# Employing ternary fission of 242Pu as a probe of very neutron rich matter

J.B. Natowitz  Cyclotron Institute, Texas A&M University, College Station,
Texas 778431  H. Pais  CFisUC, Department of Physics, University of Coimbra,
3004-516 Coimbra, Portugal  G. Röpke  Institute of Physics, University of
Rostock, 18059 Rostock, Germany

(August 27, 2024)

###### Abstract

Detailed assessments of the ability of recent theoretical approaches to
modeling existing experimental data for ternary fission confirm earlier
indications that the dominant mode of cluster formation in ternary fission is
clusterization in very neutron rich, very low density, essentially chemically
equilibrated, nucleonic matter. An extended study and comparison of these
approaches applied to ternary fission yields in the thermal neutron induced
reaction 241Pu($n_{\rm th}$,f) has been undertaken to refine the
characterization of the source matter. The resonance gas approximation has
been improved taking in-medium effects on the binding energies into account. A
temperature of 1.29 MeV, density of $6.7\times 10^{-5}$ nucleons/fm3 and
proton fraction $Y_{p}$ = 0.035 are found to provide a good representation of
yields of the ternary emitted light particles and clusters. In particular,
results for $Z=1$ and 2 isotopes are presented. Isotopes with larger $Z$ are
discussed, and the roles of medium and continuum effects, even at very low
density are illustrated.

###### pacs:

21.65.-f, 21.60.Jz, 25.70.Pq, 26.60.Kp

##  I Introduction

In the neutron induced or spontaneous ternary fission of a heavy isotope a
nucleon or light cluster is emitted perpendicular to the fission axis
determined by the two separating large fission fragments Halp ; Wage ; Mills ;
IAEA96 ; Krappe ; Shub ; Vorob ; Gusev ; Koes1 ; Koes2 ; Kopatch . 4He,
emitted in approximately 1/500 events is the most dominant charged particle
but other charged isotopes with charge number $Z=1$ up to $Z=18$ have been
observed Koes1 ; Koes2 . These light charged particles (LCP) are emitted from
the neck region at the time of scission and may be considered as signals which
describe the state of nuclear matter in the neck at that time.

To interpret the observed ternary yields, statistical models have been applied
which assume thermodynamic equilibrium at chemical freeze-out during scission
Valskii2 ; Lestone08 ; Sara . However, experimental yields for the heavier
elements are typically overestimated unless some mechanism for suppression of
higher mass products is introduced Lestone08 ; Sara . In Wuenschel et al. Sara
, chemical equilibrium is achieved in accordance with the grand canonical
ensemble only for the lightest isotopes and a time dependent nucleation
process for production of heavier LCP is introduced Schmelzer ; Demo  so that
the LCP yield becomes increasingly suppressed with increasing mass number.

Recently, this approach has been explored in more detail and used to determine
isotopic equilibrium constants for LCP emitted in the ternary fission reaction
241Pu($n_{\rm th}$,f) natowitz20 . Further investigations were then undertaken
to better characterize the ternary fissioning 242Pu source. In rnp20 , the
simple ideal model of nuclear statistical equilibrium was improved considering
medium effects and continuum correlations R20 , e.g., resonances such as 4H,
5He, 8Be (as known from the virial expansion of the nuclear matter equation of
state). A nearly perfect description of the measured yields of H and He
isotopes for 252Cf(sf) was obtained. In rnp21 , several different fission
reactions were investigated within an information entropy approach, and
isotopes up to $Z=6$ are included. These investigations showed that the
dominant mode of cluster formation in ternary fission is clusterization in
very neutron rich, very low density, essentially chemically equilibrated,
nucleonic matter at temperatures of about 1 MeV.

In the present work, an extended study and comparison of the different
approaches applied to modeling ternary fission yields in the thermal neutron
induced reaction 241Pu($n_{\rm th}$,f) Koes1 ; Koes2  is undertaken. A more
detailed exploration of the in-medium and continuum effects leads to a more
refined characterization of the source matter. We show that, in particular,
the weakly bound states are strongly influenced by in-medium effects and
provide an alternative observable to determine the density of the source
matter. We find that the neck matter at scission has a temperature $T\approx
1.29$ MeV, a nucleon density $n_{B}\approx 6.7\times 10^{-5}$ nucleons/fm3,
and a proton fraction $Y_{p}\approx 0.035$. This work provides a baseline
laboratory test of the low-density nuclear equation of state for conditions
which may be encountered in astrophysical sites such as core-collapse
supernova events in which a neo-neutron star is formed, evolves with time and
cools down to a neutron star Bezn20 . Similar neutron rich matter may also be
produced in the merging of a binary neutron star system. Both scenarios relax
to a temperature of about 1 MeV in less than a minute. In such dense, highly
excited systems, surface densities of $10^{11}$ g/cm3 are typical, but the
proton fraction may be different. In $\beta$ equilibrium, it depends upon the
neutrino density, see, e.g., Arcones08 ; Fischer . Even if the proton fraction
in astrophysical scenarios is different from the proton fraction in the neck
matter at scission, the equations of state (for a review see, e.g., Oertel17 ;
Furusawa22 ) employed over a wide region of parameter values may be checked at
the special conditions of scission. Investigations of the correct description
of correlations under scission conditions may also be applied to other regions
of the astrophysical parameter space.

##  II Yields from a quantum statistical approach

We consider different stages of the fission process. For the time evolution of
the nucleonic system up to scission we assume a quick relaxation to local
thermodynamic equilibrium. Within the statistical model framework the grand
canonical distribution at scission, where chemical freeze-out takes place,
gives the primordial (or primary) yields for the relevant species. For the
time evolution after scission we assume a reaction kinetics in which populated
excited states decay, and the kinetic energies are determined by the
interaction between the fission products. The decay of unstable nuclear states
and resonances is described as feed-down processes which transform the
primordial distribution of yields to the final observable yield distribution.
This approach to the time evolution of the fission process may be considered
as an approximation within the systematic approach of non-equilibrium
statistical operators where both stages of time evolution, the hydrodynamical
and kinetic ones, are unified within an information theoretical approach rnp20
; rnp21 . This information theoretical approach allows us to introduce
Lagrange parameters $\lambda_{T}(t),\lambda_{n}(t),\lambda_{p}(t)$ which are
the nonequilibrium generalizations of the temperature and the chemical
potentials of neutrons ($n$) and protons ($p$). They depend on time $t$ and,
in general for the hydrodynamical description, also on position.

In the present work, we focus on inferring the corresponding Lagrange
parameters $\lambda_{i}$ for the primary distribution at chemical freeze-out
from the observed final yield distribution. For this, we need an accurate
solution of the grand canonical distribution at scission. We employ and
compare three successive approximations:
(i) The resonance gas approximation (res.gas) known also as nuclear
statistical equilibrium (NSE), see Furusawa22 ; Ropke82  where the nucleonic
system is considered as an ideal mixture of nuclei in the ground state and in
(unstable) excited states and resonances. A semi-empirical improvement is the
excluded volume model Hemp1 ; Hemp2 .
(ii) The virial approximation (vir) where binary interactions between the
different constituents are taken into account considering the respective
scattering phase shifts Ropke82 ; SRS ; HS .
(iii) Accounting for in-medium corrections (medium) such as self-energy shifts
and Pauli blocking effects Ropke82 ; R09 ; Typel10 .

In particular, we investigate whether the frequently used nuclear statistical
equilibrium model is sufficient to describe nucleonic systems under the
scission conditions in the neck region or whether continuum correlations and
in medium effects must be taken into account.

In the quantum statistical approach, after the cluster decomposition of the
spectral function, the density is decomposed into partial densities of
different channels characterized by $A,Z$ rnp20 . The primordial yields, here
denoted as relevant yields $Y^{\rm rel,approx}_{A,Z}$ in the corresponding
approximation are calculated as

$\displaystyle Y^{\rm rel,approx}_{A,Z}\propto R^{\rm
approx}_{A,Z}(\lambda_{T})\,g_{A,Z}\left(\frac{2\pi\hbar^{2}}{Am\lambda_{T}}\right)^{-3/2}$
$\displaystyle\times e^{(B_{A,Z}+(A-Z)\lambda_{n}+Z\lambda_{p})/\lambda_{T}},$
(1)

where $B_{A,Z}$ denotes the (ground state) binding energy and $g_{A,Z}$ the
degeneracy Nudat . The prefactor

$R^{\rm approx}_{A,Z}(\lambda_{T})=1+\sum^{\rm
exc}_{i}[g_{AZ,i}/g_{A,Z}]e^{-E_{AZ,i}/\lambda_{T}}$ (2)

is related to the intrinsic partition function of the cluster $\\{A,Z\\}$. The
summation is performed over all excited states $i$, excitation energy
$E_{AZ,i}$ and degeneracy $g_{AZ,i}$ Nudat .

Different approximations are considered for the intrinsic partition function
as discussed above.
(i) In the the resonance gas approximation, the summation for $R^{\rm
res.gas}_{A,Z}(\lambda_{T})$ is performed over all states including resonances
as found in the nuclear data tables Nudat . The angular momentum degeneracies
are included.
(ii) In the virial approximation, $R^{\rm vir}_{A,Z}(\lambda_{T})$ includes
also the sum over all scattering states (continuum correlations) as expressed
by the Beth-Uhlenbeck formula. Expressions and interpolation formulas are
found in rnp20 ; R15 .
(iii) The expression
$R^{\rm,medium}_{A,Z}(\lambda_{T},\lambda_{n},\lambda_{p})$ takes in-medium
effects into account, in particular the shifts of binding energy values
because of self-energy, Pauli blocking effects and the modification of bound
state wave functions and scattering phase shifts. Therefore, this term is also
dependent on the densities of neutrons and protons. Expressions for the
intrinsic partition function and $R^{\rm res.gas}_{A,Z},R^{\rm
vir}_{A,Z},R^{\rm medium}_{A,Z}$ for $\lambda_{T}=1.29\,{\rm MeV}$ are given
for $Z\leq 6$ in the appendix, Tabs. 4 \- 8, see also Sec. III below.

The summation in (2) is performed over all excited states $i$ which are bound.
Also, the continuum contributions are included in the virial expression. For
instance, the Beth-Uhlenbeck formula expresses the contribution of the
continuum to the intrinsic partition function via the scattering phase shifts,
see SRS ; HS ; R20 ; R09 ; R11 ; R15 . The simple statistical equilibrium
distribution (NSE) is obtained for $R^{\rm vir}_{A,Z}(\lambda_{T})=1$, i.e.,
neglecting the contribution of all excited states including continuum
correlations.

As an example we focus on the fission reaction 241Pu($n_{\rm th}$,f) induced
by thermal neutrons where good data for the ternary fission yields are
available Koes1 ; Koes2 . The observed yields up to 20C are shown in Tabs. 1,
2 below. Instead of normalizing to the yield of total $\alpha$ particle
emission, assigned to be 10000, we employ absolute yields per fission. The
total $\alpha$ particle emission yield per fission was taken from IAEA96 .

