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?

Review Compton scattering.

You will find the equation that you need there.

To answer these questions, we can use the principles of Compton scattering, which describes the interaction between gamma rays and electrons.

a) To find the wavelength of the incident gamma rays, we can use the equation that relates the energy (E) of a photon to its wavelength (λ):

E = h*c/λ

Where:
- E is the energy of the photon (0.511 MeV in this case)
- h is Planck's constant (6.626 x 10^-34 J⋅s)
- c is the speed of light (3 x 10^8 m/s)

Rearranging the equation, we can solve for λ:

λ = h*c/E

Substituting the values, we have:

λ = (6.626 x 10^-34 J⋅s * 3 x 10^8 m/s) / (0.511 x 10^6 eV * 1.6 x 10^-19 J/eV)

λ ≈ 2.43 x 10^-12 meters (or 2.43 picometers)

b) To find the wavelength of the gamma rays scattered at 90 degrees (θ = 90°) to the incident beam, we can use the equation derived from Compton scattering:

λ' = λ + (h / (m_e * c)) * (1 - cos(θ))

Where:
- λ' is the wavelength of the scattered gamma rays
- λ is the wavelength of the incident gamma rays (from part a)
- h is Planck's constant
- m_e is the electron mass (9.109 x 10^-31 kg)
- c is the speed of light
- θ is the scattering angle (90 degrees in this case)

Substituting the known values, we have:

λ' = 2.43 x 10^-12 m + (6.626 x 10^-34 J⋅s / (9.109 x 10^-31 kg * 3 x 10^8 m/s)) * (1 - cos(90°))

λ' ≈ 2.42 x 10^-12 meters (or 2.42 picometers)

c) To find the photon energy of the rays scattered at 90 degrees, we can use the energy-wavelength relation mentioned earlier:

E' = h*c/λ'

Using the value of λ' from part b, we have:

E' = (6.626 x 10^-34 J⋅s * 3 x 10^8 m/s) / (2.42 x 10^-12 m)

E' ≈ 0.513 MeV

Therefore, the photon energy of the gamma rays scattered at 90 degrees is approximately 0.513 MeV.