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R16evised Manuscript 30 October 2017 Energy Levels of Light Nuclei A = 16 a,b a,c a,c D.R. Tilley , H.R. Weller and C.M. Cheves a Triangle Universities Nuclear Laboratory, Durham, NC 27708-0308, USA b Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, USA c Department of Physics, Duke University, Durham, NC 27708-0305, USA Abstract: An evaluation of A = 16–17 was published in Nuclear Physics A564 (1993), p. 1. This version of A = 16 differs from the published version in that we have corrected some errors discovered after the article went to press. The introduction and introductory tables have been omitted from this manuscript. Reference key numbers have been changed to the NNDC/TUNL format. (References closed December 31, 1992) This work is supported by the US Department of Energy, Ofﬁce of High Energy and Nuclear Physics, under: Contract No. DEFG05-88-ER40441 (North Carolina State University); Contract No. DEFG05-91-ER40619 (Duke University).

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Nucl. Phys. A564 (1993) 1 A = 16 Table of Contents for A = 16 Below is a list of links for items found within the PDF document. The introductory Table 2 is available on this website via the link. 16 16 16 16 16 16 16 16 16 16 16 16 16 A. Nuclides: He, Li, Be, B, C, N, O, F, Ne, Na, Mg, Al, Si B. General Tables: 16 Table 16.1: General table for C 16 Table 16.4: General table for N 16 Table 16.12: General table for O 16 16 Table 16.29: General table for F and Ne C. Tables of Recommended Level Energies: 16 Table 16.2: Energy levels of C 16 Table 16.5: Energy levels of N 16 Table 16.13: Energy levels of O 16 Table 16.30: Energy levels of F 16 Table 16.32: Energy levels of Ne D. References 16 16 16 16 E. Figures: C, N, O, F, Isobar diagram F. Erratum to the Publication: PS or PDF

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A = 16 theoretical Because of the very large body of theoretical work that has been carried out for the A = 16, and the importance of the spherical shell model in this work, a general discussion of the shell model 1 description of A = 16 nuclei is provided here. The spherical shell-model provides a complete basis for the description of nuclear states. It is convenient to use harmonic oscillator single-particle wave functions since the coordinate transfor- mations necessary to separate spurious center of mass states, to relate shell-model to cluster-model wave functions and so on can be made exactly. Conﬁgurations are classiﬁed by the number of oscillator quanta that they carry beyond the minimum allowed by the Pauli Principle as 0ℏω, 1ℏω, 2ℏω . . . excitations. Non-spurious states of A = 16 in general involve admixtures of npnh conﬁgu- π − rations but the lowest excitations of each isospin can, with the exception of the K = 0 band with 16 − −n n the O 9.58 MeV 1 state as bandhead, be thought of as dominantly p (sd) excitations. In fact, the lowest eigenstates of an nℏω calculation can usually be written economically in terms of prod- −n n uct states of low-lying p and (sd) eigenstates. In the simplest version of this weak-coupling −n n model, one identiﬁes the p and (sd) eigenstates with the physical states of the relevant nuclei and takes the diagonal expectation value of Hp + Hsd from the known masses. The contribution from the cross-shell, or particle-hole, interaction can often be quite reliably estimated by using ph 16 16 matrix elements extracted from the nominal 1p1h states of O or N. The 2p2h states with T = 0 and T = 1 cannot, in general, be described in terms of the simple weak-coupling model, although there are examples to which such a description can be applied. Shell-model calculations which use empirical interactions ﬁtted to data on 1ℏω excitations in the mass region do, however, produce 2p2h T = 1 states in one-to-one correspondence with the lowest 16 positive-parity states of N (see Table 16.5). They also produce T = 0 2p2h states starting at 16 around 12 MeV in O. In this case, the 2p2h states are interleaved with 4p4h states which begin at + lower energies. The lowest 2p2h T = 0 states can be related in energy to the 14.82 MeV 6 state 2 14 16 which is strongly populated by the addition of a stretched d pair in the N(α, d) O reaction. 5/2 The lowest six 2p2h T = 2 states can be very well described in this way. Weak-coupling ideas can be extended to the lowest 3p3h and 4p4h states. Since the 3 and 4 particle (or hole) conﬁgurations are strongly conﬁguration mixed in the jj-coupling scheme, the ph interaction is usually represented in the simple monopole form Eph = a + btp.th plus a small attractive Coulomb contribution. The ph interaction then gives a repulsive contribution of 9a and 16a to 3p3h and 4p4h conﬁgurations and separates the T = 0 and T = 1 3p3h states by b MeV. + The empirical values of a and b are a ≈ 0.4 MeV and b ≈ 5 MeV, which put the 4p4h 0 state and − the 3p3h 1 states close to experimental candidates at 6.05, 12.44 and 17.28 MeV respectively, each of which is the lowest member of a band. The weak-coupling states can be used as a basis for shell-model calculations, but the elimina- tion of spurious center-of-mass motion is approximate even within an oscillator framework; orbits outside the p(sd) space are needed and can be important components of states of physical interest. If complete nℏω spaces are used, the choice of basis can be one of computational convenience. A more physical LS-coupled basis is obtained by classifying the states according to the Wigner 1 We are very grateful to Dr. John Millener for providing these comments on the shell model for the A = 16 system. 3

