Determination of luminosity for in-ring reactions: A new approach for the low-energy domain
Y.M. Xing, J. Glorius, L. Varga, L. Bott, C. Brandau B. Bruckner, R.J. Chen, X. Chen, S. Dababneh, T. Davinson, P. Erbacher, S. Fiebiger, T. Gassner, K. Gobel, M. Groothuis, A. Gumberidze, G. Gyurky, M. Heil, R. Hess, R. Hensch, P. Hillmann, P.-M. Hillenbrand, O. Hinrichs, B. Jurado, T. Kausch, A. Khodaparast, T. Kisselbach, N. Klapper, C. Kozhuharov, D. Kurtulgil, G. Lane, C. Langer, C. Lederer-Woods, M. Lestinsky, S. Litvinov, Yu.A. Litvinov, B. Loher, N. Petridis, U. Popp, M. Reed, R. Reifarth, M. S. Sanjari, H. Simon, Z. Slavkovska, U. Spillmann, M. Steck, T. Stohlker, J. Stumm, T. Szucs, T. T. Nguyen, A. Taremi Zadeh, B. Thomas, S. Yu. Torilov, H. Tornqvist, C. Trageser, S. Trotsenko, M. Volknandt, M. Wang, M. Weigand, C. Wolf, P. J. Woods, Y.H. Zhang, X.H. Zhou
aa r X i v : . [ nu c l - e x ] A ug Determination of luminosity for in-ring reactions: A new approach for thelow-energy domain
Y.M. Xing a,b , J. Glorius b, ∗ , L. Varga b , L. Bott c , C. Brandau b,d , B. Brückner c , R.J. Chen a,b , X. Chen a , S. Dababneh e ,T. Davinson f , P. Erbacher c , S. Fiebiger c , T. Gaßner b , K. Göbel c , M. Groothuis c , A. Gumberidze b , G. Gyürky g , M.Heil b , R. Hess b , R. Hensch c , P. Hillmann c , P.-M. Hillenbrand b , O. Hinrichs c , B. Jurado h , T. Kausch c , A.Khodaparast b,c , T. Kisselbach c , N. Klapper c , C. Kozhuharov b , D. Kurtulgil c , G. Lane i , C. Langer c , C.Lederer-Woods f , M. Lestinsky b , S. Litvinov b , Yu.A. Litvinov b, ∗ , B. Löher j,b , N. Petridis b , U. Popp b , M. Reed i , R.Reifarth c , M. S. Sanjari b , H. Simon b , Z. Slavkovská c , U. Spillmann b , M. Steck b , T. Stöhlker b,k , J. Stumm c , T.Szücs g , T. T. Nguyen c , A. Taremi Zadeh c , B. Thomas c , S. Yu. Torilov l , H. Törnqvist b,j , C. Trageser b,d , S. Trotsenko b ,M. Volknandt c , M. Wang a , M. Weigand c , C. Wolf c , P. J. Woods f , Y.H. Zhang a , X.H. Zhou a a Key Laboratory of High Precision Nuclear Spectroscopy and Center for Nuclear Matter Science, Institute of Modern Physics, Chinese Academyof Sciences, Lanzhou 730000, China b GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, Darmstadt 64291, Germany c Goethe Universität, Theodor-W.-Adorno-Square 1, Frankfurt am Main 60323, Germany d Justus-Liebig Universität, Ludwigstraße 23, Gießen 35390, Germany e Al-Balqa Applied University, P.O. Box, Salt 19117, Jordan f University of Edinburgh, South Bridge, Edinburgh EH8 9YL, United Kingdom g Institute for Nuclear Research (MTA Atomki), Bem tér 18/c, Debrecen 4026, Hungary h CENBG, CNRS-IN2P3, Rue du Solarium 19, Gradignan 33170, France i Australian National University, Canberra ACT 2600, Australia j Technische Universität Darmstadt, Karolinenpl. 5, Darmstadt 64289, Germany k Helmholtz-Insitut Jena, Fröbelstieg 3, Jena 07743, Germany l St. Petersburg State University, Lieutenant Schmidt emb., 11/2, St. Petersburg 199034, Russia
Abstract
Luminosity is a measure of the colliding frequency between beam and target and it is a crucial parameter for themeasurement of absolute values, such as reaction cross sections. In this paper, we make use of experimental datafrom the ESR storage ring to demonstrate that the luminosity can be precisely determined by modelling the measuredRutherford scattering distribution. The obtained results are in good agreement with an independent measurementbased on the x-ray normalization method. Our new method provides an alternative way to precisely measure theluminosity in low-energy stored-beam configurations. This can be of great value in particular in dedicated low-energystorage rings where established methods are difficult or impossible to apply.
