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--- research:memos:inverse_compton_scattering [2022/07/24 20:49] – kobayash
+++ research:memos:inverse_compton_scattering [2022/07/28 23:29] (現在) – [General formula] kobayash
@@ 行 2: / 行 2: @@
 ==== General formula =====
+{{ :research:memos:electron_photon_scattering_20220724.png?700 |}}
 From the laws of conservation of the energy and the momentum,
 \begin{align}
-E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c\\
+E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c,\\
 \boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}} = \boldsymbol{p}_e^\prime + \boldsymbol{p}_{\mathrm{ph}}^{\prime}.\\
 \end{align}
@@ 行 25: / 行 26: @@
 \begin{align}
-p_e^{\prime 2} \cos^2 \theta_e^\prime &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} \cos^2 \theta_{\mathrm{ph}}^{\prime} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime},\\
+p_e^{\prime 2} \cos^2 \theta_e^\prime &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} \cos^2 \theta_{\mathrm{ph}}^{\prime} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\\
+\Leftrightarrow \
 p_e^{\prime 2} -p_e^{\prime 2}\sin^2 \theta_e^\prime &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} \cos^2 \theta_{\mathrm{ph}}^{\prime} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}.\\
 \end{align}
 From $0 = p_e^\prime \sin \theta_e^\prime + p_{\mathrm{ph}}^{\prime} \sin \theta_{\mathrm{ph}}^{\prime}$,
 \begin{align}
-p_e^{\prime 2} -p_{\mathrm{ph}}^{\prime 2} \sin^2 \theta_{\mathrm{ph}}^{\prime} &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} \cos^2 \theta_{\mathrm{ph}}^{\prime} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime},\\
+p_e^{\prime 2} -p_{\mathrm{ph}}^{\prime 2} \sin^2 \theta_{\mathrm{ph}}^{\prime} &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} \cos^2 \theta_{\mathrm{ph}}^{\prime} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\\
+\Leftrightarrow \
 p_e^{\prime 2}
 &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} \cos^2 \theta_{\mathrm{ph}}^{\prime} + p_{\mathrm{ph}}^{\prime 2} \sin^2 \theta_{\mathrm{ph}}^{\prime}-2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\\
@@ 行 37: / 行 40: @@
 In another way, this equation can be extracted from $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}} = \boldsymbol{p}_e^\prime + \boldsymbol{p}_{\mathrm{ph}}^{\prime}$ as follows.
 \begin{align}
-\boldsymbol{p}_e^\prime &= \boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}} - \boldsymbol{p}_{\mathrm{ph}}^{\prime},\\
+\boldsymbol{p}_e^\prime &= \boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}} - \boldsymbol{p}_{\mathrm{ph}}^{\prime}\\
+\Rightarrow \
 p_e^{\prime 2}
 &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} -2\boldsymbol{p}_{\mathrm{ph}}^{\prime}\cdot(\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}})\\
 &= |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}|^2 + p_{\mathrm{ph}}^{\prime 2} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\\
-&= p_e^2 + p_{\mathrm{ph}}^2 + 2p_e p_{\mathrm{ph}} \cos(\theta_e+\theta_{\mathrm{ph}}) + p_{\mathrm{ph}}^{\prime 2} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime},\\
+&= p_e^2 + p_{\mathrm{ph}}^2 + 2p_e p_{\mathrm{ph}} \cos(\theta_e+\theta_{\mathrm{ph}}) + p_{\mathrm{ph}}^{\prime 2} -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\\
+\Leftrightarrow \
 p_e^{\prime 2} &= p_e^2 + p_{\mathrm{ph}}^2 + p_{\mathrm{ph}}^{\prime 2} + 2p_e p_{\mathrm{ph}} \cos(\theta_e+\theta_{\mathrm{ph}}) -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime},\\
 \end{align}
-where $\theta_e$ ($\theta_{\mathrm{ph}}$) is an angle between $\boldsymbol{p}_e$ ($\boldsymbol{p}_{\mathrm{ph}}$) and $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}$, and $\theta_{\mathrm{ph}}^{\prime}$ is an angle between $\boldsymbol{p}_{\mathrm{ph}}^{\prime}$ and $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}$.
