Mapping between configuration-interaction shell-model amplitudes and experimental neutron/proton transition matrix elements in neutron-rich boron and carbon isotopes

The ratio $M_n/M_p$ of neutron to proton quadrupole transition matrix elements is the standard diagnostic for decoupled neutron and proton motion in neutron-rich light nuclei: for collective isoscalar oscillations it should track $N/Z$, and the large excesses once reported in $^{16}$C and $^{20}$C were read as evidence that valence neutrons oscillate nearly independently of the proton core. Qi (2010) showed that this diagnostic is subtly mismatched to configuration-interaction (CI) shell-model calculations: the model-space amplitudes $\mathcal{M}_{p,n}$ are related to the physical matrix elements $M_{p,n}$ only through effective charges and exact-isospin mirror-nucleus relations, so a calculated $\mathcal{M}_n/\mathcal{M}_p\sim4$ can correspond to $M_n/M_p\sim2$. The note deferred the quantitative application of this mapping: its closing sentence leaves detailed shell-model calculations on the spectroscopic properties of neutron-rich boron and carbon isotopes to future publication, and within the note the effective charges enter only through standard illustrative values while the neutron relation rests on mirror-nucleus relations that assume strictly conserved isospin. Higher-precision $B(E2)$ and $M_n/M_p$ measurements have since been reported for the carbon isotopes of this class (refs. 17, 20); the program addressed here is to connect CI amplitudes quantitatively to the measured $M_n/M_p$ systematics of the neutron-rich boron and carbon region, with explicit control of the operator conventions on which that connection depends.

Background

Toward the neutron dripline — the limit of neutron binding beyond which a nucleus cannot hold further neutrons — light nuclei such as the heavy boron and carbon isotopes can develop valence-neutron distributions that extend far beyond the proton core, the extreme case being one- or two-neutron halos. The standard observable diagnostic for such 'decoupled' neutrons is the ratio $M_n/M_p$ of the neutron and proton transition matrix elements for exciting the first $2^+$ state. The proton part is fixed electromagnetically, since $M_p^2 = B(E2;0^+_{\rm gs}\to 2^+_1)/e^2$, where $B(E2)$ is the reduced electric-quadrupole transition probability; both parts can also be expressed through neutron and proton deformation lengths $\delta_n,\delta_p$ (measures of how much each fluid is deformed in the transition) extracted from inelastic hadronic scattering. Bernstein, Brown and Madsen (1979) showed that for a collective oscillation in which neutrons and protons move together (an isoscalar transition), $M_n/M_p$ should be close to $N/Z$; a large excess over $N/Z$ signals that the valence neutrons oscillate rather independent of the proton core. In 2004 just such an excess was reported for $^{16}$C, with $M_n/M_p=7.6\pm1.7$ obtained from Coulomb-nuclear interference — the interference between the electromagnetic and nuclear parts of the proton-scattering amplitude, which lets neutron and proton excitation be separated — (Elekes et al.) together with an anomalously hindered $B(E2)\approx0.63\ e^2{\rm fm}^4$ from a lifetime measurement (Imai et al.), and in 2009 the analysis of $^{20}$C (Elekes et al.) suggested the pattern persists toward the dripline and requires polarization charges reduced by a factor of about 0.4 from their normal values.

The configuration-interaction (CI) shell model diagonalizes an effective Hamiltonian in a truncated valence model space (here the $p$ and $sd$ shells, denoted $psd$). Its E2 strength is $B(E2)=|e^n_{\rm eff}\mathcal{M}_n+e^p_{\rm eff}\mathcal{M}_p|^2/(2J_i+1)$, where $\mathcal{M}_{p,n}$ are model-space (effective) matrix elements and $e_{\rm eff}$ are effective charges — phenomenological numbers (typically $e^p_{\rm eff}\approx1.5e$, $e^n_{\rm eff}\approx0.5e$ in this region) that mimic core polarization, i.e., the polarization of the closed ${}^{4}{\rm He}$ and other core nucleons by the valence particles. Qi (2010) pointed out that $\mathcal{M}_{p,n}$ are not the physical $M_{p,n}$: comparing the two expressions gives $|eM_p|=|e^n_{\rm eff}\mathcal{M}_n+e^p_{\rm eff}\mathcal{M}_p|$, and, invoking exact isospin symmetry — the approximate charge independence of the nuclear force, under which mirror nuclei (proton and neutron numbers interchanged) obey $M_n(N,Z)=M_p(Z,N)$ — a corresponding relation follows for $M_n$, namely $|eM_n|=|e^p_{\rm eff}\mathcal{M}_n+e^n_{\rm eff}\mathcal{M}_p|$. The two ratios can differ substantially: with the standard effective charges $e^p_{\rm eff}=1.5e$ and $e^n_{\rm eff}=0.5e$ used illustratively in the note, a calculated $\mathcal{M}_n/\mathcal{M}_p\sim4$ maps to $M_n/M_p\sim2$. Claims about neutron-proton decoupling based on shell-model amplitudes therefore stand or fall with this mapping, and the note deferred exactly its quantitative application: detailed shell-model calculations on the spectroscopic properties of the neutron-rich B and C isotopes, in which the effective-charge convention and the assumption of strictly conserved isospin underlying the mirror relations directly enter the extracted $M_n/M_p$.

Problem Statement

Carry out the program deferred in Qi (2010): detailed configuration-interaction shell-model calculations of the spectroscopic properties of the neutron-rich boron and carbon isotopes that connect the model-space amplitudes $\mathcal{M}_{p,n}$ quantitatively to the physical neutron/proton transition matrix elements through the relations $|eM_p|=|e^n_{\rm eff}\mathcal{M}_n+e^p_{\rm eff}\mathcal{M}_p|$ and $|eM_n|=|e^p_{\rm eff}\mathcal{M}_n+e^n_{\rm eff}\mathcal{M}_p|$ — the neutron relation relying on the mirror-nucleus relations valid under exact isospin symmetry — and apply the resulting mapping to the measured $B(E2)$ values, deformation lengths and $M_n/M_p$ ratios reported in the cited literature for the isotopes of this class — the carbon isotopes $^{15}$C, $^{16}$C and $^{20}$C (refs. 3, 4, 5, 7, 9, 17, 20) — to determine, quantitatively, whether and to what extent their valence neutrons are decoupled from the proton core. The required scope is the class of neutron-rich boron and carbon isotopes in the $psd$ valence space, both elements included: the shell-model calculations are required across the class, while the empirical comparison is restricted to the isotopes named above, for which such measurements are cited; no boron measurement is assumed. Since the extracted ratios depend on the effective charges entering the mapping — the standard values $e^p_{\rm eff}=1.5e$, $e^n_{\rm eff}=0.5e$ used illustratively by Qi (2010) are not fixed by data — and on the validity of the isospin-symmetric mirror relations, the operator conventions and the effective-charge dependence of the extracted ratios, from the standard values to the empirically constrained charges for these isotopes, form part of the result. The scientific question is whether each deduced $M_n/M_p$, expressed in units of the isoscalar value $N/Z$, exceeds twice that value.

References

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