6.3 Radiative and Non-Radiative Transitions

Atoms can change energy states through radiative and non-radiative transitions. Radiative transitions involve photons, while non-radiative ones don't. These processes shape how atoms interact with light and each other, influencing their behavior in various environments.

Understanding these transitions is key to grasping atomic interactions. They determine excited state lifetimes, govern emission and absorption of light, and play crucial roles in phenomena like fluorescence and laser operation. Mastering this topic unlocks deeper insights into atomic physics.

Radiative vs Non-radiative Transitions

Photon Involvement

• Radiative transitions involve the emission or absorption of a photon (light particle)
• Non-radiative transitions do not involve the emission or absorption of a photon

Energy Conservation

• In radiative transitions, the energy difference between the initial and final states is equal to the energy of the emitted or absorbed photon, as described by the Planck-Einstein relation ($ΔE = hν$)
  • $h$ is Planck's constant
  • $ν$ is the frequency of the photon
• Non-radiative transitions conserve energy through various mechanisms, such as collisional excitation and deexcitation, where the energy is transferred between atoms or molecules through collisions

Excited State Lifetime

• The lifetime of an excited state is determined by the rates of both radiative and non-radiative transitions
  • Radiative transitions typically have longer lifetimes (nanoseconds to microseconds)
  • Non-radiative transitions often have shorter lifetimes (picoseconds to nanoseconds)
• The total lifetime ($τ$) of an excited state is given by the inverse of the sum of radiative ($Γ_r$) and non-radiative ($Γ_nr$) transition rates: $τ = 1/(Γ_r + Γ_nr)$

Mechanisms of Radiative Transitions

Electric Dipole Transitions

• Electric dipole transitions are the most common type of radiative transition
• Occur when there is a change in the electric dipole moment of the atom during the transition
• The transition probability is proportional to the square of the electric dipole matrix element ($μ_{fi}$), which depends on the overlap between the initial and final state wavefunctions ($ψ_i$ and $ψ_f$) and the electric dipole operator ($\hat{μ}$): $μ_{fi} = \langle ψ_f | \hat{μ} | ψ_i \rangle$

Magnetic Dipole Transitions

• Magnetic dipole transitions involve a change in the magnetic dipole moment of the atom
• Typically weaker than electric dipole transitions by a factor of $α^2$ (fine structure constant squared)
• The transition probability is proportional to the square of the magnetic dipole matrix element ($m_{fi}$), which depends on the overlap between the initial and final state wavefunctions and the magnetic dipole operator ($\hat{m}$): $m_{fi} = \langle ψ_f | \hat{m} | ψ_i \rangle$

Electric Quadrupole Transitions

• Electric quadrupole transitions involve a change in the electric quadrupole moment of the atom
• Even weaker than magnetic dipole transitions by a factor of $α^2$
• The transition probability is proportional to the square of the electric quadrupole matrix element ($Q_{fi}$), which depends on the overlap between the initial and final state wavefunctions and the electric quadrupole operator ($\hat{Q}$): $Q_{fi} = \langle ψ_f | \hat{Q} | ψ_i \rangle$

Transition Strengths and Selection Rules

• The strength of a radiative transition is determined by the transition matrix element, which depends on the overlap between the initial and final state wavefunctions and the relevant multipole operator
• The selection rules for each type of transition are determined by the conservation of angular momentum and parity
  • Selection rules dictate which transitions are allowed or forbidden based on the change in quantum numbers
  • Allowed transitions have much higher probabilities than forbidden transitions

Selection Rules for Transitions

Electric Dipole Selection Rules

• For electric dipole transitions, the selection rules are:
  • $ΔJ = 0, ±1$ (except $J = 0$ to $J = 0$)
  • $Δm_j = 0, ±1$
  • Parity must change
• These rules arise from the conservation of angular momentum and parity during the transition

Magnetic Dipole Selection Rules

• For magnetic dipole transitions, the selection rules are:
  • $ΔJ = 0, ±1$ (except $J = 0$ to $J = 0$)
  • $Δm_j = 0, ±1$
  • Parity must not change
• Magnetic dipole transitions have different parity requirements compared to electric dipole transitions

Electric Quadrupole Selection Rules

• For electric quadrupole transitions, the selection rules are:
  • $ΔJ = 0, ±1, ±2$ (except $J = 0$ to $J = 0, 1/2, 1$)
  • $Δm_j = 0, ±1, ±2$
  • Parity must not change
• Electric quadrupole transitions allow for larger changes in angular momentum compared to electric and magnetic dipole transitions

Forbidden Transitions

• Transitions that violate the selection rules are called forbidden transitions
• Forbidden transitions have much lower probabilities than allowed transitions
  • Electric dipole forbidden transitions can occur through higher-order processes (magnetic dipole or electric quadrupole)
  • Forbidden transitions often have longer lifetimes (milliseconds to seconds) compared to allowed transitions

Collisional Excitation and Deexcitation

Collisional Excitation

• Collisional excitation occurs when an atom in a lower energy state collides with another particle (electron or another atom) and gains enough energy to transition to a higher energy state
• The collision cross-section ($σ$) determines the probability of a collision resulting in a transition
  • The collision cross-section depends on the relative velocity of the colliding particles and the energy difference between the atomic states
• Collisional excitation is more likely to occur in high-density environments (plasmas or gases at high pressure)

Collisional Deexcitation

• Collisional deexcitation occurs when an atom in an excited state collides with another particle and transfers its excess energy, returning to a lower energy state
• The rate of collisional deexcitation depends on the collision cross-section and the density of the colliding particles
• Collisional deexcitation can compete with radiative decay processes in determining the lifetime of an excited state

Impact on Atomic Populations

• Collisional processes can significantly impact the population of atomic energy levels, especially in high-density environments
• The balance between collisional excitation and deexcitation rates, along with radiative transition rates, determines the equilibrium population of atomic energy levels
  • In thermal equilibrium, the population of energy levels follows the Boltzmann distribution: $N_i/N_j = g_i/g_j \exp(-ΔE_{ij}/kT)$
    • $N_i$ and $N_j$ are the populations of levels $i$ and $j$
    • $g_i$ and $g_j$ are the degeneracies of levels $i$ and $j$
    • $ΔE_{ij}$ is the energy difference between levels $i$ and $j$
    • $k$ is the Boltzmann constant, and $T$ is the temperature
• Collisional processes can lead to population inversion, a condition where higher energy levels have a greater population than lower energy levels, which is essential for the operation of lasers