Study of a data analysis method for the angle resolving silicon telescope
P. Žugec, M. Barbagallo, J. Andrzejewski, J. Perkowski, N. Colonna, D. Bosnar, A. Gawlik, M. Sabate-Gilarte, M. Bacak, F. Mingrone, E. Chiaveri, M. Šako
PPrepared for submission to JINST
Study of a data analysis method for the angle resolvingsilicon telescope
P. Žugec, a , M. Barbagallo, b , c J. Andrzejewski, d J. Perkowski, d N. Colonna, b D. Bosnar, a A. Gawlik, d M. Sabaté-Gilarte, c , e M. Bacak, c , f F. Mingrone, c E. Chiaveri c and M. Šako a onbehalf of n_TOF collaboration a Department of Physics, Faculty of Science, University of Zagreb,Bijenička cesta 32, 10000 Zagreb, Croatia b Istituto Nazionale di Fisica Nucleare, Sezione di Bari,Via Giovanni Amendola 173, 70126 Bari, Italy c European Organization for Nuclear Research (CERN),CH-1211 Geneva 23, Switzerland d Uniwersytet Łódzki,Ul. Narutowicza 68, 90-136 Łódź, Poland e Universidad de Sevilla,Calle San Fernando, 4, 41004 Sevilla, Spain f Technische Universität Wien,Karlsplatz 13, 1040 Wien, Austria
E-mail: [email protected]
Abstract: A new data analysis method is developed for the angle resolving silicon telescopeintroduced at the neutron time of flight facility n_TOF at CERN. The telescope has already beenused in measurements of several neutron induced reactions with charged particles in the exit channel.The development of a highly detailed method is necessitated by the latest joint measurement of the C( n , p ) and C( n , d ) reactions from n_TOF. The reliable analysis of these data must account forthe challenging nature of the involved reactions, as they are affected by the multiple excited statesin the daughter nuclei and characterized by the anisotropic angular distributions of the reactionproducts. The unabridged analysis procedure aims at the separate reconstruction of all relevantreaction parameters — the absolute cross section, the branching ratios and the angular distributions— from the integral number of the coincidental counts detected by the separate pairs of siliconstrips. This procedure is tested under the specific conditions relevant for the C( n , p ) and C( n , d )measurements from n_TOF, in order to assess its direct applicability to these experimental data.Based on the reached conclusions, the original method is adapted to a particular level of uncertaintiesin the input data.Keywords: Analysis and statistical methods; Detector modelling and simulations I (interaction ofradiation with matter, interaction of photons with matter, interaction of hadrons with matter, etc);Instrumentation and methods for time-of-flight (TOF) spectroscopy; Si microstrip and pad detectors Corresponding author. a r X i v : . [ phy s i c s . d a t a - a n ] F e b ontents C( n,p ) data 13 The neutron time of flight facility n_TOF at CERN is a neutron production facility aiming atmeasuring the neutron induced reactions. A massive lead spallation target irradiated by the 20 GeVproton beam from the CERN Proton Synchrotron serves as the primary source of neutrons, deliveringan extremely luminous white neutron beam spanning 12 orders of magnitude in energy — from10 meV to 10 GeV. The n_TOF facility features two experimental areas: Experimental Area 1(EAR1), horizontally placed at 185 m from the spallation target, and the Experimental Area 2(EAR2) vertically placed at 20 m above the target. While EAR1 is best adjusted to the high neutronenergy and the high resolution measurements, EAR2 excels at the measurements with small, highlyradioactive samples characterized by low cross sections for the investigated reactions. More detailson the general features of the n_TOF facility and EAR1 itself may be found in ref. [1], while thespecifics on EAR2 are addressed in Refs. [2–4]. An overview of the experimental program atn_TOF may be found in ref. [5]. A general overview of many different types of detectors used atn_TOF for the measurements of various types of the neutron induced reactions, together with thedetailed description of the procedures for the analysis of electronic signals from these detectors,can be found in ref. [6].A new, highly sophisticated silicon telescope (SITE) has recently been introduced at n_TOFfor measurements of the neutron induced reactions with charged particles in the exit channel [7].It consists of two separate and segmented layers of 16 silicon strips, 5 cm × a) (b) Figure 1 : (a) Experimental setup housing the segmented silicon telescope (SITE), originally usedfor the measurement of the Be( n , p ) reaction. (b) Top: upgraded experimental configuration usedfor the measurements of the C( n , p ) and C( n , d ) reactions, comprising two silicon telescopes (acaption from Geant4 simulations). The central object is a carbon sample, as a source of severaldisplayed proton trajectories. Bottom: closeup of a rear telescope, showing a stripped structure of ∆ E and E layers (the strips of alternating colors, separated by a very thin layer of inactive silicon).parallel between the layers. The detector is shown in figure 1a. Both layers are 5 cm × ∆ E) layer and the second (E) layer are 20 µ m and300 µ m thick, respectively. The detector allows to discriminate different types of charged particlesusing the ∆ E-E telescope method, while also offering the limited angular discrimination, governedby its geometry and the sample-relative positioning.Excellent particle discrimination capabilities of this silicon telescope have been clearly demon-strated [7] and it has already been successfully used in the challenging measurement of the Be( n , p )reaction cross section, highly relevant for the long-standing Cosmological Lithium Problem [8].This measurement has also been accompanied by the measurement of the Be( n , α ) reaction crosssection [9], employing a similar type of silicon sandwich detector [10].Rather recently an integral measurement of the C( n , p ) B reaction has been performed atn_TOF, using two liquid scintillation C D detectors for the detection of β -rays from the decay of theproduced B [11, 12]. The results of this measurement have turned out somewhat surprising, lyingentirely outside of values predicted by all earlier datasets available for this reaction (experimentalor otherwise), which are in a rather poor agreement between each other (for a concise review ofthese datasets see Refs. [11–13]). In order to resolve this conundrum a more advanced energy-differential measurement of the C( n , p ) B and C( n , d ) B reactions was proposed [13] andalready performed at EAR1 at n_TOF, using an upgraded SITE configuration displayed in figure 1b.The upgrade consisted in introducing a second telescope in order to increase the angular coverageas much as possible, while keeping both of them outside of the neutron beam. We will refer to– 2 –hese two telescopes as front and rear , the front one being parallel to the sample and coveringthe forward angles, with the rear one being parallel to the neutron beam and mostly covering thebackward angles (see figure 1b). The analysis of the experimental data on the C( n , p ) and the C( n , d ) reaction is under way, pending the development of a new analysis method for extractingthe relevant reaction parameters. Most important among these is the absolute cross section. Theangle-differential cross sections for the reaction flow via the separate excited states of a daughternucleus ( B [14] or B [15]) would also be highly desirable. However, the reliable decoupling