Examination of experimental conditions for the production of proton-rich and neutron-rich hypernuclei

1.3 MB·

by  C. Rappold

Examination of experimental conditions for the production of proton-rich and
neutron-rich hypernuclei
C. Rappold1,2,∗ and J. L´opez-Fidalgo2
1GSI Helmholtz Centre for Heavy Ion Research, Planckstrasse 1, 64291 Darmstadt, Germany
2Universidad de Castilla-La Mancha, Institute of Mathematics applied to Science & Engineering,
Avenida Camillo Jos´e Cela 3, 13071 Ciudad Real, Spain
(Dated: October 21, 2016)
Afterthedemonstrationofthefeasibilityofhypernuclearspectroscopywithheavy-ionbeams,the
HypHICollaborationwillnextfocusonthestudyofproton-andneutron-richhypernuclei. Theuse
of a fragment separator for the production and separation of rare isotope beams is a crucial aspect
to producing hypernuclei far from the stability line. Precise spectroscopy of exotic hypernuclei is
planned to be carried out at the GSI and later at the FAIR facility with the FRS and Super-FRS
fragment separators. A systematic study and an optimization analysis were performed in order
6
to determine optimal experimental conditions for producing hypernuclei with high isospin. The
1
optimal conditions are obtained based on theoretical models for the heavy-ion induced reaction
0
andhypernucleiproduction. Experimentalefficienciesfortheproductionofexoticsecondarybeams
2
werealsotakenintoaccountviaMonteCarlosimulationsofthefragmentseparator. Thedeveloped
ct methodology is presented to deduce the expected yields of 8ΛBe and subsequently other proton-rich
O and neutron-rich hypernuclei.

0 PACSnumbers: 21.80.+a,25.60.-t,25.70.Mn
2

] I. INTRODUCTION rigidityof20Tmtoperformpowerfulin-flightseparation
x
of exotic nuclei [12].
e
- The interest in strangeness production in nuclear and ThefirstexperimentsoftheHypHICollaborationtook
l
c hadron collisions has been continuously growing during place in the GSI facility in 2009 and 2010. They suc-
u thelastdecades. Inadditiontothestandardnuclearmat- ceeded in demonstrating the feasibility of performing a
n tercomposedbyordinarynucleons, formedbytripletsof precise spectroscopy of hypernuclei produced in heavy-
[
the two lightest down and up quarks, the strange (s) ion induced reactions [13–15]. This result was made

3 quark needs to be considered as well in order to under- possible by a novel experimental method, differing from
v standthepropertiesofdensematter[1,2]. Thehypernu- the typical missing mass experiments of mesons or elec-
0 cleus, a bound system of nucleons and hyperons (baryon tron beam induced reactions involved at Japans Na-
3
including at least one s-quark), has demonstrated to be tional Laboratory for High Energy Physics (KEK), the
9
a fundamental tool to study the hyperon-nucleon and Japan Proton Accelerator Research Complex (JPARC),
3
hyperon-hyperon interactions [3]. At the GSI facility [4] INFNs Double Annular φ Factory for Nice Experiments
0
. a research activity on the study of hypernuclei has been (DAΦNE), the Thomas Jefferson National Accelerator
1
carried out since 2006 by the HypHI collaboration [5]. Facility (JLab), and or the Mainz Microtron (MAMI-C)
0
The next experiments of the HypHI collaboration will accelerator [16–18]. The first experiment, the Phase 0
6
1 then proceed at the future GSI Facility for Antiproton experiment, was performed by bombarding a stable 12C
: and Ion Research (FAIR). The GSI accelerator facility target material with a 6Li beam at 2 AGeV. The main
v
provides a large variety of ion beams: stable beams from goal of the experiment was to produce, reconstruct, and
i
X proton to uranium, thanks to the 18-Tm heavy-ion syn- identify decay vertexes of Λ particle and 3H, 4H, and
Λ Λ
r chrotron (SIS18) [6, 7], and exotic beams via the frag- 5ΛHe hypernuclei [5]. The final results of the data analy-
a ment separator (FRS) [8, 9]. Ion beams with a kinetic sis show that the experimental method is viable for the
energyupto2AGeVforA/Z =2nucleicanbeprovided study of hypernuclei [13]. A second experiment with a
bythecurrentSIS18synchrotron. ThefutureFAIRfacil- 20Ne beam was then performed with similar conditions,
ity [10] is a substantial expansion of the current GSI ac- and its data analysis is ongoing.
