Probing negative-parity states of ${}^{24}\mathrm{Mg}$ probed with proton and $\alpha$ inelastic scattering

Background: The band structure of the negative-parity states of ${}^{24}\mathrm{Mg}$ has not yet been clarified. The $K^{\pi }=0^{-}$, $K^{\pi }=1^{-}$, and $K^{\pi }=3^{-}$ bands have been suggested, but the assignments have been inconsistent between experiments and theories.

Purpose: Negative-parity states of ${}^{24}\mathrm{Mg}$ are investigated by microscopic structure and reaction calculations via proton and $\alpha$ inelastic scattering to clarify the band assignment for the observed negative-parity spectra.

Method: The structure of ${}^{24}\mathrm{Mg}$ was calculated using the antisymmetrized molecular dynamics (AMD). Proton and $\alpha$ inelastic reactions were calculated using microscopic coupled-channel (MCC) calculations by folding the Melbourne $g$-matrix $NN$ interaction with the AMD densities of ${}^{24}\mathrm{Mg}$.

Results: The member states of the $K^{\pi }=0^{+}$, $K^{\pi }=2^{+}$, $K^{\pi }=0^{-}$, $K^{\pi }=1^{-}$, and $K^{\pi }=3^{-}$ bands of ${}^{24}\mathrm{Mg}$ were obtained through the AMD result. In the $\mathrm{MCC}+\mathrm{AMD}$ results for proton and $\alpha$ elastic and inelastic cross sections, reasonable agreements were obtained with existing data, except in the case of the $4_1^{+}$ state.

Conclusions: The $3^{-}$ state of the $K^{\pi }=3^{-}$ band and the $1^{-}$ and $3^{-}$ states of the $K^{\pi }=0^{-}$ bands were assigned to the $3_1^{-}$(7.62 MeV), $1_1^{-}$(7.56 MeV), and $3_2^{-}$(8.36 MeV) states, respectively. The present AMD calculation is the first microscopic structure calculation to reproduce the energy ordering of the $K^{\pi }=0^{-}$, $K^{\pi }=1^{-}$, and $K^{\pi }=3^{-}$ bands of ${}^{24}\mathrm{Mg}$.

Figure 1

The energy spectra of ${}^{24}\mathrm{Mg}$. (a) The calculated energy levels. (b) The experimental levels for the $K^{\pi }=2^{+}$ ground and the $K^{\pi }=2^{+}$ sidebands from Ref. [62], and those for the $K^{\pi }=3^{-}$, $K^{\pi }=0^{-}$, and $K^{\pi }=1^{-}$ bands assigned in Ref. [2].

Figure 2

Density distribution of intrinsic wave functions prior to the parity and total-angular-momentum projections for the $0_1^{+}(K^{\pi }=0^{+}$), $3_1^{+}(K^{\pi }=2^{+}$), $3_1^{-}(K^{\pi }=3^{-}$), $1_1^{-}(K^{\pi }=0^{-}$), and $1_1^{-}(K^{\pi }=1^{-}$) states, as obtained by AMD. The integrated density projected onto the $X-Z$, $Y-Z$, and $Y-X$ planes is plotted in the left, middle, and right panels, respectively, by contours with the interval of 0.1 ${\mathrm{fm}}^{-2}$ interval. For each state, the axes are chosen to be the principal axes of intrinsic deformation as $\langle ZZ\rangle \ge \langle YY\rangle \ge \langle XX\rangle$ and $\langle XY\rangle =\langle YZ\rangle =\langle ZX\rangle =0$. The deformation parameters $(\beta ,\gamma )$ calculated from the expectation values, $\langle ZZ\rangle$, $\langle YY\rangle$, and $\langle XX\rangle$, are shown in the left panels.

