Exercise: The Thomas-Fermi Atom

The applet models stationary states for the valence electron(s) in heavy atoms. The choice of Flex Units here means the scales for distance (d) and energy (E) can be adapted to the application at hand, but always are related as E·d² = ℏ>²÷2m. With d = a₀ = 0.529 Å (1 bohr), E becomes the Rydberg energy, 1 Ry = 13.6 eV, when m is the electron mass. The valence electrons of multi-electron atoms move in the field of a screened Coulomb potential. For heavy atoms, screening effects are incorporated using the Thomas-Fermi model; in Flex Units the model potential becomes V(r) = –2Zexp(–r/a)÷r. To avoid the singularity at r = 0, the applet uses a truncated Thomas-Fermi potential – see 1st Technical Note below. V(r) and its associated parameters for the outer electron in a gold atom are recorded on the Math tab of the applet. On the tab labeled Graphics: [r], this potential is plotted over the interval [0, 5.0 bohrs]. The listing to the right of the graph includes a placeholder for the radial wave representing the 6s valence electron in gold.

Radial waves in the applet are indexed simply by the order in which they first appear, which differs from the conventional (nlm) labeling of hydrogen-like states.

In this exercise we will find the energy of the 6s valence electron in gold and explore the wavefunction to arrive at a figure for the size of the gold atom.

1. Show the s-wave – designated g₀ in the applet – by right-clicking its placeholder in the list, clicking on the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing the OK button. The displayed waveform has no readily-discernible discontinuity, but in this case appearances are deceiving. Go to the Math tab, right-click anywhere in the value field for E₀, the energy for this s-wave, and select "Edit Parameter..." from the popup menu to activate the Energy Editor. Note the sizeable wave mismatch in the tolerance field, labeled δψ. [The wave also exhibits a clear discontinuity now because opening the editor automatically resets the match point for improved accuracy – see 2nd Technical Note below.] Reliable results require any wave mismatch to be much less than unity, so we conclude that the energy of this state (−9 Ry) is not one of the allowed energies for the electron in gold.
2. Adjust the energy of this s-wave upward (toward less negative values) to eliminate the discontinuity. Use the editor's slider to manipulate the highlighted digits in the energy field (labeled E). 'Fine tuning' is accomplished by adjusting the number of highlighted digits using the arrows to the immediate right of the energy field. The reset button () at the lower right changes the point of discontinuity, and should be used when nearing the true energy (as evidenced by a wave mismatch at the 10⁻⁴ level or below). At the 10⁻⁶ tolerance level, the energy is effectively 'allowed' and the wavefunction is one of the s-type radial waves for the electron in gold. Count the number of nodes to see which state you have found. The 6s radial wave has 5 nodes (not counting the node at the origin); if yours has fewer, continue searching at higher energies. Finish by selecting OK to end the edit session with the current settings.
   
   
   Extra digits can be added before or after the decimal by typing directly in the editor energy field, then pressing the Enter key. The sum of leading and trailing digits is limited to 9.
   
   
3. Find the most probable distance of the 6s electron from the gold nucleus. The 6s radial wave displayed in the applet is actually the effective one-dimensional matter wave g(r) = rR(r) that directly furnishes the radial probability density as P(r) = |g(r)|². Thus, the most probable location of the 6s electron is that value of r that maximizes |g₀(r)|. Locate this distance by right-clicking anywhere on the graph, selecting "Trace On/Off" from the popup menu, and surveying the results. Since the 6s electron is the outermost electron in gold, it is reasonable to equate the most probable distance of the 6s electron from the gold nucleus with the size of the gold atom. Compare this to the size of the hydrogen atom, using the same rule.

Technical Notes

• The truncated Thomas-Fermi potential used in the applet has the Thomas-Fermi form down to r = b and is constant therafter. The difference is inconsequential provided b is small. The default (b = 0.0001 bohr) is on the order of the nuclear size, below which the Thomas-Fermi form would not be expected to be accurate anyway.
• The Numerov method is used to construct stationary waves for a given (trial) energy. Using boundary conditions derived from the correct asymptotic form at r = 0, and then again at the interval endpoint, the Schrödinger equation is integrated inward to a preset match point. The right-side wave is then scaled to make its slope agree with that of the left-side wave at the match point. If the trial energy is allowed, the wavefunction values also will agree at the match point; otherwise, a discontinuity results. The [fractional] discontinuity is recorded in the editor tolerance field, labeled δψ. The technique is most accurate when the match point coincides with an extremum of the wavefunction. Each press of the reset button re-positions the match point to an extremum of the current waveform.