Keywords: luminosity, Rutherford scattering, storage ring, beam, gas target, reaction
1. Introduction
Luminosity is a key parameter used in the experiments for absolute cross section measurement [1]. To directlymeasure luminosity, detailed knowledge of beam intensity and target density is required. Beam intensity can be pre-cisely measured by a calibrated current transformer in a storage ring [2] or a beam calorimeter for stopped beams [3],while the measurement of the target density depends on experimental conditions. For a solid target, the thickness canbe well estimated to a precision below 5% [2, 4]. However, to precisely determine the effective gas target density, onehas to precisely measure the temperature and pressure as a function of the position inside the gas target [3]. Some-times, heat transfer from the intense ion beam which may influence the density of the target gas along the beam path ∗ Corresponding authors
Email addresses:
[email protected] (J. Glorius), [email protected] (Yu.A. Litvinov)
Preprint submitted to Elsevier August 28, 2020 hould also be considered [3, 5, 6]. Particularly, for the case of using an internal gas target in a storage ring, detailedknowledge of the target and beam profiles as well as the beam-target overlap [7, 8] is additionally required and anoverall uncertainty of 30% may be assumed [8].To remove this large luminosity uncertainty in reactions with an internal gas target in a storage ring, indirectmethods are widely applied. For example, the beam energy loss in the gas target has been used for the thicknessdetermination of the target as well as for the luminosity [7, 9], where a 5% precision was generally obtained. However,this precision highly depends on the accuracy of the measured revolution frequency shifts due to the beam energyloss. If the beam lifetime is too short for a reliable frequency shift measurement, the application of this method islimited. Alternatively, the luminosity can be obtained from the analysis of a reference reaction with a well-knowncross section. It is based on the precondition that the reference cross section is better known than the cross sectionto be measured. However, there are often difficulties in finding a suitable reference reaction. As an example, aparticular angle-dependent x-ray transition rate, such as the one for K-shell radiative electron capture (K-REC) isused [2, 7, 10, 11]. The counts of quasi-free proton-deuteron elastic scattering events have also been adopted in someparticular cases [12–14].In this paper, we report a method employing the Rutherford scattering distribution for the determination of thereaction luminosity in a storage ring. At low beam energy in a storage ring, the Rutherford scattering is dominantand the differential cross section is known quite well. By normalizing the simulated scattering distribution to theexperimental one, the reaction luminosity can be precisely determined. Here, we take an experiment on Xe ( p , γ ) cross section measurement [11] performed at the Experimental Storage Ring (ESR) in GSI as an example to illustratethe power of this method.
