Anthropic Importance of the “Bretscher State” in DT Fusion

Abstract

The term “Bretscher state” may not be as familiar as “Hoyle state,” but its anthropic importance cannot be overstated. In Big Bang nucleosynthesis, the deuterium-tritium (DT) fusion reaction ³H $(\mathit{d},\mathit{n}{)}^4$He, enhanced by the 3/2⁺ resonance due to the Bretscher state, is responsible for $\sim 99$% of primordial ⁴He. While this fact has been known for decades, it has not been widely appreciated, and we recently proposed that its significance be commemorated by naming the 3/2⁺ state after Egon Bretscher, its discoverer. The importance of the resonant nature of the DT fusion reaction has been amplified by recent activities related to the production and use of terrestrial fusion including recent, net gain shots at the National Ignition Facility. Here, we aim to highlight the anthropic importance of the ⁴He-producing DT reaction that plays such a prominent role in models of nucleosynthetic processes occurring in the early universe. This primordial helium serves as a source for the subsequent creation of $\ge 25$% of the carbon, ¹²C and other heavier elements that comprise a substantial fraction of the human body. Further studies are required to determine a better characterization of the amount of ¹²C than this lower limit of 25%. Some scenarios of core stellar nucleosynthetic yield of ¹²C suggest that even higher percentages of carbon from primordial helium are possible.

Keywords:

• “Bretscher state”
• anthropic importance
• Big Bang nucleosynthesis
• resonance

I. INTRODUCTION

We recently gave an account^[1,2] of the earliest deuterium-tritium (DT) fusion discoveries, including the 1945–1946 Los Alamos Laboratory measurements of the DT cross section down to energies relevant to fusion energy applications (10 keV) by Egon Bretscher.^[3,4] The ⁵He 3/2⁺ “Bretscher state” at 16.84 MeV, Fig. 1, leads to a hundredfold increase in the DT cross section. This resonant enhancement was a game-changer, opening up the potential for nuclear fusion technologies, as established by the recent inertial confinement fusion shot at the National Ignition Facility.^[5] It was in this context that we recently proposed, given the significance of the resonant enhancement in $\mathrm{DT}\to \mathit{n}+\mathit{\alpha }$ [or, equivalently, ${ }^3\mathrm{H}(\mathit{d},\mathit{n}{)}^4\mathrm{He}$], that the nuclear science community refer to ⁵He 3/2⁺ as the “Bretscher state.”^[6]

Fig. 1. Energy levels in the A = 5 system for the ${ }^3$H$(\mathit{d},\mathit{n}{)}^4$He reaction. Resonant enhancement occurs when the energy of the deuterium $\mathit{d}$ and tritium $\mathit{t}$ reactants is close to the resonance energy, here 16.84 MeV for the 3/2⁺ Bretscher state. This happens to be within $\sim 50$ keV (center-of-mass frame) of the $\mathit{dt}$ separation energy (16.79 MeV), taking the resonance peak at Bretscher’s original value.^[3] (The current value of the resonance peak is a little higher at 65 keV.) Credit: S. Tasseff.

Here, we discuss the considerations of Ref.^[6] in greater detail including the anthropic implications of cosmological nucleosynthetic processes (see Fig. 2) and the role of the resonant Bretscher state that contributed materially to the very existence of humans by its sourcing of the ⁴He fuel required for the manufacture of ¹²C. The present paper addresses events that occurred during Big Bang nucleosynthesis (BBN), the epoch covering the evolution of the early universe—from a few seconds to a few minutes after the Big Bang, some 13.8 billion years ago.

Fig. 2. A schematic of the dominant BBN pathways and the subsequent triple-alpha carbon formation in stars. The red curve shows the dominant path, which goes through the DT fusion reaction. Credit: S. Tasseff.

Work in the 1960s and 1970s by Wagoner et al.,^[7] Wagoner^[8,9] (see Fig. 3), and Peebles^[10] (extending earlier studies in Ref.^[11]) convincingly showed that—what has now become—the standard BBN model accounts for the primordial abundances of the light elements. This early model accounts fairly accurately for the light isotope abundances with roughly 75% hydrogen (¹H), 25% helium (⁴He) by mass,^[11] and trace amounts of other nuclides, all created within the first few minutes after the Big Bang by the processes depicted in Fig. 2. A more recent study at California Institute of Technology (Caltech) by Smith et al.^[12] provided a detailed sensitivity study to determine the most important nuclear reaction pathways explored in the epoch of the Big Bang, during the first few hundred seconds, using accurate representations of thermonuclear cross sections.

Fig. 3. Robert V. Wagoner, William A. Fowler, Fred Hoyle, and Donald D. Clayton in February 1967, between Sloan Laboratory and West Bridge Laboratory on the Caltech campus.