ofthese states might not be possible at the level of statistics expected from the latest measurement.The purpose of this paper is twofold. The first is to develop the necessary formalism forthe analysis of the data obtained with the multi-channel telescope (section 2). The second is toinvestigate its applicability on the artificially generated dataset resembling the first experimentaldataset from n_TOF to which the the method is to be applied at a later date: that of the C( n , p )reaction (section 3). In doing this we aim (1) to provide the future users of the method with all thenecessary steps and considerations to be taken into account in extracting the optimal set of physicalparameters from a given measurement; (2) to provide an honest assessment of the direct applicabilityof the method to a dataset of a given level of uncertainties, in particular the one expected fromthe C( n , p ) measurement, and (3) to provide alternative solutions in case the direct applicationproves to be unreliable due to the level of uncertainties in the extracted results (section 4). Sincewe develop the method having a specific C( n , p ) reaction in mind, it cannot be overemphasizedthat the procedure is aimed at and designed for the particular detector setup, based on the ∆ E-Etelescoping principle, rather than for the particular reaction of even the type of reaction. Therefore,at no point should the method be considered as limited to this specific (type of) reaction, nor shouldthe conclusions regarding a particular C( n , p ) measurement from n_TOF be misinterpreted forsome general limitation of the method itself. Let θ be the angle of proton emission in the center of mass frame (of the incoming neutron and C nucleus before the reaction, and of the outgoing proton and the B nucleus after the reaction),relative to the direction of the neutron beam. We immediately introduce: χ ≡ cos θ (2.1)as a relevant variable. For simplicity of terminology we still refer to χ as the angle of the protonemission. Let N ij be the total number of protons detected in coincidence by the ( i , j ) -pair of strips,with the first index i denoting some of the thin ∆ E-strips and the a second index j denoting some ofthe thick E-strips from any telescope (front or rear). Let E be the energy of the incident neutron.The proton produced by the neutron of sufficiently high energy might be emitted leaving the Bnucleus in any of the energetically accessible states. Thus, the proton energy is clearly contingenton the daughter nucleus’ excited states. Denoting these states by x (x = ε ij ( x , E , χ ) for the coincidental detection — by the ( i , j ) -pair of strips —of protons produced by the neutrons of energy E and emitted under an angle χ leaving the Bnucleus in a state x: ε ij ( x , E , χ ) ≡ d N ij ( x , E , χ ) d N ( x , E , χ ) , (2.2)– 3 – - - - - - ) [ % ] c ( , M e V , ij e j=i-2 j=i-1 j=i j=i+1 h_3 Figure 2 : Examples of the coincidental detection probabilities for protons from the C( n , p )reaction induced by 20 MeV neutrons, leaving the B nucleus in the ground state. The probabilitiesare shown for an arbitrary ( i -th) ∆ E-strip in coincidence with the several closest E-strips.with d N ij ( x , E , χ ) as the number of the detected protons and d N ( x , E , χ ) as the number of protonsemitted under such conditions. These probabilities may easily be obtained from the dedicatedsimulations, described in appendix A. It should be noted that they reflect the properties of theexperimental setup itself, and are independent of the angular distribution of the emitted protons.Only for illustration purposes, figure 2 shows the coincidental detection probabilities ε ij ( ,
20 MeV , χ ) for protons produced by 20 MeV neutrons, leaving the B nucleus in the groundstate (x = i -th ∆ E-strip in coinci-dence with the several closest E-strips. During the method implementation these curves, i.e. theirsmooth(ed) forms never have to be constructed, as their integrals can be calculated as the weightedsum of the simulated counts. The issue is further addressed in appendix A.The number of protons emitted under the described specific conditions may be decomposedas: d N ( x , E , χ ) = φ ( E ) µ ( E ) (cid:37) ( x , E , χ ) Σ tot ( E ) (cid:16) − e − η Σ tot ( E ) (cid:17) d E d χ, (2.3)with φ ( E ) as a time-integrated energy dependent neutron flux irradiating the sample: φ ( E ) = d Φ ( E )/ d E , d Φ ( E ) being the total number of neutrons of energy E intercepted by thesample. The multiple scattering factor µ ( E ) describes an increase in the neutron flux at an energy E due to the energy loss of higher-energy neutrons by means of the multiple scattering insidethe sample itself. With (cid:37) ( x , E , χ ) as the partial cross section for the C( n , p ) reaction, i.e. for aparticular reaction of interest, Σ tot ( E ) is the total cross section for any neutron induced reactionin the carbon sample. Finally, η is the areal density of the sample, as encountered by the neutronbeam, in the number of atoms per unit area. While the term 1 − e − η Σ tot ( E ) gives a probability forany neutron reaction to occur (the exponential term itself being the transmission probability), thedifferential ratio (cid:37) ( x , E , χ )/ Σ tot ( E ) governs the probability of that reaction being the one of interest.– 4 –he differential cross section may now be decomposed as: (cid:37) ( x , E , χ ) = σ ( E ) ρ ( x , E ) A ( x , E , χ ) , (2.4)with σ ( E ) as the total cross section for the C( n , p ) reaction, ρ ( x , E ) as the energy-dependentbranching ratios for the reaction flow via the particular excited state of B, and A ( x , E , χ ) as theangular distribution of protons specific to that state.From eq. (2.3) we now isolate all the terms that are independent of the detector setup, whilebeing available from the experiment, simulation or any evaluation database: w ( E ) ≡ − e − η Σ tot ( E ) Σ tot ( E ) φ ( E ) µ ( E ) . (2.5)The neutron flux φ ( E ) at EAR1 (as well as the flux at EAR2 [16]) is available from the dedicatedmeasurements at n_TOF [17]. Even in a general case of a thick sample, the multiple scattering factorcould be obtained from the dedicated simulations if the total cross section Σ tot ( E ) and the elasticscattering cross section Σ el ( E ) for carbon were known with sufficient precision, which they are [12].However, as the very thin carbon sample was used during the energy-differential measurement —0.25 mm [13], with the thickness of 0.35 mm, i.e. an areal density of η = × − atoms/barnbeing intercepted by the neutron beam due to the sample tilt of 45 ◦ (figure 1b) — a thin sampleapproximation becomes highly appropriate. In this approximation the deviation of the multiplescattering factor from unity is completely negligible: µ ( E ) ≈
