DOI 10.1140/epja/i2019-12672-y
Regular Article – Theoretical Physics
P
HYSICAL
J
OURNAL
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Production of Λ-hypernuclei and evaluation of their binding
energies via the double yield ratio
N. Buyukcizmeci1,a, A.S. Botvina2,3, R. Ogul1, A. Ergun1, and M. Bleicher2,4,5
1
Department of Physics, Selcuk University, 42079 Kampus, Konya, Turkey
2
FIAS and ITP J.W. Goethe University, D-60438 Frankfurt am Main, Germany
3
Institute for Nuclear Research, Russian Academy of Sciences, 117312 Moscow, Russia
4
GSI Helmholtz Center for Heavy Ion Research, 64291 Darmstadt, Germany
5
John-von-Neumann Institute for Computing (NIC), FZ J¨ulich, J¨ulich, Germany Received: 27 June 2018 / Revised: 18 November 2018
Published online: 17 January 2019 c
Societ`a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2019
Communicated by D. Blaschke
Abstract. Relativistic collisions of ions, hadrons and leptons with nuclei can produce various hypernuclei by the capture of hyperons in nuclear residues. In many cases the disintegration of such hypernuclear systems can be described with statistical approaches suggesting that the fragment production is related to the binding energies of hypernuclei. We demonstrate how the hyperon binding energies can be effectively evaluated from the yields of different hyper-isotopes using the double ratio method. Its universality and the possibility to involve many different isotopes are the advantages of this method. The same procedure can also be applied for multi-strange nuclei, for which binding energies were very difficult to measure in previous hypernuclear experiments. Modifications caused by secondary de-excitation processes in hot hypernuclei are considered additionally.
1 Introduction
In relativistic nuclear particle and ion collisions various hyperons (Λ, Σ, Ξ, Ω) and resonances are copiously pro-duced. In peripheral reactions these hyperons can be cap-tured by the projectile and target residues and hypernu-clei are formed. It is important that, compared to the time scale of a heavy ion reaction, hypernuclei can be consid-ered stable. Embedding hyperons in the nuclear matter allows to explore the many-body aspects of the strong three-flavor interaction (i.e., including u, d, and s quarks) at low energies. Hypernuclei open also opportunities to study the hyperon-nucleon and the hyperon-hyperon in-teractions. The production of hypernuclei provides com-plementary information for traditional nuclear studies and extends horizons for studying particle physics and nuclear astrophysics (see, e.g., [1–6] and references therein).
Hypernuclear physics was traditionally focused on spectroscopic information and was restricted by a quite limited set of lepton- and hadron-induced reactions [1, 2], where the hyperons are captured by nuclei in their ground states and low excited states. However, there are limita-tions on such methods, since the targets should be mainly stable (not radioactive). Many hypernuclear isotopes are
a e-mail: nihal@selcuk.edu.tr
not reachable experimentally in this case. Another possi-bility to form hypernuclei exists in the deep-inelastic re-actions leading to fragmentation processes, as they were discovered long ago [7]. One can form hypernuclei of all sizes and isospin content when the produced hyperons are captured by nucleons and nuclear fragments produced in the same reactions. Relativistic ion collisions provide promising opportunities in this respect. There are exper-imental collaborations, such as STAR at RHIC [8], AL-ICE at LHC [9], PANDA [10], CBM [11], HypHI, Super-FRS, R3B at FAIR [12, 13], BM@N, MPD at NICA [14], which plan to investigate hypernuclei and their properties in reactions induced by relativistic particles. The isospin space, particle unstable states, multi-strange nuclei and their precise lifetime can be explored in these fragmenta-tion reacfragmenta-tions. At this point, we especially emphasize the research possibilities in peripheral ion collisions, since a capture of hyperons by large nuclear projectile and target residues provides a natural way to study large chunks of hyper-matter and their evolution, for example, its disinte-gration into small fragments. It was theoretically demon-strated [5,15–21] that in nuclear collisions one can produce all kinds of hypernuclei including multi-strange and exotic ones. There also exist experimental confirmations of such processes leading to hypernuclei [12, 22, 23].
The chance to obtain various hypernuclei in the same reaction opens new opportunities for their investiga-tion in comparison with previous methods. Large multi-hypernuclear systems incorporating more than two hy-perons can be created in energetic nucleus-nucleus col-lisions [16, 19]. This is a convenient method to go beyond double hypernuclei to obtain new experimental informa-tion on properties of multi-hyperon systems. In this paper we continue the theoretical research of the fragmentation reactions, however, concentrating only on Λ-hypernuclei production, since they are mostly investigated. We sug-gest a new double ratio method, which allows to extract the hyperon binding energies (including in multi-strange nuclei) from the analysis of the relative yields of hypernu-clei.
2 Production of hypernuclei in peripheral
relativistic ion collisions
In peripheral collisions the capture of hyperons and the formation of light hypernuclei were recently observed by the HypHI Collaboration [12] at the projectile rapidities. As discussed in ref. [24] it is consistent with the capture of hyperons by projectile residues. These processes were considered theoretically in refs. [16,19], and it was demon-strated that the production of strangeness correlates with particle production. The capture of hyperons takes usu-ally place on diluted projectile and target residues, since they lose some interacting nucleons during the dynamical reaction stage. Such residues can decay afterwards and produce normal and hyper-fragments. It is very effective to treat this decay within statistical models as was proven previously: There are many confirmations of these reaction mechanisms in normal fragmentation and multifragmen-tation processes, see e.g. refs. [25–31]. The Λ-hyperon in-teractions in a nucleus are similar to normal nuclear ones, and its potential is around two-thirds of the nucleon one. A small admixture of hyperons in nuclear matter does not change the general reaction picture when the avail-able excitation energy is high. Therefore, as well as in the reactions producing normal fragments, we expect that the final cold hypernuclei will be formed as a result of the disintegration of excited hyper-residues.