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˜ supermultiplet scheme (SU4 ⊃ SU2 × SU2 symmetry [f ] in spin-isospin space) and the SU3 symmetry (λ µ) of the harmonic oscillator. States with the highest spatial symmetry [f ] maximize the number of spatially symmetric interacting pairs to take advantage of the fact that the NN inter- action is most strongly attractive in the relative 0s state and weak or repulsive in relative p states. These symmetries are broken mainly by the one-body spin-orbit interaction. In np and nh calcula- 2 tions the lowest states are dominated by the [f ] (λ µ) conﬁgurations [n] (2n 0) and [4 4 − n] (0 n) respectively (these symmetries are very good if the one-body spin-orbit interaction is turned off). In npnh calculations, the lowest states are dominated by the highest spatial symmetry allowed for given isospin T and (2n n) SU3 symmetry. These states are identical to harmonic oscillator cluster-model states with 2n quanta on the relative motion coordinate between the nh core and the np cluster. States with a large parentage to the ground state of the core should be seen strongly in the appropriate transfer reaction. In the above, a basic nℏω (mainly npnh) shell-model structure has been matched, through char- acteristic level properties and band structures, with experimental candidates. The mixing between shell-model conﬁgurations of different nℏω is of several distinct types. 2 First, there is direct mixing between low-lying states with different npnh structure; the p → 2 (sd) mixing matrix elements (SU3 tensor character mainly (4 2)) are not large (up to a few MeV) although the mixing can be large in cases of near degeneracy. A second type of mixing is more easily understood by reference to cluster models in which an oscillator basis is used to expand the relative motion wave function. To get a realistic representation of the relative motion wave function for a loosely-bound state or an unbound resonance requires many oscillators up to high nℏω excitation. A related problem, which also involves the radial structure of the nucleus, occurs for the expansion of deformed states (of which cluster states are an example) in a spherical oscillator (shell-model) basis; e.g., deformed Hartree-Fock orbits may require an expansion in terms of many oscillator shells. It is difﬁcult to accomodate this type of radial mixing in conventional shell-model calculations, but sympletic Sp(6,R) shell-models, in which the SU3 algebra is extended to include 1p1h 2ℏω monopole and quadrupole excitations, do include such mixing up to high nℏω. A third type of mixing involves the coupling of npnh excitations to high-lying (n + 2)ℏω 2 2 conﬁgurations via the strong (λ µ) = (2 0) component of the p → (sd) interaction. In the full (0 + 2 + 4)ℏω calculations, the large (30-45%) 2p2h admixtures in the ground state are mainly of the (2 0) type, which are intimately related to the ground-state correlations of RPA theory, and lead to the enhancement (quenching) of ∆T = 0, ∆S = 0 (otherwise) exitations at low momentum transfer. For most detailed structure questions, a shell-model calculation is required to include the rel- evant degrees of freedom. For example, (1990HA35) address two important problems with com- 16 − 16 plete (0 + 2 + 4)ℏω and (1 + 3)ℏω model spaces. One is the rank-zero N(0 ) → O(gs) β decay and the inverse µ capture which receive large two-body meson-exchange current contribu- 16 tions. The other is the distribution of M1 and Gamow-Teller strength based on the O ground state; this is a complicated problem which involves 2p2h . . . admixtures in the ground state which break SU4 symmetry. 4

