Atomic Spectra

Atomic spectra form one of the most important aspects of modern physics and chemistry, as they reveal the internal structure of atoms and the behavior of electrons. The study of atomic spectra not only led to the development of quantum mechanics but also provided the foundation for spectroscopy, which has wide applications in astronomy, material science, and chemistry.

This post explores types of spectra, theoretical explanations, experimental methods, calculations, and practical applications, in detail.

1. Introduction to Atomic Spectra

An atomic spectrum is a series of discrete electromagnetic radiation lines emitted or absorbed by electrons in an atom when they move between energy levels.

Key points:

Each element has a unique spectral fingerprint.
Spectra are quantized, indicating discrete energy levels in atoms.
Origin of spectral lines lies in electron transitions between energy states.

2. Types of Atomic Spectra

Atomic spectra are mainly classified into three types:

2.1 Continuous Spectrum

Produced by hot solids, liquids, or densely packed gases.
Contains all wavelengths within a region.
Example: Sunlight passing through a prism.

2.2 Emission Spectrum

Produced when excited atoms return to lower energy states, emitting photons.
Can be line or band spectrum.
Line emission spectrum: discrete wavelengths; characteristic of element.
Band spectrum: groups of closely spaced lines; found in molecules.

2.3 Absorption Spectrum

Produced when white light passes through a cooler gas.
Gas absorbs light at certain wavelengths corresponding to electron excitation.
Appears as dark lines on a continuous background (Fraunhofer lines in sunlight).

3. Historical Background

3.1 Discovery of Atomic Spectra

Fraunhofer (1814): Observed dark lines in the solar spectrum.
Bunsen and Kirchhoff (1859–1860): Emission lines linked to specific elements.
Balmer (1885): Empirical formula for hydrogen visible spectrum.

3.2 Early Theoretical Attempts

Classical physics predicted continuous radiation, unable to explain discrete spectral lines.
Bohr (1913) proposed quantized orbits, explaining hydrogen spectrum successfully.

4. Bohr Model and Atomic Spectra

Bohr’s model provides a theoretical explanation for hydrogen-like atoms:

Electrons occupy stable, quantized orbits around the nucleus.
Energy of an electron in nth orbit:

En=−13.6 eVn2,n=1,2,3,…E_n = – \frac{13.6 \text{ eV}}{n^2}, \quad n = 1, 2, 3,…En=−n213.6 eV,n=1,2,3,…

Electron transitions produce photons:

ΔE=Ei−Ef=hν=hcλ\Delta E = E_i – E_f = h\nu = \frac{hc}{\lambda}ΔE=Ei−Ef=hν=λhc

Explains Rydberg formula for hydrogen spectral lines:

1λ=RH(1nf2−1ni2)\frac{1}{\lambda} = R_H \left(\frac{1}{n_f^2} – \frac{1}{n_i^2}\right)λ1=RH(nf21−ni21)

Where RH=1.097×107 m−1R_H = 1.097 \times 10^7 \text{ m}^{-1}RH=1.097×107 m−1 (Rydberg constant).

5. Series in Hydrogen Spectrum

Hydrogen spectrum consists of several series:

5.1 Lyman Series

Transitions to n=1 from higher levels (ni>1n_i > 1ni>1)
Ultraviolet region: 10–121 nm

5.2 Balmer Series

Transitions to n=2
Visible region: 400–700 nm
Example: red line at 656 nm

5.3 Paschen Series

Transitions to n=3
Infrared region: 700–1875 nm

5.4 Brackett and Pfund Series

n=4 (Brackett), n=5 (Pfund)
Infrared lines, mostly for spectroscopy.

6. Quantum Mechanical Explanation

6.1 Schrödinger Equation

Describes electron wavefunction ψ\psiψ in an atom.
Solutions give energy eigenvalues corresponding to quantized levels:

−ℏ22m∇2ψ+Vψ=Eψ-\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi−2mℏ2∇2ψ+Vψ=Eψ

6.2 Quantum Numbers

Energy levels depend on:

Principal quantum number (n): main energy level
Azimuthal quantum number (l): orbital shape
Magnetic quantum number (m_l): orientation
Spin quantum number (m_s): electron spin

6.3 Selection Rules

For spectral lines to appear:
- Δl=±1\Delta l = \pm 1Δl=±1
- Δml=0,±1\Delta m_l = 0, \pm 1Δml=0,±1
Explains allowed and forbidden transitions.

