Assuming that the equation
Ephoton, final =
Ephoton, initial/1 + Ephoton, initial/mec2
(1 − cos θ)
can be approximately applied at all scattering angles θ, what is the minimum energy of a scattered X-ray photon if the incident X-ray energy is 76 keV? What is the largest possible energy of a scattered X-ray emitted at a scattering angle of 90°?
(a) What is the minimum energy of a scattered X-ray photon if the incident X-ray energy is 76 keV?
b) What is the largest possible energy of a scattered X-ray emitted at a scattering angle of 90°?
bji
To find the minimum energy of a scattered X-ray photon, we can use the given equation:
Ephoton, final = Ephoton, initial / (1 + (Ephoton, initial / mec^2) * (1 - cosθ))
For part (a), we need to determine the minimum energy when the incident X-ray energy is 76 keV. We can substitute the values into the equation:
Ephoton, initial = 76 keV
θ = 0° (minimum scattering angle)
Since the minimum scattering angle is 0°, the cosine of 0° is equal to 1. The equation becomes:
Ephoton, final = 76 keV / (1 + (76 keV / mec^2) * (1 - cos0°))
Now, we need to substitute the value of mec^2. mec^2 is the rest mass energy of an electron, which is approximately 0.511 MeV (Mega-electron volts).
Ephoton, final = 76 keV / (1 + (76 keV / 0.511 MeV) * (1 - 1))
Next, we can simplify the equation:
Ephoton, final = 76 keV / (1 + (76/0.511) * 0)
Ephoton, final = 76 keV / (1 + 149.11 * 0)
Ephoton, final = 76 keV / (1 + 0)
Ephoton, final = 76 keV
Therefore, the minimum energy of a scattered X-ray photon is also 76 keV when the incident X-ray energy is 76 keV.
Moving on to part (b), we need to determine the largest possible energy of a scattered X-ray emitted at a scattering angle of 90°. We can again use the given equation:
Ephoton, final = Ephoton, initial / (1 + (Ephoton, initial / mec^2) * (1 - cosθ))
This time, we need to find the largest value when θ = 90°.
Ephoton, initial = 76 keV
θ = 90° (maximum scattering angle)
The cosine of 90° is equal to 0. Substituting the values into the equation:
Ephoton, final = 76 keV / (1 + (76 keV / mec^2) * (1 - cos90°))
Again, we substitute the value of mec^2:
Ephoton, final = 76 keV / (1 + (76 keV / 0.511 MeV) * (1 - 0))
Simplifying the equation:
Ephoton, final = 76 keV / (1 + (149.11) * (1))
Ephoton, final = 76 keV / (1 + 149.11)
Ephoton, final = 76 keV / 150.11
Calculating this expression results in:
Ephoton, final ≈ 0.506 keV
Therefore, the largest possible energy of a scattered X-ray emitted at a scattering angle of 90° is approximately 0.506 keV.