Compton scattering and Klein-Nishina formula コンプトン散乱とクライン-仁科の公式 Compton scattering and Klein-Nishina formula Contents Introduction Compton scattering 56 137 Cs source Setup for measurement of Compton scattering Results Summary Shibata lab. 12_14594 Yazawa Yukitaka

1. Introduction The purpose of this research is to understand the interaction between gamma ray and matter, especially Compton scattering to verify differential cross section of Compton scattering, Klein-Nishina formula Arthur H. Compton was awarded the Nobel prize in 1927 for the discovery of Compton effect. Compton effect confirms that light also follows laws of kinematics in the same way as particles do.

2. Compton scattering Law of energy and momentum conservation 𝜃 𝜙 Recoil electron 𝐸 𝑒 − Incident photon ℎ𝜈 Electron 𝑚 𝑒 𝑐 2 Scattered photon ℎ𝜈′ : Momentum : Momentum : Energy : Lorentz factor ℎ 𝜈 ′ = ℎ𝜈 1+ ℎ𝜈 𝑚 𝑒 𝑐 2 1−cos𝜃 , Energy [keV] Energy of incident photon: 662 keV 𝐸 𝑒 − =ℎ𝜈⋅ ℎ𝜈 𝑚 𝑒 𝑐 2 1−cos𝜃 1+ ℎ𝜈 𝑚 𝑒 𝑐 2 1−cos𝜃 . Energy of scattered photon Energy of recoil electron 662 keV 𝜃 [degree]

Differential cross section electron electron p’ Differential cross section p time position Klein-Nishina formula shows differential cross section for photons scattered by single electron. photon photon k, ε(k) k’, ε’(k’) photon electron p’ k, ε(k) Feynman diagram of Compton scattering Klein-Nishina formula: electron photon p k’, ε’(k’) 𝑑𝜎 𝑑Ω = 𝑟 𝑒 2 2 1 1+𝛼 1−cos 𝜃 2 1+ cos 2 𝜃+ 𝛼 2 1−cos 𝜃 2 1+𝛼 1−cos𝜃 𝑑𝜎 𝑑𝛺 [c m 2 /str] 𝑟 𝑒 : Classical electron radius (2.82× 10 −13 cm) 𝛼= ℎ𝜈 𝑚 𝑒 𝑐 2 . ℎ𝜈: Energy of incident gamma ray 𝑚 𝑒 𝑐 2 : Electron rest mass energy 𝜃 [degree]

Photoelectric peak (662 keV) 3. 137 Cs source 55 137 Cs 56 137 Ba 662 keV 1176 keV 𝛽 − decay Beta decay process: 55 137 Cs → 56 137 Ba + 𝑒 − + 𝜈 𝑒 Decay scheme of 55 137 Cs Excited 56 137 Ba nucleus emits gamma ray (662 keV). Rate: 2.0× 10 5 Bq time position Yields (180 sec) n p u u Photoelectric peak (662 keV) d Energy spectrum of gamma ray with single NaI(Tℓ) scintillator d d u e − W − Feynman diagram of 𝛽 − decay 𝜈 𝑒 ADC channel

4. Setup for measurement of Compton scattering 10 cm NaI 1 scintillator 137 Cs source NaI(Tℓ) crystal 5 cm θ 137 Cs source NaI 1 Lead 5 cm to prevent gamma rays from going into NaI 2 25 cm NaI 2 137 Cs source emits gamma ray(662 keV). The gamma ray interacts with matter in NaI 1 scintillator by the process of Compton scattering. The scattered gamma ray then interacts with NaI 2 scintillator by the process of photoelectric absorption. This process is measured in coincidence by NaI 1 and 2 scintillator with CAMAC/NIM modules.

5. Results 5.1 Identification of the process ADC channel of NaI 2 scintillator 5.1 Identification of the process ADC channel of NaI 2 scintillator Counts 𝜃=90° c c c c c c ADC channel of NaI 1 scintillator ADC channel of NaI 1 + NaI 2 Compton scattering (NaI 1) and Photoelectric absorption (NaI 2) Sum of the energy of NaI 1 and NaI 2 is 662 keV. Peak on ADC channel of NaI 1 + NaI 2 →Determine counts and energy of the event.