The present account is based on Smith et al.’s calculations, which are supported by more recent, detailed studies such as those of Grohs et al.^[13] and Pitrou et al.^[14] In order to commence our discussion of the production of ⁴He in BBN, the dominance of the DT reaction, and the importance of the Bretscher state, we first review the nuclear reaction history of the early universe as a function of its temperature and highlighting the role of entropy in its evolution.

II. NUCLEAR REACTION HISTORY OF THE EARLY UNIVERSE

The abundance history of the light isotopes in the early universe,^a determined numerically, is shown in Fig. 4, along with the entropy per baryon. The derivative with respect to time is given as

(1) $\begin{array}{l}\frac{\mathit{d}{\mathit{s}}_{\mathit{tot}}}{\mathit{dt}}=-\frac{T_{\mathrm{cm}}^3}{2{\pi }^2\mathit{nb}}{\displaystyle \sum _{\mathit{i}=1}^6{\displaystyle {\int }_0^{\infty }\mathit{dϵ}}}{ϵ}^2{C}_{\mathit{\nu i}}[\mathit{fj}]\\ \times \left[\mathit{ϵ}\frac{{\mathit{T}}_{\mathit{cm}}}{\mathit{T}}+\mathit{ln}\left(\frac{\mathit{fi}}{1-\mathit{fi}}\right)\right],\end{array}$(1)

Fig. 4. The light isotope abundances $\mathit{Y}=Y_{\mathit{A},\mathit{Z}}$ [see Eq. (3(3) $\begin{array}{rl}& Y_{\mathit{A},\mathit{Z}}=\frac{g_A{A}^{3/2}}{2}{\left\{\frac{2\sqrt{2}{\mathit{\pi }}^{7/2}{g}_s}{45}{\left[\frac{{\mathit{T}}_{\mathit{cm}}}{m_N}\right]}^{3/2}\right\}}^{\mathit{A}-1}{Y}_p^Z{Y}_n^{A-Z}\\ & \times \frac{1}{s_{tot}^{A-1}}{e}^{{\mathit{B}}_{\mathit{A},\mathit{Z}}/T_{\mathit{cm}}},\end{array}$(3) )] (upper panel: dimensionless, measured relative nucleon number density) and the entropy per baryon $s_{\mathit{tot}}$ (lower panel: dimensionless) as a function of the temperature of the early universe covering a range of times measured from the Big Bang from $\sim 1$ to $\sim 100$ s.

where $n_b$ = baryon number density per unit volume; $f_i(\mathit{ϵ},\mathit{t})$ = neutrino $\mathit{i}=1,2,3$ and antineutrino $\mathit{i}=4,5,6$ momentum probability distribution functions; ${\mathcal{C}}_{{\nu }_i}$ = rate of collisions of neutrinos and antineutrinos, which are computed in the solution of the neutrino-antineutrino Boltzmann equations. The entropy per baryon is expressed as a function of the (comoving) temperature $T_{\mathit{cm}}$ of the early universe and $\mathit{T}$, the temperature of charged leptons and the baryons ($\mathit{n}$ and $\mathit{p}$).

The temperatures involved are very large, even by solar scales, and this necessarily implies that a very large entropy is carried by the elementary constituents of the early universe. The entropy, as a measure of the disorder of the system, is an extensive quantity, depending on the number of particles, and is therefore dominated by the photons and the neutrinos. (The nucleons collectively do not carry much of the entropy since there is only 1 nucleon for every $\sim {10}^{10}$ photons or neutrinos.) The energy units on the $\mathit{x}$-axis of Fig. 4 are proportional to temperature (via the Boltzmann constant $k_B^{-1}=11.6$ GK/MeV, where GK means ${10}^9$ kelvin). Reading the figure from left to right, the temperature decreases from a time when the age of the universe was about 1 s old with temperature $T_{\mathit{cm}}(\text{lsim }1\mathrm{s})\text{gsim}10\mathrm{MeV}\sim 100$ GK to a time when the temperature is too low for the positively charged ions to overcome the Coulomb repulsion. Here, at $T_{\mathit{cm}}(\text{gsim }100\mathrm{s})\text{lsim }0.01\mathrm{MeV}\sim 0.1$ GK, the nuclear reactions have all but ceased.

Returning to the leftmost, higher-temperature part Fig. 4, we consider the initial conditions that feed the nuclear reactions that bind neutrons and protons in nuclei. The central points are that the final, primordial abundances are limited by the number of available neutrons and that nearly all the neutrons end up being bound in ⁴He nuclei (see Table I). The first point is a consequence of the nature of the weak interactions that interconvert $\mathit{n}\leftrightarrow \mathit{p}$^[15] and the fact that neutron-destroying reactions dominate. This is mainly because the neutron mass is about 1.3 MeV larger than that of the proton, independent of the fact that it is unstable. (The neutron does not have time to decay much before the nuclear reactions begin at a few seconds after the Big Bang.) The ratio of (the number density of) neutrons to protons is governed by the weak interactions of nucleons with charged (electrons and positrons) and neutral (neutrinos and antineutrinos) leptons.

