差分
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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 00:29] – [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 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} | ||