5.2 Angle dependence of energy Channel of NaI 1 scintillator Channel of NaI 2 scintillator 𝜃=105° Channel of NaI 2 scintillator Channel of NaI 2 scintillator 𝜃=30° 𝜃=90° Channel of NaI 1 scintillator Channel of NaI 1 scintillator Theoretical energy of recoil electron Theoretical energy of scattered photon Projection to channel of NaI 1 and NaI 2 Energy measured by NaI 1 Energy[keV] Energy measured by NaI 2 Sum of energy (NaI 1+ NaI 2) 662 keV Detected energy agrees with expected value at the most scattering angles. 𝑛 𝑖 : Counts in 𝑖 𝑡ℎ bin 𝐸 𝑖 : Energy of 𝑖 𝑡ℎ bin 𝜃 [degree]

5.3 Differential cross section 𝑁 𝑑𝑒𝑡 : Detected intensity of gamma ray 𝛺: Solid angle of NaI 1 scintillator Ω’: Solid angle of NaI 2 scintillator d: Effective Thickness of NaI 1 scintillator 𝐼 𝑠 : Rate of 137 Cs radiation source 𝜌 𝑠𝑐 : Density of scattering center (density of electrons in NaI scintillator) 𝜖 𝑖𝑛𝑡 : Intrinsic detection efficiency of NaI 2 scintillator 𝛺=0.19 str 𝛺 ′ = 0.031 str 𝑑= 2.57 cm 𝑁 𝑠 =2.0× 10 5 Bq 𝜌 𝑠𝑐 =9.34× 10 23 /c m 3 𝜖 𝑖𝑛𝑡 ⋅𝛺′=2.3× 10 −3 (𝐸 𝛾 MeV ) −1.54 5.3 Differential cross section 𝑁 𝑑𝑒𝑡 = 𝛺 4𝜋 𝐼 𝑠 ⋅ 𝜌 𝑠𝑐 𝑑⋅ 𝑑𝜎 𝑑𝛺′ ⋅ 𝛺 ′ 𝜖 𝑖𝑛𝑡 Luminosity 𝑑𝜎 𝑑𝛺′ = 𝑁 𝑑𝑒𝑡 𝛺 4𝜋 𝐼 𝑠 𝜌 𝑠𝑐 𝑑⋅ 𝛺 ′ 𝜖 𝑖𝑛𝑡 𝜃=30°, 　 𝜖 𝑖𝑛𝑡 =0.141 (564 keV) 𝜃=105°, 　 𝜖 𝑖𝑛𝑡 =0.487 (252 keV) θ NaI 1 NaI 2 d 𝛺 𝛺′ Real data Klein-Nishina formula 137 Cs source Vertical bar: Statistic error Horizontal bar: Maximum range of scattering angle Rate: 𝐼 𝑠 𝑑𝜎 𝑑 Ω ′ [c m 2 /str] Real data don’t agree with Klein-Nishina formula. The reason for discrepancy is being investigated. But, angular dependence follows Klein-Nishina formula. 𝜃 [degree]

6. Summary The purpose of this experiment is to understand the interaction between gamma ray and matter, especially Compton scattering to verify differential cross section of Compton scattering, Klein-Nishina formula Energy of scattered photon depends on scattering angle. Klein-Nishina formula shows Differential cross section of Compton scattering . 137 Cs source emits 662 keV gamma ray. Gamma ray was measured in coincidence by two NaI scintillators with CAMAC / NIM modules. Gamma ray is ① scattered by NaI 1 (Compton scattering) and ② absorbed by NaI 2 (photoelectric absorption). Detected energy by NaI 1 and NaI 2 scintillator agrees with energy of recoil electrons and scattered photon by Compton scattering. There is a hint that angular dependence of detected cross section follows that of Klein- Nishina formula. Measured cross section doesn’t agree with theoretical one. Further study is needed.