TABLE I Final Primoridal Abundances (Dimensionless) Relative to Nucleon Mass Density (¹H and ⁴He) or Relative to Nucleon Number Density (²H, ³H, ³He, ⁶Li, ⁷Li, and ⁷Be)

$\mathit{A}$	Element	Relative Abundance
1	H	$0.752$
2	H	$2.61$	$\times {10}^{-5}$
3	H	$8.26$	$\times {10}^{-8}$
3	He	$1.04$	$\times {10}^{-5}$
4	He	$0.248$
6	Li	$1.13$	$\times {10}^{-14}$
7	Li	$2.87$	$\times {10}^{-11}$
7	Be	$4.11$	$\times {10}^{-10}$

The assumption of chemical equilibrium between the neutrons $\mathit{n}$ and protons $\mathit{p}$ gives an approximate expression for the ratio of their number densities: $\mathit{n}/\mathit{p}\simeq exp(-\mathit{\delta }{m}_{\mathit{np}}/T_{\mathit{cm}})$, where $\mathit{\delta }{m}_{\mathit{np}}\approx 1.293$ MeV is the mass of the neutron less the mass of the proton. At high temperature, $T_{\mathit{cm}}\gg \mathit{\delta }{m}_{\mathit{np}}$, the baryonic matter comprises equal parts neutrons and protons $\mathit{n}/\mathit{p}\approx 1$. At a lower temperature of about $T_{\mathit{cm}}\text{lsim}0.9$ MeV, i.e., the “weak freeze-out” temperature, the reactions that interconvert $\mathit{n}\leftrightarrow \mathit{p}$ cease with the $\mathit{n}/\mathit{p}$ fraction at weak freeze-out $\approx 1/6$.

The nuclear abundances (measured relative to the nucleon or baryon number density) are then set—this is the second, central point made earlier—by this initial $\mathit{n}/\mathit{p}\approx 1/6$ condition and a phase in the evolution called nuclear statistical equilibrium (NSE). The NSE obtains when all the forward (say, $\mathit{np}\to \mathit{d\gamma }$) and reverse ($\mathit{\gamma d}\to \mathit{np}$) nuclear rates are very nearly equal and very large compared to the rate of expansion of the universe. The rate of expansion is set by the Hubble parameter $\mathit{H}$ and can reach upward of 100 Hz (in inverse seconds) near $T_{\mathit{cm}}\sim 10$ MeV.

Under these extreme conditions of high temperature, the system is held in a quasi-equilibrium state by the extremely high rates of the nuclear reactions, and a condition close to detail balance holds. That is, for each nuclear reaction that fuses neutrons and protons (nucleons) into heavier nuclei with nucleon numbers $\mathit{A}>2$, another reaction immediately follows that breaks up the heavier $\mathit{A}>2$ nucleus into its component nucleons. In these near-equilibrium conditions, the Gibbs free energy $\mathit{G}={\sum }_a{\mu }_a{N}_a$ is conserved by the nuclear reactions. Here, ${\mu }_a$ and $N_a$ are the chemical potentials and numbers of species $\mathit{a}$, respectively, where $\mathit{a}$ indexes the individual nuclides ($\mathit{a}=\mathit{n},\mathit{p},\mathit{d},{}^3\mathrm{H},{}^3\mathrm{He},{}^4\mathrm{He},\dots$). These conditions (of chemical equilibrium) obtain at high temperatures while the entropy density (lower panel, Fig. 4) is nearly constant for $T_{\mathit{cm}}\text{gsim}1$ MeV and are expressed in the form

(2) ${\mu }_{\mathit{A},\mathit{Z}}=(\mathit{A}-\mathit{Z}){\mu }_n+\mathit{Z}{\mu }_p,$(2)

where $\mathit{A},\mathit{Z}$ = atomic weight and number of protons $\mathit{Z}$ of the nuclide $\mathit{A}$; ${\mu }_{\mathit{A},\mathit{Z}}$ = chemical potential of the nuclide (nucleus) with $\mathit{Z}$ protons and $\mathit{A}-\mathit{Z}$ neutrons (for example, ${ }^4\mathrm{He}$ has $\mathit{A}=4,\mathit{Z}=2,\mathit{N}=2$); μ_n = neutron chemical potential; μ_p = proton chemical potential. Equation (2)(2) ${\mu }_{\mathit{A},\mathit{Z}}=(\mathit{A}-\mathit{Z}){\mu }_n+\mathit{Z}{\mu }_p,$(2) can be used to determine the element abundances, shown in Fig. 5 as dashed curves, for high temperatures relevant for NSE. For $T_{\mathit{cm}}\text{lsim}1$ MeV, the system departs from the quasi-equilibrated NSE state as the nuclear reaction rates drop below the Hubble rate H.

