Evolution of the isoscalar giant monopole resonance in the Ca isotope chain

Background: Two recent studies of the evolution of the isoscalar giant monopole resonance (ISGMR) within the calcium isotope chain report conflicting results. One study suggests that the monopole resonance energy, and thus the incompressibility of the nucleus, $K_A$, increase with mass, which implies that $K_{\tau }$, the asymmetry term in the nuclear incompressibility, has a positive value. The other study reports a weak decreasing trend of the energy moments, resulting in a generally accepted negative value for $K_{\tau }$. Differences in the observed trends have been attributed to the use of different techniques to account for instrumental background and the physical continuum.

Purpose: An independent measurement of the ISGMR in the Ca isotope chain is provided to gain a better understanding of the origin of possible systematic trends.

Methods: Inelastically scattering of 196 MeV $\alpha$ particles were observed at small scattering angles, including $0^{\circ }$, for a range of calcium targets (${}^{40,42,44,48}\mathrm{Ca}$). The scattered $\alpha$ particles were momentum analyzed in the K600 magnetic spectrometer at iThemba LABS, South Africa. Monopole strengths spanning an excitation-energy range between 9.5 and 25.5 MeV were obtained using the difference-of-spectra (DoS) technique, adjusted to allow corrections for the variation of the angular shape of the sum of the $L>0$ multipoles as a function of excitation energy, and compared with previous results that employed the multipole-decomposition analysis (MDA) techniques.

Results: The structure of the $E0$ strength distributions of ${}^{40,42,44}\mathrm{Ca}$ agrees well with the results from the previous measurement that support a weak decreasing trend of the energy moments, while no two datasets agree in the case of ${}^{48}\mathrm{Ca}$. Despite the variation in the structural character of $E0$ strength distribution from different studies, we find for all datasets that the moment ratios, calculated from the ISGMR strength in the excitation-energy range that covers the resonance peak, display at best only a weak systematic sensitivity to a mass increase.

Conclusion: Different trends observed in the nuclear incompressibility are caused by contributions to the $E0$ strength outside of the region covering the resonance peak, and in particular for high excitation energies. While procedures exist to identify and subtract instrumental background, more work is required to characterize and subtract continuum background contributions at high excitation energies.

Figure 1

(a) The ($x_{fp},y_{fp}$) distribution of PID selected events observed in the focal plane for the ${}^{48}\mathrm{Ca}(\alpha ,{\alpha }^{\prime })$ reaction at $E_{\alpha }=196$ MeV. A display threshold of $>2$ events is imposed. In panel (b) the (solid) black line represents the focal-plane position spectrum for the vertical region from −12 mm to 20 mm. The (dashed) red line represents the smoothed background obtained when selecting $y_{fp}$ from −30 mm to −12 mm.

Figure 2

(a) Background-subtracted excitation-energy spectra for inelastic $\alpha$-particle scattering at ${\theta }_{\mathrm{lab}}=0^{\circ }$ off Mylar (short-dashed green line) and ${}^{12}\mathrm{C}$ (long-dashed blue line). The solid red line shows the difference between the Mylar and normalized ${}^{12}\mathrm{C}$ spectra, and represents the excitation-energy spectrum for the ${}^{16}\mathrm{O}(\alpha ,{\alpha }^{\prime })$ reaction. (b) Background-subtracted excitation-energy spectrum for the oxygen-contaminated ${}^{48}\mathrm{Ca}$ target (dashed blue line), the normalized ${}^{16}\mathrm{O}$ spectrum (solid red line), and oxygen-free ${}^{48}\mathrm{Ca}$ spectrum (solid black line).

Figure 3

Measured double-differential cross sections for inelastic $\alpha$ scattering from ${}^{40,42,44,48}\mathrm{Ca}$ at $E_{\alpha }=196\mathrm{MeV}$. The black (top) line in panels (a)–(d) represents data for the angle range $0^{\circ }\le {\theta }_{\mathrm{lab}}\le 2^{\circ }$, while the blue (bottom) corresponds to an angular range $2.{76}^{\circ }\le {\theta }_{\mathrm{lab}}\le 4.{13}^{\circ }$.

Figure 4

DWBA calculated differential cross sections for inelastic $\alpha$ scattering on ${}^{48}\mathrm{Ca}$ at a beam energy of $E_{\alpha }=196$ MeV for various isoscalar electric multipoles. The calculations were normalized to 100% of the EWSR for each multipole at an excitation energy of 18 MeV in ${}^{48}\mathrm{Ca}$, representing the peak of the cross section distribution.

Figure 5

Outline of the conversion process from the ${}^{48}\mathrm{Ca}$ difference cross sections (a), to the fractional EWSR strength per MeV (b), and then to the ${}^{48}\mathrm{Ca}E0$ strength distribution (c). The red dashed line in panel (a) represents the ${\theta }_{\mathrm{lab}}\le 2^{\circ }$ angle-integrated $L=0$ DWBA cross section for 100% of the $E0$ EWSR. Statistical errors are indicated in blue for the $E0$ strength distribution.

Figure 6

ISGMR strength distributions for ${}^{40,42,44,48}\mathrm{Ca}$ from this study (gray filled histograms) utilizing the basic DoS technique, compared with results from RCNP [13] (blue circles) and TAMU [10, 11, 12] (red triangles). The error bars of the present work represent both statistical and systematic uncertainties.

Figure 7

Outline of the procedure to establish an excitation-energy-dependent correction factor for the finite-angle cross sections, taking ${}^{48}\mathrm{Ca}$ as an example. The fractional EWSR strengths for different multipolarities from RCNP [13, 21] (a) are used to determine DWBA cross sections at 196 MeV representative of the zero degree (b) and the finite-angle measurements (c). The correction factors (solid line) shown in panel (d) are determined by the ratio of the $L=1,2,3$ results in panel (b) to the $L=0,1,2,3$ results in panel (c). The dashed line in panel (d) represents the correction factors when using the fractional EWSR results from TAMU [11].

Figure 8

(a) Application of the DoS method with excitation-energy-dependent correction factors for the case of ${}^{48}\mathrm{Ca}$. The finite-angle cross section spectrum (blue line) is modified with the RCNP-based excitation-energy-dependent correction factor from panel (d) in Fig. 7, resulting in the solid green line. The weak IVGDR contribution (dashed red line) is subtracted from the zero-degree spectrum (solid black line), with the resulting DoS cross section shown in panel (b). The $E0$ strength distribution in panel (c) can be compared with the basic DoS result in panel (c) of Fig. 5. Statistical errors are indicated in blue for the $E0$ strength distribution.

Figure 9

Same as Fig. 6 but using the RCNP-based excitation-energy-dependent correction factor as described in the text.

Figure 10

Moment ratios of the $E0$ strength distributions in ${}^{40,42,44,48}\mathrm{Ca}$ for the excitation-energy range 10–25.5 MeV from the present work (black circles), compared to values extracted for the TAMU [10, 11, 12] (red triangles) and RCNP [13] (blue empty circles) datasets over the same excitation-energy range.

Figure 11

Experimental results for the incompressibility of the nucleus $K_A$ for ${}^{40,42,44,48}\mathrm{Ca}$ extracted from the TAMU [10, 11, 12] (red triangles) and RCNP [13] (blue circles) datasets over the excitation-energy range 10–31 MeV accessible at RCNP.