2.3 Selection rules and spectral transitions

Quantum mechanics sets the rules for atomic transitions, determining which ones are allowed or forbidden. These selection rules govern the strength and appearance of spectral lines, crucial for understanding atomic structure and interactions with light.

Transition probabilities and intensities depend on factors like wavefunctions and energy differences. Concepts like oscillator strength and transition dipole moment help quantify transition strength, providing insights into atomic behavior and spectroscopic observations.

Transition Types and Rules

Allowed and Forbidden Transitions

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• Allowed transitions occur when selection rules are satisfied
• Allowed transitions result in strong spectral lines
• Forbidden transitions violate selection rules
• Forbidden transitions produce weak or absent spectral lines
• Forbidden transitions can occur due to magnetic dipole or electric quadrupole interactions
• Transition probability decreases significantly for forbidden transitions

Selection Rules for Atomic Transitions

• Dipole selection rules govern electric dipole transitions
• Dipole selection rules include:
  • Change in angular momentum quantum number: ΔL = ±1
  • Change in magnetic quantum number: ΔmL = 0, ±1
  • No change in principal quantum number: Δn can be any value
• Laporte rule applies to centrosymmetric molecules and atoms
• Laporte rule states transitions between states of the same parity are forbidden
• Parity refers to the symmetry of the wavefunction under inversion
• Spin selection rule restricts changes in spin quantum number
• Spin selection rule states ΔS = 0 for singlet-singlet or triplet-triplet transitions
• Transitions between singlet and triplet states (ΔS ≠ 0) are spin-forbidden

Transition Characteristics

Transition Probability and Intensity

• Transition probability measures likelihood of a spectral transition
• Transition probability depends on:
  • Initial and final state wavefunctions
  • Dipole moment operator
  • Energy difference between states
• Einstein A coefficient quantifies spontaneous emission probability
• Einstein B coefficient describes stimulated emission and absorption probabilities
• Transition intensity correlates with transition probability
• Strong transitions have high probabilities and intense spectral lines
• Weak transitions have low probabilities and faint or absent spectral lines

Oscillator Strength and Transition Dipole Moment

• Oscillator strength measures transition strength
• Oscillator strength relates to the transition dipole moment
• Oscillator strength formula: $f = \frac{2m_e\omega_{fi}}{3\hbar e^2}|\mu_{fi}|^2$
  • $m_e$ represents electron mass
  • $\omega_{fi}$ denotes transition frequency
  • $\mu_{fi}$ symbolizes transition dipole moment
• Transition dipole moment measures charge redistribution during transition
• Transition dipole moment calculation: $\mu_{fi} = \int \psi_f^* \hat{\mu} \psi_i d\tau$
  • $\psi_f$ and $\psi_i$ represent final and initial state wavefunctions
  • $\hat{\mu}$ denotes dipole moment operator
• Large oscillator strength indicates strong transition
• Small oscillator strength suggests weak or forbidden transition