3.4: Generation of Nuclear Energy

We have established that the most important source for energy in the sun results from nuclear processes. Therefore, it is time that we look more closely at the details of those processes with a view of quantifying the dependence of the energy generation rate on the local values of the state variables. During the last 50 years, great strides have been made in understanding the details of nuclear interactions. They have revealed themselves to be remarkably varied and complex. We do not attempt to delve into all these details; rather we sketch those processes of primary importance in determining the structure of the star during the majority of its lifetime. We will leave to others to describe the spectacular nuclear pyrotechnics which occur during the terminal phases of the evolution of massive stars. Indeed, the equilibrium processes that occur in the terminal phases of stellar evolution, giving rise to most of the heavier elements, are beyond the scope of this book. Nor do we attempt to develop a complete, detailed quantum theory of nuclear energy production. Those who thirst after that specific knowledge are referred to the excellent survey by Cox and Giuli² and other references at the end of this chapter. Instead, we concentrate on the physical principles which govern the production of energy by nuclear fusion.

a. General Properties of the Nucleus

The notion that the atom can be viewed as being composed of a nucleus surrounded by a cloud of electrons which are confined to shells led to a very successful theory of atomic spectra. A very similar picture can be postulated for the nucleus itself, namely, that nucleons are arranged in shells within the nucleus and undergo transitions from one excited state (shell) to another subject to the same sort of selection rules that govern atomic transitions. The origin of the shell structure of any nucleus is that nucleons are fermions and therefore must obey the Pauli Exclusion Principle, just as the atomic electrons do. Thus, only two protons or two neutrons may occupy a specific cell in phase space (protons and neutrons have the same spin as electrons, so each species can have two of its kind in a quantum state characterized by the spatial quantum numbers).

However, the nucleons are much more tightly bound in the nucleus than the electrons in the atom. Whereas the typical ionization energy of an atom can be measured in tens to thousands of electron volts, the typical binding energy of a nucleon in the nucleus is several million electron volts. This large binding energy and the Pauli Exclusion Principle can be used to explain the stability of the neutron in nuclei. Although free neutrons beta-decay to protons (and an electron and an electron antineutrino) with a half-life of about 10 min, neutrons appear to be stable when they are in nuclei. If neutrons did decay, the resulting proton would have to occupy one of the least tightly bound proton shells, which frequently costs more energy than is liberated by the beta decay of the neutron. Thus, unless the neutron decay can provide sufficient energy for the decay products to be ejected from the nucleus, the neutron must remain in the nucleus as a stable entity.

In general, for a nucleus to be stable, its mass must be less than the sum of the masses of any possible combination of its constituents. Thus, Li⁵ is not stable, whereas He⁴ is. A more detailed explanation of the reasons for the stability or instability of a particular nucleus requires a considerably more detailed discussion of nuclear interactions and nuclear structure than is consistent with the scope of this book. However, note that the instability of mass-5 nuclei posed one of the greatest barriers of the century to the understanding of the evolution of stars. The nuclear evolution beyond mass 5 was finally solved by Fred Hoyle, who showed that the triple-a process, which we consider later, could actually initiate synthesis of all the nuclei heavier than mass 12.

Before we turn to the specifics of nuclear energy production, it is worth saying something about notation. Consider the reaction where a particle a hits a nucleus \(X\), producing a nucleus ϒ and other particle(s) b. In other words, \[a+X \rightarrow Y+b\label{3.3.1}\]

Such a reaction can be written \(X(\mathrm{a}, \mathrm{~b}) \Upsilon\). Usually for such a reaction to happen, it must be exothermic. That is, the rest energy of the initial constituents of the reaction must exceed that of the products. \[E_n \equiv\left(m_a+M_X-m_b-M_Y\right) c^2>0\label{3.3.2}\]

b. The Bohr Picture of Nuclear Reactions

Although quantum mechanics formally describes the transition from the initial to the final state, it is convenient to break down the process and to say that a compound nucleus is formed by the collision and subsequently decays to the reaction products. With this assumption, a reaction can be viewed as consisting of two steps \[a+X \rightarrow C^* \rightarrow Y+b\label{3.3.3}\]

where C^* is the compound nucleus and the asterisk indicates that it is in an excited state. The compound nucleus can decay by various modes which have these convenient physical interpretations: \[\begin{array}{ll}
a+X \rightarrow C^* \rightarrow X+a & \text { elastic scattering } \\
a+X \rightarrow C^* \rightarrow X^*+a & \text { inelastic scattering } \\
a+X \rightarrow C^* \rightarrow Y+b & \text { particle emission } \\
a+X \rightarrow C^* \rightarrow C+\gamma & \text { radiative capture }
\end{array}\label{3.3.4}\]

Elastic scattering simply involves a particle "bouncing off" the nucleus in such a manner that the momentum and kinetic energy of both the constituents are conserved. However, inelastic scattering results in the nucleus being left in an excited state at the expense of the kinetic energy of the reactants. Particle emission is the process most often associated with nuclear reactions. The results of the interaction leave both reactants changed. Under certain conditions, the Bohr picture fails for these interactions since they proceed directly to the final state without the formation of a compound nucleus. In radiative capture, the compound nucleus decays from the excited state to a stable state by the emission of a photon.