As was demonstrated in our previous works [17–19], the yields of the hypernuclear residues in peripheral ion collisions saturate at energies above 3–5 A GeV (in the lab-oratory frame). As a result such deep-inelastic processes can form large hyper-residues with a very broad distribu-tion in mass and excitadistribu-tion energy [16, 18, 19]. In partic-ular, as shown in ref. [19], the saturation yields of single, double and triple hypernuclear residues in lead on lead collisions can reach around 100 mb, 1 mb, and 0.01 mb, respectively. These are quite large yields which allow for essential extension of the hypernuclei chart and for hyper-nuclear studies.
The formation and disintegration of excited nu-clear residues in high-energy nucleus-nucleus and hadron-nucleus collisions were intensively studied in connection
with fragmentation and multifragmentation processes. In particular, masses and excitation energies of the residues have been determined in experimental and theoretical works [19, 26]. At high excitation energy the dominating decay mode is a multifragmentation process [25, 28, 29]. According to the present understanding, multifragmen-tation is a relatively fast process, with a characteristic time around 100 fm/c, where, nevertheless, a high de-gree of equilibration (chemical equilibrium) is reached. This is a consequence of the strong interaction between baryons located in the vicinity of each other in the freeze-out volume. The statistical models have shown very good agreement with fragmentation and multifragmentation data [25, 26, 28, 30]. It is natural to extend the statisti-cal approach for hypernuclear systems. In the present ap-proach we assume a homogeneous distribution of the cap-tured hyperons in the nuclear residues, since, because of the strong interaction of these hyperons with nuclear mat-ter, a fast equilibration is expected. We assume also that the disintegration mechanisms with a large energy depo-sition in a big piece of nuclear matter does not change in the presence of few hyperons. The same numerical meth-ods used previously for execution of the models can also be extended.
The statistical multifragmentation model (SMM), which was very successfully applied for description of nor-mal multifragmentation processes in relativistic heavy-ion collisions [26,28,30,31], was generalized for hypernuclei in ref. [15]. In this model the break-up channels are generated according to their statistical weight. The Grand Canoni-cal approximations leads to the following average yields of individual fragments with the mass (baryon) number A, charge Z, and the Λ-hyperon number H:
YA,Z,H = gA,Z,H· Vf A3/2 λ3 T exp −1 T (FA,Z,H− μAZH) , μAZH = Aμ + Zν + Hξ. (1)
Here T is the temperature, FA,Z,H is the internal free energies of these fragments, Vf is the free volume avail-able for the translation motion of the fragments, gA,Z,H is the spin degeneracy factor of species (A, Z, H), λT = (2π2/m
NT )1/2 is the baryon thermal wavelength, mN is the average baryon mass. The chemical potentials μ, ν, and ξ are responsible for the mass (baryon) number, charge, and strangeness conservation in the system, and they can be numerically found from the corresponding conservation laws accounting for the total baryon number A0, the total charge Z0, and the total Λ-hyperon number
H0in the system. In this model the statistical ensemble
in-cludes all break-up channels composed of baryons and ex-cited fragments. The primary fragments are formed in the freeze-out volume V . We use the excluded volume approx-imation V = V0+ Vf, where V0 = A0/ρ0 (ρ0≈ 0.15 fm−3
is the normal nuclear density), and parametrize the free volume Vf = κV0, with κ ≈ 2, as extracted from
experi-ments in refs. [26,28,30]. A transition from the compound hypernucleus to the multifragmentation regime was also under investigation [5, 15].
We consider the physical conditions in the freeze-out volume which are similar to the ones established previ-ously in multifragmentation studies. In many cases nuclear clusters can be described in the liquid-drop approxima-tion: Light fragments with mass number A < 4 are treated as elementary particles with corresponding spins and translation degrees of freedom (“nuclear gas”). Their bind-ing energies were taken from experimental data [1, 2, 25]. The fragments with A = 4 are also treated as gas parti-cles with table masses, however, some excitation energy is allowed Ex= AT2/ε0 (ε0≈ 16 MeV is the inverse
vol-ume level density parameter [25]), that reflects a presence of excited states in 4He, 4ΛH, and 4ΛHe nuclei. Fragments with A > 4 are treated as heated liquid drops. In this way one can study the nuclear liquid-gas coexistence of hyper-matter in the freeze-out volume. The internal free energies of these fragments are parametrized as the sum of the bulk (FB
A), the surface (FAS), the symmetry (F
sym
AZH), the Coulomb (FC
AZ), and the hyper energy (F
hyp AH): FA,Z,H = FAB+ FAS+ F sym AZH+ F C AZ+ F hyp AH. (2) Here, the first three terms are written in the standard liquid-drop form [25]: FAB = −w0− T2 ε0 A, (3) FAS = β0 T2 c − T2 T2 c + T2 5/4 A2/3, (4) FAZHsym = γ(A− H − 2Z) 2 A− H , (5)
where w0 = 16 MeV, β0 = 18 MeV, Tc = 18 MeV and γ = 25 MeV are the model parameters which are ex-tracted from nuclear phenomenology and provide a good description of multifragmentation data [25, 26, 28, 30]. The Coulomb interaction of fragments is described within the Wigner-Seitz approximation, and FC
AZ is taken as in refs. [15, 25], FAZC (V ) = 3 5 1− V0 V 1/3 (eZ)2 r0A1/3 , (6)
where r0 = 1.2 fm and e denotes the electron charge. In
many cases the Wigner-Seitz approach is convenient and sufficient for the description of experiments and the evalu-ation of the Coulomb energy in the system. A more precise estimate of this long-range interaction can be obtained in micro-canonical approaches [32, 33]. In particular, it may lead to the anisotropy in the fragment charge distribution with the tendency to concentrate the charge in fragments on the periphery of the freeze-out volume, as known from multifragmentation studies. This should have a limited effect on the hyper-isotope distribution, since the main contributions come from the liquid-drop, symmetry and hyper-terms.
