Atomic Spectra: Models & Estimates (IPhO Modern)

Most olympiad “atomic spectra” questions are really about one idea: the hydrogenic energy scale is set by Z² and 1/n², so line wavelengths come from simple energy differences. The second idea is modeling: you are often asked to justify when a hydrogenic estimate is good (hydrogen, He⁺, high-Z ions) and when screening or structure matters (neutral multi-electron atoms).

1. Definitions (Must Know)

A. Hydrogenic energy levels (one-electron ions)

For a one-electron atom/ion with nuclear charge + Ze (hydrogenic model): Eₙ ≈ -(13.6 eV Z²)/n², n = 1,2,3,…

Ionization energy from the ground state: Eᵢₒₙ ≈ 13.6 eV Z²

B. Photon energy and wavelength

For a transition nᵢ → n_f with nᵢ > n_f: Δ E = |E_(n_f)-E_nᵢ| and the emitted photon satisfies: Δ E = h f = hc/λ

C. Rydberg formula (wavenumber form)

1/λ = RZ²(1/n_f² - 1/nᵢ²)

D. Series names (hydrogen)

• Lyman: n_f = 1 (UV)
• Balmer: n_f = 2 (visible/near-UV)
• Paschen: n_f = 3 (IR)

E. Reduced mass (isotope shift idea)

For higher precision, replace mₑ by the reduced mass μ: μ = mₑm_N/(mₑ + m_N) This slightly changes the effective Rydberg constant (and hence wavelengths).

2. Key Ideas (What Earns Marks)

• Use scaling first. If Z doubles, energy spacings scale by Z².
• Convert to wavelength late. Work in eV, then use hc ≈ 1240 eV nm at the end.
• Series limits are ionization. As nᵢ → ∞, the line approaches a sharp limit related to Eᵢₒₙ.
• Model awareness matters. Hydrogenic results work well for one-electron ions (H, He⁺, Li²⁺) and for high-n outer electrons with an effective charge Z_eff.
• Order-of-magnitude wins marks. Many questions only need the right scale and dependence.

3. Detailed Explanations

A. A fast workflow for line wavelengths

1. Identify Z, nᵢ, and n_f.
2. Compute Δ E using Eₙ ∝ -Z²/n².
3. Convert to wavelength using λ = hc/Δ E.

This is typically faster and less error-prone than plugging into the Rydberg formula directly.

B. Why the series limit is special

For fixed n_f, as nᵢ → ∞: E_nᵢ → 0 So the limit photon energy is: Δ Eₗᵢₘᵢₜ = |E_(n_f)|

For hydrogen ( Z = 1), the Balmer limit ( n_f = 2) corresponds to |E₂| = 13.6/4 eV = 3.4 eV.

C. When hydrogenic estimates fail (and how to patch them)

For neutral multi-electron atoms:

• inner electrons shield the nucleus, so outer electrons see Z_eff < Z,
• energies are not exactly -13.6Z²/n² because the potential is not purely Coulomb,
• selection rules and splitting become more important.

A standard olympiad-level patch is to treat the outer electron as hydrogenic with an effective charge Z_eff chosen to match one known line or ionization energy.

D. Line splitting scales (very rough)

You can often estimate whether a splitting is negligible by comparing with the main transition energy:

• fine structure is typically smaller by a factor of order α² (with α ≈ 1/137),
• Zeeman splitting scale is Δ E ∼ μ_B B.

4. Common Mistakes

• Forgetting the Z² factor for hydrogenic ions.
• Mixing up nᵢ and n_f (emission requires nᵢ > n_f).
• Converting eV to nm incorrectly (use hc ≈ 1240 eV nm).
• Dropping absolute values and getting a negative photon energy.
• Assuming hydrogenic formulas apply unchanged to neutral multi-electron atoms without an effective charge argument.

5. Exam Tips

• Write the scale first: “Energy levels go like Z²/n².” This earns reasoning marks even before calculation.
• Use λ = hc/Δ E with hc in eV nm to avoid unit mistakes.
• If a question asks for a limit wavelength, take nᵢ → ∞ immediately.
• If you are given one spectral line for a multi-electron atom, use it to calibrate Z_eff and then predict the next one.

6. Worked Examples

A. Balmer alpha line in hydrogen (classic)

Find the wavelength for the hydrogen transition nᵢ = 3 to n_f = 2.

B. Hydrogen-like ion scaling (He⁺ line)

For He⁺ ( Z = 2), find the photon energy and wavelength for the transition nᵢ = 2 to n_f = 1.

7. Mind Stretchers

A. Isotope shift from reduced mass (order-of-magnitude)

Estimate the fractional change in hydrogen line wavelengths when going from protium ( m_N ≈ mₚ) to deuterium ( m_N ≈ 2mₚ), using the reduced mass idea.

• The Balmer limit corresponds to ionization from n = 2. Why does the line crowding occur near the limit?
• In multi-electron atoms, why do outer electrons often behave hydrogen-like for high n (Rydberg states)?

8. Practice