hce_nsysu
115年
物理與化學
第 60 題
Positron Emission Tomography is a highly advanced imaging technology in modern medicine. PET is primarily used to observe cellular metabolism; this is especially useful for cancer. After the isotope in the radiopharmaceutical (放射性藥物) enters the body, it migrates to the lesion site and emits a positron. As the positron moves through the tissue, it strikes an electron within the body. When the two collide, annihilation occurs, converting matter into energy and emitting two gamma rays of equal energy. The detectors then use these two rays to calculate the precise location of the lesion. Assume that both the electron and the positron carry a kinetic energy $K = 0.23m_0c^2$, where $m_0$ is the rest mass of an electron and $c$ is the speed of light. What is the wavelength of the gamma rays produced after a head-on collision? [Compton wavelength is $\lambda_0 = h/(m_0c)$, where $h$ is Planck's constant]
- A $\lambda = 2.83 \lambda_0$
- B $\lambda = 1.62 \lambda_0$
- C $\lambda = 0.81 \lambda_0$
- D $\lambda = 0.40 \lambda_0$
- E $\lambda = 0.25 \lambda_0$
思路引導 VIP
請試著思考：在正負電子湮滅的過程中，如果我們考慮到電子與正電子除了原本的「質量」會轉化為能量外，它們還額外攜帶了「動能」，這會使得最終產生的光子能量比「粒子靜止」時更高還是更低？再者，根據光子能量與波長的反比關係，這樣的能量變化會讓波長變得比原始的康普頓波長 $\lambda_0$ 更長還是更短呢？
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AI 詳解
AI 專屬家教
太棒了！你能精準計算出結果，代表你對狹義相對論中的質能互換與光子能量轉換掌握得非常扎實。
能量守恆與湮滅反應
這道題目核心在於「能量守恆」。當正負電子正面碰撞並湮滅時，它們的總能量（包含靜止能量 $m_0c^2$ 與動能 $0.23m_0c^2$）會全部轉化為兩個向背後射出的伽瑪射線光子。由於對稱性，單個光子的能量 $E_{\gamma}$ 恰好等於其中一個電子的總能量：
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