips that did not register the γ -flash, or the regis-tered pulse was rejected for any reason during the data analysis,one simply takes A ns = A ( s )1 ≡ (cid:80) N n = A ns and A ( s )2 ≡ (cid:80) N n = A ns , (3)with N as the total number of proton pulses delivered, we mayexpress the weighted averaged o ff set for the s -th strip as:¯ ∆ s ≡ (cid:80) N n = A ns ∆ ns A ( s )1 , (4)together with the unbiased estimator of its weighted variance:Var s ∆ = (cid:16)(cid:80) N n = A ns ∆ ns (cid:17) − A ( s )1 ¯ ∆ s A ( s )1 − A ( s )2 / A ( s )1 . (5)In that, Var s ∆ is the sample variance (related to the width of thedistribution of ∆ ns ), as opposed to the variance of the mean ¯ ∆ s ,which equals: Var s ¯ ∆ = Var s ∆ × A ( s )2 / (cid:0) A ( s )1 (cid:1) .The black plot from Fig. 3 shows a total distribution of o ff sets ∆ ns from all 64 silicon strips, related to the Li( n , t ) calibrationmeasurement. One can clearly observe that the average o ff setbetween the WCM and SITE signals is approximately 350 ns.The root-mean-square (RMS) of the distributions for particularstrips varies between 30 ns and 280 ns. The rest of the plotsfrom Fig. 3 will be discussed later.From the definition of o ff sets ∆ ns by Eq. (2) and the centralrelation from Eq. (1) it is evident that the mean o ff sets ¯ ∆ s pro-vide a set of estimators for the sought o ff sets τ s : τ s (cid:39) ¯ ∆ s . (6)This is a first set of equations for τ s , one that apparently im-mediately provides the entire set of τ s . However, we shallsoon observe that this set of conditions is of insu ffi cient qual-ity. The reason is precisely the occasional inaccuracy in the γ -flash pulse recognition within some of the silicon strips. Inthe presence of competing pulses in the immediate vicinity ofthe supposed γ -flash pulse, an erroneous pulse may be assigned [ns] s t - ns D - - - C oun t s =0) s t No sync. ( ) s D = s t Partial sync. ( ) m = t Full sync. (M h6_2
Figure 3: Overall distribution of di ff erences between the γ -flash instants asregistered by a particular silicon strip and as registered by an external timingdevice (WCM). The synchronization procedure aims at identifying the set ofo ff sets τ s such that the condition from Eq. (6) is satisfied, i.e. that the mean ofthe corrected spectra is as close to zero as possible. The full synchronizationprocedure takes into account further requirements from Eqs. (10) and (13). a status of a γ -flash pulse. These sporadic misidentificationspropagate into the calculation of the average ¯ ∆ s , thus makingthese estimators prone to a certain degree of error. We willtherefore use additional sources of information. In conjunctionwith these, the conditions from Eq. (6) will be shown to behaveonly as the good initial estimates for τ s . The additional constraints upon the set of o ff sets τ s may beobtained by observing the time di ff erences between the pairsof pulses from any pair of strips. From entirely uncorrelatedpulses one would expect a flat or featureless contribution to thespectrum of time di ff erences. On the other hand, coincidentalpulses gather around a well defined value, corresponding to arelative o ff set between the strips, thus forming a recognizablespectral peak.Let α and β denote the two strips either from the same orseparate ( ∆ E or E ) silicon layer. We now consider the time dif-ferences between all the pulses registered during the particularmeasurement. In this case the index n from Eq. (1) denotes alldetected counts – as opposed to the sole γ -flash pulses fromSection 3.1 – while T n corresponds to an instant of the n -thdetected event as it would have been registered by an externaltiming device (WCM) if this device were used for the detectionof these events. Simply put, it is the detection time accordingto an external clock. Defining the time di ff erence between thecoincidental counts from strip α and strip β : δ n αβ ≡ t n α − t n β , (7)one immediately observes that it will be invariant of T n and willprovide an estimator for the relative o ff set τ α − τ β between thetwo strips.It should be noted that by plotting the distribution of time4 [ns] ba n d - - - C oun t s E E = 2 (F) ba E E D = 4 (R) ba E E = 5 (R) ba E E = 10 (F) ba E E D = 6 (F) ba Figure 4: Distribution of the registered-time di ff erences between the counts de-tected in coincidence by several arbitrarily selected ( α, β )-pairs of strips eitherfrom the same ( ∆ E or E ) or separate ( ∆ E and E ) silicon layers. In legend thedesignation F / R indicates either the front (F) or rear (R) telescope (see the topscheme from Fig. 1). Numbers are strip designations from a given layer (1–16),with the layer itself being identified by the superscript ∆ E or E . di ff erences between any and all pairs of pulses from a par-ticular pair of strips, one in general observes the di ff erences t n α − t m β between the separate ( n -th and m -th) events. How-ever, by recognizing and selecting only the counts from thecoincidental spectral peak, one ensures that both time instantsfrom Eq. (7) belong to the same ( n -th) event. Of course, whenthere are reasonable indications for the expected relative o ff -set τ α − τ β , one does not need to consider all the possiblepairs of pulses from the separate strips. Otherwise, the pro-cedure is apt to become extremely computationally ine ffi cient,especially when the recorded data-stream (signal waveform)is much longer than the expected o ff set between the strips,which is certainly the case at n TOF ( ∼
