Whitaker Correction

Whitaker, in 1999, published a paper in which the Bohr, hydrogenic model, and the Moseley model have been reconciled to one another and to our modern understanding of quantum mechanics. In our derivation of the hydrogenic model we solve the Schrodinger equation for the binding energy for the quantum number, n.

$$E_{n}=\frac{1}{n^{2}}Z^{2}\left(\frac{m e^{4}}{2 \hbar}\right)=\frac{1}{n^{2}}Z^{2}Er$$ (6)

The energy of the emitted photon is determined by the change in energy for a photon dropping from a looser bound state into a tight bound state.

$$E_{photon}=\Delta E=E_{n_{2}}-E_{n_{1}}=\left(\frac{1}{n_{2}^{2}}-\frac{1}{n_{1}^{2}}\right)Z^{2}Er$$ (7)

This leads to equation 1 in section 1.1.

Moseley introduced a shielding term inside the square. The shielding term depends on the energy level. Since according to quantum theory we are working with two different electron levels we in fact have two different shielding terms, not just one, as proposed by Moseley's work. Equation 7 becomes

$$E_{photon}=\left(\frac{1}{n_{2}^{2}}(Z-\phi_{2})^{2}-\frac{1}{n_{1}^{2}}(Z-\phi_{1})^{2}\right)Er$$ (8)

where $\phi_{1}$ and $\phi_{2}$ are shielding terms for electron states $n_{1}$ and $n_{2}$, and $\phi_{1}~=\phi_{2}$. In addition, these shielding terms not only account for shielding from electrons which lie between the nucleus and the electron state in question (interior shielding) but also effects from exterior electrons. It is demonstrated by Whitaker that indeed, neglecting the exterior electron shielding will result in flawed results. Whitaker derives the shielding conditions in section 6 of his paper. The derivation will not be discussed in this exam.

The conclusion of Whitaker's paper is that the energy of photons emitted due to a L to K transition (when $Z \geq 10$)

$$E_{KL}=\frac{1}{4}\alpha(3 Z^{2}-6 Z + 14)$$ (9)

where $\alpha$ is the energy of the Bohr ground state

$$\alpha=-\frac{1}{2}\alpha_{o}^{2}m c^{2}Z^{2}$$ (10)

and $\alpha_{o}$ is the fine-structure constant so $\alpha_{o}=\frac{1}{137}$.

He then asserts that we can calculate more accurate the $\sigma_{K}$ term, Moseley's $\sigma$ for the K transition, by

$$\sigma_{K} = Z -Q_{K}$$ (11)

for

$$Q_{K}=\left( (Z-1)^{2}+\frac{11}{3} \right)^{0.5}$$ (12)

Application of Whitaker's work for the K$\alpha$ series via the correction for $\sigma_{K}$ are found in figure 9.

Figure 9: Whitaker's Correction for K$\alpha$ X-rays

Whitaker's work appears to match Moseley's closely. Figure 10 shows the percent error of Whitaker's model as compared to Moseley's.

Figure 10: Comparison of Whitaker's and Moseley's K$\alpha$ Model to Measurement

In his paper, Whitaker shows that the error of the Moseley's law will continue to increase as Z increases, because of relativistic effects are not accounted for in Moseley's Law. Hence, for characteristic $K\alpha$ x-rays, Whitaker's modifications to Moseley's constant produced superior results for Z $>$ 25.

Whitaker give a correction term for the L series as well, for Z $>$ 30. As we see in figure 11, although his theory is sound, the results are, in his words, ``less startling than for the previous [K$\alpha$] case,'' but, ``Nevertheless, agreement is good enough for us to conclude that the model is basically correct.'' He gives no suggestions for why his model is less accurate than Moseley's aside from possible errors in outer shielding. Within the scope of this exam, I do not believe I can make a correction for this work.

Figure 11: Comparison of Whitaker's and Moseley's L Model to Measurement