Bibliography 大学院物理基本実験Ⅰ テーマB 「NaIシンチレータによるガンマ線 の測定」 物理学実験第一 テキスト 物理学実験第一　テキスト 長島順清 (1998) 「朝倉物理学体系3 素粒子物理学の基礎Ⅰ」　朝 倉書店 長島順清 (2008) 「朝倉物理学体系3 素粒子物理学の基礎Ⅱ」　朝 倉書店 Richard B. Firestone (1999), Table of Isotopes

Appendix A: Intrinsic Detection Efficiency 𝛺: Solid angle(distance 25 cm) 𝜖 𝑖𝑛𝑡 ⋅𝛺 Energy of gammra ray [MeV]

Appendix B: Energy calibration Energy calibration by 3 peaks. Gamma radiation sources: 11 22 Na (511 keV, 1275 keV), 56 137 Cs (662 keV). Calibration of NaI 2 scintillator Calibration of NaI 1 scintillator ADC Channel ADC Channel Energy [keV] Energy [keV] Energy [keV] = 1.5 × Channel - 41 Energy [keV] = 1.5 × Channel - 51

Appedix B: Energy Calibration 2 Energy calibration by 6 peaks. Gamma radiation sources: 11 22 Na (511 keV, 1275 keV), 56 137 Cs (662 keV), 27 60 Co (1177 keV, 1333keV), 56 133 Ba (356 keV). NaI scintillator 1 NaI scintillator 2 ADC channel ADC channel Energy of gamma ray [keV] Energy of gamma ray [keV] Energy [keV] = 1.3 × Channel – 4.4 Energy [keV] = 1.3 × Channel + 38

Energy in case of new energy calibration (6 peaks) Theoretical energy of recoil electron Theoretical energy of scattered photon Energy measured by NaI 1 Energy [keV] Energy measured by NaI 2 Sum of energy (NaI 1+ NaI 2) 662 keV Sum of energy agrees with theoretical one (662 keV). But, Energy doesn’t agree with theoretical one especially in case of forward scattering (𝜃=30°). 𝜃 [degree] 16

Appendix C: Reason of discrepancy (Energy) Deviation of scattering angle 𝜃 NaI 1 scintillator ① Compton scattering (𝜃) NaI 1 scintillator 56 137 Cs source 𝜃 ′ (>𝜃) 56 137 Cs source 𝜃 ② Compton forward scattering → Gamma ray loses energy. Scattering angle becomes larger when the gamma ray is scattered above and below the center of NaI 1 scintillator. NaI 2 scintillator

Appendix D: 2 dimensional plot 𝜃=30° 𝜃=45° 𝜃=60° 𝜃=90° 𝜃=105° ADC channel of NaI 2 scintillator ADC channel of NaI 1 scintillator

Appendix E: Histgram of ADC1 + ADC2 channel 𝜃=30° 𝜃=45° 𝜃=60° Counts 𝜃=90° 𝜃=105° Sum of ADC channel (ADC 1 + ADC 2)

Δ 2 = 𝑖 𝑛 𝑖 𝜕𝐸 𝜕 𝑛 𝑖 2 = 𝑖 𝑛 𝑖 ⋅ 𝐸 𝑖 2 𝑖 𝑛 𝑖 2 = 𝑖 𝑛 𝑖 ⋅ 𝐸 𝑖 − 𝐸 𝑚𝑒𝑎𝑛 2 𝑖 𝑛 𝑖 2 +2 𝑖 𝑛 𝑖 𝐸 𝑖 𝐸 𝑚𝑒𝑎𝑛 𝑖 𝑛 𝑖 2 − 𝐸 𝑚𝑒𝑎𝑛 2 𝑖 𝑛 𝑖 2 = 𝑖 𝑛 𝑖 ⋅ 𝐸 𝑖 − 𝐸 𝑚𝑒𝑎𝑛 2 𝑖 𝑛 𝑖 2 + 𝐸 𝑚𝑒𝑎𝑛 2 𝑖 𝑛 𝑖 2

質疑応答 Q1.シミュレーションをしないとガンマ線がどこでコンプトン散乱したのかわ からないのではないか？ A. シミュレーションも今後検討中だが、鉛をおいて立体角を絞ることによっ ても計測できるのでそちらでも実験したい。 Q2. エネルギーでNaIシンチレータで計測した値と理論式が一致している のはなぜか？ A. 理論式はコンプトン散乱での散乱されたガンマ線と反跳電子のエネル ギーであることと、NaI1が反跳電子をNaI2がガンマ線を測定していること を再度説明した。 Q3. エネルギーの角度依存性が前方散乱でずれているのはなぜか？ A. 現在考察中。原因としてはコンプトン散乱する位置がNaIの中心とずれ ていることだと考えられると、バックアップのスライドを使いつつ説明した。