Isospin-forbidden electric dipole capture and the α(d, γ)⁶Li reaction

Since the energies of the emitted photons are usually not large at astrophysical energies, their wavelengths are large with respect to typical dimensions of the system and the photon wavenumbers

$$\begin{eqnarray}&&{k}_{\gamma }={E}_{\gamma }/{\hslash }c\end{eqnarray} \tag{ 1 }$$

can be considered as small. The long-wavelength approximation can be used. Let  r_j be the coordinate of the jth nucleon. At the long-wavelength approximation, the translation-invariant electric transition operators of multipolarity λ are given to a good approximation by

$$\begin{eqnarray}&&{{ \mathcal M }}_{\mu }^{E\lambda }=e\displaystyle \sum _{j=1}^{A}(\tfrac{1}{2}-{t}_{j3}){r}_{j}^{{\prime} \lambda }{Y}_{\lambda \mu }({{\rm{\Omega }}}_{j}^{{\prime} }{}_{}),\end{eqnarray} \tag{ 2 }$$

where t_j3 is the third component of the isospin operator ${{\boldsymbol{t}}}_{j}$ of the jth nucleon related to its charge by $e(\tfrac{1}{2}-{t}_{j3})$, and

$$\begin{eqnarray}&&{{\boldsymbol{r}}}_{j}^{{\prime} }={{\boldsymbol{r}}}_{j}-{{\boldsymbol{R}}}_{\mathrm{cm}}\end{eqnarray} \tag{ 3 }$$

is its coordinate with respect to the center-of-mass

$$\begin{eqnarray}&&{{\boldsymbol{R}}}_{\mathrm{cm}}=\displaystyle \frac{1}{A}\displaystyle \sum _{j=1}^{A}{{\boldsymbol{r}}}_{j}\end{eqnarray} \tag{ 4 }$$

of the A-nucleon system. The functions Y_λμ (${{\rm{\Omega }}}_{j}^{{\prime} }$) are spherical harmonics depending on the angular part of ${{\boldsymbol{r}}}_{j}^{{\prime} }=(r{{\prime} }_{j},{\rm{\Omega }}{{\prime} }_{j})$.

The orbital angular momentum with respect to the center-of-mass and spin of nucleon j are denoted as ${\boldsymbol{L}}{{\prime} }_{j}$ and ${{\boldsymbol{S}}}_{j}$, respectively. The total orbital momentum operator of the system is ${\boldsymbol{L}}={\sum }_{j=1}^{A}{\boldsymbol{L}}{{\prime} }_{j}$, the total spin is ${\boldsymbol{S}}={\sum }_{j=1}^{A}{{\boldsymbol{S}}}_{j}$ and the total angular momentum is ${\boldsymbol{J}}={\boldsymbol{L}}+{\boldsymbol{S}}$. The total isospin operator of the system is ${\boldsymbol{T}}={\sum }_{j=1}^{A}{{\boldsymbol{t}}}_{j}$.

The operators defined by equation (2) contain isoscalar (IS) and isovector (IV) parts. At the long-wavelength approximation, the E1 operator is special. It mainly contains an isovector component,

$$\begin{eqnarray}&&{{ \mathcal M }}_{\mu }^{E1}\approx {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}=-e\displaystyle \sum _{j=1}^{A}{t}_{j3}{r}_{j}^{{\prime} }{}_{}{Y}_{1\mu }({\rm{\Omega }}{{\prime} }_{j}).\end{eqnarray} \tag{ 5 }$$

The lowest-order term of the isoscalar part vanishes since ${\sum }_{j=1}^{A}{{\boldsymbol{r}}}_{j}^{{\prime} }=0$. This operator connects eigenstates of the total isospin operator with initial and final isospin quantum numbers differing by one unit, ${T}_{{\rm{f}}}=| {T}_{{\rm{i}}}\pm 1|$. It also connects states with T_i = T_f, but only for $N\ne Z$. Transitions from T_i = 0 to T_f = 0 are forbidden.

The isoscalar part of the E1 operator is however not exactly zero. It might play a non-negligible role in some cases. The first non-vanishing term reads using the Siegert theorem [31]

$$\begin{eqnarray}{{ \mathcal M }}_{\mu }^{E1,\mathrm{IS}} & \approx & -\displaystyle \frac{1}{60}{{ek}}_{\gamma }^{2}\displaystyle \sum _{j=1}^{A}{r}_{j}^{{\prime} 3}{Y}_{1\mu }({\rm{\Omega }}{{\prime} }_{j})\\ & & +\displaystyle \frac{e{\hslash }{k}_{\gamma }}{8{m}_{{\rm{p}}}c}\displaystyle \sum _{j=1}^{A}{r}_{j}^{{\prime} }{}_{}[{\boldsymbol{L}}{Y}_{1\mu }]({\rm{\Omega }}{{\prime} }_{j})\cdot \left[\displaystyle \frac{2}{3}{{\boldsymbol{L}}}_{j}^{{\prime} }{}_{}+({g}_{{\rm{p}}}+{g}_{{\rm{n}}}){{\boldsymbol{S}}}_{j}\right],\end{eqnarray} \tag{ 6 }$$

where m_p is the proton mass, and g_p and g_n are the proton and neutron gyromagnetic factors, respectively. The vector function $[{\boldsymbol{L}}{Y}_{1\mu }]({\rm{\Omega }})$ is the result of the action of the orbital momentum operator on the spherical harmonics Y_1μ (Ω) with l = 1. This operator connects components with the same initial and final isospins, T_i = T_f. When it acts on a wave function with a largely dominant component with zero total orbital momentum and small intrinsic spin, the first term of equation (6) should give a reasonable approximation.

We consider transitions in N = Z systems between an initial scattering state and a final bound state with dominant zero-isospin components. Their wave functions can be written symbolically as

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{i,f}^{{JM}}={{\rm{\Psi }}}_{i,f}^{{JM};0}+{{\rm{\Psi }}}_{i,f}^{{JM};1}.\end{eqnarray} \tag{ 7 }$$

The T = 1 components ${{\rm{\Psi }}}_{i,f}^{{JM};1}$ are much smaller than the T = 0 components ${{\rm{\Psi }}}_{i,f}^{{JM};0}$. Possible admixtures of larger isospin values are neglected.