celerator, where additional synchrotron rings of 100 Tm Several hypernuclear bound states were identified in
(SIS100)and300Tm(SIS300)areplannedtobeaddedto the same data sample due to the open geometry of the
theSIS18ofGSI.Newexperimentalapparatusesisunder experimental setup, which is a common characteristic of
construction for different research programs on nuclear inclusiveexperiments. Inaddition,anindicationofapos-
and hadron physics [11]. One of them is the NUSTAR sible new bound state has been found: the association of
program which foresees the construction of a supercon- twoneutronsandaΛ0 hyperon,forminganeutralhyper-
ductingfragmentseparator(Super-FRS)withamagnetic nucleus[14]. Ontheotherhand,exclusivemeasurements
could provide more precise information on the hypernu-
clear structure. Consequently, the future HypHI experi-
∗ [email protected] mentsareplannedtobeperformedasexclusivemeasure-
2
ments in the fragment separators, FRS or Super-FRS.
Main separator
pre-separator
Forinstance,recenttheoreticalcalculationsdisprovedthe
existence of the 3n bound state [19–23], which could be
Λ FMF2
either confirmed or denied by a precise exclusive mass
measurement. Thefuturephases,namely,Phases1and2
oftheHypHIproject,focusonthestudyofexotichyper- secondary target:
primary target: hypernuclear production
nuclei toward the proton and neutron drip lines. It will
exotic beam production
necessarily involve the use of rare-isotope beams, aim-
ing to extend the hypernuclear chart to the proton drip primary beam 50 m
line up to 22Si hypernuclei and the neutron drip line up
to 14Li hypΛernuclei. A large charge symmetry breaking FIG.1. (Coloronline)Super-FRSfragmentseparatorlayout.
Λ The primary beam from the synchrotron, represented by the
effect may be expected in proton- and neutron-rich hy-
yellow arrow on the beam line, bombards the primary target
pernuclei. A difference between Λ-proton and Λ-neutron
to produce the exotic beam of interest. The pre-separator of
interactionsmayinduceashiftofthedrip-linepositions.
Super-FRS,highlightedinred,isusedtoselectandtoproduce
For this purpose a new experimental apparatus is under ahighqualityexoticbeam. Themainseparator,filledinblue,
development in order to exploit the rare isotope beam acts as the high-resolution forward spectrometer. A second
provided by the FRS in the actual GSI facility or by the target is installed for the production of hypernuclei at the
Super-FRS of the future FAIR facility. The Super-FRS focal plane of the main separator FMF2.
fragment separator is then crucial to the future phases
of the HypHI project at FAIR for studying proton- and
neutron-rich hypernuclei. forthedecayfragmentoriginatedfromthemesonic-weak
Theproductionofexotichypernucleicanbeinfluenced decayofthehypernucleusofinterest. Thefragmentsepa-
by the high isospin of the beam projectile as it will be ratorwillbesettomeasureaspecificmomentumrangeor
shown in this article. The study reported in this article magnetic rigidity of this decay fragment. This approach
aimed at determining which experimental conditions are will allow a more precise invariant mass measurement of
necessary for the production of proton-rich or neutron- the reconstructed hypernucleus thanks to a momentum
richhypernuclei. Thefeasibilitystudydemonstratingthe resolvingpowerp/∆p=1500oftheFRSandSuper-FRS
possibilityofoperatingtheSuper-FRSatenergiesaround spectrometers[12,24]. Therefore,thefragmentseparator
2 AGeV is presented. The primary beam and target iso- settingandtheexperimentalconditionshavetobedeter-
topes have to be chosen to obtain the exotic beam of mined beforehand and a method of data-processing was
interest at 2 AGeV. The selected exotic beam then im- developed to determine the optimal experimental condi-
pingesonthesecondaryproductiontargettoproducethe tionsfortheproductionofaspecificexotichypernucleus.
exotic hypernucleus of interest. A first systematic study was performed based on the
First, the different models and simulations used for phenomenological empirical parametrization of fragmen-
this purpose will be presented. The description of the tation cross sections (EPAX) model [25–27]. EPAX
methoddevelopedforcombiningalltheinformationinto calculations offer an energy-independent description of
a multivariate data set follows. This allows us to extract the fragmentation cross section at relativistic energy in
the optimal experimental conditions for any possible hy- heavy-ion reactions by means of a universal analytical
pernucleus of interest. formula. This phenomenological formula arises from the
experimental data sets of the fragmentation reactions of
medium-andheavy-ionprojectiles. Itresultsinareason-
II. SIMULATION PROCESSES able estimation of the production cross section of exotic
or stable nuclei for a given collision system.
