Unfolding method - Data analysis - in Concrete and Steel for Shielding Design of High Energy Ha

2.2 Data analysis

2.2.4 Unfolding method

0 500 1000 1500 2000 2500 3000 3500 4000

QDC [ch]

1 10 102

103

104

Counts

(b) 100 MeV/u Si ions TTY

0 500 1000 1500 2000 2500 3000 3500 4000

QDC [ch]

1 10 102

103

104

Counts

(a) 100 MeV/u Si ions DDX

Figure 2.14: QDC distribution of total gate for neutron events for (a) thin and

0 500 1000 1500 2000 2500 3000 3500 4000

QDC [ch]

1 10 102

103

104

Counts

(d) 290 MeV/u Si ions TTY

0 500 1000 1500 2000 2500 3000 3500 4000

QDC [ch]

1 10 102

103

104

Counts

Figure 2.15: QDC distribution of total gate for neutron events for (a) thin and (b) thick silicon from 290 MeV/nucleon Si ion incidence

0 50 100 150 200 250 300 350 400 450 500

QDC [ch]

1 10 102

103

104

105

106

Counts

Co-60 Am-Be

60Co ± 3%

241Am-Be ± 3%

Compton peak

Half maximum

Figure 2.16: QDC distributions of total gate measured by⁶⁰Co and for²⁴¹ Am-Be sources

Half maximum, 1/2 N

₀

2/3 N

₀

1/3 N

₀

Maximum, N

₀

± 3%

Compton peak X

₀

C o u n ts (l o g sca le )

QDC [ch]

Figure 2.17: The uncertainty determination of the half maximum position X0

in schematic Compton spectrum

The peak of cosmic-ray muons in the natural background was also used as a calibration point of for 20.5 _±1.0 MeVee of the NE213 with a thickness of 12.7 cm and a diameter of 12.7 cm ⁽³⁶⁾. Fig 2.18 shows the QDC distri-bution for the natural background obtained after the experiment. To reduce statistical fluctuation, data for more than 100 events at the peak channel were accumulated. The peak position was determined by fitting the spectrum with the Gaussian function. The uncertainty for cosmic-ray muon peak was estimated to be 5.8 % from error propagation with independent uncertainty values for light output of muon and peak determination. The uncertainty of light output of muon was 4.9 % ⁽³⁶⁾. The uncertainty of peak determination was estimated from intrinsic energy resolution to be 3.2 % by interpolating widths of Compton edges of the two _γ-ray sources and the peak of maximum energy deposition protons at 124 MeV.

In addition to these two calibration methods, recoil proton edges can be used as calibration points for this dataset since the TOF data were available.

Fig 2.19 shows a two-dimensional plot of TDC value against QDC one with the total gate for neutron events of 290 MeV/nucleon ²⁸Si ion data. The red curved line in the figure indicates recoil proton edges that correspond to the maximum energy of recoil protons for neutron energies. The neutron ener-gies were derived from the TOF method. Thus, the light output, L_e (MeVee), for the channel of recoil proton edges can be obtained using theequation 2.3.

Eighteen points were selected from 1 MeVee to 70 MeVee for the 290 MeV/nu-cleon ²⁸Si ion dataset.

The events for maximum energy deposition of protons can be used for cali-bration point when high energy protons are observed in the dataset. Fig 2.20 shows a two-dimensional QDC plot of the total gate and veto detector for the dataset of 290 MeV/nucleon²⁸Si ions. The boundary around 4240 ch indicates

0 500 1000 1500 2000 2500 3000

QDC [ch]

1 10 10

10 Counts

Muon peak point

(1017 ch) Gaussian fitting

Cosmic-ray muon peak

Figure 2.18: QDC distribution of total gate measured by cosmic-ray muon sources

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TDC [ch]

0 1000 2000 3000 4000 5000

QDC [ch]

hist20

Entries 4545 Mean x 4879 Mean y 392.3 Std Dev x 733.1 Std Dev y 457.6

0 2 4 6 8 10 12 14 16 18 hist20

Entries 4545 Mean x 4879 Mean y 392.3 Std Dev x 733.1 Std Dev y 457.6

Recoil proton edge

Figure 2.19: TOF vs. QDC total gate histogram of neutron events from the thick ²⁸Si target bombarded by 290 MeV/nucleon

maximum energy deposition was calculated to be 124 MeV using the SRIM code ⁽³⁷⁾ for NE213 scintillator with a thickness of 12.7 cm. The light output of the maximum energy deposition became to be 98 MeVee by equation 2.3.

0 1000 2000 3000 4000 5000 6000

QDC total gate [ch]

0 1000 2000 3000 4000 5000

QDC veto gate [ch]

Maximum energy deposition point

(4238 ch)

Figure 2.20: Two-dimensional QDC plot of the total gate and the veto detec-tor in the case of 290 MeV/nucleon²⁸Si incidence on a silicon thick target

The QDC value of this event was obtained from the half maximum of the proton edge in the QDC distribution of the total gate. The uncertainty was to be estimated at 6.5 % which were derived from standard deviation of the Gaussian function for proton edges. As shown in Figs 2.14 and 2.15, the significant amount of neutron components were observed from the data of DDXs than TTYs. Thus, the effects of light output calibration on the data were checked in two ways.