The Koester data do not contain values for scission nucleon ($n$ or $p$)
emission. Determination of those yields is experimentally very challenging
Vorob ; Gusev ; Shub . Experimental scission proton yields are very small and
careful experiments have revealed that secondary processes dominate the
apparent yields reported Shub . In our opinion the best available constraint
on the scission proton yield is that of reference Shub , where only an upper
limit of 2.9 to $4.0\times 10^{-5}$ is deduced. Interestingly, although the
determination of scission neutron emission yield in the presence of a much
larger yield of secondary neutrons evaporated from the separated fission
fragments is inherently even more difficult, very precise measurements and
analyses of the neutron energy and angular distributions have been carried out
and lead to the conclusion that the scission neutron yield is 0.107 per
fission, approximately one thirtieth of the total neutron yield Vorob ; Gusev
. We have adopted this number for our analysis. It is worth noting that, in an
equilibrium picture, the relative scission neutron and proton yields implied
by these results indicate that the neck matter at scission is extremely
neutron rich. That this must be the case has previously been inferred from the
fact that 3He has not been detected in ternary fission experiments while the
isotope 3H, with a similar binding energy is the second most abundant ternary
charged fragment observed Mills ; IAEA96 .

isotope | $A$ | $Z$ | $Y^{\rm obs}_{A,Z}$ | $R_{A,Z}^{\rm res.gas}(1.29)$ | $Y_{A,Z}^{\rm rel,res.gas}$ | $Y_{A,Z}^{\rm final,res.gas}$ | $X^{\rm res.gas}_{A,Z}$
---|---|---|---|---|---|---|---
1n | 1 | 0 | 0.107 | 1 | 0.1098 | 0.1098 | 0.9215
1H | 1 | 1 | - | 1 | 4.298E-6 | 4.298E-6 | 0.00003414
2H | 2 | 1 | 8.463E-6 | 1 | 8.889E-6 | 8.889E-6 | 0.000146
3H | 3 | 1 | 1.584E-4 | 1 | 1.212E-4 | 1.398E-4 | 0.00359
4H* | 4 | 1 | - | 1.579 | 1.855E-5 | [$\to^{3}$H] | -
3He | 3 | 2 | - | 1 | 2.549E-9 | 2.817E-9 | 6.163E-8
4He | 4 | 2 | 2.015E-3 | 1 | 1.449E-3 | 1.911E-3 | 0.06529
5He* | 5 | 2 | - | 1 | 3.999E-4 | [$\to^{4}$He] | -
6He0 | 6 | 2 | 5.239E-5 | 1 | 4.322E-5 | 5.916E-5 | 0.003232
6He* | 6 | 2 | - | 1.242 | 5.407E-5 | [$\to^{4}$He] | -
7He* | 7 | 2 | - | 1.156 | 1.594E-5 | [$\to^{6}$He] | -
8He | 8 | 2 | 3.022E-6 | 1 | 2.605E-6 | 2.898E-6 | 0.0002249
8He* | 8 | 2 | - | 0.452 | 1.193E-6 | [$\to^{4}$He] | -
9He* | 9 | 2 | - | 1.426 | 2.929E-7 | [$\to^{8}$He] | -
6Li | 6 | 3 | - | 1 | 4.059E-8 | 4.059E-8 | 2.085E-6
6Li* | 6 | 3 | - | 0.5471 | 2.46E-8 | [$\to^{3}$H] | -
7Li | 7 | 3 | 1.35E-6 | 1.345 | 2.207E-6 | 2.545E-6 | 0.0001591
8Li | 8 | 3 | 8.463E-7 | 1.281 | 1.357E-6 | 1.357E-6 | 0.00009983
8Li* | 8 | 3 | - | 0.3151 | 3.374E-7 | [$\to^{7}$Li] | -
9Li | 9 | 3 | 1.672E-6 | 1.062 | 2.181E-6 | 2.335E-6 | 0.0002005
10Li* | 10 | 3 | - | 1 | 1.531E-7 | [$\to^{9}$Li] | -
11Li0 | 11 | 3 | 9.068E-10 | 1 | 2.785E-8 | 2.962E-8 | 3.312E-6
12Li* | 12 | 3 | - | 1 | 1.773E-9 | [$\to^{11}$Li] | -
7Be | 7 | 4 | - | 1.359 | 2.382E-11 | 2.382E-11 | 1.401E-9
8Be* | 8 | 4 | - | 1.477 | 1.59E-6 | [$\to^{4}$He] | -
9Be | 9 | 4 | 8.866E-7 | 1 | 1.616E-6 | 1.616E-6 | 0.0001325
9Be* | 9 | 4 | - | 0.5628 | 1.244E-6 | [$\to^{4}$He] | -
10Be | 10 | 4 | 9.269E-6 | 1.367 | 1.112E-5 | 1.51E-5 | 0.001435
10Be0 | 10 | 4 | - | 0.0789 | 6.553E-7 | [$\to^{10}$Be] | -
11Be0 | 11 | 4 | 1.189E-6 | 1 | 2.422E-6 | 4.313E-6 | 0.0004634
11Be0 | 11 | 4 | - | 0.7803 | 1.891E-6 | [$\to^{11}$Be] | -
11Be* | 11 | 4 | - | 1.343 | 3.287E-6 | [$\to^{10}$Be] | -
12Be | 12 | 4 | 5.642E-7 | 2.149 | 2.781E-6 | 3.73E-6 | 0.0004532
12Be0 | 12 | 4 | - | 0.3657 | 7.655E-7 | [$\to^{12}$Be] | -
13Be* | 13 | 4 | - | 1 | 1.84E-7 | [$\to^{12}$Be] | -
14Be | 14 | 4 | 5.441E-10 | 1 | 3.606E-8 | 4.117E-8 | 6.237E-6
15Be* | 15 | 4 | - | 1 | 5.114E-9 | [$\to^{14}$Be] | -
Table 1: Observed yields per fission of ternary fission of 241Pu($n_{\rm th}$,f), including 0.107 for scission neutrons (col. 4), are compared to a final state distribution, calculated in the resonance gas approximation. The Lestone fit metric Lestone08  is calculated for the isotopes with $Z\leq 2$. The minimum is found for the parameter values given at the end of table. $Y_{A,Z}^{\rm rel,res.gas}$ denotes primordial yield, $Y_{A,Z}^{\rm final,res.gas}$: final yield, $X^{\rm res.gas}_{A,Z}$: mass fraction. Units: MeV, fm. isotope | $A$ | $Z$ | $Y^{\rm obs}_{A,Z}$ | $R_{A,Z}^{\rm res.gas}(1.29)$ | [$Y_{A,Z}^{\rm rel,res.gas}$ ] | [$Y_{A,Z}^{\rm final,res.gas}$] | [$X^{\rm res.gas}_{A,Z}$]
---|---|---|---|---|---|---|---
10B | 10 | 5 | - | 1.363 | 2.484E-9 | 2.565E-9 | 2.295E-7
10B0 | 10 | 5 | - | 0.0443 | 8.17E-11 | [$\to^{10}$B] | -
11B | 11 | 5 | 3.224E-7 | 1.175 | 8.876E-7 | 8.876E-7 | 0.00009106
12B | 12 | 5 | 2.015E-7 | 2.251 | 1.703E-6 | 1.835E-6 | 0.0002126
12B0 | 12 | 5 | - | 0.1715 | 1.32E-7 | [$\to^{12}$B] | -
13B | 13 | 5 | - | 1 | 4.367E-6 | 4.91E-6 | 0.0006411
14B0 | 14 | 5 | 2.62E-8 | 1 | 1.123E-6 | 1.504E-6 | 0.0002177
14B0 | 14 | 5 | - | 0.3381 | 3.809E-7 | [$\to^{14}$B] | -
14B* | 14 | 5 | - | 0.4803 | 5.426E-7 | [$\to^{13}$B] | -
15B | 15 | 5 | 9.269E-9 | 1 | 7.523E-7 | 7.691E-7 | 0.0001235
16B* | 16 | 5 | - | 1 | 1.685E-8 | [$\to^{15}$B] | -
17B | 17 | 5 | - | 1 | 2.021E-8 | 2.26E-8 | 4.397E-6
18B* | 18 | 5 | - | 1 | 2.389E-9 | [$\to^{17}$B] | -
13C | 13 | 6 | - | 1.358 | 2.079E-6 | 2.079E-6 | 0.0002587
14C | 14 | 6 | 2.539E-6 | 1.069 | 4.479E-5 | 5.129E-5 | 0.007177
15C | 15 | 6 | 8.665E-7 | 1 | 2.08E-5 | 5.606E-5 | 0.008647
15C0 | 15 | 6 | - | 1.69 | 3.527E-5 | [$\to^{15}$C] | -
15C* | 15 | 6 | - | 0.3074 | 6.501E-6 | [$\to^{14}$C] | -
16C | 16 | 6 | 1.008E-6 | 2.272 | 6.149E-5 | 9.799E-5 | 0.01218
16C0 | 16 | 6 | - | 0.885 | 2.425E-5 | [$\to^{16}$C] | -
17C0 | 17 | 6 | 1.29E-7 | 1 | 1.816E-5 | 4.693E-5 | 0.008776
17C0 | 17 | 6 | - | 1.582 | 2.877E-5 | [$\to^{17}$C] | -
17C* | 17 | 6 | - | 0.6677 | 1.225E-5 | [$\to^{16}$C] | -
18C | 18 | 6 | 5.642E-8 | 3.172 | 3.515E-5 | 4.42E-5 | 0.009127
19C0 | 19 | 6 | 5.038E-10 | 5.112 | 1.662E-5 | 1.662E-5 | 0.003715
19C* | 19 | 6 | - | 2.776 | 9.051E-6 | [$\to^{18}$C] | -
20C | 20 | 6 | 7.254E-10 | 2.426 | 3.743E-6 | 3.743E-6 | 0.0009117
$\lambda_{T}$ |  |  |  | 1.2897 |  |  |
$\lambda_{n}$ |  |  |  | -3.1486 |  |  |
$\lambda_{p}$ |  |  |  | -16.273 |  |  |
volume |  |  |  | 1859.4 |  |  |
$n_{B}$ |  |  |  | 0.000064 |  |  |
$Y_{p}$ |  |  |  | 0.03486 |  |  |
fit metric |  |  |  | 0.005485 |  |  |
Table 2: Continuation of Tab. 1. Observed yields per fission of ternary
fission of 241Pu($n_{\rm th}$,f) (col. 4) are compared to a final state
distribution, calculated in resonance gas approximation, including 0.107 for
scission neutrons. The Lestone fit metric is calculated for the isotopes with
$Z\leq 2$. The minimum is found for the parameter values given at the end of
table. Units: MeV, fm.

We are interested in a quantum statistical description of dense matter. The
standard nuclear statistical equilibrium (NSE) approach describes matter in
the low-density limit where interaction between the components may be
neglected (ideal gas of nucleons and nuclei in ground and excited states). We
are interested in a consistent quantum statistical description of interacting
components as done, for instance, for the equation of state Qin12 . In
particular we take into account continuum correlations and in-medium effects.

##  III Extraction of Lagrange parameters from observed yields

In this section we demonstrate how the Lagrange parameters $\lambda_{i}$ may
be extracted from the observed yields. For simplicity, we use in this example
only the simplest approximation for the intrinsic partition function, i.e.,
the ideal resonance gas approximation. $R_{A,Z}^{\rm res.gas}(\lambda_{T})$ is
calculated according (2) where for isotopes $\\{A,Z\\}$ all excited states $i$
are summed over, the excitation energies $E_{AZ,i}$ and angular momentum
degeneracies $g_{AZ,i}=2J_{AZ,i}+1$ are taken from the nuclear data tables
Nudat . In the Appendix we give these values for $Z\leq 6$ as well as the
ground state binding energies and the threshold energies for the continuum of
scattering states $E_{A,Z}^{\rm thresh}$. In Tabs. 4 \- 8, we divide the
intrinsic partition function into different parts with respect to the
contribution to the final yields: The summation over all excited states above
the threshold energy is marked by an asterisk. After freeze-out, these excited
states are assumed to decay and feed the yields of daughter isotopes observed
in the final distribution. This process is also explained in Tabs. 1, 2, where
in col. 7 the respective feed-down channel is indicated. The other excited
states are assumed to gamma decay to the ground state and remain in the same
isotope channel. In Tabs. 4 \- 8, we indicate by the superscript ”0” the
states which have a binding energy smaller than 1 MeV below the continuum
threshold. These weakly bound states are of special interest when in-medium
effects are considered below in Sec. IV.3.

We use this subdivision of the intrinsic partition function to model the
kinetic stage of evolution, i.e. the transition from the primary distribution
to the final distribution considering only decay processes of the excited
nuclei. In a more general approach, this sharp subdivision should be replaced
by branching ratios describing the feed-down to the final yields or using
reaction networks.

For the resonance gas approximation and the virial approximation the $R$
factors are functions of the temperature-like parameter $\lambda_{T}$. If in-
medium effects are taken into account, the $R$ factor depends in addition on
the chemical potentials $\lambda_{n},\lambda_{p}$. Within our fit procedure
described below, they are determined self-consistently. Values for the $R$
factors for $\lambda_{T}=1.29$ MeV are given in the appendix, Tabs. 4 \- 8,
for the resonance gas and virial approximations. In Tabs. 9, 10 values
$R_{A,Z}^{\rm medium}(\lambda_{T},\lambda_{n},\lambda_{p})$ are shown for
$\lambda_{T}=1.29$ MeV, $\lambda_{n}=-3.09$ MeV, and $\lambda_{p}=-16.19$ MeV,
which take into account in-medium effects. With a given set of Lagrange
parameters, we are able to calculate the primary yields and the final yields.
This is shown for the resonance gas approximation in Tabs. 1, 2 as well as in
Tabs. 9, 10 where in-medium effects are taken into account. In each case the
calculated final yields are compared with the observed yields.

The task is then to find values for the Lagrange parameters which reproduce
the observed yield in an optimum way. We have previously observed Sara ; rnp20
; rnp21  that yields of isotopes with large mass number are suppressed because
of nucleation kinetics or size effects. For the fission process 241Pu($n_{\rm
th}$,f) this suppression with respect to the grand canonical distribution is
observed for $A>10$. To avoid this effect we use only the charged particles up
to $Z=2$, i.e., the observed yields of 1n, 2H, 3H, 4He, 6He, 8He to find
values for the Lagrange parameters.

