Gamma rays of photon energy 0.511 MeV are directed onto an aluminum target and are scattered in various directions by loosely bound electrons there.a) What is the wavelength of the incident gamma rays?
b) What is the wavelength of the gamma rays scattered at 90 degrees to the incident beam?
c) What is the photon energy of the rays scattered in this direction?
To answer these questions, we need to use the formula for the energy of a photon:
E = hc/λ
where E is the photon's energy, λ is its wavelength, and h is Planck's constant (6.626 x 10^-34 J·s) and c is the speed of light (2.998 x 10^8 m/s).
a) What is the wavelength of the incident gamma rays?
Given: E = 0.511 MeV (convert into Joules: 1 MeV = 1.602 x 10^-13 J)
To find the wavelength (λ) of the incident gamma rays, we can rearrange the formula to solve for λ:
λ = hc/E
Substituting the values:
λ = (6.626 x 10^-34 J·s * 2.998 x 10^8 m/s) / (0.511 x 1.602 x 10^-13 J)
Calculating this expression will give us the wavelength of the incident gamma rays in meters.
b) What is the wavelength of the gamma rays scattered at 90 degrees to the incident beam?
The scattered gamma rays will experience a change in wavelength due to the Compton effect. However, the change in wavelength depends on the scattering angle and the initial energy of the gamma rays. In this case, the scattering angle is given as 90 degrees.
To calculate the wavelength of the scattered gamma rays at 90 degrees, we use the Compton scattering formula:
Δλ = λ' - λ = (h / mc) * (1 - cosθ)
where Δλ is the change in wavelength, λ' is the wavelength of the scattered gamma rays, λ is the wavelength of the incident gamma rays, θ is the scattering angle, h is Planck's constant, m is the mass of the electron, and c is the speed of light.
Given that the incident gamma rays are scattered at 90 degrees, we can substitute the values into the formula:
λ' = λ + Δλ = λ + (h / mc) * (1 - cos90)
Calculating this expression will give us the wavelength of the gamma rays scattered at 90 degrees.
c) What is the photon energy of the rays scattered in this direction?
After finding the wavelength (λ') of the gamma rays scattered at 90 degrees in part b), we can use the photon energy formula stated earlier to find their energy:
E' = hc/λ'
Substituting the values into the formula will give us the photon energy of the rays scattered at 90 degrees.
Note: It's important to ensure that all units are in the correct form (e.g., energy in Joules, wavelength in meters) to obtain accurate results.