The free hyper-energy FAHhyp is a new term in this ap-proach. We assume that it is determined only by the bind-ing energy of hyper-fragments. Presently, only few ten
masses of single hypernuclei (mostly light ones) are ex-perimentally established [1, 2], and only few single-event measurements of double hypernuclei exist. Still, there are theoretical estimations of their masses based on this lim-ited amount of available data. In ref. [15] we have sug-gested a liquid-drop hyper-term:
FAHhyp= (H/A)· (−10.68A + 21.27A2/3) MeV. (7) In this formula the binding energy is proportional to the fraction of Λ-hyperons in matter (H/A). The second part is the volume contribution minus the surface one: It is a standard liquid-drop parametrization assuming satura-tion of the nuclear interacsatura-tion. The linear dependence at small H/A is in agreement with theoretical predictions [34] for hyper-matter. As was demonstrated in refs. [5, 15] this parametrization of the hyperon binding energy de-scribes available experimental data quite reasonably. It is important that two boundary physical effects are cor-rectly reproduced: The binding energies of light hypernu-clei (if a hyperon substitutes a neutron) can be lower than in normal nuclei, since the hyperon-nucleon potential is smaller than the nucleon-nucleon one. However, since the hyperon can take the lowest s-state, it can increase the nu-clear binding energies for large nuclei. Also different phe-nomenological hyper-formulae (e.g., [35]) have been sug-gested in the literature, and their sensitivity for fragment production was under investigations [15].
In fig. 1 we show calculations with this model, which are important for understanding our following analysis. The top panel presents the mass distribution of all frag-ments and hyper-fragfrag-ments following the disintegration of a hypernuclear residue containing A0 = 200 baryons
with charge Z0 = 80, and with H0 = 3 captured
hy-perons. Such a system can be produced after the initial dynamical stage of relativistic heavy-ion collisions with a non-negligible probability [19]. The two bottom panels give the isotope distributions for produced Mg and Sn elements, including elements with captured hyper-ons. A realistic temperature of T = 4 MeV was assumed. In this case the hyperons are kept mostly by heavy nu-clei because of their large binding energy and the yield of lightest hypernuclei are essentially low. One can see a very broad mass distribution and isotope distributions for the produced fragments, as we expect to obtain in exper-iments too. Namely, these fragment yields can be used to back-trace the information about unknown properties of nuclei.
3 Evaluation of the hyperon binding energies
via hypernuclei yields
Conceptually our approach depends on parameters of thermalized systems produced after the dynamical stage, but not directly on projectiles initiating this stage. Also, the important advantage of the method introduced below is that it does not depend on the theoretical assumptions on the hypernuclei masses and it depends mainly on the
Fig. 1. Characteristics of all fragments (solid lines) and hyper-fragments (dashed lines, with Λ hyperons) produced after dis-integration of an excited hypernuclear system with initial mass (baryon) number A0 = 200, charge Z0 = 80, temperature
T = 4 MeV, and containing three Λ hyperons (H0 = 3). Top
panel: Yields (per disintegration event) of fragments versus their mass number. Bottom panels: Isotope yields for magne-sium (Mg, Z = 12) and tin (Sn, Z = 50) elements versus the neutron number.
fragment yields obtained in the multifragmentation reac-tions [36]. One can assume both hot and cold fragments in the statistical freeze-out state, and these both possibili-ties depend on the production mechanisms. As established in the analysis of normal multifragmentation experiments, there are reliable methods to identify the processes of the fragment production in selected reaction events. Within the statistical approaches we demonstrate below how the hypernuclei yields can be used for addressing the hyperon binding energies.