100 ms waveform vs.max αβ | τ α − τ β | ≈
100 ns). Instead, one just considers the pairsof pulses within the appropriate time window. Alongside theresponse to the γ -flash, the physical cause for the coincidencesbetween the strips from separate ( ∆ E and E ) silicon layers isself-evident, as it constitutes the working principle of the sili-con telescope – it is the detection of a charged particle passingthrough both layers (tritons from the Li( n , t ) reaction in caseof the energy calibration data used in this work). Moreover,even for the certain immediate pairs of strips from the same sil-icon layer there are available coincidental counts, caused eitherby the γ -flash or by the signal separation between the neigh-boring strips due to the charge particle passing close to theirshared boundary. This is certainly a source of information tobe further exploited. Figure 4 shows the coincidental peaks inthe distribution of time di ff erences δ n αβ for several arbitrarilyselected ( α, β )-pairs of strips either from the same or separatesilicon layers. Among the coincidences within the same layer,those from E -layer are more frequent than those from ∆ E -layerdue to the thinner strips’ increased insensitivity to the γ -flash.Similarly to the calibration relative to the external timing de-vice (WCM), we will consider the mean value ¯ δ αβ as the rele- vant estimator for the inter-strip o ff set. However, this time weadopt a simple unweighted mean:¯ δ αβ = (cid:80) N αβ n = δ n αβ N αβ , (8)with N αβ as the total number of coincidental counts detected bythe ( α, β )-pair of strips. This selection is motivated by the co-incidental counts caused by the detection of charged particlesproduced in the sample. They are characterized by an exten-sive deposited-energy spectrum, a ff ected by the intrinsic spec-trum of the measured nuclear reaction(s) and by the interactionof charged particles with the silicon detector. The amplitudeof these counts is thus a consistent and physically meaningfulquantity, rather than the measure of the reliability of the de-tector response. As such, these amplitudes may no longer beconsidered as the weighting factors a ff ecting the significance ofparticular inputs δ n αβ to Eq. (8). In that, the unbiased estimatorof the sample variance:Var αβ δ = (cid:16)(cid:80) N αβ n = δ n αβ (cid:17) − N αβ ¯ δ αβ N αβ ( N αβ −
1) (9)is also directly related to the variance of the mean value fromEq. (8) as: Var αβ ¯ δ = (Var αβ δ ) / N αβ .From Eqs. (1) and (7) it is now evident that the averages ¯ δ αβ serve as the inter-strip o ff set estimators: τ α − τ β (cid:39) ¯ δ αβ , (10)thus providing a second set of constraints upon the sought o ff -sets τ s , in addition to the one from Eq. (6). One is, of course,well advised to use only the constraints from those pairs ofstrips that feature a statistically significant number N αβ of co-incidental counts and / or acceptably low variance Var αβ ¯ δ of theobtained mean values. If necessary, one can also attempt to construct additional lin-early independent constraints, alongside those from Eqs. (6)and (10). We illustrate here one such example that may be offurther help in obtaining as accurate values of τ s as possible.For the total of S available silicon strips we define a weightedaverage ¯ T n of already synchronized γ -flash instants, as regis-tered by separate strips:¯ T n ≡ (cid:80) Ss = A ns ( t ns − τ s ) (cid:80) Ss = A ns (11)and demand that their average deviation from the γ -flash in-stants registered by an external timing device (WCM) vanishes: (cid:80) N n = ( ¯ T n − T n ) (cid:39) , (12)with N as the total number of proton pulses, just as in Sec-tion 3.1. The black plot from Fig. 5 shows a distributionof γ -flash instants without correction for time o ff sets, aver-aged over all available strips, i.e. the distribution of terms5 [ns] n - T s t - ns t - - - - - C oun t s =0) s t No sync. ( ) s D = s t Partial sync. ( ) m = t Full sync. (M h7_1
