Investigation of ${}^{11}\mathrm{B}$ and ${}^{40}\mathrm{Ca}$ levels at 8–9 MeV by nuclear resonance fluorescence

We report on the measurement of ${}^{11}\mathrm{B}$ and ${}^{40}\mathrm{Ca}$ levels between excitation energies of 8 and 9 MeV using nuclear resonance fluorescence (NRF). The experiment was carried out with nearly monoenergetic and linearly polarized photon beams provided by the High-Intensity $\gamma$-ray Source ($\mathrm{HI}\gamma \mathrm{S}$) facility at the Triangle Universities Nuclear Laboratory (TUNL). States in ${}^{11}\mathrm{B}$ are important for calibrations of NRF measurements, while the properties of ${}^{40}\mathrm{Ca}$ levels impact potassium nucleosynthesis in globular clusters. For ${}^{40}\mathrm{Ca}$, we report on improved excitation energies and an unambiguous $2^{-}$ assignment for the state at 8425 keV. For ${}^{11}\mathrm{B}$, we obtained improved values for $\gamma$-ray multipolarity mixing ratios and branching ratios of the 8920-keV level.

Figure 1

Setup used for the present experiment. The incident $\gamma$-ray beam moves inside a plexiglass vacuum tube and impinges on a sample (boron powder or CaO). The direction of the linear polarization of the beam points parallel to the horizontal plane. The dewars of the two 60% (D1 and D2) and the 100% (D3) HPGe detectors are colored green and blue, respectively. The front face of each detector is covered by a passive shield (yellow) to reduce backgrounds. For reference, the pairs of polar and azimuthal angles, ($\theta$, $\varphi$), are depicted for each detector. The gray parts indicate the mechanical support structure of the setup.

Figure 2

Excitation energies of ${}^{40}\mathrm{Ca}$ levels observed in the present work (red circles) as compared to the previously evaluated results [17] (blue squares). For better comparison, the ordinate refers to the difference relative to the previous mean value. The state labels on the abscissa correspond to those listed in column 1 of Table 1.

Figure 3

Pulse height spectra measured at an incident $\gamma$-ray energy of 8400 keV. The spectra of the vertical (D1), horizontal (D2), and out-of-plane (D3) HPGe detectors are depicted in red, blue, and green, respectively. See also Fig. 1. The peaks are labeled according to the initial and final state energies (given in units of keV) of the transition. The spectrum in green was scaled down by a factor of $\approx 2$ to account for the ratio of detection efficiencies; detectors D1 and D2 had similar efficiencies. Left panel: Energy region of the secondary decay from the $E_x=3737$ keV level in ${}^{40}\mathrm{Ca}$ to the ground state, 3737 keV $\to 0$ keV. Right panel: Energy region of the primary transition from the $E_x=8425$ keV level to the $E_x=3737$ keV state, 8425 keV $\to 3737$ keV.

Figure 4

Ratio vs analyzing power for the deexcitation of the 8425-keV level in ${}^{40}\mathrm{Ca}$. Top: Primary transition, 8425 keV $\to 3737$ keV. Bottom: Secondary transition, 3737 keV $\to 0$ keV, with the primary transition unobserved. The measured values are shown in black, where both the error bars and the dashed black line (error ellipse) correspond to 68% coverage probability. Different colors refer to theoretical values based on given choices of $J^{\pi }$ for this level. The purple and green lines correspond to the theoretical results for a range of multipolarity mixing ratios of the measured transitions. Colored circles, squares, and triangles indicate theoretical values for mixing ratios of $\delta =-10$, 0, and $+10$, respectively. For $J^{\pi }=1^{-}$ and $1^{+}$, only the values corresponding to $\delta =0$ are indicated (see text). Notice that, in the top panel, the purple and green squares (signifying $\delta =0$) are covered by the yellow and red ones, respectively.

