Rydberg Atoms and the Quantum Defect

A Rydberg atom is an atom with a single valence electron in a state with a very large principal quantum number n. The many core electrons in a Rydberg atom effectively shield these valence electrons from the electric field of the nucleus. The outer electron generally "sees" a nucleus with only one proton and will behave much like the electron of a Hydrogen atom. High energy levels in the Hydrogen atom can be modeled by the Rydberg equation:

, Where the Rydberg constant R = 1.097*10⁵ cm^-1. 

E_n is the energy above the ground state. T is the ionization limit.

Rydberg atoms will closely fit this equation, but they will deviate from the relation because they do not have circular orbits.  Orbits of electrons in a high n-state will pass through the core of shielding electrons, so the electron will occasionally “see” the whole nucleus. To adjust the relation for this penetration of the inner core electrons, we introduce a correction term called the "quantum defect."  The Rydberg relation is modified to include the quantum defect(s) of the element being studied;

, where d is the quantum defect.

The quantum defect is different for different angular momentum states. For S states, having zero angular momentum, the quantum defect is roughly on the order of 5 to 7. An electron with no angular momentum essentially passes clean through the core of shielding electrons.  For low angular momentum states, the shielding effect discussed earlier is diminished.  Since shielding produces the similarity to Hydrogen atoms required for the Rydberg relation, the high quantum defect accounts for that.  For D states, which have higher angular momentum, the quantum defect should be significantly smaller because the shielding effect is maintained.  Its orbit will not pass through the core so deeply.

Using the ionization spectrum to calculate energy levels of various transitions, Kverno and Nolen were able to calculate the quantum defect of D and S states in Cesium.  The statistical fluctuation in the quantum defects is due to a systematic error in the data.

For the s-states, the quantum defect is in a narrow range, hovering around 5.10.  The quantum defect for the d-states is lower, as expected, and is near 2.50.  The defects given in the American Journal of Physics are 4.06 for s-states and 2.46 for d-states ².

Their graphs and calculations are on the following page.

d_s= 5.12            d_d = 2.52

A graph of the energy versus 1/(n-d)² should yield a straight line of slope R, if the Rydberg relation is correct. The graphs closely fit straight lines with slopes near R.  The intercepts should be at the energy corresponding to the ionization limit (31398), yet they are slightly above.   This, too, was expected since the systematic error in the data left all experimental values slightly higher than book values.

Those graphs are on the following pages.

S-States: 

D-States: