Precision neutron interferometric measurement of the $n\text{−}{}^3\mathrm{He}$ coherent neutron scattering length

A measurement of the $n\text{−}{}^3\mathrm{He}$ coherent scattering length using neutron interferometry is reported. The result, $b_c=(5.8572\pm 0.0072)\mathrm{fm}$, improves the measured precision of any single measurement of $b_c$ by a factor of eight; the previous world average, $b_c=(5.74\pm 0.04)\mathrm{fm}$, now becomes $b_c=(5.853\pm 0.007)\mathrm{fm}$. Measurements of the $n\text{−}p$, $n\text{−}d$, and $n\text{−}{}^3\mathrm{He}$ coherent scattering lengths have now been performed using the same technique, thus allowing one to extract the scattering length ratios: parameters that minimize systematic errors. We obtain values of $b_{n{}^3\mathrm{He}}∕b_{np}=(-1.5668\pm 0.0021)$ and $b_{nd}∕b_{np}=(-1.7828\pm 0.0014)$. Using the new world average value of $b_c$ and recent high-precision spin-dependent scattering length data also determined by neutron optical techniques, we extract new values for the bound singlet and triple scattering lengths of $b_0=(9.949\pm 0.027)\mathrm{fm}$ and $b_1=(4.488\pm 0.017)\mathrm{fm}$ for the $n\text{−}{}^3\mathrm{He}$ system. The free nuclear singlet and triplet scattering lengths are $a_0=(7.456\pm 0.020)\mathrm{fm}$ and $a_1=(3.363\pm 0.013)\mathrm{fm}$. The coherent scattering cross section is ${\sigma }_c=(4.305\pm 0.007)\mathrm{b}$ and the total scattering cross section is ${\sigma }_s=(5.837\pm 0.014)\mathrm{b}$. Comparisons of $a_0$ and $a_1$ to the only existing high-precision theoretical predictions for the $n\text{−}{}^3\mathrm{He}$ system, calculated using a resonating group technique with nucleon-nucleon potentials incorporating three-nucleon forces, have been performed. Neutron scattering length measurements in few-body systems are now sensitive enough to probe small effects not yet adequately treated in present theoretical models.

Figure 1

A schematic view of the $\mathrm{Si}$ perfect crystal neutron interferometer. Parameters associated with the neutron optics are discussed in the text.

Figure 2

A typical pair of interferograms with ${}^3\mathrm{He}$ present in the cell. The oscillations arise from the change in path lengths created as the phase shifter is rotated (see Fig. 1). Data are shown for both the cell in and out of the interferometer.

Figure 3

Measurements of the coherent scattering length $b_c$ plotted on a run-by-run basis. Each data point represents approximately $42\min$ of data. The solid line is the weighted average of the data, $b_c=(5.8572\pm 0.0072)\mathrm{fm}$. The estimated uncertainty for one of the data points is shown.

Figure 4

Measured values of the ${}^3\mathrm{He}$ neutron coherent scattering length including the present measurement [26, 27, 28, 29, 30]. The dashed line denotes the weighted average of all measurements, $b_c=(5.853\pm 0.007)\mathrm{fm}$.

Figure 5

Comparison of the experimental and theoretical determinations of the free-atom singlet scattering length $a_0$. The theoretical values were calculated using the $\mathrm{AV}18+\mathrm{UIX}$ and $\mathrm{AV}18+\mathrm{UIX}+V_3^{*}$ potential models [32]. Note that the experimental uncertainty on the ${}^3\mathrm{He}$ binding energy is negligibly small $E_{bind}=-7.718109\pm 0.000010\mathrm{MeV}$ [36].

Figure 6

Comparison of the experimental and theoretical determinations of the free-atom triplet scattering length $a_1$. The theoretical values were calculated using the $\mathrm{AV}18+\mathrm{UIX}$ and $\mathrm{AV}18+\mathrm{UIX}+V_3^{*}$ potential models [32].