To a good approximation, three types of matrix elements must be calculated. Two of them involve an isovector transition, i.e., between the dominant T_i = 0 component in the initial scattering state and the T_f = 1 admixture in the final bound state

$$\begin{eqnarray}&&\langle {{\rm{\Psi }}}_{f}^{J^{\prime} M^{\prime} ;1}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| {{\rm{\Psi }}}_{i}^{{JM};0}\rangle ,\end{eqnarray} \tag{ 8 }$$

and between the T_i = 1 admixture in the initial scattering state and the dominant T_f = 0 component in the final bound state

$$\begin{eqnarray}&&\langle {{\rm{\Psi }}}_{f}^{J^{\prime} M^{\prime} ;0}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| {{\rm{\Psi }}}_{i}^{{JM};1}\rangle .\end{eqnarray} \tag{ 9 }$$

An isoscalar transition is also possible, essentially between the dominant components,

$$\begin{eqnarray}&&\langle {{\rm{\Psi }}}_{f}^{J^{\prime} M^{\prime} ;0}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IS}}| {{\rm{\Psi }}}_{i}^{{JM};0}\rangle .\end{eqnarray} \tag{ 10 }$$

The E1 transition matrix element is the coherent sum of these three contributions.

To fix ideas we consider the α(d, γ)⁶Li reaction. We use the notation of the resonating-group method (RGM) [32, 33]. This notation is also valid for ab initio descriptions. We limit ourselves to α + n + p configurations. Realistic calculations might also include ³H + ³He configurations, for example, that we neglect to simplify the presentation. The wave functions that we now describe display the main components expected to play a significant role in E1 transitions. Many other smaller components are of course possible.

In the RGM, a partial wave of the initial scattering wave function (7) is written as

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{i}^{{JM}\pi }={ \mathcal A }{\phi }_{\alpha }^{00+}{[{\phi }_{d}^{1+}\otimes {Y}_{L}({{\rm{\Omega }}}_{R})]}^{{JM}}{g}_{i}^{J\pi }(R),\end{eqnarray} \tag{ 11 }$$

where ${ \mathcal A }$ is the six-nucleon antisymmetrizer and ${\boldsymbol{R}}=(R,{{\rm{\Omega }}}_{R})$ is the relative coordinate between the centers of mass of the α and deuteron clusters. The functions ${\phi }_{\alpha }^{00+}$ and ${\phi }_{d}^{1m+}$ are translation-invariant internal wave functions of the ground states of the ⁴He nucleus with angular momentum 0 and positive parity and of the deuteron with angular momentum 1 and positive parity, respectively. The ⁴He wave function depends on three internal coordinates. The deuteron wave function depends on the relative coordinate ${\boldsymbol{r}}=(r,{{\rm{\Omega }}}_{r})$ between the proton and neutron. The total parity π is equal to ( − 1)^L. The ⁴He ground state internal wave function may contain a small T = 1 admixture

$$\begin{eqnarray}&&{\phi }_{\alpha }^{00+}={\phi }_{\alpha }^{00+;0}+{\phi }_{\alpha }^{00+;1}.\end{eqnarray} \tag{ 12 }$$

The T = 1 component is mainly due to the Coulomb interaction between the protons. The neutron–proton mass difference and isospin non-conserving terms in the nuclear force also contribute but to a lesser extent. The deuteron ground state wave function is purely T = 0. In reactions of α particles with heavier N = Z nuclei, a T = 1 admixture also appears in the second cluster.

Various corrections may also appear in the scattering wave function to take distortion of the initial state at short distances into account. They may involve sums over pseudo-states of the deuteron and/or of the α particle. The most important ones should arise from deuteron pseudo-states which can simulate its Coulomb polarizability [15]. They may also include additional shell-model-like ⁶Li terms [32]. We do not display these corrections here to simplify the discussion but they can be treated in the same way as similar terms displayed below in the final state.

Under some simplifying assumptions, the main components of the final bound state wave function of the 1⁺ ground state of ⁶Li can be approximated as

$$\begin{eqnarray}{{\rm{\Psi }}}_{f}^{1M+} & = & { \mathcal A }{\phi }_{\alpha }^{00+}{[{\phi }_{d}^{1+}\otimes {Y}_{0}({{\rm{\Omega }}}_{R})]}^{1M}{g}_{f}^{1+}(R)\\ & & +\displaystyle \sum _{n}{ \mathcal A }{\phi }_{\alpha }^{00+}{[{\phi }_{{d}^{* }n}^{1{\pi }_{n};{T}_{n}}\otimes {Y}_{{L}_{n}}({{\rm{\Omega }}}_{R})]}^{1M}{g}_{{d}^{* }n}^{1+}(R)\\ & & +\displaystyle \sum _{I,n}{ \mathcal A }{[{[{\phi }_{{\alpha }^{* }n}^{1-;1}\otimes {\phi }_{d}^{1+}]}^{I}\otimes {Y}_{1}({{\rm{\Omega }}}_{R})]}^{1M}{g}_{{\alpha }^{* }{In}}^{1+}(R).\end{eqnarray} \tag{ 13 }$$

The ${\phi }_{{d}^{* }n}^{1{\pi }_{n};{T}_{n}}$ with T_n = 0 or 1 are excited pseudo-states of the deuteron. The relative orbital momentum is L_n = 0 for π_n = + and L_n = 1 for π_n = −. The ${\phi }_{{\alpha }^{* }n}^{1-;1}$ are excited pseudo-states of the ⁴He nucleus with angular momentum 1 and isospin 1. The channel spin I can take the values 0–2.

