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| 両方とも前のリビジョン前のリビジョン次のリビジョン | 前のリビジョン | ||
| research:memos:inverse_compton_scattering [2022/07/25 00:12] – [General formula] kobayash | research:memos:inverse_compton_scattering [2022/07/28 23:29] (現在) – [General formula] kobayash | ||
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| 行 6: | 行 6: | ||
| From the laws of conservation of the energy and the momentum, | From the laws of conservation of the energy and the momentum, | ||
| \begin{align} | \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}.\\ | \boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}} = \boldsymbol{p}_e^\prime + \boldsymbol{p}_{\mathrm{ph}}^{\prime}.\\ | ||
| \end{align} | \end{align} | ||
| 行 49: | 行 49: | ||
| 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}, | 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} | \end{align} | ||
| - | where $\theta_e$ ($\theta_{\mathrm{ph}}$) is an angle between $\boldsymbol{p}_e$ ($\boldsymbol{p}_{\mathrm{ph}}$) and $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}$, | + | where $\theta_e$ ($\theta_{\mathrm{ph}}$) is angle between $\boldsymbol{p}_e$ ($\boldsymbol{p}_{\mathrm{ph}}$) and $\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}$, |
| Then, from the law of energy conservation $E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c$, | Then, from the law of energy conservation $E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c$, | ||
| 行 59: | 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.\\ | 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} | \end{align} | ||
| - | By using $E^2 = m^2c^4 + p^2c^2$, | + | By using $E_e^2 = m_e^2c^4 + p_e^2c^2$ |
| \begin{align} | \begin{align} | ||
| - | m_e^2c^4 | + | m_e^2c^4 |
| \Leftrightarrow \ | \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.\\ | 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.\\ | ||
| 行 88: | 行 88: | ||
| \end{align} | \end{align} | ||
| - | If unit vectors $\widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}} = \boldsymbol{p}_{\mathrm{ph}}^{\prime}/ | + | If unit vectors $\widehat{\boldsymbol{p}_{\mathrm{ph}}^{\prime}} = \boldsymbol{p}_{\mathrm{ph}}^{\prime}/ |
| \begin{align} | \begin{align} | ||
| |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime} | |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| \cos \theta_{\mathrm{ph}}^{\prime} | ||
| 行 157: | 行 157: | ||
| By eliminating $p_{\mathrm{ph}}^{\prime}$ from the two equations, | By eliminating $p_{\mathrm{ph}}^{\prime}$ from the two equations, | ||
| \begin{align} | \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 + 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 | ||
| \end{align} | \end{align} | ||
| By making the square of both sides, | By making the square of both sides, | ||
| 行 166: | 行 167: | ||
| Then, | Then, | ||
| \begin{align} | \begin{align} | ||
| - | 2p_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.\\ | + | 2p_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\\ |
| - | | + | \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} | \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 | ||
| + | &= - \frac{1}{2c}\left[\frac{m_{e}^2c^4 | ||
| + | &= - \frac{1}{2c}\left[\frac{m_{e}^2c^4 | ||
| + | &= - \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 $E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c$, | From $E_e + p_{\mathrm{ph}}c = E_e^\prime + p_{\mathrm{ph}}^{\prime}c$, | ||
| \begin{align} | \begin{align} | ||
| 行 182: | 行 202: | ||
| \begin{align} | \begin{align} | ||
| - | m_e^2c^4 | + | m_e^2c^4 |
| + | \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.\\ | 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} | \end{align} | ||
| 行 196: | 行 217: | ||
| \begin{align} | \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^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 | - 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} | + | \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} |
| + | \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} & | p_{\mathrm{ph}}^{\prime} & | ||
| \end{align} | \end{align} | ||
| 行 216: | 行 241: | ||
| \end{align} | \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 | + | |
| - | &= - \frac{1}{2c}\left[\frac{m_{e}^2c^4 | + | |
| - | &= - \frac{1}{2c}\left[\frac{m_{e}^2c^4 | + | |
| - | &= - \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, | From the general formula, | ||
| \begin{align} | \begin{align} | ||
| 行 242: | 行 248: | ||
| For head-on scattering, | For head-on scattering, | ||
| \begin{align} | \begin{align} | ||
| - | \theta_e+\theta_{\mathrm{ph}} &= \pi,\\ | + | \theta_e |
| + | \theta_{\mathrm{ph}} &= \pi,\\ | ||
| |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| &= p_e - p_{\mathrm{ph}}, | |\boldsymbol{p}_e + \boldsymbol{p}_{\mathrm{ph}}| &= p_e - p_{\mathrm{ph}}, | ||
| \theta_\mathrm{ph}^{\prime} &= 0.\\ | \theta_\mathrm{ph}^{\prime} &= 0.\\ | ||