Fig. 5. The light isotope abundance $\mathit{Y}$, relative to the nucleon number density, calculated numerically (solid curves) compared to the NSE values (dashed curves) for isotopes with charge $\le 4$, shown with the entropy per baryon (lower panel), as in Fig. 4. The dashed and solid curves overlap when NSE obtains.

Before discussing the details of the nuclear reactions that give rise to the abundances in Figs. 4 and 5, we point out some general features of the abundance history graph.^b

During the phase of NSE, Eq. (2)(2) ${\mu }_{\mathit{A},\mathit{Z}}=(\mathit{A}-\mathit{Z}){\mu }_n+\mathit{Z}{\mu }_p,$(2) ^c may be solved to give

(3) $\begin{array}{rl}& Y_{\mathit{A},\mathit{Z}}=\frac{g_A{A}^{3/2}}{2}{\left\{\frac{2\sqrt{2}{\mathit{\pi }}^{7/2}{g}_s}{45}{\left[\frac{{\mathit{T}}_{\mathit{cm}}}{m_N}\right]}^{3/2}\right\}}^{\mathit{A}-1}{Y}_p^Z{Y}_n^{A-Z}\\ & \times \frac{1}{s_{tot}^{A-1}}{e}^{{\mathit{B}}_{\mathit{A},\mathit{Z}}/T_{\mathit{cm}}},\end{array}$(3)

where the abundance $Y_{\mathit{A},\mathit{Z}}$ is the ratio of the number density of nuclide $(\mathit{A},\mathit{Z})$ relative to the nucleon (or baryon) number density. The quantities $g_A=2J_A+1$, where $J_A$ is the spin of nuclide of mass $\mathit{A}$ and $g_s=10.75$ count the number of effective spin degrees of freedom of the nuclide of mass $\mathit{A}$ and the plasma constituents. (See Ref.^[18], p. 87ff, for a detailed discussion.) Suffice it to say here that this factor is only weakly dependent on both the properties ($\mathit{A},\mathit{Z}$) of the nuclei and the temperature $T_{\mathit{cm}}$. The factor in the second line of Eq. (3(3) $\begin{array}{rl}& Y_{\mathit{A},\mathit{Z}}=\frac{g_A{A}^{3/2}}{2}{\left\{\frac{2\sqrt{2}{\mathit{\pi }}^{7/2}{g}_s}{45}{\left[\frac{{\mathit{T}}_{\mathit{cm}}}{m_N}\right]}^{3/2}\right\}}^{\mathit{A}-1}{Y}_p^Z{Y}_n^{A-Z}\\ & \times \frac{1}{s_{tot}^{A-1}}{e}^{{\mathit{B}}_{\mathit{A},\mathit{Z}}/T_{\mathit{cm}}},\end{array}$(3) ), $s_{tot}^{1-A}exp(B_{\mathit{A},\mathit{Z}}/T_{\mathit{cm}})$, is however very strongly dependent on the temperature $T_{\mathit{cm}}$ and the atomic weight $\mathit{A}$. It represents a competition between the energetics of nuclear binding and the disordering effect of the high-entropy environment created by the huge number of photons at high temperature that we will call the photon bath.

The strong dependence on $T_{\mathit{cm}}$ is owed to the exponential factor that features the nuclear binding energy (see Table II) $B_{\mathit{A},\mathit{Z}}=\mathit{Z}{m}_p+(\mathit{A}-\mathit{Z})m_n-m_{\mathit{A},\mathit{Z}}$, where $m_p,m_n,\mathrm{and}{m}_{\mathit{A},\mathit{Z}}$ are the mass of the proton, neutron, and nucleus with atomic weight $\mathit{A}$ and charge $\mathit{Z}$. The strong dependence on $\mathit{A}$ is owed to the fact that the entropy per baryon $s_{\mathit{tot}}$ is a very large (dimensionless) number.^d The entropy in the early universe, dominated by contributions from the electroweak components (photons, charged leptons, and neutrinos), is of order $s_{\mathit{tot}}\sim {10}^{10},$ as seen in the lower panel of Fig. 4. The origin of the exponential factor, which favors nuclides with larger total binding energies, is a consequence of the Boltzmann factor $exp(-\mathit{E}/\mathit{T})$, which arises in the (canonical) thermal average in the determination of the nuclide density. The entropic effect, which suppresses nuclides with higher numbers $\mathit{A}$ of nucleons, is due to the influence of the photon bath at high temperature ($\text{gsim}1$ MeV) and the tendency for this entropy to be transferred to the forming nuclear system thereby reducing the likelihood that larger $\mathit{A}$ systems will form.

TABLE II Binding Energy of Some Light Isotopes*

$\mathit{A}$	Element	$B_{\mathit{A},\mathit{Z}}$ (MeV)
2	H	2.22
3	H	8.48
3	He	7.72
4	He	28.30
6	Li	31.99
7	Li	39.25
7	Be	37.60

*Binding energy is in units of mega-electron-volts. Reference 19.

