Spin dependence in the $p$-wave resonance of $\bm{{{{}^{139}\overrightarrow{\rm{La}}}}+\overrightarrow{n}}$

The experiment was performed with a pulsed epi-thermal neutron beam at the RADEN beamline of the Material and Life Science Experimental Facility (MLF) at the Japan Proton Accelerator Research Complex (J-PARC) Shinohara et al. (2020). The experimental setup is depicted in Fig. 1. The La target is placed 23.0 m from the moderator surface. A 2.0 cm cubic lanthanum metal cooled with a dilution refrigerator was used as the target. A 6.8 Tesla transverse magnetic field was applied using a superconducting magnet to polarize the target nuclei. The neutron beam, collimated to a 3 cm by 3 cm size, was stripped of thermal neutrons using a cadmium filter upstream of the beamline to reduce the heat load on the La target induced by the neutron beam. The beam was polarized with a neutron polarizer using polarized ³He gas (³He spin filter), located 4.3 m upstream of the polarized target. The ³He spin filter was polarized using the spin exchange optical pumping method (SEOP) with a 110 W laser system constructed outside of the beamline and then installed on the beamline with a coil and a double magnetic shield to maintain the ³He polarization Okudaira et al. (2020). The ³He cell was 45 mm in diameter by 70 mm in length and the pressure was 0.31 MPa. The neutron beam, longitudinally polarized by the ³He spin filter, was guided using a guide magnet. The spin direction was adiabatically rotated to the transverse direction utilizing the stray magnetic field of the superconducting magnet. The neutron spin was flipped every 30 minutes by flipping the spin of ³He gas using adiabatic fast passage (AFP) NMR. The loss of the ³He polarization was $4\times 10^{-5}$ per flip, which was negligibly small. Transmitted neutrons were recorded in list mode using a 256-pixel lithium glass scintillator detector located at 24.71 m from the moderator surface Satoh (2015). Downstream of the La target, another collimator was installed to reduce the beam divergence. The neutron energy $E_{n}$ was determined using the neutron time-of-flight (TOF) and the flight path length. The proton beam power was 750 kW during the experiment.

Figure 2 illustrates the configuration around the La target. The La target was held in place between upper and lower copper holders, fastened using copper screws. The upper holder was connected to the cold head of the dilution refrigerator, enabling cooling of the La target through thermal conduction. The temperature $T$ was monitored with a ruthenium oxide thermometer installed in the cold head. We performed the experiment in two conditions: (a) low temperature condition ($T=67~{}\rm{mK}$ ) and (b) high temperature condition ($T=1~{}\rm{K}$). The measurement times were 22 hours and 6 hours, respectively. In the condition (a), the temperature increase by the beam irradiation to the La target was approximately 1 mK, indicating that the temperature difference between the cold head and the La target can be considered negligible. The temperature fluctuation was also around 1 mK, which was caused by beam interruptions in the accelerator due to malfunctions.

The cross sections for parallel and antiparallel polarized neutron and nucleus can be written with the spin-independent cross section $\sigma_{0}$ and spin-dependent cross section $\sigma_{\rm S}$ as

where ${+}$ and ${-}$ denote the parallel and antiparallel spins, respectively. The asymmetry of neutron counts for parallel and anti-parallel spins transmitted through the polarized lanthanum target, defined as

where $N_{-}$ and $N_{+}$ are the neutron counts for parallel and anti-parallel spins, was measured. The neutron counts $N_{\pm}$ are expressed using the neutron polarization $P_{n}$ and nuclear vector polarization $P_{I}$ as

where $N$, $\epsilon$, $\rho$, and $d$ are number of incident neutrons, detection efficiency of a neutron detector, number density of the nuclear target, and the thickness of the nuclear target, respectively. The spin-dependent asymmetry $\varepsilon_{\rm S}$ can be described using Eq.3 as

In this paper, the measurement and analysis were performed using resonance parameters of La+n reactions listed in Table 1, which were recently measured by Endo et al. Endo et al. (2023) using both neutron transmission and ($n$, $\gamma$) reaction with an intense pulsed neutron beam at J-PARC.

Figure 3 shows the TOF spectra of the transmitted neutrons and asymmetry $\varepsilon_{\rm S}$ in conditions (a) and (b). We observed a significant asymmetry in condition (a), corresponding to a high nuclear polarization, while the asymmetry disappeared in condition (b) due to the lower nuclear polarization. The peak and dip structures were observed at the 2.99 eV and 0.75 eV resonances. The global structure observed less than 0.3 ms is attributed to the spin-dependent cross section of the negative $s$-wave resonance.