As in Ref. Sara , we use for the fit metric that of Lestone Lestone08 ,
defined by

$M^{2}=\sum_{A,Z}^{n}\left(\ln[Y_{A,Z}^{\rm final,x}]-\ln[Y_{A,Z}^{\rm
obs}]\right)^{2}/n$ (3)

where $n$ is the number of fitted experimental data points.

As shown in Tab. 2, the values $\lambda_{T}=1.29$ MeV, $\lambda_{n}=-3.149$
MeV, and $\lambda_{n}=-16.273$ MeV are obtained from the minimum of the fit
metric for the resonance gas approximation. The fit metric is also given in
Tab. 2. From the yields per fission, we can also derive a volume. The baryon
density $n_{B}$ is obtained from the observed yields per fission
$n_{B}=\sum_{A,Z}AY^{\rm obs}_{A,Z}$ divided by the volume, it is mainly
determined by the neutron density. The proton fraction $Y_{p}$ is obtained as
$\sum_{A,Z}ZY^{\rm obs}_{A,Z}/n_{B}$, it is mainly determined by the yield of
4He. There is an uncertainty because the free proton density is not included
in the fit, but the calculated values $Y_{1,1}$ are small so that no large
effect is expected if the contribution of 1H is dropped.

The observed values 1nobs, 2Hobs, 3Hobs, 4Heobs, 6Heobs, and 8Heobs are well
reproduced. Below, in the next Section, we will also discuss the isotopes with
$Z>2$.

Whereas the values for temperature and the proton fraction are expected and
understood, the value of the optimum baryon density is to be determined. We
calculate the composition assuming different values for the density to show
that there is a strong variation of the composition and a sharp minimum of the
fit metric.

A main result is the low value of the density $n_{B}=6.4\times 10^{-5}$/fm3 in
resonance-gas approximation. We performed the fit considering the isotopes
with $Z\leq 2$. Larger LCP (”metals”) will be included below in the following
section IV. To show how sharp the fit value for the density is, we performed
calculations for the composition, mass fraction
$\tilde{X}_{A,Z}=AY_{A,Z}/\sum_{A^{\prime},Z^{\prime}\leq
2}A^{\prime}Y_{A^{\prime},Z^{\prime}}$, for fixed values of $\lambda_{T}$ and
proton fraction $\tilde{Y}_{p}=\sum_{A,Z\leq 2}ZY_{A,Z}/\sum_{A,Z\leq
2}AY_{A,Z}$ but variable density $\tilde{n}_{B}=\sum_{A,Z\leq 2}AY_{A,Z}$,
divided by the volume, see Fig. 1, where we compare the calculated values
(final,res.gas) with the observed values (obs). (Note that the densities and
proton fractions are calculated in the present section from the yields of $n$,
H, and He isotopes, dropping the contributions of the metals.)

As indicated above, all calculations presented in this section were performed
for the ideal resonance gas approximation, the simplest of the three cases
considered. The same calculations can be performed for the other
approximations of the intrinsic partition function as shown below. With
respect to the results discussed in the present section, no significant
changes in the Lagrange parameters will be observed. However the successive
approaches do reveal evidence for continuum and medium effects that even at
this very low density, particularly for weakly bound and/or very neutron rich
isotopes.

Figure 1:  Calculated light isotope yields (resonance-gas approximation) as a
function of baryon density $n_{B}$ (fixed temperature $T=1.289$ MeV and proton
fraction $Y_{p}=0.0349$) for $Z=1,2$ isotopes (represented by symbols) are
compared with the observed experimental yields represented by horizontal
dotted lines. Optimum agreement based on a fit metric proposed by J. Lestone
(3) is found at a density of $n_{B}=6.4\times 10^{-5}$ fm-3 (dashed line). See
Figure 2.

The quality of the fit is expressed by the Lestone fit metric (3), see Fig. 2.

Figure 2:  Fit metric (3) vs. baryon density $n_{B}$ for $T=1.289$ MeV and
$Y_{p}=0.0349$.

##  IV Inclusion of heavier isotopes

We now consider the yields of isotopes with $2<Z\leq 6$. For these elements,
accurate values for observed yields of ternary fission are available for of
241Pu($n_{\rm th}$,f) Koes1 ; Koes2 . For heavier elements $Z>6$ calculations
can be performed, but observed data become incomplete and less accurate.

For this purpose, we again calculate the relevant, primary distribution within
a quantum statistical approach using the several different approximations
proposed. In a first approximation of the ideal resonance gas, for each $A,Z$
not only the ground state, but also all excited states are considered which
will give the intrinsic partition function. We take the excited states from
the data tables Nudat  together with their degeneracy, see the appendix, Tabs.
4 \- 8. The possible decay to other isotopes is also indicated. For this, the
threshold energy $E^{\rm thresh}_{AZ}$ is given where the decay channels open,
in general the separation energy $S_{n}$ for neutrons, but also other possible
decay channels such as for 7Li to 4He + 3H or 8Be to 2$\times^{4}$He. In this
resonance gas approximation, all known excited states are considered, and
states above the threshold energy are assumed to decay and to feed other
isotopes in the final distribution.

The second approximation takes the states in the continuum more accurately
into account. The virial expansion implements continuum correlations in a
systematic way. This virial approximation is also known from the nuclear
matter equation of state Ropke82 ; SRS ; HS .

In the third approximation, in-medium modifications are taken into account.
These are the shifts of binding energies owing to Pauli blocking and possible
dissolution of bound states if they are shifted to the continuum R09 ; Typel10
; R20 .

###  IV.1 Excited states and resonance gas approximation

isotope | $A$ | $Z$ | $\frac{B_{A,Z}}{A}$ | $g_{A,Z}$ | $E^{\rm thresh}_{AZ}$ | $Y^{\rm obs}_{A,Z}$ |  $Y^{\rm obs}_{A,Z}$/$Y^{\rm final,res.gas}_{A,Z}$ |  $Y^{\rm obs}_{A,Z}$/$Y^{\rm final,vir}_{A,Z}$ |  $Y^{\rm obs}_{A,Z}$/$Y^{\rm final,medium}_{A,Z}$
---|---|---|---|---|---|---|---|---|---
1n | 1 | 0 | 0 | 2 | - | 0.107 | 0.9742 | 0.9555 | 0.9551
1H | 1 | 1 | 0 | 2 | - | - | - | - | -
2H | 2 | 1 | 1.112 | 3 | 2.224 | 8.463E-6 | 0.952 | 0.92 | 0.917
3H | 3 | 1 | 2.827 | 2 | 6.257 | 1.584E-4 | 1.133 | 1.179 | 1.223
4He | 4 | 2 | 7.073 | 1 | 20.577 | 2.015E-3 | 1.055 | 1.049 | 1.042
6He | 6 | 2 | 4.878 | 1 | 0.975 | 5.239E-5 | 0.8856 | 0.8753 | 0.8903
8He | 8 | 2 | 3.925 | 1 | 2.125 | 3.022E-6 | 1.043 | 1.04 | 1.03
7Li | 7 | 3 | 5.606 | 4 | 2.461 | 1.35E-6 | 0.5305 | 0.4964 | 0.4523
8Li | 8 | 3 | 5.160 | 5 | 2.038 | 8.463E-7 | 0.6235 | 0.5669 | 0.5211
9Li | 9 | 3 | 5.038 | 4 | 4.062 | 1.672E-6 | 0.7164 | 0.6363 | 0.5814
11Li | 11 | 3 | 4.155 | 4 | 0.396 | 9.068E-10 | 0.03061 | 0.02936 | 0.2992
9Be | 9 | 4 | 6.462 | 4 | 1.558 | 8.866E-7 | 0.5488 | 0.4908 | 0.3191
10Be | 10 | 4 | 6.497 | 1 | 6.497 | 9.269E-6 | 0.6149 | 0.5343 | 0.5285
11Be | 11 | 4 | 5.953 | 2 | 0.502 | 1.189E-6 | 0.2756 | 0.2584 | 0.4202
12Be | 12 | 4 | 5.721 | 1 | 3.17 | 5.642E-7 | 0.1513 | 0.1362 | 0.1385
14Be | 14 | 4 | 4.994 | 1 | 1.264 | 5.441E-10 | 0.01321 | 0.01326 | 0.0121
11B | 11 | 5 | 6.928 | 4 | 8.674 | 3.224E-7 | 0.3632 | 0.3015 | 0.2257
12B | 12 | 5 | 6.631 | 3 | 3.369 | 2.015E-7 | 0.1098 | 0.09418 | 0.08358
14B | 14 | 5 | 6.102 | 5 | 0.97 | 2.62E-8 | 0.01742 | 0.01552 | 0.01793
15B | 15 | 5 | 5.880 | 4 | 2.78 | 9.269E-9 | 0.01205 | 0.01005 | 0.008624
14C | 14 | 6 | 7.520 | 1 | 8.176 | 2.539E-6 | 0.0495 | 0.04598 | 0.02229
15C | 15 | 6 | 7.100 | 2 | 1.218 | 8.665E-7 | 0.01546 | 0.01353 | 0.01464
16C | 16 | 6 | 6.922 | 1 | 4.25 | 1.008E-6 | 0.01028 | 0.009781 | 0.007721
17C | 17 | 6 | 6.558 | 4 | 0.734 | 1.29E-7 | 0.002748 | 0.00241 | 0.005271
18C | 18 | 6 | 6.426 | 1 | 4.18 | 5.642E-8 | 0.001276 | 0.001078 | 0.0006891
19C | 19 | 6 | 6.118 | 2 | 0.58 | 5.038E-10 | 0.00003031 | 0.00002647 | 0.0002309
20C | 20 | 6 | 5.961 | 1 | 2.98 | 7.254E-10 | 0.0001937 | 0.0001578 | 0.0001285
$\lambda_{T}$ | - | - | - | - | - | - | 1.2897 | 1.2913 | 1.2924
$\lambda_{n}$ | - | - | - | - | - | - | -3.1486 | -3.1425 | -3.087
$\lambda_{p}$ | - | - | - | - | - | - | -16.273 | -16.236 | -16.189
volume | - | - | - | - | - | - | 1859.4 | 1876.4 | 1796.3
$n_{B}$ | - | - | - | - | - | - | 0.000064 | 0.0000645 | 0.00006741
$Y_{p}$ | - | - | - | - | - | - | 0.03486 | 0.03486 | 0.03479
fit metric | - | - | - | - | - | - | 0.005485 | 0.009384 | 0.009574
Table 3: Observed yields per fission of ternary fission of 241Pu($n_{\rm
th}$,f) are compared to a final state distribution, obtained from feeding of a
relevant (primary) distribution. The primary distribution was calculated in
the ideal resonance gas approximation (res.gas), as well as for the virial
form (vir) and including medium effects (medium). Lagrange parameters and
other properties are given at the bottom of the Table, the fit metric refers
to all isotopes with $Z\leq 2$. Units: MeV, fm. Figure 3:  Ternary fission of
241Pu($n_{\rm th}$,f): Ratios $Y^{\rm obs}_{A,Z}$/Y${}^{\rm
final,res.gas}_{A,Z}$ (bold symbols) and $Y^{\rm obs}_{A,Z}$/Y${}^{\rm
final,vir}_{A,Z}$ (full lines) as function of the mass number $A$. Isotopes
with $Z\leq 6$ are shown. Data from Tab. 3.

Data for excited states are shown in the appendix, Tabs. 4 to 8. In addition,
the threshold energy $E^{\rm thresh}_{AZ}$ is shown for the continuum of
scattering states. As indicated above, this is in general the neutron
separation energy $S_{n}$, but other decay channels (e.g. triton, $\alpha$)
are also possible. For each isotope of the primary distribution, different
final states are possible if excited, unstable states are considered. A
branching ratio would indicate the ratio to final state transitions during the
expansion after freeze out.

We use the simple approximation that all excited states below the continuum
edge decay to the ground state of the same isotope. States above the continuum
edge decay to other final isotopes. The same happens also with the unbound
nuclei like 4H, 5He. These states which feed down to other isotopes are marked
with an asterisk *, and the process is indicated ($\to^{3}$H for 4H, etc.).
Weakly bound states within 1 MeV below the threshold energy are marked with
”0”. In the present approximation of the ideal resonance gas, we assume that
these states contribute to the ground state final yield of the same isotope
after de-excitation. The corresponding factors $R_{A,Z}^{\rm
res.gas}(\lambda_{T})$ which are related to the intrinsic partition function
are calculated for $\lambda_{T}=1.29$ MeV in Tabs. 4 to 8. These partition
function multipliers are also shown in Tabs. 1, 2, together with the relevant,
primary yields $Y^{\rm rel,res.gas}_{A,Z}$, and the final yields $Y^{\rm
final,res.gas}_{A,Z}$.

