Phy5646/Group5RelativisticProb

Practice with relativistic 4-vectors, presented by group #5: (Anthony Kuchera, Jeff Klatsky, Chelsey Morien)

Consider the collision of an energetic positron with an electron at rest in the laboratory frame. The collision is so violent that the electron-positron pair gets converted into a pair of muons. Compute the minimum kinetic energy of the positron in the laboratory frame for the reaction to proceed.

The reaction we want to study is: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{+} + e^{-} \rightarrow \mu^{+} + \mu^{-} }

In the center of mass frame, the total linear momentum is zero, which means the two muons that are produced are at rest.

The 4-momentum of the electron and positron in the COM frame are given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^{\mu}_{e^{+}} = (\frac{E}{c}, +p) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^{\mu}_{e^{-}} = (\frac{E}{c}, -p) }

By energy conservation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = m_e\gamma c^2 = m_\mu c^2 }

From this equation we find that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{m_\mu}{m_e} } . Recall that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma } is defined by: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{1}{\sqrt{1 - \beta ^2}} } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta = \frac{v}{c} } , where v is the velocity if the particle we are observing.

Now that we have derived this relation, we must shift back into the laboratory frame, in which the electron is at rest. Do do this we compute the 4-momentum in a reference frame moving with the electron velocity (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta } to the left). Define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^{\mu}_{e^{+}} } as the 4-momentum of the positron in the laboratory frame. Recall that in relativistic mechanics the following matrix (called the boost matrix) can be used to transform between frames of reference: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{ll} \gamma & \beta \gamma \\ \beta \gamma & \gamma \end{array}\right) }

Therefore we have:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^{\mu}_{e^{+}} = \left(\begin{array}{ll} \gamma & \beta \gamma \\ \beta \gamma & \gamma \end{array}\right) P_{e^+}^{\mu}} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{ll} \gamma & \beta \gamma \\ \beta \gamma & \gamma \end{array}\right) \left(\begin{array}{l} m_e\gamma c \\ m_e \gamma v \end{array}\right)} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_e\gamma c \left(\begin{array}{ll} \gamma & \beta \gamma \\ \beta \gamma & \gamma \end{array}\right) \left(\begin{array}{l} 1 \\ \beta \end{array}\right)} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_e\gamma c \left(\begin{array}{l} \gamma (1 + \beta ^2) \\ 2\beta \gamma \end{array}\right)}

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^{\mu}_{e^{+}} = \left(\begin{array}{l} E_{lab}^{min} \\ pc \end{array}\right)} we can write the equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{lab}^{min}= m_e c^2 \gamma ^2 (1+ \beta ^2) = m_e c^2 \gamma ^2 \frac{2\gamma ^2 - 1}{\gamma ^2} = m_e c^2 (2 \gamma^2 - 1) } .

Therefore the minimum kinetic energy is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{lab}^{min} = 2m_e c^2 \left[ \left( \frac{m_{\mu}}{m_e} \right)^2 - 1\right]}