Given the angular momentum and parity of the final state, the initial state for E1 transitions corresponds to J = 0–2 and a negative parity. This is realized by choosing L = 1 in equation (11). Within these assumptions, let us write the various matrix elements. Matrix element (8) reads for an initial wave with L = 1,

$$\begin{eqnarray} & & \langle {{\rm{\Psi }}}_{f}^{1M^{\prime} +;1}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| {{\rm{\Psi }}}_{i}^{{JM}-;0}\rangle \\ & & \quad =\langle { \mathcal A }{\phi }_{\alpha }^{00+;1}{[{\phi }_{d}^{1+}\otimes {Y}_{0}]}^{1M^{\prime} }{g}_{f}^{1+}(R)| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| { \mathcal A }{\phi }_{\alpha }^{00+;0}{[{\phi }_{d}^{1+}\otimes {Y}_{1}]}^{{JM}}{g}_{i}^{J-}(R)\rangle \\ & & \quad +\displaystyle \sum _{n}{ \mathcal A }{\phi }_{\alpha }^{00+;0}{[{\phi }_{{d}^{* }n}^{1{\pi }_{n};{T}_{n}}\otimes {Y}_{{L}_{n}}]}^{1M^{\prime} }{g}_{{d}^{* }n}^{1+}(R)| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| { \mathcal A }{\phi }_{\alpha }^{00+;0}{[{\phi }_{d}^{1+}\otimes {Y}_{1}]}^{{JM}}{g}_{i}^{J-}(R)\rangle \\ & & \quad +\displaystyle \sum _{I,n}{ \mathcal A }{[{[{\phi }_{{\alpha }^{* }n}^{1-;1}\otimes {\phi }_{d}^{1+}]}^{I}\otimes {Y}_{1}]}^{1M^{\prime} }{g}_{{\alpha }^{* }{In}}^{1+}(R)| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| { \mathcal A }{\phi }_{\alpha }^{00+;0}{[{\phi }_{d}^{1+}\otimes {Y}_{1}]}^{{JM}}{g}_{i}^{J-}(R)\rangle \end{eqnarray} \tag{ 14 }$$

and matrix element (9) reads

$$\begin{eqnarray} & & \langle {{\rm{\Psi }}}_{f}^{1M^{\prime} +;0}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| {{\rm{\Psi }}}_{i}^{{JM}-;1}\rangle \\ & & \quad =\langle { \mathcal A }{\phi }_{\alpha }^{00+;0}{[{\phi }_{d}^{1+}\otimes {Y}_{0}]}^{1M^{\prime} }{g}_{f}^{1+}(R)| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| { \mathcal A }{\phi }_{\alpha }^{00+;1}{[{\phi }_{d}^{1+}\otimes {Y}_{1}]}^{{JM}}{g}_{i}^{J-}(R)\rangle ,\end{eqnarray} \tag{ 15 }$$

where J can be equal to 0–2. Other contributions appear when the initial state is distorted. Matrix element (10) reads

$$\begin{eqnarray} & & \langle {{\rm{\Psi }}}_{f}^{1M^{\prime} +;0}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IS}}| {{\rm{\Psi }}}_{i}^{{JM}-;0}\rangle \\ & & \quad =\langle { \mathcal A }{\phi }_{\alpha }^{00+;0}{[{\phi }_{d}^{1+}\otimes {Y}_{0}]}^{1M^{\prime} }{g}_{f}^{1+}(R)| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IS}}| { \mathcal A }{\phi }_{\alpha }^{00+;0}{[{\phi }_{d}^{1+}\otimes {Y}_{1}]}^{{JM}}{g}^{J\pi }(R)\rangle .\end{eqnarray} \tag{ 16 }$$

As the operator is much smaller here, only the dominant T = 0 components are kept.

We now consider the three-body α + n + p model. The ⁴He nucleus is treated as a structureless particle. Its properties appear in the interaction with the nucleons. They may also appear in some parameters of the model.

Let us start from the isovector microscopic operator (5). Let us assume that the first four coordinates ${{\boldsymbol{r}}}_{j}$ correspond to the α particle and that the last two correspond to the deuteron. In vector notation, operator (5) reads

$$\begin{eqnarray}&&{{\boldsymbol{ \mathcal M }}}^{E1,\mathrm{IV}}=-e\displaystyle \sum _{j=1}^{6}{t}_{j3}({{\boldsymbol{r}}}_{j}-{{\boldsymbol{R}}}_{\mathrm{cm}}).\end{eqnarray} \tag{ 17 }$$

The deuteron internal coordinate is

$$\begin{eqnarray}&&{\boldsymbol{r}}={{\boldsymbol{r}}}_{5}-{{\boldsymbol{r}}}_{6}\end{eqnarray} \tag{ 18 }$$

and the α-deuteron relative coordinate is given by

$$\begin{eqnarray}&&{\boldsymbol{R}}={{\boldsymbol{R}}}_{\mathrm{cm}}^{\alpha }-\tfrac{1}{2}({{\boldsymbol{r}}}_{5}+{{\boldsymbol{r}}}_{6}),\end{eqnarray} \tag{ 19 }$$

where ${{\boldsymbol{R}}}_{\mathrm{cm}}^{\alpha }=\tfrac{1}{4}{\sum }_{j=1}^{4}{{\boldsymbol{r}}}_{j}$ is the center-of-mass coordinate of the α particle.

Then, the E1 operator can be rewritten as

$$\begin{eqnarray}&&{{\boldsymbol{ \mathcal M }}}^{E1,\mathrm{IV}}={{\boldsymbol{ \mathcal M }}}_{\alpha }^{E1,\mathrm{IV}}-\tfrac{1}{2}e({t}_{\mathrm{5,3}}-{t}_{\mathrm{6,3}}){\boldsymbol{r}}-\tfrac{1}{3}e({T}_{\alpha 3}-2{T}_{d3}){\boldsymbol{R}},\end{eqnarray} \tag{ 20 }$$

where the first term

$$\begin{eqnarray}&&{{\boldsymbol{ \mathcal M }}}_{\alpha }^{E1,\mathrm{IV}}=-e\displaystyle \sum _{j=1}^{4}{t}_{j3}({{\boldsymbol{r}}}_{j}-{{\boldsymbol{R}}}_{\mathrm{cm}}^{\alpha })\end{eqnarray} \tag{ 21 }$$

is the E1 operator for the α particle. The second term is the E1 operator for the deuteron and the last term corresponds to the relative motion. The operators ${{\boldsymbol{T}}}_{\alpha }={\sum }_{j=1}^{4}{{\boldsymbol{t}}}_{j}$ and ${{\boldsymbol{T}}}_{d}={{\boldsymbol{t}}}_{5}+{{\boldsymbol{t}}}_{6}$ are the isospin operators of the α particle and deuteron, respectively. Hence, in multipolar form, one has

$$\begin{eqnarray}&&{{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}={{ \mathcal M }}_{\alpha ,\mu }^{E1,\mathrm{IV}}-\tfrac{1}{2}e({t}_{\mathrm{5,3}}-{t}_{\mathrm{6,3}}){{ \mathcal Y }}_{1\mu }({\boldsymbol{r}})-\tfrac{1}{3}e({T}_{\alpha 3}-2{T}_{d3}){{ \mathcal Y }}_{1\mu }({\boldsymbol{R}})\end{eqnarray} \tag{ 22 }$$

with

$$\begin{eqnarray}&&{{ \mathcal Y }}_{\lambda \mu }({\boldsymbol{x}})={x}^{\lambda }{Y}_{\lambda \mu }({{\rm{\Omega }}}_{x}).\end{eqnarray} \tag{ 23 }$$

