Nuclear Astrophysics Physics and Astronomy Department at the University of Washington

Nucleosynthesis in Supernovae

The explosion of a core-collapse supernova leads to ejection of the star's mantle, and thus to substantial enrichment of the interstellar medium with the major burning products of hydrostatic equilibirium: He, C, O, Ne, etc. As was described in the first lecture of this course, these are among the most plentiful elements in nature. However neither this mechanism nor any other process we have discussed so far describes how many other elements found in nature are synthesized. In this chapter several mechanisms associated with the explosive conditions of a supernova - explosive nucleosynthesis, the s- and r-processes, and the neutrino process - will be described.

5.1 Explosive nucleosynthesis and the iron peak

The creation of elements by the explosion itself - e.g., the high temperatures associated with passage of the shock wave - is called explosive nucleosynthesis. The properties of this process are tied to those of the explosion, which we have seen are still poorly understood. But the observation that the kinetic energy of a supernova explosion is typically in the range of (1-4) ergs, provides an important constraint.

Estimates of this synthesis depend on a number of issues:

Description of the presupernova model. The conventional approach is to evolve a star of given initial composition (e.g., metal content) through the various burning stages. The result is influenced by the assumed initial metallicity, the nuclear cross sections adopted, and the phyical mechanisms modeled, such as mass loss and convection.
The galactic model. Presumably the abundances we see today result from integrating over a large number of events. Thus one needs to know the characteristics of typical supernovae, e.g., the distribution of Type II supernovae over possible masses. And one has to take account of evolutionary effects: are the number of large stars in the early history of our galaxy similar to today? How does the supernova rate evolve? Was there an early ``bright phase" in our galaxy's evolution? Do overall changes in our galaxy, such as its metallicity, have an effect on supernova characteristics?
Reaction rates. Our information about exotic nuclear reactions - reactions involving excited states, unstable nuclei that have not been studied in the laboratory, or simply reactions that have not been measured - often is too limited. Thus there is always some change being made due to new information from laboratory measurements.

For a mass point well away from the neutrinosphere - perhaps 30,000 kilometers from the center of the star - it is a reasonable approximation to assume the density and temperature of the matter are changed little during the explosion until the shock wave arrives at that point. There is an approximate expression for the time of shock arrival

where is the radius of point in km, is the energy of the explosion in ergs, solar masses is the mass of the newly formed neutron star, and is the Lagrangian mass coordinate of the shell in question. Under the asumption that the part of the supernova behind the shock front is approximately isothermal and that the energy is contained in the radiation field, one might expect the peak temperature produced by the shock wave to be

where a is the radiation constant (a = 7.56 ergs/cmK and is related to the Stefan Boltzmann constant by ). The equation above says that the energy density of radition () times the volume gives the explosion energy. Numerically a relation is obtained that is quite compatible with this simple picture

For a 20 solar mass star (from Woosley et al.) one gets the following results

The numerical values are for the centers of each shell except for carbon, where the values are for the inner part of that shell. The difference between and T shows the elevated temperature that results from shock wave passage.

Note that reaches several in units of in the inner (silicon, oxygen, neon) shells. The resulting soup of photons, s, and nucleons of several hundred keV can clearly process material in these shells. In the silicon shell significant production of iron-peak elements results: Fe, Co, Ni, Cu, Zn. Other important productions includes Ca, Ti, etc. The pattern is quite similar in the oxygen shell. The somewhat lower characterizing the neon shell shifts the synthesis to somewhat lighter nuclei, e.g., Si, S, Ar, K, Ca, etc. Note many of these specifies can be formed by capture reactions that are facilitated by the higher temperatures. Outside the neon shell, very little explosive synthesis occurs.

It is quite possible that individual regions of the ejected mantle may remain more or less intact on ejection, thereby allowing observers to study the processes that occurred in each shell individually. Around 300-year-old supernova remnant Cas A regions have been found that are strongly overabundant in elements such as S, Ar, and Ca. Another exciting possibility - discussed in a recent Science magazine article - is to use the composition of individual stellar grains to determine not only the conditions under which specific isotopes are synthesized, but also the specific chemistry connected with the ejection and cooling of the material from which these grains condensed.

5.2 Abundances above the iron peak and neutron capture

The figure shows the abundance pattern found in our solar system. We see abundant light nuclei, especially the H, He, and light elements. An abundance peak near the iron isotopes is seen. Then there are lower abundances for heavier isotopes, but also interesting structure: mass peaks are seen around A 130 and 190. The low-mass structure (at and below iron) reflects a general tendancy for Coulomb barriers to inhibit synthesis of increasingly heavier nuclei, with the iron group being exceptional because it is favored by its strong binding.

