Search for the $1/2^{+}$ intruder state in ${}^{35}\mathrm{P}$

The excitation energy of deformed intruder states (specifically the 2p2h bandhead) as a function of proton number $Z$ along $N=20$ is of interest both in terms of better understanding the evolution of nuclear structure between spherical ${}^{40}\mathrm{Ca}$ and the Island of Inversion nuclei, and for benchmarking theoretical descriptions in this region. At the center of the $N=20$ Island of Inversion, the $n\mathrm{p}n\mathrm{h}$ (where $n=2,4,6$) neutron excitations across a diminished $N=20$ gap result in deformed and collective ground states, as observed in ${}^{32}\mathrm{Mg}$. In heavier isotones, $n\mathrm{p}n\mathrm{h}$ excitations do not dominate in the ground states but are present in the relatively low-lying level schemes. With the aim of identifying the expected $2\mathrm{p}2\mathrm{h}\otimes {\mathrm{s}}_{1/2^{+}}$ state in ${}^{35}\mathrm{P}$, the only $N=20$ isotone for which the neutron 2p2h excitation bandhead has not yet been identified, the ${}^{36}\mathrm{S}(d,{}^3\mathrm{He}){}^{35}\mathrm{P}$ reaction has been revisited in inverse kinematics with the HELical Orbit Spectrometer (HELIOS) at the Argonne Tandem Linac Accelerator System (ATLAS). While a candidate state has not been located, an upper limit for the transfer reaction cross section to populate such a configuration within a 2.5 to 3.6 MeV energy range provides a stringent constraint on the wave function compositions in both ${}^{36}\mathrm{S}$ and ${}^{35}\mathrm{P}$.

Figure 1

Experimental (solid black lines) and calculated (dashed orange lines and dotted blue lines) 2p2h bandheads for the $N=20$ isotones between $Z=12$ and $Z=16$. The orange dashed lines represent shell-model calculations performed with the SDPF-U-MIX effective interaction [4, 5, 6], while the blue dotted lines are the results of MCSM calculations [7, 8, 9]. The black solid lines represent data from Refs. [4, 10, 11, 12].

Figure 2

Schematic representation of the experimental setup installed in HELIOS. The incoming beam (${}^{36}\mathrm{S}$) hits a deuterated-plastic target placed in the center of the solenoid. The heavy reaction products are measured with the recoil detector or stopped in the (inactive) beam stop. The light particles (${}^3\mathrm{He}$) spin in the magnetic field until they hit the silicon array installed behind the recoil detector.

Figure 3

A representation of the two main analysis cuts used to filter events. The cyclotron period is proportional to the mass over charge ratio of the light particle and the area between the two horizontal lines is the location of ${}^3\mathrm{He}$ particles. The recoil energy loss is proportional to the heavy particles $Z$ and the two vertical lines select $Z=15$. The color scale represents the number of particles in the region below 6 MeV excitation energy. The red squares indicate the areas used to estimate the background component in the center gate. The background counts were weighted by a factor of $\frac{1}{3}$ to compensate for the larger coverage of the background gate.

Figure 4

Center-of-mass angle (in degrees) vs excitation energy for all events corresponding to detection of ${}^3\mathrm{He}$ and a $Z=15$ heavy fragment. The color scale shows the number of observed events; the white lines represent the location of the states calculated from simulations of the setup.

Figure 5

The excitation energy measured in the ${}^{36}\mathrm{S}(d,{}^3\mathrm{He}){}^{35}\mathrm{P}$ reaction. The blue area indicates the region of interest for a potential 2p2h bandhead (bounded by measurements in neighboring isotones). The vertical axis of the top panel is logarithmic. The bottom panel shows a linear scale, with the background defined in Fig. 3 subtracted.

Figure 6

Angular distributions for the three lowest energy states (black or gray error bars) are compared with DWBA calculations. Data and models are normalized to the ground-state distribution of Ref. [17] which is shown with orange crosses. The black dashed line is the DWBA calculation presented in Ref. [17]. The blue lines illustrate ptolemy DWBA results using the possible combinations of optical potentials [23, 24, 25, 26, 27, 28, 29]. The points marked with gray error bars were not used in the fit so that all states were fit over a similar angular range.

Figure 7

(top panel) number of counts in a peak in the region of interest that would be required to reject the null result (no additional peak present) with a given confidence. The background subtracted spectrum is shown to guide the eye. (bottom panel) Exclusion curves for the $C^{2}S$ ratio between the ground state and first-excited $1/2^{+}$ state.

Figure 8

Sensitivity analysis in the $2\times 2$ model. (left panel) Amplitude squared, ${\alpha }^2$, of the 2p2h component in the $0_2^{+}$ state of ${}^{36}\mathrm{S}$, derived from the ${}^{36}\mathrm{S}(d,p){}^{37}\mathrm{S}$ reaction (black line and gray bands) and the $C^{2}S$ ratio from Eq. (3) (dashed line). (right panel) Lower limits on the 2p2h excitation amplitude, $A^2$, in ${}^{35}\mathrm{P}$, derived from the experimental 90% confidence sensitivity as a function of the expected energy of the excited state. The amplitude ${\alpha }^2$ is shown at the energy of the $0_2^{+}$ in ${}^{36}\mathrm{S}$ (blue circle) together with that for ${}^{34}\mathrm{Si}$ [4] (green square).