Figure 5: Overall distribution of averaged γ -flash instants from Eq. (11), as reg-istered by separate silicon strips. Already the partially synchronized spectrum– accounting only for the set of conditions from Eq. (6) – satisfies to a highdegree the expectation from Eq. (12). Full synchronization considers this as anexplicit requirement, alongside Eqs. (6) and (10). (cid:80) Ss = A ns t ns (cid:14) (cid:80) Ss = A ns , serving as the basis of Eq. (11). The restof the plots will be discussed later.One needs to be mindful of the reliability of conditionslike Eq. (12), since there are sporadic misidentifications in the γ -flash instants t ns extracted from particular strips (see Sec-tion 3.1). However, a massively decreased width of the dis-tributions from Fig. 5 (RMS of 22 ns), relative to the ones fromFig. 3 (RMS for particular strips from 30 ns to 280 ns) con-firms that the misidentified pulses are of low amplitude. Thus,their contribution to the average ¯ T n is heavily suppressed by theweighting procedure.Equation (12) may now be rewritten as: (cid:80) Ss = w s τ s (cid:39) ¯ τ, (13)with the following terms, easily obtained upon its careful rear-rangement: w s = N N (cid:88) n = A ns (cid:80) S σ = A n σ , (14)¯ τ = N N (cid:88) n = (cid:80) S σ = A n σ ∆ n σ (cid:80) S σ = A n σ . (15)Evidently, Eq. (13) is an additional constraint upon the setsought τ s , linearly independent of Eqs. (6) and (10). While theindependence from the set of of equations (10) is self-evident,as they refer to a di ff erent dataset, the independence from theset of equations (6) is demonstrated in Appendix A. By Eqs. (6), (10) and (13) we constructed an overdetermined set of constraints for the set of sought o ff sets τ s . This entire system of linear equations may be put into a matrix form: M (cid:126)τ = (cid:126)µ (16)and it is easily solved in a least-squares sense, by weighted fit-ting. Let us illustrate the structure of the design matrix M onan artificial example of a single SITE comprising three strips in ∆ E -layer ( s = , ,
3) and three strips in E -layer ( s = , , − − − −
11 0 0 − − − − −
10 0 1 0 − − w w w w w w τ τ τ τ τ τ = ¯ ∆ ¯ ∆ ¯ ∆ ¯ ∆ ¯ ∆ ¯ ∆ ¯ δ ¯ δ ¯ δ ¯ δ ¯ δ ¯ δ ¯ δ ¯ δ ¯ δ ¯ δ ¯ δ ¯ τ . The vertical dashed line separates the matrix coe ffi cients for thestrips from the opposing layers ( ∆ E -layer on the left, E -layeron the right). The first horizontal block is formed by a set ofequations from Eq. (6). The lowest horizontal block incorpo-rates Eq. (13). The middle horizontal block (bounded by dou-ble dashed lines) subsumes the constraints from Eq. (10), withthe first sub-block corresponding to the coincidences within thesame ( ∆ E or E ) layer and the second sub-block to the coinci-dences between the pairs of strips from opposing layers. Thestructure of vector (cid:126)µ from Eq. (16) is also evident.In order to perform a weighed fitting, one needs to constructthe appropriate weight matrix W . We use a simple diagonal ma-trix and consider the reliability of specific constraints to be de-termined by the sample variance of the components contribut-ing to (cid:126)µ , i.e. by their distribution widths (Figs. 3 and 4). Thereason is that the variance of the mean is sensitive to the amountof statistics (going as N − / with the number of counts in theunweighted case). On the other hand, the severity of the timeo ff sets between the particular strips is independent of the ac-cumulated statistics, which is better reflected through a samplevariance, it being an intrinsic property of the relevant distribu-tions. Relying on Eqs. (5) and (9), while using a notation W kk ( · )in a sense that the k -th diagonal element corresponds to the ar-6 trip ID (s) [ n s ] s t - - - - - ) s D = s t Partial sync. ( ) m = t Full sync. (M h_dummy
Figure 6: Set of o ff sets between each particular silicon strip and an external tim-ing device (WCM), obtained either by the partial synchronization from Eq. (6),or the full synchronization accounting for Eqs. (6), (10) and (13). gument in parentheses, we thus define the weighting factors as: W kk ( ¯ ∆ s ) = / Var s ∆ , (17) W kk (¯ δ αβ ) = / Var αβ δ, (18) W kk (¯ τ ) = (cid:14) (cid:80) Ss = w s Var s ∆ . (19)For simplicity, in Eq. (19) we use the variance estimate basedon the coupling of Eqs. (6) and (13). In the actual implemen-tation one wishes to keep only the statistically significant andotherwise reliable constraints – based, for example, on the ob-servation of the total number of events ( N or N αβ ) formingthe underlying distributions and / or the variance of their means(Var s ¯ ∆ or Var αβ ¯ δ ).Finally, defining the covariance matrix V = (cid:0) M (cid:62) W − M (cid:1) − ,the solution to the weighted fitting may be expressed as: (cid:126)τ = VM (cid:62) W − (cid:126)µ, (20)together with accompanying variances Var τ s = V ss , thus fullyresolving a synchronization problem.Figure 6 shows the di ff erence between the set of o ff sets ob-tained by a full synchronization method from Eq. (20) and thoseobtained directly from Eq. (6). We refer to the former set of o ff -sets as fully synchronized and to the latter as partially synchro-nized . Earlier Figs. 3 and 5 also show their respective spectraafter being corrected by either of these two sets of o ff sets. Thecorrected spectra for each particular strip, contributing to theoverall distributions from Fig. 3, are expected to satisfy the re-quirement ¯ ∆ s − τ s (cid:39) ff erence between the corrected ) [ns] b t - a t - ( ba n d - - - - C oun t s · =0) s t No sync. ( ) s D = s t Partial sync. ( ) m = t Full sync. (M h_org - - - -
