Potassium's single valence electron sits outside a closed core
PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure · One-electron atoms and the origins of atomic structure
Reading the Spectrum, Ch Radial functions, orbital size and Rydberg states
Problem
Potassium's single valence electron sits outside a closed core, just as sodium's does, and its levels follow the same hydrogen-shaped term formula with one fitted defect per l: T=RK/n∗2, n∗=n−δl. Potassium's reduced-mass Rydberg constant is RK=109735.8cm−1 (found as in Chapter 2, from potassium's atomic mass), and its ionisation energy is 35009.8cm−1 (NIST ASD), which is also the term value of the 4s ground level. Two members of potassium's principal series (4s→np), each the fine-structure-averaged wavenumber of its doublet, are measured at 13023.7cm−1 for n=4 and 24713.9cm−1 for n=5 (NIST ASD). Neither term value, T(4p) or T(5p), is given outright; only the ionisation energy and the two lines are.
((a)) Find the term value and the quantum defect δp of 4p and 5p. Is δp the same at the two values of n?
((b)) Take the defect fitted at n=5 as the better estimate of the value that governs larger n, and use it to predict the term value of 6p and the wavenumber of the 4s→6p line.
((c)) A further absorption line is measured at 29004.9cm−1 (NIST ASD). One line may have been missed while the spectrum was recorded, so this could be either 4s→6p or 4s→7p. Using the defect from part (b), decide which it is, and give the residual ∣ν~measured−ν~predicted∣, in cm−1, for each candidate. Carry n∗ to at least five decimal places through the calculation and round only the final residuals, since the two predicted wavenumbers are each a small difference of much larger numbers.