+where $\theta_e$ ($\theta_{\mathrm{ph}}$) is angle between $\boldsymbol{p}_e$ ($\boldsymbol{p}_{\mathrm{ph}}$) and $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}$, and $\theta_{\mathrm{ph}}^{\prime}$ is angle between $\boldsymbol{p}_{\mathrm{ph}}^{\prime}$ and $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}$.
 Then, from the law of energy conservation $E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c$,
@@ 行 54: / 行 59: @@
 E_e^{\prime 2} = E_e^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2.\\
 \end{align}
-By using $E^2 = m^2c^4 + p^2c^2$,
+By using $E_e^2 = m_e^2c^4 + p_e^2c^2$ and $E_e^{\prime 2} = m_e^2c^4 + p_e^{\prime 2}c^2$
 \begin{align}
-m_e^2c^4  + p_e^{\prime 2}c^2 &= m_e^2c^4  + p_e^{\prime 2}c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2,\\
+m_e^2c^4  + p_e^{\prime 2}c^2 &= m_e^2c^4  + p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2\\
+\Leftrightarrow \
 p_e^{\prime 2}c^2 &= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2.\\
 \end{align}
 By using $p_e^{\prime 2} = p_e^2 + p_{\mathrm{ph}}^2 + p_{\mathrm{ph}}^{\prime 2} + 2p_e p_{\mathrm{ph}} \cos(\theta_e+\theta_{\mathrm{ph}}) -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}$,
 \begin{align}
-\left[p_e^2 + p_{\mathrm{ph}}^2 + p_{\mathrm{ph}}^{\prime 2} + 2p_e p_{\mathrm{ph}} \cos(\theta_e+\theta_{\mathrm{ph}}) -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\right]c^2 &= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2,\\
+\left[p_e^2 + p_{\mathrm{ph}}^2 + p_{\mathrm{ph}}^{\prime 2} + 2p_e p_{\mathrm{ph}} \cos(\theta_e+\theta_{\mathrm{ph}}) -2 p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}\right]c^2 &= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2\\
-p_e p_{\mathrm{ph}} c \cos(\theta_e+\theta_{\mathrm{ph}}) - p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime} &= E_e p_{\mathrm{ph}} - E_e p_{\mathrm{ph}}^{\prime} -p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c,\\
+\Leftrightarrow \
-\left[ E_e + p_{\mathrm{ph}} c - |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime}\right]p_{\mathrm{ph}}^{\prime} &= \left[E_e -p_e c \cos(\theta_e+\theta_{\mathrm{ph}})\right]p_{\mathrm{ph}},\\
+p_e p_{\mathrm{ph}} c \cos(\theta_e+\theta_{\mathrm{ph}}) - p_{\mathrm{ph}}^{\prime} |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime} &= E_e p_{\mathrm{ph}} - E_e p_{\mathrm{ph}}^{\prime} -p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c\\
+\Leftrightarrow \
+\left[ E_e + p_{\mathrm{ph}} c - |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime}\right]p_{\mathrm{ph}}^{\prime} &= \left[E_e -p_e c \cos(\theta_e+\theta_{\mathrm{ph}})\right]p_{\mathrm{ph}}\\
+\Rightarrow \
 p_{\mathrm{ph}}^{\prime} &= \frac{E_e -p_e c \cos(\theta_e+\theta_{\mathrm{ph}})}{E_e  + p_{\mathrm{ph}} c - |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| c\cos \theta_{\mathrm{ph}}^{\prime}}p_{\mathrm{ph}}.\\
 \end{align}
@@ 行 79: / 行 88: @@
 \end{align}