For more general clusters with mass numbers A₁ and A₂, the factor in front of $-e{{ \mathcal Y }}_{1\mu }({\boldsymbol{R}})$ in the last term becomes $({A}_{2}{T}_{{A}_{1}3}-{A}_{1}{T}_{{A}_{2}3})/A$. Its eigenvalue contains the factor ${Z}_{1}/{A}_{1}-{Z}_{2}/{A}_{2}$ mentioned in the introduction.

In a similar way, the first term of the isoscalar E1 operator (6) becomes

$$\begin{eqnarray}{{ \mathcal M }}_{\mu }^{E1,\mathrm{IS}} & = & {{ \mathcal M }}_{\alpha ,\mu }^{E1,\mathrm{IS}}-\displaystyle \frac{1}{60}{k}_{\gamma }^{2}\left\{\displaystyle \frac{5}{9}e(4{R}_{\alpha }^{2}-{R}^{2}-{r}^{2}){{ \mathcal Y }}_{1\mu }({\boldsymbol{R}})\right.\\ & & \left.-\displaystyle \frac{\sqrt{32\pi }}{9}(2{[{{ \mathcal Y }}_{1}({\boldsymbol{R}})\otimes {{ \mathcal M }}_{\alpha }^{E2,\mathrm{IS}}]}_{1\mu }-e{[{{ \mathcal Y }}_{1}({\boldsymbol{R}})\otimes {{ \mathcal Y }}_{2}({\boldsymbol{r}})]}_{1\mu })\right\},\end{eqnarray} \tag{ 24 }$$

where ${R}_{\alpha }^{2}=\tfrac{1}{4}{\sum }_{j=1}^{4}{({{\boldsymbol{r}}}_{j}-{{\boldsymbol{R}}}_{\mathrm{cm}}^{\alpha })}^{2}$, and the E2 operator reads

$$\begin{eqnarray}{{ \mathcal M }}_{\mu }^{E2} & = & {{ \mathcal M }}_{\alpha ,\mu }^{E2}+\displaystyle \frac{\sqrt{120\pi }}{9}{[{{ \mathcal Y }}_{1}({\boldsymbol{R}})\otimes {{ \mathcal M }}_{\alpha }^{E1,\mathrm{IV}}]}_{2\mu }+\displaystyle \frac{1}{9}e(6-{T}_{\alpha 3}-4{T}_{d3}){{ \mathcal Y }}_{2\mu }({\boldsymbol{R}})\\ & & +\displaystyle \frac{1}{4}e(1-{T}_{d3}){{ \mathcal Y }}_{2\mu }({\boldsymbol{r}})+\displaystyle \frac{\sqrt{120\pi }}{9}e({t}_{\mathrm{5,3}}-{t}_{\mathrm{6,3}}){[{{ \mathcal Y }}_{1}({\boldsymbol{R}})\otimes {{ \mathcal Y }}_{1}({\boldsymbol{r}})]}_{2\mu }.\end{eqnarray} \tag{ 25 }$$

In the simplest version of a three-body model, the α particle is in its ground state ${\phi }_{\alpha }^{00+}$. Effective multipole operators are obtained by taking the mean value of the above expressions,

$$\begin{eqnarray}&&{\widetilde{{ \mathcal M }}}_{\mu }^{E\lambda }=\langle {\phi }_{\alpha }^{00+}| {{ \mathcal M }}_{\mu }^{E\lambda }| {\phi }_{\alpha }^{00+}\rangle .\end{eqnarray} \tag{ 26 }$$

The eigenvalue of T_α3 is zero, as well as the mean value of ${{ \mathcal M }}_{\alpha ,\mu }^{E\lambda }$. The eigenvalue of T_d3 vanishes for the neutron–proton system. Hence, for E1, one obtains from (22) and (24), with the neutron as particle 5 and the proton as particle 6,

$$\begin{eqnarray}&&{\widetilde{{ \mathcal M }}}_{\mu }^{E1,\mathrm{IV}}=-\displaystyle \frac{1}{2}e{{ \mathcal Y }}_{1\mu }({\boldsymbol{r}})\end{eqnarray} \tag{ 27 }$$

and

$$\begin{eqnarray}&&{\widetilde{{ \mathcal M }}}_{\mu }^{E1,\mathrm{IS}}=-\displaystyle \frac{1}{60}{{ek}}_{\gamma }^{2}\left\{\displaystyle \frac{5}{9}(4{r}_{\alpha }^{2}-{R}^{2}-{r}^{2}){{ \mathcal Y }}_{1\mu }({\boldsymbol{R}})+\displaystyle \frac{\sqrt{32\pi }}{9}{[{{ \mathcal Y }}_{1}({\boldsymbol{R}})\otimes {{ \mathcal Y }}_{2}({\boldsymbol{r}})]}_{1\mu }\right\},\end{eqnarray} \tag{ 28 }$$

where ${r}_{\alpha }^{2}=\langle {\phi }_{\alpha }^{00+}| {R}_{\alpha }^{2}| {\phi }_{\alpha }^{00+}\rangle$ is the mean square radius of the α particle. With (25), the E2 operator is given by

$$\begin{eqnarray}&&{\widetilde{{ \mathcal M }}}_{\mu }^{E2}=\displaystyle \frac{2}{3}e{{ \mathcal Y }}_{2\mu }({\boldsymbol{R}})+\displaystyle \frac{1}{4}e{{ \mathcal Y }}_{2\mu }({\boldsymbol{r}})+\displaystyle \frac{\sqrt{120\pi }}{9}e{[{{ \mathcal Y }}_{1}({\boldsymbol{R}})\otimes {{ \mathcal Y }}_{1}({\boldsymbol{r}})]}_{2\mu }.\end{eqnarray} \tag{ 29 }$$

This expression can also be deduced from equation (B2) of [29]. The first two terms are also derived in [22].