IntheforthcomingphasesoftheHypHIproject,within Numerous beam-target combinations were calculated
theFRSandSuper-FRSfragmentseparators, theexper- as an initial data set, establishing the normalized yields
imental apparatus will focus on exclusive measurements of exotic beams per centimeter of production target
ofspecifichypernuclei. Theproductionofthedesiredhy- length. The most interesting beam-target combinations
pernucleusdependsespeciallyonwhichexoticsecondary were then preselected. All possible exotic isotopes from
beam has to be produced and impinges on a secondary hydrogentoscandiumhavebeencalculatedfromallpos-
target. Figure 1 shows the layout of the Super-FRS sep- sible combinations of stable isotopes up to 40Ca. Only a
arator: the primary beam, provided by the synchrotron, subset of isotopes are usable within the FRS and Super-
bombardstheprimarytarget,thusproducingexoticfrag- FRS at 2 AGeV because of their maximum acceptable
ments. The exotic beam of interest is then selected and magnetic rigidities: 18 and 20 Tm respectively.
purified by the pre-separator, indicated in red in Fig. 1. The exotic isotopes yield estimations are integrated
The exotic beam is delivered to the focal plane, FMF2, intotheMOCADIcodefortheMonteCarlosimulationof
of the main separator where a second target for produc- the ion-optic transmission [28, 29]. The availability of a
tion of hypernuclei is located. The second half of the givenexoticsecondarybeamatrare-isotopeseparatorfa-
main separator is used as a high-resolution spectrometer cilitiessuchasthecurrentFRSorfutureSuper-FRScan
3
rigidity are transported from the production target loca-
tionuptotheFMF2experimentalareaoftheSuper-FRS,
shown in Fig. 1. In the FMF2 experimental area, a sec-
ondaryproductiontargetwillbeplacedforthehypernu-
eV2000 clei production. The systematic study then includes the
M secondary beam yield for each set of beam and target
10 species at several target thicknesses. Within the MO-
s / CADI simulations, an optimization procedure was im-
nt1000 plemented to find and set the optimal parameters of the
u
o ion-opticalelementsoftheSuper-FRSseparatorwiththe
C
aim of obtaining the highest intensity of the secondary
beam of interest at FMF2 of the Super-FRS. The Monte
Carlo study was performed to achieve 1% of systematic
0
1.2 1.4 1.6 1.8 2 uncertainties. Figure 2 shows the results obtained from
Exotic Beam Energy (GeV) the MOCADI simulations. In Fig. 2(a) and 2(b) the
distribution of the kinetic energy of the fragment beam
1 and the transmission between the production target and
theFMF2areaasafunctionofthekineticenergyarere-
0.8 ported,respectively,forallsimulatedfragments. Abeam
n energy as close as possible to 2 AGeV is necessary for
o
si 0.6 maximizing the hyperon production while keeping a rea-
s sonable transmission to the FMF2 experimental area.
mi
s
n 0.4
a
r
T
0.2
s107
01.2Exo1t.i4c Be1a.6m En1e.8rgy (2GeV)2.2 nsity /106 111224CCN+++199BB2Cee
FIG. 2. Beam kinetic energy distribution and transmission m inte105 111476NOO+++1192B2CCe
distribution of the fragments propagated in the MOCADI ea104 18O+9Be
sCimanudlaletiopnlost. o(fat)hPerboejeacmtedenberegaymdeisnterribgyutidoinstraisbuatiofunn.cti(obn) 0C b103 3306SSi++91B2Ce
of the transmission. For each transmission bin, the box rep- 1 41K+12C
resents the underlying distributions: the bold line represents 102
thequantileat50%,theleftandrightsidesoftheboxarethe 5 10 15 20 25 30 35
75% and 25% quantile. The maximum and the minimum of Target thickness (g/cm2)
the distributions are represented by the vertical segments of
the dotted line. The open circle is the position of the mean
FIG. 3. (Color online) Intensity of the secondary beam 10C
value of the distributions.
at the FMF2 experimental area of the Super-FRS as a func-
tion of the production target thickness, the primary beam,
and the target isotope. Only ten entries of the results of the
bethendetermined. OnlytheuseoftheSuper-FRSsep- MOCADIsimulationsaredisplayedoutofthewholedataset.
arator was accounted for in the experimental efficiency The intensity of the primary beam was set to 5×109 ions/s,
estimations in the following optimization process. The which will be available at the Super-FRS. The legend is or-
MOCADIsimulationsconsistoftracingtheionsthrough dered from highest to lowest intensity.
