Moseley's law

Moseley's law is an empirical law concerning the characteristic electromagnetic spectrum that is emitted or absorbed by atoms. It is historically important as being on equal footing with Rutherford and Bohr's early work in quantitatively justifying the conception of the nuclear model of the atom, with all or nearly all positive charges of the atom located in the nucleus, and associated on an integer basis with atomic number.

History

Using x-ray diffraction techniques in the 1910s, Henry Moseley found that the most intense short-wavelength line in the x-ray spectrum of a particular element was related to the element's atomic number, $Z$. Moseley found that this relationship could be expressed by a simple formula, later called Moseley's Law.

${\sqrt {f}}=k_{1}\cdot (Z-k_{2})$

where:

$f\$ is the frequency of the main x-ray emission line

$k_{1}\$ and $k_{2}\$ are constants that depend on the type of line

For example, the values for $k_{1}\$ and $k_{2}\$ are the same for all $K_{\alpha }$ lines (in Siegbahn notation), so the formula can be rewritten thus:

$f=(2.48*10^{15})\cdot (Z-1)^{2}$ Hz

Derivation and justification from the Bohr model of the Rutherford nuclear atom

Moseley derived his formula empirically by plotting the square root of X-ray frequencies against a line representing atomic number. However, it was immediately noted that his formula could be explained in terms of the newly postulated 1913 Bohr model of the atom (see for details of derivation of this for hydrogen), if certain reasonable extra assumptions about atomic structure in other elements were made.

The 19th century empirically-derived Rydberg formula for spectroscopists is explained in the Bohr model as describing the transitions or quantum jumps between one energy level and another in a hydrogen atom. When the electron moves from one energy level to another, a photon is given off. Using the derived formula for the different 'energy' levels of hydrogen one may determine the energy or frequencies of light that a hydrogen atom can emit.

The energy of photons that a hydrogen atom can emit in the Bohr derivation of the Rydberg formula, is given by the difference of any two hydrogen energy levels:

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

$m_{e}\,$ = mass of an electron

$q_{e}\,$ = charge of an electron (1.60 × 10⁻¹⁹ Coulombs)

$n_{f}\,$ = final energy level

$n_{i}\,$ = initial energy level

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

For hydrogen, the quantity $q_{e}^{2}q_{Z}^{2}=q_{e}^{4}\,$ because Z (the nuclear positive charge, in fundamental units of the electron charge q_e ) is equal to 1. That is, the hydrogen nucleus contains a single charge. However, for hydrogenic atoms (those in which the electron acts as though it circles a single structure with effective charge Z), Bohr realized from his derivation that an extra quantity Z² would need to be added to the conventional q_e⁴ , in order to account for the extra pull on the electron, and thus the extra energy between levels, as a result of the increased nuclear charge.

Moseley and Bohr realized that Moseley's formula could be adapted from Bohr's, if two assumptions were made. The first was that the electron responsible for the brightest spectral line (K-alpha) which Moseley was investigating from each element, results from a transition between the K and L shells of the atom (i.e., from the nearest to the nucleus and the one next farthest out), with energy quantum numbers corresponding to 1 and 2. Finally, the Z in Bohr's formula, though still squared, required diminishment by 1 to calculate K-alpha, apparently to account for the screening effect on full nuclear charge Z by the remaining electron left in the K shell (later this would be seen as a single 1s electron). With this assumption, it was the quantity (Z-1) which required squaring. Thus, Bohr's formula for Moseley's K-alpha X-ray transitions became:

$E=h\nu =E_{i}-E_{f}={\frac {m_{e}q_{e}^{4}(Z-1)^{2}}{8h^{2}\epsilon _{0}^{2}}}\left({\frac {1}{1^{2}}}-{\frac {1}{2^{2}}}\right)\,$

or (dividing both sides by h to convert E to f):

$f=\nu ={\frac {m_{e}q_{e}^{4}}{8h^{3}\epsilon _{0}^{2}}}\left({\frac {3}{4}}\right)(Z-1)^{2}=(2.48*10^{15}Hz)(Z-1)^{2}\,$

Collection of the constants in this formula into a single frequency scaling-constant gives a frequency equivalent to about 3/4 of the 13.6 eV ionization energy (see Rydberg constant for hydrogen = 3.28 x 10¹⁵ Hz), with the final value of 2.46 x 10¹⁵ Hz in good agreement with Moseley's empirically-derived value of 2.48 x 10¹⁵ Hz. This fundamental frequency is the same as that of the hydrogen Lyman-alpha line, because the 1s to 2p transition in hydrogen is responsible for both Lyman-alpha line in hydrogen, and also the K-alpha lines in X-ray spectroscopy for elements beyond hydrogen, which are described by Moseley's law. Moseley was fully aware that his fundamental frequency was Lyman-alpha, the fundamental Rydberg frequency differing by the Rydberg-Bohr factor of 3/4 (see his original papers below).

Historical importance

See the biographical article on Henry Moseley for more. Moseley's formula, by Bohr's later account, not only established atomic number as a measurable experimental quantity, but gave it a physical meaning as the positive charge on the atomic nucleus (number of protons). This in turn was able to produce quantitative predictions for spectral lines in keeping with the Bohr/Rutherford semi-quantum model of the atom, which assumed that all positive charge was concentrated at the center of the atom, and that all spectral lines result from changes in total energy of electrons circling it as they move from one permitted level of angular momentum and energy to another. The fact that Bohr's model of the energies in the atom could be made to calculate X-ray spectral lines from aluminum to gold in the periodic table, and that these depended reliably and quantitatively on atomic number, did a great deal for the acceptance of the Rutherford and Bohr view of the structure of the atom. When later quantum theory essentially also recovered Bohr's formula for energy of spectral lines, Moseley's law became incorporated into the full quantum mechanical view of the atom, including the role of the single 1s electron which remains in the K shell of all atoms after another K electron is ejected, according to the Schroedinger equation prediction.

References

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