In the present α + n + p three-body model, the initial scattering wave function is defined by coupling the ground state deuteron wave function with partial waves describing the relative motion. The polarizability of the deuteron and other distortion effects of the initial wave are thus neglected. The deuteron wave function is defined as a pure s state (except in section 5.2 below) by

$$\begin{eqnarray}&&{\phi }^{{lSjm}}({\boldsymbol{r}})={[{Y}_{l}({{\rm{\Omega }}}_{r})\otimes {\chi }^{S}]}^{{jm}}{r}^{-1}{u}^{{lSj}}(r)\end{eqnarray} \tag{ 30 }$$

with l = 0 and S = j = 1. The spinor χ^S is the total spin state of the neutron and proton. The initial scattering functions for partial wave L read

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{i}^{{JM}\pi }({\boldsymbol{r}},{\boldsymbol{R}})={[{\phi }_{d}^{011+}({\boldsymbol{r}})\otimes {{\rm{\Phi }}}^{L\pi }({\boldsymbol{R}})]}^{{JM}}\end{eqnarray} \tag{ 31 }$$

with π = (−1)^L and

$$\begin{eqnarray}&&{{\rm{\Phi }}}^{{Lm}\pi }({\boldsymbol{R}})={Y}_{{Lm}}({{\rm{\Omega }}}_{R}){g}_{i}^{L\pi }(R),\end{eqnarray} \tag{ 32 }$$

since the α particle has spin 0 and positive parity.

The final ⁶Li(1⁺) ground state is described by a three-body wave function defined in the hyperspherical basis as

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{f}^{1M+}({\boldsymbol{r}},{\boldsymbol{R}})={\rho }^{-5/2}\displaystyle \sum _{\gamma ,K}{\chi }_{\gamma K}(\rho ){{ \mathcal Y }}_{\gamma K}^{1M}({{\rm{\Omega }}}_{5}),\end{eqnarray} \tag{ 33 }$$

where $\rho =\sqrt{\tfrac{1}{2}{r}^{2}+\tfrac{4}{3}{R}^{2}}$ is the hyperradius and Ω₅ represents five angles, the orientations Ω_r of ${\boldsymbol{r}}$ and Ω_R of ${\boldsymbol{R}}$, and the hyperangle $\alpha =\arctan (\sqrt{8/3}\,R/r)$ (see [29, 30] for details). Number K is the hypermomentum. Notation γ represents the other quantum numbers of the problem, i.e., the orbital momentum l and spin S of the proton–neutron pair, and the orbital momentum L of the α − (n + p) relative motion. The functions ${{ \mathcal Y }}_{\gamma K}^{{JM}}({{\rm{\Omega }}}_{5})$ are hyperspherical harmonics and the functions χ_γK(ρ) are hyperradial functions. The positive parity requires l + L even.

Thanks to the antisymmetry of the deuteron wave function, it is possible to associate an isospin to the different parts of the three-body wave function,

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{f}^{1M+}({\boldsymbol{r}},{\boldsymbol{R}})={{\rm{\Psi }}}_{f}^{1M+;0}({\boldsymbol{r}},{\boldsymbol{R}})+{{\rm{\Psi }}}_{f}^{1M+;1}({\boldsymbol{r}},{\boldsymbol{R}}).\end{eqnarray} \tag{ 34 }$$

For the neutron–proton system in the isospin formalism, antisymmetry imposes that l + S + T must be odd. Hence it is possible to perform the separation (34) of the final wave function according to the deuteron isospin T. The component with l + S odd corresponds to T_f = 0 while the component with l + S even corresponds to T_f = 1. The wave function (33) can be interpreted as corresponding to the first two terms of equation (13). Indeed, while the α particle is frozen in its ground state, the deuteron can be fully distorted or excited and T_f = 1 admixtures can appear in the neutron–proton system.

Matrix element (14) becomes with (20)

$$\begin{eqnarray}&&\langle {{\rm{\Psi }}}_{f}^{1M^{\prime} +;1}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| {{\rm{\Psi }}}_{i}^{{JM}-;0}\rangle =\langle {{\rm{\Psi }}}_{f}^{1M^{\prime} +;1}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| {[{\phi }_{d}^{011+}\otimes {{\rm{\Phi }}}^{1-}]}^{{JM}}\rangle ,\end{eqnarray} \tag{ 35 }$$

where J can be equal to 0–2. Matrix element (9) vanishes,

$$\begin{eqnarray}&&\langle {{\rm{\Psi }}}_{f}^{1M^{\prime} +;0}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IV}}| {{\rm{\Psi }}}_{i}^{{JM}-;1}\rangle =0.\end{eqnarray} \tag{ 36 }$$

Matrix element (10) reads

$$\begin{eqnarray}&&\langle {{\rm{\Psi }}}_{f}^{1M^{\prime} +;0}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IS}}| {{\rm{\Psi }}}_{i}^{{JM}-;0}\rangle =\langle {{\rm{\Psi }}}_{f}^{1M^{\prime} +;0}| {{ \mathcal M }}_{\mu }^{E1,\mathrm{IS}}| {[{\phi }_{d}^{011+}\otimes {{\rm{\Phi }}}^{1-}]}^{{JM}}\rangle .\end{eqnarray} \tag{ 37 }$$

When comparing with the microscopic expressions, one observes that important components are missing in the α + n + p model. The last term of equation (14) suggests that the transition matrix elements involving a virtual excitation of the α particle described by

$$\begin{eqnarray}&&\langle {\phi }_{{\alpha }^{* }n}^{1-;1}| {{ \mathcal M }}_{\alpha ,\mu }^{E1,\mathrm{IV}}| {\phi }_{\alpha }^{00+;0}\rangle \end{eqnarray} \tag{ 38 }$$

could play a significant role. Indeed, such a matrix element is related to the giant dipole resonance of the α particle. This effect occurs for an initial relative orbital momentum L = 1.