The neutron polarization was obtained using the ³He polarization of the ³He spin filter. The ³He polarization was determined with the ratio of the transmitted neutrons for polarized and unpolarized ³He spin filter. The ratio of the transmitted neutrons is described as

where, $P_{\rm{He}}$, $\sigma_{\rm{He}}$, and $\rho_{\rm{He}}d_{\rm{He}}$ are the ³He polarization, neutron absorption cross section of ³He, and areal density of ³He gas, respectively. Here, $N_{\rm{pol}}$ is defined as $N_{+}+N_{-}$ for cancelling the spin-dependent asymmetry derived from the polarization of the La target. The areal density $\rho_{\rm{He}}d_{\rm{He}}$ was obtained from the measurement of the ratio of transmitted neutrons for unpolarized ³He spin filter and empty glass cell as 21.4 atm$\cdot$cm. The ³He polarization was obtained for each flip by fitting the TOF dependence of $N_{\rm{pol}}/N_{\rm{unpol}}$ using Eq. 5 with a fit parameter of $P_{\rm{He}}$ as shown in Fig 4. Figure 5 shows the time dependence of the ³He polarization. The relaxation time of the ³He polarization $\tau$, which was obtained by fitting with $P_{\rm{He}}(t)=P_{\rm{He}}(0)\exp(-t/\tau)$, was 161 h. The averaged ³He polarization $\bar{P}_{\rm{He}}$ during the measurement was $(68\pm 1)$%.
The neutron polarization $P_{n}$ transmitted through the ³He spin filter is determined as

Figure 6 shows an averaged neutron polarization $\bar{P}_{n}$ as a function of the neutron energy calculated from the averaged ³He polarization. The averaged neutron polarization at 0.75 eV was $(36.1\pm 0.5)$%.

The ¹³⁹La nuclear polarization was determined utilizing the spin-dependent asymmetry at the 2.99 eV $s$-wave resonance of ¹³⁸La. The spin-dependent asymmetry at the 2.99 eV resonance, after subtracting of the negative $s$-wave component, was obtained as

The spin-dependent cross section of the 2.99 eV resonance $\sigma^{\rm theo}_{{\rm S},s}$ can be theoretically described using the resonance parameters listed in Table 1 as

where $E_{s}$, $\Gamma^{n}_{s}$, and $\Gamma_{s}$ are the resonance energy, neutron width, and total width of the 2.99 eV s-wave resonance, respectively. The nuclear polarization of ¹³⁸La can be determined using Eqs. 4, 7 and 8, taking into account its natural abundance and the neutron polarization at 2.99 eV, yielding a value of 4.9$\pm$0.7%. The target temperature $T_{\rm La}$ was calculated based on a Boltzmann distribution and using the magnetic moment and nuclear spin listed in Table 2, resulting in $T_{\rm{La}}=75.7^{+10.2}_{-8.9}~{}\rm{mK}$, which is consistent with the temperature measured at the cold head of 67 mK. Under the assumption that the spin temperature of ¹³⁹La is the same as that of ¹³⁸La, the corresponding ¹³⁹La nuclear polarization was determined to be $3.9\pm 0.5\%$.

The experimental value of the spin-dependent cross section $\sigma^{\rm exp}_{\rm S}$ was obtained from the asymmetry $\varepsilon_{\rm S}$ using Eq. 4. The resonance component $\sigma^{\rm{exp}}_{{\rm{S}},r}$ was isolated by fitting the global structure attributed to the negative $s$-wave component with a third order polynomial function. The resonance regions listed in the Table 1 are excluded from the fitting. Figure 7 shows the TOF dependence of $P_{I}\sigma^{\rm exp}_{\rm S}$ and $P_{I}\sigma^{\rm exp}_{{\rm{S}},r}$. Note that Fig. 7 was calculated using the areal density of ¹³⁹La. A $p$-value, defined as $p=(1-\rm{C.L.})/2$, where C.L. is the confidence level of the non-zero asymmetry, is also depicted to show the significance of $P_{I}\sigma^{\rm exp}_{{\rm{S}},r}$ in Fig. 7. The p-value indicates the probability to observe a non-zero value of $P_{I}\sigma^{\rm exp}_{{\rm{S}},r}$ in the hypothesis of no asymmetry. A confidence level of over 99.7% corresponds to a p-value less than $1.35\times 10^{-3}$. The spin-dependence cross section was first observed at the $p$-wave resonance with over 99.7% C.L. as shown in Fig 7.