To compare with the observed yields, results for the ratio $Y^{\rm
obs}_{A,Z}$/$Y^{\rm final,res.gas}_{A,Z}$ for this resonance gas approximation
are shown in Tab. 3 as well as in Fig. 3. The global behavior can be described
as follows: Up to $A=10$ the observed yields are rather well reproduced by the
grand canonical equilibrium calculations, for larger $A$ we see a strong
suppression, as has already been discussed in Sara ; rnp20 ; rnp21 .
Individual isotopes show deviations from the global behavior which may be
caused by the approximations in calculating the final distribution, i.e.,
neglecting interaction effects. This clearly bears further investigation.

###  IV.2 Virial expansion and continuum correlations

As previously indicated we have improved the equilibrium calculations in two
successive steps. First we consider the virial expansion which accounts for
continuum contributions. In the subsequent section, in-medium effects are also
taken into account. Results for the virial approximation are shown in Tab. 3
and Fig. 3.

The cluster-virial expansion considers continuum correlations of two
constituents of nuclear matter in terms of the scattering phase shifts, as
obtained in correspondence to the Beth-Uhlenbeck formula. For instance, 4H is
considered as resonance in the $t-n$ channel, where scattering phase shifts
have been measured, see R20 ; HS . For details see the appendix C. Data for
$R_{A,Z}^{\rm vir}(T)$ are shown in the appendix, Tabs. 4 to 8. The resulting
ratios $Y^{\rm obs}_{A,Z}$/$Y^{\rm final,vir}_{A,Z}$ are also shown in Tab. 3.

We see that the changes are small in general, so that the influence of the
continuum correlations on the final yield distribution is not essential. This
may be attributed to the low temperature so that the contribution of
scattering states is small for binding energies larger than $T$. At higher
$T$, the contribution of scattering states would become more important. The
most important changes are seen for 4H so that the feed-down to the final 3H
is reduced. Some He isotopes are also reduced. The optimum fit of Lagrange
parameters is also slightly changed. However, for several of the isotopes
considered, strong deviations from the global behavior remain.

As discussed in the case of 6He rnp21 , weakly bound states are more sensitive
to medium effects. The threshold energy of 11Li is low, and the yield is
strongly overestimated within the ideal gas and virial approaches. Obviously,
the measured yield of 11Li is not well described in the approximations where
the interaction between the components is neglected.

Low yields of other ’exotic’ nuclei (19C etc.) are also known, and are
observed in ternary fission of other actinides (Am, Cf, etc.). For the carbon
isotopes, the spectrum of excited states is rather complex and feeds different
final states. In particular, isotopes with excited states above the neutron
separation energy feed the yield of isotopes with $A-1$. Calculations of the
yields, see Figs. 3 and 4 in Koestera , cannot explain the low yield for 19C.
As seen there, the authors offer no explanation for the low yield of these
exotic neutron rich, halo-like nuclei.

###  IV.3 In-medium effects

Figure 4:  Ternary fission of 241Pu($n_{\rm th}$,f): Ratios $Y^{\rm
obs}_{A,Z}$/Y${}^{\rm final,res.gas}_{A,Z}$ (bold symbols) and $Y^{\rm
obs}_{A,Z}$/Y${}^{\rm final,medium}_{A,Z}$ (full lines) as function of the
mass number $A$. Isotopes with $Z\leq 6$ are shown. Data from Tabs. 3, 9, 10.

We now consider two medium effects which might modify the yield distributions:
Self-energy shifts and Pauli blocking. Self-energy shifts act on all nucleons
and can be accounted for by a renormalization of the chemical potentials
$\tilde{\lambda}_{n},\tilde{\lambda}_{p}$ if they are not momentum dependent.
Ratios of yields are not influenced by these shifts. In contrast, Pauli
blocking acts individually for each isotope and leads to strong deviations of
the yield distribution.

Self-energy of nucleons has been treated, for instance, within the
relativistic mean-field approximation. The well-established parametrization of
DD2-RMF Typel  gives for the parameter values $T=1.29$ MeV, $n_{B}=0.000067$
fm-3, $Y_{p}=0.035$ the self-energy shifts $\Delta_{n}^{\rm SE}=-0.03934$ MeV,
$\Delta_{p}^{\rm SE}=-0.09936$ MeV. For a cluster $\\{A,Z\\}$ the self-energy
shift is $\Delta E^{\rm SE}_{A,Z}=(A-Z)\Delta_{n}^{\rm SE}+Z\Delta_{p}^{\rm
SE}$. We then have $\tilde{\lambda}_{n}=\lambda_{n}-\Delta_{n}^{\rm
SE},\tilde{\lambda}_{p}=\lambda_{p}-\Delta_{p}^{\rm SE}$.

In all our earlier fits rnp21 , the yields of 6He are overestimated, whereas
the yields of 8He are underestimated. One possible explanation is that 6He is
only weakly bound ($E^{\rm thresh}=0.975$ MeV) compared to 8He ($E^{\rm
thresh}=2.125$ MeV). In the dense medium, binding energies are shifted, and
weakly bound states are more affected than strongly bound states.

More visible in Fig. 3 is the strong suppression of weakly bound isotopes such
as 11Li and 19C. These isotopes are near to the continuum edge so that the
shift owing to Pauli blocking may lead to dissolution (the Mott effect Ropke82
). This suggests the use of yields of weakly bound states as test probes to
infer the free neutron density. The observation of such Mott effects would
provide us with an independent test of the free neutron density at scission.

To quantify this effect, we consider the bound state contribution to the
intrinsic partition function which contains the factor $e^{-E^{0}_{A,Z}/T}-1$
rnp20 . If the bound state energy is shifted owing to in-medium effects, this
factor is also changed and goes to zero at the Mott density where bound states
disappear.

The Pauli blocking shift of bound states in dense matter and the Mott effect
has previously been considered in detail for individual isotopes Ropke82 ; R09
; R20 . We will give here only a general estimate applicable to bound states
of all nuclei. The shift of bound state energies
$E_{A,Z}(T,n_{B},Y_{p})-E^{0}_{A,Z}$ in a dense medium (Pauli blocking) has
been estimated in R20 , Eq. (22), for nucleons in the $1s$ and $2p$ orbits as

$\Delta E_{A,Z}^{\rm Pauli}=1064(A-Z)n_{B}e^{-0.0513/T}{\rm MeV\,fm}^{3}$ (4)

where we assume that the proton contribution is negligible because of its very
low density. With the baryon density $n_{B}=6.7\times 10^{-5}$ fm-3 (which
should be determined self-consistently), the shifts of the binding energy of
the different isotopes are given in Tabs. 9 and 10. These shifts may become
important for weakly bound states. For instance, for 11Li the shift is larger
than the threshold energy so that the bound state is dissolved. This is also
reflected in the low observed yield of this isotope. In a more detailed
calculation R11  not considered here, the Pauli blocking shift depends on the
center-of-mass momentum $P$ of the cluster. Because the shift becomes smaller
for larger total momentum $P$, and continuum correlations may be present, a
small yield of this isotope remains. The same happens also with 19C where the
shift is also larger than the threshold energy so that the Mott effect leads
to a significant reduction of the observed yields. This is also seen in Fig.
4. The yields of other isotopes with a low threshold energy ($E^{\rm
thresh}_{AZ}<1$ MeV) will be significantly influenced too. The excited states
11Be0, 14B0, 15C${}^{0}-^{17}$C0 become dissolved because of Pauli blocking.
The corresponding calculations are seen in Tabs. 9, 10.

As shown from Tab. 3 and Fig. 4, the observed yields are, in fact, sensitive
to in-medium effects. Suppression effects owing to Pauli blocking seem to be
visible. This allows an independent second determination of the baryon
density, and the densities at which such isotopes which are dissolved are
consistent with the inferred value $n_{B}=6.7\times 10^{-5}$ fm-3. Note that a
fully ab initio calculation of the Pauli blocking effects is very involved,
and we gave here only exploratory calculations to illustrate the effect of
suppression.

##  V Generalized relativistic mean-field approach

As discussed above, important in medium effects are self-energy shifts and
Pauli blocking. If the self-energy shifts are not dependent on the nucleon
momentum, they can be absorbed into the chemical potential thus creating an
effective chemical potential. Different approximations my be used to describe
the in-medium self-energy shifts, for instance the Skyrme or the relativistic
mean-field approaches.

In Ref. Pais1 , Pais et al. reported a generalized relativistic mean-field
(RMF) approach formulated for the study of in-medium modifications on light
cluster properties. Explicit binding energy shifts and a modification of the
scalar cluster-meson coupling were introduced in order to take these medium
effects into account. The interactions of the clusters $i=\,^{2}$H, 3H, 3He,
4He with the surrounding medium are described with a phenomenological
modification, $x_{s}$, of the coupling constant to the $\sigma$ meson,
$g_{\sigma_{i}}=x_{s}A_{i}g_{\sigma}$, where $g_{\sigma}$ is the nucleon
scalar coupling, and $A_{i}$ the number of nucleons in cluster $i$. Using the
FSU Gold EoS FSU , and requiring that the cluster fractions exhibit the
correct behavior in the low-density virial limit virial ; Horowitz06 , they
obtained a universal scalar cluster-meson coupling fraction,
$x_{\sigma_{i}}=0.85\pm 0.05$, which could reproduce both this limit and the
equilibrium constants extracted from reaction ion data Qin12  reasonably well.
Later this work was generalized to include clusters up to $A=12$ Pais3 .

In a more recent work PaisExp , Pais et al. compared results of an analysis,
where in-medium effects are included, to experimental equilibrium constants
measured in intermediate energy Xe + Sn collisions. This comparison required a
higher scalar cluster-meson coupling constant $x_{\sigma_{i}}=0.92\pm 0.02$.
With this higher assumed value of the coupling constant, the in-medium effects
are reduced, and the clusters melt at larger densities.

In the top panel of Fig. 5, we show the mass fractions of the clusters with
$Z=1,2$ and also the fraction of the free neutron gas, calculated using the
RMF formalism with $x_{s}=0.85$ , as a function of the density, for the
temperature and proton fraction found in the QS approach with in-medium
effects described in the previous subsection. In the bottom panel, the mass
fractions of light particles are plotted against the density, for different
values of the scalar cluster-meson coupling. We observe that the abundances
shown in the top panel are quite similar to the ones in the resonance gas
approximation (see Fig. 1). Also, from the bottom panel, we see that the
medium effects are almost negligible because the densities are very small. As
it was shown in Tab. III, the effect of the medium is more relevant when
considering clusters with higher $Z$.

---
Figure 5: (Top) Mass fractions as a function of the density for the $Z=1,2$
isotopes and the free neutron gas in a RMF calculation with in-medium effects
($x_{s}=0.85$) for the FSU model with a fixed temperature of $T=1.29$ MeV, and
a proton fraction of $Y_{p}=0.035$. The observed experimental yields are
represented by horizontal dotted lines. (Bottom) Mass fractions of the light
particles as a function of the density in a RMF calculation employing
different values of the scalar-cluster-meson coupling constant.

In Fig. 6, we present the calculated fit metrics derived with different values
of the scalar cluster-meson coupling constant, and we observe a minimum at
about the same density as in the QS approach, $n_{B}\sim 6.7\times 10^{-5}$
fm-3.

Figure 6:  Fit metric (3) vs. baryon density for $T=1.29$ MeV and
$Y_{p}=0.035$, using different values of the scalar cluster-meson coupling.

##  VI Conclusions

We have considered the modeling of ternary fission yields within a systematic
quantum statistical approach and a generalized relativistic mean field
approach. We consider our quantum statistical approach as a first step in a
strict non-equilibrium approach. As an example, we have analyzed the observed
yields for 241Pu($n_{\rm th}$,f).

We assume that the light charged fragments are emitted from the neck region at
scission. We have first considered partial chemical equilibrium for clusters
$Z\leq 2$. During the further expansion, feed-down processes were considered.
This leads to the final yields which are compared to the observed yields.
Higher mass clusters are suppressed by nucleation kinetics and/or size
effects.

We are interested in an optimal description of the primary equilibrium
distribution at scission. Within a quantum statistical approach, excited bound
states and continuum correlations are taken into account. In particular, we
have searched for in-medium effects going beyond a simple statistical model of
non-interacting components (ideal gas). These in-medium effects are determined
by the thermodynamic parameters of matter in the scission region. For all
nuclei $Z\leq 2$, the Lagrange parameter values are:
$\lambda_{T}=1.29,\,\,\lambda_{n}=-3.09,\,\,\lambda_{p}=-16.19$ MeV, volume
1796 fm3, neutron density $n_{n}=6.7\times 10^{-5}$ fm-3. We showed that for
those parameter values, density effects are expected and are manifested by the
experimental data.