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Many interesting structure problems remain. A detailed understanding of the shapes and mag- nitudes of inelastic form factors is lacking, particularly the shapes at momentum transfers beyond −1 ′ ′ 2 fm . Even in the relatively simple case of M4 excitations, much studied via (e, e ), (p, p ) ′ and (π, π ) reactions, a rather low value of the oscillator parameter b is required to describe the − form factor. Also, the conﬁguration mixing which splits the 4 ; T = 0 strength into two major components and causes isospin mixing has not been satisfactorily described by a shell-model cal- culation. Similar interesting problems occur for isospin-mixed negative-parity states near 13 MeV excitation energy. It is worth noting that, to avoid some serious consistency problems, the large shell-model calculations have omitted orbits outside the p(sd) space except in as much to cleanly separate spurious center-of-mass states. A consistent treatment of 1p1h and 2p2h correlations in multi-ℏω shell-model spaces remains a challenging question. 16 He (not illustrated) This nucleus has not been observed. See (1982AV1A, 1983ANZQ, 1986AJ04). 16 Li (not illustrated) π This nucleus has not been observed. Shell model studies (1988POZS) are used to predict J and the magnetic dipole moment. 16 Be (not illustrated) This nucleus has not been observed. Its atomic mass is calculated to be 59.22 MeV. It is 14 then unstable with respect to breakup into Be + 2n by 2.98 MeV. See (1974TH01, 1986AJ04, π + + + 1987SA15). The ﬁrst three excited states with J = 2 , 4 , 4 are calculated to be at 1.90, 5.08, and 6.51 MeV using a (0 + 1)ℏω space shell model (1985PO10). 16 B (not illustrated) This nucleus has not been observed in the 4.8 GeV proton bombardment of a uranium target. It is particle unstable. Its mass excess is predicted to be 37.97 MeV; it would then be unstable 15 with respect to decay into B + n by 0.93 MeV. See (1985WA02, 1986AJ04). The ground state π − is predicted to have J = 0 and the ﬁrst three excited states are predicted to lie at 0.95, 1.10, π − − − and 1.55 MeV [J = 2 , 3 , 4 ] in a (0 + 1)ℏω space shell model calculation. See (1983ANZQ, 1985PO10, 1986AJ04). Predicted masses and excitation energies for higher isospin multiplets for 9 ≤ A ≤ 60 are included in the compilation (1986AN07). An experiment (1985LA03) involving 40 18 16 in-ﬂight identiﬁcation of fragments from 44 MeV/u Ar found no trace of B or B and provides 16 strong evidence that B is particle-unstable. 5

