To calculate the values of the characteristic wavelengths λKα and λSWL for an X-ray tube with a molybdenum (Mo) target and an acceleration potential of 50 keV, we can make use of two key formulas.
1. For the Kα line wavelength:
The Kα line wavelength (λKα) corresponds to the transition between the K-shell and the L-shell. This can be calculated using Moseley's law:
λKα = λK / (Z - eff)^2
where λK is the K-shell constant equal to 0.71 pm (picometers), Z is the atomic number of the target material, and eff is the effective nuclear charge.
For molybdenum, Z = 42. The effective nuclear charge (eff) is usually estimated by the formula:
eff = Z - σ
where σ is the screening constant. For molybdenum, σ is approximately 1.
Using these values, we can calculate λKα:
λKα = (0.71 pm) / (42 - 1)^2
2. For the shortest wavelength (λSWL):
The shortest wavelength (λSWL) corresponds to the highest energy X-rays emitted by the X-ray tube and can be calculated using the formula:
λSWL = hc / eV
where h is Planck's constant (6.626 x 10^-34 J s), c is the speed of light (3 x 10^8 m/s), and eV is the energy in electron volts.
Converting the acceleration potential of 50 keV to joules:
50 keV = 50 * 1000 eV = 50,000 eV = 50,000 * 1.602 x 10^-19 J
Now, we can calculate λSWL:
λSWL = (6.626 x 10^-34 J s * 3 x 10^8 m/s) / (50,000 * 1.602 x 10^-19 J)
Calculating these values will give you the accurate wavelengths λKα and λSWL in meters.