Preamble

New Algorithms for Particle Mass Number Identification and Time-of-Flight Calibration for Secondary Beams Produced via Projectile Fragmentation Hu-Wei Xie, Ya-Zhou Sun, Wei-Ping Lin, Zhi-Yu Sun, 2, 3, Shu-Ya Jin, Xue-Heng Zhang, Shu-Wen Tang, Duo Yan, Yu-Hong Yu, Fang Fang, Yong-Jie Zhang, Xuan Jiang, Xiao-Bao Wei, Fen-Hua Lu, Zhi-Yao Li, Lin-Feng Wan, and Shi-Tao Wang 1 Key Laboratory of Radiation Physics and Technology of the Ministry of Education, Institute of Nuclear Science and Technology, Sichuan University of Chengdu 610064, China State Key Laboratory of Heavy Ion Science and Technology, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China Department of Physic, Southern University of Science and Technology, Shenzhen 518005, China College of Physics, Centre for Theoretical Physics, Henan Normal University, Xinxiang 453007, China This paper presents new data analysis algorithms for identifying the mass number and calibrating the time of flight ( ) for secondary beams produced via projectile fragmentation at incident energies of several hundred MeV per nucleon. The algorithms need only conventional measurements of in the beam line, the energy deposit ( ), and the hit positions in the entering and exiting foci of the beam line for unambiguous identification of the mass number of the secondary cocktail beams, and also the calibration of with a precision of about . The algorithms are implemented by fitting the transfer matrix elements under linear beam optics to extract the central magnetic rigidity with the assumed mass is the atomic mass unit) of the nuclei. This procedure is implemented for each chosen nuclide in the secondary beam respectively so that a scatter plot is obtained. The deviation of the assumed mass and the real value are evaluated by fitting the scatter plot with a function , from which one obtains . The calibration of follows a similar method. The algorithms are tested to give a satisfactory and consistent particle identification for fragments of

350 MeV

nucleon Kr in a beam test conducted at the External Target Facility (ETF) of the second Radioactive Ion Beam Line in Lanzhou (RIBLL2) in HIRFL-CSR.

Keywords

Particle identification algorithm, Projectile fragmentation, Radioactive ion beam line

INTRODUCTION

Radioactive ion beams (RIBs) produced via projectile frag- mentation at several hundred MeV per nucleon have been em- ployed as a sensitive and potent probe to systematically study the single-particle structure of unstable nuclei since the 1980s ], and opened a vast new testing ground for the nuclear structure models and reaction theories to develop [ ]. The particle identification (PID) of the secondary beams (frag- ments) is the premise for their applications to further physical studies. The method is prevalent for this pur- pose, which is adopted in major international radioactive ion beam lines, such as A1900 [ ], BigRIPS [ ], FRS ], and RIBLL2-ETF [ ]. The particle positions at the focal planes of the RIB line are recorded to help extract the trajectory radius and the magnetic rigidity . The time of flight ( ) through the RIB line is measured to obtain

the velocity v = βc ( c is the light speed in vacuum) and the 17

Lorentz factor . The mass-to-charge ratios of the beam Supported by the National Key R&D Program of China (Grant No. 2023YFA1606400), the National Natural Science Foundation of China (Grants No. 12305133, No. 12205340, and No. 12475129), China Postdoctoral Science Foundation (Grant No. 2023M733575), and Longyuan Youth Talent Program. particles are obtained following Eq.

Bρ = p q = Au

Q βcγ

β = v c = L c · TOF , (2) 22

Here Au is the ion mass in kg , with u = 23

1 . 66053906660(59) × 10 − 27 kg the atomic mass unit. 24

is the elementary charge, and the flight length. represents the charge carried by the ion, with denoting the charge number, differing from the atomic number

a prime. For fully stripped ions, we have Z ′ = Z . The 28

last step of Eq. 1 is for a convenient numerical evaluation 29

of Bρ , where uc/e = 3 . 1071299 · · · T m is a constant, and 30

is dimensionless. The particle charge is given by the energy deposit and is in many cases much easier to be identified than e.g., by a careful calibration of the detector. In com- parison, according to Eq. , the identification entails accurate information, which are susceptible to offsets. offsets to nominal values may come from small deviations of beam-line working conditions from design val- ues in a certain run, particularly for beam lines composing of many magnets. On the other hand, the fragment velocity

carries a considerable dispersion width, and usually the esti- mation of its central value relies on simulation. The simulated may be even more unreliable when new timing config- urations (e.g., timing thresholds in leading edge discrimina-

tion) are used in the detector electronics of the experiment. 45

Moreover, it is difficult to directly use secondary beams for their own calibration as their velocities are not known.

