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Chapter 7: Problem 7

An experiment is to be designed to measure the differential cross-section for elastic pion-proton scattering at a CM scattering angle of \(70^{\circ}\) and a pion CM kinetic energy of \(490 \mathrm{keV}\). (The electron-volt \((\mathrm{eV})\) is the atomic unit of energy.) Find the angles in the Lab at which the scattered pions, and the recoiling protons, should be detected, and the required Lab kinetic energy of the pion beam. (The ratio of pion to proton mass is \(1 / 7\).)

Short Answer

Expert verified
Based on the given information and the calculations, determine the lab angles for scattered pions and recoiling protons, and the lab kinetic energy of the pion beam.

Step by step solution

01

Calculate the energy and momentum in the CM system

Using the given CM kinetic energy, we can find the energy of pions and protons in the CM system: $$E_{\pi, CM} = \frac{1}{2} m_{\pi} v_{\pi, CM}^2 \Rightarrow v_{\pi, CM} = \sqrt{\frac{2 E_{\pi, CM}}{m_{\pi}}}$$ $$E_{p, CM} = \frac{1}{2} m_{p} v_{p, CM}^2 \Rightarrow v_{p, CM} = \sqrt{\frac{2 E_{\pi, CM}}{m_{p}}}$$ Using the conservation of momentum, we have: $$m_\pi v_{\pi, CM} = m_p v_{p, CM}$$
02

Calculate the lab frame velocity of the center of mass

We can find the lab frame velocity of the center of mass using the mass ratio and the given CM kinetic energy as: $$v_{CM} = \frac{v_{\pi, CM} m_\pi}{m_\pi + m_p}$$
03

Find the final momentum of pions and protons in the lab frame

We have the final CM momenta of pions and protons given by: $$\vec{p}_{\pi, CM} = m_\pi v_{\pi, CM} \hat{p}_{\pi, CM}$$ $$\vec{p}_{p, CM} = m_p v_{p, CM} \hat{p}_{p, CM}$$ We now need to change the reference frame from CM to LAB. We'll use the Galilean transformation: $$\vec{p}_{\pi, LAB} = \vec{p}_{\pi, CM} + m_\pi \vec{v}_{CM}$$ $$\vec{p}_{p, LAB} = \vec{p}_{p, CM} + m_p \vec{v}_{CM}$$
04

Calculate the lab angles of the scattered pions and recoiling protons

Using the transformed momenta in the lab frame, we can find the angles \(\theta_{\pi, LAB}\) and \(\theta_{p, LAB}\): $$\cos{\theta_{\pi, LAB}} = \frac{\vec{p}_{\pi, LAB} \cdot \hat{p}_0}{|\vec{p}_{\pi, LAB}|}$$ $$\cos{\theta_{p, LAB}} = \frac{\vec{p}_{p, LAB} \cdot \hat{p}_0}{|\vec{p}_{p, LAB}|}$$ where \(\hat{p}_0\) is the unit vector in the direction of the initial pion momentum.
05

Determine the required lab kinetic energy of the pion beam

Finally, using the momentum in the lab frame for the pion, we can calculate the required lab kinetic energy of the pion beam: $$E_{\pi, LAB} = \frac{1}{2} m_\pi |\vec{p}_{\pi, LAB}|^2$$

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Most popular questions from this chapter

A double star is formed of two components, each with mass equal to that of the Sun. The distance between them is \(1 \mathrm{AU}\) (see Chapter 4 , Problem 2). What is the orbital period?

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