2. Experiment
Taking into account the revolution frequency of the beam of several hundred kHz, the storage ring can increasethe beam utilization efficiency by several orders and has been proved to be a facility that is suitable for cross sectionmeasurements [10, 15]. In the experiment performed at ESR, a
Xe beam was used to measure the proton capturecross section at low energies. The experiment and the results have been reported in Ref. [11]. Here we only give abrief introduction.Firstly, the
Xe ions were accelerated to about 100 MeV/u by the UNIversal Linear ACcelerator (UNILAC) andSchwerIonenSynchroton (SIS18), and then were extracted to the transfer beam line, completely stripped, injected andstored in the ESR. The beam with an intensity of about 10 to 10 per spill was cooled by the electron cooling systemand decelerated to the desired energy of a few MeV/u. After that, the internal hydrogen gas target with a diameterof about 5 mm and density reaching about 10 atoms/cm [16] was switched on. The beam passed through the gastarget with a revolution frequency of about 300 kHz. The ( p , γ ) reaction products Cs as well as the scattered beamions were detected with a double-sided silicon strip detector (DSSSD) mounted inside the dipole behind the gas target. double-sidedsilicon strip detscatteringdistribution
Figure 1: (Colour online) Schematic view of the experimental setup at the ESR from the gas target to the next dipole magnet.
Figure 1 shows the schematic illustration of the setup at the ESR including the detector system and the internalgas target. The active area of the DSSSD detector was 4.95 cm × ◦ ,60 ◦ and 90 ◦ with respect to the beam direction. 2 . Simulation of the scattering distribution In the experiment a large part of the ions scattered off the target were measured by the DSSSD detector. Toreproduce the scattering distribution on the detector, the Monte-Carlo (MC) code MOCADI [17] has been used tosimulate the transport of reaction products through the ion-optical system.In the simulation, according to Ref. [18], an estimated beam emittance ε =0.5 mm × mrad and a relative momen-tum spread δ p / p = × − were adopted. In addition, four different beam energies 5.47 MeV/u, 5.95 MeV/u, 6.65MeV/u and 6.96 MeV/u have been used. These energies are just around the non-Rutherford threshold energy ( ∼ ◦ . However, considering: (1) most of the ions de-tected by the DSSSD detector have much smaller scattering angles well below 100 ◦ and (2) the ion scattering througha smaller scattering angle has higher threshold energy, these energies are roughly in a domain where Rutherford scat-tering is dominant. Thus, the pure Rutherford scattering was presumed in the simulation. However, computationalefficiency for these simulations is challenging, because of the steepness of the cross section which spans many ordersof magnitude for range of scattering angles covered in the experiment. In this study, MOCADI was only used to sim-ulate scattering kinematics and beam optics adopting a uniform angular distribution. As a result, a model distributionof the scattered ions on the detector was obtained. Afterwards a transformation of the distribution by adding properweights to the events was introduced to obtain a correct scattering distribution. q d q DW
124 54+ Xe proton Figure 2: (Colour online) The schematic view of the scattering kinematics. The red and black solid lines represent the track lines of two Xe + ions with scattering angel θ and θ + d θ , respectively. A schematic view of the scattering kinematics is shown in Fig. 2. In the experiment, the target is H . However,compared with the beam energy, the binding energy of hydrogen atoms and electrons is negligible. So the proton isactually used as the target in our simulation. For the solid angle ∆Ω = π sin θ d θ , presented with the shadowed area,the corresponding cross section is σ ( θ ) = d σ d Ω ( θ ) ∆Ω , where d σ d Ω ( θ ) is the Rutherford differential cross section: d σ d Ω ( θ ) = (cid:16) πε Z Z e E (cid:17) sin ( θ ) . (1)Here, E is the kinetic energy in centre of mass system, Z and Z are the atomic numbers of the target and projectilenuclei respectively. If the number of the scattered ions within the solid angle ∆Ω is M (M>0), a definition of theweight w ( θ ) is w ( θ ) = σ ( θ ) Lt / M . (2)with L being the average luminosity during the experimental time t . It is straightforward, that w ( θ ) is the scaling factorto be applied to each ion in the simulation within scattering angles θ and θ + d θ . By introducing the corresponding w for all ions in the simulation, the model scattering distribution is transformed into the realistic Rutherford one. Inour simulation, the d θ was set to be 0.01 rad.