ion-optical elements in which high-order aberrations of
the magneticfield are taken intoaccount. The whole ex- Additionally, Fig. 3 shows a summary of the 10C
perimental equipment of a fragment separator system is secondary beam production as a function of the tar-
simulated within the MOCADI code. In addition, the get thickness and of the primary-beam and production-
nuclear interactions with the material of the detectors target combination. Simulations show that the reaction
are simulated in order to allow direct comparison with between(C,N,O)beamisotopesand(Be,C)targetiso-
high-resolution experimental measurements. topes gives the highest 10C beam intensity up to several
Thetransmissionandyieldofeachpossibleexoticsec- million ions/s for a primary beam intensity of 5×109
ondary beam with the Super-FRS apparatus are esti- ions/s. Figure 3 shows the possible optimal case when
mated with this framework. The secondary beam of in- the target thickness is solely considered for the optimal
terest and other exotic isotopes with a similar magnetic search. However, other parameters such as the contam-
4
ination of other produced exotic isotopes, the beam en- For instance, the hypernuclei production cross section is
ergy, or the secondary reaction for the hypernuclei pro- then reduced by 73% or 46% if the energy of the exotic
duction have been considered. beamisdecreasedto1.9AGeVor1.8AGeVrespectively.
The production of the exotic hypernuclei for each sec- A multivariate data set was then created in order to
ondary beam of interest can then be studied thanks to find the optimal set of experimental parameters to max-
the theoretical model of Ref. [30]. It is an hybridization imize the production of a hypernucleus of interest. It
between the transport model DCM-QGSM (Dubna cas- gathers the results of those theoretical calculations and
cade model–quark-gluon string model), which simulates Monte Carlo simulations. This parameter set is defined
thecollisionbetweenthebeamandthetarget,andasta- by the primary beam, the production target, its thick-
tisticalapproachoftheFermibreakupmodeltodescribe ness, and the exotic secondary beam that are optimal to
the de-excitation of spectators. The hypernuclei produc- producethehypernucleusofinterest. Agenericapproach
tion was investigated for each nucleus-nucleus collision was developed, so that the optimal experimental setup
of exotic beam and target species at 2 AGeV. For each can be determined for each hypernucleus of interest.
theoretical calculation a suitable number of events were
simulated in order to keep the systematic uncertainty on
the hypernuclei production cross sections to the level of III. MULTIVARIATE ANALYSIS
0.1µb. Moreover,thetheoreticalcalculationfor6Li+12C
collisions at 2 AGeV was performed and compared with It is merely impossible to simply plot the multivariate
the published experimental results [15]. The theoretical data set and estimate the best case by compiling all the
estimations of the hypernuclei production cross section possible combinations and permutations of the isotopes
were compatible with the experiment, validating the cal- species. An optimization procedure was used to find the
culations of the other colliding systems. Theoretical cal- optimalcase. Thisarticlefocusesfirstontheproduction
culationsforthehypernucleiproductionwerecarriedout of the proton-rich hypernucleus 8Be, yet other hyper-
Λ
accordingtotheexoticbeam,fromLitoNeisotopes,col- nuclei are considered afterwards since the optimization
liding on a 12C or 9Be target at 2AGeV. The obtained procedure does not depend on the hypernuclear species.
results were gathered into a data set, which can be or- Duringthemultivariateanalysis,theproductionofthe
dered by colliding isotopes or by produced hypernuclei. hypernucleus of interest is taken into account: all pos-
sible secondary beams on Beryllium or Carbon targets
Figure4showstheΛ-hypernucleiproductioncrosssec-
were considered in the theoretical calculations of the hy-
tion in µb as a function of the neutron and proton num-
bers of the core for a proton-rich 9C and a neutron-rich pernuclei production. The secondary beam selected by
15C secondary beam on a 12C target. A clear difference theprocedureisusedtofindtheoptimalconditionsofthe
is observed with respect to the case of the 12C+12C col- Super-FRS,whichallowustocalculatethehypernuclear
yieldpersecondfora4-centimetersecondaryproduction
lision,wheretheexoticcarbonbeamsenhanceexotichy-
target. The secondary target thickness was selected to
pernucleiproductionfrom1.2to3times. Concerningthe
matchtheexperimentalconditionofthepreviousHypHI
productionofneutron-richhypernuclei,higherincreaseof
theproductioncrosssectionisobservedinthe12Bexotic experiments. Thishypernuclearyieldestimationincludes
beam compared to the 12C beam as shown in Fig. 4. the experimental efficiency of the ion-optic transmission
and the exotic beam intensity obtained from a primary
Choosing the proton-rich exotic beam clearly favors the
beam of 5 × 109 ions/s. Additionally, the simple case
production of proton-rich hypernuclei, and reciprocally
of using a stable beam for the hypernuclei production is
a neutron-rich beam to produce a neutron-rich hypernu-
also included in the data set. An intensity of 107 ions/s
clei.