These tendencies are clearly visible in the plot of the nuclear abundances in both Figs. 4 and 5. At temperatures $T_{\mathit{cm}}$ far above 1 MeV, the entropic effect suppresses each nuclide in NSE by powers of $\mathit{A}$: $Y_{{ }^4\mathrm{He}}<<Y_{{ }^3\mathrm{H}{,}^3\mathrm{He}}<<Y_{{ }^2\mathrm{H}}$. The rate of increase of the abundances near $T_{\mathit{cm}}\text{lsim}1$ MeV is governed by the value of $B_{\mathit{A},\mathit{Z}}$ with the rate of ⁴He production largest.

Moving to the right-hand side of the figure, at temperatures $T_{\mathit{cm}}\text{lsim }30$ keV, the abundances are essentially constant except for that of the neutron, which falls off due to free-neutron decay with a half-life of about 10 min.^e This is a consequence of the complete freeze-out of all of the electroweak and nuclear reactions, a result of the Hubble expansion cooling the reactive plasma, which transmute nuclides into one another. The final, unchanging values of the nuclear abundances determine what is referred to as the primordial abundances. The most abundant primordial nuclide (with $\mathit{A}>1$) is ⁴He, as expected from the foregoing discussion of the role of its large binding energy, making up about one-fourth of the baryonic matter by mass. Most of the rest of the mass fraction is accounted for by hydrogen.

The remaining trace nuclides are essentially the ash left in the thermonuclear furnace of the early universe. The primordial deuterium abundance, with number fraction $Y_{{ }^2\mathrm{H}}$ (of baryonic matter), is about 1 part in ${10}^5$, and the mass 3 nuclei are down about one to several orders of magnitude below ²H. The ⁷Li nuclide,^f infamous because of a persistent discrepancy between the abundance predicted in the standard picture (given here) and observations that find a factor value of between two to five times smaller, is about 1 in ${10}^{10}$. This discrepancy indicates either a problem with the observations or a physical mechanism that destroys the mass 7 nuclides (⁷Li and ⁷Be) but has only marginal effect on the other well-measured nuclides [²H and ⁴He]; see Ref.^[20] for a recent discussion.

In the intermediate-temperature region, from 1 MeV $\text{gsim}{T}_{\mathit{cm}}\text{gsim }30$ keV, starting at the higher-temperature end of this region, the evolution is determined by a sequence of partial nuclear reaction freeze-outs, which are detectable by the change of the curvature of the abundance curves from positive to negative and back to positive. The first of these regions occurs for the ⁴He abundance at about $T_{\mathit{cm}}\simeq 0.6$ MeV. It is at this point that the NSE abundance (dashed curve in Fig. 5) diverges from the numerical simulation and corresponds to a point in the evolutionary history of the early universe when the weak interactions that kept the neutrons and protons close to equilibrium become slow enough that the NSE no longer obtains. At this point, the ⁴He abundance drops off its NSE curve and begins to more closely track the slower mass 3 nuclides (³H and ³He), which are the source for reactions that produce the ⁴He. Subsequent curvature transitions are visible in the numerical simulation and correspond to similar rate freeze-out effects.

We now turn to the individual production and destruction mechanisms responsible for the major isotopes produced in Big Bang (or primordial) nucleosynthesis.

III. INDIVIDUAL PRODUCTION AND DESTRUCTION MECHANISMS

III.A. BBN Deuterium Production Via p(n,γ)²H

At the early stages of the Big Bang, as previously mentioned, the temperature was sufficiently high for neutron and proton number densities to be equal since $\mathit{n}\leftrightarrow \mathit{p}$ transmutations, mediated by the weak interaction, maintained their equilibrium. Owing to the high-entropy, energetic photon bath environment of the early universe (photons outnumber nucleons by about ${10}^{10}$ to 1), the electromagnetic reaction $\mathit{n}(\mathit{p},\mathit{\gamma }{)}^2\mathrm{H}$ and its reverse ${ }^2\mathrm{H}(\mathit{\gamma },\mathit{n})\mathit{p}$ remain in equilibrium for temperatures down to about $T_{\mathit{cm}}\simeq 70$ keV, far below that expected simply on the basis of the binding factor $exp{B}_{2,1}/T_{\mathit{cm}}$ in Eq. (3)(3) $\begin{array}{rl}& Y_{\mathit{A},\mathit{Z}}=\frac{g_A{A}^{3/2}}{2}{\left\{\frac{2\sqrt{2}{\mathit{\pi }}^{7/2}{g}_s}{45}{\left[\frac{{\mathit{T}}_{\mathit{cm}}}{m_N}\right]}^{3/2}\right\}}^{\mathit{A}-1}{Y}_p^Z{Y}_n^{A-Z}\\ & \times \frac{1}{s_{tot}^{A-1}}{e}^{{\mathit{B}}_{\mathit{A},\mathit{Z}}/T_{\mathit{cm}}},\end{array}$(3) . This is visible in Fig. 5 when the dashed, blue NSE curve for ²H diverges from the solid, blue numerical curve. At this stage, the free neutron supply (see Fig. 2 of Ref.^[12]) is being rapidly depleted by other, highly efficient, reactions that ultimately pack the neutrons predominantly into ⁴He nuclei, and the deuteron ²H production begins to fall below its NSE track. The ²H destructive reactions, such as ${ }^2\mathrm{H}(\mathit{\gamma },\mathit{n})\mathit{p}$, ${ }^2\mathrm{H}(\mathit{d},\mathit{n}{)}^3\mathrm{He}$, ${ }^2\mathrm{H}(\mathit{d},\mathit{p}{)}^3\mathrm{H},$ and perhaps most importantly ${ }^3\mathrm{H}(\mathit{d},\mathit{n}{)}^4\mathrm{He},$ ablate some of the previously produced deuterium, reducing its abundance (or mass fraction) by two or three orders of magnitude compared to its peak value near $T_{\mathit{cm}}\sim 60$ keV.^g