The RIB experiments will benefit from a convenient and ef- 48

fective method for identifying the mass number of the sec- ondary beams via projectile fragmentation, with is easy to calibrate for each experiment run with their own

electronics configurations. 52

identification for fragments of very light projec-

tiles (typically A < 20 ) is trivial. Firstly, with Z around 54

10 and below, and at fragmentation energies of several hun- per nucleon, these RIBs are usually fully stripped.

Since the product nucleus species are so few and the PID spectrum possesses so distinctive characteristics (e.g., the ab- sence of certain particle-unstable products such as B and 8 Be, and specific features of the relative positions of

such as the A/Z = 2 vertical line in the A/Z - Z PID spec- 61

trum), that the ( A, Z ) of the fragments are definitely resolved 62

]. Other assignments by adding or removing the same number ( ) of neutrons from all the product nuclei re- sult in disparate PID spectra that contradict with physical re- ality. For a typical example the readers are referred to Fig. 9 [FIGURE:9] of Ref. [ As the projectiles come from a heavier region in the nu-

clear chart, the aforementioned advantages soon vanish. The 69

limited dynamical range of the detector restricts the min- that can be measured while keeping the high in its measurable range for fully stripped beams. This results in a much higher starting in the PID spectrum, easily exceeding B and Be isotopes. The PID spectra obtained by applying a mass number offset to the real PID as formerly mentioned are not guaranteed to be so non-physical to rule themselves

out. A definitive A identification in this scenario usually re- 77

quires the measurement of characteristic -rays emitted from

the known isomer of a specific nucleus, a technique known as 79

isomer tagging [ Given that -ray detectors are cumbersome to construct and operate, an identification method independent of any ray detector is highly appealing for practical applications. On the other hand, uncalibrated may easily distort the PID spectrum so much that it may affect our judgment in iden- tification. So the calibration is necessary with the ab- sence of a -ray detector.

This paper introduces a fragment mass number identifica- tion algorithm using only as input the conventional measure- ments of and positions at the focal planes of the beam line. Another algorithm for the calibrations of is also given. They are formulated in detail in the chapter to follow. An application of the algorithms to beam test data analysis of 350- nucleon Kr impinging on a 10- beryllium target are presented afterwards.

THE ALGORITHMS To facilitate the formulation of the algorithms, we assume that the beams are fully stripped in the following text. The issues involving different charges states of not-fully-stripped (or the so-called hydrogen-like) ions will be discussed sepa- rately in the end of Sec.

III A The Fragment Mass Number Identification We first introduce the algorithm for the identification of fragment mass number . According to linear beam optics ], the transverse motion of the beam in horizontal plane can be expressed as

x ( s ) = C ( s ) x i + S ( s ) x ′ i + D ( s )∆ p p 0 (3) 107

is the arc length of the reference orbit, and the trans- verse position of the beam particle in the horizontal plane rel-

ative to the reference orbit. x ′ = d x/ d s represents the mo- 110

mentary angle in the horizontal plane. x i and x ′ i are the initial 111

values of these optical coordinates at s = s i . p 0 is the mo- 112

mentum of the reference particle, and ∆ p = p − p 0 is the 113

momentum deviation of the beam particle from . For beam transfer from focal plane 0 ( ) to focal plane 1 ( ) realizing

point-to-point image, the angular dependence S = 0 , as in 116

the case of beam transfer from the dispersive focal plane to the ETF [ ] terminal of RIBLL2 [ ]. We have in this situation

x 1 = Cx 0 + D ∆ p p 0 = Cx 0 + DBρ − Bρ 0 Bρ 0 (4) 120

where Eq. has been applied in the last step, with magnetic rigidity of the reference particle, or the central mag- netic rigidity of beam. Then can be expressed as a linear combination of , and

Bρ ( x 0 , x 1 ) = Bρ 0 + ax 0 + bx 1 . (5) 125

It is sometimes necessary to include higher-order depen- dence in Eq. so that are accurate to our satisfac-

tion. Using A and A a = A + δA as the real and the assumed 128

mass number of a certain fragment nuclide respectively, we can get the assumed magnetic rigidity following Eqs.