Figure 5

Pulse height spectra measured at an incident $\gamma$-ray energy of 8920 keV. The spectra of the vertical (D1), horizontal (D2), and out-of-plane (D3) HPGe detectors are depicted in red, blue, and green, respectively. See also Fig. 1. The peaks are labeled according to the initial and final state energies (given in units of keV) of the transition. The spectrum shown in green was scaled down by a factor of $\approx 2$ to account for the ratio of detection efficiencies; detectors D1 and D2 had similar efficiencies. Left panel: Energy region of the primary transition to the $E_x=4445$ keV level in ${}^{11}\mathrm{B}$, 8920 keV $\to 4445$ keV, and the subsequent secondary transition, 4445 keV $\to 0$ keV. Right panel: Energy region of the primary transition to the ${}^{11}\mathrm{B}$ ground state, 8920 keV $\to 0$ keV.

Figure 6

Ratio vs analyzing power for the deexcitation of the 8920-keV ($5/2^{-}$) level to the ${}^{11}\mathrm{B}$ ground state ($3/2^{-}$). Top: The measured value is depicted in black. The purple line depicts the theoretical solutions for the full range of mixing ratios. Purple circles, squares, and triangles indicate theoretical values of $\delta =-10$, 0, and $+10$, respectively. The regions between pairs of green asterisks correspond to the four possible solutions from our reanalysis of the data by Ref. [5] (see Appendix pp2). Bottom: Expanded view of the top panel. Both the error bars and the dashed black line (error ellipse) refer to 68% coverage probability.

Figure 7

Ratio vs analyzing power for the deexcitation of the 8920-keV ($5/2^{-}$) level in ${}^{11}\mathrm{B}$. Measured values are depicted in black, where the dashed black lines indicate 68% error ellipses. The purple lines depict the theoretical solutions for a range of mixing ratios. Purple circles, squares, and triangles indicate theoretical values of ${\delta }_2=-10$, 0, and $+10$, respectively. Top: Primary 8920 keV $\to 4445$ keV transition. The purple line was obtained with the present value of the mixing ratio for the transition exciting the 8920-keV level (see Table 2 and Sec. 4a) and a range of mixing ratios, ${\delta }_2$, for the primary decay to the 4445-keV state. The region between the green asterisks or red crosses corresponds to the two possible solutions from our reanalysis of the data by Ref. [5] (for ${\delta }_1=-0.001$; see Appendix pp2 and Table 3). Bottom: Secondary 4445 keV $\to 0$ keV transition, with the primary transition unobserved. The purple line was obtained assuming the present value of the mixing ratio for the transition exciting the 8920-keV level, a mixing ratio of $|{\delta }_u|=0.72$ for the unobserved intermediate transition (see Table 2), and a range of mixing ratios, ${\delta }_2$, for the secondary decay of the 4445-keV state.

Figure 8

Determination of the multipolarity mixing ratio, $\delta$, for the sequence 0 keV ($3/2^{-}$) $\to$ 4445 keV ($5/2^{-}$) $\to 0$ keV ($3/2^{-}$) in ${}^{11}\mathrm{B}$ from the experimental asymmetry, $a=-A(\theta ={90}^{\circ })$, using the “min-max method.” The solid blue curve indicates the theoretical dependence of the asymmetry on the mixing ratio. The dashed red curve is instead obtained, if the erroneous mixing-ratio phase convention of Ref. [5] is adopted. Both curves take into account the correction for the finite opening angle of the detectors (factor 0.95 [5]). The asymmetry measured by Ref. [5] is shown as a solid gray horizontal region. The possible solutions for the multipolarity mixing ratio, $\delta$, are depicted by vertical bars (solid blue and dashed red for the correct and erroneous phase convention, respectively).

Figure 9

Energy residuals (i.e., the difference with respect to the best fit line) versus peak centroid (in channels) for detector 2 (D2; see Fig. 1). The four data points refer to the four ${}^{11}\mathrm{B}$ calibration energies (see text). The red lines show credible lines for different steps of the Markov chains. The two blue lines enclose a coverage probability of 68%. The region of interest for the ${}^{40}\mathrm{Ca}$ levels measured in the present work is located between channels 13 000 to 14 000. Only 300 out of 50 000 regression lines are plotted for the purpose of illustration.