III.B. BBN Tritium and ³He Production Via DD

The NSE curves in Fig. 5 for ³H and ³He are coincident with their numerically determined values until $T_{\mathit{cm}}\sim 0.2$ MeV where the mass 3 destructive reactions begin to overcome those that produce mass 3. The more tightly bound ³H (compared to³He; see Table II) gives it an advantage during NSE, as seen in the exponential factor in Eq. (3)(3) $\begin{array}{rl}& Y_{\mathit{A},\mathit{Z}}=\frac{g_A{A}^{3/2}}{2}{\left\{\frac{2\sqrt{2}{\mathit{\pi }}^{7/2}{g}_s}{45}{\left[\frac{{\mathit{T}}_{\mathit{cm}}}{m_N}\right]}^{3/2}\right\}}^{\mathit{A}-1}{Y}_p^Z{Y}_n^{A-Z}\\ & \times \frac{1}{s_{tot}^{A-1}}{e}^{{\mathit{B}}_{\mathit{A},\mathit{Z}}/T_{\mathit{cm}}},\end{array}$(3) . The two branches of the mass 3 producing deuterium-deuterium (DD) fusion reactions, i.e., ${ }^2\mathrm{H}(\mathit{d},\mathit{p}{)}^3\mathrm{H}$ and ${ }^2\mathrm{H}(\mathit{d},\mathit{n}{)}^3\mathrm{He}$, shown in Fig. 2, produce neutrons, protons, and the mass 3 nuclides. These fusion reactions occurred at nearly equal rates (a consequence of isospin symmetry) producing tritium (T or ³H) and ³He in nearly equal amounts. Again, ³H has an advantage over the latter, ³He nuclide, since it is more readily transmuted by the neutron-induced, charge neutral ${ }^3\mathrm{He}(\mathit{n},\mathit{p}{)}^3\mathrm{H}$ reaction than its reverse ${ }^3\mathrm{H}(\mathit{p},\mathit{n}{)}^3\mathrm{He},$ which is Coulomb suppressed. These same DD reactions were first observed in the laboratory in 1934 by Oliphant, Harteck, and Rutherford, and the cross sections were measured at the Metallurgical Laboratory in Chicago (1942–1944) and at Los Alamos Laboratory by Bretscher.^[1,2]

III.C. BBN ⁴He Production Via DT

The ⁴He mass fraction in Fig. 5 (black curves), previously discussed in terms of NSE and the partial freeze-out mechanism, is seen to fall out of equilibrium with the mass 3 nuclides near $T_{\mathit{cm}}\approx 150$ keV and subsequently to enjoy an explosive acceleration in its production near $T_{\mathit{cm}}\simeq 65$ keV. This distinctive and perhaps surprising feature occurs at precisely the same energy where the temperature-dependent reactivity $⟨\mathit{\sigma v}⟩$ reaches its maximum; see Fig. 6 in the companion paper in this special issue.^[2] This accelerated production is due to resonance enhancement of ⁴He production by the $3/2^{+}$ Bretscher state in the ⁵He compound system, which has a corresponding peak cross section of 5 b at 65 keV. We might pause to emphasize that the large magnitude of the cross section, compared to other light nuclear reactions, is unparalleled and clearly the dominant effect in the production of ⁴He.

Fig. 6. Egon Bretscher (foreground, left) and Robert Serber (foreground, right) at the 1946 Nuclear Physics Conference held in Los Alamos.

