Rydberg Atoms and the Quantum Defect
A Rydberg atom is an atom whose valence electrons are in states with a very large
principal quantum number n. In such an atom, the many core electrons effectively shield
the outer electron from the electric field of the nucleus. As a result, 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 fit the Rydberg
, where Rydberg constant R =
En is the energy above the ground state. T is the ionization
Rydberg atoms will nearly fit this equation due to the shielding of the nucleus, but
they will deviate from the relation because their orbits are not circular. Even for
electrons in a high n-state, their orbits will pass through the inner core of shielding
electrons and the Rydberg relation will not work. To adjust the relation for the electrons
penetration of the inner core electrons, we introduce a correction term called the
"quantum defect." The Rydberg relation for such atoms becomes:
, where d
is the quantum defect.
This quantum defect should be different for different angular momentum states. For S
states, having zero angular momentum, this quantum defect should be relatively large, on
the order of 5 to 7. An electron with zero angular momentum essentially passes clean
through the core of shielding electrons. With low angular momentum states, the
shielding effect breaks down a bit . . . the electron energy levels can not mimic those of
hydrogen without the shielding. Hence, there is a higher quantum defect for low
angular momentum states. For D states, having higher angular momentum, the quantum
defect should be relatively small, as its orbit will not pass through the core so deeply--
Using the ionization spectrum to calculate energy levels of various transitions, we
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 our data.
You can see that for the s-states, the quantum defect is in a narrow range, hovering
around 5.10. The quantum defect for the d-statas is lower, as expected, and is near
2.50. These are very close to the defects given in the American Journal of Physics:
4.06 for s-states and 2.46 for d-states (Hazel 329).
ds= 5.12 dd
Graphing the energy versus 1/(n-d)2 should
yield a straight line of slope R, if the Rydberg relation is correct. As you can see, the
graphs are indeed 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.
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