1, while the full fractional term fromeq. (2.5) approximates to η due to η Σ tot ( E ) (cid:28) w ( E ) ≈ ηφ ( E ) . (2.6)Using Eqs. (2.4) and (2.5), eq. (2.3) may now be rewritten as:d N ( x , E , χ ) = w ( E ) σ ( E ) ρ ( x , E ) A ( x , E , χ ) d E d χ. (2.7)We now take into account that due to the energy spread of the neutron beam the experimental datamust be analyzed within the energy intervals of finite width. We use the following notation for onesuch interval: E ≡ [ E min , E max ] , (2.8)meaning that all the later quantities denoted by E are either integrals or averages over E , or that theymay be separately and independently selected for each such energy interval. Since any particularmethod implementation requires a weighted averaging over w ( E ) , we immediately introduce thefollowing norm: W E ≡ ∫ E w ( E ) d E . (2.9)Returning to the differential number of protons d N ij ( x , E , χ ) detected by a particular pair of stripsand recalling that there may be multiple excited states of the daughter nucleus contributing to thereaction, we may write the expression for the total number of protons detected by the ( i , j )-pair ofstrips: N ( E ) ij = X E (cid:213) x = ∫ E d E ∫ − d χ d N ij ( x , E , χ ) d E d χ , (2.10)– 5 –ith X E as the highest excited state affecting the data from the energy interval E . It should be notedthat the total detected counts N ( E ) ij taken for analysis will also be dependent on the energy depositioncuts imposed on the experimental data. We will consider this dependence implicitly absorbed within the terms N ( E ) ij and, consequently, the corresponding detection probabilities ε ij ( x , E , χ ) .We now define an arbitrary bijective mapping: ( i , j ) (cid:55)→ α, (2.11)allowing us to write eq. (2.10) in a single index, which will soon become useful in bringing thesystem to an appropriate matrix form. This bijection never needs to be explicitly constructed. Usingthis formal manipulation in conjunction with applying Eqs. (2.2) and (2.7) to eq. (2.10), we arriveat the master equation: N ( E ) α = X E (cid:213) x = ∫ E d E ∫ − d χ ε α ( x , E , χ ) w ( E ) σ ( E ) ρ ( x , E ) A ( x , E , χ ) . (2.12)Our goal is now to bring this equation into the matrix form: (cid:174) N ( E ) = E E (cid:174) P ( E ) (2.13)by constructing the vector (cid:174) N ( E ) of total detected counts N ( E ) α , by designing an appropriate matrix E E and by isolating the sought parameters of the partial cross section within the vector (cid:174) P ( E ) . Wewill obtain this matrix form by decomposing the angular distributions into partial waves (Legendrepolynomials).Before proceeding further let us put forth the tools and considerations common to any particularimplementation of the method. Let us denote by R E the number of relevant pairs of strips composingthe experimental dataset from (cid:174) N ( E ) and by P E the number of the partial cross section parametersfrom (cid:174) P ( E ) . Then we can select at most R E parameters to reconstruct: R E ≡ dim (cid:2) (cid:174) N ( E ) (cid:3) P E ≡ dim (cid:2) (cid:174) P ( E ) (cid:3) (cid:41) ⇒ P E ≤ R E . (2.14)When P E < R E , the best solution to this system may be found by means of the weighted leastsquares method [18]: (cid:174) P ( E ) = (cid:0) E (cid:62) E V − E E E (cid:1) − E (cid:62) E V − E (cid:174) N ( E ) , (2.15)with V E are the covariance matrix of the input data, allowing for the propagation of experimentaluncertainties in order to obtain the covariance matrix V E of the reconstructed parameters and theirrespective uncertainties δ P ( E ) β as: V E = (cid:0) E (cid:62) E V − E E E (cid:1) − ⇒ δ P ( E ) β = (cid:113) [ V E ] ββ . (2.16)From the raw results obtained from eq. (2.15) we will have to calculate various consequent quantities— the total reaction cross section, branching ratios and the angular distribution parameters — whiledealing with the high uncertainties and possible correlations in the reconstructed (cid:174) P ( E ) . Therefore,we are well advised to take into account the effects of the full covariance matrix upon the propagation– 6 –f uncertainties. Let any of these quantities be the scalar function of (cid:174) P ( E ) that we generally denoteas: F E ≡ F (cid:0) (cid:174) P ( E ) (cid:1) . Then the respective uncertainty δ F E may be expressed as: δ F E = (cid:113) J F V E J (cid:62) F = (cid:118)(cid:117)(cid:117)(cid:116) P E (cid:213) β = P E (cid:213) β (cid:48) = ∂ F E ∂ P ( E ) β ∂ F E ∂ P ( E ) β (cid:48) [ V E ] ββ (cid:48) , (2.17)with J F indicating the conventionally defined Jacobian matrix of the function F . We start by decomposing the angular distributions into the selected number of partial waves, i.e.Legendre polynomials P l ( χ ) : A ( x , E , χ ) ≈ L E ( x ) (cid:213) l = a l ( x , E ) P l ( χ ) , (2.18)with the maximum wave number L E ( x ) freely adjustable to any given excited state, within theconstraints of the total number R E of the available data points. Eq. (2.12) may now be rewritten as: N ( E ) α ≈ X E (cid:213) x = L E ( x ) (cid:213) l = ∫ E w ( E ) σ ( E ) ρ ( x , E ) a l ( x , E ) d E ∫ − ε α ( x , E , χ ) P l ( χ ) d χ = W E X E (cid:213) x = L E ( x ) (cid:213) l = (cid:10) σ ρ a l (cid:10) ε α (cid:11) l (cid:11) E , (2.19)where we recognize the appearance of the weighted averages (cid:104)·(cid:105) E and (cid:104)·(cid:105) l , with w ( E ) and P l ( χ ) as the respective weighting functions. We now approximate the average product by the product ofaverages: (cid:10) σ ρ a l (cid:10) ε α (cid:11) l (cid:11) E ≈ ¯¯ ε ( E ) α x l ¯ σ ( E ) ¯ ρ ( E ) x ¯ a ( E ) x l , (2.20)with the single and double averages appearing as:¯ σ ( E ) ≡ W E ∫ E w ( E ) σ ( E ) d E , (2.21)¯ ρ ( E ) x ≡ W E ∫ E w ( E ) ρ ( x , E ) d E , (2.22)¯ a ( E ) x l ≡ W E ∫ E w ( E ) a l ( x , E ) d E , (2.23)¯¯ ε ( E ) α x l ≡ W E ∫ E w ( E ) d E ∫ − ε α ( x , E , χ ) P l ( χ ) d χ. (2.24)In analogy to eq. (2.11) we introduce another arbitrary bijective mapping: ( x , l ) (cid:55)→ β, (2.25)never having to be explicitly constructed, but allowing for a unique-index labeling. In that, β spansthe range of all free parameters, i.e. the total number of coefficients ¯ a ( E ) x l : β = , ..., P E . As it holds:– 7 – E = (cid:205) X E x = [L E ( x ) + ] , from eq. (2.14) we have the following constraint upon the distribution ofLegendre coefficients among the relevant exited states:1 + X E + X E (cid:213) x = L E ( x ) ≤ R E . (2.26)Equation (2.19) is now recast as: N ( E ) α ≈ W E X E (cid:213) x = L E ( x ) (cid:213) l = ¯¯ ε ( E ) α x l ¯ σ ( E ) ¯ ρ ( E ) x ¯ a ( E ) x l = W E P E (cid:213) β = ¯¯ ε ( E ) αβ ¯ σ ( E ) ¯ ρ ( E ) β ¯ a ( E ) β , (2.27)having thus been brought into the matrix form from eq. (2.13), with the appropriate definitions: (cid:2) E E (cid:3) αβ ≡ W E ¯¯ ε ( E ) αβ , (2.28)P ( E ) β ≡ ¯ σ ( E ) ¯ ρ ( E ) β ¯ a ( E ) β . (2.29)The entire solution (cid:174) P ( E ) and the corresponding uncertainties are now easily found from Eqs. (2.15)and (2.16).Applying the normalization condition ∫ − A ( x , E , χ ) d χ = a ( x , E ) = / ⇒ ¯ a ( E ) x0 = / , (2.30)i.e. all the 0 th terms are fixed and carry the entire angular distribution norm. Plugging this resultinto eq. (2.29): P ( E ) x0 = ¯ σ ( E ) ¯ ρ ( E ) x / (cid:205) X E x = ¯ ρ ( E ) x = σ ( E ) = X E (cid:213) x = P ( E ) x0 . (2.31)The next step consists of identifying the branching ratios as:¯ ρ ( E ) x = ( E ) x0 ¯ σ ( E ) = P ( E ) x0 (cid:205) X E y = P ( E ) y0 , (2.32)culminating in the separation of the angular coefficients:¯ a ( E ) x l = P ( E ) x l ¯ σ ( E ) ¯ ρ ( E ) x =
12 P ( E ) x l P ( E ) x0 . (2.33)The uncertainties δ ¯ σ ( E ) , δ ¯ ρ ( E ) x and δ ¯ a ( E ) x l follow directly from eq. (2.17), according to the fullcovariance matrix V E from eq. (2.16) .When the total number of the relevant excited states becomes so large that the total number P E of required parameters becomes comparable to the total number R E of available data points,and/or when these points are affected by large uncertainties, the coefficients ¯ a ( E ) x l exhibit substantialuncertainties themselves and the contributions from the particular excited states can not be reliably– 8 –eparated. In this case one may attempt to reconstruct the "global" partial wave coefficients averagedover the excited states: ¯¯ a ( E ) l = X E (cid:213) x = ¯ ρ ( E ) x ¯ a ( E ) x l f L E ( x )− l = (cid:205) X E x = P ( E ) x l f L E ( x )− l (cid:205) X E y = P ( E ) y0 , (2.34)hoping for a manageable uncertainty in the total contribution to a given partial wave. The factors f (cid:96) ,defined as: f (cid:96) ≡ (cid:40) (cid:96) <
