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Chapter 34: Problem 51

A 150 -pm X-ray photon Compton-scatters off an electron and emerges at \(135^{\circ}\) to its original direction. Find (a) the wavelength of the scattered photon and (b) the electron's kinetic energy.

Short Answer

Expert verified
The answers to the exercise are: (a) the scattered photon's wavelength (\(\lambda'\)) can be found using the Compton Scattering formula, and (b) the kinetic energy of the electron can be calculated using the difference between initial and final photon energies.

Step by step solution

01

Part (a): Compton Scattering Calculation

To begin, we need to use the Compton Scattering formula: \( \lambda' - \lambda = \frac{h}{m_ec}(1 - \cos \theta) \), where \(\lambda\) is the initial wavelength, \(\lambda'\) is the final wavelength, \(h\) is Planck鈥檚 constant, \(m_e\) is the electron mass, \(c\) is the speed of light, and \(\theta\) is the scattering angle. Substituting the given initial wavelength and scattering angle, and the known values of Planck's constant, speed of light, and electron mass, we can solve for \(\lambda'\).
02

Part (b): Kinetic Energy Calculation

The kinetic energy of the recoiling electron can be calculated using energy conservation. The initial photon energy \(E_{i} = \frac{hc}{\lambda}\) got transformed into the final photon energy \(E_{f} = \frac{hc}{\lambda'}\) and kinetic energy of the electron. So the electron's kinetic energy can be calculated as: \(KE = E_i - E_f\).

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Most popular questions from this chapter

A photon undergoes a \(90^{\circ}\) Compton scattering off a stationary electron, and the electron emerges with total energy \(\gamma m_{e} c^{2}\) where \(\gamma\) is the relativistic factor introduced in Chapter \(33 .\) Find an expression for the initial photon energy.

How might our everyday experience be different if Planck's constant had the value \(1 \mathrm{J} \cdot \mathrm{s} ?\)

Imagine an atom that, unlike hydrogen, had only three energy levels. If these levels were evenly spaced, how many spectral lines would result? How would their wavelengths compare?

How many spectral lines are in the entire Balmer series?

What would the constant in Equation 34.2 be if black body radiance were defined for fixed intervals of frequency rather than wavelength? (Hint: Use \(\lambda=c / f\) to express the radiance as \(R(f, T),\) then differentiate to find the maximum, and solve the resulting relation numerically. Express your answer in a form like Equations \(34.2 \mathrm{a}\) and \(\mathrm{b} .\) )
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