The spin-dependent cross section in the $p$-wave resonance region of $E_{p}-3\Gamma_{p}<E_{n}<E_{p}+3\Gamma_{p}$ after the subtraction of the negative $s$-wave component, defined as $\sigma^{\rm exp}_{{{\rm{S}},p}}$, is obtained using the nuclear polarization in Section II.4 as

where $E_{p}$ and $\Gamma_{p}$ are the resonance energy and total width of the $p$-wave resonance, shown in Table 1. Here, the total width is defined as $\Gamma_{p}=\Gamma_{p}^{\gamma}+\Gamma_{p}^{n}$. The asymmetry of the spin-dependent cross section relative to the spin-independent cross section of the $p$-wave component was also obtained as

The spin-independent cross section $\sigma^{\rm theo}_{{\rm 0},p}$ was theoretically calculated with a Breit-Wigner formula, defined as,

When using the nuclear polarization calculated from the temperature measured at the cold head, the differences of $\sigma^{\rm exp}_{{\rm{S}},p}$ and $A_{\rm S}$ from the values in Eq. 9 and Eq. 10 were $+0.03~{}\rm{barn}$ and $+0.04$, respectively. These differences were smaller than the statistical error.

The partial neutron width can also be extracted from the angular correlations of $\gamma$-rays emitted from $p$-wave resonances, which arise from interference between $s$- and $p$-wave amplitudes Okudaira et al. (2018, 2021); Yamamoto et al. (2020); Koga et al. (2022); Endo et al. (2022). The advantage of using the spin-dependent cross section is that the neutron partial width can be directly determined without assuming the interference between partial amplitudes and the final state spin after the $\gamma$ decay.

The spin-dependent cross section at the $p$-wave resonance can be calculated using the explicit theoretical expression of $B^{\prime}$ as Gudkov and Shimizu (2020)

where $x_{s}$ and $y_{s}$ are ratios of the neutron partial width of the channel spin, defined as

The neutron partial widths of the neutron total angular momentum $j=1/2$ and $3/2$ components, denoted as $\Gamma^{n}_{p,j=1/2}$ and $\Gamma^{n}_{p,j=3/2}$, are expressed by $x$ and $y$ defined as

where $x$ and $y$ satisfy $x^{2}+y^{2}=1$. The corresponding mixing angle $\phi$ can be defined as

as discussed in Ref Gudkov and Shimizu (2020). The broadening effect by the pulse shape of the neutron beam at 0.75 eV was negligibly small compared with the total width of the $p$-wave resonance and the statistical error, and therefore, the spin-dependent cross section obtained in Eq. 9 can be directly compared with the theoretical calculation. By calculating the Breit-Wigner function over the region $E_{p}-3\Gamma_{p}<E_{n}<E_{p}+3\Gamma_{p}$ in Eq. 13, we obtained the following equation.

Using Eq. 17 and Eq. 18, we find the solutions for $\phi$ as

The corresponding $x$ and $y$ values are also obtained as

The visualization of $\phi$ is shown in Fig. 8. Equation 18 is described as the curved line in the $xy$ plane. The intersections of the curved lines and unit circle show the solutions of $\phi$.

The above analysis was also performed using the resonance parameters reported by other groups in Appendix B. The differences in the analysis results that arose from differences in the resonance parameters were within the statistical error. We confirmed that these differences stemming from the resonance parameters do not affect the conclusions of this paper.

For the $s$-wave resonances, the spin $J$ can be directly determined from the asymmetry. The positive (negative) sign of the asymmetry indicates that neutrons with parallel (anti-parallel) spin are likely to be absorbed by nuclei. The sign of the asymmetry in Fig. 7 (a) and (b) implies: $J=4$ for the negative $s$-wave resonance of ¹³⁹La, whose spin is $7/2$; $J=11/2$ for the 2.99 eV $s$-wave resonance of ¹³⁸La, whose spin is 5; and $J=3$ for the 72.3 eV $s$-wave resonance of ¹³⁹La. The spins of the $s$-wave resonances determined in this experiment are consistent with the reference values in Table 1.