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16 C (Figs. 1 and 5) GENERAL: See Table 16.1. 16 − 16 1. C(β ) N Qm = 8.012 16 16 π − The half life of C is 0.747 ± 0.008 sec. It decays to N*(0.12, 3.35, 4.32) [J = 0 , + + 1 , 1 ]: see Table 16.3 and (1993CH06). See also (1986AJ04) and see (1986KI05, 1988WA1E, + − 1992WA1L) for theoretical discussions of extended shell-model calculations of 0 → 0 transi- tions and determination of the mesonic enhancements εmec of the time-like component of the axial 16 current. See also (1992TO04) and see N, reaction 1. 14 16 2. C(t, p) C Qm = −3.013 16 States of C observed in this reaction are displayed in Table 16.2. See also Table 16.3 of (1982AJ01). 16 − + 16 3. O(K , π ) C Σ (1985BE31) used negative kaons of 450 MeV/c to produce Σ hypernuclear states, which they − 16 interpreted as Σ particles in the p3/2 and p1/2 orbits of the Σ C hypernucleus. Their energy − splitting was used to constrain the Σ spin-orbit coupling. (1986HA26) performed a systematic shell-model analysis of Σ-hypernuclear states, in which they deduced a ΣN-spin-orbit interaction about twice as strong as the one for the nucleon. (1986MA1J) Σ reached a similar conclusion after extracting the one-particle spin-orbit splitting εΣ = ε p1/2 − Σ ε p3/2. (1987WU05) used the continuum shell-model to study competition between resonant and quasi-free Σ-hypernuclear production. The observed structures in the excitation spectra are essentially accounted for by the quasi-free mechanism alone. (1989DO1I) perform a series of shell model calculations of energy spectra of p-shell Σ hypernuclei, starting with several different parametrizations of the ΣN effective interaction. Production cross sections are estimated using DWBA. They suggest experiments to resolve open questions regarding the ΣN and Σ-nucleus in- teractions. (1989HA32) uses the recoil continuum shell model to calculate in-ﬂight Σ hypernuclei production of this reaction (and others). They needed to modify the ΣN central interaction to ﬁt data. Coupled channels (CC) calculations for Σ-hypernuclear spectra give an energy integrated cross section which is about 1.7 times the experimental value (1987HA40). (1988HA44) report CC calculations emphasizing the proper treatment of the Σ continuum states. They ﬁnd that a weak Σ central potential and a comparable ΣΛ conversion potential are required to describe experiment. 6

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16 Fig. 1: Energy levels of C. For notation see Fig. 2. 7

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16 Table 16.1: C – General Reference Description Complex Reactions 1986BI1A Heavy ion secondary beams - Results from GANIL 14 222−224 232 1987GU04 Exotic emission of C & other heavy clusters in the fragmentation of Ra & U 1987RI03 Isotopic distributions of fragments in intermediate energy heavy ion reactions 1987SA25 The LISE spectrometer at GANIL (secondary radioactive beam production) 1987SN01 Partitioning of a 2-component particle system & isotope distribution in fragmentation 16 232 1987VI02 Anisotropies in transfer-induced ﬁssion of O + Th 1988RU01 Dynamic treatment of ternary ﬁssion - calculates light charged particle formation 1989SA10 Total cross sections of reactions induced by neutron-rich light nuclei (exp. results) Hypernuclei 1987FA1A Review of International Conference on a European Hadron Facility − 16 1988MA09 Hypernucleus production by K capture at rest on O targets 1989BA92 Strangeness production by heavy ions Other Topics 1986AN07 Predicted masses & excitation energies in higher isospin multiplets for 9 ≤ A ≤ 60 1987BL18 Calc. ground state energy of light nuclei (& excited states for N = Z) using H-F method 1989PO1K Exotic light nuclei and nuclei in the lead region + + 1989RA16 Predictions of B(E2; 0 –2 ) values for even-even nuclei 1 1 Ground State Properties 1987BL18 Calculated ground state energies using Gogny’s effective interaction and HF method 1987SA15 Hartree-Fock calculations of light neutron-rich nuclei using Skyrme interactions 1988POZS Shell model study of light exotic nuclei - compares calculated ground state prop. to data + + 1989RA16 Predictions of B(E2; 0 –2 ) values for even-even nuclei 1 1 1989SA10 Total cross sections of reactions induced by neutron-rich light nuclei 8