01 if (cid:96) ≥ th terms are fixed by eq. (2.30), we immediately have ¯¯ a ( E ) = / δ ¯¯ a ( E ) = We illustrate the implementation of the method by applying it to the C( n , p ) data artificiallygenerated by the Geant4 simulations. We consider here the data from 1 MeV wide energy range E = [ . , . ] , approximately where this reaction’s cross section is expected to reachits maximum. The branching ratios and the angular distributions for each relevant excited state werearbitrarily constructed. The excited states contributing to the reaction within the given neutron energy range E need tobe clearly identified, as the method requires them to be known in advance. For the C( n , p )reaction, there are total of 15 states in the B daughter nucleus with the energy threshold E thr belowthe upper limit of the considered neutron energy range ( E thr < . Q -values and the energy thresholds in the laboratory frameare listed in table 1. While all these states contribute to the reaction, not all of them necessarilycontribute to the totality of the detected counts, especially those very close to the reaction threshold.The reason is threefold: (1) the very low reaction cross section close to the threshold; (2) thepronounced forward boost of the produced protons in the laboratory frame, making them miss mostof the detection setup; (3) the low proton production energy causing them to be stopped by thesample itself, never reaching the detectors at all. Therefore, it needs to be estimated in advancewhich states may be excluded from the analysis of the experimental data. As the exact evaluationof the expected amount of the detected counts from each state is, of course, impossible withoutthe prior knowledge of the partial cross sections for each separate state (their branching ratios andangular distributions), one needs to rely on some reasonable estimate. One such useful figure ofmerit is the approximate probability estimator ˜ ε ( E ) x for the coincidental detection by any pair of the ∆ E-E strips: ˜ ε ( E ) x ≡ W E (cid:213) α ∫ E w ( E ) d E ∫ − ε α ( x , E , χ ) d χ, (3.1)constructed by assuming — in the absence of any prior information — the isotropic angulardistribution of protons in the center of mass frame: A ( x , E , χ ) ≈ /
2, and applying the same energy– 9 – able 1 : States in the B nucleus relevant for the selected demonstration example. The table liststheir excitation energies E x [15], the corresponding Q -values and the energy thresholds E thr for the C( n , p ) reaction in the laboratory frame. x E x [ MeV ] Q [ MeV ] E thr [ MeV ] B states. Although the portion N ( E ) x of the produced protons still remainsentirely unknown, the observed decrease in ˜ ε ( E ) x together with the expected decrease in N ( E ) x forthe higher states allows one to make informed estimates about the relevance of the expected partialcontributions N ( E ) x to the detected counts: N ( E ) x ≈ ˜ ε ( E ) x N ( E ) x . From these considerations applied tofigure 3 we elect to include only the first 11 states (up to the 10 th excited one, i.e. X E =
10) forfurther analysis. The artificial data to be analyzed were, of course, simulated by including all 15states with the energy thresholds below the upper limit of the considered neutron energy range.It must be pointed out that this exclusion of higher states from the analysis may, in principle,affect the cross section normalization, as the branching ratios of the excluded states become unob-tainable. However, as already discussed, the cross sections around the energy thresholds for thesestates are expected to be negligible and so is their impact upon the total reaction cross section. Still,if there were reasonable indications to the contrary, one should be aware that the reconstructed crosssection ¯ σ ( E ) is only partially contributed by those states that were kept for the analysis. The highest wave numbers L E ( x ) for each excited state are evidently the method’s adjustableparameters. For the total of R E relevant pairs of strips from eq. (2.14), there is a total of (cid:0) R E X E + (cid:1) selections of L E ( x ) satisfying the constraint from eq. (2.26), with (cid:0) ·· (cid:1) denoting a binomial factor. For R E =
60, as used later, and X E =
10 this amounts to approximately 3 . × combinations. If wewere to impose some maximum admissible wave number L E that may be assigned to any particularstate — implying, for purpose of these simple estimates, that the selection of L E itself must be such– 10 – [ % ] x ε ∼ ( 𝔼 ) blank Figure 3 : Figure of merit: estimated probabilities for the coincidental detection of protons fromthe C( n , p ) reaction by any pair of ∆ E-E strips, dependent on the excited state x that the daughternucleus B was left in. The considered neutron energy range is E = [ . , . ] .that ( L E + )(X E + ) ≤ R E , in order for each of X E + L E + L E ( x ) reduces to ( L E + ) X E + . For example, the maximumvalue L E = R E =
60 and X E =
10 yields approximately 4 . × combinations.However, the following physical argument helps us in reducing the number of possible combinationseven further, by keeping only the physically sensible selections of L E ( x ) . We consider that closeto the reaction threshold the nuclear reactions are expected to be isotropic (in the center-of-massframe), while the anisotropy is expected to appear (and possibly intensify) with increasing incidentparticle energy. This suggests that the higher excited states — characterized by a higher threshold— should not be assigned more partial waves than the lower states, i.e.: L E ( x ) ≥ L E ( x ) for x < x . (3.2)For the maximum admissible wave number L E , the number of combinations consistent with thisconstraint is now reduced to (cid:0) L E + X E + X E + (cid:1) . For example, the maximal value L E = R E =
60 and X E =
10 leaves the total of 1365 combinations. All we need now is the algorithm forconstructing such combinations. For the maximum wave number L E to be assigned to any state,the particular combination of nonincreasing L E ( x ) values may be uniquely defined by the set of L E states Λ (cid:96) ( (cid:96) = , . . . , L E ) at which the maximum wave number L E ( x ) increases by 1. In otherwords, Λ (cid:96) form a set of states such that L E ( x ) = (cid:96) ends at x = Λ (cid:96) , i.e.: L E ( x ) = (cid:96) for Λ (cid:96) + < x ≤ Λ (cid:96) , (3.3)with additional fixed boundaries Λ = X E and Λ L E + = −