3.1 Standard multifragmentation scenario
As the standard case we can use the formulas (1) and (2). It is convenient to rewrite the above statistical expressions in order to show separately the binding energy EAbhof one hyperon at the temperature T inside a hypernucleus with
(A, Z, H):
EAbh= FA,Z,H− FA−1,Z,H−1. (8) Since Λ-hyperons are usually bound, this value is nega-tive. Then the yield of hypernuclei with an additional Λ hyperon can be recursively written by using the former yields:
YA,Z,H = YA−1,Z,H−1· CA,Z,H· exp −1 T EAbh− μ − ξ , (9) where CA,Z,H = (gA,Z,H/gA−1,Z,H−1)· (A3/2/(A− 1)3/2) depends mainly on the ratio of the spin factors of (A, Z, H) and (A− 1, Z, H − 1) nuclei, and very weakly on A. Since in the liquid-drop approximation we assume that the frag-ments with A > 4 are excited and do populate many states above the ground state according to the given tempera-ture dependence of the free energy, we take gA,Z,H = 1. Within SMM we can connect the relative yields of hyper-nuclei with the hyperon binding energies. It is interesting that in this mathematical formulation one can use other parametrizations to describe nuclei in the freeze-out. This statistical approach is quite universal, and only small cor-rections, like the known spins and energies, may be re-quired for more extensive considerations.
We propose the following receipt for obtaining infor-mation on the binding energies of hyperons inside nu-clei. Let us take two hypernuclei with different masses, (A1, Z1, H) and (A2, Z2, H), together with nuclei which
differ from them only by one Λ hyperon. We con-sider the double ratio (DR) of YA1,Z1,H/YA1−1,Z1,H−1 to
YA2,Z2,H/YA2−1,Z2,H−1. One can obtain, from the above
formulas, DRA1A2 = YA1,Z1,H/YA1−1,Z1,H−1 YA2,Z2,H/YA2−1,Z2,H−1 = αA1A2exp −1 T ΔEAbh1A2 , (10) where ΔEAbh1A2 = EAbh1− EAbh2, (11) and the ratio of the C-coefficients is denoted by
αA1A2 = CA1,Z1,H/CA2,Z2,H. (12)
Within this model we see that the double ratio depends on the temperature of the system and the difference between the hyperon separation energies of the fragments, and it is independent from the fragment description. We can also take into account a small uncertainty coming from the Coulomb interaction of fragments in the freeze-out, for example, by varying the coordinate volume, see discussion in subsect. 3.4.
3.2 Freeze-out state scenario for cold fragments
In the first statistical approaches, the production of final (cold) fragments in the freeze-out volume were considered
(see, e.g., [37]). In this case the sophisticated description of the hot fragments is omitted, and we consider only fixed fragment binding energies without a temperature depen-dence. This physical condition may still be adequate for the formation of lightest fragments in the high-energy nu-clei collisions with a large energy deposition. Then, af-ter disintegration of nuclear systems, the grand-canonical yields of a normal nucleus in the ground state can be writ-ten as YA,Z= gA,Z· Vf A3/2 λ3 T exp −1 T EA,Zb − μA,Z , μA,Z= Aμ + Zν, (13)
where A is the nucleon number, gA,Z is the standard spin factor, and EA,Zb is the nucleus ground state bind-ing energy.
It is obvious that this case can be easily generalized for hypernuclei, with the same expression (1), if we take into account the binding energy of hyperons and introduce the total (temperature-independent) nucleus binding energy Eb
A,Z,H instead of FA,Z,H. Then all our formulae (8)–(12) remain similar to the standard multifragmentation, how-ever, with new spin factors.
For all physical cases one can simply deduce from eq. (10) that the logarithm of the double ratio is directly proportional to the difference of the hyperon binding ener-gies in A1 and A2 hypernuclei, ΔEbhA1A2, divided by
tem-perature. Therefore, we can finally rewrite the relation between the hypernuclei yields and the hyperon binding energies as
ΔEAbh1A2= T · [ln(αA1A2)− ln(DRA1A2)] . (14)
Sometimes we expect a large difference in hyperon bind-ing energy in both nuclei. For example, accordbind-ing to the liquid-drop approach (see eq. (2)), it can happen when the difference between A1 and A2 is essential (i.e., the
mass number A2 is much larger than A1). The influence
of the pre-exponential α coefficients is small and can be directly evaluated, depending on the selected hypernuclei. This opens a possibility for the explicit determination of the binding energy difference from experiments. Within this method, it is necessary to measure a certain number of the hypernuclei in one reaction and select the corre-sponding pairs of hypernuclei. One has to identify such hypernuclei, for example, by the correlations and vertex technique. However, there is no need to measure very pcisely the momenta of all particles produced in the re-action (including after the week decay of hypernuclei) to obtain their binding energy, as must be done if one uses the direct processes of the hyperon capture in the ground and slightly excited states of the target nuclei (e.g., in missing mass experiments [2, 38]). Therefore, our proce-dure perfectly suits for investigation of hypernuclei in the high-energy deep-inelastic–hadron and ion–induced reac-tions.
