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Calculation of X-ray wavelengths

If electrons are accelerated to a velocity v by a potential difference V and then allowed to collide with a metal target, the maximum frequency of the X-rays emitted is given by the equation:

mv2= eV = hf


X ray frequency (f) = eV/h

This shows that the maximum frequency is directly proportional to the accelerating voltage.

Example problem
Calculate the minimum wavelength of X-rays emitted when electrons accelerated through 30 kV strike a target.

f = [1.6 x 10-19 x 3 x 104]/ 6.63 x 10-34 = 7.2x1018 Hz

Therefore the wavelength l (= c/f) is 0.41 x 10-10 m = 0.04 1 nm (compared with some 600 nm for yellow light).

Moseley's law

In 1914 Moseley proposed a law showing how the X-ray frequency can be related to the proton (atomic) number Z of the target material. If f is the X-ray frequency, then:

X ray frequency (f) = k(Z- b)2

where k and b are constants, k having a value of 2.48x1015.

Plotting a graph of Z against √f will give a straight line as shown in the diagram, and in fact Moseley predicted the existence of elements 43, 61, 72 and 75 by the gaps that he found in his original version of the graph.

Electrons falling to the lowest level (or K-shell) in the atom from other excited levels give out X-rays in a series of wavelengths like an optical spectrum. This is known as the K-series, and individual lines are denoted by Ka, Kb and so on. Electron transitions ending on the second level are known as the L-series.

The following table shows the wavelengths of the Ka lines for some elements:

Element Proton number Wavelength (nm) Element Proton number Wavelength (nm)
Aluminium 13 0.823 Copper 29 0.139
Calcium 20 0.335 Bromine 35 0.104
Manganese 25 0.210 Silver 47 0.056
Iron 26 0.194 Tungsten 74 0.021
Cobalt 27 0.179 Uranium 92 0.017
Nickel 28 0.166      

© Keith Gibbs 2020