1, defined for the convenience of theimplementation. The algorithm now reduces to generating all combinations ( L E -tuples) of Λ (cid:96) suchthat: Λ (cid:96) + ≤ Λ (cid:96) with Λ (cid:96) ∈ (cid:8) − , . . . , X E (cid:9) for (cid:96) = , . . . , L E . (3.4)– 11 –t is easy to confirm that if Λ (cid:96) = − (cid:96) , then L E ( x ) = Λ (cid:96) = X E for all (cid:96) , meaning that L E ( x ) = L E forall x, i.e. all the states are assigned the maximum allowed number of partial waves. The obvious question now is how to select an optimal combination of the wave numbers L E ( x ) .We propose here a simple — and by no means unique — selection principle. As the variationsin L E ( x ) directly affect the number of the model parameters: P E = (cid:205) X E x = [L E ( x ) + ] , the reducedchi-squared estimator X lends itself easily to a quick and efficient evaluation of the goodness ofthe fit:X = (cid:0) (cid:174) N ( E ) − E E (cid:174) P ( E ) (cid:1) (cid:62) V − E (cid:0) (cid:174) N ( E ) − E E (cid:174) P ( E ) (cid:1) R E − P E (cid:39) R E − P E R E (cid:213) α = (cid:0) N ( E ) α − (cid:205) P E β = (cid:2) E E (cid:3) αβ P ( E ) β (cid:1) (cid:0) δ N ( E ) α (cid:1) . (3.5)The rightmost expression holds when the covariance matrix V E of the input data is diagonal, i.e.when the correlations between the components of (cid:174) N ( E ) are negligible. As opposed to the goodnessof fit — which will for large R E systematically improve by increasing the number of partial waves,as long as P E does not closely approach R E — the reliability of the fit, reflected through theuncertainties in the reconstructed (cid:174) P ( E ) , rapidly degrades with increasing number of parameters. Forestimating this reliability we propose a simple calculation of the uncertainty δ X in the chi-squaredvalue from eq. (3.5) by means of eq. (2.17), since X is sensitive to all the fitted parameters —unlike, for example, the reconstructed cross section ¯ σ ( E ) from eq. (2.31). In the context of ourproblem the minimization of X and its uncertainty δ X seem to be opposing objectives. Therefore,we propose to minimize their product X δ X as the simplest estimator that should at its minimumprovide the optimal tradeof between the goodness and the reliability of the fit.There are additional issues to consider. For the number of partial waves too inadequate fora given set of the experimental data, some of the branching ratios ¯ ρ ( E ) x from eq. (2.32) may turnout to be negative or greater than unity — a clear signature of the badness of the fit, going beyondthe particular values of X . These fits should be immediately rejected as physically unsound, i.e.disqualified from any kind of optimization procedure, be it the minimization of X δ X or somealternate technique.Yet another quality control mechanism consists of checking if the reconstructed angular distri-butions for each angular state:¯ A ( E ) x ( χ ) ≡ L E ( x ) (cid:213) l = ¯ a ( E ) x l P l ( χ ) = L E ( x ) (cid:213) l = P ( E ) x l P ( E ) x0 P l ( χ ) (3.6)become negative at any point. If so, such fits may also be immediately rejected, regardless oftheir goodness. One should be wary, however, in making such rejections when there are priorindications that some states may indeed feature the very low branching ratios or highly anisotropicangular distributions that locally come close to zero. In this case any statistical fluctuation in theinput data may easily drive the reconstructed results toward the negative values, while the resultsdo remain reasonably reliable representations of the true reaction parameters. It should be notedthat the reconstructed branching ratios may discard all fits as unphysical, if every combination of– 12 –ave numbers L E ( x ) produces at least one negative ¯ ρ ( E ) x . On the other hand, the isotropic angulardistributions will always pass the negativity test, so that the fully isotropic fit ( L E ( x ) = , δ X or any consequent quantity to be used in judging the suitabilityof the fitted result, one may also consider manually eliminating from the set of fitted parametersthose P ( E ) x l that, according to eq. (2.33), yield the angular coefficients too small ( | ¯ a ( E ) x l | (cid:28)
1) orunreasonably large ( | ¯ a ( E ) x l | (cid:29)
1) in magnitude. For the associated β , this is most easily done bysetting P ( E ) β = [ V E ] ββ (cid:48) = [ V E ] β (cid:48) β = β (cid:48) within the covariance matrix from eq. (2.16).This procedure helps in regularizing the fit, as the exceedingly small | ¯ a ( E ) x l | are commonly thesporadic results caused by the finite precision data, while the distinctly large | ¯ a ( E ) x l | are expected toappear as the consequence of overfitting the statistical fluctuations in the input data. One should, ofcourse, be prepared for the closer inspection and the critical evaluation of the results if the optimalset of parameters happens to be precisely thus manipulated set. However, what is expected fromthis procedure is the artificially induced increase in the fit suitability estimator X δ X , such thatsome alternative set of parameters takes precedence as the optimal one.In summary, we propose to identify the optimal combination of the maximum wave numbers L E ( x ) by minimizing the product X δ X — or any such estimator balancing between the goodnessand the reliability of the fit — while taking into account the physical soundness of the results,whether by immediately rejecting those physically inadmissible or by appropriately penalizingthem during the optimization procedure. C( n,p ) data We now test the method on a particularly challenging example of the artificially generated C( n , p )data, as means of appraising its applicability to the experimental data from n_TOF. The simulateddataset — the set of counts N ( E ) α detected by a particular pair of strips — was obtained from thesame Geant4 simulations as used for obtaining the coincidental detection probabilities, i.e. thecentral design matrix E E . The neutron energies were sampled within the 1 MeV wide interval E = [ . , . ] , all 15 states from table 1 were used in constructing the dataset, whileonly the first 11 states from figure 3 were considered for the reconstruction. For the buildup ofthe test counts an arbitrarily constructed branching ratios ρ ( x , E ) for each of the 15 states wereused (represented by later figure 5), together with the angular distributions A ( x , E , χ ) arbitrarilyconstructed for each state, which were all designed from the three lowest Legendre polynomials( P , P , P ).Figure 4 shows the relevant set of coincidental counts recorded by different ∆ E-E pairs ofstrips, ordered by magnitude. While there are (16 E-strips) × (16 ∆ E-strips ) × (2 telescopes) = 512possible pairs of strips in the used SITE configuration from figure 1b, one can see from figure 4that only the tenth of those are characterized by a sufficient coincidental detection probability tobe considered for analysis. It should be noted that the counts from figure 4 were constructed froman exceedingly large dataset, featuring the negligible statistical fluctuations. In order to easilygenerate the statistical variations in the dataset to be taken for analysis, we first scale these counts toa desired level of statistics (thus constructing their statistically expected values) and then generatethe appropriate Poissonian fluctuations. For purposes of this demonstration we keep only those– 13 – α N / m a x α N ( 𝔼 )( 𝔼 ) Graph