4. Luminosity determination
To determine L and its uncertainty, the maximum likilihood estimation and the χ minimization method have beenused [20]. Since the two methods gave quite similar results, here we present only the results of the χ minimizationmethod. 3 e n t r a l o r b i t ... ... ... ... D SSSD x yz y β α ( , , )Dx Dy Dz Figure 3: The geometrical layout of the DSSSD detector. The detector is equipped with 16 ×
16 silicon strips. The central point of the detector isset as (Dx,Dy,Dz). The two angles determining the detector orientation are defined as α and β . Normally, to get the maximized effective detectingarea, both α and β would be set to 90 ◦ in the experiment. However, α was set about 45 ◦ due to the limited size of the vacuum pipe where thedetector inside the dipole magnet was installed. c oun t s xs t r i p xs t r i p c oun t s y s t r i p y s t r i p Experiment Simulation
Figure 4: (Colour online) Left: The experimental scattering distribution detected by the DSSSD detector for the beam energy 5.95 MeV/u. Thesmall region marked with the yellow circle indicates the peak from the ( p , γ ) products. Right: The corresponding simulated scattering distributionon the detector. ×
16 bins. We use i and j as the two-dimensional index of the bins (1 ≤ i ≤
16, 1 ≤ j ≤
16) and the χ is defined as: χ = ∑ i ∑ j ( N expi j − N simi j ) σ i j , (3)where N simi j and N expi j are the ion numbers detected in the bin ( i , j ) from simulation and experiment, respectively, σ i j isthe uncertainty of ( N expi j − N simi j ) which was taken as q N simi j [20].If N is the total number of bins considered in the χ calculation and M is the number of free parameters in thesimulation, the minimum χ , χ min , obeys the standard chi-square distribution with N − M degrees of freedom [21].Since all free parameters are adjusted at the same time, the luminosity is determined when χ reaches χ min . Thegoodness-of-fit test can easily be done by calculating the Q value, which is the probability that the observed χ exceeds the χ min with N − M degrees of freedom [20].In the simulation, besides the luminosity, various parameters have been used, such as the beam parameters in-cluding the initial beam energy, emittance and momentum spread, the magnetic fields of the quadrupole and dipolemagnets, the detector parameters including the detector position and space orientation, etc. It is not realistic to keepall these parameters free due to the limitation of computer capabilities. Nevertheless, a sensitivity test to check theimportance of these parameters in a reasonable range is meaningful. In the test, all parameters were initially set tobest estimated values. Then a chosen parameter for the test was varied and the luminosity value corresponding to theminimum χ value was recorded. In this way, the sensitivity and importance of the parameters for the luminositydetermination was checked. For example, the beam emittance was increased by 3 times or the energy of the beam wasvaried by ±
20 keV (the expected energy uncertainty) compared with the set value. No effect (less than 1%) on theluminosity determination has been observed. Thus, the beam energy was fixed in the simulation to a nominal valuegiven by the electron cooling system. For many other parameters, similar behavior has been observed. Finally, thegeometrical orientation of the DSSSD detector which was not well known in the experiment was found to play thekey role in the final luminosity determination.Figure 3 shows the specific geometry of the detector as used in the simulation. The coordinates of the detectorcenter were defined as (Dx,Dy,Dz), where the x, y, z directions are the direction pointing to the inner side of the dipole,along the central orbit of the beam and vertically upward, respectively. For the orientation of the detector, tilting angles α and β were defined as shown in Fig. 3. In the simulation, together with the luminosity, these parameters were takenas free parameters.To find χ min , the experimental scattering distribution without contributions of other reaction channels is needed.In the present experiment, the ( p , γ ) products were distributed around the centre of the DSSSD detector (see the binsin the yellow circle in Fig. 4) [11] and had to be excluded from the calculation.Eventually, a reduced chi-square χ red = χ min / ( N − M ) of 1.24, 1.13, 1.11 and 1.38, was obtained for the beamenergies of 5.47 MeV/u, 5.95 MeV/u, 6.65 MeV/u and 6.96 MeV/u, respectively. The corresponding Q values are0.03, 0.13, 0.19 and 0.01. According to Ref. [21], a model is roughly acceptable if Q>0.001, so the Q values showthe credibility of the simulation. As an example, a qualitative comparison of the measured and modelled scatteringdistributions for the beam energy of 5.95 MeV/u is shown in Fig. 4.Since the parameters (including the luminosity) controlling the simulated Rutherford scattering distribution whichis used for the χ calculation are coupled in the simulation, to determine the uncertainty of the extracted luminosity,the following approach has been used. If ν parameters from the total of M free parameters ( ν < M ) are fixed and theremaining M − ν parameters are varied to minimize χ , this minimum value is called χ ν ( χ ν > χ min ). As shown inRef. [21], ∆χ ≡ χ ν − χ min is distributed as a chi-square distribution with ν degrees of freedom. This connects theprojected ∆χ region with the confidence interval. For example, for ν = ∆χ < .