In particular we find evidence for the Mott effect for the weakly bound states
11Li and 19C, and also the suppression of the yields for weakly bound nuclei
6He, 11Be, 14B, and 15C $-^{17}$C (excited states). This shows the consistency
of our approach: the nucleon density $6.7\times 10^{-5}$ fm-3 inferred from
composition is also seen in the in-medium shifts, the weakly bound clusters
serving as test probes. We identify the strong suppression of 11Li and 19C as
a signature of the Mott effect and find that other isotopes are not dissolved
so that we find an estimate for the Pauli blocking effect and the
corresponding interval of density.

We also considered a generalized relativistic mean-field approach to study the
in-medium modifications on light cluster properties, introducing a
modification on the scalar cluster-meson coupling. At the low nucleon
densities considered here, no significant changes in the yield distribution of
isotopes and the nuclear-matter parameter values have been observed. This
underlines the validity of the parameter values for temperature, nucleon
density and proton fraction, presented in this article, to describe nuclear
matter in the neck region at scission.

The improved description of nuclear matter at scission conditions is of
interest for the properties of neutron rich astrophysical systems Bezn20 ;
Fischer ; Furusawa22 ; Gulm ; Lim ; Du ; Sedr , for instance the equation of
state or the neutrino opacity, at similar thermodynamic parameter values.

## ACKNOWLEDGMENTS

This work was supported by the United States Department of Energy under Grant
No. DE-FG03-93ER40773. It was partly supported by the FCT (Portugal) Project
No. UID/FIS/04564/2020. H.P. acknowledges the grant CEECIND/03092/2017 (FCT,
Portugal). G.R. acknowledges support by the German Research Foundation (DFG),
Grant # RO905/38-1.

##  Appendix A Excited state multiplier and intrinsic partition function

The contribution of excited states to the intrinsic partition function of a
particular channel $\\{A,Z\\}$ which after expansion (feed-down) decays to a
different final isotope, observed in experiment (branching ratio), is denoted
by *. We separate also the weakly bound states which are within 1 MeV below
the edge of continuum. We denote it by ”0” below. Sometimes we separate the
ground state so that ”0” appears twice.

The threshold energy $E^{\rm thresh}_{AZ}$ is generally given by the neutron
separation energy, but lower threshold energies may appear for other
decompositions, for instance 6Li $\to\alpha+d$ and 7Li $\to\alpha+t$, see
Jesinger05 . This defines feed down channels and the corresponding
subdivisions of the intrinsic partition function.

The corresponding partition function multipliers $R^{\rm
res.gas}_{AZ}(\lambda_{T})$ and $R^{\rm vir}_{AZ}(\lambda_{T})$ are calculated
for $\lambda_{T}=1.29$ MeV. The resulting yields $Y^{\rm rel,vir}_{A,Z}$ are
given in Tabs. 4 \- 8 for the elements up to carbon, see also the Supplemental
material of Ref. rnp21 .

###  A.1 $Z\leq 2$: n, H, He

isotope | feeddown | $A$ | $Z$ | $B_{A,Z}/A$ | $g_{A,Z}$ | $E^{\rm thresh}_{AZ}$ | $E_{i}\,\,[g_{i}]$ | $R_{A,Z}^{\rm res.gas}(1.29)$ | $R_{A,Z}^{\rm vir}(1.29)$ | $R_{A,Z}^{\rm medium}(1.29)$
---|---|---|---|---|---|---|---|---|---|---
1n | - | 1 | 0 | 0 | 2 | - | - | 1 | 1 | 1
1H | - | 1 | 1 | 0 | 2 | - | - | 1 | 1 | 1
2H | - | 2 | 1 | 1.112 | 3 | 2.224 | - | 1 | 0.974 | 0.9306
3H | - | 3 | 1 | 2.827 | 2 | 6.257 | - | 1 | 0.999 | 0.9138
4H* |  $\to^{3}$H | 4 | 1 | 1.720 | 5 | -1.6 | 0.31 [3], 2.08 [1], 2.83 [3] | 1.579 | 0.09404 | 0.03564
3He | - | 3 | 2 | 2.573 | 2 | 5.494 | - | 1 | 0.9979 | 0.9545
4He | - | 4 | 2 | 7.073 | 1 | 20.577 | 20.21 [1] | 1 | 1 | 0.915
5He* |  $\to^{4}$He | 5 | 2 | 5.512 | 4 | -0.735 | - | 1 | 0.6906 | 0.5029
6He0 | [$\to^{4}$He] | 6 | 2 | 4.878 | 1 | 0.975 | - | 1 | 0.9343 | 0.7723
6He* |  $\to^{4}$He | 6 | 2 | 4.878 | 1 | 0.975 | 1.797 [5] | 1.242 | 0.7919 | 0.4617
7He* |  $\to^{6}$He | 7 | 2 | 4.123 | 4 | -0.410 | 2.92 [6] | 1.156 | 0.9313 | 0.6577
8He | - | 8 | 2 | 3.925 | 1 | 2.125 | - | 1 | 0.972 | 0.7383
8He* |  $\to^{4}$He | 8 | 2 | 3.925 | 1 | 2.125 | 3.1 [5] | 0.452 | 0.2334 | 0.06923
9He* |  $\to^{8}$He | 9 | 2 | 3.349 | 2 | -1.25 | 1.1 [2] | 1.426 | 0.364 | 0.04882
Table 4: Data of nucleons/nuclei $Z\leq 2$ [units: MeV, fm]. Feed-down to
other final isotopes indicated by $\to^{A}$Z. Mass number $A$, charge number
$Z$. Ground state binding energy $B_{A,Z}$ and degeneracy $g_{A,Z}=2J+1$,
continuum threshold energy $E^{\rm thresh}_{AZ}$, excitation energy $E_{i}$
and degeneracy $g_{i}$ according Nudat . Partition function multiplier
$R_{A,Z}^{\rm res.gas}(T)$ according (11). The contribution of excited states
above the continuum threshold is denoted by ”*”, the contribution of states
within 1 MeV below the threshold is denoted by ”0” (weakly bound states).

Excited states and intrinsic partition function multipliers $R$ of H and He
isotopes are shown in Tab. 4. The contribution of excited states which after
freeze-out decays to a different channel $\\{A,Z\\}$ (feed-down) is denoted by
∗. At low temperatures considered in this work, the subnuclear excitations of
the nucleons $n,p$ are neglected so that $R_{1,0}=R_{1,1}=1$.

###  A.2 Lithium

isotope | feeddown | $A$ | $Z$ | $B_{A,Z}/A$ | $g_{A,Z}$ | $E^{\rm thresh}_{AZ}$ | $E_{i}\,\,[g_{i}]$ | $R_{A,Z}^{\rm res.gas}(1.29)$ | $R_{A,Z}^{\rm vir}(1.29)$ | $R_{A,Z}^{\rm medium}(1.29)$
---|---|---|---|---|---|---|---|---|---|---
6Li | - | 6 | 3 | 5.332 | 3 | 1.475 | - | 1 | 0.9544 | 0.83
6Li* |  $\to^{3}$H | 6 | 3 | 5.332 | 3 | 1.475 | 2.186 [7], 3.563 [1], 4.312 [5], 5.366 [5], 5.65 [3] | 0.5471 | 0.3847 | 0.2842
7Li | - | 7 | 3 | 5.606 | 4 | 2.461 | 0.478 [2] | 1.345 | 1.316 | 1.097
8Li | - | 8 | 3 | 5.160 | 5 | 2.038 | 0.980 [3] | 1.281 | 1.242 | 0.9877
8Li* |  $\to^{7}$Li | 8 | 3 | 5.160 | (5) | 2.038 | 2.255 [7], 3.210 [3], 5.4 [3] , 6.1 [7] | 0.3151 | 0.2646 | 0.1975
9Li | - | 9 | 3 | 5.038 | 4 | 4.062 | 2.691 [2], (4.301) [?] | 1.062 | 1.055 | 0.8069
10Li* |  $\to^{9}$Li | 10 | 3 | 4.531 | (3) | -0.032 | - | 1 | 0.863 | 0.5866
11Li0 | [$\to^{9}$Li] | 11 | 3 | 4.155 | 4 | 0.396 | (1.266) [?] | 1 | 0.9003 | 0.6026
12Li* |  $\to^{11}$Li | 12 | 3 | 3.792 | (3) | -0.201 | - | 1 | 0.8422 | 0.4697
Table 5: Data of lithium nuclei [units: MeV, fm]. Mass number $A$, charge
number $Z$. Ground state binding energy $B_{A,Z}$ and degeneracy $g_{A,Z}$,
excitation energy $E_{i}$ and degeneracy $g_{i}$ as well as continuum
threshold energy $E^{\rm thresh}_{AZ}$ according Nudat . Partition function
multiplier $R_{A,Z}^{\rm res.gas}(\lambda_{T})$ for $\lambda_{T}=1.29$ MeV.
7Li: $E^{\rm thresh}_{AZ}$ for decay to 4He+3H; excited states at 4.630 MeV
[8], 6.680 MeV [6] decay this way, but also IT.

The yields of Li isotopes are shown in Tab. 5. We consider $6\leq A\leq 12$
The degeneracy is not known for 10Li and 12Li, the value 3 was assumed, but is
not of relevance for the discussion here. The same holds also for some excited
states. The threshold energy $E^{\rm thresh}_{AZ}$ is given in general by the
neutron separation energy, but lower threshold energies appear for 6Li
$\to\alpha+d$ and 7Li $\to\alpha+t$, see Jesinger05 .

###  A.3 Beryllium

isotope | feeddown | $A$ | $Z$ | $B_{A,Z}/A$ | $g_{A,Z}$ | $E^{\rm thresh}_{AZ}$ | $E_{i}\,\,[g_{i}]$ | $R_{A,Z}^{\rm res.gas}(1.29)$ | $R_{A,Z}^{\rm vir}(1.29)$ | $R_{A,Z}^{\rm medium}(1.29)$
---|---|---|---|---|---|---|---|---|---|---
7Be | - | 7 | 4 | 5.372 | 4 | 1.585 | 0.429 [2] | 1.359 | 1.302 | 1.1324
8Be* |  $\to^{4}$He | 8 | 4 | 7.062 | 1 | -0.088 | 3.03 [5] | 1.477 | 1.266 | 1.0192
9Be | - | 9 | 4 | 6.462 | 4 | 1.558 | - | 1 | 0.9572 | 0.7585
9Be* |  $\to^{4}$He | 9 | 4 | 6.462 | (4) | 1.558 | 1.684 [2], 2.429 [6], 2.78 [2], 3.049 [6] | 0.5628 | 0.4796 | 0.3629
10Be | - | 10 | 4 | 6.497 | 1 | 6.497 | 3.368 [5] | 1.367 | 1.366 | 1.0462
10Be0 | [$\to^{9}$Be] | 10 | 4 | 6.497 | (1) | 6.497 | 5.958 [5], 5.959 [3] | 0.0789 | 0.07181 | 0.05348
11Be0 | [$\to^{10}$Be] | 11 | 4 | 5.953 | 2 | 0.502 | - | 1 | 0.9076 | 0.6422
11Be0 | [$\to^{10}$Be] | 11 | 4 | 5.953 | (2) | 0.502 | 0.320 [2] | 0.7803 | 0.6894 | 0.4802
11Be* |  $\to^{10}$Be | 11 | 4 | 5.953 | (2) | 0.502 | 1.783 [6], 2.654 [4], 3.4 [4], 3.889 [4], 3.955 [4] | 1.343 | 0.308 | 0.03957
12Be | - | 12 | 4 | 5.721 | 1 | 3.170 | 2.109 [5], 2.251 [1] | 2.149 | 2.122 | 1.4798
12Be0 | [$\to^{10}$Be] | 12 | 4 | 5.721 | (1) | 3.170 | 2.715 [3] | 0.3657 | 0.3307 | 0.222
13Be* |  $\to^{12}$Be | 13 | 4 | 5.241 | 2 | -0.510 | - | 1 | 0.7806 | 0.3144
14Be | - | 14 | 4 | 4.994 | 1 | 1.264 | - | 1 | 0.9468 | 0.5897
15Be* |  $\to^{14}$Be | 15 | 4 | 4.541 | 6 | -1.800 | - | 1 | 0.02241 | 0.0005243
Table 6: Data of beryllium nuclei [units: MeV, fm]. Mass number $A$, charge
number $Z$, observed yields $Y^{\rm obs}_{A,Z}$ Koes1 . Ground state binding
energy $B_{A,Z}$ and degeneracy $g_{A,Z}$, excitation energy $E_{i}$ and
degeneracy $g_{i}$ as well as continuum threshold energy $E^{\rm thresh}_{AZ}$
according Nudat . Excited state multiplier $R_{A,Z}^{\rm
excited}(\lambda_{T})$ for $\lambda_{T}=1.29$ MeV. Weekly bound isotopes
10Be0, 11Be0, and 12Be0 are separated, they will merge with the continuum at
higher densities.