Simulating the effect of matrix element (38) is not possible in the present three-body model. Indeed, while the value of matrix element (38) might be estimated, the radial component ${g}_{{\alpha }^{* }{In}}^{1+}(R)$ of the relative wave function in equation (14) is unknown.

The determination of the final ⁶Li(1⁺) ground state wave function in a variational calculation is explained in [29]. The central Minnesota NN potential is employed as neutron–proton interaction [34]. For the α + N nuclear interaction, the potentials of Voronchev et al [35] and of Kanada et al [36] are employed. They are slightly renormalized by respective scaling factors 1.014 and 1.008 to reproduce the experimental binding energy 3.70 MeV of ⁶Li with respect to the α + n + p threshold. The Coulomb interaction between α and proton is taken as 2e² erf(0.83 R)/R [37]. The coupled hyperradial equations are solved with the Lagrange-mesh method [29, 38]. The hypermomentum expansion includes terms up to K_max = 24, which ensures a good convergence of the energy and of the T = 1 component of ⁶Li. The ground state is essentially S = 1 (96%). The matter r.m.s. radius of the ground state (with 1.4 fm as α radius) is found as $\sqrt{{r}^{2}}\approx 2.25$ fm with the potential of [35] or 2.24 fm with the potential of [36], i.e. values slightly lower than the experimental value 2.32 ± 0.03 fm [39]. The isotriplet component in the ⁶Li ground state has a squared norm 5.3 × 10⁻³ with the potential of [35] and 4.2 × 10⁻³ with the potential of [36].

For the initial scattering waves, the radial wave function u⁰¹¹(r) of the deuteron is the ground state solution of the Schrödinger equation with the Minnesota potential with ${{\hslash }}^{2}/2{m}_{N}=20.7343\,\mathrm{MeV}$ fm². The Schrödinger equation is solved by using the Lagrange–Laguerre mesh method [38]. The converged deuteron energy is E_d = −2.202 MeV with 40 mesh points and a scaling parameter h_d = 0.40. The scattering wave functions ${g}_{i}^{L\pi }(R)$ of the α + d relative motion are calculated with the deep potential of [19] adapted from the potential of [40].

The astrophysical S factor for multipolarity Eλ is defined in terms of the cross section σ_Eλ(E) as [41]

$$\begin{eqnarray}&&{S}_{E\lambda }(E)=E{\sigma }_{E\lambda }(E)\exp (2\pi \eta ),\end{eqnarray} \tag{ 39 }$$

where η is the Sommerfeld parameter.

First, we evaluate the role of the two contributions to S_E1 that are calculable in the present model, i.e. the isovector transition involving operator (27) from the L = 1 initial partial wave to the T_f = 1 component of the ⁶Li ground state and the isoscalar transition involving operator (28) to the T_f = 0 component. These two contributions add coherently. The transition operator given by the first term of equation (28) differs from the ones studied in several earlier works [6, 14, 15]. Indeed, it is argued in [31] that a neglected term in the matrix element may be rather large in these works. In the isoscalar operator (6) based on a Siegert transformation from which expression (28) is deduced, the second term should be negligible in the present case. The resulting difference is that the coefficient of the first term of equation (6) is smaller by a factor 4 than in the operators considered in [6, 14, 15].

In table 1, the resulting isovector and isoscalar ${S}_{E1}^{\mathrm{IV}+\mathrm{IS}}$ factor is compared at three energies with the purely isovector ${S}_{E1}^{\mathrm{IV}}$ factor. The isoscalar correction represents about 2%. It can be neglected as long as the isovector part is not better known. Notice that the isoscalar correction should be more important in the d(d, γ)⁴He capture reaction since the photon wavenumber k_γ is much larger at low scattering energy.

Table 1.  E1 astrophysical S factor with the isovector (IV) and isovector + isoscalar (IV+IS) models. The α + N interaction of [35] is used.

E (MeV)	${S}_{E1}^{\mathrm{IV}}$ (MeV b)	${S}_{E1}^{\mathrm{IV}+\mathrm{IS}}$ (MeV b)
0.01	6.38 × 10−10	6.23 × 10−10
0.1	1.17 × 10−9	1.15 × 10−9
1	1.45 × 10−8	1.41 × 10−8

With the α + N potential of [35], the present IV+IS S_E1 is represented in figure 1 as a dotted line. We have reanalyzed S_E2 calculated with the E2 operator of equation (29) within the three-body model of [24], depicted as a dashed line in figure 1. At low energies, the cross section is very sensitive to the asymptotic behavior of the overlap integrals between the deuteron and the three-body wave functions for partial waves L = 0 and 2,

$$\begin{eqnarray}&&{I}_{L}(R)=\langle {[{\phi }^{011}\otimes {Y}_{L}({{\rm{\Omega }}}_{R})]}^{1M}| {{\rm{\Psi }}}_{f}^{1M+}\rangle ,\end{eqnarray} \tag{ 40 }$$

up to large α − d distances R. In the model of [24], I_L(R) follows over the interval [5, 10] fm the expected asymptotic behavior ${C}_{L}{W}_{-{\eta }_{b},L+1/2}(2{k}_{b}R)/R$, where η_b and k_b are the Sommerfeld parameter and wavenumber calculated at the separation energy 1.474 MeV of the ⁶Li bound state into α and d. The L = 0 asymptotic normalization coefficient is C₀ ≈ 2.05 fm^−1/2 in reasonable agreement with the value C₀ ≈ 2.30 fm^−1/2 extracted in [42] from experimental data on α + d scattering. However, beyond about 10 fm, the absolute value of I_L(R) decreases faster than the correct asymptotics. Hence, within that model, S_E2 is underestimated at low collision energies. To solve this problem, beyond R₀ = 7.5 fm, we replace I_L(R) by the exact asymptotic expression with C_L calculated at 7.5 fm. This corrected S factor is denoted as S_E2c and is represented as a full line in figure 1. It is significantly larger than S_E2 because the cross section is sensitive to R values up to about 50 fm at E = 0.1 MeV. From now on, we only use S_E2c. Around the resonance, the S factor is dominated by E2 transitions. Dipole transitions should be dominant below about 0.1 MeV.