was selected in those cases to estimate the hypernuclear
When the results are ordered by the produced hyper-
yield per second.
nucleus, Fig. 5 shows the production cross section of
In an optimization problem, a cost function has to be
proton-richhypernucleus8Beandneutron-richhypernu-
Λ definedbetweenthedifferentvariablesandparametersin
clei11Beat2AGeVdependingontheneutronandproton
Λ order to find the optimal set which maximizes or mini-
numbersoftheexoticbeamreactingona12Ctarget. The
mizes this cost function. The cost function was defined
secondary beam isotope that maximizes the production
as follow for our multivariate data set:
of each exotic hypernucleus can be then identified. How-
ever, this maximum is not necessarily the optimal since
F (C,E,T,I)=αC +βE −γT +δI, (1)
α,β
the production of the exotic beam has to be also con-
sidered. Moreover, the hypernuclei production cross sec- inwhichthevariablesC,E,T,andI refertothehyper-
tions need to be adjusted since the kinetic energy of the nuclear yield, secondary beam energy, production target
exoticbeammayvaryfrom2AGeV.Theparametrization thickness, and intensity of the secondary beam of inter-
[31]usedtofittheworlddatasetoftotalproductioncross est,respectively. Theparametersα,β,γ andδdefinedin
sectionsofpp→pK+Λ[32]canbeemployedtoscalethe the cost function F are the weight coefficients that con-
theoretical calculations. The parametrization from [32] nect the different variables. The weight coefficient δ for
was used. The value at 2 AGeV was used as a normal- theintensityparameterwasfixedto1/2forachievingthe
izationfactortoobtainthescalingfunctioninthisstudy. numericalstabilityoftheconvergenceofthecostfunction
5
9 12
6 7
8
5 0.4 1.8 1.6 7 6 1.3 0.6 10
6 5 1 1.9 4.1 4.7 2.9 8
4 1.8 5.1 5.6 1.4
5 4 0.6 2.1 5.9 9.3 9.1 5.5 2.1
Z 3 2.8 8.2 6.6 1.8 Z 6
4 3 1.8 6 8.6 11.210.4 5.4 1.2
2 4.6 7.5 5.8 1.4 3 2 3.4 8.7 10.510.8 6.3 3.1 0.7 4
1 5.3 3.6 0.9 2 1 8.9 9.6 6.2 3.1 1.2 2
1
0 1.1 0 3.7 1.7 0.8
0 0
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
N N
12
7 6 10
6 0.4 0.6 0.7 10 5 0.5 0.8 1.2 1.3
8
5 0.8 1.6 4.5 4.4 2.1 8
4 0.6 2.9 4.1 6.3 5.2 2.9
4 0.5 2.6 4.9 10.1 9.9 5.4 1.4 6
Z 6 Z 3 0.6 2.4 6 8.6 9.6 8.6 5.4 0.6
3 1.2 4.5 8.8 10.6 9.5 5.9 1.6
2 2.4 8.5 11.1 7.8 7.1 3.7 0.6 4 2 1.6 6.2 9 10.6 9.8 7 3.9 4
1 9.9 8.6 6 3.3 0.9 2 1 7.7 8.8 8.8 4.9 2.9 1.2 0.4 2
0 4.8 2.1 0.6 0.4 0 5.8 2.9 1.4 1
0 0
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
N N
FIG. 4. (Color online) Production cross section in µb of Λ-hypernuclei from the collision of (a) 9C+12C at 2 AGeV, (b)
12C+12C,(c)15C+12C,and(d)12B+12C.TheneutronandprotonnumbersoftheΛ-hypernucleuscorearerepresentedbythe
horizontal and vertical axis respectively. The systematic uncertainty of the production cross sections is estimated to the level
of 0.1 µb.