30 occurs 68.3% (corresponds to1 σ for normal distribution), ∆χ < .
18 occurs 95.4% (2 σ ), ∆χ < . σ ).Figure 5 shows an example of the determination of the luminosity uncertainty. Here the luminosity and the tiltingangle α which is defined in Fig. 3 are chosen as the fixed parameters ( ν = ∆χ as a function of them. As stated above, the luminosity uncertainties can be determined when ∆χ is less than a certainvalue. For example, the 2 σ and 3 σ uncertainties are determined when ∆χ is 6.18 and 11.8, respectively. In thiswork, we adopted a conservative approach by choosing 3 σ uncertainty. It is shown with the black solid line in Fig. 5.5 able 1: The luminosities and the errors determined by the x-ray measurements ((from three independent x-ray detectorsat different observation angles as shown in Fig. 1) and this work. The last column shows the relative deviation defined asthe ratio between the luminosity difference L − L K and the combined error δ L which is the root-mean-square of the errorsfrom L K and L . Energy [ MeV / u ] time [ s ] ◦ ◦ ◦ Average L K [ barn / s ] This work L [ barn / s ] Relative deviation ( L − L K ) / δ LL K [ barn / s ] L K [ barn / s ] L K [ barn / s ] T il t i n g a n g l e α ( ° ) Luminosity (barn/s) s2s2s s Dc Figure 5: (Colour online) The variation of ∆χ with tilting angle α and luminosity for the beam energy 5.95 MeV/u. The 3 σ uncertainty ( ∆χ = .
8) is adopted for the error estimation and shown with black solid line. The determined tilting angle α within 3 σ uncertainty is in agreementwith the expected value of 45 . ◦ . The determined α = . ◦ ( . ◦ ) is consistent with the experimental arrangement, in which the angle is expected tobe around 45 ◦ . From the figure, it is seen that the luminosity determination is very sensitive to α . This is reasonablebecause the titling angle α ≈ ◦ and if it is slightly changed, the effective area of the DSSSD detector projected on thez direction is considerably changed. If in future experiments, the tilting angle α is determined precisely (uncertaintyof less than 0 . ◦ ), the uncertainty of the luminosity determination will be significantly improved.The luminosities and the estimated errors for different beam energies are listed in the penultimate column inTable 1. We can see that for different beam energies, the luminosity does not change significantly. However, thecorresponding errors differ considerably. This is mainly because the magnitude of the error is highly dependent onthe accumulated statistics which was different in all cases. For example, for the case of 6.65 MeV/u, the experimenttime was much shorter than in other cases and the detector performance degraded due to an increased ion dose level.As a result, the accumulated statistics of scattering events was the lowest and hence the obtained error is the largest.For another case of 6.96 MeV/u, the yield of ( p , γ ) product is much higher than in other cases mainly because ofthe largest ( p , γ ) cross section [11]. Accordingly, the area dominated by ( p , γ ) becomes larger, reducing the effectivedata set for the Rutherford scattering model. At this or even higher energy, the ( p , n ) products and maybe also thenon-Rutherford scattering can contribute to the scattering events which increase the luminosity uncertainty.6 . The verification of this method As stated in the introduction, the luminosity can also be determined through the K-REC x-ray measurement. Thisprovides a good opportunity to check the validity of this work. In this case, the luminosity (noted as L K ) can beexpressed as L K = N K ε ( d σ K / d Ω ) ∆Ω t (4)where N K is the number of K-REC x-rays, ε is the intrinsic efficiency of the Ge detector, ∆Ω is the solid angle spannedby the Ge detector and d σ K / d Ω is the theoretical differential K-REC cross section. The individual efficiency-correctedK-REC counts per steradian N K / ε∆Ω and the effective theoretical d σ K / d Ω can be found in Ref. [11] for each beamenergy. Based on these values and the measurement duration time t listed in Table 1, L K from three different anglesand the average L K were calculated.The relative deviation between our present results and the K-REC method are listed in the last column of Table 1.All the deviation relative to the combined error δ L are positive and well below 2. This systematic deviation may beinduced by some unconsidered factors in the simulation of the experiment. This method was not foreseen to be usedin this experiment [11]. Thus, since no dedicated efforts were undertaken to determine relevant parameters, the goodagreement between two methods is remarkable and indicates the power of this new approach.The averaged errors obtained by the K-REC method are listed in Table 1. For the cases with high statistics, theerrors from the two methods are comparable.
6. Summary and outlook
In the direct measurement of absolute reaction cross sections, the luminosity is a critical quantity, which is hardto determine precisely. This is especially true for experiments using thin gas targets. Just like the x-ray emissions, thescattering distribution itself reflects the collision frequency and is useful for the determination of the luminosity.As a proof of concept, this work has shown the feasibility to use the elastic scattering distribution to preciselyobtain the luminosity. A weighting method is introduced for the simulation of the scattering distribution on thedetector. By taking the
Xe + H experiment performed at the ESR storage ring as an example [11], the luminosityis determined via a χ minimization approach. Although, a small systematic offset to the established K-REC methodis found, there is still agreement within the error bars demonstrating the validity of the method. The uncertaintiesdetermined by the two independent and complementary methods are comparable.This new method will become indispensable in the future experiments as planned in the dedicated low-energystorage rings [22, 23] with light beams at much lower energies, e.g., to reach the Gamow window for astrophysicalreaction studies [24].In the future, with a precisely defined geometry of the detector, the uncertainty of the determined luminosity isexpected to be significantly reduced. Furthermore, if the measurement of the scattering distribution can be performedjust before the dipole, the simulation would be simplified as the transport of the ions through magnetic system is notneeded, and the determination of the luminosity will become more accurate.It should be mentioned that when the beam energy is high enough, the non-Rutherford scattering should be ac-counted for and the optical model can be used for the calculation of the corresponding differential cross section [25].However, with the higher energy, the reaction products from nuclear channels like ( p , n ) and ( p , α ) may mask scat-tering events. In such situation, a target recoil measurement close to the gas target can be considered [10, 26].Although this method is based on the experiment performed at a storage ring, it can also be employed to anyreaction measurement on a thin target, where the luminosity needs to be precisely measured. Acknowledgments
This work is supported in part by the European Research Council (ERC) under the European Union’s Horizon 2020research and innovation programme (Grant Agreement No. 682841 "ASTRUm"), the NSFC (Grants No. 11905261),the Key Research Program of Frontier Sciences of CAS (Grant No. QYZDJ-SSW-S), the National Key R&D Programof China (Grant No. 2016YFA0400504 and No. 2018YFA0404400) and the Helmholtz-CAS Joint Research Group7Grant No. HCJRG-108). Y.M.X. thanks for support from CAS "Light of West China" Program. Y.A.L. acknowledgessupport by the CAS President’s International Fellowship Initiative (Grant No. 2016VMA043).
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