Figure 3

Density difference between the positive-parity and negative-parity components in the intrinsic states of the bandhead states; (a) $3_1^{-}(K^{\pi }=3^{-})$, (b) $1_1^{-}(K^{\pi }=0^{-})$, and (c) $1_2^{-}(K^{\pi }=1^{-})$. For each state, The intrinsic density of the positive-parity component is subtracted from that of the negative-parity component, and the difference of the integrated densities is projected onto the $Z-X$, $Y-Z$, and $Y-X$ planes, as shown in the left, middle, and right of the figure, respectively. The intrinsic axes are chosen to for consistency with Fig. 2. The contour interval is 0.003 ${\mathrm{fm}}^{-2}$, and the red (blue) color map indicate positive (negative) values.

Figure 4

The matter densities of ${}^{24}\mathrm{Mg}$. The Fermi density ${\rho }_{\text{Fermi}}^{}\left(r\right)={\rho }_0{[1+exp\left(\frac{r-c}{t/4.4}\right)]}^{-1}$ with $c=2.876$ fm and $t=2.333$ fm is also shown in (a).

Figure 5

Square of the charge-form factors of the elastic and inelastic processes for the positive-parity states of ${}^{24}\mathrm{Mg}$. For the calculated result, the square of the renormalized form factors $F\left(q\right)$ multiplied by the $f^{\text{tr}}$ values in Table 3 are plotted. The experimental data were measured by electron scattering [4, 6, 7, 8, 9]. In Refs. [4, 6, 8] for the $2_2^{+}$(4.24 MeV) state, the $4_1^{-}$(4.12 MeV) contributions were not separated.

Figure 6

Same as Fig. 5 but for the negative-parity states. The experimental data are taken from Refs. [7, 10]. In the data from Ref. [7] for the $3_1^{-}$(7.62 MeV) and $3_2^{-}$(8.36 MeV) states, the $1_1^{-}$(7.56 MeV) and $1_2^{-}$(8.44 MeV) contributions were not separated.

Figure 7

The isoscalar components (average of the proton and neutron components) of the transition densities of ${}^{24}\mathrm{Mg}$. The renormalized transition densities multiplied by the $f^{\text{tr}}$ values in Table 3 are plotted. For transitions to the $2^{+}$, $4^{+}$, and $3^{-}$ states, the collective-model-transition density of the Fermi-type Tassie form ${\rho }_{\text{Tassie}}^{\text{tr}}\left(r\right)\propto r^{\lambda -1}\partial {\rho }_{\text{Fermi}}^{}\left(r\right)/\partial r$ is also shown for comparison. ${\rho }_{\text{Tassie}}^{\text{tr}}\left(r\right)$ is normalized to fit the $E\lambda$-transition strengths from the $0_1^{+}$ state to the $2_1^{+}$, $4_2^{+}$, and $3_1^{-}$ states.

Figure 8

Cross sections of proton scattering off ${}^{24}\mathrm{Mg}$ at incident energies of $E_p=40$, 49, 65, and 100 MeV, as calculated with $\mathrm{MCC}+\mathrm{AMD}$ (solid lines with label “CC”) and DWBA (dotted lines with label “DWBA”). For the $4_1^{+}$ state, the DWBA cross sections are two orders smaller than the CC cross sections. Experimental data are cross sections at $E_p=40$ MeV [15, 16], 49 MeV [13], 65 MeV [17, 70], and 100 MeV [18, 70].

Figure 9

Cross sections of $\alpha$ scattering off ${}^{24}\mathrm{Mg}$ at incident energies of $E_{\alpha }=104$, 120 MeV, 130 MeV, and 386 MeV, as calculated with $\mathrm{MCC}+\mathrm{AMD}$ (solid lines labeled “CC”) and DWBA (dotted lines labeled “DWBA”). For the $4_1^{+}$ state, the DWBA cross sections are two-order smaller than the CC cross sections. The experimental data are cross sections at $E_{\alpha }=104$ MeV [27, 70], 120 MeV[28], 130 MeV [29], and 386 MeV [29].