152 CHAPTER 7. THE 16O NUCLEUS Differently from the 3+1 state, the path traced by the three components of the 3−1 line is smooth and possesses almost the expected shape in the case of exact restoration of rotational symmetry. In particular, at N = 25, the energies of 3−A
2, 3−T
1 and 3−T
2 overlap within 0.96 MeV.
Concerning the 4−1 level, itsT2component displays some deviations atN ∼23, already detected in the same region in the 2−T2 multiplet (cf. figs.7.9and 7.11). However, it is possible to infer from fig. 7.17, that the 4−T
1 and 4−T
2 at N = 25 reach the multiplet-averaged eigenvalue from above, whereas the 4−A1 and 4−E approach the latter from below, as observed for the 4+2 state of 8Be in fig.5.13.
FIGURE 7.18 – Average value of the squared angu-lar momentum for the multiplets of the 3−1 state as a function of the lattice size fora ≈0.59 fm. As in fig. F-7-04, the multiplet average ofL2 is denoted by a red line. The dashed curves denote the ex-pected behaviour ofL2 in the largeN regime, but their predictive power is limited by discretization errors increasing withN.
FIGURE 7.19 – Average value of the squared angu-lar momentum for the multiplets of the 4−1 state as a function of the lattice size for a ≈ 0.59 fm.
The blue line represents the multiplet average of L2, while the dashed curves denote the expected behaviour of the squared angular momentum in the large N region, but increasing discretization effects withN are present.
We pursue our analysis of finite-volume effects by computing the average values of the total squared angular momentum operator (cf. sec.4.3.3). Differently from the 3+1 line of the previous section, the values of L2 for the 3−1 multiplets follow a smoother path, even in the region of the maximum. Besides a shift of ≈5.5ℏ2 with respect to the exact eigenvalue, the curves overlap at at N = 25 region within ≈ 1.5 units of ℏ2 and seem to tolerate better the augmented discretization errors in dashed part of the curves, see fig. 7.18.
On the other hand, the average values of L2 for the 4−1 multiplets follow a separate paths in almost all the considered box size interval, except for the neighbourhood ofN ≈13, where the four curves intersect and theN ≲5 region. It is evident that the discrepancies for N ≲13 are essentially genuine finite-volume effects, whereas the visible separation between the curves in the asymptotic region is prompted by discretization artifacts.
The average values of the inter-α distance, dispalyed in figs. 7.20 and 7.21 confirm some of the trends observed in the L2 plots in figs.7.18and 7.19). The multiplet average of R for the 3−A
2, 3−T
1 and 3−T
2 states, in fact, reaches smoothly a value of 3.95 fm at N = 25 and the three individual values for the interparticle distance overlap within ≈ 0.15 fm in that limit. The components of the 4−1 level, on the other hand, reveal a more oscillatory behaviour, albeit not comparable with the one in fig. 7.19.
7.4. THE 3−1 AND41− ENERGY LEVELS 153
FIGURE 7.20 – Behaviour of the average inter-particle distance for the 3−1 multiplets as a function of the lattice size. The multiplet average ofR for the 3−1 state is denoted with a green line, while the dashed curves represent the behaviour ofRin the N >10 region and are subject to larger discretiz-ation errors, increasing as 1/a.
FIGURE 7.21 – Behaviour of the average inter-particle distance for the 4−1 multiplets as a function of the lattice size. The multiplet average ofR for the 4−1 state is denoted with a blue line, while the dashed lines represent the expected behaviour of R in the large N regime and are affected by dis-cretization errors increasing with the inverse of the lattice spacing.
In particular, the 4−T
2multiplet displays some disturbance aroundN = 15, then follows smoothly the A1 and E multiplets until N = 25, where it reaches 4.51 fm. The curve of the 4−T1 state, instead, gives rise to a shallow minimum at N ∼21, then increases until 4.64 fm atN = 25.
Finally, we conclude the chapter with the behaviour of the energy eigenvalues of the 31− and 4−1 states as a function of the lattice spacing.
FIGURE 7.22 – Behaviour of the energy of the 3−1 multiplet eigenstates as a function of the lattice spa-cing for Na ≥20 fm (solid lines). The multiplet-averaged 3−1 energy (green line) quenches the de-viations in the path towards the continuum and infinite-volume counterpart, while the dashed lines denote the behaviour ofR in theN >10 region.