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The total S factors S_E1 + S_E2c calculated with the potentials of [35] (Model A) and [36] (Model B) are presented in figure 2. They are compared with the direct data of [8] above the resonance (triangles), of [9] on resonance (open circles), and of [12] around 0.1 MeV (full circles). The indirect breakup data of [10] are indicated as squares. At low energies, the total S factor obtained in Model A (full line) nicely agrees with the LUNA data. The total S factor in Model B (dotted line) is lower by about 35% than in Model A (full line) but remains within the experimental error bars. This relative smallness is related with a smaller T_f = 1 component in Model B.

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Despite that several possibly important T = 1 contributions are not included in the present discussion, i.e. mainly the whole T_i = 1 component in the initial wave and the T_f = 1 dipole excitation of the α core in the final wave function, one may nevertheless conjecture that a consistent treatment of all isovector E1 transitions can explain the low-energy experimental data. One can assume that the T_i = 1 component in the initial wave has the same order of magnitude as the one of the final bound state. Even a rough estimation of the role of the T_f = 1 dipole excitation of the α core is not possible because of a lack of knowledge of the functions ${g}_{{\alpha }^{* }{In}}^{1+}(R)$ appearing in equation (13). The present conjecture assumes, however, that the different contributions do not interfere destructively.

The developments of the previous sections now allow us to discuss the validity of the exact-masses prescription for reactions between N = Z nuclei. We have seen that one can conjecture that isovector E1 transitions are able to explain the low-energy S factor with a good accuracy. This is incompatible with the exact-masses prescription as we now show.

To simplify the discussion, let us consider E1 transitions in the two-cluster case. When the isospin formalism is not used, the E1 operator reads

$$\begin{eqnarray}&&{{ \mathcal M }}_{\mu }^{E1}=e\displaystyle \sum _{j=1}^{Z}{r}_{{pj}}^{{\prime} }{Y}_{1\mu }({{\rm{\Omega }}}_{{pj}}^{{\prime} }{}_{}),\end{eqnarray} \tag{ 41 }$$

where ${{\boldsymbol{r}}}_{{pj}}^{{\prime} }$ is the coordinate of proton j with respect to the center-of-mass of the system. For a system of two clusters with masses A₁m_N and A₂m_N, where ${m}_{N}=\tfrac{1}{2}({m}_{n}+{m}_{p})$ is the nucleon mass, and charges Z₁e and Z₂e, this operator can be rewritten as [43]

$$\begin{eqnarray}&&{{ \mathcal M }}_{\mu }^{E1}={{ \mathcal M }}_{1\mu }^{E1}+{{ \mathcal M }}_{2\mu }^{E1}+e\,\displaystyle \frac{{A}_{1}{A}_{2}}{A}\left(\displaystyle \frac{{Z}_{1}}{{A}_{1}}-\displaystyle \frac{{Z}_{2}}{{A}_{2}}\right){{ \mathcal Y }}_{1\mu }({\boldsymbol{R}}).\end{eqnarray} \tag{ 42 }$$

For the two structureless clusters of the potential model, the internal E1 operators ${{ \mathcal M }}_{1\mu }^{E1}$ and ${{ \mathcal M }}_{2\mu }^{E1}$ do not contribute and matrix elements are obtained from the final term. It vanishes if Z₁/A₁ = Z₂/A₂, and in particular in the case of two N = Z clusters.

If the masses of the clusters are M₁ and M₂, the center-of-mass coordinate ${{\boldsymbol{R}}}_{\mathrm{cm}}$ replacing (4) becomes

$$\begin{eqnarray}&&{{\boldsymbol{R}}}_{\mathrm{cm}}=\displaystyle \frac{1}{M}({M}_{1}{{\boldsymbol{R}}}_{\mathrm{cm}}^{1}+{M}_{2}{{\boldsymbol{R}}}_{\mathrm{cm}}^{2})\end{eqnarray} \tag{ 43 }$$

as a function of the center-of-mass coordinates ${{\boldsymbol{R}}}_{\mathrm{cm}}^{1}$ and ${{\boldsymbol{R}}}_{\mathrm{cm}}^{2}$ of the clusters, with M = M₁ + M₂. When this expression is introduced in (41), one recovers (42) except for the dimensionless factor multiplying ${{ \mathcal Y }}_{1\mu }({\boldsymbol{R}})$ which is exactly replaced by

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{1}{M}_{2}}{M}\left(\displaystyle \frac{{Z}_{1}}{{M}_{1}}-\displaystyle \frac{{Z}_{2}}{{M}_{2}}\right).\end{eqnarray} \tag{ 44 }$$

The exact-masses prescription consists in using this factor in (42), where M₁ and M₂ are the experimental masses of the colliding nuclei. For N = Z nuclei, this factor does not vanish in reactions between different nuclei. While this prescription is exact in the context of the two-cluster model with masses M₁ and M₂ differing from A₁m_N and A₂m_N, it is in contradiction with the treatment of a collision between N = Z nuclei within the isospin formalism as we show below.

The factor (44) is sometimes also justified as a relativistic correction [26]. If one replaces the center-of-mass coordinates of the clusters by center-of-energy coordinates, the electric dipole moment becomes closer to expression (44). Though it is true that relativistic corrections could play a role, the argument is weakened by the fact that the original factor ${Z}_{1}/{A}_{1}-{Z}_{2}/{A}_{2}$ is based on a microscopic description in terms of nucleons while the center-of energy argument is based on a two-cluster structure. Consistent relativistic corrections should also be based on nucleons.

Let us first notice that the factor (44) still vanishes in collisions between identical nuclei. It predicts that E1 transitions are strictly forbidden in collisions between two identical N = Z nuclei, in contrast with microscopic and ab initio calculations [3, 4]. It would for example be ineffective to try to describe the forbidden E1 deuteron-deuteron capture.