3.3 Opportunities of isobar double ratios
Another interesting way for this study is to use the double ratios of yields with the same mass numbers for light and heavy pairs. For example, the so-called strangeness popu-lation factor S was introduced in ref. [39] for interpretation of light hypernuclei production in relativistic heavy-ion collision (at momenta of 11.5 A GeV/c):
S =Y3HΛ/Y3He
YΛ/YP
. (15)
Generally, if we involve the pairs of nuclei which dif-fer by one proton instead of Λ-hyperon, we can write the isobar double ratio:
DRIA1A2 = YA1,Z1,H/YA1,Z1+1,H−1 YA2,Z2,H/YA2,Z2+1,H−1 = αIA1A2exp −1 T ΔEXb , (16) where αAI1A2 = gA1,Z1,H/gA1,Z1+1,H−1 gA2,Z2,H/gA2,Z2+1,H−1 , (17)
and the binding energy difference between 4 fragments ΔEXb =EAb1,Z1,H− EAb2,Z2,H
−EAb1,Z1+1,H−1− EAb2,Z2+1,H−1 . (18) The expression (16) cannot be factorized into the parts related to the binding energies of nuclei with A1 and A2
and the parts related only to the hyperon binding (as was possible by using the formula (8)), since it includes also the difference of the hyperon binding in hypernuclei with Z+1. Therefore, such an extraction of the hyperon binding energy would require a complicated solution of the coupled equations and extra experimental isobar measurements. Still, the convenient application of DRI can be found for single hypernuclei with H = 1, when for the pair nuclei (at H− 1 = 0 and Z + 1) there are only normal nuclei with known binding energies. In this case one can rewrite the formula (14) as
ΔEAbh1A2 = T·ln(αIA1A2)− ln(DRIA1A2) +ΔEAGS1A2, (19) where ΔEGS
A1A2is the difference of the ground state binding
energies of non-strange nuclei:
ΔEAGS1A2 =EAb1,Z1+1− EbA2,Z2+1 −EAb1−1,Z1− E b A2−1,Z2 . (20)
If the binding energies of hypernuclei taken for the dou-ble ratio are known, formula (19) can also be used to evalu-ate the chemical temperature relevalu-ated to the statistical for-mation of hypernuclei. Such temperature may essentially be lower than the baryon kinetic energies which are deter-mined by the collision dynamics. Within our approach the initial assumption is that all baryons (including strange ones) are produced primarily at the dynamical stage of
the reaction. In the following the fragment formation can be described as manifestation of the liquid-gas–type phase transition in hyper-matter. Such an approach is fully con-sistent with the limited equilibrium of fragments reported previously even for central heavy-ion collisions [40].
3.4 Analysis and advantages of the method
It is clear, from subsects. 3.1 and 3.2, that the suggested double ratio approach can be applied to hypernuclei with any number of hyperons: As is obvious, eqs. (1), (10) and (14) can be used for H > 1. In heavy-ion nuclear reactions one can obtain a multi-strange residues with a quite large probability [19], and a very wide mass/isospin range will be available for examination. As a result, one can get direct experimental evidences for hyperon bind-ing energies in double/triple hypernuclei and on influence of the isospin on hyperon interactions in multi-hyperon nuclear matter.
The connection between the relative hyperon bind-ing energies ΔEbhA1A2 and its absolute values can be done straightforward: It will be sufficient to normalize to the binding energy of one known hypernucleus (e.g., A2)
ob-tained with a different technique. However, even relative values are extremely important, when we pursue the goal to investigate the trends of the hyperon interaction in ex-otic nuclear surroundings, e.g., rich or neutron-deficient ones, and investigate multi-strange hypernuclei.
To illustrate the last point, in fig. 2 we present the dependence of the difference in hyperon binding ener-gies ΔEAbh1A2 (the notation is shortened to ΔEbh),
di-vided by the temperature, versus the mass number dif-ference ΔA = A2− A1 of isotopes. The calculations were
performed with the hyper-SMM version (see refs. [5, 15]) outlined in sect. 2, for the system with baryon number A0 = 200, charge Z0 = 80, and containing H0 = 3
Λ hyperons. Formulae (10) and (14) are applied to ex-tract this difference. We have plotted seven various dou-ble ratios of isotopes by suggesting hyper-25Mg nucleus
as A1. Other nuclei (A2 = 33, 41, 50, 60, 81, 101, 125) are
selected in order to provide a broad and representa-tive range of ΔA. In particular, we consider the yield ratios of 33P N Λ/32P(N−1)Λ, 41SN Λ/40S(N−1)Λ, 50CaN Λ/ 49Ca (N−1)Λ, 60CrN Λ/59Cr(N−1)Λ, 81GeN Λ/80Ge(N−1)Λ, 101Zr N Λ/100Zr(N−1)Λ,125SnN Λ/124Sn(N−1)Λ, to25MgN Λ/ 24Mg
(N−1)Λ, where N = 1, 2, 3. We investigate the
sensi-tivity of our results to the primary excitation of the sys-tem by assuming sys-temperatures T = 2, 3, 4, and 6 MeV, which cover the expected temperature range in fragmen-tation and multifragmenfragmen-tation reactions. One can see that the extracted ΔEbh/T increases regularly with ΔA, as
fol-lows from the hyper-term (7) adopted in the model. It is interesting that in our case this difference in the multi-strange hypernuclei is close to single hypernuclei. This is an obvious consequence of the model formula for hyper-ons in nuclei. Naturally, this function might be a different one reflecting modified hyperon binding energies, and it could be investigated via the double ratios from experi-mental data. In such a way we can get also an interesting
Fig. 2. (Color online) The difference of binding energies of hy-perons in nuclei (ΔEbh) divided by the temperature T versus
the mass number difference of these nuclei ΔA as calculated with the statistical model at different temperatures relevant for multifragmentation reactions. Baryon composition and tem-peratures (for groups of curves) of the initial system are given in the figure. The results for involved isotopes (see the text) are demonstrated by different color symbols connected with lines: Circles (solid lines) are for single hypernuclei; inverse trian-gles (dotted lines) are for double hypernuclei; squares (dashed lines) are for triple hypernuclei.
possibility to improve phenomenological formulae for hy-pernuclei according to the observed trends.