2
11 11
10 1.8
10 8.89.79.77.58.58.2 7 7.77.6 10 1.4 1.3 1.3 1.11.6 1.3 1.4 1.5 1.7
1.6
9 7.58.86.88.58.26.5 8 9 1.3 0.8 0.7 1.4 1.6 1.5 1.4
8 8.98.77.68.2 8.1 9 8.8 8 1.1 1 0.8 1.1 1.7 1.2 1.7 1.2
6
Z 7 118.58.89.36.97.28.4 Z 7 1.3 1.4 1.2 1.9 1.11.7 1
6 1.46.28.59.38.68.110.17.78.2 4 6 2.1 2.2 1.3 1.4 1.4 1.5 0.8
0.6
5 2.1 4 4.34.1 5 2.9
0.4
4 0.70.71.2 1 2 4 0.1 0.9
0.2
3 3
0 0
2 3 4 5 6 7 8 9 10111213141516 4 5 6 7 8 9 10 11 12 13 14 15 16
N N
FIG. 5. (Color online) Production cross section (in µb) of (a) 8Be and (b) 11Be hypernuclei, according to the collision of
Λ Λ
differentexoticsecondarybeamsZ+NZona12Ctargetat2AGeV.Thesystematicuncertaintyoftheproductioncrosssections
is estimated to the level of 0.1 µb.
to its optimum. Besides, the sum of the squared weights is set to 1, α2 +β2 +γ2 +δ2 = 1, resulting in the cor-
6
(cid:112)
responding coefficient γ being equal to 3/4−α2−β2 and 7(b), while the target thickness in centimeters is de-
and in a constraint for α and β being within a circle of picted in Fig. 7(c). The secondary beam that should be
(cid:112)
radius 3/4. In the cost function, the different param- selectedfortheoptimalproductionof8Beanditsresult-
Λ
eter distributions were normalized to be within a [0, 1] ingkineticenergyforeach(α,β)areshowninFigs. 7(d)
interval in order to keep the weight coefficients within and 7(e) respectively. The yield per second of 8Be, pro-
Λ
the unit interval. The parameters α and β are set arbi- ducedbythecollisionoftheoptimalsecondarybeamand
trarily depending on the weight to be associated to each a 4-centimeter 12C target, is finally shown in Fig. 7(f)
variableofthedataset. Thiscostfunctionisbuiltinten- as a function of α and β. The evolution of the weight
(cid:112) (cid:112)
tionally to maximize the production of the hypernucleus parameters α and β within [− 3/4, 3/4] represents
of interest, such as 8Be, by obtaining an optimal energy theaimofminimizingormaximizingtheinfluenceofthe
Λ
and intensity of the secondary beam while minimizing variables Cs and E in the cost function. For instance,
the thickness of the production target. when α is negative the overall goal is to minimize the
hypernuclear yield or when β is negative the intent is
to minimize the kinetic energy of the optimal secondary
1 1.6 beam. It is useful to calculate the optimal conditions
between the limits of the weights since another decision
1.4
criterion can be considered instead of the maximax ap-
0.5 1.2 proach of Eq. 2.
Inthecaseof8Be,showninFig. 7,thesystematicpro-
1 Λ
ceduregivesthefollowingsetofexperimentalparameters:
β0 0.8 withaprimarybeamof14Nimpingedona5.5-centimeter
0.6 9Betarget,anexoticbeamof12Nshouldbeselectedand
transported to bombard a 12C target. The intensity of
−0.5 0.4 the12Nsecondarybeamisabout5.1×106ionspersecond
0.2 withaprimarybeamof5×109 ionspersecond, whichis
within the expected intensity that the new FAIR facility
−1 0
−1 −0.5 0 0.5 1 will provide at the entrance of the Super-FRS. Subse-
α quently, under those conditions the 8Be yield is about
Λ
4.0 hypernuclei produced per second for a 4-centimeter
secondary target.
FIG. 6. (Color online) The evolution of the cost function
After reviewing the details for the 8Be case, one can
maximumasafunctionoftheαandβ weightcoefficientsfor Λ
the multivariate data set optimization of the studied case of proceed similarly for other hypernuclei. Table I gathers
8Be. The red dot represents the position of the maximum of the results of the optimization for Λ-hypernuclei up to
Λ
the maxima within the α-β space. Carbon hypernuclei. The reaction necessary for the op-
timal exotic beam production is reported with its target
thickness. The selected exotic beam is then mentioned
A search for the maximum of the cost function is per-
withitsoptimalkineticenergyandintensity. Finallythe
formed with fixed weight coefficients α and β to deter-
yield per second of the hypernucleus of interest is given
mine the optimal parameter set:
for a production on a 4-centimeter 12C target. Several
argmax{F (C,E,T,I)}. cases in which a stable beam provides a higher hypernu-
α,β
C,E,T,I clearyieldcanbenotedinTableI.Thoseoptimalexperi-
mentalconditionswillbeusefulforconceivingthefuture
The evolution of the obtained optimal variables accord- hypernuclearexperimentswithintheSuper-FRSofFAIR
ing to those weight coefficients can then be investigated. facility.
