§2.1 Nuclear burning through resonances

The general expression for a reaction rate per particle pair in a stellar environment is [Rolfs and Rodney 1988]:

$⟨ σ ν ⟩ = \int_{0}^{\infty} φ (ν) ν σ (ν) d ν$	(2.1)

where ν is the relative velocity of the particle pair, σ(ν) is the velocity dependent cross-section for the reaction, and φ(ν) is the velocity distribution of the particles.

Consider a nuclear reaction, I+a→F+b, where I and F are the initial and final “heavy nucleus”, while a and b are the incoming and outgoing particles, which here may be α’s, p’s or γ’s. The total reaction rate is:

$r = N_{I} N_{a} ⟨ σ ν ⟩$	(2.2)

where the abundances of the two types of particles are denoted N_I and N_a.

The lifetime of I in the burning environment due to the given reaction is:

$τ (A) = \frac{1}{N_{I} ⟨ σ ν ⟩}$	(2.3)

At high enoug temperatures, both the reaction and its inverse may occur, I+a↔F+b. The convention here is that the forward reaction, I+a→F+b, is the exothermic reaction. The net energy generation, usually expressed in erg⋅g^-1⋅s^-1, is then:

$ϵ_{net} = ϵ_{Ia} + ϵ_{Fb} = (r_{Ia} - r_{Fb}) Q / ρ$	(2.4)

In equation (2.4), Q is the energy liberated in the forward reaction in ergs, and ρ is the density in g⋅cm^-3.

For the temperatures and densities of interest here, we will assume that isolated and narrow resonances dominate the reaction rate (which is not always true, see §5.1.4). Formally, this means Γ ≪ E_R, where E_R is the resonance energy and Γ is its width. For a Maxwell-Boltzmann velocity distribution, the narrow-resonance reaction rate per pair is expressed, using equation (2.1):

$⟨ σ ν ⟩ = {(\frac{8}{π μ})}^{1 / 2} \frac{1}{{(kT)}^{3 / 2}} E_{R} \exp (\frac{- E_{R}}{kT}) \int_{0}^{\infty} σ_{BW} (E) dE$	(2.5)

In equation (2.5), T is the temperature, μ=m_Im_a/(m_I + m_a) is the reduced mass of the interacting nuclei, and σ_BW is the Breit-Wigner expression for the energy-dependent cross section through a resonance. Substituting the Breit-Wigner expression into (2.5), we obtain:

$⟨ σ ν ⟩ = {(\frac{2 π}{μ kT})}^{3 / 2} ℏ^{2} {(ω γ)}_{R} \exp (\frac{- E_{R}}{kT}) f$	(2.6)

where:

$ω = \frac{2 J + 1}{(2 J_{I} + 1) (2 J_{a} + 1)} (1 + δ_{Ia})$	(2.7)
$γ = \frac{Γ_{a} Γ_{b}}{Γ}$	(2.7)

The factor, f, is an electron screening factor and reflects the enhancement of the rate by the Coulomb potential. It can be calculated using atomic physics considerations [Salpeter 1954]. All the nuclear properties are contained in the resonance energy ( E_R) and the resonance strength (ωγ). The resonance strength contains two factors, the first of which is just statistical, dependent on the spins of the incoming particles and on the spin of the resonance. (The δ_ab term takes care of the case of identical particles.) The second factor summarizes the nuclear structure information in terms of the partial widths, Γ_a and Γ_b, and the total width of the resonance, Γ. If a and b are the only open channels, then Γ=Γ_a+Γ_b; furthermore, if Γ_a≪Γ_b, then ωγ≈ωΓ_a and the uncertainty in Γ_a dominiates the uncertainty in resonance strength.

As described by Iliadis [Iliadis 1997], a single-particle partial width may be parameterized as follows:

$Γ_{c} = 2 P_{c} (\frac{ℏ^{2}}{μ_{c} a_{c}^{2}}) C^{2} S θ_{sp}^{2}$	(2.8)
$θ_{sp}^{2} = \frac{a_{c}}{2} ϕ_{l}^{2} (a_{c})$	(2.8)

The subscript c refers to the single-particle channel (the single particle having a mass number A_c) in or out of the nucleus with mass number A. Technically, the α-channel is not single-particle, but a similar formula applies for that case. P_c contains the Coulomb and centrifugal penetrability; μ_c is the reduced mass, and a_c ≡ a₀ (A^1/3 + A_c^1/3) is the channel radius, where some value around 1.2 fm is chosen for a₀. C is the isospin Clebsch-Gordan coefficient for coupling the nuclear state to the decay channel. S is the single-particle spectroscopic factor, which may be roughly understood as the overlap integral between the actual state wavefunction and a coupled-channel wavefunction describing the initial nucleus plus c [Brussard and Glaudemans 1977]. The final factor θ_sp² is the dimensionless single-particle reduced width and is calculated by evaluating the single-particle radial wave function of the l orbit at a_c. This wave function is normalized to unity inside the channel radius. θ_sp² may vary between zero and one, one being the Wigner limit [Iliadis 1997].

It is immediately apparent from (2.6) that the reaction rate depends most strongly on the resonance energy. There have been many experiments that have precisely located important resonances in ¹⁹Ne. Next in importance is the linear dependence on the resonance strength. The experiment described in this thesis measures proton- or alpha-decays from excited states. The measured decays are normalized to the total detected population of the state, so that we actually measure the branching ratios. It is possible to rewrite the expression for γ:

$γ = (\frac{Γ_{a}}{Γ}) (\frac{Γ_{b}}{Γ}) Γ$	(2.9)

This new form shows proportionality to 3 factors. The first two factors are the dimensionless branching ratios.

In addition to measuring the branching ratios, the experiment is sensitive to the angular distribution of the decays. In certain cases, this information may be used to determine the spin and parity of the states in ¹⁹Ne.

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