7. Fine Structure of Spectral Lines

Splitting of spectral lines due to:

Spin-orbit coupling: interaction between electron spin and orbital motion.
Relativistic effects: faster electrons experience mass variation.
Lamb shift: quantum electrodynamics effect.

Observed as closely spaced doublets or multiplets.

8. Hyperfine Structure

Caused by interaction between nuclear spin and electron magnetic field.
Responsible for splitting of spectral lines into finer components.
Example: Hydrogen 21 cm line in astronomy.

9. Zeeman Effect

Splitting of spectral lines in a magnetic field.
Observed in sunspots and laboratory experiments.
Normal Zeeman effect: simple splitting
Anomalous Zeeman effect: includes electron spin effects

10. Stark Effect

Splitting of spectral lines in an electric field.
Explains field-induced energy level changes.
Useful in plasma diagnostics and spectroscopy.

11. Experimental Techniques

11.1 Prism Spectroscopy

Light passed through a prism to disperse wavelengths.
First method to observe discrete spectral lines.

11.2 Diffraction Grating

Produces sharper spectral lines using interference.
Widely used in modern spectrometers.

11.3 Flame Test

Excites atoms in a flame; emitted light analyzed.
Simple method to identify elemental composition.

11.4 Photographic Plate Spectroscopy

Historical method to record emission lines.
Led to discovery of new elements and isotopes.

12. Applications of Atomic Spectra

12.1 Element Identification

Every element has a unique spectrum.
Basis for analytical chemistry and spectroscopy.

12.2 Astrophysics

Determine composition, temperature, and movement of stars.
Use Doppler shift in spectral lines to measure velocities.

12.3 Plasma Diagnostics

Determine ionization states, temperature, and density of plasma.

12.4 Laser Technology

Electron transitions are controlled to produce coherent light.
Basis for lasers in medicine, communication, and industry.

12.5 Quantum Mechanics Validation

Spectral lines confirm energy quantization and Bohr-Sommerfeld predictions.

13. Mathematical Examples

13.1 Wavelength of Balmer Line

Transition ni=3→nf=2n_i=3 \to n_f=2ni=3→nf=2: 1λ=RH(122−132)\frac{1}{\lambda} = R_H \left(\frac{1}{2^2} – \frac{1}{3^2}\right)λ1=RH(221−321) 1λ=1.097×107(14−19)\frac{1}{\lambda} = 1.097 \times 10^7 \left(\frac{1}{4} – \frac{1}{9}\right)λ1=1.097×107(41−91)

λ≈656 nm\lambda \approx 656 \text{ nm}λ≈656 nm (red line)

13.2 Energy Difference

E=hν=13.6(122−132) eV≈1.89 eVE = h \nu = 13.6 \left(\frac{1}{2^2} – \frac{1}{3^2}\right) \text{ eV} \approx 1.89 \text{ eV}E=hν=13.6(221−321) eV≈1.89 eV

14. Hydrogen-like Ions

Single-electron ions: He+,Li2+He^+, Li^{2+}He+,Li2+
Energy levels scaled by nuclear charge (Z):

En=−13.6Z2n2 eVE_n = -13.6 \frac{Z^2}{n^2} \text{ eV}En=−13.6n2Z2 eV

Spectra shifted to higher energy and shorter wavelength.

15. Molecular Spectra (Brief Overview)

Molecules show rotational and vibrational spectra.
Leads to band spectra, slightly more complex than atomic lines.
Useful in chemistry and atmospheric science.

16. Summary of Key Points

Feature	Atomic Spectra
Origin	Electron transitions between discrete energy levels
Types	Continuous, emission, absorption
Hydrogen Series	Lyman, Balmer, Paschen, Brackett, Pfund
Fine Structure	Spin-orbit and relativistic splitting
Hyperfine Structure	Nuclear spin-electron interaction
Zeeman Effect	Magnetic field splitting
Stark Effect	Electric field splitting
Applications	Spectroscopy, astrophysics, plasma diagnostics, lasers