Bρ ( β ) a = uc

= A a A a − δA · uc

= A a A a − δA · ( Bρ 0 + ax 0 + bx 1 )

A) [Tm]

Bρ 0 ( δA ) as a function of A a with a series of δA , using Bρ 0 = 7 T m .

shows that the magnetic rigidity calculated with the as- sumed fragment mass number has -dependent coefficients. is estimated by fitting experimental with Eq. , which then gives . It turns out in practice that is the best option to do this job. are not recommended because their absolute values are so small that the change in values brought about by the factor is easily overwhelmed by their fluctu- ations (e.g., due to difference in statistics) among different . As a result, the error of the fitted is too large for

identification. Bρ 0 ( δA ) with a series of δA and using Bρ 0 = 144

with δA ̸ = 0 exhibit trends distinct from that with δA = 0 . 146

Another noteworthy feature is that the resolution ability

will be significantly improved as the collection of nuclides in 148

the secondary beam starts from a lighter nuclide. As the extraction of needs to be done for each fragment respectively, one firstly has to separate the fragment nuclides from each other. This is achieved by computing pre- liminary via Eq. , with calculated by Eq. , for which the preliminary assignments of the transfer matrix el- ements are needed. They can be made from beam optics calculations, beam line design parameters, or fitted from experiment data [ ]. We recommend ted from experiment data, because it is both reliable [ ] and self-contained. All fittings in this paper are implemented by

minimizing the fitting χ 2 . 160

The fitting of follows the steps below:

1. Select a

in the fragments with large statistics.

2. Fix

by narrowing it down to a small range. plot for the fragments of Kr impinging on a 10- thick beryllium target at

350 MeV

nucleon in a beam test con- ducted at RIBLL2-ETF. We have fixed for the fragments.

3. Draw the

plot, where we can find several dis- crete stripes. Equating Eq. with Eq. we have

βγ = 1 uc/e · A/Z ( Bρ 0 + ax 0 + bx 1 ) (8) 166

which shows that takes discrete values correspond- ing to discrete , and are fixed. So each of the stripes corresponds to an isotope. An example of such a plot is shown in Fig.

4. Keep in mind that

is a constant and fit one of the stripes with Eq. to get is also ob- tained by the way. Note that here has to be roughly assumed (namely, has to be given a value) to obtain . Practically it is not required that be very close to , as long as the fragments are clearly separated so that the fitting can be implemented on each nuclides without contamination from each other. as large as 5 has been tested to work fine. Still we do recommend making the assignment with the help of extra infor- mation, e.g., nominal provided by the accelerator crew or measurements of and central , so that close to is fitted similarly.

Inserting the preliminary thus obtained into Eq. and equating it with Eq. for all the fragments, we can extract and get an PID plot, where different fragment isotopes are separated.

After the separation of the fragment isotopes, the fitting of for each fragment isotope follows a simpler procedure

2. Fix

by narrowing it down to a small range.

3. Draw the

plot, with evaluated by the first line of Eq. . Fit the plot to get . An example is illustrated in Fig. of Sec.

III A , where a second- order term of is also added. is fitted similarly (namely by fixing and fitting the plot).

5. The

is then evaluated according to Eq.

Bρ 0 ( δA ) = Bρ ( β ) a − ( ax 0 + bx 1 ) (9) 201

using the fitted values.

6. Finally we get a

data point. The above procedure is repeated for all fragments, so that plot is obtained. It is then fitted with the first formula of Eq. , with as the fitting parameters.

It is worth noting that this identification algorithm is ro- bust against moderate offsets in . It has been tested that the above procedure is still effective with (unknown) system- atic error in of the order of several . This is explained in Sec.