The validity of this two-stage process, due to Neils Bohr, depends on the lifetime of the compound nucleus C*. The duration of a nuclear collision can be characterized by the time it takes for the colliding particle to cross the nucleus. For typical nuclear radii and relative collision speeds of, say 0.1c, this is about 10-²¹ s.

If the lifetime of the compound nucleus is long compared to this crossing time, you may assume that the nucleons of the compound nucleus have undergone many "collisions" and that the interaction energy has been statistically redistributed among them. In short, the compound nucleus will have reached statistical equilibrium and reside in a well defined state. In some sense, the compound nucleus can be said to exist. This effectively separates the details of the C* → ϒ + b reaction from those of the a + X → C* reaction. One might say that C* will have 'forgotten' about its birth.

More properly, the statistical equilibrium state of C* is independent of the approach to that state. This was the case in Chapter 1 where we considered the establishment of statistical equilibrium for a variety of gases. It will also be the case when we consider the details of absorption and reemission of photons by atoms much later. Another way of stating this condition is to say that the average distance between collisions with the nucleons (the mean free path) is much less than the size of the nucleus. Experimentally, this appears to be true for collision energies below 50 Mev. Thus, if the energy is shared among more than a half dozen nucleons, any given nucleon will not have sufficient energy to exceed the binding energy and escape. The result is the formation of a stable nucleus by means of radiative capture.

Figure 3.1 shows a typical damping, or dispersion profile. A marked increase in the interaction probability occurs in the vicinity of the resonance energy \(\overline{\mathrm{E}}\). The width of the curve is characterized by the damping constant \(\Gamma\).

By analogy to the photoexcitation of atoms, called bound-bound transitions, there exist resonances for nuclear reactions, particularly at low energy. A resonance is an enhancement in the probability that a nuclear reaction will take place. Classically, one may view these as collision energies which excite particular nucleon shell transitions within the nucleus. These energies will be particularly favored for interactions and are known as the resonance energies.

The probability density distribution with energy is characterized by a function known as a damping, or dispersion, profile whose form we will derive in some detail when we consider the formation of spectral lines in Chapter 13. All that need be understood is the general topological shape (see Figure 3.1) and the fact that the width of the probability maximum can be characterized by a width in energy usually denoted by \(\Gamma\). As long as the resonance is a simple one and not blended with others, the energy at which the peak of the probability distribution occurs is known as the resonance energy.

c. Nuclear Reaction Cross Sections

The words cross section have come to have a somewhat generic meaning in nuclear physics as a measure of the likelihood of a particular reaction taking place, in the sense that the larger the cross section, the greater the probability that the reaction will happen. The simplest way to visualize a reaction cross section is to consider the classical notion of a collision cross section. If you were to shoot a bullet through a swarm of hornets, the probability of hitting a particular hornet would be proportional to the cross-sectional area of the hornet as seen by the bullet. Of course, the cross-sectional area of the bullet will also play a role in determining the likelihood of hitting the hornet. The combined effect of these two cross-sectional areas is said to represent the geometric cross section of the collision. In a similar manner, one may interpret a nuclear reaction cross section as the "effective" geometric cross-sectional area of a collision between the particle and the nucleus. Remember that this is not a simple geometric cross section unless you are comfortable with the notion that the nucleus appears to have very different "sizes", as seen by the colliding particle, depending on the particle's energy.