If we take into account the temperature we can get very practical curves of ΔEbh versus ΔA shown in fig. 3. For
simplicity, only the results obtained via the double ratio of single hypernuclei and normal nuclei are shown. However, as is clear from fig. 2, an involvement of multiple hypernu-clei lead to similar trends. The demonstrated regularities are again consistent with the adopted hyper-mass formula (see also fig. 1 in ref. [5]). For a detailed comparison, with the solid dark curve connecting the star symbols we show the results for the differences in hyperon binding ener-gies between the selected isotopes obtained directly from this mass formula. One can see within SMM that with decreasing temperature we approach the formula values. The physical reason of the deviations is in the temperature corrections of the bulk and surface fragment energies (in the liquid-drop approximation, see sect. 2). Nevertheless, they decrease with temperature and are under control in the present model. Another reason of the deviation is the Coulomb interaction of fragments in the freeze-out vol-ume, see eq. (6). This influences the fragment yields and their double ratios, especially if large fragments are in-volved. At this point the correct estimate of the Coulomb energy should be implemented. In the model, when we increase this volume our results for small T become very
Fig. 3. (Color online) The difference of binding energies of hy-perons in nuclei (ΔEbh) versus the mass number difference of
these nuclei ΔA for single hypernuclei. The statistical calcula-tions are performed involving the double ratio yields shown in the figure, and for temperatures T = 2 MeV (dashed line), 4 MeV (thin solid line, circle symbols), and 6 MeV (dotted line). The stars (thick solid line) are the direct calculation of
ΔEbh according to the adopted hyper-fragment formulas (2)–
(7) at T = 0 and V → ∞. The initial parameter of the hyper-nuclear system are as in fig. 2.
close to the mass formula ones. The results at all tempera-tures are regular and close to each other within 10%. This gives us a confidence that the method is reliable.
The comprehensive analysis of double hypernuclei and multi-hyperon nuclei is possible within this approach, and it seems a realistic way to experimentally address the hy-peron binding energy also in multi-strange nuclei. This is an important advantage over the standard hypernuclear measurements. Actually, the disintegration of hot hyper-residues suits best for this analysis since all kinds of nor-mal and hyper-fragments can be formed within the same statistical process. As was previously established in mul-tifragmentation studies [41, 42], the selection of adequate reaction conditions can be experimentally verified.
4 Processes following the statistical
fragmentation of hyper-residues
The concept of the statistical formation of fragments in the freeze-out volume suggests the existence of important parameters, e.g., the temperature. Also it may suggest
some phenomena, as secondary processes, which can fi-nally change the baryon composition of fragments after they leave the freeze-out volume. All these effects were under careful examination previously in multifragmenta-tion reacmultifragmenta-tions in normal nuclei. We outline how they could be taken into account in the hypernuclear case.
4.1 Estimates for freeze-out temperature
In order to extract ΔEAbh1A2 from experiment within the double ratio approach, we should determine the tem-perature T of the disintegrating hypernuclear system. This quantity was under intensive investigation in recent years in connection with multifragment formation. Vari-ous methods were suggested: using kinetic energies of frag-ments, excited states population, and isotope thermome-ters [29,43,44]. Usually, all evaluations give a temperature around 4–6 MeV in the very broad range of the excitation energies (at E∗ > 2–3 MeV per nucleon), providing so-called a plateau-like behavior of the caloric curve [25, 29]. The isotope thermometer method is the most promising one, since it allows for involving a large number of normal measured isotopes in the same reactions which produce hypernuclei. The corresponding experimental and theo-retical research were performed over the last years to in-vestigate better the temperature and isospin dependence of the nuclear liquid-gas–type phase transition [44–47].
The connection between the temperature and excita-tion energy (so-called “caloric curve”) is one of the impor-tant characteristics of the system which reflects its evolu-tion in disintegraevolu-tion into fragments. In fig. 4, we show the caloric curve for the nuclear system with the mass num-ber of 200 and the charge of 80. The calculations were done in the framework of the above-formulated statisti-cal model [5, 15]. Because of the multifragmentation sstatisti-cal- scal-ing [25] the caloric curves are similar for the systems with sizes A0 50. The cases with the initial hyperon numbers
of 0, 1, and 3 are considered. One can see that a small hy-peron admission in the system does not change practically the caloric curve: only the temperature becomes a little bit lower. We have calculated also the so-called helium-lithium temperature (THeLi) within the normal
multifrag-mentation model. This temperature was found from the standard prescription for isotope yields, as was suggested in ref. [29], and as performed in many other works. Such an isotope temperature follows quite reasonably the normal temperature in the most important for fragment produc-tion region of excitaproduc-tion energy from 2 to 6 MeV per nu-cleon. Therefore, with the correction of 20% to 30% THeLi
can be used for the determination of temperatures in the systems.
In addition to measuring isotopes and hyper-isotopes it would be instructive to select the reaction conditions lead-ing to similar freeze-out states. The freeze-out restoration methods were extensively tested previously: In particu-lar, the masses and excitation energies of the hypernuclear residues can be found with a sufficient precision within the methods developed in refs. [41, 42]. In the future one can analyze the subsequent ranges of the excitation energy of
Fig. 4. (Color online) The temperature versus the excitation energy for the disintegration of the hypernuclear system with parameters given in the figure and the density of ρ = ρ0/3.