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16 Table 16.2: Energy Levels of C π Ex (MeV ± keV) J ; T τ1/2 (sec) or Γ (keV) Decay Reactions + − 0 0 ; 2 τ1/2 = 0.747 ± 0.008 β 1, 2 + 1.766 ± 10 2 γ 2 + 3.027 ± 12 (0 ) (γ) 2 3.986 ± 7 2 γ 2 (+) 4.088 ± 7 3 γ 2 + 4.142 ± 7 4 γ 2 + − + 6.109 ± 15 (2 , 3 , 4 ) Γ ≤ 25 2 − 16 Table 16.3: The β decay of C 16 π Decay to N* (MeV) J Branch (%) log ft − +0.09 a +0.07 0.120 0 0.68 6.70 −0.11 −0.05 − b 0.298 3 < 0.5 > 6.83 − a 0.397 1 < 0.1 > 7.46 + b 3.35 1 84.4 ± 1.7 3.551 ± 0.012 + b 4.32 1 15.6 ± 1.7 3.83 ± 0.05 a (1983GA03). See also (1984GA1A). b (1976AL02). 9

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16 N (Figs. 2 and 5) GENERAL: See Table 16.4. 16 16 For a comparison of analog states in N and O, see (1983KE06, 1983SN03). 16 − 16 1. N(β ) O Qm = 10.419 16 The half-life of N is 7.13 ± 0.02 sec: see Table 16.3 in (1971AJ02). From the unique ﬁrst- 16 π − forbidden character of the β decay [see Table 16.25 and (1984WA07)], N must have J = 2 : 16 see O, reaction 39. See also (1985HE08, 1988BA15). 16 π − The β-decay of N*(0.12) [J = 0 ] has been measured (1983GA18, 1985HA22); adopted −1 16 − 16 − value: λβ = 0.489±0.020sec (1985HE08). The relationship of this rate to that for O(µ , ν) N(0 ) [see reaction 18] and the fact that the large values of these rates support the prediction (1978GU05, 1978GU07, 1978KU1A) of a large ( ≈ 60%) enhancement over the impulse approximation (e.g., εmec = 1.60) has been the subject of a great deal of theoretical study, see, e.g. (1981TO16, 1986KI05, 1986TO1A, 1988WA1E, 1990HA35). The work of (1990HA35, 1992WA1L) is a cul- mination of present knowledge on the determination and interpretation of εmec. See also (1992TO04). − − − + A branching ratio R(0 → 1 )/(0 → 0 ) = 0.09± 0.02 has been reported (1988CH30), imply- − − 16 ing log ft = 4.25 ± 0.10 for the 0 → 1 transition to the O 7.12-MeV level. 7 11 16 2. Li( B, pn) N Qm = 2.533 Gamma rays with Eγ = 120.42± 0.12, 298.22± 0.08 and 276.85± 0.10 keV from the ground 16 state decays of N*(0.12, 0.30) and the decay of the state at 397.27± 0.10 keV to the ﬁrst excited 16 state have been studied. τm for N*(0.30, 0.40) are, respectively, 133 ± 4 and 6.60 ± 0.48 psec. 7 11 See (1986AJ04). Cross section measurements for Li + B at E(c.m.) = 1.45–6.10 MeV have been reported (1990DA03). 9 7 15 3. (a) Be( Li, n) N Qm = 18.082 Eb = 20.572 9 7 14 (b) Be( Li, 2n) N Qm = 7.249 9 7 13 (c) Be( Li, t) C Qm = 8.179 9 7 12 (d) Be( Li, α) B Qm = 10.461 9 7 8 8 (e) Be( Li, Li) Be Qm = 0.368 7 ◦ At incident Li energies of 40 MeV, neutron yields at 0 for reactions (a) and (b) are 50 to 70 9 times smaller than for 40 MeV deuteron-induced reactions on Be (1987SC11). For reactions (c, d, e) see (1982AJ01). 10

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