A blowup of the pattern of heavy elements shows a clear structure associated with the closed neutron shells in nuclear physics: the stablest configurations are at the closed shells N =50, 82, and 126. There is a splitting of the abundance peaks, suggesting that perhaps there are two processes of interest. One can also see that the integrated abundance above the iron peak is not large, comparable to about 3% of the iron peak. Thus the processes responsible can be reasonably rare.

This synthesis is associated with the neutron-capture reaction . There are sources of neutrons in stellar interiors, and neutron capture cross sections on heavy nuclei can be quite large. We will also see that the observed shell structure is natural for such a process. Unstable but long-lived neutron-capture products, such as technetium, are seen in the atmospheres of red giants, indicating that neutron-induced synthesis is occurring in the cores of existing stars (and then dredged up to the surface).

The nuclear physics for neutron capture follows directly from our earlier work on charged particle reactions, if we set the charge of the initiating particle to zero. Recall for the compound nuclear reaction

where

for neutrons. It follows that

Therefore

Since this quantity is independent of energy

The conclusion from these considerations is that the thermally averaged rate should depend on energy only implicitly through the nuclear structure: the result would be given by the average cross section times the density of such resonances near threshold. Such an average cross section is shown in the figure. Its dominant feature is the dips that occur at the closed neutron shells N = 8, 20, 28, 50, 82, and 126. This reflects the very low level density in the vicinity of the closed shells: the shell gaps produce a low cross section.

5.3 The s-process

The s- and r-processes are mechanisms for synthesizing heavy nuclei by capturing neutrons one at a time. We consider the s-process first.

The following diagram of the (N,Z) plane shows the process of neutron capture in a plasma containing neutrons and heavy seed nuclei. The assumption made is that the neutron capture rate is much slower than the typical decay rate, which then has several consequences:
1) The weak interactions are then fast and maintain the Z-N equilibrium: Every time a neutron is captured, the resuilting system of A+1 nucleons has an opportunity to decay to a nucleus of greater stability, if such a nucleus exists.
2) The rate of synthesis is then proportional to the rate of neutron capture: this controls the ``mass flow" to heavier nuclei.
3) The path of nucleosynthesis, due to point 1) above, is thus along the so-called ``valley-of-stability." These are the familiar nuclei we study in laboratories, about which we know a great deal.

One consequence of the mass flow along the valley of stability is that a number of stable nuclei are avoided. One common situation is illustrated in the figure: frequently nuclei (N,Z) and (N+2,Z-2), when N and Z are even, are stable to decay, while the odd-odd nucleus (N+1,Z-1) is unstable. The odd-odd nucleus has an unpaired proton and an unpaired neutron, accounting for its unfavorable ground state energy. The (N+2,Z-2) nucleus can be ``shielded" from production in the s-process, as illustrated in the figure. Thus the existence of such isotopes with significant abundances indicates a second process, other than the slow- or s-process, must also occur.

One can also quantity the ``slowness" of the s-process. decay rates along the valley of stability are in the range of seconds to years. If one takes an average cross section of 0.1b at 30 KeV (corresponding to a neutron velocity of 0.008c), the reaction rate per particle pair is

and the capture rate per heavy nucleus is obtained by multiplying this by the neutron number density. Thus if we require

Such a neutron density, if maintained for 2000 years, could synthesize A 200 nuclei from iron group seeds.

One can also offer arguments to place a lower bound on the required neutron density. No stable nucleus exists at N=61: there is, however, a long-lived isotope Pd with y. Thus if the neutron capture rate is too slow, Pd will decay to Ag, and neutron capture will then produce the stable nucleus Ag. The nucleus will be bypassed. This nucleus cannot be made in the r-process (to be discussed below) because Pd is shielded on the neutron-rich side by the stable nucleus Cd. We conclude that neutron capture must be fast enough to synthesize Pd in the s-process,

This neutron number density then permits Pd to be produced by guaranteeing neutron capture is faster than the decay in this case.

The general equation describing the mass flow in the s-process is

where is the neutron density at time t and is the decay rate. Clearly this is one of a couple set of equations, complicated to solve in that the initial conditions would have to be fully specified, and the time evolution of given. The equation allows for destruction of mass number A by either neutron capture or decay, but the usual case is that the decay dominates if that channel is open, and otherwise the neutron capture occurs. If we take the later case and envision a constant neutron exposure, so that it makes sense to define an average cross section

over the Maxwell-Boltzmann distribution of relative velocities, then

If equilibrium has been achieved in the mass flow the LHS is zero and

That is, the abundance achieved is inversely proportional to the neutron cross section: if the capture rate is slow, then mass piles up at that target number. Of course, the same argument goes through if decay is the destruction channel (and the decay rate is presumably fast). The low neutron capture cross sections at the closed shells should result in mass peaks, just as observation shows. It also follows that equilibrium will set in most quickly in the broad plateaus between the mass peaks: mass must pile up at the closed shells before the closed shell is breached. Thus if the neutron flux is prematurely ended, the synthesis may not have yet gone beyond, for example, the N 82 peak.

