Compton scattering and Klein-Nishina formula
Contents
1. Introduction
2. Compton scattering
3. 56137Cs source
4. Setup for measurement of Compton scattering
5. Results
6. Summary
Shibata lab.12_14594Yazawa Yukitaka
コンプトン散乱とクライン-仁科の公式
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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
ℎ𝜈′ =ℎ𝜈
1 +ℎ𝜈𝑚𝑒𝑐2 1 − cos𝜃
,
𝐸𝑒− = ℎ𝜈 ⋅
ℎ𝜈𝑚𝑒𝑐2 1 − cos𝜃
1 +ℎ𝜈𝑚𝑒𝑐2 1 − cos𝜃
.
2. Compton scatteringLaw of energy and momentum conservation
: Momentum
: Momentum
: Energy
: Lorentz factor
𝜃
𝜙
Recoil electron𝐸𝑒−
Incident photonℎ𝜈
Electron𝑚𝑒𝑐2
Scattered photonℎ𝜈′
Energy of scattered photonEnergy of recoil electron
Energy [keV]
𝜃 [degree]
Energy of incident photon: 662 keV
662 keV 3
k, ε(k)
p
p’
k’, ε’(k’)
k’, ε’(k’)
p
k, ε(k)
p’
electron
electron
photon
photon
photon
electron
electron
photon
Klein-Nishina formula:
𝑑𝜎
𝑑Ω=𝑟𝑒2
2
1
1 + 𝛼 1 − cos 𝜃
2
1 + cos2𝜃 +𝛼2 1 − cos 𝜃 2
1 + 𝛼 1 − cos𝜃
𝑟𝑒 : Classical electron radius (2.82 × 10−13cm)
𝛼 =ℎ𝜈
𝑚𝑒𝑐2.
ℎ𝜈: Energy of incident gamma ray𝑚𝑒𝑐2: Electron rest mass energy
Klein-Nishina formula shows differential cross section for photons scattered by single electron.
Feynman diagram of Compton scattering
time
position
𝜃 [degree]
𝑑𝜎
𝑑𝛺[cm2/str]
Differential cross section
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3. 137Cs source
ud
u
d du
pn
W−
𝜈𝑒
e−
55137Cs
56137Ba
662 keV
1176 keV
𝛽− decay
Rate: 2.0 × 105 Bq
Beta decay process:
55137Cs → 56
137Ba + 𝑒− + 𝜈𝑒
Excited 56137Ba nucleus emits gamma ray (662 keV). Decay scheme of 55
137Cs
Feynman diagram of 𝛽− decay
ADC channel
Yields(180 sec)
Photoelectric peak (662 keV)
Energy spectrum of gamma ray with single NaI(Tℓ) scintillator
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time
position
4. Setup for measurement of Compton scattering
1. 137Cs source emits gamma ray(662 keV).2. The gamma ray interacts with matter in NaI 1 scintillator by the process of Compton scattering.
3. The scattered gamma ray then interacts with NaI 2 scintillator by the process of photoelectric absorption.
10 cm
NaI 1scintillator137
Cs source
NaI(Tℓ)crystal
5 cm
θ
137Cs source NaI 1
Lead
5 cm
5 cm
5 cm
to prevent gamma rays from going into NaI 2
25 cm
NaI 2
5 cm
5 cm This process is measured in coincidence by NaI 1 and 2 scintillator with CAMAC/NIM modules.
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Compton scattering (NaI 1) and Photoelectric absorption (NaI 2)
Sum of the energy of NaI 1 and NaI 2 is 662 keV.
ADC channel of NaI 1 scintillator
5. Results
ADC channel of NaI 2 scintillator
ADC channel of NaI 1 + NaI 2
𝜃=90°
Counts
Peak on ADC channel of NaI 1 + NaI 2→Determine counts and energy of the event.
ADC channel of NaI 2 scintillator
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5.1 Identification of the process
𝜃=90°
Channel of NaI 2 scintillator
Channel of NaI 1 scintillator Channel of NaI 1 scintillator
Channel of NaI 2 scintillator
𝜃=105°
Theoretical energy of recoil electronTheoretical energy of scattered photon
Energy[keV]
𝑛𝑖: Counts in 𝑖𝑡ℎ bin
𝐸𝑖: Energy of 𝑖𝑡ℎ bin
Energy measured by NaI 2
Energy measured by NaI 1
Sum of energy (NaI 1+ NaI 2)
662 keV
Detected energy agrees with expected value at the most scattering angles.