In fact, the calculations of Smith et al. (see Fig. 7a in Ref.^[12]) show that $\sim 99$% of the ⁴He was produced via ${ }^3\mathrm{H}(\mathit{d},\mathit{n}{)}^4\mathrm{He}$, labeled “DT fusion” in Fig. 2. The remaining 1% of primordial ⁴He came from the ${ }^2\mathrm{H}{(}^3\mathrm{He},\mathit{p}{)}^4\mathrm{He}$ reaction, which benefits from the same mirror 3/2⁺ resonance but is suppressed by the larger Coulomb repulsion between D and ³He. As we pointed out previously in Sec. III.B. on mass 3 production, much of this ³He was rapidly converted to tritium by the $(\mathit{n},\mathit{p})$ reaction, which becomes another source for ⁴He via the very fast DT fusion reaction.^[20] The importance of DT in BBN was noted by Hupin et al.^[21] in their interesting paper on ab initio cross-section calculations. A central point of our current considerations is that ⁴He, the fuel for further nucleosynthesis of heavier elements in stars, which constitutes about 25% by mass of the baryonic matter in the universe, was dominantly made via resonant DT fusion in BBN.

Fig. 7. DT and DD cross sections in the incident deuteron projectile laboratory frame, from EDA R-matrix code analyses, including a thought experiment DT calculation in which the A = 5 $3/2^{+}$ Bretscher state is removed.

III.D. Stellar Nucleosynthesis of ⁴He

In his Nobel Prize–awarded work, “Energy Production in Stars,”^[22] Bethe laid out that the only two reaction chains, consistent with known nuclear cross sections, that convert four protons and two electrons to a helium nucleus are the proton-proton (p-p) chain and the carbon-nitrogen-oxygen (CNO) cycle. The former is important in less massive stars (below 1.2 M_⊙), and the latter is important in heavier ones (above 1.2 M_⊙). Both have the effect of transmuting hydrogen to helium without changing the chemical composition of other, heavier, elements. In the CNO cycle, some ¹²C must be present to seed the reaction chain, and the ¹²C nucleus is reproduced at the end of the cycle by the reaction ${ }^{15}\mathrm{N}(\mathit{p},\mathit{\alpha }{)}^{12}\mathrm{C}$.

Our current, anthropic considerations lead us to ask the following question: In an idealized scenario where stars are formed only from primordial material (75% hydrogen and 25% helium), what proportion of the resulting helium produced by whatever stellar nucleosynthetic mechanism comes from BBN or stellar processing? The then ensuing process of igniting, in sufficiently massive stars, the ⁴He in triple-alpha burning reaction takes place resulting in the production of carbon, the foundation of all organic chemical compounds on our planet. Such sufficiently massive stars are twice as large as the sun as they evolve into red giants, when their cores contract and heat further, to about six times that of the sun.^[23,24]

Similar to the DT reaction that produces ⁴He, the triple-alpha reaction is resonantly enhanced. In the triple-alpha process, two alpha particles form the short-lived ($\sim {10}^{-16}$ s) ⁸Be compound nucleus. Should a third alpha particle find the ephemeral ⁸Be before it decays, the reaction ${ }^8\mathrm{Be}(\mathit{\alpha },2\mathit{\gamma }{)}^{12}\mathrm{C}$ occurs with branching ${\mathrm{\Gamma }}_{\mathit{rad}}/\mathrm{\Gamma }\sim 4.1(1)\times {10}^{-4}$ dominated by the ¹²C resonance with excitation energy 7.65 MeV and spin parity $0^{+}$, the ¹²C Hoyle state.^[25] Then, more ⁴He was catalyzed by the CNO cycle, and heavier element nucleosynthesis could ensue through progressive captures of $\mathit{\alpha }$ particles on the chain of nuclei starting with ¹²C. Although the CNO stellar nucleosynthesis reactions increased the ⁴He content in the universe, the increase is smaller, compared with the initial BBN amount,^[18] even in the case of complete hydrogen burning to helium.

It is interesting to ask what percentage of the ⁴He that is converted to carbon and heavier elements came from BBN-produced helium compared to stellar nucleosynthesis–produced helium, where the hydrogen was transmuted to helium through p-p chain and/or CNO processes. Complete hydrogen burning in stellar nucleosynthesis places a rigorous lower bound of 25% ⁴He from BBN, required by the fact that the number of neutrons and protons is conserved. This important point bears reiteration: If the primordial material from which carbon is synthesized had all of its hydrogen converted to helium, then one would have the lower limit, 25% BBN helium. Otherwise, with incomplete burning of the primordial hydrogen—which is likely—the percentage of ⁴He would be higher. A more precise estimate requires future study via detailed simulations.^[23]

Many features of the universe we inhabit were therefore produced through the resonance-enhanced DT fusion reaction, notably 25% of the universe’s mass being He. Also, at least 25% of the mass in elements comprising 90% of our body’s mass (the nonhydrogen component) was created in BBN DT fusion through the Bretscher state—a substantial fraction!