In practice, the nuclear cross section will depend on all the quantities that govern the interaction between the colliding particles and the nucleons in the shell structure of the nucleus. The detailed calculation is usually very complicated, depending on the approximate wave function of the nucleus and the wave function of the colliding particle. A common approximation formula for nuclear cross sections known, as the Breit-Wigner 1-level dispersion formula, is \[\sigma(a, b)=(2 l+1) \pi ƛ^2 \omega T_l(a) Y(E) S G(b)\label{3.3.5}\]

where

\[\begin{aligned}
&ƛ=\frac{\lambda}{2 \pi}=\frac{\cancel{h}}{p}=\frac{\cancel{h}}{\sqrt{2 m_a E}}\\
&L=\cancel{h}[l(l+1)]^{1 / 2}=\text { orbital angular momentum of } m_a \text { about } X\\
&T_l(a)=\text { transmission function of particle } a\\
&\omega=\frac{2 J+1}{\left(2 I_a+1\right)\left(2 I_x+1\right)} \sim 1\\
&I_a=\text{spin of particle }a\\
&I_x=\text{nuclear spin of particle }X\\
&\vec{J}=\overline{I}_a+\vec{I}_x\\
&Y(E) \text{ allows for resonances}\\
&S=1\text{ or }2 \text{ depending on particle degeneracy}\\
&G(b) \sim \frac{\Gamma(b)}{\Gamma}=\text { branching ratio for } b\\
& \Gamma_\gamma=\text { damping constant for radiative capture (i.e., particle } b \text { is a photon) } \\
& \Gamma_i=\text { all other possible decay damping constants }
\end{aligned}\label{3.3.6}\]

We will make no attempt to derive this result. However, we do try to show that the result at least contains the right sort of terms and is reasonable. The term \(\pi ƛ^2\) is essentially the geometric cross section of the colliding particle as it is related to the particle's de Broglie wavelength. The angular momentum term \((2\ell+1)\) is a measure of the impact parameter and the energy. As \(\ell\) increases, so does the impact parameter. For constant angular momentum, an increasing impact parameter will mean a decreasing collision energy, implying a net increase of the collision probability. However, as the impact parameter increases and the collision energy drops, the probability that the colliding particle will be able to overcome the coulomb barrier decreases drastically. Thus, we need be concerned only with \(\ell=0\), or 1. The term transmission function of particle a includes the probability that the particle will penetrate the coulomb barrier of the nucleus. The parameter ω allows for the spin-spin interactions of the nucleus and the particle and is of the order unity. Function ϒ(E) includes the effects of resonances and from the dispersion curve in Figure 3.1 can clearly be a very strong function of collision energy E. The spin degeneracy parameter \(\boldsymbol{S}\) is generally 1 except when a and \(X\) are the same kind of particle and also have zero spin; then \(\boldsymbol{S}=2\). Finally, G(b) is a measure of the probability that particle b will be created from the compound nucleus as opposed to some other possibility. Now that we have the nuclear reaction cross sections, we have to determine the rate at which collisions will occur. Then we will be able to find the energy produced by stellar material.

d. Nuclear Reaction Rates

The reaction cross section of the previous section can be measured as a function of the collision energy (and some atomic constants) alone and therefore can be written as a function of the particle's velocity v relative to the target. By resurrecting the geometric interpretation of the cross section, the number of particles crossing an area (colliding with the target) per unit time is just Nσ(v)v where N is the density of colliding particles (see Figure 3.2)

Consider collisions between two different kinds of particles with a number density in phase space of dN₁ and dN₂. To obtain the number of collisions per second per unit volume, we must integrate over all available velocity space. That is, we must sum over the collisions between particles so that the collision rate r is \[r=\iint v \sigma(v) d N_1\left(\vec{v}_1\right) d N_2\left(\vec{v}_2\right)\label{3.3.7}\]

Figure 3.2 is a schematic representation of a collision between particle a and a target with a geometrical cross section σ.

Let us assume that the velocity distributions of both kinds of particles are given by maxwellian velocity distributions \[d N=N\left(\frac{m}{2 \pi k T}\right)^{3 / 2} e^{-m v^2 /(2 k T)} d v\label{3.3.8}\]

so that equation \ref{3.3.7} becomes \[r=N_1 N_2\left(\frac{m_1}{2 \pi k T}\right)^{3 / 2}\left(\frac{m_2}{2 \pi k T}\right)^{3 / 2} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \exp \left(\frac{-m_1 v_1^2}{2 k T}-\frac{m_2 v_2^2}{2 k T}\right) v \sigma(v) d \vec{v}_1 d \vec{v}_2\label{3.3.9}\]

If we transform to the center-of-mass coordinate system, assuming the velocity field is isotropic so that the triple integrals of equation \ref{3.3.9} can be written as spherical "velocity volumes", then we can rewrite equation \ref{3.3.9} in terms of the center of mass velocity v0 and the relative velocity v as \[\begin{aligned}
r= & N_1 N_2\left(\frac{m_1}{2 \pi k T}\right)^{3 / 2}\left(\frac{m_2}{2 \pi k T}\right)^{3 / 2} \int_0^{\infty} \int_0^{\infty} \exp \left[\frac{\left(m_1+m_2\right) v_0^2+\tilde{m} v^2}{2 k T}\right] \\
& \cdot v \sigma(v) 4 \pi v_0^2\left(4 \pi v^2\right) d v_0 d v
\end{aligned}\label{3.3.10}\]

where \[\begin{aligned}
\tilde{m} & =\frac{m_1 m_2}{m_1+m_2} \\
\vec{v} & =\vec{v}_1-\vec{v}_2 \\
\vec{v}_0 & =\frac{m_1 \vec{v}_1+m_2 \vec{v}_2}{m_1+m_2}
\end{aligned}\label{3.3.11}\]

