ASTR 498N                     Lecture 14                           Nuclear burning II

 

(online at www.astro.umd.edu/~drabin/)

 

We do not argue with the critic who urges that stars are not hot enough for this process; we tell him to go and find a hotter place.

                A. Eddington

               

 

Hydrogen burning and the proton-proton chain

 

In the last lecture, we saw that the first conundrum of stellar nuclear fusionthe apparently impossibility of overcoming the Coulomb barrier at achievable temperaturesis solved by quantum tunneling.

 

The second conundrum arose from the inconvenient fact that the nuclear force between two protons is not quite strong enough to produce a bound state of ²He, so that the most obvious way to kick off hydrogen burning doesn’t work:

 

 

 

H. Bethe found the way around this in 1939:  start with the weak interaction

                   

                               

 

to create deuterium.  The bare reaction  is always endothermic because , but the binding energy of  ²H more than compensates and allows the pp reaction to be exothermic.  Nevertheless, it is very slow:  a proton in the center of the Sun has to wait almost 10¹⁰ y on average before fusing with another proton.  This sets the timescale for hydrogen burning.

 

             Q   What happens to the positron produced by the pp reaction?

 

Once deuterons are formed, much faster reactions lead to the production of  ⁴He through the proton-proton chain, summarized below:

 

 

 

 

 

 

 

The second reaction in the proton-proton chain occurs 17 orders of magnitude faster than the first!  In particular, it takes ~10¹⁰ y to produce a deuteron but only ~1 s to destroy it.  This leads to a very low equilibrium concentration of deuterium, about 1 deuteron for every 10¹⁸ protons.  In contrast, the terrestrial abundance of  ²H  is much higher, about 0.015% of all hydrogen.  We conclude that terrestrial deuterium was not produced in stars like the Sun (we think that almost all deuterium was produced in the big bang).

 

Some of you may find the following representation to be visually memorable (I see a guy waving at me):

 

 

 

 

 

The relative importance of the three branches of the proton-proton chain is significantly temperature dependent, as shown below:

 

 

 

Although PPIII produces a tiny fraction of the helium in the Sun, the  reaction produces the most energetic neutrinos and dominates the neutrino signal measured by detectors based on the reaction .  We will return to the solar neutrino problem in another lecture.

 

 

 

The CNO cycle

                             

There is a second, independent set of reactions that can produce ⁴He from H.  The so-called “CNO bi-cycle” is in fact dominated by the simpler carbon-nitrogen cycle, illustrated below:

 

 

 

The carbon-nitrogen cycle is not an important source of energy production in the Sun (the split is about 98% proton-proton, 2% carbon-nitrogen) but becomes increasingly important on the upper main sequence.  This is because the carbon-nitrogen cycle involves heavy elements, which have high Coulomb barriers.  The difficulty of penetrating such barriers leads to fusion rates that increase rapidly with temperature.

 

The carbon-nitrogen cycle is also important in determing the relative abundances of ¹³C, ¹⁴N, and ¹⁵N.  In chemical equilibrium, the relative abundances of these three nuclei are inversely proportional to their fusion rates.  Since the  reaction is the slowest, the relative abundance of ¹⁴N is the largest.  Nitrogen in the solar system was primarily produced by the operation of the carbon-nitrogen cycle in earlier generations of nearby stars.

 

The full CNO cycle includes a branch that only occurs about 0.04% of the time:

 

                               

 

 

 

The following figure compares the rate of energy generation as a function of temperature for the two types of hydrogen burning, PP and CNO:

 

 

Over a limited range of temperature, the energy generation rate for each of the processes can be represented by a power law,

 

                               

 

The following table shows the temperature dependence of ε₀ and ν as a function of temperature for the PP, CNO, and triple-alpha (helium burning) processes:

 

 

The temperature dependence of the CNO cycle and triple-alpha process is usually quite steep.  Note the general trend, for all processes, that the temperature dependence flattens with increasing temperature while the efficiency ε₀  increases.  

 

 

Helium burning

 

Carbon-based, oxygen-breathing life forms owe a considerable debt to helium burning in stars, which produces both elements.  As in the case of hydrogen burning, the most obvious reaction,

 

                               

 

is unstable, with a lifetime of only   yet another conundrum.  Indeed, there are no stable nuclei with A = 8 or A = 5, which foiled many attempts to find chains of light-particle reactions that would build up heavier elements.  The solution came in 1952, when E. Salpeter realized that the brief ⁸Be lifetime is nevertheless long compared to the mean collision (scattering) time of alpha particles at T ~ 10⁸  K.  Thus, even with only one out of 10⁹ nuclei being ⁸Be, there is a non-negligible probability that an alpha particle will collide with the transient nucleus to produce carbon:

 

                               

 

However, there is still a problem:  the rate for this reaction appears to be too low to enable helium burning in a red giant with a central temperature not much above 10⁸ K.  On that basis, F. Hoyle predicted that the reaction must be enhanced by a resonance between the collision energy and an internal energy level of ⁸Be.  Hoyle was even able to predict the energy of the resonance, and it was found almost exactly where predicted.

