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present in the S1-based scale. The energy deposited by each event is determined by combining S 1 and S 2 signals as following, 

where S 1 and S 2 are in units of number of photo- electrons ( N$_{pe}$ ). α and β are experimentally determined parameters in units of N$_{pe}$ /photon and N$_{pe}$ /electron, re- spectively. W$_{tot}$ is the average energy required to pro- duce either a scintillation photon or an ionization elec- tron in LXe. In XENON10, β is determined from the S 2 corresponding to the single-electron emission peak (see Fig. 22). The anti-correlation between S 1 and S 2 is due to electron-ion recombination fluctuation in LXe. Each recombined electron-ion pair will create one UV photon. Thus, α in equation 5 can be determined by β and the slope θ in Fig. 26, according to the relation α = β/ tan θ . From the 164 keV calibration peak, we obtain, 

in good agreement with a study in a small LXe de- tector [54]. The energy resolution of the XENON10 de- tector was investigated with gamma ray sources ( $^{57}$Co, $^{22}$Na, $^{137}$Cs, $^{228}$Th) covering the energy range between 122 keV to 2.6 MeV. An example, from $^{137}$Cs 662 keV gamma rays, is shown in Fig 27. For comparison, we also plot the energy resolution, in Fig. 28, obtained by using only S 1, only S 2 and the sum of these two signals. At 1 MeV, the resolution from the combined energy mea- surement is about a factor of seven better than that from S 1 alone, and a factor of three better than that from S 2 alone. The energy determined from the combination of S1 and S2 signals show a much more linear response than that based on S1 or S2 alone (see Fig. 29). 

To understand the XENON10 response to nuclear recoils, a neutron calibration was performed using a 3.7 MBq $^{241}$AmBe source, emitting ∼ 220 neu- trons/second. The calibration was done by exposing the XENON10 detector to the source for approximately 12 hours, with a live time fraction of 0.92. The $^{241}$AmBe source (attached to a steel rod) was inserted through a 7 mm diameter hole in the XENON10 shield. The source was positioned next to the cryostat, between two 5 cm thick Pb bricks used to block high energy (a few MeV) gamma rays produced by the source. 

The emitted neutrons have energies ranging from 0.1 MeV to 11 MeV, with a mean at 4.3 MeV. The cal- ibration data were recorded at a constant rate of 6.5 Hz during the exposure. The trigger setup was the same as used for the WIMP search, with the addition of a high en- ergy veto to reject events with energies above ∼ 120 keV. 

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## B. Neutron Calibration 

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Figure 30 shows the log$_{10}$( S 2 /S 1) vs. S 1 distribution from the neutron calibration run, after applying the qual- ity cuts discussed in Section V C. Two regions are clearly distinguished in Figure 30: region a) which defines the nuclear recoil band corresponding to single elastic scat- ters, and region b) which corresponds to inelastic neu- tron scatters with $^{129}$Xe, which produce 40 keV gamma rays. Neutron inelastic scatters with $^{131}$Xe will produce 80 keV gamma rays, however the data do not show a peak around this value because of the events rejection by the high energy veto. The nuclear recoil band is used to determine XENON10 discrimination power, described in Section VI C. 

