J. Astrophys. Astr. (2009) 30, 165–175
Rapid Neutron Capture Process in Supernovae and Chemical
Rulee Baruah1,∗, Kalpana Duorah2& H. L. Duorah2
1Department of Physics, HRH The Prince of Wales Institute of Engineering and Technology,
Jorhat 785 001, India.
2Department of Physics, Gauhati University, Guwahati 781 014, India.
Received 2007 May 28; accepted 2009 September 29
major nucleosynthesis processes responsible for the synthesis of heavy
nuclei beyond iron. Isotopes beyond Fe are most exclusively formed in
neutron capture processes and more heavier ones are produced by the
r-process. Approximately half of the heavy elements with mass number
A ? 70andalloftheactinidesinthesolarsystemarebelievedtohavebeen
produced in the r-process. We have studied the r-process in supernovae for
the production of heavy elements beyond A = 40 with the newest mass
values available. The supernova envelopes at a temperature ?109K and
sites for the r-process. The primary goal of the r-process calculations is to
fit the global abundance curve for solar system r-process isotopes by vary-
ing time dependent parameters such as temperature and neutron density.
This method aims at comparing the calculated abundances of the stable
isotopes with observation. We have studied the r-process path correspond-
ing to temperatures ranging from 1.0 × 109K to 3.0 × 109K and neutron
density ranging from 1020cm−3to 1030cm−3. With temperature and den-
sity conditions of 3.0 × 109K and 1020cm−3a nucleus of mass 273 was
theoretically found corresponding to atomic number 115. The elements
obtained along the r-process path are compared with the observed data at
all the above temperature and density range.
The rapid neutron capture process (r-process) is one of the
Burbidge et al. (1957), in their seminal paper, outlined the rapid neutron capture
process in the supernova envelope at a high neutron density and a temperature of 109
degrees. According to them, this mode of synthesis is responsible for the produc-
tion of a large number of isotopes in the range 70 ≤ A ≤ 209, and also for synthe-
sis of uranium and thorium. This would explain the abundances of the neutron rich
nuclei in the periodic table. Major advances have been made in calculating r-process
nucleosynthesis in supernovae (Woosley et al. 1992) and in using a wide range of
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model parameters to obtain yields that approximate the solar r-process abundances
(Kratz et al. 1993). Studies of galactic chemical evolution (Mathews & Cowan 1992)
show that the enrichment of the r-process elements in the galaxy is consistent with low
mass type II supernovae being the r-process sites. In the usual picture the r-process
stops when the neutron supply ceases (freeze-out). The produced very neutron rich
progenitor nuclei then undergo a series of β-decays until they reach a stable nucleus
whose calculated abundance can then be compared with observation. It was recog-
attainable only in dynamical events, i.e., supernovae.
The essential feature of the r-process is that a large flux of neutrons becomes avail-
able in a short time interval for addition to elements of the iron group, or perhaps, in
nuclei such as Ne22. So we started our analysis with A = 40 and obtained the abun-
dances beyond that. We have summarised our calculations within a site-independent,
classical approach based on neutron number density nnand temperature T9, defin-
ing the neutron binding (separation) energy Qnof the path, where the waiting point
approximation, i.e., (n, γ) ↔ (γ, n) equilibrium could be applied. The dependence
on nuclear masses enters via Qn.
We choose supernova as the site for r-process because the supernova light curves
show the presence of98Cf254. We have considered the r-process in supernovae for
the production of heavy elements, under extreme conditions of temperature and den-
sity. For our purpose, the most interesting evolution occurs as the temperature falls
favours the assemblage of nucleons into α-particles and heavy nuclei. As the tem-
perature drops below about 5.0 × 109K, the reactions responsible for converting
α-particles back into heavy nuclei begin to fall out of equilibrium. By 3.0 × 109K,
the charged particle reactions freeze out. Below this temperature, the r-process occurs
until the temperature reaches (1–2) × 109K, where the neutron reactions also cease
as the neutrons are depleted (Woosley et al. 1994). Using new mass tables of Audi
et al. (2003) we have calculated the average excess neutron binding energy to nuclei
with neutron number which is then used in the calculation of neutron capture chain.
We start with a temperature of 1.0 × 109K and neutron number density of 1020cm−3
as these are the conditions prevailing in supernova envelopes during the eventually
expanding stages. In our present paper, we emphasize only on the r-process path to
obtain the elements in our astrophysical conditions considered and consequent build-
up to heavier nuclei. In our next paper, we propose to present the abundances of these
elements along the path.