Fig 2.21 shows a relationship between the QDC value and light output for the data points obtained using the four methods; Compton edges of _γ-rays, peak of cosmic-ray muons, recoil proton edges and maximum energy deposi-tion of protons. The black line (a) was obtained from the two data points for Compton edges of ⁶⁰Co and ²⁴¹Am-Be sources. The data point of cosmic-ray muon peak was not used in determination of the black line. As can be seen in Fig 2.21, the black line is in agreement with the point of muon within the uncertainty limit. The red line (b) was obtained by connecting 3 % larger QDC value of the Compton edge of the _γ-ray from ⁶⁰Co and 3 % lower QDC value of that from ²⁴¹Am-Be as shown in inner panel of Fig 2.21. The blue line (c) was obtained in the same manner as the red line using the opposite side of uncertainties. The area surrounded by the (b) and (c) lines is probable area of the calibration line (a) connecting QDC values of Compton edges from both _γ-ray sources.

Light output [MeVee]

Compton edge of sources Cosmic muon

Recoil proton edge from TOF

Maximum energy deposition of a proton case (a)

case (b) case (c) case (d)

100 120 140 160 180 200 220 240 260 280 300 QDC [ch]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Light output [MeVee]

Compton edge of sources Cosmic muon Recoil proton edge from TOF upper fitting center fitting lower fitting Recoil proton edge fitting

100 120 140 160 180 200 220 240 260 280 300 QDC [ch]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Light output [MeVee]

Compton edge of sources Cosmic muon Recoil proton edge from TOF upper fitting center fitting lower fitting Recoil proton edge fitting

100 120 140 160 180 200 220 240 260 280 300 QDC [ch]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Light output [MeVee]

Compton edge of sources Cosmic muon Recoil proton edge from TOF upper fitting center fitting lower fitting Recoil proton edge fitting

100 120 140 160 180 200 220 240 260 280 300 QDC [ch]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Light output [MeVee]

Compton edge of sources Cosmic muon Recoil proton edge from TOF upper fitting center fitting lower fitting Recoil proton edge fitting

100 150 200 250 300

QDC [ch]

0 1 2 3 4 5

Light output [MeVee] ⁶⁰Co ± 3%

Compton edge of sources Cosmic muon

Recoil proton edge from TOF

Maximum energy deposition of a proton case (a)

case (b) case (c) case (d)

100 120 140 160 180 200 220 240 260 280 300 QDC [ch]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Light output [MeVee]

Compton edge of sources Cosmic muon Recoil proton edge from TOF upper fitting center fitting lower fitting Recoil proton edge fitting

100 120 140 160 180 200 220 240 260 280 300 QDC [ch]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Light output [MeVee]

Compton edge of sources Cosmic muon Recoil proton edge from TOF upper fitting center fitting lower fitting Recoil proton edge fitting

100 120 140 160 180 200 220 240 260 280 300 QDC [ch]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Light output [MeVee]

Compton edge of sources Cosmic muon Recoil proton edge from TOF upper fitting center fitting lower fitting Recoil proton edge fitting

100 120 140 160 180 200 220 240 260 280 300 QDC [ch]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Light output [MeVee]

Compton edge of sources Cosmic muon Recoil proton edge from TOF upper fitting center fitting lower fitting Recoil proton edge fitting

100 150 200 250 300

QDC [ch]

0 1 2 3 4 5

Light output [MeVee] ⁶⁰Co ± 3%

241Am-Be ± 3%

Muon

Figure 2.22: Calibration lines of QDC values vs. light output for thin silicon targets data

The DDXs and TTYs were derived fromequations 2.7 and2.8. The neu-tron energy spectrumΦ(E_i) at i-th energy bin (E_i) was obtained by unfolding the neutron light output spectrum using the response functions. The N(L_j) is expressed using a simple equation 2.9as follows:

N^¡L_j^¢₌^X

R_{i j}Φ(E_i) , (2.9)

where N(L_j) is the recorded count in thej-th light output (L_j) on the neutron light output spectrum. R_{i j} is the response ati-th energy bin and thej-th light output bin. The distribution of N(L_j) is the neutron light output spectrum, which was obtained by analyzing event-by-event data.

For the unfolding process, the iterative Bayesian algorithm in the RooUn-fold package ⁽³⁴⁾ was used. The response (R_{i j}) can be represented as a con-ditional probability P(L_j_|E_i). The Bayesian formula is the following equa-tion 2.10by conditional probability equation P(L_j_|E_i) :

P^¡L_j_|E_i^¢₌ P^¡E_i_|L_j^¢P(E_i) PⁿE

i=1P^¡E_i_|E_j^¢P(E_i), (2.10) where P(Ei) is the probability of the incident neutron number in the i-th en-ergy bin to that in all enen-ergy bins while nE is the number of energy bins. The regularization parameter, i.e., iteration number, was chosen in a way that fulfills the condition _χ² _≈ 0.9 (100 MeV/nucleon) and 0.1 (290 MeV/nucleon) for best fit with TOF results. The regularization parameters were 10 (100 MeV/nucleon) and 50 (290 MeV/nucleon).

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