The integral over \(\mathrm{v_0}\) is analytic and is \[\int_0^{\infty} e^{-\left(m_1+m_2\right) v_0^2 /(2 k T)} 4 \pi v_0^2 d v_0=\left(\frac{2 \pi k T}{m_1+m_2}\right)^{3 / 2}\label{3.3.12}\]

which reduces equation \ref{3.3.10} to \[r=N_1 N_2\left(\frac{\tilde{m}}{2 \pi k T}\right)^{3 / 2} \int_0^{\infty} e^{-\tilde{m} v^2 /(2 k T)} v \sigma(v)\left(4 \pi v^2\right) d v\label{3.3.13}\]

Since the relative kinetic energy in the center of mass system is \(\mathrm{E}=\frac{1}{2} \tilde{\mathrm{~m}} \mathrm{v}^2\), we can rewrite equation \ref{3.3.13} in terms of an average reaction cross section \(<\sigma(\mathrm{v}) \cdot \mathrm{v}>\) so that \[r=N_1 N_2\langle\sigma(v) v\rangle\label{3.3.14}\]

where \[\langle\sigma(v) v\rangle=\frac{2}{\sqrt{\pi}}\left(\frac{1}{k T}\right)^{3 / 2} \int_0^{\infty} e^{-E /(k T)} \sigma(v) v E^{1 / 2} d E=\int_0^{\infty} n(E) v \sigma(v) d E\label{3.3.15}\]

Thus \(<\sigma(\mathrm{v}) \mathrm{v}>\) is the "relative energy" weighted average of the collision probability of particle 1 with particle 2. When this average cross section is written, the explicit dependence on velocity is usually omitted, so that \[\langle\sigma v\rangle \equiv\langle\sigma(v) v\rangle\label{3.3.16}\]

If the collisions involve identical particles, then the number of distinct pairs of particles is N(N-1)/2 so the factor of N₁N₂ in equation \ref{3.3.14} is replaced by N²/2.

If we call the energy produced per reaction Q, we can write the energy produced per gram of stellar material as \[\epsilon=\frac{r_{a, X} Q}{\rho}=\frac{N_a N_X\langle\sigma v\rangle Q}{\rho}\label{3.3.17}\]

The number densities can be replaced with the more common fractional abundances by mass to get \[\epsilon=\left[N_0^2 Q\langle\sigma v\rangle\left(\frac{X_X}{m_X}\right)\left(\frac{X_a}{m_a}\right)\right] \rho\label{3.3.18}\]

where N₀ is Avogadro's number. Since <σv> is a complicated function of temperature and must be obtained numerically, equation \ref{3.3.18} is usually approximated numerically as \[\epsilon \approx \epsilon_0 \rho T^v\label{3.3.19}\]

where \[\epsilon_0=\frac{N_0^2\left(X_a / m_a\right)\left(X_X / m_X\right)\left(\left.Q\langle\sigma v\rangle\right|_{T_0}\right)}{T_0^v}\label{3.3.20}\]

where \(v\) itself is very weakly dependent on the temperature. Most of the important energy production mechanisms have this form. Equation \ref{3.3.19} expresses the energy generated for a specific energy generation mechanism in terms of the state variables T and p. This is what we were after. Formulas such as these, where ε0 has been determined, will enable us to determine the energy produced throughout the star in terms of the state variables. Before turning to the description of processes which impede the flow of this energy, let us consider a few of the specific nuclear reactions for which we have expressions of the type given by equation \ref{3.3.20}.

e. Specific Nuclear Reactions

The nuclear reactions that provide the energy for main sequence stars all revolve on the conversion of hydrogen to helium. However, this is accomplished by a variety of ways. We may divide these ways into two groups. The first is known as the proton-proton cycle (p-p cycle) and it begins with the conversion of two hydrogen atoms to deuterium. Several possibilities occur on the way to the production of ⁴He. These alternate options are known as P2-P6 cycles. In addition to the proton-proton cycle, a series of nuclear reactions involving carbon, nitrogen, and oxygen also can lead to the conversion of hydrogen to helium with no net change in the abundance of C, N, and O. For this reason, it is known as the CNO cycle. These reactions and their side chains as given by Cox and Giuli² are given in Table 3.3