2. Source of neutron flux
For the r-process nucleosynthesis in supernovae, the existence of enormous neutron
as13C(α,n)16O can produce free neutrons in red giants, but the number of these free
neutrons is also small. It is possible to circumvent this problem by having the only
charged particles accompanying the neutrons be alphas. Single alpha particles do not
capture neutrons. It is proposed that (Schramm 1973) at high temperatures associated
with the collapse of massive iron core in type II supernovae, iron will photo-dissociate
Neutron Capture and Chemical Elements
into alphas and neutrons as:
56Fe → 13α + 4n.
As the material expands and cools from these photo-dissociation conditions, the
tion is hampered by the fact that alphas only systhesize heavy elements via three-body
interactions. Thus there will be a time during which a few iron peak seed nuclei have
been produced in a sea of alphas and neutrons. The ratio of neutrons to seed will be
large, so that an r-process can take place.
Another set of conditions where large number of free neutrons exist is when
the temperature and density get sufficiently high that the reaction p + e−→ n + νe
dominates over n + e+→ p + ¯ νe(Arnett 1979). Thus neutronisation refers to elec-
tron capture driven by high electron Fermi energy (i.e., high density). Subsequently,
p + e−→ n + νeat high density. Moreover, nuclei resist electron capture because of
the large threshold energies required as they become more neutron rich. Also electron
capture on free protons is limited by the small abundance of free protons. These prob-
lems are eased by higher density and higher temperature, so neutronization speeds up
as collapse continues. Once collapse begins, neutronization becomes the dominant
mode of neutrino productions, overwhelming thermal processes.
According to Mukhopadhyay (2007), the neutrino–antinutrino oscillation under
gravity explains the source of abnormally large neutron abundance to support the
r-process nucleosynthesis in astrophysical site, e.g., supernova. He also proposed two
n + νe→ p + e−;
p + ¯ νe→ n + e+
as given by Arnett (1979). If ¯ νeis over-abundant than νe, then, from this expression,
neutron production is expected to be more than proton production into the system.
Therefore,thepossibleconversionofν to ¯ νeduetogravityinducedoscillationexplains
the over-abundance of neutron.
flow and the track followed in the (A,Z) plane are the (n, γ) and (γ, n) reactions, beta
decay and at the end of the track the neutron induced fission. The timescale τnfor a
heavy nucleus to capture an additional neutron is rapid on the competing timescale τβ
for it to undergo beta decay. Whereas τβdepends only on nuclear species, τndepends
critically on the ambient neutron flux.
λn> λβ(τn< τβ).
In rapid process, a sufficient flux of neutrons makes τnmuch shorter than τβ. Then
neutron capture will proceed into the very neutron rich and unstable regions far from
the valley of beta stability. Once the neutron flux is exhausted, the unstable nuclei
produced by the r-process will beta decay to the valley of stability to form the stable
3. Nuclear physics considerations and the r-process path
To illustrate the significant differences of the astrophysical conditions during
the r-processing, we refer to the classical quantity, namely, the neutron binding
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(separation Sn) energy Qn, that represents the r-process path in the chart of nuclides
once the specific values of neutron density nnand the temperature T are assigned.
The Qnvalues vary in time as well as in space along with the dynamical evolution of
our astrophysical environment.
3.1 Dynamical evolution of the neutrino heating phase in type-II supernovae
We first summarize the type-II supernova explosion scenario according to the current
understanding. We emphasize some characteristic features on the hydrodynamical
evolution of the neutrino wind phase. During the final stages of the evolution of a
massive (8 ∼25M?) star, an ‘iron’ core forms in its central region and subsequently
undergoes gravitational collapse. When the central density reaches nuclear matter
density, the collapse stops abruptly to cause a ‘core bounce’. A hydrodynamical shock
wave is created and starts to propagate outward. According to calculations (Bruenn
1989a), this shock wave loses its entire kinetic energy within a few milliseconds to
to about half a second, the neutrinos streaming out from the new born neutron star can
deposit energy behind the standing accretion shock at high enough a rate to revive its
outward motion and initiate the final explosion of the star. This is the neutrino-driven
‘delayed’ explosion mechanism originally suggested by Wilson et al. (1986).
The neutron star releases its gravitational binding energy of several 1053erg in the
form of neutrino radiation. A region of net energy deposition by neutrinos (‘neutrino
of temperature with increasing radius. The energy is transferred to the stellar gas
predominently by absorption of electron neutrinos (νe) on neutrons and electron anti-
neutrinos (¯ νe) on protons. About one percent of the neutron star’s binding energy is
sufficient to drive a powerful shock into the overlaying stellar mantle. Behind the
shock, an extended and rapidly expanding region of low density and relatively high
temperature develops and is further energized by neutrino heating.