Figure 4 : Artificial set of the coincidentally detected counts obtained from an exceedingly largedataset generated by Geant4 simulations, virtually unaffected by the statistical fluctuations. Thenumbers of counts are ordered by their magnitude and scaled relative to their maximum value(from the most efficient ∆ E-E pair of strips). The values for the analysis are constructed by firstscaling these counts to a desired level of statistics and then generating the appropriate Poissonianfluctuations. Only the counts above 5% of the maximum value (the dashed threshold) are kept forthe analysis.coincidental counts N ( E ) α that are higher than 5% of the maximum value (the dashed threshold fromfigure 4). Depending on a particular realization of the Poissonian fluctuations, around R E = V E from Eqs. (2.15) and(2.16) to: [ V E ] αα = N ( E ) α .In order to vary the maximum wave numbers L E ( x ) for each excited state we follow theprocedure from section 3.2, adopting the maximum supported value L E =
4. We choose the numberof counts from the most efficient pair of strips to be: max (cid:2) N ( E ) α (cid:3) = , making the total numberof counts detected across all kept pairs: (cid:205) R E α = N ( E ) α = × . The reason behind this selection israther simple and carries the critical repercussions for the analysis of the experimental data fromn_TOF: at lower statistics basically all the fits are discarded due to the appearance of the negativebranching ratios. In other words, for almost all generated instances of Poissonian fluctuations allthe fits (for any combination of state boundaries Λ (cid:96) ) produce at least one negative ¯ ρ ( E ) x . One must becareful at this point not to confuse max (cid:2) N ( E ) α (cid:3) = with some minimum intrinsic level of reliablestatistics. Instead, it reflects an amount of excited states at play: a high number of states naturallyrequires high statistics if they were to be reliably disentangled one from the other.We now appraise the method based on the accuracy and uncertainty of the reconstructedparameters. For a condensed demonstration of the results on the reconstructed angular distributions,– 14 – [ % ] x ρ ( 𝔼 ) Generated Reconstructed blank0
Figure 5 : Typical example of the reconstructed set of branching ratios, recovered by an optimalset of wave numbers assigned to each excited state. Only the first 11 states were considered for thereconstruction, as the rest of them hardly contribute to the detected counts or not at all.we we use the overall distribution A E ( χ ) , averaged over all excited states: A E ( χ ) = W E X + E (cid:213) x = ∫ E w ( E ) ρ ( x , E ) A ( x , E , χ ) d E (cid:39) X E (cid:213) x = ¯ ρ ( E ) x L E ( x ) (cid:213) l = ¯ a ( E ) x l P l ( χ ) = (cid:205) X E x = (cid:205) L E ( x ) l = P ( E ) x l P l ( χ ) (cid:205) X E x = P ( E ) x0 . (3.7)The reference distribution stems from the arbitrarily constructed distributions A ( x , E , χ ) for eachof the 15 states contributing to the reaction ( X + E =
14; see table 1). The reconstructed distribution,as denoted by (cid:39) , is contributed by the reduced number of states taken for the analysis ( X E = A E ( χ ) and, in general, the reconstructedcross section ¯ σ ( E ) , reflecting the absolute normalization of the data. As such, they should indeedbe immediately rejected. Among the physically admissible fits (if there are any at all) the onesidentified as optimal do seem to reasonably reconstruct both the overall angular distribution andthe cross section, at least under the level of statistics adopted here out of necessity. However, theset of reconstructed branching ratios themselves most often seems to be unrepresentative of thetrue results, as illustrated by a typical example from figure 5. The example from figure 5 alsoshows that their uncertainties may also be grossly underestimated and unrepresentative of theirerror. Therefore, the reconstructed branching ratios should be taken with maximum caution.At the adopted level of statistics most often there seems to be little difference in the resultsobtained by minimizing X or the proposed product X δ X , as the physicality of the branchingratios serves as the main discriminator of the unreliable fits. Figure 6 shows an example when thedifference in the reconstructed overall angular distributions obtained by minimizing X or X δ X – 15 – − − − − − ) χ A ( 𝔼 Generated Minimized X X δ Minimized X blank1
Figure 6 : Overall angular distribution recovered from an optimal set of wave numbers for eachexcited state, obtained by minimizing either X (the goodness of the fit) or X δ X (the tradeoffbetween the goodness and the reliability).turns out to be appreciable. This example clearly illustrates the superiority of optimizing the tradeoffbetween he goodness and the reliability of the fit. The power of this procedure lies not in reducingthe uncertainties per se , but in penalizing the overfitting, i.e. in rejecting the sporadic parametersthat unnecessarily and disproportionately increase the uncertainties in all other parameters, besidesintroducing their own excessive ones. Indeed, while the reference angular distribution from figure 6was constructed as a linear combination of the 3 lowest Legendre polynomials, the one identified byminimizing X allows for 5 of them (the maximum amount supported by L E =
4; a clear symptomof overfitting), while the minimization of X δ X finds the combination of 4 partial waves as theoptimal one.Let us recall that with so many exited states at play, the physicality of the branching ratiosserves as the primary discriminator of unreliable fits. For a significantly reduced number of states,this method of assessment becomes much more insensitive or even entirely unavailable in caseof a single relevant, ground state. In that case the quality tradeoff between the goodness and thereliability of the fit remains the crucial, if not the only available method for identifying the optimalset of the fit parameters.Finally, at the adopted level of statistics the relative uncertainty in the reconstructed cross section¯ σ ( E ) appears to be around 10%. As the statistically expected uncertainty scales as (cid:0) N ( E ) tot (cid:1) − / withthe total number N ( E ) tot of the detected counts, one can easily estimate the expected level of uncertaintyat any level of statistics, provided that the available data produce any acceptable fit in the first place.Considering that the experimental n_TOF data on the C( n , p ) reaction are expected to provide 4 to5 orders of magnitude less statistics than adopted for this demonstration [13], even if they could befitted without all fits failing the physicality test, the uncertainty in ¯ σ ( E ) is thus expected to be at leastan order of magnitude grater than reconstructed cross section itself! Hence, the direct applicationof the full reconstruction method presented up to this point is ill-adjusted to these experimental– 16 –ata, due to the particularly unfavorable combination of the available statistics and the amount ofexcited states at play. This outcome should not be confused with some intrinsic shortcoming of themethod itself, as there is a limit to the quality of the results that could be extracted from the data ofa finite statistical precision. Fortunately, this eventuality was foreseen in advance of the experimentand the experimental setup was specially optimized so as to minimize the systematic effects dueto the alternative approach to the analysis of these data. This approach consists of utilizing thereduced variant of the method, by adopting a priori information on the branching ratios and theangular distributions from an outside source — such as the TALYS theoretical model [19], adjustedto the preexisting experimental data — and aiming solely at the reconstruction of the absolute crosssection ¯ σ ( E ) . This reduced variant is addressed in the following section. Even when the full unfolding procedure may not