-If unit vectors $\widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}} = \boldsymbol{p}_{\mathrm{ph}}^{\prime}/p_{\mathrm{ph}}^{\prime}, \widehat{\boldsymbol{p}_{\mathrm{ph}}} = \boldsymbol{p}_{\mathrm{ph}}/p_{\mathrm{ph}}$, and $\widehat{\boldsymbol{p}_e} = \boldsymbol{p}_e/p_e$ is defined,
+If unit vectors $\widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}} = \boldsymbol{p}_{\mathrm{ph}}^{\prime}/p_{\mathrm{ph}}^{\prime},\ \widehat{\boldsymbol{p}_{\mathrm{ph}}} = \boldsymbol{p}_{\mathrm{ph}}/p_{\mathrm{ph}}$, and $\widehat{\boldsymbol{p}_e} = \boldsymbol{p}_e/p_e$ are defined,
 \begin{align}
 |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime}
@@ 行 109: / 行 118: @@
 E_\gamma = \frac{E_l(1 - \beta \cos\theta)}{1 - \beta \cos\theta_\gamma + \frac{E_l(1 - \cos\eta)}{E_e}},\\
 \end{align}
-where $E_\gamma$ ($E_l$) is an energy of the scattered photon (incident lasesr photon), $\beta$ is a ratio of speed of the incident electron to that of light, $\theta$ ($\theta_\gamma$) is an angle between the inicident electron direction and incident laser photon (scattered photon), and $\eta$ is an angle between the incident laser photon and scattered photon. The definition of the angles can be seen in Fig. 1 of the paper.
+where $E_\gamma$ ($E_l$) is energy of the scattered photon (incident lasesr photon), $\beta$ is ratio of speed of the incident electron to that of light, $\theta$ ($\theta_\gamma$) is angle between the inicident electron direction and incident laser photon (scattered photon), and $\eta$ is angle between the incident laser photon and scattered photon. The definition of the angles can be seen in Fig. 1 of the paper.
-{{:research:memos:1-s2.0-s0168900222002625-gr1_lrg.jpg?400|}}
+{{ :research:memos:1-s2.0-s0168900222002625-gr1_lrg.jpg?600 |}}
 From the above general formula,
 \begin{align}
-p_{\mathrm{ph}}^{\prime} = \frac{1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})}{1 - \beta_e \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e} + \frac{p_{\mathrm{ph}} c}{E_e} (1 - \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}})}p_{\mathrm{ph}},\\
+p_{\mathrm{ph}}^{\prime} &= \frac{1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})}{1 - \beta_e \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e} + \frac{p_{\mathrm{ph}} c}{E_e} (1 - \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}})}p_{\mathrm{ph}},\\
-p_{\mathrm{ph}}^{\prime}c = \frac{p_{\mathrm{ph}}c\left[1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})\right]}{1 - \beta_e \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e} + \frac{p_{\mathrm{ph}} c(1 - \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}})}{E_e}}.\\
+\Leftrightarrow\ p_{\mathrm{ph}}^{\prime}c &= \frac{p_{\mathrm{ph}}c\left[1 - \beta_e \cos(\theta_e+\theta_{\mathrm{ph}})\right]}{1 - \beta_e \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_e} + \frac{p_{\mathrm{ph}} c(1 - \widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}}\cdot\widehat{\boldsymbol{p}_{\mathrm{ph}}})}{E_e}}.\\
 \end{align}
 According to definitions of the slant scattering paper,
@@ 行 148: / 行 157: @@
 By eliminating $p_{\mathrm{ph}}^{\prime}$ from the two equations,