Janka (1993) performed hydrodynamical simulations of the formation and evolu-
tion of the neutrino-wind phase of a type II supernova with a proper description
of the neutrino physics and an adequate representation of the equation of state.
The hydrodynamical investigations were carried on from an initial configuration
made available by Wilson. From Wilson’s post-collapse model the radial profiles of
density, temperature, electron concentration, composition and velocity were taken
to specify the initial conditions for the set of partial differential equations, which
was integrated in time to follow the gas composition and the evolution of the fluid
flow in spherical symmetry. The equation of state for the stellar gas contained the
contributions from nucleus, α-particles, and a representative typical heavy nucleus in
nuclear statistical equilibrium. The model evolved under the influence of the neutrino
fluxes from the protoneutron star at the center. Since all hydrodynamical and thermo-
dynamical quantities were determined from the numerically solved set of equations,
the effects of the particular choice of the initial model configuration were not cru-
cial and became even less relevant as time went on. The most important parameter
of the input model to influence the simulated evolution was the mass of the central
neutron star (Witti et al. 1994). However, the hydrodynamical evolution in the range
of temperatures below T9= 2 is not very fast (Takahashi et al. 1994).
Neutron Capture and Chemical Elements
3.2 The r-process network and the waiting-point approximation
Supernova is a dynamical event. When a constant Sn(nnand T) is assumed over a
duration time τ, then the nuclei will still be existent in the form of highly unstable
isotopes, which have to decay back to β-stability. In reality nnand T will be time
T) is important. The abundance flow from each isotopic chain to the next is governed
by beta decays. The waiting point approximation is only valid for high temperatures
and neutron number densities of the gas. If not, the flow of nuclei towards higher
neutron number N for a given proton number Z is steadily depleted by beta decay.
As a result only a small fraction of the flow can easily reach a waiting point. Cameron
et al. (1983b) found that for temperatures of 2.0 × 109K and higher, the waiting point
approximation was valid for neutron number densities as low as 1020cm−3. For lower
temperature (T < 109K) even with high values of nn≈ 1025cm−3, the waiting point
approximation is not valid. The r-process path requires a synthesis time of the order
of seconds to form the heaviest elements such as thorium, plutonium and uranium.
The r-process network includes radiative neutron capture, i.e., (n, γ) reactions, the
inverse photo-disintegration, i.e., (γ, n) reactions, β-decay, i.e., (β, γ) processes and
β-delayed neutron emission, i.e., (β, n) processes. If the neutron density is very high,
successive (n, γ) reactions may produce very neutron rich isotopes out of the limited
α-process network in a ‘mini r-process’. The (n, γ) and (γ, n) reactions are then
much faster than β-decays. Therefore, as soon as the ‘proper’ r-process is started, the
isotopic abundances, stuck at the most neutron-rich isotopes included in the α-process
network, will quickly be redistributed according to the (n, γ) ↔ (γ, n) equilibrium
(Takahashi et al. 1994). In our model, the neutron number densities are so high that
an equilibrium between the (n, γ) and (γ, n) reactions is quickly established.
Inan(n, γ) ↔ (γ, n)equilibrium(thewaiting-pointapproximation),themaximum
abundances in isotopic chains occur at the same neutron separation energy, which is
up of heavy nuclei is governed by the abundance distribution in each isotopic chain
from (n, γ) ↔ (γ, n) equilibrium and by effective decay rates λZ
After charged particle freeze-out, when only (n, γ) ↔ (γ, n) equilibrium remain
in place, matter can progress to heavier nuclei via β-decays between isotopic chains,
which is modelled by the r-process network to follow further evolution (Freiburghaus
et al. 1999).
βof isotopic chains.
4. Calculation of the r-process path
A nucleus of fixed Z cannot add neutrons infinitely even in the presence of an intense
neutron flux. The binding energy of each successive neutron becomes progressively
weaker as more and more neutrons are added until ultimately the binding falls to zero,
which sets an upper limit to neutron addition at fixed Z. The nucleus then waits until
β-decay allows it to move onto the next nucleus. Thus in a rapid process two inverse
reactions n+(Z,A) ↔ (Z,A+1)+γ come to an equilibrium. This balance governs
the equilibrium distribution of isotope abundances for a given Z. The maximum abun-
dance along an isotope chain is determined by the temperature and neutron density.