Second, let us show that there is no relation of this factor with isospin mixing. The mass of a nucleus ${}_{Z}^{A}$X_N can be written as

$$\begin{eqnarray}&&M={{Am}}_{N}+(N-Z)\tfrac{1}{2}({m}_{n}-{m}_{p})-B(A,Z)/{c}^{2},\end{eqnarray} \tag{ 45 }$$

where B(A, Z) is the binding energy. As the binding energy per nucleon is small with respect to the nucleon mass energy, factor (44) can be approximated for a capture involving nuclei with N = Z as

$$\begin{eqnarray}{m}_{N}\left(\displaystyle \frac{{Z}_{1}}{{M}_{1}}-\displaystyle \frac{{Z}_{2}}{{M}_{2}}\right) & \approx & \displaystyle \frac{{Z}_{1}}{{A}_{1}}\left(1+\displaystyle \frac{B({A}_{1},{Z}_{1})}{{A}_{1}{m}_{N}{c}^{2}}\right)-\displaystyle \frac{{Z}_{2}}{{A}_{2}}\left(1+\displaystyle \frac{B({A}_{2},{Z}_{2})}{{A}_{2}{m}_{N}{c}^{2}}\right)\\ & = & \displaystyle \frac{1}{2{m}_{N}{c}^{2}}\left(\displaystyle \frac{B({A}_{1},{Z}_{1})}{{A}_{1}}-\displaystyle \frac{B({A}_{2},{Z}_{2})}{{A}_{2}}\right).\end{eqnarray} \tag{ 46 }$$

This correction is small since the binding energy per nucleon does not vary much from one nucleus to another. In the α + d case, it is about 4 × 10⁻⁴. This factor is quite small and is fortuitously able to reproduce a plausible order of magnitude of forbidden E1 transitions. However, there is no physical relation between this correction and the dominant isovector transitions when the E1 transition is isospin-forbidden. Indeed, the binding energy per nucleon of an N = Z nucleus mainly depends on the dominant T = 0 component of its ground state. It is in no appreciable way sensitive to T = 1 admixtures as E1 matrix elements describing an isospin-forbidden capture should be.

Moreover, for the α + d capture, the prescription involves a matrix element which is sensitive to the long distance α + d tail of the ⁶Li ground state wave function. This is not the case in the isospin formalism where the main part of isospin mixing occurs when the two clusters overlap.

Can the exact-masses prescription give a realistic energy dependence of the S factor below the 711 keV resonance? Since the dominant initial orbital momentum is L = 1, the low-energy dependence of the initial relative scattering wave (equation (11)) is close to the dependence of the regular Coulomb function F₁ (see equation (7) of [44]),

$$\begin{eqnarray}&&{g}_{i}^{J-}(R)\approx {E}^{1/4}[{f}_{0}(R)+{f}_{1}(R)E+...]\exp (-\pi \eta ).\end{eqnarray} \tag{ 47 }$$

In any model, the coefficients f_i(R) are calculable functions of R. For Coulomb waves, they are given by equation (22) of [44]. The integral M(E) over R appearing in matrix element (35) and its various corrections can thus be written at very low energies as

$$\begin{eqnarray}&&M(E)\propto {E}^{1/4}({M}_{0}+{M}_{1}E+...)\exp (-\pi \eta ),\end{eqnarray} \tag{ 48 }$$

where coefficient M_i is an integral involving f_i(R), the radial operator R, and the overlap integral I_L(R) of the bound state wave function with the internal cluster wave functions (such as equation (40) in the three-body case). This last factor is quite different in the exact-masses prescription (where it is just given by the final bound state wave function with T_f = 0) and in isovector matrix elements (where it corresponds to a small T_f = 1 admixture of the final wave function). In particular, it is quite different at large distances since the T_f = 1 admixture does not have an α + d asymptotic behavior. Hence M₀ and M₁ may be quite different in both descriptions.

The low-energy behavior of the S factor is given by the expansion

$$\begin{eqnarray}&&S(E)=S(0)(1+{s}_{1}E+...),\end{eqnarray} \tag{ 49 }$$

where the slope s₁ depends on the ratio of M₁ and M₀ [44, 45]. At sufficiently low energies, this ratio computed with the exact-masses prescription is not related to the one in the isovector transition picture. We have checked numerically that the S factors of both descriptions have different low-energy dependences. The prescription does not reproduce the physical energy slope of S_E1 near zero energy.

The E1 S factor which is dominant below about 0.1 MeV decreases with decreasing energy since it is due to a transition from an initial P wave. As transitions from S waves have an almost flat energy dependence at very low energies, an energy (possibly very low) may exist where transitions from an initial S wave dominate.

The E2 capture cross section mainly corresponds to a transition between an initial D-wave and the ⁶Li ground state. In the present α + n + p model, an E2 capture from an initial S wave exists but is smaller than the other E2 contributions by several orders of magnitude in the energy range of figures 1 and 2 [24]. However, other transitions starting from the S wave are possible, which are not considered here. Since the ⁶Li, ⁴He, and ²H ground states contain a D-wave component due to the NN tensor force, several types of E2 transition from an initial S wave can contribute. As the energy dependence of transition matrix elements from an initial S wave is much weaker than for a D-wave, this contribution could become dominant below some low-energy. This mechanism is well illustrated by the d(d, γ)⁴He capture reaction [3, 4]. The main contribution to the capture at low energies is due to the small D-wave components of the α particle and of the deuterons. For ⁴He(d, γ)⁶Li, earlier works indicate that this component is small [13, 22] but they are restricted to energies above the 711 keV resonance. It is thus not possible for the moment to estimate the energy below which this mechanism would be important nor the order of magnitude of its contribution to the cross section at low energies.

We have performed a partial test within the α + n + p three-body model by including a D-wave component in the initial deuteron wave function. With the full deuteron wave function obtained with the soft-core potential of [46], the S wave contribution to S_E2 is negligible above 10 keV. The resulting S wave capture remains very small in agreement with previous studies. Full confirmation requires a calculation taking simultaneous account of the ⁶Li, ⁴He, and ²H D components. Such a calculation requires extensions of the three-body model but is within the reach of present-day ab initio calculations.

The magnetic dipole capture is another case where capture from the S wave can occur. The microscopic M1 operator can be written as a sum of a term proportional to the total angular momentum and a residual spin term. The matrix elements of the first term must vanish in any model because of the orthogonality between the initial and final wave functions [26, 27, 47]. It is thus meaningless to evaluate M1 capture in models (like the present one) where the initial scattering partial waves and the final bound state wave function are not derived from the same Hamiltonian. When the matrix element of the residual spin term is small, M1 transitions are strongly hindered. The energy below which M1 transitions might dominate E1 transitions must be very small.