III.E. The Bretscher State

Because the ⁵He 3/2⁺ 16.84-MeV state discovered by Bretscher^[3,4] is responsible for the fast rate of the DT reaction and the concomitant implications described in this paper, we propose that the nuclear fusion community refer to it as the Bretscher state, in analogy with the ¹²C Hoyle state. The magnitude and importance of the DT resonant enhancement effect is evident in Fig. 7, where we have used an R-matrix reaction analysis code (the Los Alamos National Laboratory EDA code^[26]) to calculate the DT cross section with and without the 3/2⁺ Bretscher state. If there was no such state, DT would be similar to the DD cross section.

IV. CONCLUSION

Given the obscurity of the role of the Bretscher state and its singular anthropic importance, in closing, we should briefly revisit the history of its discovery by Bretscher and collaborators. The Los Alamos National Laboratory archives, housed in the National Security Research Center, hold a variety of original materials available nowhere else, including monthly progress reports from F-Division, the division led by Enrico Fermi during the Manhattan Project. These reports record that Bretscher and French obtained their first DT data (with tritons impinging on a heavy ice target) in July 1945 (for a center-of-mass energy of 16 keV). A detailed characterization of the DT resonance was given in Bretscher’s Los Alamos Laboratory report LA-582, dated November 15, 1946.^[3] An image, extracted from this report, is given in Fig. 8, and one can see that this document is identical to the 1949 version published later in Physical Review.^[4] (This 1946 Los Alamos Laboratory report was initially classified as a secret document; its open publication was delayed until 1949.) The report establishes the resonant nature of the ²H(t,n)⁴ (T on D) reaction and provides two approaches to determine the resonance energy. One (see Fig. 8) gave the resonance energy in terms of the incident triton energy $E_t\approx 124$ keV. The other method gave it as $E_t\approx 343$ keV. In the center of mass, these values are equivalent to 50 and 137 keV, respectively, spanning the known value today of 65 keV. Although Bretscher was the first to document and quantify the resonance’s energy, we think it very likely that Bethe would have recognized the likely presence of a resonance as early as 1943 when the first DT cross-section data were obtained at Purdue University^[1,2] (though at energies above the resonance’s peak). The spin of the resonance was correctly identified by 1952; see our discussion on the evolving understanding of the 3/2⁺ resonance in Ref.^[2].

Fig. 8. An extract from Bretscher and French’s paper LA-582,^[3] November 15, 1946, providing the first identification of the DT resonance.

While Bretscher is generally credited with correctly anticipating the properties of ²³⁹Pu,^[2,27] his discovery of the DT resonance has gone largely unappreciated. His 1945–1946 DT measurements, which were published in Physical Review in 1949,^[4] have been cited only 54 times, compared with the 460 citations of Hoyle’s paper. This is—perhaps strangely—similar to the fact that the first-ever observation of DT fusion, in 1938 by Ruhlig,^[28] had never been cited until our recent paper.^[1] It is within this context that we propose the naming of the 3/2⁺ DT resonance as the Bretscher state, in the spirit of its being anthropically important, like the Hoyle state.

In Ref. ^[2], we present a brief biography of Bretscher, which includes his early work at Cambridge; his time at Los Alamos National Laboratory; his friendship with Oliphant, Bloch, and Staub; and his later leadership roles in nuclear energy research at Harwell.

Acknowledgments

We acknowledge useful discussions with Michael Smith, Jonathan Katz, Joyce Guzik, Michael Bernardin, Evan Grohs, Gerry Hale, George Fuller, Craig Carmer, and Tom Kunkle.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the U.S. Department of Energy through the Office of Science [13313917] and Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001).

Notes

^a The abundance history is determined from a system of coupled nonlinear ordinary differential equations in time that account for the general-relativistic expansion of the universe, its thermodynamics, and the strong and electroweak nuclear reaction rates.^[8,9]

^b We thank George Fuller for emphasizing to us the importance of the role of entropy in the state of NSE.^[16,17]

^c In the context of atomic and plasma physics, this relation is known as the Saha equation and allows one to compute the ionization state of the gas as a function of temperature in terms of the energy levels of the atom.

^d We might qualify use of the phrase “a very large number” as the black hole entropy, given by the Bekenstein-Hawking relation, has the much higher value of ~10⁷⁷ for a one solar mass object.

^e The time after the Big Bang is inversely proportional to the square of the temperature: $\mathit{t}\sim T_{cm}^{-2}.$

^f We sum the abundances of the mass 7 nuclides, ⁷Li and ⁷Be, since ⁷Be decays with half-life ~53 days due to ${ }^7\mathrm{Be}(e^{-},{\nu }_e{)}^7\mathrm{Li}$, in order to obtain the ⁷Li primordial abundance.

^g Perhaps it is worth noting that BBN production of deuterium differs from its production in stars where there are no free neutrons. Stellar nucleosynthesis of deuterium relies on the relatively much slower weak interaction $\mathit{p}+\mathit{p}{\to }^2\mathrm{H}+e^{+}+{\nu }_e$.