FIGURE 7.23 – Behaviour of the averageα − α dis-tance of the 3−1 eigenstates as a function of the lattice spacing for Na ≥ 20 fm. As before, the multiplet average ofR for the 3−1 state is denoted with a green line, while the dashed lines consti-tute the expected behaviour of R in the large N regime.
154 CHAPTER 7. THE 16O NUCLEUS We first concentrate on the 3−1 multiplets and adopt for solid and dashed lines the same conventions as in secs.7.2and7.3. The curves of the three multiplets overlap with appreciable precision along all the path from 5.0 fm to the dashed part of the cuve, except in the regions of the two minima at a ≈ 2.2 and 3.0 fm. At the endpoint (a ≈ 1.0 fm), the superposition between the three curves is reached withnin ≈1.42 MeV precision, see fig. 7.22. Analogously as in the 2−1 case, the two minima seem correlated with the ones at ≈2.2 and 2.9 fm of the potential energy, V, which, in turn, are in quite good agreement with the two minima of the α-αseparation at ≈2.15 and 2.80 fm in fig.7.23. Therefore, the connection between the max-ima of the PDFs for the 3−A
2, 3−T
1 and 3−T
2 multiplets and the minima of Er and R can be still established. Noteworthy is also the overlap between the three 3−1 curves ata ≈1.0 fm, equal to ≈0.16 fm.
FIGURE 7.24 – Behaviour of the energy of the 4−1 multiplet eigenstates as a function of the lat-tice spacing for Na ≥ 20 fm (solid lines). Al-though the multiplet-averaged 4−1 energy (blue line) quenches the fluctuations towards the con-tinuum and infinite-volume counterpart, the two cubic group multiplets almost follow the same path. The dashed lines denote the behaviour of Rin theN >10 regime and are affected by larger finite-volume errors.
FIGURE 7.25 – Behaviour of the averageα − α dis-tance of the 4−1 eigenstates as a function of the lattice spacing for Na ≥20 fm. The multiplet av-erage ofRfor the 4−1 state is denoted with a blue line, while the dashed lines reproduce the beha-viour ofR in the largeN regime, but are affected by finite-volume errors increasing with 1/a.
Regarding the the 4−1 line, the four curves for the energy eigenvalue in fig.7.24follow separate paths, except for theA−1 andT1−multiplets. Additionally, not all the minimum points are found in the same position. While the latter two multiplets display common minima at a ≈2.2 and 3.0 fm, the T2− extrema appear weakly pronounced and uncertain, whereas of the minima of the 4−E is found at a slightly shifted position (a ≈ 2.75 fm). The average values of the α-α separation, instead, highlight more agreement among the four multiplets in the region of the two minimum point, loacted at a ≈2.1 and 2.75 fm.
Despite the sizable shifts between the paths until a ≈ 2.25 fm, in the asymptotic region the energy eigenvalues of the four cubic-group multiplets overlap in good approximation, over-lapping within 1.3 MeV at a ≈1. Similarly, the dotted part of theR curves in fig. 7.25display negligible discrepancies and reach a value of≈4.28 fm with discrepancy of≈0.10 fm, despite the finite-volume errors.
CHAPTER 8
VARIATIONAL CALCULATION OF THE STRENGTH PARAMETER OF THE 3 α GAUSSIAN POTENTIAL FOR
16O
As shown in fig. 8.1, the M-α Hamiltonian with a superposition of the isotropic Ali-Bodmer potential (cf. eq. (4.2)) with the Coulomb (cf. eq. (4.3)) and 3αGaussian potentials (cf. eq. (4.3)) with the parameters displayed in sec.4.2.1underbinds the 16O nucleus by circa 11 MeV.
Without resorting to the addition of further interacting terms in the Hamiltonian, in this section we aim at re-adjusting the parameterV3 of the 3α potential so that the energy eigenvalue of the ground state concides with the opposite of the 4αdissociation energy gap (≈14.42 MeV).
Knowing both the three potentials, we perform a linear variational calculation based on ref. [5]
in a model space made of four-body harmonic-oscillator functions. The latter are completely symmetric under particle exchange, translationally invariant and possess a well-defined orbital angular momentum. We extend the calculation by anchoring the V3 parameter to theα+12C threshold at ≈ 7.16 MeV. Additionally, we perform the variational calculation in the case of Dirac-delta distribution of the electric charge, in which the Coulomb potential assumes its original form for pointlikeαparticles. The results are presented in secs.8.5.1and8.5.2in the optimized and non-optimized approach with respect to the frequency 2πω of the harmonic oscillator respectively.