The statistical calculations including different initial numbers of hyperons (0, 1, and 3) are shown by different symbols and lines. (For better view the symbols are shifted slightly along the abscissa axis being at the same E∗.) The helium-lithium isotope temperature (see the text) calculated within the stan-dard multifragmentation model are presented by diamonds.
sources (from low to very high ones) to investigate the evolution of the hypernuclei yields with the temperature and the phase transition in hyper-matter. It is especially interesting to move into the neutron-rich domain of the nuclear chart, by selecting neutron-rich target or projec-tiles. This is also important because of the problem of the neutron-rich hyper-matter discussed for the neutron-star structure.
4.2 Yields corrections after secondary de-excitation
In few cases, the cold fragments can be considered as being produced finally in the freeze-out volume with the statis-tical mechanism. However, taking into account the expe-rience accumulated in nuclear reactions, we may expect that the primary excited fragments and hyper-fragments (specially, large ones) can be formed. Therefore, they will quickly decay after escaping the freeze-out stage. For low excited sources this fragment excitation energy should roughly correspond to the compound nucleus tempera-ture. As was established in theory and multifragmentation experiments [48], the internal fragment excitations can reach around 2 MeV per nucleon and even more for highly excited residue sources. The secondary de-excitation in-fluences all 4 fragments entering the double ratio and the fragments will lose few nucleons and, possibly, hyperons.
Fig. 5. (Color online) Influence of the secondary de-excitation on the difference of binding energies of hyperons in nuclei ΔEbh
as a function of their mass number difference ΔA, by taking single hypernuclei (which are the same as in fig. 3). The calcu-lations of double ratio yields for primary hot nuclei shown for temperature 4 MeV (dashed line, color circle symbols). The triangles, squares, and stars are the calculations with modi-fied double ratios after the secondary de-excitation (via nu-clear evaporation) of primary nuclei at excitation energies of 1.5, 2.0, and 3.0 MeV/nucleon, respectively. The same color symbols show the evolution of the results corresponding to the nuclei evolution during de-excitation. The statistical errors are within the symbol sizes.
Previous investigations of similar nuclear decay processes of excited nuclei in normal multifragmentation reactions tell us that this is a process of the continuous modifica-tion of initial (mother) nuclei into final (daughter) ones by emitting particles. Following this de-excitation the mass numbers will change and we expect a smooth transfor-mation of ΔEbh
A1A2 versus the variation of mass difference
ΔA = (A2− A1). Therefore, modified yields and mass
numbers should be possible to use for the final estimate. In order to evaluate this effect we have adopted the nu-clear evaporation model generalized for hypernuclei [49]. In this case the fragments at the freeze-out are described as discussed in sect. 2. It was demonstrated in ref. [49] that mostly neutrons and other light normal particles will be emitted from hot large hyper-fragments, since the hy-perons have a larger binding energy because of the occu-pation of the lowest (s-) states. In figs. 5 and 6 we show how the secondary de-excitation can modify the results on ΔEbh
A1A2 (in the figures noted again as ΔEbh) versus
ΔA, respectively, for single and double hypernuclei. For clarity, we use the same mother nuclei as presented in figs. 2 and 3, which are noted in subsect. 3.4. The typical temperature of T = 4 MeV is taken for initial fragment
Fig. 6. (Color online) The same as in fig. 5 but for double hypernuclei (see the text).
yields to be consistent with the previous figures. As can be seen in fig. 3, the results obtained from double ra-tios of primary yields of nuclei do change very little with variation of the temperature in the multifragmentation region. The realistic internal excitation energies of these fragments E∗= 1.5, 2 MeV per nucleon, and a highest es-timate E∗= 3 MeV per nucleon were assumed. The latest value were taken to see the trend and fluctuations caused by this secondary process. To calculate the double ratios after the evaporation we have determined the daughter nuclei with the maximum yield. In this case, for example, after de-excitation of24Mg,25MgΛ, 124Sn, and125SnΛ at E∗ = 3 MeV/nucleon we obtain 20Ne, 21Ne
Λ, 99Pd, and
100Pd
Λ nuclei, respectively. As we know, the same daugh-ter nuclei can be produced afdaugh-ter evaporation of other nu-clei with close A and Z, and we have calculated the de-excitation of such nuclei too. For example, at the highest excitation energy few tens of mother nuclei can result into a specific daughter nucleus. We have taken into account the weight of all primary nuclei after multifragmentation in the freeze-out volume and evaluated their contribution in the final yields of daughter ones. Afterwards, accord-ing to the formula (14) we have found new ΔEbh. The
results of the calculations including the evaporation are given by triangle, square and star symbols corresponding to the above-mentioned internal excitations.
To see clearly the general transformation of the initial function, in figs. 5 and 6 the values of ΔA and ΔEbh are
shown with the same color symbols for the mother and daughter isotopes. This connection can also be seen by the corresponding groups in the ΔA-axis: After evaporation there is a regular shift of ΔA values to smaller ones as a result of nucleon loses. That can be distinguished by comparing the primary circles with the triangles, squares
and stars. This shift is especially prominent in big nuclei because of their total excitation energies are higher.