 

The triple-alpha process therefore involves the production of an excited state of carbon (¹²C^*) and the decay of a small fraction of the excited nuclei to the ground state (most of them just decay back to ⁸Be and ⁴He):

 

 

 

Once some ¹²C has built up, oxygen and neon can be produced during helium burning by alpha capture:   and (to a minor extent) .  Further alpha capture is inhibited by the ever-growing Coulomb barrier of the heavier nuclei.

 

Again from the perspective of carbon-based life forms, we note that relative abundance of carbon, oxygen and neon produced by helium burning is sensitive to seemingly arbitrary details of nuclear structure.  Were the ¹²C^* resonance a bit less favorably placed, the triple-alpha process would be slower and the  reaction would predominate, converting most of the carbon to oxygen.  Were an excited state of ²⁰Ne of the correct nuclear parity, as well as being at a resonant energy (which it is), the  would be enhanced and most of the oxygen would be converted to neon.  These what-if speculations can fuel interesting musings on the anthropic principle.

 

 

 

 

 

Advanced stages of nuclear burning

 

Sufficiently massive stars (M  8M) can evolve beyond helium burning to burn heavier elements. 

 

Carbon burning

Carbon burning begins at  and .  Carbon burning produces neon, sodium, and magnesium through the reactions

                                      Q = 13.9 MeV

                                                 Q = 2.2 MeV
                                              Q = 4.6 MeV

                                                Q = –2.6 MeV

                                       Q = –0.1 MeV

 

Oxygen burning

Reactions between ¹²C and ¹⁶O are not important.  At carbon-burning temperatures, the higher Coulomb barrier between carbon and oxygen (Z₁Z₂ = 48) relative to carbon–carbon (Z₁Z₂ = 36) makes the carbon–oxygen rate much slower.  By the time the temperature increases to the point where the rate is significant, carbon is almost entirely exhausted.  Therefore, the next major stage of nuclear burning, commencing at T  10⁹ K, is ¹⁶O reacting with itself through the channels

 

                                         Q = 16.5 MeV

                                                    Q = 7.7 MeV

                                                    Q = 1.5 MeV

                                                Q = 9.6 MeV

                                    Q = –0.4 MeV

 

In both carbon and oxygen burning, the main reactions result in protons and alpha particles that will quickly react (low Coulomb barrier) in a network of secondary reactions.  These secondary reactions are important in determining isotopic abundances (for example,  ) and also significant for the net energy generation rates (both through exothermic reactions and neutrino losses).  Free neutrons play a key role in the synthesis of heavy elements, as we shall discuss later.

 

Photodisintegration and neon burning

At T  10⁹ K, a new type of nuclear reaction becomes possible:  the disintegration of nuclei by thermal photons.  Nuclear photodisintegration is the nuclear analogue of the photoionization of atoms and has nothing to do with spontaneous nuclear fission.  Since nuclear binding energies are typically 10⁶ times greater than atomic binding energies, photodisintegration becomes important at a few times 10⁹ K instead of a few times 10³ K.  The first major disintegration reaction is

 

                               

 

which competes with the alpha-capture production of neon:

 

                               

 

For , the disintegration reaction progressively depletes ²⁰Ne.  Some of the remaining ²⁰Ne can capture an alpha particle to produce ²⁴Mg:

 

                               

 

Silicon burning

We might suppose that, if the temperature is high enough, two magnesium nuclei could fuse to create chromium, , or similarly  for silicon and iron.  In fact, however, another process takes place before the extreme temperatures necessary to overcome the MgMg or SiSi Coulomb barrier are reached.  At , thermal photons begin to have sufficient energy to initiate the photodisintegration of silicon,

 

                                               Q ≤ –10.0 MeV

 

The ⁴He nuclei thus released can then fuse with other ²⁸Si nuclei to start a sequence of reactions that build up heavier elements by alpha capture:

 

                               

                               

                               

                              …

                               

 

Many other photodisintegration reactions are possible, including the photoejection of protons, neutron, and alpha particles from nuclei lighter than silicon.  All these reactions proceed much faster than the initial photodisintegration of silicon, which therefore sets the timescale for the whole process.  The bidirectional arrows indicate that the reactions are nearly in thermodynamic equilibrium.  In near-equilibrium, the relative abundances of different nuclei can be estimated by the familiar Saha equation.  For example, for ²⁸Si and ³²S,

 

                               

 

where μ is the reduced mass of the ²⁸Si–⁴He system (3.5 amu) and Q is the energy needed to release an alpha particle from a ³²S nucleus,

 

                               

 

