Theory
For a gas mixture, the differential cross-section of Migdal effect from neutron–nucleus scattering in the soft limit is given by26,27,57
$$\frac{{{\rm{d}}}^{2}\sigma }{{\rm{d}}{E}_{{\rm{R}}}{\rm{d}}{E}_{{\rm{e}}}}=\sum _{i}\frac{{\rm{d}}{\sigma }_{{\rm{s}}}^{i}}{{\rm{d}}{E}_{{\rm{R}}}}\sum _{n\kappa }\frac{{\rm{d}}{p}_{\upsilon }^{i}(n\kappa \to {E}_{{\rm{e}}})}{{\rm{d}}{E}_{{\rm{e}}}}$$ (2)
where the summation over i accounts for each species in the gas mixture. \({\sigma }_{{\rm{s}}}^{i}\) denotes the scattering cross-section, which includes contributions from elastic (n, n), inelastic (n, n′), fission (n, 2n) and radiative capture (n, γ) processes and can be obtained from the ENDF database ENDF/B-VIII.0 (ref. 58). The term \({p}_{\upsilon }^{i}(n\kappa \to {E}_{{\rm{e}}})\) is the transition probability for ionization of an electron from an initial state nκ to a final state with energy E e . This probability can be obtained from first principles using the Dirac–Hartree–Fock method27,57. As we use a 2.5-MeV D–D neutron beam and require an approximately 50 keV NR threshold in our analysis, the chemical bonds of C–H and C–O (about 10 eV) are easily broken at our recoil energies. Accordingly, it is reasonable to approximately treat the C, H and O as the free atoms in our NR energy regime and calculate the likelihood of electrons from Migdal scattering as the sum of the individual Migdal transition probabilities for C, H and O, respectively. Note that we include all final states with at least one electron above the energy threshold of detector, \({E}_{{\rm{e}}}^{\mathrm{th}}\). This is particularly relevant as it allows for a precise description of the Migdal effect in the experiment.
To compare with the elastic scattering cross-section, we calculate the total Migdal cross-section by integrating over the kinematically allowed range of NR energies and the energy spectrum of emitted electrons in equation (2):
$${\sigma }_{\text{Migdal}}={\int }_{\text{max}[{E}_{{\rm{R}}}^{\text{min}},{E}_{{\rm{R}}}^{{\rm{th}}}]}^{{E}_{{\rm{R}}}^{\text{max}}}{\int }_{{E}_{{\rm{e}}}^{{\rm{th}}}}^{{E}_{{\rm{e}}}^{\text{max}}}\frac{{\rm{d}}{\sigma }_{{\rm{s}}}^{i}}{{\rm{d}}{E}_{{\rm{R}}}}\sum _{n\kappa }\frac{{\rm{d}}{p}_{\upsilon }^{i}(n\kappa \to {E}_{{\rm{e}}})}{{\rm{d}}{E}_{{\rm{e}}}}{{\rm{d}}E}_{{\rm{e}}}{{\rm{d}}E}_{{\rm{R}}}$$ (3)
where \({E}_{{\rm{N}}}^{\max }\) can be expressed as \({E}_{{\rm{R}},{\rm{m}}{\rm{a}}{\rm{x}}}=\frac{4{m}_{{\rm{n}}}{m}_{{\rm{T}}}{E}_{{\rm{n}}}}{{({m}_{{\rm{n}}}+{m}_{{\rm{T}}})}^{2}}\), where m n is the mass of the incident neutron, m T is the mass of the target nucleus and E n is the kinetic energy of incident neutron. \({E}_{{\rm{e}}}^{\max }=10\,\mathrm{keV}\). \({E}_{{\rm{n}}}^{\mathrm{th}}\) and \({E}_{{\rm{e}}}^{\mathrm{th}}\) are the thresholds of detector for nucleus and electron recoils, respectively. The ratio \(r={\sigma }_{\text{Migdal}}/{\sigma }_{\text{Elastic}}\) will offer a direct measure to assess the impact of the Migdal effect in dark matter detection experiments. Extended Data Fig. 1e compares the theoretically calculated Migdal differential probabilities for the gas mixture with the experimentally measured results (more detailed information of the theoretical cross-section calculations can be found in Extended Data Fig. 1 and Supplementary Information Note 1). The integrated theoretical probability in the 5–10 keV range is 3.9 × 10−5, which is consistent with the experimental result of \({(4.9}_{-1.9}^{+2.6})\times {10}^{-5}\) within the margin of error.
Detector assembly
The detector unit is sealed using brazing and laser welding to ensure high gas tightness and good mechanical properties. The main detector components—ceramics, Kovar alloys, beryllium and lead glass—have a low out-gas rate, which greatly reduces the pollution of other impurities in the gas. To fill the detector with working gas, the detector has a gas pipe that is brazed to the cathode. By using brazing technology, the metal ceramic tube shell is constructed of three layers of ceramic rings and four layers of Kovar alloy rings, with a ceramic ring placed between every two layers of Kovar alloy rings. The ceramic layer is used for insulation and positioning between the Kovar alloy layers. The Kovar alloy rings at both ends of the cermet tube shell are used to seal the connection with the cathode and the base by laser welding. The middle two Kovar alloy rings are used to install the gas microchannel plate (GMCP) as the exit electrode of the GMCP. The GMCP is installed in a cermet tube, and the support ring is extruded to fix the GMCP. The two electrodes of the GMCP are electrically connected by the Kovar alloy on the metal cermet tube. The ceramic pedestal is also composed of ceramic and Kovar alloy. A pixel chip is mounted on the ceramic pedestal and electrically connected to the ceramic pedestal using gold wire. The power supply of the pixel chip and information transmission are realized by 24 pins on the ceramic pedestal. The basic performance of the detector is tested59, the parameters and performance of GMCP are provided60, detailed parameters of the Topmetal-II chip are described52,61 and the electronic information of the detector is given62. The structure of the detector and its geometric parameters are shown in Extended Data Fig. 2a.
Electronic system and data acquisition
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