91Ó°ÊÓ

Using the Compton scattering formula, the change in the scattered photon wavelength is given by: \(\Delta \lambda = \lambda' - \lambda = \dfrac{h}{m_e c}(1 - \cos\theta)\) , where \(\lambda'\) is the scattered photon wavelength, \(\theta\) is the scattering angle, \(h\) is the Planck's constant, \(= 6.63 × 10^{-34}\,\text{Js}\), \(m_e\) is the electron mass, \(= 9.11 × 10^{-31}\,\text{kg}\) \(c\) is the speed of light, \(= 3 \times 10^8\,\text{m/s}\). We are given that the scattered photon energy is half the incident photon energy, so \(\dfrac{hc}{\lambda'} = \dfrac{1}{2}\dfrac{hc}{\lambda}\). Rearranging for the ratio of wavelengths, we have \(\dfrac{\lambda'}{\lambda} = 2\). At a scattering angle of \(90^\circ\), we plug in the values: \(\Delta \lambda = \dfrac{h}{m_e c}(1 - \cos{90^\circ}) = \dfrac{h}{m_e c}\). So, \(\lambda' = 2\lambda = \lambda + \dfrac{h}{m_e c}\). Now we can find the incident wavelength \(\lambda\): \(\lambda = \dfrac{h}{m_e c} = \dfrac{6.63 \times 10^{-34}}{9.11 \times 10^{-31} \times 3 \times 10^8} \,\text{m} \approx 2.43 \times 10^{-12}\,\text{m}\).

We now need to find the wavelength for a scattering angle of \(180^\circ\): \(\Delta \lambda_{180} = \dfrac{h}{m_e c}(1 - \cos{180^\circ}) = 2\dfrac{h}{m_e c}\). Thus, the backscattered wavelength \(\lambda_{180}' = \lambda + 2\dfrac{h}{m_e c} = 2\lambda + \dfrac{h}{m_e c}\). Plugging in the values for \(\lambda\), we get \(\lambda_{180}' = 2(2.43 \times 10^{-12}) + 2.43 \times 10^{-12} \,\text{m} = 7.29 \times 10^{-12}\,\text{m}\).

In the Compton scattering formula, there are no material-dependent factors. The scattering depends on the incident photon wavelength, scattering angle, and electron properties. As a result, there would be no change in the scattering pattern if a copper target were used instead of aluminum.