The Saha equation shows that the relative abundance of ²⁸Si and ³²S is governed by the temperature and the concentration of ⁴He produced by the silicon burning.  Because of the Boltzmann factor, more tightly bound nuclei will always be favored.  In fact, silicon burning is often described as a nucelar rearrangement process in which alpha particles are preferentially transferred from less to more tightly bound nuclei.  If we remember that the curve of binding energy per nucleon rises monotonically from the light elements to a peak at atomic number 56, it is evident that silicon burning tends toward the production of iron-group nuclei (A = 56), i.e., isotopes of Cr, Mn, Fe, Co, and Ni.  Elements heavier than this are not formed during silicon burning.  The production of the iron group can be thought of as a small net flow (“leakage”) superposed on the much faster inverse (bidirectional) reactions illustrated above.

 

Iron photodisintegration

At even higher temperatures, , even the tightly bound ⁵⁶Fe can be photodisintegrated, with the net result

 

                               

 

thereby undoing almost the entire nucleosynthesis process and aborbing ~100 MeV per reaction of energy from the radiation field.   As the temperature rises above , helium becomes more abundant than iron, and eventually even helium is disintegrated into protons and neutrons, again draining energy (28 MeV per reaction) from the photon bath.  This energy (to which must be added neutrino losses) can only be supplied from gravitational contraction.  Numerically, the drain is so fierce that the stellar core must be essentially in free fall.  We have definitely left the arena of quasistatic nuclear burning and entered the cataclysmic realm of the supernova.

 

Pair production

A photon with energy  can create an e⁺e^– pair.  Although  occurs at , photons on the high-energy tail of the Planck function produce a non-negligible population of positrons at temperatures as low as a few times 10⁹ K.  This is analogous to the presence of ionized hydrogen at temperatures considerably below kT = 13.6 eV (corresponding to  ).

 

 

Summary of nuclear burning stages

 

 

The table reminds us that sharply higher temperatures are necessary to burn anything beyond hydrogen, and that the liberated energy per nucleon is at least an order of magnitude lower.  Another dramatic aspect of nuclear burning is highlighted by the following table:

 

 

On paper, the various reaction chains and cycles have a certain generic similarity.  But contrast the stately pace of hydrogen burning (some 10 million years even in a massive star) with the unimaginable fury of silicon burning:  1 day!  It’s no surprise that such a star may be headed for Big Trouble.

 

 

Synthesis of elements beyond iron

 

The creation of elements heavier than iron is a vast and complex topic, with many unsolved problems.  Also, a meaningful comparison of theoretical with observed abundances is tied inextricably to an understanding of mass loss in the late stages of stellar evolution and of supernova events—for only those elements that escape into the interstellar medium can be incorporated in later generations of stars.  Here we address only the most basic underlying question:  given the peak in binding energy per nucleon at the iron group, how is it possible to form heavier elements at all?

 

The answer is neutron capture.  So far we have considered the interactions of charged nuclei, which must overcome a Coulomb barrier, and photodisintegration, which is effective only at very high temperatures.  However, free neutrons produced during carbon, oxygen, and silicon burning can be captured by heavy nuclei at relatively low temperatures because of the absence of a Coulomb barrier.  Because we can’t give a simple answer to the key question of how many free neutrons are available, we confine ourselves to exposing the types of interactions that are possible.

 

The element  can capture a neutron to create a heavier isotope of the same element,

 

                               

 

If this isotope is stable, the process can be repeated to form  and so on.  If instead the neutron-rich nucleus is unstable to beta decay, a different element can be produced through

 

                               

 

The new element may in turn be stable, in which case it can capture one or more neutrons, or unstable, in which case it can undergo one or more successive beta decays.

 

The processes just described involve two types of reactions—neutron captures and beta decays—and two types of nuclei—stable and unstable.  Stable nuclei can undergo neutron capture only.  Unstable nuclei are susceptible to both neutron capture and beta decay; the outcome depends on the characteristic timescale for neutron capture relative to the radioactive half-life.  The beta-decay half life is a nuclear characteristic, independent of external physical conditions.  The probability of neutron capture, on the other hand, does depend on the prevailing temperature and neutron density.  Thus, neutron capture may proceed either more slowly or more rapidly than beta-decay, depending on physical conditions.  The resulting chains of reactions and nuclear products also differ.  In a seminal paper—known universally as B²FH—Burbidge, Burbidge, Fowler and Hoyle (1957, Rev. Mod. Phys. 29, 547) referred to the two regimes of neutron capture as s-process (slow) or r-process (rapid).  The figure below illustrates how the competing processes of neutron capture and beta decay define reaction paths in the periodic table and allow a nucleus to be labelled s, r, or (s,r) depending on whether it can be produced through the s-process, the r-process, or both (a few proton-rich nuclei are bypassed by both processes).   In this way, nuclei heavier than the iron peak can be produced.