Characteristic X-rays are produced when an inner-shell vacancy in an atom is filled by an electron from a higher shell.
Because the energy difference between inner shells is large, the emitted radiation lies in the X-ray range. One of the
most important historical discoveries in atomic spectroscopy was Moseley’s law, which showed that the
frequencies of characteristic X-ray lines vary systematically with the atomic number \(Z\). This provided strong
evidence that atomic number, not atomic mass, is the fundamental ordering principle of the periodic table.
Moseley’s law
In its standard linear form, Moseley’s law is written as
Linear Moseley relation.
\[
\begin{aligned}
\sqrt{f} &= a\,(Z-b),
\end{aligned}
\]
where \(f\) is the frequency of the characteristic X-ray line, \(a\) is a line-dependent constant, and \(b\) is a
screening constant. The screening constant represents the fact that inner electrons partially shield the nucleus, so the
transitioning electron feels an effective charge smaller than the full nuclear charge \(Z\).
Squaring the equation gives
\[
\begin{aligned}
f &= a^2 (Z-b)^2.
\end{aligned}
\]
This immediately explains why a plot of \(\sqrt{f}\) versus \(Z\) is approximately a straight line. That is exactly the
pattern Moseley found experimentally.
K-series lines
The most common characteristic X-ray lines in an introductory treatment are the K-series lines, which terminate in the
\(n=1\) shell. Two especially important examples are:
| Line |
Transition |
Factor \(F\) |
Meaning |
| Kα |
\(2 \to 1\) |
\(\frac{3}{4}\) |
L shell to K shell |
| Kβ |
\(3 \to 1\) |
\(\frac{8}{9}\) |
M shell to K shell |
In a hydrogenic shell picture, these factors come from the energy-level expression
\[
\begin{aligned}
E_n &= -\frac{13.6\,Z_{\mathrm{eff}}^2}{n^2}\ \mathrm{eV},
\qquad
Z_{\mathrm{eff}} = Z-b.
\end{aligned}
\]
The energy difference between the two shells then becomes
\[
\begin{aligned}
\Delta E &= 13.6\,Z_{\mathrm{eff}}^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\ \mathrm{eV}.
\end{aligned}
\]
For Kα, with \(n_2=2\) and \(n_1=1\), the factor is \(1 - \frac{1}{4} = \frac{3}{4}\). For Kβ, with \(n_2=3\) and
\(n_1=1\), the factor is \(1 - \frac{1}{9} = \frac{8}{9}\). Converting energy into frequency using \(E = hf\) gives a
theoretical basis for the line-specific constant in Moseley’s law.
Frequency, wavelength, and photon energy
Once the frequency is known, two standard conversions follow immediately:
Conversions.
\[
\begin{aligned}
\lambda &= \frac{c}{f}, \\
E_\gamma &= hf.
\end{aligned}
\]
These relations allow you to express the same characteristic X-ray line as a frequency, a wavelength, or a photon
energy. In practice, X-ray spectroscopy commonly uses frequencies, wavelengths in nanometers or angstroms, and energies
in electron-volts or kilo-electron-volts.
Sample calculation: copper Kα
A standard example is copper with atomic number \(Z=29\). If we use the Kα line and a simple screening constant
\(b \approx 1\), then the effective charge is
\[
\begin{aligned}
Z_{\mathrm{eff}} &= 29 - 1 = 28.
\end{aligned}
\]
In the Kα case, the transition factor is \(\frac{3}{4}\), so the frequency is approximately
\[
\begin{aligned}
f &\approx cR_\infty \left(\frac{3}{4}\right)(28)^2.
\end{aligned}
\]
Numerically, this gives a characteristic frequency of about
\[
\begin{aligned}
f &\approx 1.9 \times 10^{18}\ \mathrm{Hz},
\end{aligned}
\]
which is in excellent agreement with the expected X-ray scale. The corresponding wavelength is around
\[
\begin{aligned}
\lambda &\approx 0.155\ \mathrm{nm},
\end{aligned}
\]
and the photon energy is about
\[
\begin{aligned}
E_\gamma &\approx 8\ \mathrm{keV}.
\end{aligned}
\]
This is why copper is such a common textbook example in characteristic X-ray spectroscopy.
Why Moseley’s law matters
Moseley’s law was historically important because it revealed a direct connection between spectral measurements and
nuclear charge. Instead of treating \(Z\) as just a label in the periodic table, the law showed that the X-ray spectra
depend on \(Z\) in a precise and predictable way. This helped settle ambiguities in the ordering of the elements and
supported the modern idea that atomic number determines chemical identity.
At a higher university level, one can refine the model by considering L-series and M-series lines, more accurate
shell-specific screening constants, relativistic corrections, and many-electron atomic structure. But the simple
K-series Moseley relation remains the most useful first approximation and the clearest introduction to characteristic
X-ray spectroscopy.
| Concept |
Main relation |
Meaning |
| Moseley law |
\(\sqrt{f} = a(Z-b)\) |
Linear relation between X-ray frequency and atomic number |
| Frequency form |
\(f = a^2(Z-b)^2\) |
Characteristic line frequency |
| Effective charge |
\(Z_{\mathrm{eff}} = Z-b\) |
Charge felt after screening |
| Kα factor |
\(\frac{3}{4}\) |
Transition \(2 \to 1\) |
| Kβ factor |
\(\frac{8}{9}\) |
Transition \(3 \to 1\) |
| Photon relations |
\(\lambda = c/f,\ E_\gamma = hf\) |
Convert frequency into wavelength and energy |