Results for Be isotopes are shown in Tab. 6. For the relevant, primary
distribution we have to consider also the unstable, excited states. Excited
states above the continuum edge which emit a neutron or other nuclei are taken
separately, their contribution to the intrinsic partition function is denoted
by an asterisk. Also excited states little below the continuum edge (1 MeV)
may become dissolved in a dense medium if the binding energy is reduced owing
to Pauli blocking, their contribution is denoted by ”0”.

###  A.4 Boron

isotope | feeddown | $A$ | $Z$ | $B_{A,Z}/A$ | $g_{A,Z}$ | $E^{\rm thresh}_{AZ}$ | $E_{i}\,\,[g_{i}]$ | $R_{A,Z}^{\rm res.gas}(1.29)$ | $R_{A,Z}^{\rm vir}(1.29)$ | $R_{A,Z}^{\rm medium}(1.29)$
---|---|---|---|---|---|---|---|---|---|---
10B | - | 10 | 5 | 6.475 | 7 | 4.466 | 0.718 [3], 1.740 [1], 2.154 [3] | 1.363 | 1.357 | 1.086
10B0 | - | 10 | 5 | 6.475 | (7) | 4.466 | 3.587 [5] | 0.0443 | 0.04117 | 0.0324
11B | - | 11 | 5 | 6.928 | 4 | 8.674 | 2.124 [2], 4.445 [6], 5.020 [4], 6.741 [8] | 1.175 | 1.175 | 0.9
12B | - | 12 | 5 | 6.631 | 3 | 3.369 | 0.953 [5], 1.674 [5] | 2.251 | 2.227 | 1.626
12B0 | [$\to^{11}$B] | 12 | 5 | 6.631 | 3 | 3.369 | 2.621 [3], 2.723 [1] | 0.1715 | 0.1582 | 0.1127
13B | - | 13 | 5 | 6.496 | 4 | 4.879 | 3.482 [?], 3.535 [?], 3.681 [?], 3.712 [?], 4.131 [?] | 1 | 0.9966 | 0.6976
14B0 | - | 14 | 5 | 6.102 | 5 | 0.970 | - | 1 | 0.9341 | 0.6065
14B0 | [$\to^{13}$B] | 14 | 5 | 6.102 | (5) | 0.970 | 0.740 [3] | 0.3381 | 0.3001 | 0.188
14B* |  $\to^{13}$B | 14 | 5 | 6.102 | (5) | 0.970 | 1.38 [7] | 0.4803 | 0.387 | 0.1801
15B | - | 15 | 5 | 5.880 | 4 | 2.78 | (?) | 1 | 0.983 | 0.6246
16B* |  $\to^{15}$B | 16 | 5 | 5.507 | 1 | -0.082 | (?) | 1 | 0.8574 | 0.4311
17B | - | 17 | 5 | 5.270 | 4 | 1.39 | (?) | 1 | 0.9515 | 0.5399
18B* |  $\to^{17}$B | 18 | 5 | 4.977 | 5 | -0.005 | (?) | 1 | 0.8659 | 0.3832
Table 7: Data of boron nuclei [units: MeV, fm]. Mass number $A$, charge number
$Z$. Ground state binding energy $B_{A,Z}$ and degeneracy $g_{A,Z}$,
excitation energy $E_{i}$ and degeneracy $g_{i}$ as well as continuum
threshold energy $E^{\rm thresh}_{AZ}$ according Nudat . Excited state
multiplier $R_{A,Z}^{\rm excited}(\lambda_{T})$ for $\lambda_{T}=1.29$ MeV.

Results for B isotopes are shown in Tab. 7. As before, we denoted the weakly
bound part of the intrinsic partition function with 10B0, 12B0, and 14B0. 16B
feeds 15B, and 18B feeds 17B. It is expected that continuum correlations and
in-medium effects may give further corrections to describe the final yields of
the isotopes.

###  A.5 Carbon

isotope | feeddown | $A$ | $Z$ | $B_{A,Z}/A$ | $g_{A,Z}$ | $E^{\rm thresh}_{AZ}$ | $E_{i}\,\,[g_{i}]$ | $R_{A,Z}^{\rm res.gas}(1.29)$ | $R_{A,Z}^{\rm vir}(1.29)$ | $R_{A,Z}^{\rm medium}(1.29)$
---|---|---|---|---|---|---|---|---|---|---
13C | - | 13 | 6 | 7.469 | 2 | 4.946 | 3.089 [2], 3.684 [4], 3.853 [6] | 1.358 | 1.353 | 0.9905
14C | - | 14 | 6 | 7.520 | 1 | 8.176 | 6.093 [3], 6.589 [7] | 1.069 | 1.069 | 0.7491
15C | - | 15 | 6 | 7.100 | 2 | 1.218 | - | 1 | 0.945 | 0.6169
15C0 | [$\to^{14}$C] | 15 | 6 | 7.100 | (2) | 1.218 | 0.740 [6] | 1.69 | 1.532 | 0.9775
15C* |  $\to^{14}$C | 15 | 6 | 7.100 | (2) | 1.218 | 3.103 [2], 4.220 [6], 4.657 [4], 4.780 [4] | 0.3074 | 0.004482 | 0.000206
16C | - | 16 | 6 | 6.922 | 1 | 4.250 | 1.766 [5] | 2.272 | 2.26 | 1.445
16C0 | [$\to^{15}$C] | 16 | 6 | 6.922 | (1) | 4.250 | 3.986 [5], 4.089 [7], 4.142 [9] | 0.885 | 0.7875 | 0.4683
17C0 | [$\to^{16}$C] | 17 | 6 | 6.558 | 4 | 0.734 | - | 1 | 0.9218 | 0.5379
17C0 | [$\to^{16}$C] | 17 | 6 | 6.558 | (4) | 0.734 | 0.217 [2], 0.332 [6] | 1.582 | 1.438 | 0.8288
17C* |  $\to^{16}$C | 17 | 6 | 6.558 | (4) | 0.734 | 2.15 [8], 2.71 [2], 3.085 [10] | 0.6677 | 0.09067 | 0.002612
18C | - | 18 | 6 | 6.426 | 1 | 4.18 | 1.588 [5], 2.515 [5] | 3.172 | 3.153 | 1.843
19C0 | [$\to^{18}$C] | 19 | 6 | 6.118 | 2 | 0.580 | 0.209 [4], 0.282 [6] | 5.112 | 4.665 | 2.431
19C* |  $\to^{18}$C | 19 | 6 | 6.118 | (2) | 0.580 | 0.653 [6], 1.46 [6] | 2.776 | 2.383 | 0.9985
20C | - | 20 | 6 | 5.961 | 1 | 2.98 | 1.618 [5] | 2.426 | 2.391 | 1.268
Table 8: Data of carbon nuclei [units: MeV, fm]. Mass number $A$, charge
number $Z$. Ground state binding energy $B_{A,Z}$ and degeneracy $g_{A,Z}$,
excitation energy $E_{i}$ and degeneracy $g_{i}$ as well as continuum
threshold energy $E^{\rm thresh}_{AZ}$ according Nudat . Excited state
multiplier $R_{A,Z}^{\rm vir}(\lambda_{T})$ for $\lambda_{T}=1.29$ MeV.

For carbon, several states are close to the edge of the continuum, see Tab. 8.
In particular, 19C is weakly bound, as is 17C, with the threshold of the
continuum below 1 MeV. We expect that these weakly bound nuclei are stronger
influenced by in-medium modifications, in particular Pauli blocking.

##  Appendix B Estimate for the Pauli blocking shift

There are two contributions to the in-medium shift of quasiparticles, the
self-energy contribution and the Pauli blocking effects. Self-energy shifts
are calculated, e.g., by the relativistic mean-field approximation. If the
momentum dependence of the shift is neglected, each nucleon suffers the same
shift so that it can be implemented in the chemical potential. The composition
is not influenced. For a more detailed discussion see R09 ; R11 .

We give an estimate for the Pauli blocking $\Delta E^{\rm Pauli}_{A,Z}({\bf
P};T,n_{B},Y_{p})$ as function of total momentum $\hbar{\bf P}$ and the
thermodynamic variables $T,\mu_{n},\mu_{p}$ respectively $T,n_{B},Y_{p}$ as

$\displaystyle\Delta E^{\rm Pauli}_{A,Z}({\bf P};T,n_{B},Y_{p})$
$\displaystyle=$ $\displaystyle
e^{-\frac{\hbar^{2}P^{2}}{2A^{2}mT}}2\,(Nn_{n}+Zn_{p})$ (5)
$\displaystyle\times a_{A,Z}e^{-b_{A,Z}T}$

according to Eq. (21) in Ref. R20 , where also values for the parameters
$a_{A,Z},b_{A,Z}$ are given. An average value for $10\leq A\leq 16$ is
$\bar{a}=532.0$ MeV fm3 and $\bar{b}=0.05103$ MeV-1. The $\bf P$ dependence is
according Eq. (43) of Ref. R09 .

Within an exploratory calculation, we use the average value to estimate the
Pauli blocking shift. Values for $\Delta E^{\rm Pauli}_{A,Z}({\bf
P}=0;T,n_{B},Y_{p})=\Delta E^{\rm Pauli}_{A,Z}(0)$ are shown in Tabs. 9, 10.
The Mott condition where the Pauli blocking shift exceeds the binding energy
is fulfilled for 11Li and 19C, but also for excited states of 11Be, 14B, 15C,
16C, and 17C. Further ground states and excited states, denoted by ”0”, are
also approaching the continuum edge and partially suppressed.

The Mott condition does not mean that all clusters of the corresponding state
are dissolved. The transition to the continuum gives also a contribution like
a resonance, see the case of 2H. An important point is that the Pauli blocking
depends on the total momentum $\hbar{\bf P}$, see R11 . For $\Delta E^{\rm
Pauli}_{A,Z}(0)\geq B_{A,Z}$, the Mott momentum follows as

$P^{\rm Mott}=\left\\{2A^{2}\frac{Tm}{\hbar^{2}}\ln\left[\frac{\Delta E^{\rm
Pauli}_{A,Z}(0)}{B_{A,Z}}\right]\right\\}^{1/2}.$ (6)

Not only the high-momentum bound states but also part of the continuum states
may evolve to the ground state in the final distribution. However, this leads
to the discussion of branching ratios for the reaction network describing the
evolution from freeze-out to the final observed yields, what is beyond the
scope of the present work.

##  Appendix C Evaluation of the intrinsic partition function multipliers $R$:
Virial approximation

In (1), the intrinsic partition function multiplier was introduced as a
correction to the simple ideal gas approximation where for each isotope
$\\{A,Z\\}$ only the ground state was considered. However, for each channel we
have to consider the intrinsic partition function, in particular the account
of all excited states. The resonance gas approximation (2) considers all
excited, including unstable, states, also resonances in the continuum.

A systematic description of correlations in the continuum is given by the
virial approximation. We repeat: we perform a cluster expansion for the
interacting system R20 , and partial densities of different clusters
$\\{A,Z\\}$ are introduced. The relevant yields $Y^{\rm rel,vir}_{A,Z}$ in the
virial approximation are calculated as

$\displaystyle Y^{\rm rel,vir}_{A,Z}$ $\displaystyle\propto$ $\displaystyle
g_{A,Z}\left(\frac{2\pi\hbar^{2}}{Am\lambda_{T}}\right)^{-3/2}\times$ (7)
$\displaystyle e^{(B_{A,Z}+(A-Z)\lambda_{n}+Z\lambda_{p})/\lambda_{T}}\,R^{\rm
vir}_{A,Z}(\lambda_{T})$

(nondegenerate limit), where $B_{A,Z}$ denotes the (ground state) binding
energy and $g_{A,Z}$ the degeneracy Nudat . The factor

$\displaystyle R^{\rm vir}_{A,Z}(\lambda_{T})=1+\sum^{\rm
exc}_{i}[g_{AZ,i}/g_{A,Z}]e^{-E_{AZ,i}/\lambda_{T}}$ (8)

is related to the intrinsic partition function of the cluster $\\{A,Z\\}$. The
index $i$ characterizes further quantum numbers of the excited state such as
angular momentum. The summation is performed over all excited states of
excitation energy $E_{AZ,i}$ and degeneracy $g_{AZ,i}$ Nudat .

