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# 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*10^{5 }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:

*Derek Kverno and Jim
Nolen produced all images, tables, and data presented here. *