An X-ray photon scatters from a free electron with negligible initial kinetic energy at an angle of 121.2 degrees relative to the incident direction.

 
(Take the Compton wavelength of the electron to be 2.43 pm, and all other constants/conversions to three accepted sig digs.)
 
a)  If the scattered photon has a wavelength of 0.0690 nm, what is the wavelength of the incident photon?  (in nm)

b)  What is the kinetic energy of the recoil electron in electron volts?
 
(Because the Compton shift is so small, make sure to retain at least 4 sig digs in intermediate .)

c)  What is the speed of the recoil electron expressed as a percentage of the speed of light?
 
(Because the Compton shift is so small, make sure to retain at least 4 sig digs in intermediate calcs.)

To answer these questions, we will use the principles of Compton scattering. Compton scattering is the interaction between a photon and an electron, resulting in a change in the photon's wavelength and momentum. In this case, we are given the angle of scattering and the wavelength of the scattered photon, and we need to find the wavelength of the incident photon, the kinetic energy of the recoil electron, and the speed of the recoil electron.

a) To find the wavelength of the incident photon, we can use the Compton scattering formula:

Δλ = λ' - λ = λc(1 - cosθ)

Where Δλ is the change in wavelength, λ' is the wavelength of the scattered photon, λc is the Compton wavelength of the electron, and θ is the scattering angle.

Plugging in the given values:
Δλ = 0.0690 nm - λ = (2.43 pm)(1 - cos121.2°)

To convert the Compton wavelength to nm, we divide it by 10:
λ = 0.0690 nm - λ = (0.243 nm)(1 - cos121.2°)

Now we can solve for λ:
λ = 0.0690 nm - λ = (0.243 nm)(1 - cos121.2°)
0.9310 nm = 0.243 nm - λcos121.2°
λcos121.2° = 0.243 nm - 0.9310 nm = -0.6880 nm
λ = -0.6880 nm / cos121.2°

Calculating this gives us the wavelength of the incident photon in nm.

b) To find the kinetic energy of the recoil electron, we can use the energy-momentum conservation in Compton scattering:

hv + mc² = h'v' + √((pc)² + (mc²)²)

Where h is the Planck's constant, v is the frequency of the incident photon, m is the electron's mass, c is the speed of light, h' is the Planck's constant, v' is the frequency of the scattered photon, p is the momentum of the recoil electron.

The momentum and energy of a photon are related by E = pc, and the energy of a photon is given by E = hv. We can substitute these relations into the conservation equation and solve for the recoil electron's kinetic energy.

c) To find the speed of the recoil electron expressed as a percentage of the speed of light, we divide the speed of the recoil electron by the speed of light and multiply by 100.

Using these methods, we can find the answers to all the three questions.