A gamma ray with 700KeV energy is Compton-Scattered from an electron. Find the energy of the photon scattered at 110 degrees, the energy of the scattered electron, and the recoil angle of the electron.
To find the energy of the photon scattered at 110 degrees and the energy of the scattered electron, we can use the principles of Compton scattering.
Compton scattering is an inelastic scattering process between a photon and a charged particle, usually an electron. In this process, the incident photon transfers some of its energy and momentum to the electron, resulting in a scattered photon with reduced energy and a recoiling electron with increased energy.
The energy of the scattered photon, or the energy loss (∆E) of the incident photon in Compton scattering, can be calculated using the Compton wavelength shift equation:
∆E = E_initial - E_final = (h * c) / λ
Where:
- ∆E is the energy loss/scattered photon energy
- E_initial is the initial energy of the incident photon
- E_final is the final energy of the scattered photon
- h is the Planck's constant (6.62607015 × 10^-34 J·s)
- c is the speed of light (299,792,458 m/s)
- λ is the wavelength shift caused by the scattering
The wavelength shift λ can be calculated using the Compton scattering formula:
λ = λ_initial - λ_final = h / (m_e * c) * (1 - cos(θ))
Where:
- λ_initial is the initial wavelength of the incident photon
- λ_final is the final wavelength of the scattered photon
- m_e is the rest mass of an electron (9.10938356 × 10^-31 kg)
- c is the speed of light (299,792,458 m/s)
- θ is the scattering angle
To find the energy of the scattered electron, we can use the conservation of energy equation:
E_initial_photon = E_final_photon + E_scattered_electron
Given the incident photon's energy is 700KeV (700 * 10^3 eV), let's calculate the energy of the scattered photon and the scattered electron.
Step 1: Calculate the wavelength shift (λ)
λ_initial = (hc) / E_initial_photon
Substituting the known values:
λ_initial = (6.62607015 × 10^-34 J·s * 299,792,458 m/s) / (700 * 10^3 eV * (1.602176634 × 10^-19 J/eV))
Step 2: Calculate the final wavelength (λ_final)
λ_final = λ_initial - h / (m_e * c) * (1 - cos(θ))
Substituting the known values:
λ_final = λ_initial - (6.62607015 × 10^-34 J·s) / (9.10938356 × 10^-31 kg * 299,792,458 m/s) * (1 - cos(110 degrees))
Step 3: Calculate the energy loss and final energy of the scattered photon (∆E and E_final_photon)
∆E = (h * c) / λ_final
E_final_photon = E_initial_photon - ∆E
Substituting the known values in the equation:
∆E = (6.62607015 × 10^-34 J·s * 299,792,458 m/s) / λ_final
E_final_photon = E_initial_photon - ∆E
Step 4: Calculate the energy of the scattered electron (E_scattered_electron)
E_scattered_electron = E_initial_photon - E_final_photon
Now, let's calculate the values using the equations above.