Calibration calibration follows a similar procedure. Using similar notations, by denoting the real and the assumed time of flight by respectively, we

β a (∆ t ) c ≡ L TOF a = L TOF + ∆ t,

Bρ ( β a ) = uc

= β a (∆ t ) γ a (∆ t ) βγ · uc

= β a (∆ t ) γ a (∆ t ) βγ · ( Bρ 0 + ax 0 + bx 1 )

where will be used to fit , due to similar reasons with Sec. are those correspond to with a series

of ∆ t and using Bρ 0 = 7 T m and L = 30 m are drawn in 222

Like the situation in Sec. , we also observe the dis-

tinct trends of Bρ 0 (∆ t ) with ∆ t ̸ = 0 compared with that 225

of ∆ t = 0 , although they are not as prominent. The overall 226

spread of the fragments is of the order of, say, 20 ns The relative change, as can be told from Fig. , is only around 1%, for changing from 120 ns 140 ns , for

∆ t = 3ns . This imposes a Bρ 0 dispersion of at most the 230

same magnitude to the plot, because for data points with the same , their can vary at most from 120 ns 140 ns in the case discussed here. The relative change of

is about 1 to 2 orders of magnitude smaller than that with 234

changing from 7 to 100 for nonzero , as can be told from Fig. . Moreover, moderate nonzero (e.g., cussed here) has a very limited effect on the overall trend of plot, since while is positively related to barely changes with it (see Eqs. ). So it is feasible and suggested to kick start the fragment mass number identifi- cation algorithm using (un-calibrated) estimated from, e.g., simulation or simple calculations, given that small give sharp plots. However, the converse is not valid.

Non-zero will impose dispersion in that easily overwhelms the trends brought about by and spoils Fig.

It is recommended to substitute . It is told from Eqs. is positively related to

TOF = L

spread for a certain nuclide could be omitted if is fixed, or limited in a small range compared with the variation of (around 7%) for different nuclides, which is usually the case due to the limited momentum acceptance (around 1%) of beam lines. This substitution facilitates the calibration process because it is trivial to get

A/Z data points from the corresponding Bρ 0 ( δA = 0) - A 255

data points in the fragment mass identification procedure, which has been implemented in the first place. We only have to replace for each data point.

Note that since changes (although slightly) with (thus also with ) for moderate nonzero , the

eraged over all the Bρ 0 ( δA = 0) - A obtained in the fragment 261

mass identification procedure could be used for the . By iteration of the two algorithms, the dispersion of

Bρ 0 due to ∆ t will be rapidly diminished along with the min- 264

imization of The extraction of data points is some- what cumbersome. is calculated via Eq. factor given by , where the are calculated separately for each nuclide with their . All the events are then accumulated to constitute the final scatter plot. calibration algorithm is summarized as follows:

1. Complete the

identification following the procedure

given in Sec. II A to get Bρ 0 with δA = 0 for each A , 274

which will then be used as

2. Draw the

) scatter plot.

3. Extract

by fitting the scatter plot with Eq. . Or fit the plot with Eq.

t) [Tm]

Bρ 0 (∆ t ) as a function of TOF a with a series of ∆ t , using Bρ 0 = 7 T m and flight length L = 30 m .

and Eq. , and using fitted and averaged from

4. Apply the

to get calibrated new

5. Go to step

for another iteration. Usually satisfactorily converges after two iterations.

THE BEAM TEST The above algorithms for fragment mass number identifica- tion and calibration are tested in data analysis of a beam test conducted at the RIBLL2-ETF. A schematic layout of the experiment setup is illustrated in Fig. . The primary beam of 78 Kr was accelerated in the Cooler Storage main Ring (CSRm ]) to

350 MeV

nucleon and fragmented on a primary tar- get of 10- thick beryllium at F0. The fragments were transported to ETF through RIBLL2 with its centered around 2.3. of the secondary beams were measured by the time difference of two plastic scintillators start ] and ] installed at the dispersive focal plane F1 and the achromatic focal plane ETF. A position detector was installed at F1 next to start measurement, which is composed of 50 vertical 2- wide and 1- thick plastic scintillator strips, covering an effective area of 100 mm ]. The of the fragments were measured by a MUlti-Sampling