be meaningfully applied due to the uncertaintiesin the input data limiting the usefulness of the output results, the method formalism from section 2still remains relevant, as it clearly establishes the connection between the measured observablesand the underlying reaction parameters. Furthermore, the coincidental detection probability ofthe experimental setup must be characterized — most appropriately by means of the dedicatedsimulations described in appendix A — regardless of the particular approach to the data analysis.Starting from eq. (2.12), one may derive any particular variant of the unfolding procedure, be itthe reduction of the one from section 2.1 or even some further extension, shortly addressed inappendix B. Motivated by the status of the experimental n_TOF data on the C( n , p ) reaction, weconsider here the adoption of a priori information on the branching ratios and angular distributions,aiming solely at the reconstruction of the absolute cross section. Assuming that information to beavailable from an independent source, eq. (2.12) may be linearized as: (cid:174) N ( E ) ≈ (cid:174) (cid:15) ( E ) ¯ σ ( E ) , (4.1)with the vector (cid:174) (cid:15) ( E ) (as a matrix of a reduced dimensionality) standing in place of the design matrix E E from eq. (2.13) and the single unknown ¯ σ ( E ) replacing the previous set (cid:174) P ( E ) of underlyingreaction parameters. While the definition of ¯ σ ( E ) stays the same as in eq. (2.21), (cid:174) (cid:15) ( E ) is now definedby components as: (cid:15) ( E ) α ≡ X E (cid:213) x = ∫ E d E ∫ − d χ w ( E ) ρ ( x , E ) A ( x , E , χ ) ε α ( x , E , χ ) , (4.2)where the branching ratios ρ ( x , E ) and angular distributions A ( x , E , χ ) are taken from an outsidesource of information. Applying Eqs. (2.15) and (2.16) — while taking the covariance matrix V E to be diagonal and composed of the uncertainties δ N ( E ) α in the detected counts: [ V E ] αα = (cid:0) δ N ( E ) α (cid:1) — the final solution for the sought cross section may now be written in a rather simple closed form:¯ σ ( E ) = (cid:0) δ ¯ σ ( E ) (cid:1) R E (cid:213) α = (cid:15) ( E ) α N ( E ) α (cid:0) δ N ( E ) α (cid:1) , (4.3)– 17 –ith the associated uncertainty: δ ¯ σ ( E ) = (cid:169)(cid:173)(cid:171) R E (cid:213) α = (cid:32) (cid:15) ( E ) α δ N ( E ) α (cid:33) (cid:170)(cid:174)(cid:172) − / . (4.4)It should be noted that this procedure still makes full use of all the experimentally availableinformation from separate pairs of ∆ E-E strips. This feature is in clear opposition with themore extreme reduction of the method, taking only the total number N ( E ) tot = (cid:205) R E α = N ( E ) α of coin-cidental counts detected across an entire detection setup, in conjunction with its total detectionprobability (cid:15) ( E ) tot = (cid:205) R E α = (cid:15) ( E ) α in order to obtain the absolute cross section overly simplistically as¯ σ ( E ) = N ( E ) tot / (cid:15) ( E ) tot , thus defeating any benefit of having used a high-end telescope — in particular, itssophisticated dissociation into multiple strips.Evidently, the main challenge with thus reduced method is the estimation of the systematicuncertainties brought on by the out-of-necessity adopted branching ratios and angular distributions.An indication of those uncertainties — and a conservative one, at that — may be obtained byadopting all the involved angular distributions as isotropic: A ( x , E , χ ) = /
2, and recalculating¯ σ ( E ) . The difference between the externally provided and all-isotropic distributions is to be takenas representing the extreme case of the possible disparity with the true angular distributions.Another possibility is taking among the externally provided distributions only the branching ratiosor the angular distributions as given, and unfolding the data with the other type of distributionsunconstrained. Comparing these alternative results for ¯ σ ( E ) allows for an informed estimate of thesystematic uncertainties. A new angle resolving stripped silicon telescope (SITE) has recently been introduced at the neutrontime of flight facility n_TOF at CERN for the measurements of the neutron induced reactions withthe charged particles in the exit channel. Its outstanding detection properties have already beendemonstrated in the challenging measurement of the Be( n , p ) reaction, relevant for the famousCosmological Lithium Problem. The joint energy-differential measurement of the C( n , p ) and C( n , d ) reactions has also been recently performed at n_TOF, using the upgraded and speciallyoptimized detector configuration consisting of the two separate silicon telescopes. As the nature ofthese reactions poses significant challenges for the meaningful data analysis — being affected bythe multiple excited states in the daughter nuclei and featuring the anisotropic angular distributionsof the reaction products — we have established a clear and detailed formalism behind the measuredobservables: the total number of the coincidental counts detected by any combination of ∆ E-E pairsof silicon strips. From this formal connection we have developed and tested the unfolding procedurefor the reconstruction of the underlying reaction parameters, consisting of the absolute reaction crosssection, the branching ratios and the angular distributions of the reaction products for each excitedstate in the daughter nucleus. We have also addressed the finer points of the method implementation,thus providing the consistent and reliable methodology for obtaining the optimal set of the outputparameters. Though the method may, in principle, reconstruct all these quantities separately, itsperformance may be severely limited by the amount of parameters — determined by the number of– 18 –xcited states and the level of anisotropy — as well as the level of uncertainties in the input data.By testing the method on the artificially generated dataset resembling the n_TOF measurementthe of the C( n , p ) reaction, we have found little hope that the full unfolding procedure could bemeaningfully applied to these particular experimental data, precisely due to the highly unfavorablecombination of the large number of the excited states and the reduced level of statistics expected fromthe experiment. This unfortunate outcome should not be misinterpreted for the inherent deficiencyof the method itself, as at some point all considered reaction parameters must be fully taken intoaccount if the experimental data are to be properly described and reliably analyzed. Precisely theclarity of the formalism behind the method allows for its many alternative variants to be developed.One of these, to be applied to the measured C( n , p ) and C( n , d ) data, is the reduced procedurerelying on the independent source of information on the branching ratios and angular distributions,aiming at the reconstruction of the absolute cross section as the central reaction parameter ofinterest. It is worth noting that thus reduced method still takes advantage of the distribution of thedetected counts across the separate ∆ E-E pairs of strips, as opposed to considering only the totalnumber of counts across all of them. Thus retained angular sensitivity opens the possibility for theestimation of the systematic effects due to the adopted outside information (branching ratios and/orangular distributions), allowing for the informed assessment of the systematic uncertainties in thefinal results.
