Question

Photons of energy \(E=4.000 \mathrm{keV}\) undergo Compton scattering. What is the largest possible change in photon energy, measured as a fraction of the incident photon's energy \(\left(E-E^{\prime}\right) / E ?\)

Short answer

Answer: The largest possible change in photon energy as a fraction of the incident photon energy after Compton scattering is approximately 1.46%.

Step-by-step solution

Recall the Compton Scattering Formula

The Compton scattering formula relates the initial photon energy \(E\), the scattered photon energy \(E'\), the rest mass energy of the electron \(m_ec^2\), and the scattering angle \(\theta\): \(E' = \frac{E}{1 + \frac{E}{m_ec^2}(1-\cos{\theta})}\) Here, \(E\) is the initial photon energy, \(E'\) is the scattered photon energy, \(m_e\) is the mass of the electron, \(c\) is the speed of light, and \(\theta\) is the scattering angle.

Determine the maximum energy change

To find the maximum energy change, we need to find the minimum scattered photon energy, \(E'_{min}\). This occurs when \(\theta = 180^{\circ}\) or \(\pi\) radians. Substitute this value into the Compton scattering formula: \(E'_{min} = \frac{E}{1 + \frac{E}{m_ec^2}(1-\cos{\pi})}\) Since \(\cos{\pi}=-1\), the formula simplifies to: \(E'_{min} = \frac{E}{1 + 2\frac{E}{m_ec^2}}\) Now, we can find the maximum energy change, \(E - E'_{min}\).

Calculate the energy change as a fraction of the initial energy

We are asked to find the largest possible change in photon energy as a fraction of the incident photon energy \(\left(E-E^{\prime}\right) / E\). We will now plug the values we have into this expression. Given that \(E = 4.000\,keV\) and \(m_ec^2 \approx 511\,keV\): \(\frac{\left( E - E'_{min} \right)}{E} = \frac{ E - \frac{E}{1 + 2\frac{E}{m_ec^2}}} {E} \) Next, divide both the numerator and denominator by \(E\): \(\frac{\left( E - E'_{min} \right)}{E} = \frac{1 - \frac{1}{1 + 2\frac{E}{m_ec^2}}} {1} \) Substitute the given values: \(\frac{\left( E - E'_{min} \right)}{E} = \frac{1 - \frac{1}{1 + 2\frac{4.000}{511}}} {1} \approx 0.0146 \) Thus, the largest possible change in photon energy, measured as a fraction of the incident photon's energy, is approximately \(1.46\%\).