If a 6.0keV photon scatters from a fee proton at rest, what is the change in the photon's wavelength if the photon recoils at 90 degrees?
To determine the change in the photon's wavelength when it scatters from a free proton, we can use the concept of Compton scattering. Compton scattering is an inelastic scattering phenomenon involving the interaction of photons with charged particles, such as electrons or protons.
The change in wavelength (Δλ) can be calculated using the following formula:
Δλ = λ' - λ
where λ' is the final wavelength of the scattered photon and λ is the initial wavelength of the incident photon.
In Compton scattering, the change in wavelength is related to the scattering angle (θ) and the rest mass of the particle (m) by the Compton wavelength shift equation:
Δλ = λc * (1 - cos(θ))
where λc = h / (m * c) is the Compton wavelength, h is Planck's constant, and c is the speed of light.
For a photon scattering from a free proton, we can approximate the scattering event as an elastic collision, taking into account the conservation of energy and momentum. In this case, the energy of the photon is given by:
E = (hc) / λ
where h is Planck's constant and c is the speed of light.
By conservation of energy, the energy of the scattered photon (E') can be calculated using the equation:
E' = E / (1 + (E / (m * c^2)) * (1 - cos(θ)))
In this equation, m is the rest mass of the proton.
Finally, we can calculate the final wavelength (λ') of the scattered photon using the relation:
λ' = (hc) / E'
Now, we can substitute the known values into the equations above and calculate the change in the photon's wavelength.
Given:
Initial photon energy, E = 6.0 keV = 6.0 x 10^3 eV
Scattering angle, θ = 90 degrees
Rest mass of the proton, m = 938.27 MeV/c^2
First, we convert the initial energy from eV to joules:
E = 6.0 x 10^3 eV = 6.0 x 10^3 x 1.6 x 10^-19 J ≈ 9.6 x 10^-16 J
Next, we calculate the Compton wavelength:
λc ≈ h / (m * c) = (6.626 x 10^-34 J s) / (938.27 x 10^6 eV / c^2) ≈ 2.43 x 10^-12 m
Now, we can substitute the values into the formula:
Δλ = λc * (1 - cos(θ))
= (2.43 x 10^-12 m) * (1 - cos(90°))
= 2.43 x 10^-12 m
Therefore, the change in the photon's wavelength when it scatters from a free proton at rest, recoiling at 90 degrees, is approximately 2.43 x 10^-12 meters.