Coulomb excitation of ${}^{44}\mathrm{Ca}$ and ${}^{46}\mathrm{Ar}$

The reduced transition probabilities $B(E2;0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+})$ of the ${}^{46}\mathrm{Ar}$ and ${}^{44}\mathrm{Ca}$ nuclei were studied using the Coulomb excitation technique at intermediate energy at the LISE/GANIL facility. The in-flight $\gamma$ rays, emitted after the Coulomb excitation of their first $2^{+}$ states, were detected in an array of $64{\mathrm{BaF}}_2$ crystals. The present $\text{B}\left(\mathrm{E}2\uparrow \right)$ value for ${}^{44}\mathrm{Ca}$, 475(36) ${\mathrm{e}}^2{\mathrm{fm}}^4$, agrees well with the value of 495(35) ${\mathrm{e}}^2{\mathrm{fm}}^4$ obtained by averaging results of previous experiments. Consistent $B(E2;0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+})$ values of 225(29) ${\mathrm{e}}^2{\mathrm{fm}}^4$ and 234(19) ${\mathrm{e}}^2{\mathrm{fm}}^4$ have been obtained for ${}^{46}\mathrm{Ar}$ from an absolute and a relative measurement, normalized to the ${}^{44}\mathrm{Ca}$ value. Both results agree with the ones obtained with the same experimental technique at the NSCL facility but are a factor of 2 smaller than the shell model predictions. The drop in $B(E2;0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+})$ in the Ar chain at $N=28$, confirmed in this experiment, shows that ${}^{46}\mathrm{Ar}$ is sensitive to the $N=28$ shell closure.

Figure 1

Schematic picture of the experimental setup.

Figure 2

Simulated efficiency of the DSSSD detector for ${}^{44}\mathrm{Ca}$ ions. Angles are given in the laboratory frame.

Figure 3

Left: The Doppler corrected and background subtracted $\gamma$-ray spectra are displayed as a function of the ${}^{44}\mathrm{Ca}$ scattering angles, in ranges of $1^{\circ }$. Right: Same for scattering angles between ${\theta }_{\text{lab}}=1.0^{\circ }$ and ${\theta }_{\text{lab}}^{\text{max}}=4.0^{\circ }$.

Figure 4

Comparison between the calculated angular distribution in laboratory frame (red line) to experimental angular distribution (black crosses), obtained with a $B(E2;0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+})$ value of $475{\mathrm{e}}^2{\mathrm{fm}}^4$ and neglecting nuclear excitations. To describe the experimental angular distribution over the whole angular range, a value of ${\beta }_N=0.35$(4) was implemented for the nuclear contribution (blue line), keeping the Coulomb contribution constant.

Figure 5

Left: Doppler-corrected background subtracted $\gamma$-ray spectra displayed as a function of the scattering angle of ${}^{46}\mathrm{Ar}$, cast in fixed angular ranges of $1^{\circ }$. Right: Same between scattering angles of ${\theta }_{\text{lab}}=1^{\circ }$ and ${\theta }_{\text{lab}}=3.{25}^{\circ }$.

Figure 6

Comparison between the calculated angular distribution in laboratory frame (red line) to the experimental angular distribution (black crosses) obtained by using a $B(E2;0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+})$ value of $225{\mathrm{e}}^2{\mathrm{fm}}^4$ and by neglecting nuclear excitations. To describe the experimental angular distribution over the whole angular range, a value of ${\beta }_N=0.25$(6) was implemented for the nuclear contribution (blue line), keeping the Coulomb contribution unchanged.

Figure 7

Measured and calculated $B(E2;0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+})$ values for the Ar isotopes using SDPFK, SDPFMU, and SDPFU interactions with standard polarization charges ${\delta }_n={\delta }_p=0.5e$. The $B(E2;0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+})$ values are taken from Ref. [36] for ${}^{36}\mathrm{Ar}$ and from Ref. [53] for ${}^{48}\mathrm{Ar}$. The experimental value presented for ${}^{46}\mathrm{Ar}$ corresponds to the weighted average value from Refs. [2, 20] and from the present work.

Figure 8

Square of the phenomenological proton (continuous red line) and neutron (dotted blue line) transition matrix elements. The $M_p^2$ values for for ${}^{36-44}\mathrm{Ar}$ [36], for ${}^{46}\mathrm{Ar}$ extracted from the average $B(E2;0_{\mathrm{g}.\mathrm{s}.}^{+}\to 2_1^{+})$ value obtained from the Coulomb excitation experiments, and for ${}^{48}\mathrm{Ar}$ [53].

Figure 9

Comparison of the square of the quadrupole matrix elements (given in ${\mathrm{e}}^2{\mathrm{fm}}^4)$ deduced for protons and neutrons using the standard ${\delta }_n={\delta }_p=0.5$ polarization charges (solid line) and the ${\delta }_n=0.8,{\delta }_p=0.2$ polarization charges proposed by Riley et al. [51] (dotted line) with those calculated within the shell model using the SDPFU interaction (dashed line).