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Given that A/(A + 1) ≈ 1, the abundance maxima in each isotopic chain are deter-
mined by the neutron number density nnand temperature T. The maximum value of
the abundance occurs at neutron separation energy Snwhich is same for all isotopic
chains irrespective of Z. Approximating abundances Y(Z,A + 1)/Y(Z,A) ≈ 1 at
the maximum and keeping all other quantities constant, the neutron separation energy
Snhas to be the same for the abundance maxima in all isotopic chains.
The condition for the dynamical equilibrium between (n, γ) and (γ, n) reactions for
nucleus X(A,Z) is expressed as (Burbidge et al. 1957):
X(A,Z) + n ⇐⇒ X(A + 1,Z) + γ + Qn(A,Z),
Writing n(A,Z) and nnfor the number densities of the nuclei (A,Z) and neutrons
respectively, the statistical balance in this reaction is expressed by (Burbidge et al.
log(n(A + 1,Z)/n(A,Z)) = log nn− 34.07 − (3/2)log T9+ (5.04/T9)Qn
T9being temperature in units of 109degrees.
Using the condition that in equilibrium, n(A + 1,Z) ≈ n(A,Z) we obtain Qnas:
Qn= (T9/5.04)(34.07 + (3/2)log T9− log nn).
A rough estimate of Qnvalues that are preferred for explaining the r-process abun-
abundance peaks and the neutron magic numbers. The prominent peaks at A ≈ 130
and A ≈ 195 are correlated with the nuclear shell effects of their precursor nuclei near
the neutron magic numbers 82 and 126 respectively. With the aid of nuclear mass for-
mula, one finds from the abundance peaks that the Qnvalue is most likely somewhere
considered here to range from
T = 1.0 × 109K to 3.0 × 109K
The variation of Qnvalues with temperature and neutron number densities is shown
in Table 1.
data alone. First, we consider the determination of Qn(A,Z) on the basis of smooth
Weizsacker atomic mass formula given by equation (5) neglecting shell, pairing and
quadrupole deformation effects:
nn= 1020cm−3to 1030cm−3.
Mw(A,Z) = (A − Z)Mn+ ZMp− (1/c2)[αA − β(A − 2Z)2/A
− γA2/3− ?Z(Z − 1)/A1/3],
whereMnandMparemassesoftheneutronandprotonandα,β,γ and? areconstants
in energy units, which represent volume, isotopic, surface and coulomb energy para-
meters respectively, the values being taken from Burbidge et al. (1957). With these we
modify the expression for M(A,Z) as:
M(A,Z) = Mw(A,Z) − (1/c2)[f(N) + g(Z)],
Neutron Capture and Chemical Elements
Table 1. Variation of Qnvalues with temperature and density.
where Mw(A,Z) represents the Weizsacker expression given by equation (5). With
this we calculate the neutron binding energy as:
Qn(A,Z) = Bn(A + 1,Z) = c2[M(A,Z) + Mn− M(A + 1,Z)].
We note that Qnfor nucleus (A,Z) is equal for the neutron binding energy Bn(taken
positive) in nucleus (A + 1,Z). These functions of N and Z separately takes into
account the important effects on nuclear masses of:
• neutron and proton shell structure
• spheroidal quadrupole deformation of partially filled shells and
• pairing of neutrons and pairing of protons.
numbers for N and Z respectively. The sign is taken negative so that f(N) and g(Z)
as positive quantities, decrease the mass and add to the stability of the nucleus.
We now obtain:
Qn(A,Z) = f(A,Z) + f?(A − Z)
putting M(A,Z) from equation (6) and using:
f?(A − Z) = f?(N) = df(N)/d(N) = f(A + 1 − Z) − f(A − Z).
On simplification and on putting Z = A − N, we rewrite equation (8) as:
Qn(A,N) = f(A,N) + f?(N).
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by the smooth Weizsacker atomic mass formula.
The average excess neutron binding energy vs. neutron number N, over that given
conditions considered, showing the schematic view of the isotopes produced.
The average r-process path in the (A,Z) plane at all the temperature and density
Here f?(N) is the excess neutron binding energy to nuclei with a specified N over
of the shell in which N lies. We have used the mass tables of Audi et al. (2003) and
for various N, the resulting average values have been plotted against N. Another form
of averaging is then affected by drawing a smooth curve through the points obtained
in this way and plotted in Fig. 1. These equations are then solved for fixed values of
Z, to obtain the corresponding values of A by trial and error, at different temperature
and density conditions which specify Qnin equation (8). The neutron capture paths
so obtained are then plotted in Fig. 2.