Channel of NaI 1 scintillator
Channel of NaI 2 scintillator
𝜃=30°
Projection to channel of NaI 1 and NaI 2
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𝜃 [degree]
5.2 Angle dependence of energy
𝑑𝜎
𝑑Ω′
[cm2/str]
𝑑𝜎
𝑑𝛺′=
𝑁𝑑𝑒𝑡𝛺4𝜋 𝐼𝑠 𝜌𝑠𝑐𝑑 ⋅ 𝛺
′𝜖𝑖𝑛𝑡
5.3 Differential cross section𝑁𝑑𝑒𝑡: Detected intensity of gamma ray𝛺: Solid angle of NaI 1 scintillatorΩ’: Solid angle of NaI 2 scintillatord: Effective Thickness of NaI 1 scintillator
𝐼𝑠: Rate of 137Cs radiation source𝜌𝑠𝑐: Density of scattering center (density of electrons in NaI scintillator)𝜖𝑖𝑛𝑡: Intrinsic detection efficiency of NaI 2 scintillator
𝛺 = 0.19 str𝛺′ = 0.031 str𝑑 = 2.57 cm𝑁𝑠 = 2.0 × 10
5 Bq𝜌𝑠𝑐 = 9.34 × 10
23 /cm3
𝜖𝑖𝑛𝑡 ⋅ 𝛺′ = 2.3 × 10−3(𝐸𝛾 MeV )
−1.54
θ
NaI 1
NaI 2
d𝛺
𝛺′
𝑁𝑑𝑒𝑡 =𝛺
4𝜋𝐼𝑠 ⋅ 𝜌𝑠𝑐𝑑 ⋅
𝑑𝜎
𝑑𝛺′⋅ 𝛺′𝜖𝑖𝑛𝑡
Vertical bar: Statistic errorHorizontal bar: Maximum range of
scattering angle
Real data don’t agree with Klein-Nishina formula.The reason for discrepancy is being investigated.But, angular dependence follows Klein-Nishina formula.
Real data
Klein-Nishina formula
𝜃 [degree]
137Cssource
Rate: 𝐼𝑠
Luminosity
𝜃 = 30°, 𝜖𝑖𝑛𝑡 = 0.141 (564 keV)
𝜃 = 105°, 𝜖𝑖𝑛𝑡 = 0.487 (252 keV)
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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 .
• 137Cs 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. 10
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
•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
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Appendix A: Intrinsic Detection Efficiency
Energy of gammra ray [MeV]
𝜖𝑖𝑛𝑡 ⋅ 𝛺
𝛺: Solid angle(distance 25 cm)
Appendix B: Energy calibration
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Calibration of NaI 1 scintillator
Energy [keV]
ADC Channel
Calibration of NaI 2 scintillator
Energy [keV]
ADC Channel
Energy calibration by 3 peaks.
Gamma radiation sources: 1122Na (511 keV, 1275 keV), 56
137Cs (662 keV).
Energy [keV] = 1.5 × Channel - 41Energy [keV] = 1.5 × Channel - 51
Appedix B: Energy Calibration 2
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Energy of gamma ray [keV]
NaI scintillator 1
ADC channel
NaI scintillator 2
Energy of gamma ray [keV]
ADC channel
Energy calibration by 6 peaks.
Gamma radiation sources: 1122Na (511 keV, 1275 keV), 56
137Cs (662 keV), 2760Co (1177 keV, 1333keV), 56
133Ba (356 keV).
Energy [keV] = 1.3 × Channel – 4.4 Energy [keV] = 1.3 × Channel + 38
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𝜃 [degree]
Energy [keV]
Energy in case of new energy calibration (6 peaks)
Theoretical energy of recoil electronTheoretical energy of scattered photon
Energy measured by NaI 2
Energy measured by NaI 1
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°).
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Appendix C: Reason of discrepancy (Energy)
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Deviation of scattering angle 𝜃
56137Cs source
𝜃
𝜃′(> 𝜃)
NaI 1 scintillator
NaI 2 scintillator
Scattering angle becomes larger when the gamma ray is scattered above and below the center of NaI 1 scintillator.
56137Cs source
① Compton scattering (𝜃)
② Compton forward scattering→ Gamma ray loses energy.
NaI 1 scintillator
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ADC channel of NaI1 scintillator
ADC channel of NaI 2 scintillator
𝜃 = 30° 𝜃 = 60°𝜃 = 45°
𝜃 = 90° 𝜃 = 105°
Appendix D: 2 dimensional plot
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𝜃 = 30° 𝜃 = 60°𝜃 = 45°
𝜃 = 90° 𝜃 = 105°
Appendix E: Histgram of ADC1 + ADC2 channel
Sum of ADC channel (ADC 1 + ADC 2)
Counts
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Δ2 =
𝑖
𝑛𝑖𝜕𝐸
𝜕𝑛𝑖
2
= 𝑖 𝑛𝑖 ⋅ 𝐸𝑖
2
𝑖 𝑛𝑖2
= 𝑖 𝑛𝑖 ⋅ 𝐸𝑖 − 𝐸𝑚𝑒𝑎𝑛
2
𝑖 𝑛𝑖2 + 2
𝑖 𝑛𝑖𝐸𝑖 𝐸𝑚𝑒𝑎𝑛
𝑖 𝑛𝑖2 −
𝐸𝑚𝑒𝑎𝑛2
𝑖 𝑛𝑖2
= 𝑖 𝑛𝑖 ⋅ 𝐸𝑖 − 𝐸𝑚𝑒𝑎𝑛
2
𝑖 𝑛𝑖2 +
𝐸𝑚𝑒𝑎𝑛2
𝑖 𝑛𝑖2
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。
