Hydrogenic Model

There are three important features that should be noticed in the plot of the error analysis of the Hydrogenic model, figure 4. These are the sign of the error, the magnitude of the error, and the change in error with change in atomic number. All three of these can be explained by electron shielding of the nucleus.

The sign of all error, for the hydrogenic model, is positive. This indicates that the calculate wavelengths are less than the actual wavelengths.

$$\lambda_{calculated} < \lambda_{measured}$$

because

$$\lambda = \frac{h c}{E}$$

and c and h are constant so

$$\frac{1}{E_{calculated}} < \frac{1}{E_{measured}}$$

which means

$$E_{measured} < E_{calculated}$$

This energy is actually the change in energy associated with an electron dropping from a loosely bound b state into a higher bound a state so

$$E_{b\ measured} - E_{a\ measured} < E_{b\ calculated} - E_{a\ calculated}$$

We assume that the hydrogenic model is more accurate in describing the binding energy of the inner most electron state so we can say that approximately

$$E_{a\ measured} = E_{a\ calculated}$$

Which means that

$$E_{b\ measured} < E_{b\ calculated}$$

The actually binding energy of state b is lower than the calculated binding energy. This is because the amount of positive charge, $Z_{effective}$, involved in the attractive forces on the electrons, is actually lower than the value used in the hydrogenic model, Z. As electrons are added to the atom, they act to shield one another from feeling the total positive charge in the nucleus.

The percent error for K$\alpha$ radiation is the smallest, K$\beta$ is the next largest, and L is the largest. Characteristic K$\alpha$ radiation is caused by electrons dropping from the 2p shell into the 1s shell. Assuming that interior electrons are the dominant contributors to shielding, the error introduced is from the electrons in the 1s and 2s orbitals, shielding the 2p shell. K$\beta$ radiation is caused by electrons in the 3p and 3d orbitals dropping into the 1s shell. The error is from the 1s, 2s, 2p, and 3s orbitals shielding the 3p and 3d orbitals. Hence, for k$\beta$, the effective Z, which binds the outer shell electrons, is reduced further and the error is greater. The L radiation comes from the electrons dropping from the 3s, 3p, and 3d shells into the 2p and 2s shells. The shielding error is introduced both at the initial electron state and the final.

The percent error decreases with increasing Z. This is because as the atomic number increases, the percent of the positive charge shielded decreases. Consider a simplistic view that each interior electron can shield one positive charge. So for an atom with and atomic number, Z, the electrons in the 2p orbital would not feel the attraction from Z but rather would feel $Z_{eff}$=Z=-2-2, because the 2 electrons in the 1s and the 2 electrons in the 2s orbitals would block a total of 4 positive charges. Hence as Z increase so does $Z_{eff}$ and as $Z->\inf$, $Z_{eff}->Z$. As the atomic number increases, the effect of electron shielding, from the interior electron shells, decreases and the hydrogenic model becomes more accurate.

In this this section a rather naive picture of the shielding was presented. In section 3 of this exam a more correct view is presented.