14.5: Totally Elastic Collision - Compton Scattering
- Page ID
- 17455
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As a final example of a collision in special relativity, we consider the totally elastic case: a collision in which the total momentum, total kinetic energy, and the mass of all particles are conserved. An example of such a collision is Compton scattering: the collision between a photon and an electron, resulting in a transfer of energy from one to the other, visible in a change of wavelength of the photon. For our example, we’ll take the electron to be initially stationary, and the photon to be coming in along the \(x\)-axis; after the collision, both particles have nonzero velocities in both the \(x\) and \(y\) directions (see Figure 14.4.1).
The four-momenta of the electron and photon before and after the collision are given by:
\[\overline{\boldsymbol{p}}_{\mathrm{e}, \mathrm{i}}=\left( \begin{array}{c}{m_{\mathrm{e}} c} \\ {0} \\ {0} \\ {0}\end{array}\right), \quad \overline{\boldsymbol{p}}_{\gamma, \mathrm{i}}=\frac{E_{\mathrm{i}}}{c} \left( \begin{array}{c}{1} \\ {1} \\ {0} \\ {0}\end{array}\right), \quad \overline{\boldsymbol{p}}_{\mathrm{e},\mathrm{f}}=\left( \begin{array}{c}{E_{\mathrm{e},\mathrm{f}} / c} \\ {p_{\mathrm{e}, \mathrm{f}} \cos \phi} \\ {p_{\mathrm{e}, \mathrm{f}} \sin \phi} \\ {0}\end{array}\right), \quad \overline{\boldsymbol{p}}_{\gamma, \mathrm{f}}=\frac{E_{\mathrm{f}}}{c} \left( \begin{array}{c}{1} \\ {\cos \theta} \\ {-\sin \theta} \\ {0}\end{array}\right) \label{14.4.1}\]
We can now solve for the energy \(E_{\mathrm{f}}\) of the outgoing photon (and thus its wavelength) in terms of that of the incoming photon (\(E_{\mathrm{i}}\)) and the scattering angle \(\theta\). There are again (at least) two ways to do this. One is to compare the components of the initial and final energy-momentum four-vector term by term. The other is to again use the fact that we know about the length of the four-vector to immediately eliminate the scattering angle \(\phi\) of the electron. To do so, we first rewrite the conservation of energy-momentum equation, \(\overline{\boldsymbol{p}}_{\mathrm{e}, \mathrm{i}}+\overline{\boldsymbol{p}}_{\gamma, \mathrm{i}}= \overline{\boldsymbol{p}}_{\mathrm{e}, \mathrm{f}}+\overline{\boldsymbol{p}}_{\gamma, \mathrm{f}}\) to isolate the term of the outgoing electron, and then take the square, to get:
\[\left(\overline{\boldsymbol{p}}_{\mathrm{e}, \mathrm{i}}+\overline{\boldsymbol{p}}_{\gamma, \mathrm{i}}-\overline{\boldsymbol{p}}_{\gamma, \mathrm{f}}\right)^{2}=\overline{\boldsymbol{p}}_{\mathrm{e}, \mathrm{f}}^{2} \label{14.4.2}\]
\[\overline{\boldsymbol{p}}_{\mathrm{e}, \mathrm{i}}^{2}+\overline{\boldsymbol{p}}_{\gamma, \mathrm{i}}^{2}+\overline{\boldsymbol{p}}_{\gamma, \mathrm{f}}^{2}+\overline{\boldsymbol{p}}_{\mathrm{e}, \mathrm{i}} \cdot \overline{\boldsymbol{p}}_{\gamma, \mathrm{i}}^{2}-2 \overline{\boldsymbol{p}}_{\mathrm{e}, \mathrm{i}} \cdot \overline{\boldsymbol{p}}_{\gamma, \mathrm{f}}-2 \overline{\boldsymbol{p}}_{\gamma, \mathrm{i}} \cdot \overline{\boldsymbol{p}}_{\gamma, \mathrm{f}}=\overline{\boldsymbol{p}}_{\mathrm{ef}}^{2}\label{14.4.3}\]
\[m_{e}^{2} c^{2}+0+0+2 m_{\mathrm{e}} E_{\mathrm{i}}-2 m_{\mathrm{e}} E_{\mathrm{f}}-2 \frac{E_{\mathrm{i}} E_{\mathrm{f}}}{c^{2}}(1-\cos \theta)=m_{e}^{2} c^{2}\label{14.4.4}\]
from which we can solve for \(E_{\mathrm{f}}\). Rewriting to wavelengths (through\(E=h f=h c / \lambda\)), we get
\[\lambda_{\mathrm{f}}=\lambda_{\mathrm{i}}+\frac{h}{m_{\mathrm{e}} c}(1-\cos \theta)\label{14.4.5}\]