First, in Fig. 6 the distribution of the maximum of the
cost functionas afunction ofthe weights α and β is pre-
sented. Additionally Fig. 7 shows the different results of
IV. CONCLUSION
the optimization as a function of α and β. Each value
of this distribution is a possible optimal condition set.
A general procedure was developed in order to deter-
To determine the best optimal condition that should be
minetheoptimalexperimentalconditionsfortheproduc-
considered, a maximax criterion was exploited. This cri-
tionofexotichypernucleiwithintheSuper-FRSfragment
terion is defined as follows:
separatorofthenewFAIRfacility,inwhichfuturehyper-
nuclear spectroscopy experiments will take place. This
argmax max {F (C,E,T,I)}. (2)
α,β
α,β C,E,T,I optimization process includes the results from several
theoretical models. The production of hypernuclei and
The evolutions of the variable set according to α and of exotic beams from the fragmentation of the primary
β are presented Fig. 7. The optimal isotope species of beamontheproductiontargetwereestimated. Thepro-
the primary beam and target are shown in Figs. 7(a) cedurealsoincludesMonteCarlosimulationsofthebeam
7
1
1
20Ne 1 20
12C 18
0.5
27Al 19F 14N 0.5 11B 9Be 0.5 16
23Na 14
β0 13C β0 12C 27Al 12
15 30Si β0 10
N
21Ne 8
−0.5 17O −0.5 −0.5 46
16O 10B 2
−1−1 −0.5 0 0.5 1 −1−1 −0.5 0 0.5 1 −1−1 −0.5 0 0.5 1 0
α α α
1
1 2 1 4
17F
10C 1.95
0.5 3.5
12B 12N 0.5 1.9 0.5 3
14O 1.85
β0 2.5
β0 1.8 β0
18Ne 2
1.75
−0.5 1.5
−0.5 1.7 −0.5
1.65 1
−1−1 −0.5 0 0.5 1 −1−1 −0.5 0 0.5 1 1.6 −1−1 −0.5 0 0.5 1 0.5
α α α
FIG.7. (Coloronline)Resultsofthemultivariatedatasetoptimizationforthestudiedcaseof8Beasafunctionofthetarget
Λ
thickness, beam energy, beam intensity, hypernuclear yield, exotic beam, primary beam, and target species. Figures (a) to (f)
represent the evolution of the variables, the primary beam and target species, the thickness in centimeters, the selected exotic
beamspeciesanditskineticenergyinAGeV,andthe8Beyieldpersecond,respectively. Eachreddotrepresentstheposition
Λ
of the overall maximum within the α-β space.
transportation to estimate the experimental efficiency of consisting of the detector setups and data acquisition,
theseparator. Thoseefficienciescorrespondtothetrans- willprovidetheestimatedhypernuclearcountrateinthe
port from the secondary beam production site to the recorded data. In addition, the cases of all hypernuclei
experimental area where both the hypernuclei produc- up to carbon hypernuclei were optimized and reported.
tion and spectroscopy will take place. The optimization Thisinformationwillbevaluableforfurtherexperiments
procedure combined those different results and tried to onproton-richorneutron-richhypernuclei. Thoseresults
provide the best conditions for any considered hypernu- were achieved by a particular quasi-convex combination
cleus. Thebestexperimentalrequirementswereobtained of the variables that were optimized. There are other
by the maximization of the cost function. The optimal possibilities to be explored, especially depending on the
conditions of a 8Be hypernucleus were determined. The weight definition in the cost functions. Furthermore the
Λ
use of a secondary beam of 12N from the fragmentation maximaxcriterionwasusedanddifferentcriteriacanalso
of a 14N primary beam on a 9Be target was found to be beappliedtotunetheweights. Additionalconsiderations
optimal. Aroundfour8Bewouldbeproducedpersecond could provide new perspectives on the data set.
Λ
ona4-centimeter12Ctarget,withanestimated8Bepro-
Λ
duction cross section of 11 µb. Under those conditions,
andconsideringtheefficiencyofthepreviousexperiment
[13], about 345×103 hypernuclei per day are expected.