Table 3.3 Common Nuclear Reations

Besides the steps marked with asterisks, which denote reactions that occur by spontaneous decay and do not depend on local values of the state variables, the steps that are the important contributors to the energy supply have their contribution (their Q value) indicated. The energy of the neutrinos has not been included since they play no role in determining the structure of normal stars. When the p-p cycle dominates on the lower main sequence, most of the energy is produced by means of the P1 cycle. The neutrino produced in the fifth step of the P2 cycle is the high energy neutrino which has been detected, but in unexpectedly low numbers, by the neutrino detection experiment of R. Davis in the Homestake Gold Mine. In general, the relative importance of the P1 cycle relative to P2 and P3 is determined by the helium abundance, since this governs the branching ratio at step 3 in the p-p cycle. If ⁴He is absent, it will not be possible to make ⁷Be by capture on ³He.

Virtually all the energy of the CNO cycle is produced by step 6 as the production of ¹²C from ¹⁵N is strongly favored. However, all the higher chains close with only the net production of ⁴He. The first stage of the P4 cycle is endothermic by 18 keV so unless the density is high enough to produce a Fermi energy of 18 keV, the reaction does not take place. This requires a density of \(\rho>2 \times 10^4 \mathrm{~g} / \mathrm{cm}^3\) and so will not be important in main sequence stars. Once ³H is produced it can be converted to ⁴He by a variety of processes given in step 4. The last two are sometimes denoted P5 and P6, respectively, and are rare.

While the so-called triple-α process is not operative in main sequence stars, it does provide a major source of energy during the red-giant phase of stellar evolution. The extreme temperature dependence of the triple-a process plays a crucial role in the formation of low-mass red giants and, we shall spend some time with it later. The ⁸Be* is unstable and decays in an extremely short time. However, if during its existence it collides with another ⁴He nucleus, ¹²C can form, which is stable. The very short lifetime for ⁸Be* basically accounts for the large temperature dependence since a very high collision frequency is required to make the process productive.

The exponent of the temperature dependence given in equation \ref{3.3.20} and the constant ε₀ both vary slowly with temperature. This dependence, as given by Cox and Giuli² (p. 486), is shown in Table 3.4.

Table 3.4 Temperature Dependence of \(\nu\) and \(\epsilon_0\)

Proton-Proton			CNO Cycle		Triple-\(\alpha\) Process
\(T_6\)	\(\epsilon_0(cgs)\)	\(\nu\)	\(\epsilon_0(cgs)\)	\(\nu\)	\(T_8\)	\(\epsilon_0(cgs)\)	\(\nu\)
10 20 40 80 100	7 x 10^-2 1 9 43 __	4.60 3.54 2.72 2.08 __	3 x 10^-4 4.5 x 10^2 3 x 10^7 2 x 10^11 2 x 10^12	22.9 18 14.1 11.1 10.2	0.8 1.0 2.0 3.0 4.0	2 x 10^-12 4 x 10^-8 15 6 x 10^3 10^5	49 41 19 12 7.9

The temperature \(\mathrm{T}_6\) in Table 3.4 is given in units of \(\mathrm{T}_6\). Thus \(\mathrm{T}_6=1\) is \(10^6\mathrm{K}\). It is a general property of these types of reaction rates that the temperature dependence "weakens" as the temperature increases. At the same time the efficiency ε₀ increases. In general, the efficiency of the nuclear cycles rate is governed by the slowest process taking place. In the case of p-p cycles, this is always the production of deuterium given in step 1. For the CNO cycle, the limiting reaction rate depends on the temperature. At moderate temperatures, the production of ¹⁵O (step 4) limits the rate at which the cycle can proceed. However, as the temperature increases, the reaction rates of all the capture processes increase, but the steps involving inverse β decay (particularly step 5), which do not depend on the state variables, do not and therefore limit the reaction rate. So there is an upper limit to the rate at which the CNO cycle can produce energy independent of the conditions which prevail in the star. However, at temperatures approaching a billion degrees, other reaction processes not indicated above will begin to dominate the energy generation and will circumvent even the beta-decay limitation.

We have now determined the various sources of energy that are available to a star so that it can shine. Clearly the only viable source of that energy results from nuclear fusion. The condition for the production of energy by nuclear processes can occur efficiently only under conditions that prevail near the center of the star. From there, energy must be carried to the surface in some manner in order for the star to shine. In the next chapter we investigate how this happens and describe the mechanisms that oppose the flow.