Physical Approximations and Domains of Validity
Approximation |
Mathematical form |
Domain of validity |
Failure mode |
|---|---|---|---|
Within-Shell Isolation (\(\Delta n = 0\)) |
\(V_E = \mathrm{diag}_n(V_E)\), no \(n \to n\!\pm\!1\) coupling |
Stark shift \(\ll\) shell spacing \(2Z^2\mathrm{Ry}/n^3\) |
Quadratic Stark; Inglis-Teller merging at \(N_e \gtrsim 10^{24}\) m-3. |
Quasi-Static Ion Microfields |
\(\vec{F}_\mathrm{ion} = \mathrm{const}\) |
Fluctuation rate \(\nu_i \ll\) Stark width \(\Delta E_S\) |
Ion dynamics / motional narrowing; use FFM. |
Semi-Classical Electron Impact |
Lorentzian with GBK width \(\gamma_e\) |
\(\lambda_{dB} \ll\) impact parameter \(\rho\) |
Quantum close-coupling needed for cold dense plasmas. |
Dipole Approximation |
\(V_\mathrm{rad} \propto \vec{r}\cdot\vec{E}\) |
\(\lambda \gg \langle r\rangle_n \sim n^2 a_0/Z\) |
Quadrupole/octupole for high-\(n\) Rydberg states. |