As already discussed for H, He, we have also to consider the scattering
states. This leads to a change in the contributions obtained for bound states,
and, in particular, for the unbound states. We use the Beth-Uhlenbeck
relation, see rnp20  to perform the sum over the continuum states,

$\displaystyle\\!\\!\\!\\!\\!\\!C^{\rm vir}_{AZ,i}(\lambda_{T})=[1-e^{-(E^{\rm
thresh}_{A,Z}-E_{AZ,i})/\lambda_{T}}]\Theta(E^{\rm thresh}_{A,Z}-E_{AZ,i})$
$\displaystyle\\!\\!\\!\\!\\!\\!+\frac{1}{\pi\lambda_{T}}e^{-(E^{\rm
thresh}_{A,Z}-E_{AZ,i})/\lambda_{T}}\int_{0}^{\infty}dEe^{-E/\lambda_{T}}\delta_{AZ,i}(E)),$
(9)

as prefactor for the contribution of the different channels $\\{A,Z,i\\}$,
degeneracy $g_{AZ,i}$. $E^{\rm thresh}_{A,Z}$ is the difference of the binding
energy of the ground state and the binding energy of the components of the
lowest continuum, the $\Theta$ function is 1 if a bound state occurs. If a
channel has no common bound state, we have $E_{AZ,i}=0$, all excited states
are in the continuum. The intrinsic partition function multiplier $R^{\rm
vir}_{A,Z}(\lambda_{T})$ (8) can be rewritten as

$\displaystyle\\!\\!\\!R^{\rm vir}_{A,Z}(\lambda_{T})=\sum^{\rm
exc}_{i}\frac{g_{AZ,i}}{g_{A,Z}}e^{-E_{AZ,i}/\lambda_{T}}C_{A,Z}^{\rm
vir}(\lambda_{T},E_{AZ,i}).$ (10)

As in the case of the resonance gas approximation, the sum over all states $i$
can be subdivided into different contributions $\alpha$,

$\displaystyle R^{\rm vir}_{A,Z}(\lambda_{T})=\sum_{\alpha}R^{{\rm
vir},\alpha}_{A,Z}(\lambda_{T}).$ (11)

We divided it into the contribution of well-bound nuclear states with binding
energy larger than 1 MeV ($R$) and weakly bound states, ($R^{0}$), both
decaying to the ground state, and in unbound states ($R^{*}$) which feed down
to other channels. The same subdivision has been performed also for the
intrinsic partition function multiplier $R^{\rm res.gas}_{A,Z}(\lambda_{T})$.
We perform this subdivision to calculate the feed-down process from freeze-out
to the final yields.

We are not able here to evaluate the integral over the scattering phase shifts
for all channels of interest. We give only an estimate in analogy of the cases
2H, 4H, 5He which have been discussed in Ref. R20 . The relation $R^{\rm
vir}_{AZ}(T)=e^{(E^{\rm thresh}_{A,Z}-E^{\rm eff}(T))/T}$ was given there. We
use the effective energy $E^{\rm eff}(T)$ given there and have for 4H
($S_{n}=-1.6$ MeV) the value $R^{\rm vir}_{{}^{4}{\rm H}}(1.29)=0.057$, and
for 5He for $S_{n}=-0.735$ MeV the value $R^{\rm vir}_{{}^{5}{\rm
He}}(1.29)=0.731$. Together with the known virial coefficient in the deuteron
channel, in rnp20  the interpolation formula (in units of MeV)

$\displaystyle C_{A,Z}^{\rm vir}(\lambda_{T},E_{AZ,i})$ $\displaystyle=$
$\displaystyle\frac{1}{e^{-(E^{\rm
thresh}_{A,Z}-E_{AZ,i}+1.129)/0.204}+1}\times$ (12)
$\displaystyle\frac{1}{e^{-(E^{\rm
thresh}_{A,Z}-E_{AZ,i}+2.45)/\lambda_{T}}+1}$

has been introduced which reproduces the values for 2H, 4H, and 5He at
$\lambda_{T}=1.29$ MeV. If we have multiple contributions within a group
$\alpha$ we replace $C_{AZ,i}^{\rm vir}(\lambda_{T},E_{AZ,i})$ by its value
for the lowest excitation energy $E_{AZ,\alpha}$ of the group, and can extract
this factor from the summation so that

$\displaystyle R^{{\rm vir},\alpha}_{A,Z}(\lambda_{T})=C_{A,Z}^{\rm
vir}(\lambda_{T},E_{AZ,\alpha})R^{{\rm res.gas},\alpha}_{A,Z}(\lambda_{T}).$
(13)

We used this expression to calculate the multipliers $R_{A,Z}^{{\rm
vir},\alpha}(\lambda_{T})$ shown in the Tables.

##  Appendix D Evaluation of the intrinsic partition function multipliers $R$:
Medium modifications

The virial expression uses the bound state energies and scattering phase
shifts of the free clusters, neglecting interaction effects. The most
important in-medium effects are self-energy and Pauli blocking. Self-energy
effects are well known and are parametrized, for instance, as Skyrme forces or
relativistic mean-field approximation. If we neglect the momentum dependence
of the single-particle mean-field shift, this self-energy shift $\Delta^{\rm
SE}_{\tau}$ for neutron and protons, which occurs also in the bound states may
be included in the effective chemical potentials
$\tilde{\lambda}_{\tau}=\lambda_{\tau}-\Delta^{\rm SE}_{\tau}$. Values for
$\Delta^{\rm SE}_{\tau}$ have been given in Sec. IV.3. Thus, it has no
influence on the composition at given temperature and densities.

In contrast, the Pauli blocking shift occurs only for interacting nucleons in
a cluster and depends strongly on the quantum state, as discussed above.
Therefore we take this shift of energy levels explicitly into account. In
principle, also the scattering phase shifts are modified. See SRS  for the
two-nucleon case, where a generalized Beth-Uhlenbeck approach is given which
accounts for in-medium effects.

The Pauli blocking has different consequences. (i) In (7), (8), the energy
shift of the bound states $\Delta E^{\rm Pauli}_{A,Z,i}$ has to be added to
the binding energies $B^{\rm medium}_{A,Z}=B_{A,Z}-\Delta E^{\rm Pauli}_{A,Z}$
for the ground state, but also for all excitation energies $E^{\rm
medium}_{AZ,i}=E_{AZ,i}+\Delta E^{\rm Pauli}_{AZ,i}-\Delta E^{\rm
Pauli}_{A,Z}$ which are taken relatively to the ground state. In principle,
the Pauli blocking depends on the wave function of the bound state as well as
the total momentum $\bf P$ of the cluster, but for simplicity we assume that
we can neglect this and approximate these different Pauli blocking shifts by
$\Delta E^{\rm Pauli}_{A,Z}$. Then, a common factor can be extracted so that
$e^{-\Delta E^{\rm Pauli}_{A,Z}/\lambda_{T}}R_{A,Z}^{\rm vir}(\lambda_{T})$
appears.

(ii): The medium modifications will also result in a density dependence of the
intrinsic partition function or the corresponding factors $R^{{\rm
medium},\alpha}_{A,Z}(T,n_{B},Y_{p})$ in the decomposition corresponding to
Eq. (13). We use the virial expression (13) but replace in $C_{A,Z}^{\rm
vir}(\lambda_{T},E_{AZ,\alpha})$ the excitation energy $E_{AZ,\alpha}$ by
$E_{AZ,\alpha}+\Delta E^{\rm Pauli}_{A,Z}$,

$\displaystyle R^{{\rm
medium},\alpha}_{A,Z}(\lambda_{T},\lambda_{n},\lambda_{p})$
$\displaystyle=C_{AZ}^{\rm vir}(\lambda_{T},E_{AZ,\alpha}+\Delta E^{\rm
Pauli}_{A,Z})\times$ (14) $\displaystyle R^{{\rm
res.gas},\alpha}_{A,Z}(\lambda_{T})e^{-\Delta E^{\rm
Pauli}_{A,Z}/\lambda_{T}}.$

The shift of the bound states gives a decrease of the binding energy which may
disappear (Mott effect). The dissolution of bound states opens additional
channels for feed-down processes.

We discuss three different cases: Well bound states with binding energy larger
than $T\approx 1$ MeV are shifted but not dissolved. Because the temperature
is low, the effect of continuum correlations and the threshold energy is small
for the strongly bound isotopes so that, in addition to the self-energy shift
$\Delta^{\rm SE}_{A,Z}$, the factor $\exp[-\Delta E_{A,Z}^{\rm
Pauli}/\lambda_{T}]$ determines the influence of the medium. The contribution
of scattering states has been approximated by Eq. (12), replacing $E_{AZ,i}$
by $E_{AZ,i}+\Delta E^{\rm Pauli}_{AZ,i}$.

Of special interest are the weakly bound states where the shift brings it
above the continuum edge, $E_{AZ,i}+\Delta E^{\rm Pauli}_{A,Z}-E^{\rm
thresh}_{AZ}>0$ so that the bound state will be dissolved and feed down other
final states. We estimate these contributions also according to Eq. (12) as
done above for the scattering states. Instead to feed down the ground state
after de-excitation, we assume that they feed down other isotopes as indicated
in Tabs. 9, 10.

This simplified picture to describe the reaction processes after freeze-out
has to be improved in future treatments within a systematic nonequilibrium
approach. As known from reaction networks describing expanding matter in
astrophysics, branching ratios may be introduced to describe the evolution of
the primary distribution to the final distribution, which allow for different
final states of an excited state. For our considerations this is not of
relevance as long as the contribution of excited states (including continuum
states) is small, and the main contribution to the yield of isotopes is
determined by the strongly bound states. Another situation appears for the
weakly bound final states like 11Li and 19C in our calculation, see Tabs. 9,
10. Because they are shifted to the continuum, in our simple treatment of the
feed down process they will feed other isotopes, and the final yield for both
isotopes will be zero. A more detailed description would allow that some of
the primary states may remain in the bound state. For instance, the Pauli
blocking is momentum dependent, and clusters with large momentum remain bound
because Pauli blocking is effective only within the Fermi sphere, as discussed
in Sec. B and Sec. IV.3. There is also a finite probability that continuum
contributions according to the Beth-Uhlenbeck formula will de-excite to the
ground state according to a coalescence model. For this one has to introduce
branching ratios which allow feed-down to different final states. Even being
small, special feed-down processes become visible if they are the only process
to form a final bound state.

We are not able in this work to derive a microscopic expression for the
branching ratios during expansion after freeze-out. They are of relevance when
they are the only process to populate the final yield, for instance, of the
dissolved isotopes 11Li and 19C in Tabs. 9, 10. To give an example, we assumed
in both cases a branching ratio 0.1 that the primary yield remains at the same
isotope to form the final yield. As shown in Fig. 4, a branching ratio would
nicely fit the general behavior of the calculated yields. This example is only
for illustration, the derivation of the branching ratio goes beyond the frame
of the present work.

