understand the appearance and formation of emission and absorption line spectra

Energy Levels in Atoms and Line Spectra

Learning Objective

Understand the appearance and formation of emission and absorption line spectra.

1. Quantised Energy Levels

Atoms possess a set of discrete energy states. An electron can only occupy these allowed levels, denoted by the principal quantum number $n = 1,2,3,\dots$.

• Each level has a specific energy $E_n$.
• For a hydrogen‑like atom the Bohr model gives $$E_n = -\frac{Z^2 R_H}{n^2},$$ where $Z$ is the atomic number and $R_H = 13.6\ \text{eV}$.
• More accurate quantum‑mechanical treatment introduces orbital ($l$), magnetic ($m_l$) and spin ($m_s$) quantum numbers, but the principal quantum number still determines the main 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:

$$\Delta E = E_j - E_i = hu = \frac{hc}{\lambda}$$

where $h$ is Planck’s constant, $u$ the frequency and $\lambda$ the wavelength of the photon.

3. Emission Line Spectra

Emission occurs when an electron in an excited state ($E_j$) drops to a lower state ($E_i$). The released photon produces a bright line at wavelength $\lambda$ in the spectrum.

• Spontaneous emission: occurs without external influence.
• Stimulated emission: an incoming photon of the same energy induces the transition (basis of lasers).

4. Absorption Line Spectra

If a beam of continuous radiation passes through a cool gas, photons whose energies match a possible transition are absorbed, creating dark lines (absorption lines) in the otherwise continuous spectrum.

• The gas must have atoms in the lower energy state $E_i$.
• After absorption the electron is promoted to $E_j$.

5. Spectral Series

For a given atom, transitions that share a common lower (or upper) level form a series of lines. In hydrogen the most important series are:

1. Lyman series: $n_{\text{final}} = 1$ (ultraviolet)
2. Balmer series: $n_{\text{final}} = 2$ (visible)
3. Paschen series: $n_{\text{final}} = 3$ (infra‑red)

6. Example – Balmer Series of Hydrogen

The Balmer series corresponds to transitions from $n \ge 3$ down to $n = 2$. Using the Rydberg formula:

$$\frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{n^2} \right), \qquad n = 3,4,5,\dots$$

Transition	Upper level $n$	Wavelength $\lambda$ (nm)	Colour (perceived)
H$_\alpha$	3 → 2	656.3	Red
H$_\beta$	4 → 2	486.1	Blue‑green
H$_\gamma$	5 → 2	434.0	Violet
H$_\delta$	6 → 2	410.2	Violet‑ultraviolet

7. Factors Influencing Line Intensity

• Population of the lower level: governed by Boltzmann distribution $N_i \propto g_i e^{-E_i/kT}$.
• Transition probability (Einstein A coefficient): larger $A_{ji}$ gives a stronger emission line.
• Selection rules: only transitions satisfying $\Delta l = \pm 1$, $\Delta m_l = 0, \pm1$, etc., are allowed.
• Instrumental factors: resolution and detector sensitivity affect observed line strength.

8. Visualising Energy Levels

9. Summary

Atoms have quantised energy levels. Photons are emitted or absorbed when electrons transition between these levels, producing discrete lines in a spectrum. Emission lines appear bright against a dark background, while absorption lines appear dark against a continuous spectrum. The pattern of lines (spectral series) is characteristic of each element and forms the basis of spectroscopic identification.