Error Analysis 04 September

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

3 Or, Matrix form:

Statistical Error 4

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

6 Where:

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

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

Systematic Error 9

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 (%) This is the cross section dependence on energy Accounted for dN ij due to energy error when calculating dy i

11 With: Then:

12 Where:

Systematic Error Propagation (2) 13

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

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

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, %)

Electron Beam K. E. Cross Section (nb) Stat Error (no bg, %) Stat Error (with bg, %) Electron Beam K. E. Cross Section (nb) Sys Error (Energy, %) Sys Error (Total, %)

Electron Beam K. E. Gamma Energy (MeV) E CM (MeV) Cross Section (nb) S E1 Factor (keV b) Stat Error (%) Sys Error (%)

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