Moseley's law

Henry Moseley

The historic periodic table was roughly ordered by increasing atomic weight, but in a few famous cases the physical properties of two elements suggested that the heavier ought to precede the lighter. An example is cobalt having a weight of 58.9 and nickel having an atomic weight of 58.7.

Henry Moseley and other physicists used x-ray diffraction to study the elements, and the results of their experiments led to organizing the periodic table by proton count.

Since the spectral emissions for the heavier elements would be in the soft X-ray range (absorbed by air), the spectrometry apparatus had to be enclosed inside a vacuum.[4] Details of the experimental setup are documented in the journal articles "The High-Frequency Spectra of the Elements" Part I[1] and Part II.[2]

Moseley found that the ${\displaystyle K_{\alpha }}$ lines (in Siegbahn notation) were indeed related to the atomic number, Z.[2]

Following Bohr's lead, Moseley found that for the spectral lines, this relationship could be approximated by a simple formula, later called Moseley's Law.

${\displaystyle \nu =A\cdot \left(Z-b\right)^{2}}$[2]

where:

${\displaystyle \nu \ }$ is the frequency of the observed x-ray emission line

${\displaystyle A\ }$ and ${\displaystyle b\ }$ are constants that depend on the type of line (that is, K, L, etc. in x-ray notation)

${\displaystyle A=\left({\frac {1}{1^{2}}}-{\frac {1}{2^{2}}}\right)\cdot }$ Rydberg frequency and ${\displaystyle b\ }$= 1 for ${\displaystyle K_{\alpha }}$ lines, and ${\displaystyle A=\left({\frac {1}{2^{2}}}-{\frac {1}{3^{2}}}\right)\cdot }$ (Rydberg frequency) and ${\displaystyle b\ }$= 7.4 for ${\displaystyle L_{\alpha }}$ lines.[2]

Moseley derived his formula empirically by line fitting the square roots of the X-ray frequencies plotted by atomic number[2], and his formula could be explained in terms of the Bohr model of the atom.

${\displaystyle E=h\nu =E_{\text{i}}-E_{\text{f}}={\frac {m_{\text{e}}q_{\text{e}}^{2}q_{Z}^{2}}{8h^{2}\varepsilon _{0}^{2}}}\left({\frac {1}{n_{\text{f}}^{2}}}-{\frac {1}{n_{\text{i}}^{2}}}\right)\,}$

in which

${\displaystyle \varepsilon _{0}}$ is the permittivity of free space

${\displaystyle m_{\text{e}}}$ is the mass of an electron

${\displaystyle q_{\text{e}}}$ is the charge of an electron

${\displaystyle n_{\text{f}}}$ is the quantum number of final energy level

${\displaystyle n_{\text{i}}}$ is the quantum number of initial energy level

It is assumed that the final energy level is less than the initial energy level.

Considering the empirically found constant that approximately reduced (or apparently "screened") the energy of the charges, Bohr's formula for Moseley's ${\displaystyle K_{\alpha }}$ X-ray transitions became:

${\displaystyle E=h\nu =E_{\text{i}}-E_{\text{f}}={\frac {m_{\text{e}}q_{\text{e}}^{4}}{8h^{2}\varepsilon _{0}^{2}}}\left({\frac {1}{1^{2}}}-{\frac {1}{2^{2}}}\right)(Z-1)^{2}\,=\left({\frac {3}{4}}\right)(Z-1)^{2}\times 13.6\ \mathrm {eV} }$[2]

or (dividing both sides by h to convert E to ${\displaystyle \ \nu }$):

${\displaystyle \nu ={\frac {E}{h}}={\frac {m_{\text{e}}q_{\text{e}}^{4}}{8h^{3}\varepsilon _{0}^{2}}}\left({\frac {3}{4}}\right)(Z-1)^{2}=(2.47\cdot 10^{15}\ \mathrm {Hz} )(Z-1)^{2}\,}$

The coefficient in this formula simplifies to a frequency of 3/4h Ry, with an approximate value of 2.47×10¹⁵ Hz.

A simplified explanation for the effective charge of a nucleus being one less than its actual charge is that an unpaired electron in the K-shell screens it.[5][6] An elaborate discussion criticizing Moseley's interpretation of screening can be found in a paper by Whitaker[7] which is repeated in most modern texts.

A list of experimentally found X-ray transitions is available at NIST.[8] Theoretical energies can be computed to a much greater accuracy than Moseley's law using a method based on Dirac-Fock.[9]

• Auger electron spectroscopy, a similar phenomenon with increased x-ray yield from species of higher atomic number

• Oxford Physics Teaching - History Archive, "Exhibit 12 - Moseley's graph" (Reproduction of the original Moseley diagram showing the square root frequency dependence)