Neutron Capture and Chemical Elements
5. Existence of chemical elements along the r-process path
In our classical condition we notice an element of mass 273 corresponding to atomic
number 115. Experimentally some new elements were synthesized at the Lawrence
Berkley Laboratory, e.g., elements with Z = 116, 118, etc. (Swiatecki et al. 2005).
Also theories have long predicted the island of stability for nuclei with approximately
114 protons and 184 neutrons. Thus we conclude that a nuclei with mass 273 is a
possibility. The seed nuclei in the neutrino driven wind are produced early in the
density become low in a short dynamic time scale and the charged particle reactions
almost cease, the r-process starts from these seed nuclei. So we start our calculation
at20Ca40and obtain the neutron capture path beyond that.
We notice that at densities >1030cm−3, the r-process chain does not show the ele-
ments as seen in data of Audi et al. (2003). As we try to obtain the elements at
lower densities in our analysis, we find them more prominently as we go from high to
low density site. Most of our observed elements are seen in the range of neutron num-
ber density 1020cm−3to 1024cm−3and temperature from 2.0×109K to 3.0×109K.
For example, at densities 1028cm−3, 1026cm−3, etc., the r-process chain does not give
us all the observable elements. But at condition of density 1020cm−3and temperature
T9= 2.0, that path contains all the elements as was given in the experimental data of
Audi et al. (2003).
We tabulate some of the elements (experimental) obtained along the r-process path
We note that the element98Cf254shown by the supernova light curves is found
in our classical astrophysical condition of temperature T9 = 1.9 and neutron num-
ber density nn = 1020cm−3. We also note that the double magic nucleus28Ni78
obtained at T9= 1.0 and nn= 1020cm−3; T9= 1.1 and nn= 1022cm−3; T9= 1.2
and nn= 1024cm−3; T9= 1.4 and nn= 1026cm−3; T9= 2.0 and nn= 1028cm−3;
all of these conditions correspond to Qnvalue ≈ 2.5Mev. Another double magic
Table 2. Chemical elements at the r-process site.
For double magic nuclei
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and nn= 1022cm−3; these correspond to Qnvalue ≈ 4.5Mev.
82is found in our analysis at T9= 1.7 and nn= 1020cm−3; T9= 1.9
6. Discussion and conclusion
Whenever and however the r-process operates, it appears to be very uniform and well
confined in astrophysical parameter space. The temperature, density and neutron flux
at r-process sites vary over a small range. This means that only a small minority of
type II supernovae produces r-process elements. The beta decay lifetimes, separation
are built in to the equations, which are, then numerically solved to determine the chain
for various separation energies. The neutrino winds drive out the r-process elements
which then decay to the lines nearer to the beta-stable valley, and, they are ready for
comparison with observation.
to 3.0 × 109K and neutron number densities ranging from 1020cm−3to 1030cm−3.
We mostly concentrate our analysis at energies greater than 2Mev as this is the condi-
tion prevailing in the supernova envelopes and neutron capture occurs during the later
expanding stage. We have used the mass table of Audi et al. (2003) for the calculation
of the average excess neutron binding energy which is obtained by the normalization
at the magic neutron numbers 20, 50, 82, 126. It has been found that a nucleus is stable
if the number of neutrons or protons in it is equal to the magic number, and it can-
not capture further neutrons because the shells are closed and they cannot contain an
extra neutron. With the subsequent addition of neutrons at fixed Z, correspondingly
the binding falls and ultimately falls to zero. At this point, the nucleus undergoes a
β-decay and gets converted to the next element. This r-process path is shown in Fig. 2
by corresponding relations between Z and A.
As the high density conditions do not show much of the experimentally observed
elements, we propose that the heavy elements which must have been produced during
the high density and temperature situation. Only in the later expansion stages after the
explosion, where the neutron density supposedly falls, the r-process nucleosynthesis
produces the heavy elements which subsequently β-decays and the r-process path
in the later expansion stages they were distributed all over the universe. In supernova
during the expansion stage if the ejected matter flow reaches the waiting point nuclei
associated with the magic neutron numbers at rather small radii above the neutron
star, neutrino induced charged current reactions can compete with the β-decays of the
longest lived waiting point nuclei and thus speed up the matter flow to heavier nuclei.
We tried to get our abundances with respect to all the nuclei whose β-decay lifetimes
are considerably higher. We conclude that our theoretical model will be successful in
providing new light to solve some problems in the r-process and the corresponding
build-up to heavier nuclei.
The authors are sincerely thankful to the Government of Assam for giving the per-
mission to carry out the research work. Grateful thanks are due to B. K. Rajkhowa of
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POWIET and Dr S. M. Hazarika of Tezpur University for their valuable help during
the preparation of the manuscript.
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