There are also modifications of ΔEbh because of the
isotope yield variations after secondary de-excitation. For instance, nuclear shell effects in N e (daughter) nuclei in-fluence the process, therefore, the emission of nucleons correlate with a particular excitation energy. Nevertheless, we can control it with the model calculations, and, as can be seen from the figures, all possible deviations are within 10%. The reason of such a stability of results is that af-ter the evaporation calculations the double ratios change small even for new conditions of primary fragments: As is well known, the nuclei initially close in mass number A and charge Z de-excite in a similar way. Also averaging over the neighbor primary nuclei smoothens the possible fluctu-ations of the final nuclei yield, which can occur in the case of separate nuclei. Other uncertainties of the secondary de-excitation may be related to our poor knowledge of the freeze-out volume state, reflected in the primary fragments weights. To analyze it, we have performed calculations of primary fragments for the most probable freeze-out den-sity range, namely, from 1/3 to 1/6 of the normal nuclear densities, which result also in a smaller Coulomb inter-action in the freeze-out volume and which give different weights for primary fragments. We have found that the influence of all effects on ΔEbh is rather moderate (the
maximum uncertainties are again around 10%). Finally, after the secondary processes, the whole curve of ΔEbh
versus ΔA may look a little bit shifted. However, the gen-eral form of this dependence does not change.
If we knew more precisely the freeze-out conditions from experiments the model calculations could be used for adequate corrections of the experimentally extracted ΔEbh. It is similar to the procedures elaborated
previ-ously, for example, in normal multifragmentation studies for evaluating the isotope temperatures [29]. We conclude, from figs. 5 and 6, that the initial difference in the hyperon binding energies of hypernuclei can be extracted by gener-ating similar plots, even after the secondary de-excitation.
5 Conclusion
Because of the special requirements on targets in hadron and lepton induced reactions, the traditional hypernuclear methods (e.g., involving the missing mass spectroscopy) can address only a small number of isotopes. Also the development of the detectors for measuring practically all produced particles with their exact kinetic energies is very complicated and expensive. All these factors do not favor a rapid increase of the number of measured hypernuclei, including their binding energies. The experimental infor-mation on hypernuclei is very limited, therefore, there is an urgent need to increase the number of known hypernu-clei by involving new reactions for experimental measure-ments. The deep-inelastic fragmentation and multifrag-mentation reactions are very promising since they lead to the production of both strangeness and large fragments. The peripheral relativistic ion-ion and hadron-ion colli-sions produce all kinds of hypernuclei with sufficiently
large cross-sections. These hypernuclei can be produced in the projectile and target spectator regions under con-ditions of chemical equilibrium. In this case one can es-sentially extend hypernuclear studies and the suggested double ratio method can be used. Only the identification of hypernuclei is required, and, as demonstrated in recent ion experiments of searching for strange particles, there are effective ways to perform it. The experimental extrac-tion of the difference in the hyperon binding energies be-tween hypernuclei (ΔEbh
A1A2) via their yields is a novel
and practical way to pursue hypernuclear studies. The advantage of this method over the traditional hypernu-clear ones is that the exact determination of all produced particles parameters (with their decay products) is not necessary. Only relative measurements are necessary for this purpose. Therefore, one can use similar weak-decay chains and their products for comparison of hypernuclei. For example, if we take pairs of large hyper isotopes, they undergo weak decay in a non-mesonic channel that can be found by products far from the collision point with the vertex technique. The correlation of the produced isotopes and particles is sufficient information to employ the dou-ble ratio method. Nevertheless, we should note that this approach can address a so-called average binding energy, which includes the occupation of all hyper-states (s-, p-state, and others) inside nucleus. This is important for nuclear matter studies. However, the detail structure in-formation on hypernuclei may be obtained with the tra-ditional measurements.
It is interesting and important that with this method one can also determine the difference of hyperon binding energies in double and multi-hypernuclei. This knowledge may help to study the hyperon-hyperon interactions by involving standard analytic prescriptions for finite nuclei. It is very difficult to measure the hypernuclei binding en-ergy for neutron-rich/-poor nuclear species within tradi-tional hypernuclear experiments, but hypernuclei with ex-treme isospin can be easily obtained in high-energy deep-inelastic fragmentation reactions. Most of their yields can be described by a statistical approach, and the suggested method opens an effective way to extend the hypernuclear chart. We believe, with the double ratio method, novel conclusions can be obtained concerning neutron-rich and neutron-deficient hypernuclei. On the other hand, these nuclei can be considered as bulks of nuclei matter, there-fore, within our approach we can address the properties of multi-hyperon and exotic matter: It will be possible to extract the isospin influence on the hyperon interac-tion in matter (revealing in the hyperon binding ener-gies) directly from experimental data. The measurements of multi-strange and neutron-rich nuclear systems would be important for addressing the problem of the hyperon matter in neutron stars [50, 51], since they can give in-formation on evolution of the hyper-matter properties de-pending on strangeness and isospin.
The new generation of ion accelerators at intermediate energies, such as FAIR (Darmstadt) and NICA (Dubna), will make possible this research. It is promising that new advanced experimental installations for the fragment de-tection can be available soon [52–55].
The authors thank Ch. Scheidenberger and J. Pochodzalla for stimulating discussions on hypernuclear studies. AS Botv-ina acknowledges the support of Bundesministerium f¨ur Bil-dung und Forschung (BMBF), Germany. The support of GSI, FIAS, and HIC for FAIR is appreciated. NB acknowledges the Scientific and Technological Research Council of Turkey (TUBITAK) support under Project No. 118F111. The work of MB and NB has been performed in the framework of COST Action CA15213 THOR.
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