In fact, if equilibrium were always reached over the entire range of the isotopes, then all the abundances would be in a proportion that tracks the inverse of their neutron capture cross sections. This is not what is observed: there is a drop in the abundance beyond each mass peak. Careful investigations indicate the s-process distribution observed in nature is the result of a series of neutron exposures, e.g., total neutron fluences

This has units of cm, that is, of flux times time. In fact, it has been concluded that two types of exposures, one involving a smaller fluence and the second one about four times larger, are required to produce the observed distribution.

Several sites have been suggested for the s-process, but one well accepted site is in the helium-burning shell of a red giant, where temperatures are sufficiently high to liberate neutrons by the reaction Ne(Mg, where the Ne is produced from helium burning on the elements the CNO cycle left after hydrogen burning.

Finally, we note the s-process cannot proceed beyond Bi: neutron capture on this isotope leads to a decay chain that ends with emission. This is a gap the s-process cannot cross. It follows that the tranuranic elements must have some other origin.

5.4 The r-process

The plot of the s-process path in the (N,Z) plane, shown in the previous subsection, demonstrates that certain nuclei on the neutron-rich side of the valley of stability will be missed in the s-process. This indicates a second mechanism for synthesizing heavy nuclei is needed. More convincing evidence is shown in the figure, where the mass peaks at A 130 and A 190 are shown to split into two components, one corresponding to the expected s-process closed-neutron-shell peaks at N 82 and N 126, and the second shifted to lower N, 76 and 116.

This second process is the r- or rapid-process, characterized by:
The neutron capture is fast compared to decay rates.
Thus the equilibrium maintained in : neutron capture fills up the available bound levels in the nucleus until this equilibrium sets in. The new Fermi level depends on the temperature and the relative abundance.
The nucleosynthesis rate is thus controlled by the decay rate: each capture coverting n p opens up a hole in the neutron Fermi sea allowing another neutron to be captured.
The nucleosynthesis path is along exotic, neutron-rich nuclei that would be highly unstable under normal laboratory conditions.
In analogy with the s-process calculation we did (where the production and destruction of a given isotope depended on the rate-controlling production and destruction neutron-capture cross sections and ), the r-process abundance A(Z,N) [(Z,N)] (the new rate-controlling reaction) for constant neutron exposure and equilibrated mass flow.

Let's first explore the equilibrium condition, which requires that the rate for balances that for for an average nucleus. So consider the formation cross section

This is an exothermic reaction, as the neutron drops into the nuclear well. Our averaged cross section, assuming a resonant reaction (the level density is high in heavy nuclei) is

where E 0 is the resonance energy. Thus the rate is

This has to be compared to the rate.

The reaction requires the photon number density in our gas. This is the Bose Einstein

The high-energy tail of the normalized distribution can thus be written

Although we will not be needing it, the approximate expression for the total photon density (gotten by integrating the distribution over all energies) is

In the two expressions above we have set .

Now we need the resonant cross section in the direction. For photons the wave number is proportional to the energy, so

As the velocity is c =1,

We evaluate this in the usual way for a sharp resonance, remember that the energy integral over just the denominator above (the sharply varying part) is :

So that the rate becomes

Equating the and rates and taking then yields

where the s and cs have been properly inserted to give the right dimensions. Now is esssentially the binding energy. So plugging in the conditions /cm and , we find that the binding energy is 2.4 MeV. Thus neutrons are bound by about 30 times kT, a value that is still small compared to a typical binding of 8 MeV for a normal nucleus. (In this calculation I calculated the neutron reduced mass assuming a nuclear target with A=150.)

Now I should stress that here, as before in these lectures, I have neglected the spins of the particles. This clearly comes in because, when one uses the distribution for photons mentioned earlier, the two helicity states of the photon have been counted. Thus we should have carefully averaged over initial photon spins, etc., in doing our cross sections. Our earlier work on and other reactions had the same defects. Thus don't take factors of 2 seriously. Also note that the derivation done above can be run through for any other reaction, like , which is needed in the home work.

We mentioned before that gaps existed at the shell closures, at N 82 and 126. When a shell gap is reached in the r-process (I'll illustrate this on the board), the neutron number of the nucleus remains fixed until the nucleus can change sufficiently to overcome the gap, i.e., bring another bound neutron quantum level below the continuum. Thus N remains fixed while successive decays occur. In the (N,Z) trajectory, the path is along increasing Z with fixed N: every decay is followed by an reaction to fill the open neutron hole, but no further neutrons can be captured until the gap is overcome.

The path of the r-process is shown in the accompanying figure. The closed neutron shells are called the waiting points, because it takes along time for the successive decays to occur to allow progression through higher N nuclei. The decays are slow at the shell closures. Just as in the s-process, the abundance of a given isotope is inversely proportional to the decay lifetime. Thus mass builds up at the waiting points, forming the large abundance peaks seen in the figure.

