16 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. 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. B. Neutron Calibration E = ( S 1 α + S 2 β ) · W$_{tot}$ (5) W$_{tot}$ = 14 . 0 eV (6) FIG. 26: (Color online) Anticorrelation between S 1 and S 2 for 164 keV and 236 keV γ rays from activated xenon isotopes ( $^{131m}$Xe and $^{129m}$Xe). FIG. 27: Combined energy spectrum for single-scatter events from 662 keV gamma rays interacting within the 5.4-kg fidu- cial mass of XENON10. Insets: S1 and S2 distributions for 662 keV photo-absorped events (events within 2- σ around the 662 keV peak in the combined energy are selected). The en- ergy resolution at 662 keV is 10.7%, 5.4% and 2.5% for S1, S2 and the combined energy, respectively. S1 [pe] 0 200 400 600 800 1000 1200 S2 [pe] 20 40 60 80 100 120 140 160 3 10 ! Events/bin 50 100 150 200 250 θ Energy [keV] 0 100 200 300 400 500 600 700 800 900 1000 Counts 200 400 600 800 1000 S2 [pe] 0 50 100 150 200 250 300 350 400 450 500 3 10 ! Counts 200 400 600 800 1000 S1 [pe] 0 500 1000 1500 2000 2500 3000 Counts 50 100 150 200 250 300 350 400 450 σ/E =2 . 5% < S 1 > = 1530 σ/ < S 1 > = 10 . 7% < S 2 > =2 . 76 × 10 5 σ/ < S 2 > =5 . 4% 662 keV