understand the appearance and formation of emission and absorption line spectra

Energy Levels in Atoms and Line Spectra

Learning objectives (AO1‑AO3)

• Explain how the quantised energy levels of atoms give rise to emission and absorption line spectra (link to AO1 22.1 – “Energy levels are introduced in the Quantum Physics section”).
• Use the photon‑energy relation \(hf = E_j-E_i\) to convert between wavelength, frequency and energy.
• State and apply the Rydberg formula for hydrogen‑like atoms.
• Identify the main spectral series (Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys) and calculate line positions.
• Interpret factors that affect line intensity (population, transition probability, selection rules, instrumental effects).

1. Quantised energy levels

• Electrons can occupy only a set of discrete energy states characterised by the principal quantum number \(n = 1,2,3,\dots\).
• For a hydrogen‑like atom (one electron) the Bohr model gives \[ E_n = -\frac{Z^{2}R_{\!H}}{n^{2}}\qquad\text{with }R_{\!H}=13.6\;\text{eV}, \] where \(Z\) is the atomic number.
• In the full quantum‑mechanical description each level is split further by the orbital (\(l\)), magnetic (\(m_l\)) and spin (\(m_s\)) quantum numbers, but the principal quantum number determines the dominant energy spacing.

2. Photon emission and absorption

When an electron moves between two levels \(i\) and \(j\) the atom either emits or absorbs a photon whose energy equals the difference between the two levels:

\[ \Delta E = E_j-E_i = hu = \frac{hc}{\lambda}=hf \]

• \(h = 6.626\times10^{-34}\;\text{J s}\) – Planck’s constant
• \(u\) – frequency (Hz); \(f\) is often used interchangeably
• \(\lambda\) – wavelength (m)
• The compact form \(hf = E_j-E_i\) is frequently required in exam questions.

3. Types of spectra

Spectrum	How it is produced	Appearance on a detector
Continuous (black‑body)	Thermal radiation from a hot, dense source (filament, stellar surface)	Smooth curve containing all wavelengths
Emission line	Low‑density gas of excited atoms relax to lower levels, emitting photons	Bright, narrow lines on a dark background
Absorption line	Continuous radiation passes through a cooler gas; photons matching allowed transitions are removed	Dark lines superimposed on a continuous spectrum

4. Emission lines

• Spontaneous emission: an excited electron decays without external influence.
• Stimulated emission: an incoming photon of the same energy induces the transition (principle of lasers).
• The wavelength of the emitted photon is given by the relation in Section 2.

5. Absorption lines

• Only atoms that are already in the lower state \(E_i\) can absorb a photon of energy \(E_j-E_i\).
• After absorption the electron is promoted to the upper state \(E_j\).
• Because the background source is continuous, the removed photons appear as dark lines.

6. Spectral series (hydrogen‑like atoms)

Transitions that share a common lower (or upper) level form a series, identified by the final principal quantum number \(n_f\).

Series	Lower level \(n_f\)	Region of the spectrum	Typical transition (example)
Lyman	1	Ultraviolet (≈ 90–400 nm)	n=2→1 (121.6 nm)
Balmer	2	Visible (≈ 400–700 nm)	n=3→2 (656.3 nm, Hα)
Paschen	3	Infra‑red (≈ 800–2500 nm)	n=4→3 (1875 nm)
Brackett	4	Infra‑red (≈ 1.5–4 µm)	n=5→4 (4051 nm)
Pfund	5	Infra‑red (≈ 3–7 µm)	n=6→5 (7460 nm)
Humphreys	6	Far‑infra‑red (≈ 12–30 µm)	n=7→6 (12 µm)

7. Rydberg formula and the origin of \(R_{\!H}\)

For a hydrogen‑like ion the wavenumber of a spectral line is

\[ \frac{1}{\lambda}=R_Z\!\left(\frac{1}{n_f^{2}}-\frac{1}{n_i^{2}}\right),\qquad n_i>n_f, \] where \(R_Z = R_{\!H}Z^{2}\) and \(R_{\!H}=1.097\,\times10^{7}\;\text{m}^{-1}\).