 \begin{align}
-E_e + 2p_{\mathrm{ph}}c - p_{e}c = E_e^\prime - p_e^{\prime}c\\
+E_e + 2p_{\mathrm{ph}}c - p_{e}c &= E_e^\prime - p_e^{\prime}c\\
-E_e + 2p_{\mathrm{ph}}c - p_{e}c + p_e^{\prime}c = \sqrt{p_e^{{\prime}2}c^2+m_{e}^2c^4}.\\
+\Rightarrow \
+E_e + 2p_{\mathrm{ph}}c - p_{e}c + p_e^{\prime}c &= \sqrt{p_e^{{\prime}2}c^2+m_{e}^2c^4}.\\
 \end{align}
 By making the square of both sides,
@@ 行 157: / 行 167: @@
 Then,
 \begin{align}
-p_e^{{\prime}}c\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right) = m_{e}^2c^4 -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)^2.\\
+p_e^{{\prime}}c\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right) = m_{e}^2c^4 -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)^2\\
- p_e^{{\prime}} = \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)\right].\\
+\Rightarrow \
+p_e^{{\prime}} = \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)\right].\\
 \end{align}
-=== Energy of the scattered photon $E_{\mathrm{ph}}^{\prime}$ (Solution A)===
+=== Energy of the scattered photon $E_{\mathrm{ph}}^{\prime}$ (Solution A) ===
+By using the above $E_{e}^{\prime}$,
+\begin{align}
+p_{\mathrm{ph}}^{\prime}
+&= p_e - p_{\mathrm{ph}} -p_e^\prime \\
+&= p_e - p_{\mathrm{ph}} - \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)\right] \\
+&= - \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right) -2c\left(p_e - p_{\mathrm{ph}}\right) \right] \\
+&= - \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + p_{e}c\right) \right] \\
+&= - \frac{1}{2c}\left[\frac{m_{e}^2c^4  -\left(E_e + p_{e}c\right)\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\
+&= - \frac{1}{2c}\left[\frac{m_{e}^2c^4  -2p_{\mathrm{ph}}c\left(E_e + p_{e}c\right)-\left(E_e^2 - p_{e}^2c\right)}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\
+&= - \frac{1}{2c}\left[\frac{m_{e}^2c^4  -2p_{\mathrm{ph}}c\left(E_e + p_{e}c\right)-m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\
+&= - \frac{1}{2c}\left[\frac{-2p_{\mathrm{ph}}\left(E_e + p_{e}c\right)}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\
+&= p_{\mathrm{ph}}\frac{E_e + p_{e}c}{E_e + 2p_{\mathrm{ph}}c - p_{e}c}\\
+\end{align}
+Then,
+\begin{align}
+p_{\mathrm{ph}}^{\prime} = \frac{E_e + p_{e}c}{E_e - p_{e}c + 2p_{\mathrm{ph}}c}p_{\mathrm{ph}}\\
+\end{align}
+=== Energy of the scattered photon $E_{\mathrm{ph}}^{\prime}$ (Solution B) ===
 From $E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c$,
 \begin{align}
@@ 行 173: / 行 202: @@
 \begin{align}
-m_e^2c^4  + p_e^{\prime 2}c^2= m_e^2c^4  + p_e^{\prime 2}c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2,\\
+m_e^2c^4  + p_e^{\prime 2}c^2= m_e^2c^4  + p_e^{\prime 2}c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2\\
+\Leftrightarrow \
 p_e^{\prime 2}c^2= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2.\\
 \end{align}
@@ 行 187: / 行 217: @@
 \begin{align}
 (p_e^2 + p_{\mathrm{ph}}^2 + p_{\mathrm{ph}}^{\prime 2} - 2 p_e p_{\mathrm{ph}} - 2 p_e p_{\mathrm{ph}}^{\prime} + 2 p_{\mathrm{ph}}p_{\mathrm{ph}}^{\prime})c^2