Afterwards, the design of the experimental apparatus,
8
TABLE I. Summary of the results from the optimization procedure. All Λ-hypernuclei up to carbon isotopes were considered,
and for each one, the optimal experimental conditions are reported: the reaction necessary to produce the exotic beam, the
targetthickness,theexoticbeamselectedtoproducethehypernucleiofinterestona4-centimeter12Ctarget,theexoticbeam
kinetic energy and the intensity, and the resulting hypernuclear yield.
Reaction Target 2nd beam E I Yield
k Reaction Target 2nd beam E I Yield
(cm) (AGeV) (106/s) (/s) k
(cm) (AGeV) (106/s) (/s)
8C 14N+9Be 5.5 12N 1.94 5.1 0.2
Λ 8Li 20Ne+9Be 2 17F 1.97 5.7 3.7
9C 14N+9Be 5.5 12N 1.94 5.1 0.8 Λ
Λ 9Li 16O+9Be 5.5 14O 1.93 5.5 2.2
10C 14N+9Be 5.5 12N 1.94 5.1 1.5 Λ
Λ 10Li 23Na+11B 15.5 12B 1.79 11.5 1.1
11C 14N+9Be 5.5 12N 1.94 5.1 0.9 Λ
Λ 11Li 23Na+11B 15.5 12B 1.79 11.5 0.12
7B 14N+9Be 5.5 12N 1.94 5.1 0.7 Λ
Λ 3He 14N+9Be 5.5 12N 1.94 5.1 1.8
8B 14N+9Be 5.5 12N 1.94 5.1 2.7 Λ
Λ 4He stable beam 14N 2. 10. 4.1
9B 14N+9Be 5.5 12N 1.94 5.1 3.5 Λ
Λ 5He stable beam 20Ne 2.0 10. 5.2
10B 14N+9Be 5.5 12N 1.94 5.1 2.5 Λ
Λ 6He stable beam 12C 2. 10. 4.8
11B 20Ne+9Be 2 17F 1.97 5.7 1.2 Λ
Λ 7He 20Ne+9Be 2 17F 1.97 5.7 2.9
5Be 14N+9Be 5.5 12N 1.94 5.1 0.6 Λ
Λ 8He 20Ne+9Be 2 17F 1.97 5.7 1.4
6Be 14N+9Be 5.5 12N 1.94 5.1 1.9 Λ
Λ 9He 23Na+11B 15.5 12B 1.79 11.5 0.8
7Be 14N+9Be 5.5 12N 1.94 5.1 3.9 Λ
Λ 3H stable beam 16O 2. 10. 5.1
8Be 14N+9Be 5.5 12N 1.94 5.1 4.0 Λ
Λ 4H stable beam 20Ne 2. 10 4.5
9Be stable beam 16O 2. 10. 4.4 Λ
Λ 5H stable beam 14N 2. 10. 3.1
10Be stable beam 14N 2. 10 3.1 Λ
Λ 6H 20Ne+9Be 2 17F 1.97 5.7 1.5
11Be 23Na+11B 15.5 12B 1.79 1.2 0.6 Λ
Λ 7H 20Ne+9Be 2 17F 1.97 5.7 0.5
4Li 20Ne+9Be 2 17F 1.97 5.7 1.1 Λ
Λ 8H 23Ne+9Be 15.5 12B 1.79 11.5 0.3
5Li 12C+9Be 6 10C 1.94 5.1 2.5 Λ
Λ 3n 20Ne+9Be 2 17F 1.97 5.7 2.1
6Li 14N+9Be 5.5 12N 1.94 5.1 4.3 Λ
Λ 4n 20Ne+9Be 2 17F 1.97 5.7 1.0
7Li stable beam 14N 2. 10. 5.2 Λ
Λ
V. ACKNOWLEDGMENTS Castilla-La Mancha “Ayudas para estancias de investi-
gadores invitados en la UCLM para el an˜o 2015” and
FEDER 2014-2020. The second author has been spon-
This work has been supported by the HypHI project sored by Ministerio de Economa y Competitividad and
funded by the Helmholtz association as Helmholtz- fondos FEDER MTM2013-47879-C2-1-P. A part of this
University Young Investigators Group VH-NG-239 at workwascarriedoutontheHIMSTERhighperformance
GSI, and the German Research Foundation (DFG) computing infrastructure provided by the Helmholtz-
under contract number SA 1696/1-1 and EU FP7 Institute Mainz. We would like to thank to A. Botv-
HadronPhysics-2SPHERE.Thisworkhasalsobeensup- ina, H. Geissel, T. Saito and C. Scheidenberger for the
ported by the co-funded program of the University of involved discussions.
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