Derivation from the Bohr model (useful for AO1 22.1):

\[ R_{\!H}= \frac{m_e e^{4}}{8\varepsilon_0^{2}h^{3}c} \]

• \(m_e\) – electron mass
• \(e\) – elementary charge
• \(\varepsilon_0\) – permittivity of free space
• \(h\) – Planck’s constant
• \(c\) – speed of light

This expression shows that the Rydberg constant arises from the fundamental constants that appear in the Bohr model of the hydrogen atom.

8. Converting between wavelength, frequency and energy

Quantity	Formula	Useful form for calculations
Energy ↔ Frequency	\(E = hf\)	\(E\,[\text{eV}] = 4.1357\times10^{-15}\,f\,[\text{Hz}]\)
Energy ↔ Wavelength	\(E = \dfrac{hc}{\lambda}\)	\(E\,[\text{eV}] = \dfrac{1240}{\lambda\,[\text{nm}]}\)
Frequency ↔ Wavelength	\(c = \lambda f\)	\(f\,[\text{Hz}] = \dfrac{c}{\lambda\,[\text{m}]}\)

9. Factors influencing line intensity (AO2)

• Population of the lower level – given by the Boltzmann distribution \[ N_i \propto g_i\,e^{-E_i/kT} \] where \(g_i\) is the statistical weight, \(k\) the Boltzmann constant and \(T\) the temperature.
• Transition probability – characterised by the Einstein \(A_{ji}\) coefficient; larger \(A_{ji}\) → stronger emission.
• Selection rules (electric‑dipole) \[ \Delta l = \pm1,\qquad \Delta m_l = 0,\pm1,\qquad \Delta s = 0 \] Only transitions obeying these rules appear as observable lines.
• Instrumental factors – spectral resolution, slit width, detector efficiency and Doppler/pressure broadening can alter the observed line shape and strength.

10. Worked examples (AO2)

Example 1 – Identify a Balmer line and find its energy

Question: A photon of wavelength \(410\;\text{nm}\) is observed in the emission spectrum of hydrogen. Identify the transition and calculate the photon energy in electron‑volts.

1. Use the Rydberg formula for the Balmer series (\(n_f=2\)): \[ \frac{1}{\lambda}=R_{\!H}\!\left(\frac{1}{2^{2}}-\frac{1}{n_i^{2}}\right) \] Substituting \(\lambda=410\times10^{-9}\,\text{m}\) gives \(n_i=6\). Transition: \(6\rightarrow2\) (H\(_\delta\)).
2. Energy: \[ E=\frac{hc}{\lambda} =\frac{(6.626\times10^{-34})(3.00\times10^{8})}{410\times10^{-9}} =4.85\times10^{-19}\,\text{J} \] \[ E=\frac{4.85\times10^{-19}}{1.602\times10^{-19}}\approx3.03\;\text{eV} \]

Example 2 – Convert a line’s wavelength to frequency and energy

Question: The H\(_\alpha\) line of hydrogen has \(\lambda = 656.3\;\text{nm}\). Determine (a) its frequency and (b) its energy in joules and electron‑volts.

1. Frequency: \[ f = \frac{c}{\lambda} = \frac{3.00\times10^{8}}{656.3\times10^{-9}} = 4.57\times10^{14}\;\text{Hz} \]
2. Energy: \[ E = hf = (6.626\times10^{-34})(4.57\times10^{14}) = 3.03\times10^{-19}\;\text{J} \] \[ E = \frac{3.03\times10^{-19}}{1.602\times10^{-19}} \approx 1.89\;\text{eV} \]

11. Visualising energy‑level transitions

12. Summary (AO1)

• Atoms possess discrete energy levels; transitions between them produce photons with energy \(hf = E_j-E_i\).
• Emission lines appear as bright, narrow features on a dark background; absorption lines appear as dark gaps in a continuous spectrum.
• The Rydberg formula (derived from the Bohr model) predicts the wavelength of every line for hydrogen‑like atoms; the constant \(R_{\!H}\) can be expressed in terms of fundamental constants.
• Each element has a characteristic set of spectral series (Lyman, Balmer, …) that act as a unique fingerprint.
• Line intensity depends on the population of the lower level (Boltzmann distribution), the Einstein \(A_{ji}\) coefficient, selection rules, and the experimental setup.