After the r-process finishes (the neutron exposure ends) the nuclei decay back to the valley of stability by decay. This can involve some neutron spallation (-delayed neutrons) that shift the mass number A to a lower value. But it certainly involves conversion of neutrons into protons, and that shifts the r-process peaks at N 82 and 126 to a lower N, off course. This effect is clearly seen in the abundance distribution: the r-process peaks are shifted to lower N relative to the s-process peaks.

It is believed that the r-process can proceed to very heavy nuclei (A 270) where it is finally ended by -delayed and n-induced fission, which feeds matter back into the process at an A A/2. Thus there may be important cycling effects in the upper half of the r-process distribution.

What is the site(s) of the r-process? This has been debated many years and still remains a controversial subject.
The r-process requires exceptionally explosive conditions (n) cm   T K   t 1sec both primary and secondary sites proposed primary: requiring no preexisting metals
secondary: neutron capture occurs on s-process seeds
different evolution with galactic metalicity suggested primary sites:
) neutronized atmosphere above proto-neutron star in a Type II supernova
) neutron-rich jets from supernovae or neutron star mergers
) inhomogeneous big bangs

secondary sites (where (n) can be lower)
) He/C zones in Type II supernovae
) red giant He flash
) spallation neutrons in He zone

The balance of evidence favors a primary site, so one requiring no preenrichment of heavy s-process metals. Among the evidence:
1) HST studies of very-metal-poor halo stars: The most important evidence are the recent HST measurements of Sneden et al. of very metal-poor stars ([Fe/H] -1.7 to -3.12) where an r-process distribution very much like that of our sun has been seen for Z 56. Furthermore, in these stars the iron content is variable. This suggests that the "time resolution" inherent in these old stars is short compared to galactic mixing times (otherwise Fe would be more constant). The conclusion is that the r-process material in these stars is most likely from one or a few local supernovae. The fact that the distribution matches the solar r-process (at least above charge 56) strongly suggests that there is some kind of unique site for the r-process: the solar r-process distribution did not come from averaging over many different kinds of r-process events. Clearly the fact that these old stars are enriched in r-process metals also strongly argues for a primary process: the r-process works quite well in an environment where there is little initial s-process metals.

2) There are also fairly good theoretical arguments that a primary r-process occurring in a core-collapse supernova might be viable. First, galactic chemical evolution studies indicate that the growth of r-process elements in the galaxy is consistent with low-mass Type II supernovae in rate and distribution. More convincing is the fact that modelers have shown that the conditions needed for a r-process (very high neutron densities, temperatures of 1-3 billion degrees) might be realized in a supernova. The site is the last material blown off the supernova, the material just above the mass cut. When this material is blown off the star initially, it is a very hot neutron-rich, radiation-dominated gas containing neutrons and protons, but an excess of the neutrons. As it expands off the star and cools, the material first goes through a freezeout to particles, a step that essentially locks up all the protons in this way. Then the s interact through reactions like



to start forming heavier nuclei. Note, unlike the big bang, that the density is high enough to allow such three-body interactions to bridge the mass gaps at A = 5,8. The capture continues up to heavy nuclei, to A 80-100, in the network calculations. This was a surprising results of the network calculations that were performed. The net result is a small number of "seed" nuclei, a lot of s, and left over excess neutrons. These neutrons preferentially capture on the heavy seeds to produce an r-process. Of course, what is necessary is to have 100 excess neutrons per seed in order to successfully synthesis heavy mass nuclei. Some of the modelers find conditions where this almost happens.

There are some very nice aspects of this site: the amount of matter ejected is about 10 solar masses, which is just about what is needed over the lifetime of the galaxy to give the integrated r-process metals we see, taking a reasonable supernova rate. But there are also a few problems:
The calculated entropies, neutron fractions are a bit too low to produce a successful A 190 peak.
, chronometers argue for two distinct types of r-process events, with the A 130 associated with rarer, larger events and the A 190 with more frequent, smaller events. It has been suggested that these might be supernovae leading to neutron stars vs. those leading to black holes, respectively.

There are some interesting neutrino physics issues that I'll mention briefyly which depend on the characteristics of the supernova (or "hot bubble") r-process: r-process T: 3 k k
freezeout radius 600-1000 km
L (0.015-0.005) ergs/(100km)s
3 sec Thus the neutrino fluence after freezeout (when the temperature has dropped below 10K and the r-process stops) is (0.045-0.015) ergs/(100km) the ejection of r-process material occurs
in an intense neutrino flux This brings up the question of whether the neutrino flux could have any effect on the r-process. This is actually a more general issue about a nucleosynthesis mechanism called the neutrino process that we will now discuss.