-&= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2,\\
+&= p_e^2c^2 + p_{\mathrm{ph}}^2c^2 + p_{\mathrm{ph}}^{\prime 2}c^2 + 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2\\
+\Leftrightarrow \
 - 2 p_e p_{\mathrm{ph}}c^2 - 2 p_e p_{\mathrm{ph}}^{\prime}c^2 + 2 p_{\mathrm{ph}}p_{\mathrm{ph}}^{\prime}c^2
-&= 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2,\\
+&= 2E_e p_{\mathrm{ph}}c - 2E_e p_{\mathrm{ph}}^{\prime}c -2p_{\mathrm{ph}} p_{\mathrm{ph}}^{\prime}c^2\\
-E_e p_{\mathrm{ph}}^{\prime}  -  p_e p_{\mathrm{ph}}^{\prime}c + 2 p_{\mathrm{ph}}p_{\mathrm{ph}}^{\prime}c &= E_e p_{\mathrm{ph}}  + p_e p_{\mathrm{ph}}c,\\
+\Leftrightarrow \
-\left[ E_e -  p_e c + 2 p_{\mathrm{ph}}c \right]p_{\mathrm{ph}}^{\prime} &= \left[E_e + p_e c\right]p_{\mathrm{ph}},\\
+E_e p_{\mathrm{ph}}^{\prime}  -  p_e p_{\mathrm{ph}}^{\prime}c + 2 p_{\mathrm{ph}}p_{\mathrm{ph}}^{\prime}c &= E_e p_{\mathrm{ph}}  + p_e p_{\mathrm{ph}}c\\
+\Leftrightarrow \
+\left[ E_e -  p_e c + 2 p_{\mathrm{ph}}c \right]p_{\mathrm{ph}}^{\prime} &= \left[E_e + p_e c\right]p_{\mathrm{ph}}\\
+\Rightarrow \
 p_{\mathrm{ph}}^{\prime} &=\frac{E_e + p_{e}c}{E_e - p_{e}c + 2p_{\mathrm{ph}}c}p_{\mathrm{ph}}\\
 \end{align}
@@ 行 207: / 行 241: @@
 \end{align}
-=== Energy of the scattered photon $E_{\mathrm{ph}}^{\prime}$ (Solution B)===
+=== Energy of the scattered photon $E_{\mathrm{ph}}^{\prime}$ (Solution C) ===
-By using the above $E_{e}^{\prime}$,
-\begin{align}
-p_{\mathrm{ph}}^{\prime}
-&= p_e - p_{\mathrm{ph}} -p_e^\prime \\
-&= p_e - p_{\mathrm{ph}} - \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)\right] \\
-&= - \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right) -2c\left(p_e - p_{\mathrm{ph}}\right) \right] \\
-&= - \frac{1}{2c}\left[\frac{m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} -\left(E_e + p_{e}c\right) \right] \\
-&= - \frac{1}{2c}\left[\frac{m_{e}^2c^4  -\left(E_e + p_{e}c\right)\left(E_e + 2p_{\mathrm{ph}}c - p_{e}c\right)}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\
-&= - \frac{1}{2c}\left[\frac{m_{e}^2c^4  -2p_{\mathrm{ph}}c\left(E_e + p_{e}c\right)-\left(E_e^2 - p_{e}^2c\right)}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\
-&= - \frac{1}{2c}\left[\frac{m_{e}^2c^4  -2p_{\mathrm{ph}}c\left(E_e + p_{e}c\right)-m_{e}^2c^4}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\
-&= - \frac{1}{2c}\left[\frac{-2p_{\mathrm{ph}}\left(E_e + p_{e}c\right)}{E_e + 2p_{\mathrm{ph}}c - p_{e}c} \right] \\
-&= p_{\mathrm{ph}}\frac{E_e + p_{e}c}{E_e + 2p_{\mathrm{ph}}c - p_{e}c}\\
-\end{align}
-Then,
-\begin{align}
-p_{\mathrm{ph}}^{\prime} = \frac{E_e + p_{e}c}{E_e - p_{e}c + 2p_{\mathrm{ph}}c}p_{\mathrm{ph}}\\
-\end{align}
-=== Energy of the scattered photon $E_{\mathrm{ph}}^{\prime}$ (Solution C)===
 From the general formula,
 \begin{align}
@@ 行 233: / 行 248: @@
 For head-on scattering,
 \begin{align}
-\theta_e+\theta_{\mathrm{ph}} &= \pi,\\
+\theta_e &= 0,\\
+\theta_{\mathrm{ph}} &= \pi,\\
 |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| &= p_e - p_{\mathrm{ph}},\\
 \theta_\mathrm{ph}^{\prime} &= 0.\\