Ionization Chamber MUSIC0 [ 37 ]. The beam position x 1 at 302

ETF is given by drift chambers DCTaU0,1. The Data analysis is completed using code ETFAna The Fragment Mass Number Identification Following the algorithm in Sec. , we have selected 76 nuclides in the secondary beams with appropriate statistics from an experiment run of 242,919 events. Since the exper- imental has shown an appreciable parabolic de- pendence on , we have added an term to Eq. , which then reads As an example, Fig. shows the fitting of for one

specific case of 44 Ca in the 76 fragment nuclides. The units 314

of the fitted parameters in the figure are deduced when

is in T m and x 0 , 1 in mm for practical convenience in data 316

analysis. scatter plots with a series of are pre- sented in Fig. , and fitted with the first formula of Eq.

the region of A a ∈ (20 , 62) . The fitted Bρ 0 and δA are listed 320

in Table is the integer nearest to the fitted . The values are cross-checked to be compatible with beam infor- mation from the accelerator, which will be explained later. It shows that the of each curve in Fig. are unambiguously solved with the algorithm formulated in Sec.

It is also shown in Fig. and also Fig. that as the frag- ments start from a heavier region, e.g., by using a degrader for beam purification [ ], the distinctions between the trends of different will shrink and higher resolution (higher resolution and position resolution) are expected to com- pensate for the decrease in the resolving power of . The fitted being close to an integer and differing by 1 for adja- should be used for the estimation of the applicability of the algorithm.

The deviation of the fitted from its true value are directly related to those (off-line) data points that go downwards off the curve constituted by the main body of the data points. The fitted thus tends to suppress the

increasing its value to minimize the fitting χ 2 . Those off- 339

line data points may be attributed to low statistics, and/or too narrow distributions in , which deteriorate the fitting

process and result in significant error in the extracted Bρ 0 . 342

Still, the deviation of around 0.1 as seen in Table is neg-

ligibly small in determining δA i . The robustness of the al- 344

gorithm against systematic error in is also tested using

∆ t = ± 3 ns . The fitted δA turn out to have smaller deviations 346

(around 0.03) from their corresponding δA i for ∆ t = +3 ns , 347

as the positive decreases , reducing the need to in-

crease δA . The ∆ t = − 3 ns expectedly gives larger δA de- 349

viation of around , which is still acceptable. Besides, by

minimizing δA and ∆ t through iteratively calling of the algo- 351

rithms, the trouble of non-zero could be solved once and for all.

The PID spectrum with zero and calibrated is pre-

sented in Fig. 7 [FIGURE:7] , with the N = Z + x ( x = 0 , 1 , 2 , · · · ) lines 355

superimposed for reference. here is the neutron number.

The exact values of from AME2020 database [ are also marked in the figure for those nuclides selected in fitting. The axis represents the charge of the beam particle when they reach MUSIC0. Its calculation only in- volves the energy deposit in the detector plus some ver- tical position-dependence corrections [ ]. The axis is re- lated to the mass over charge ratios of the beams in the beam line, which is totally determined by and horizontal po-

RIBLL2 Kr at

350 MeV

nucleon (a) and (b) for the fragment Ca in the Kr beam test, with the conditions of 25)mm 39)mm , respectively. units of the fitted parameters are deduced when is in for convenience. fitted from data in Fig. . The are in is the integer nearest to the fitted . The uncertainties ( the fitted parameters are given in the brackets. data points of 76 nuclides selected from the secondary beam with five different assignments. The five lines corresponds to values as indicated in the figure, and have been fitted with Eq. is the integer nearest to the fitted sitions. So the two variables in Fig. are independently mea- sured. As far as we are concerned, it is necessary to discuss the different charge states for the same particle in the beam line from tstart tstop (denoted as ), and in MUSIC0 (denoted as ). For clarity we first talk about the cases of capturing or stripping of one electrons with respect to fully- stripped ions:

1. Q

decremented by 1. Then the new spot falls from N = 377

Z + x to N = Z + x + 2 line. Each fragment isotope 378

suffering this issue should exhibit an spot at the point two line rightward and one element down. There is no perceptible presence of such spots in Fig. . Since it

is the fragment with maximal that is most attractive to electrons, the outlier of Kr is an optimal example.

