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Chapter 39: Problem 30

Some particle detectors measure the total number of particles integrated over part of a sphere of radius \(R\) from the target. Assuming symmetry about the axis of the incoming particle beam, use the Rutherford scattering formula to obtain the total number of particles detected as a function of the scattering angle \(\theta\).

Short Answer

Expert verified
Answer: The total number of particles detected is determined by integrating the Rutherford scattering formula over the azimuthal and scattering angles and multiplying the resulting total cross-section with the number of incident particles. The relationship between the total number of particles detected and the maximum scattering angle is given by: \(N_\text{detected} = N_\text{incident} \int_{\theta_\text{min}}^{\theta_\text{max}} \frac{2\pi k^2 \sin \theta}{R^2 \sin^4(\theta / 2)} d\theta \)

Step by step solution

01

Convert the differential cross-section formula into polar coordinates

To make the integration easier, we need to convert the Rutherford scattering formula from Cartesian coordinates to polar coordinates. Recall that: \(d\Omega = \sin \theta d\theta d\phi\) where \(\theta\) and \(\phi\) are polar and azimuthal angles, respectively. Substitute this into the Rutherford scattering formula, we get: \(d\sigma = \frac{k^2 \sin \theta}{R^2 \sin^4(\theta / 2)} d\theta d\phi\)
02

Integrate the cross-section formula over the azimuthal angle

Now we need to integrate the Rutherford scattering formula over the azimuthal angle \(\phi\), which goes from \(0\) to \(2\pi\). So: \(\int_{0}^{2\pi} d\sigma d\phi = \int_{0}^{2\pi} \frac{k^2 \sin \theta}{R^2 \sin^4(\theta / 2) } d\theta d\phi\) First, integrate with respect to \(\phi\): \(d\sigma_\theta = \frac{2\pi k^2 \sin \theta}{R^2 \sin^4(\theta / 2)} d\theta\)
03

Integrate the cross-section formula over the scattering angle

Now, we need to integrate the cross-section formula over the scattering angle \(\theta\), which goes from the minimum scattering angle \(\theta_\text{min}\) to the maximum scattering angle \(\theta_\text{max}\). So: \(\Sigma(\theta_\text{max}) = \int_{\theta_\text{min}}^{\theta_\text{max}} \frac{2\pi k^2 \sin \theta}{R^2 \sin^4(\theta / 2)} d\theta \)
04

Find the total number of particles detected

To find the total number of particles detected, we need to multiply the total cross-section with the number of incident particles \(N_\text{incident}\): \(N_\text{detected} = N_\text{incident} \Sigma(\theta_\text{max})\) This equation gives the total number of particles detected as a function of the maximum scattering angle \(\theta_\text{max}\).

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Most popular questions from this chapter

A Geiger-Marsden experiment, where \(\alpha\) particles are scattered off of a thin gold film, yields an intensity of particles of \(I\left(90^{\circ}\right)=100 .\) counts/s at a scattering angle of \(90^{\circ} \pm 1^{\circ} .\) What will be the intensity of particles at a scattering angle of \(60^{\circ} \pm 1^{\circ}\) if the scattering obeys the Rutherford formula?

Which of the following formed latest in the universe? a) quarks b) protons and neutrons c) hydrogen atoms d) helium nuclei e) gluons

The text describes and sketches the basic Feynman diagram for the fundamental process involved in the decay of the free neutron: One of the neutron's \(d\) -quarks converts to a \(u\) -quark, emitting a virtual \(W^{-}\) boson, which decays into an electron and an electron anti-neutrino (the only decay energetically possible). Similarly describe and sketch the basic (tree-level) Feynman diagram for the fundamental process involved in each of the following decays: a) \(\mu^{-} \rightarrow e^{-}+\nu_{\mu}+\bar{\nu}_{e}\) b) \(\tau^{-} \rightarrow \pi^{-}+\nu_{\tau}\) c) \(\Delta^{++} \rightarrow p+\pi^{+}\) d) \(K^{+} \rightarrow \mu^{+}+\nu_{\mu}\) e) \(\Lambda \rightarrow p+\pi\)

Consider a hypothetical force mediated by the exchange of bosons that have the same mass as protons. Approximately what would be the maximum range of such a force? You may assume that the total energy of these particles is simply the rest-mass energy and that they travel close to the speed of light. If you do not make these assumptions and instead use the relativistic expression for total energy, what happens to your estimate of the maximum range of the force?

Does the proposed decay \(n \rightarrow p+\pi^{-}\) violate any conservation rules?
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