200 150 100 50 0 50 100 150 200
Figure 7: Overall distribution of time di ff erences between coincidental pulsesfrom the measurement of the Li( n , t ) reaction, registered by any pair of siliconstrips. The partial synchronization based solely on Eq. (6) exacerbates the timeo ff sets between the strips. The inset shows the plots in the logarithmic scale,facilitating the selection of ±
100 ns as a window width for the identification ofcoincidental pules. spectra from Figs. 3 and 5 is inconclusive. This is preciselydue to the fact that requirements from Eqs. (6) and (13) are notof su ffi cient quality by themselves , thus having to be comple-mented by further rigorous constraints from Eq. (10). In that,Fig. 7 shows the overall distribution of time di ff erences betweenthe pairs of coincidental pulses detected by any pair of siliconstrips. These spectra clearly show the e ff ects and the quality oftwo considered types of synchronization, confirming that theseare the relevant spectra for the assessment of the synchroniza-tion procedure. The uncorrected spectrum is, in essence, anoverlap of spectra from Fig. 4, but taking into account all avail-able pairs of strips. The spectrum obtained by a partial syn-chronization not only reconfirms that conditions from Eq. (6)are insu ffi cient, but reveals that they are even inadequate bythemselves , as the o ff sets between the strips are further exacer-bated. The full synchronization expectedly manages to achievean optimal adjustment between the pulses (i.e. the entire signalwaveforms) from the separate strips, as per explicit requirementfrom Eq. (10). It should be noted that the width of the fully syn-chronized distribution from Fig. 7 is the basis for selecting thecoincidental window width for the identification of coinciden-tal pulses during the analysis of the experimental data from thelatest joint measurement of the C( n , p ) and C( n , d ) reactionsfrom n TOF. We select it as ±
100 ns.
4. Conclusions
We presented a simple method for a simultaneous o ff -line synchronization of the digital data-streams from a multi-channel silicon telescope (SITE). An absolute synchronizationis performed relative to the external timing device. A Wall Cur-rent Monitor (WCM) – used for the detection of an instanta-neous proton beam from the CERN Proton Synchrotron – is7dopted as the external timing device at n TOF. The proton-pulse-coincidental pulses from SITE strips are caused by theirresponse to an intense γ -flash caused by the instantaneous burstof γ -rays and ultrarelativistic particles from a spallation pro-cess producing the neutron beam. A procedure for obtaining theconstraints upon the WCM-relative strip o ff sets was described,revealing that the minimal necessary subset of these conditionsis insu ffi cient for a quality synchronization. Therefore, a rel-ative synchronization between the separate silicon strips wasalso included into procedure. The relative synchronization isbased on observing the coincidental pairs of pulses caused bythe detection of charged particles from the neutron induced re-actions, alongside the strips’ response to the γ -flash. The fullsynchronization method was thus expanded to account for themaximum achievable amount of information, in order to con-strain the sought o ff sets as well as possible. Upon completion,a successful synchronization procedure provides an objectiveevaluation of the coincidental window width to be used in iden-tifying the coincidental pulses during an o ff -line analysis of theexperimental data.. Acknowledgements
This work was supported by the Croatian Science Foundationunder Project No. 8570.
Appendix A. Linear independence of constraints
We demonstrate here the linear independence of con-straint (13) from the set of constraints (6). It should be notedthat the requirement of their independence is not related to theinvertibility of Eq. (16), as this system of equations is alreadyoverdetermined, i.e. carries su ffi cient information for findingthe set of time o ff sets τ s . In other words, introducing the lin-early dependent term into Eq. (16) does not make the matrix M singular because M is larger than the square matrix (carryingminimal amount of information) necessary for obtaining (cid:126)τ , sothat the concept of singularity is not even strictly applicable.Rather, linear dependence must be avoided because it acts as arepeated inclusion of already accounted constraints.Let us first define a set of terms T s , obtained by extractingthe common denominator from Eq. (6): T s ≡ (cid:80) N n = A ns ( ∆ ns − τ s ) . (A.1)The set of constraints (6) is then equal to T s (cid:39)