1. Error Analysis 04 September 2013 1

2. Penfold-Leiss Cross Section Unfolding Measure Yields at: where, The solution can be written in two forms: 2

3. 3 Or, Matrix form:

4. Statistical Error 4

5. Statistical Error Propagation (1) 5 With: Then: Note: In case of background Subtraction

6. 6 Where:

7. Statistical Error Propagation (2) 7 For mono- chromatic beam Although,

8. Statistical Error Propagation (Wrong) 8 Then: This is equivalent to: Wrong

9. Systematic Error 9

10. Systematic Error Propagation (1) 10 For absolute beam energy uncertainty of δE (= 0.1%): E i (MeV)dy i /y i (%)dσ i /σ i (%) 7.917.412.6 8.012.310.5 8.110.09.1 8.28.67.1 8.37.66.3 8.46.85.8 8.56.15.2 This is the cross section dependence on energy Accounted for dN ij due to energy error when calculating dy i

11. 11 With: Then:

12. 12 Where:

13. Systematic Error Propagation (2) 13

14. Other Systematic Errors 14 Beam Current, δI/I3% Photon Flux, δφ/φ5% Radiator Thickness, δR/R3% Bubble Chamber Thickness, δT/T3% Bubble Chamber Efficiency, ε5% Then: Simulation

15. Results 15 I.Radiator Thickness = 0.02 mm II.Bubble Chamber Thickness = 3.0 cm III.Number of 16 O nuclei = 3.474e22 /cm 2 IV.Background subtraction of 18 O(γ,α) 14 C V. 17 O(γ,n) 16 O: Still to do

16. Electron Beam K. E. Beam Current (µA) Time (hour) yiyi dy i (no bg) dy i /y i (no bg, %) dy i (with bg) dy i /y i (with bg, %) 7.9100 545234.213424.6 8.010020581244.17713.3 8.18010852293.4607.0 8.22010634253.9406.3 8.310 812283.4394.8 8.4410746273.6364.8 8.5210763283.7324.2 16

17. Electron Beam K. E. Cross Section (nb) Stat Error (no bg, %) Stat Error (with bg, %) 7.90.0464.424.5 8.00.1856.020.7 8.10.586.314.7 8.21.538.213.8 8.33.499.113.3 8.47.210.613.8 8.513.612.214.8 17 Electron Beam K. E. Cross Section (nb) Sys Error (Energy, %) Sys Error (Total, %) 7.90.04617.419.5 8.00.18518.121.5 8.10.5818.422.9 8.21.5318.924.9 8.33.4919.727.2 8.47.220.630.0 8.513.621.633.0

18. Electron Beam K. E. Gamma Energy (MeV) E CM (MeV) Cross Section (nb) S E1 Factor (keV b) Stat Error (%) Sys Error (%) 7.97.850.690.04662.224.519.5 8.07.950.790.18548.720.721.5 8.18.050.890.5841.814.722.8 8.28.150.991.5335.513.824.9 8.38.251.093.4932.013.327.2 8.48.351.197.228.813.830.0 8.58.451.2913.626.314.833.0 18

19. 12 C(α, γ) 16 O S-Factor  Statistical Error: dominated by background subtraction from 18 O(γ,α) 14 C (depletion = 5,000)  Systematic Error: dominated by absolute beam energy (δE = 0.1%) 19 S E1 (300 keV) = 74±21 keV b