There is no discernible trace of its spot around (2 . 4 , 35) 384

in Fig. . So the possibility of this case is deduced to be negligibly small.

2. Q

3. Q

0 = Z − 1 , Q 1 = Z : Similar to case 2 , Fig. 7 is short 393

-like spots, which should appear next to the right side of the original fully-stripped A/Z, Z confirming that this case could also be safely neglected.

Following the above discussion, the capturing or stripping of more electrons involves more abnormal spots in the PID spectrum, which Fig. turns out to do not show in perceptible

significance. Moreover, they are naturally more difficult to 400

happen than that involving one electron. So Fig. is taken to be predominantly fully-stripped ions.

The conspicuous outlier of Kr is inferred to be unreacted nuclide from the primary beam, as it is seldom possible that the cross section is so large for Kr capturing one specific number of neutrons, and at the same time almost zero for the production of the neighboring nuclides [ ]. Moreover, the momentum spread of this outlier is nearly an order of mag-

nitude smaller than other fragments, also supporting the no- 409

tion that it is the unreacted primary beam. So it is not with- out surprise that our fragment mass identification algorithm suggests that it is Kr, instead of Kr. With as the primary beams that were accelerated in the CSRm, it is still possible that the outlier is contamination in the ion source. Since the mass over charge ratios of

78 Kr

are almost identical (differ only by 0.003%), these two ion species appear almost the same as far as the accel- erator and RIBLL2 concern, according to the equations of motion of beam optics [ ]. It is highly possible that were accelerated and delivered to ETF altogether.

The suspicion of ion source contamination by Kr is corrob- orated by the much higher natural abundance of Kr (57.0%) Kr (0.4%). It is unnecessarily and prohibitively expen- sive if ever feasible to get rid of every Kr atom from the 78 Kr ion source. Finally, there exist small deviations of the

experimental data spots from the nominal N = Z + x lines 426

and the values from AME2020 in Fig. for some nu- clides. This may be due to the time walk effect in the leading edge timing, which varies for different , introducing pendence in offset constants. This is not taken care of in the calibration algorithm in this work.

Calibration scatter plot following procedures de- picted in Sec. are extracted from the experiment data as Kr fragmented on a beryllium target at

350 MeV

nucleon . The cyan lines draw the functions with · · · . The little red dots mark the exact from AME2020 database [ ] for those nuclides selected for fitting. shown by Fig. . The five datasets with distinct colors are for five preset values as indicated in the figure. Specifi- cally speaking, for each of the values, we added it artifi- cially to the real TOF of each of all the particles, and see how well the algorithm could reproduce them. The fitted 2.83(1), 0.98(1), 0.00(1), -1.02(2) and -3.18(2) for preset

∆ t = 3 , 1 , 0 , − 1 , − 3 ns , respectively. The deviation of the 441

fitted from their preset values are less than , and in- creasingly small as approaches zero. This is rather mean- ingful since we are most interested in eliminating so that is without a constant offset. method shows comparable olution.

The experimental scatter plot is shown in Fig. . Similarly we used the same preset values, and get fitted of 2.807(1), 0.947(1), 0.035(1), -1.055(2)

and -3.126(1) ns for the preset ∆ t = 3 , 1 , 0 , − 1 , − 3 ns , re- 450

spectively.

SUMMARY

New algorithms for mass number identification and the re- lated calibration are introduced. The algorithms make use of the fact that the central magnetic rigidity the secondary beam exhibits dependence on mass number

t) [Tm] scatter plot for the 76 nuclides selected from the secondary beam. The five lines corresponds to different preset values as indicated in the figure. They are fitted with using expressed by according to Eq. , and represented by the solid lines. t) [Tm] scatter plot for the 76 nuclides selected from the secondary beam. The five bulks correspond to data with different preset values as indicated in the figure. They are fitted with Eq. as represented by the solid lines.

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