Another issue which needs further considerations is the threshold energy
$E^{\rm thresh}_{A,Z}$ which is taken in this work from the data tables. It is
possible that this quantity is also changed in a dense medium. For this, a
nonequilibrium approach to the reaction processes is necessary which finally
will provide us with in-medium branching ratios discussed above. Our
description of the reactions during the expansion process after freeze-out
remains a semi-empirical one with a simplified reaction network as shown in
Tabs. 9, 10.

isotope | $A$ | $Z$ | $\frac{B_{A,Z}}{A}$ | $g_{A,Z}$ | $E^{\rm thresh}_{AZ}$ | $R_{A,Z}^{\rm vir}(1.29)$ |  $\Delta E^{\rm Pauli}_{A,Z}$ [MeV] | $R_{A,Z}^{\rm medium}(1.29)$ | $Y_{A,Z}^{\rm rel,medium}$ | $Y_{A,Z}^{\rm final,medium}$ |  $Y^{\rm obs}_{A,Z}$/$Y^{\rm final,medium}_{A,Z}$
---|---|---|---|---|---|---|---|---|---|---|---
1n | 1 | 0 | 0 | 2 | - | 1 | - | 1 | 0.11203 | 0.11203 | 0.9551
1H | 1 | 1 | 0 | 2 | - | 1 | - | 1 | 4.563E-6 | 4.563E-6 | -
2H | 2 | 1 | 1.112 | 3 | 2.224 | 0.9739 | 0.057 | 0.9306 | 9.226E-6 | 9.226E-6 | 0.9173
3H | 3 | 1 | 2.827 | 2 | 6.257 | 0.9988 | 0.114 | 0.9138 | 1.289E-4 | 1.342E-4 | 1.223
4H∗ | 4 | 1 | 1.720 | 5 | -1.6 | 0.09404 | 0.172 | 0.03564 | 5.146E-6 | [$\to^{3}$H] | -
3He | 3 | 2 | 2.573 | 2 | 5.494 | 0.9979 | 0.057 | 0.9545 | 2.948E-9 | 2.948E-9 | -
4He | 4 | 2 | 7.073 | 1 | 20.577 | 1 | 0.114 | 0.915 | 1.639E-3 | 1.935E-3 | 1.042
5He∗ | 5 | 2 | 5.512 | 4 | -0.735 | 0.6906 | 0.172 | 0.5029 | 2.622E-4 | [$\to^{4}$He] | -
6He0 | 6 | 2 | 4.878 | 1 | 0.975 | 0.9343 | 0.2291 | 0.7723 | 4.575E-5 | 5.884E-5 | 0.8903
6He∗ | 6 | 2 | 4.878 | 1 | 0.975 | 0.7919 | 0.2291 | 0.4617 | 2.735E-5 | [$\to^{4}$He] | -
7He∗ | 7 | 2 | 4.123 | 4 | -0.410 | 0.9313 | 0.2864 | 0.6577 | 1.309E-5 | [$\to^{6}$He] | -
8He | 8 | 2 | 3.925 | 1 | 2.125 | 0.972 | 0.3437 | 0.7383 | 2.919E-6 | 2.935E-6 | 1.029
8He∗ | 8 | 2 | 3.925 | 1 | 2.125 | 0.2332 | 0.3437 | 0.06923 | 2.737E-7 | [$\to^{4}$He] | -
9He∗ | 9 | 2 | 3.349 | 2 | -1.25 | 0.364 | 0.4009 | 0.04882 | 1.605E-8 | [$\to^{8}$He] | -
6Li | 6 | 3 | 5.332 | 3 | 1.475 | 0.9545 | 0.172 | 0.83 | 4.78E-8 | 4.78E-8 | -
6Li∗ | 6 | 3 | 5.332 | 3 | 1.475 | 0.3847 | 0.172 | 0.2842 | 1.417E-8 | [$\to^{3}$H] | -
7Li | 7 | 3 | 5.606 | 4 | 2.461 | 1.3159 | 0.2291 | 1.0972 | 2.66E-6 | 2.985E-6 | 0.4523
8Li | 8 | 3 | 5.160 | 5 | 2.038 | 1.2424 | 0.2864 | 0.9877 | 1.624E-6 | 1.624E-6 | 0.5211
8Li∗ | 8 | 3 | 5.160 | (5) | 2.038 | 0.2647 | 0.2864 | 0.1975 | 3.247E-7 | [$\to^{7}$Li] | -
9Li | 9 | 3 | 5.038 | 4 | 4.062 | 1.0553 | 0.3437 | 0.8069 | 2.694E-6 | 2.876E-6 | 0.5814
10Li∗ | 10 | 3 | 4.531 | 3 | -0.032 | 0.863 | 0.4009 | 0.5866 | 1.539E-7 | [$\to^{9}$Li] | -
11Li+ | 11 | 3 | 4.155 | 4 | 0.396 | 0.9003 | 0.4582 | 0.6026 | 3.03E-8$\times 0.1$ | 3.03E-9 | 0.2992
11Li0 | 11 | 3 | 4.155 | 4 | 0.396 | 0.9003 | 0.4582 | 0.6026 | 3.03E-8$\times 0.9$ | [$\to^{9}$Li] | -
12Li∗ | 12 | 3 | 3.792 | 3 | -0.201 | 0.8422 | 0.5155 | 0.4697 | 1.585E-9 | [$\to^{9}$Li] | -
7Be | 7 | 4 | 5.372 | 4 | 1.585 | 1.3015 | 0.172 | 1.1324 | 3.058E-11 | 3.058E-11 | -
8Be∗ | 8 | 4 | 7.062 | 1 | -0.088 | 1.2657 | 0.2291 | 1.0192 | 1.722E-6 | [$\to^{4}$He] | -
9Be | 9 | 4 | 6.462 | 4 | 1.558 | 0.9572 | 0.2864 | 0.7585 | 2.03E-6 | 2.778E-6 | 0.3191
9Be∗ | 9 | 4 | 6.462 | 4 | 1.558 | 0.4796 | 0.2864 | 0.3629 | 9.712E-7 | [$\to^{4}$He] | -
10Be | 10 | 4 | 6.497 | 1 | 6.497 | 1.366 | 0.3437 | 1.0462 | 1.464E-5 | 1.752E-5 | 0.5285
10Be0 | 10 | 4 | 6.497 | (1) | 6.497 | 0.07181 | 0.3437 | 0.05348 | 7.482E-7 | [$\to^{9}$Be] | -
11Be | 11 | 4 | 5.953 | 2 | 0.502 | 0.9076 | 0.4009 | 0.6422 | 2.829E-6 | 2.829E-6 | 0.4202
11Be0 | 11 | 4 | 5.953 | 2 | 0.502 | 0.6894 | 0.4009 | 0.48015 | 1.141E-6 | [$\to^{10}$Be] | -
11Be∗ | 11 | 4 | 5.953 | (2) | 0.502 | 0.308 | 0.4009 | 0.03957 | 3.341E-9 | [$\to^{10}$Be] | -
12Be | 12 | 4 | 5.721 | 1 | 3.170 | 2.1223 | 0.4582 | 1.4798 | 3.957E-6 | 4.074E-6 | 0.1385
12Be0 | 12 | 4 | 5.721 | (1) | 3.170 | 0.3307 | 0.4582 | 0.222 | 5.937E-7 | [$\to^{10}$Be] | -
13Be∗ | 13 | 4 | 5.241 | 2 | -0.510 | 0.7806 | 0.5155 | 0.3144 | 1.164E-7 | [$\to^{12}$Be] | -
14Be | 14 | 4 | 4.994 | 1 | 1.264 | 0.9468 | 0.5728 | 0.5897 | 4.497E-8 | 4.498E-8 | 0.01209
15Be∗ | 15 | 4 | 4.541 | 6 | -1.800 | 0.02241 | 0.6301 | 0.0005243 | 5.994E-12 | [$\to^{14}$Be] | -
Table 9: Calculated yields per fission of ternary fission of 241Pu($n_{\rm th}$,f), in-medium effects included. For the fit of the 241Pu($n_{\rm th}$,f) data, the Lagrange parameter values are $\lambda_{T}=1.29,\,\,\tilde{\lambda}_{n}=-3.09,\,\tilde{\lambda}_{p}=-16.19$ MeV, volume 1796 fm3, fit metric 0.0095 for all nuclei $Z\leq 2$. Baryon density $n_{B}=6.7\times 10^{-5}$, $Y_{p}=0.035$. Bold figures: Mott effect, the Pauli blocking shift exceeds the ground state binding energy so that the bound state is dissolved. For illstration of this effect, a branching ratio 0.1 is assumed to remain as final yield (denoted by superscript ”+”) whereas 0.9 is assumed to feed down the yields of other isotopes. isotope | $A$ | $Z$ | $\frac{B_{A,Z}}{A}$ | $g_{A,Z}$ | $E^{\rm thresh}_{AZ}$ | $R_{A,Z}^{\rm vir}(1.29)$ |  $\Delta E^{\rm Pauli}_{A,Z}$ [MeV] | $R_{A,Z}^{\rm medium}(1.29)$ | $Y_{A,Z}^{\rm rel,medium}$ | $Y_{A,Z}^{\rm final,medium}$ |  $Y^{\rm obs}_{A,Z}$/$Y^{\rm final,medium}_{A,Z}$
---|---|---|---|---|---|---|---|---|---|---|---
10B | 10 | 5 | 6.475 | 7 | 4.466 | 1.357 | 0.2864 | 1.086 | 3.547E-9 | 3.547E-9 | -
11B | 11 | 5 | 6.928 | 4 | 8.674 | 1.175 | 0.3437 | 0.9 | 1.261E-6 | 1.428E-6 | 0.2258
12B | 12 | 5 | 6.631 | 3 | 3.369 | 2.227 | 0.4009 | 1.626 | 2.411E-6 | 2.411E-6 | 0.08358
12B0 | 12 | 5 | 6.631 | (3) | 3.369 | 0.1582 | 0.4009 | 0.1127 | 1.671E-7 | [$\to^{11}$B] | -
13B | 13 | 5 | 6.496 | 4 | 4.879 | 0.9966 | 0.4582 | 0.6976 | 6.209E-6 | 7.096E-6 | -
14B | 14 | 5 | 6.102 | 5 | 0.970 | 0.9341 | 0.5155 | 0.6065 | 1.461E-6 | 1.461E-6 | 0.01793
14B0 | 14 | 5 | 6.102 | (5) | 0.970 | 0.3001 | 0.5155 | 0.188 | 4.53E-7 | [$\to^{13}$B] | -
14B∗ | 14 | 5 | 6.102 | (5) | 0.970 | 0.387 | 0.5155 | 0.1801 | 4.338E-7 | [$\to^{13}$B] | -
15B | 15 | 5 | 5.880 | 4 | 2.78 | 0.9829 | 0.5728 | 0.6246 | 1.058E-6 | 1.075E-6 | 0.008624
16B∗ | 16 | 5 | 5.507 | 1 | -0.082 | 0.8574 | 0.6301 | 0.4311 | 1.723E-8 | [$\to^{15}$B] | -
17B | 17 | 5 | 5.270 | 4 | 1.39 | 0.9515 | 0.6873 | 0.5399 | 2.723E-8 | 2.964E-8 | -
18B∗ | 18 | 5 | 4.977 | 5 | -0.005 | 0.8659 | 0.7446 | 0.3832 | 2.407E-9 | [$\to^{17}$B] | -
13C | 13 | 6 | 7.470 | 2 | 4.946 | 1.353 | 0.4009 | 0.9905 | 3.134E-6 | 3.137E-6 | -
14C | 14 | 6 | 7.520 | 1 | 8.176 | 1.069 | 0.4582 | 0.7491 | 6.769E-5 | 1.139E-4 | 0.02229
15C | 15 | 6 | 7.100 | 2 | 1.218 | 0.9449 | 0.5155 | 0.6169 | 2.916E-5 | 5.92E-5 | 0.01464
15C0 | 15 | 6 | 7.100 | (2) | 1.218 | 1.5315 | 0.5155 | 0.9775 | 4.621E-5 | [$\to^{14}$C] | -
15C∗ | 15 | 6 | 7.100 | (2) | 1.218 | 0.004482 | 0.5155 | 0.000206 | 9.728E-9 | [$\to^{14}$C] | -
16C | 16 | 6 | 6.922 | 1 | 4.250 | 2.259 | 0.5728 | 1.445 | 9.266E-5 | 1.305E-4 | 0.007721
16C0 | 16 | 6 | 6.922 | (1) | 4.250 | 0.7875 | 0.5728 | 0.4683 | 3.004E-5 | [$\to^{15}$C] | -
17C | 17 | 6 | 6.558 | 4 | 0.734 | 0.9218 | 0.6301 | 0.5379 | 2.447E-5 | 2.447E-5 | 0.005271
17C0 | 17 | 6 | 6.558 | (4) | 0.734 | 1.438 | 0.6301 | 0.8288 | 3.77E-5 | [$\to^{16}$C] | -
17C∗ | 17 | 6 | 6.558 | (4) | 0.734 | 0.09067 | 0.6301 | 0.002612 | 1.188E-7 | [$\to^{16}$C] | -
18C | 18 | 6 | 6.426 | 1 | 4.18 | 3.153 | 0.6873 | 1.843 | 5.329E-5 | 8.188E-5 | 0.0006891
19C+ | 19 | 6 | 6.118 | 2 | 0.580 | 4.665 | 0.7446 | 2.431 | 2.181E-5$\times 0.1$ | 2.181E-6 | 0.0002309
19C0 | 19 | 6 | 6.118 | 2 | 0.580 | 4.665 | 0.7446 | 2.431 | 2.181E-5$\times 0.9$ | [$\to^{18}$C] | -
19C∗ | 19 | 6 | 6.118 | (2) | 0.580 | 2.383 | 0.7446 | 0.9985 | 6.858E-6 | [$\to^{18}$C] | -
20C | 20 | 6 | 5.961 | 1 | 2.98 | 2.391 | 0.8019 | 1.268 | 5.646E-6 | 5.646E-6 | 0.0001285
Table 10: Continuation of Tab. 9.

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