A Detection probability simulations
We describe the detection probability simulations and the use of the simulated data in the construc-tion of the design matrix E E from eq. (2.28). For specificity, we again keep the C( n , p ) reactionin mind. The reaction details — except for its basic kinematics — are assumed unknown. Thereaction itself or its basic details may not even be (properly) implemented in the used simulationpackage. Therefore, the simulations need to start by generating the exit products (protons), basedon the energy and the spatial distribution of the primary reaction-inducing particles (neutrons).For each separate excited state x in the daughter nucleus ( B; see table 1) the neutron energy E is sampled from some preselected energy distribution ˆ ϕ E ( x , E ) , where we use the hat-notation ˆ · to indicate the simulated (as opposed to the later determined, experimental) quantities. Thesedistributions are best selected as uniform or isolethargic, for the simplicity of later analysis. Theproduced proton direction in the center-of-mass frame is then sampled from a preselected angulardistribution ˆ A ( x , E , χ ) , which is best selected as isotropic. The proton energy in the center-of-mass frame is calculated based on the Q -value for a particular excited state. The proton energyand direction are then boosted into the laboratory frame by the proper (in our case relativistic)transformations. As for the initial proton position, its radial distribution relative to the direction ofthe neutron beam must be sampled according to the known neutron beam profile; alternatively, thedata need to be properly reweighed according to the same beam profile during the later constructionof the E E matrix. The sampling (or the later data reweighing) of the initial proton position along theneutron beam direction depends on the properties of the simulated sample and may vary betweenextremely simple and rather involved. In case of the thin sample — implying the combination ofthe geometric thickness and the total cross section such that η Σ tot ( E ) (cid:28)
1, as discussed in a contextof eq. (2.6) — the longitudinal proton distribution may be sampled uniformly, as the neutron beam– 19 –ttenuation along the sample is negligible. This was the case with our setup. Otherwise, if thebeam losses are known to be considerable, then the relative proton production probability alongthe sample must be properly accounted for. In case of the homogeneous and geometrically regularsample this correction amounts to the factor 1 − e − η Σ tot ( E )× d / D with d as the proton productiondepth and D as the sample thickness along the neutron beam direction ; however, this procedure stilldoes not take into account the multiple scattering effects. For more complex samples the correctioninvolves its full spatial characterization.Each coincidental proton detection by any pair of ∆ E-E strips needs to be recorded by outputtingthe relevant physical parameters of that particular event. The necessary data consist of: (1) theprimary neutron energy E ; (2) the proton emission angle χ from the center-of-mass frame ; (3) theunique designation of the activated ∆ E-E pair of strips; (4) the energy deposited in those strips.In addition, the excited state x, the sampled neutron energy distribution ˆ ϕ E ( x , E ) and the protonangular distribution ˆ A ( x , E , χ ) , together with the total number ˆ N E ( x ) of generated protons withinthe sampled neutron energy interval E also have to be documented for a complete and meaningfulutilization of the simulated data. By the virtue of eq. (2.2), the elements of the design matrix E E from eq. (2.28) may be treated as the integrals over the detected counts, so that by identifying theamount d ˆ N ( x , E , χ ) of generated protons as:d ˆ N ( x , E , χ ) = ˆ N E ( x ) ˆ ϕ E ( x , E ) ˆ A ( x , E , χ ) d E d χ (A.1)we may write: (cid:2) E E (cid:3) αβ = ∫ E ∈ E ∫ χ ∈[− , ] w ( E ) P l ( χ ) ˆ ϕ E ( x , E ) ˆ A ( x , E , χ ) d ˆ N α ( x , E , χ ) ˆ N E ( x ) . (A.2)This formalism allows us to construct the sought integrals directly on a count-by-count basis,without ever having to build the full coincidental detection probability distributions ε α ( x , E , χ ) ,such as those shown in figure 2. This is achieved simply by taking a weighted sum of all detectedcounts: (cid:2) E E (cid:3) αβ (cid:39) N E ( x ) ˆ N ( E ) α x (cid:213) q = w ( E q ) P l ( χ q ) ˆ ϕ E ( x , E q ) ˆ A ( x , E q , χ q ) . (A.3)Here (cid:39) symbolically denotes the representation of the integrals from eq. (A.2), with the index q enumerating all the appropriately detected counts: ˆ N ( E ) α x of them caused by the protons leaving thedaughter nucleus in the excited state x and being coincidentally detected by the α -th pair of strips.In exactly the same manner, the design vector elements from eq. (4.2) may be expressed as: (cid:15) ( E ) α = X E (cid:213) x = ∫ E ∈ E ∫ χ ∈[− , ] w ( E ) ρ ( x , E ) A ( x , E , χ ) ˆ ϕ E ( x , E ) ˆ A ( x , E , χ ) d ˆ N α ( x , E , χ ) ˆ N E ( x ) (A.4)and thus constructed on a count-to-count basis: (cid:15) ( E ) α (cid:39) X E (cid:213) x = N E ( x ) ˆ N ( E ) α x (cid:213) q = w ( E q ) ρ ( x , E q ) A ( x , E q , χ q ) ˆ ϕ E ( x , E q ) ˆ A ( x , E q , χ q ) , (A.5)– 20 –here the branching ratios ρ ( x , E ) and angular distributions A ( x , E , χ ) are now taken to be knownfrom an independent source of information.We remind that the energy deposition cuts used in the analysis of the experimental data areto be implemented precisely at this point, in the construction of the matrix E E or the vector (cid:174) (cid:15) ( E ) ,thus directly affecting the numbers ˆ N ( E ) α x of the acceptable counts. It is also worth noting thatthe weighting function w ( E ) is determined by the actual experimental conditions, as opposed tothe arbitrary sampling distributions ˆ ϕ E ( x , E q ) and ˆ A ( x , E q , χ q ) . In that, it is evident that both thesimulations and the computational procedures from Eqs. (A.3) and (A.5) are immensely simplifiedwhen the uniform neutron energy distributions ˆ ϕ E ( x , E ) = /| E | and the isotropic proton angulardistributions ˆ A ( x , E , χ ) = / B Method extension
We shortly comment on the possibility of the further method generalization that may be applicableunder specific conditions, namely the high statistics and at least a partial separation of the excitedstates in the deposited energy spectra. For the silicon telescope consisting of ∆ E and E-layers, theentire two-dimensional ∆ E-E spectra may be considered in the most general case, as we do here. Forsimplicity, figure 7 illustrates the basic idea on the schematic example of the one-dimensional, e.g. ( E + ∆ E ) -spectrum. Evidently, if the excited states are sufficiently far apart in energy (as determinedby the detector resolution), the spectra shapes may serve as an additional source of information tobe exploited. In this case one defines the differential coincidental detection probability: ξ ij ( x , E , χ, E ( ) , E ( ) ) ≡ d N ij ( x , E , χ, E ( ) , E ( ) ) d N ( x , E , χ ) dE ( ) dE ( ) , (B.1) [a.u.] dep E C oun t s den s i t y [ a . u .]
1. state 2. state 3. state Total h0 Figure 7 : Illustrative example: the reaction products leaving the daughter nucleus in any of itsexcited states may leave a clear signature in the deposited energy spectrum if the energy separationof the excited states is sufficient. – 21 –tarting from the number of counts d N ij ( x , E , χ, E ( ) , E ( ) ) characterized by the energy E ( ) de-posited in the i -th ∆ E-strip and the energy E ( ) deposited in the j -th E-strip. The master equationfor the total number of counts N ( E ) ij ı detected within the ı-th E ( ) -interval of width E ( ) ı and the -thE ( ) -interval of width E ( ) is easily rewritten as: N ( E ) ij ı = X E (cid:213) x = ∫ E d E ∫ − d χ ∫ E ( ) ı dE ( ) ∫ E ( ) dE ( ) × ξ ij ( x , E , χ, E ( ) , E ( ) ) w ( E ) σ ( E ) ρ ( x , E ) A ( x , E , χ ) (B.2)where the binning of the deposited-energy distributions, i.e. the set of bin widths E ( ) ı and E ( ) isentirely arbitrary and may, in the most general case, depend on the particular ( i , j ) -pair of strips andthe neutron energy interval E . In place of the earlier bijective mapping from eq. (2.11), an entireset of indices i , j , ı , is now to be mapped onto the unique index α : ( i , j , ı , ) (cid:55)→ α, (B.3)allowing the extended design matrix E E : (cid:2) E E (cid:3) αβ ≡ ∫ E d E ∫ − d χ ∫ E ( ) ı dE ( ) ∫ E ( ) dE ( ) × ξ ij ( x , E , χ, E ( ) , E ( ) ) w ( E ) P l ( χ ) (B.4)to be used in bringing eq. (B.2) to the matrix form from eq. (2.13), with (cid:174) P ( E ) staying the same as ineq. (2.29). Acknowledgments
This work was supported by the Croatian Science Foundation under Project No. 8570.
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