13.2 Atomic spectroscopy and term diagrams

Atomic spectroscopy reveals the inner workings of atoms through their interaction with light. By studying emission and absorption spectra, we can uncover electronic configurations, energy levels, and allowed transitions between states.

Term diagrams visually represent these energy levels and transitions, helping us understand atomic structure. Using quantum numbers and selection rules, we can predict and interpret spectral lines, unlocking the secrets of atomic behavior.

Atomic Term Symbols and Quantum Numbers

Describing Electronic States with Term Symbols

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• Atomic term symbols describe the electronic states of atoms, providing information about the angular momentum and spin of the electrons
• Term symbols are written in the form $2S+1L_J$, where $S$ is the total spin angular momentum, $L$ is the total orbital angular momentum, and $J$ is the total angular momentum
• The value of $L$ is represented by a letter: S ($L=0$), P ($L=1$), D ($L=2$), F ($L=3$), G ($L=4$), H ($L=5$), and so on
• The value of $J$ ranges from $|L-S|$ to $L+S$ in integer steps

Multiplicity and Parity

• The multiplicity ($2S+1$) represents the number of possible orientations of the total spin angular momentum
  • A singlet state ($2S+1 = 1$) has anti-parallel electron spins
  • A triplet state ($2S+1 = 3$) has parallel electron spins
• The parity of the electronic state is denoted by a superscript "o" for odd parity (odd sum of $l$ quantum numbers) or no superscript for even parity (even sum of $l$ quantum numbers)
  • Example: $^3P_2^o$ represents a triplet state with $L=1$, $J=2$, and odd parity
  • Example: $^1S_0$ represents a singlet state with $L=0$, $J=0$, and even parity

Term Diagrams for Multi-Electron Atoms

Hund's Rules for Determining Ground State

• Hund's rules are guidelines used to determine the ground state electronic configuration and term symbol of a multi-electron atom based on minimizing the total energy of the system
• Hund's first rule (maximum multiplicity): The term with the highest spin multiplicity ($2S+1$) has the lowest energy, maximizing the total spin angular momentum ($S$)
  • Example: For the electronic configuration $1s^22s^22p^2$, the term with the highest multiplicity is $^3P$
• Hund's second rule (maximum orbital angular momentum): For a given multiplicity, the term with the largest total orbital angular momentum ($L$) has the lowest energy, maximizing the total orbital angular momentum
  • Example: For the electronic configuration $1s^22s^22p^2$, the term with the largest $L$ is $^3P$
• Hund's third rule (spin-orbit coupling): For a given multiplicity and total orbital angular momentum, the term with the lowest total angular momentum ($J$) has the lowest energy for a less than half-filled shell, while the term with the highest $J$ has the lowest energy for a more than half-filled shell
  • For atoms with a less than half-filled shell, $J = |L-S|$
  • For atoms with a more than half-filled shell, $J = L+S$
  • Example: For the electronic configuration $1s^22s^22p^2$, the ground state term symbol is $^3P_0$

Constructing Term Diagrams

• Term diagrams visually represent the relative energies of electronic states in a multi-electron atom
• States are arranged vertically according to their energy, with the ground state at the bottom
• States with the same $L$ and $S$ values are grouped together, with the $J$ values increasing from left to right
• The energy spacing between states is determined by the strength of the spin-orbit coupling interaction
  • Larger spin-orbit coupling leads to greater energy separation between states with different $J$ values
  • Example: In the carbon atom ($1s^22s^22p^2$), the term diagram shows the $^3P$ states ($^3P_0$, $^3P_1$, $^3P_2$) at lower energies than the $^1D_2$ and $^1S_0$ states

Allowed Transitions and Selection Rules

Electric Dipole Selection Rules

• Selection rules determine which transitions between atomic energy levels are allowed based on the conservation of angular momentum and parity
• Allowed transitions satisfy the selection rules, while forbidden transitions violate them
• The electric dipole selection rules for LS coupling are:
  • $\Delta S = 0$ (spin multiplicity must not change)
  • $\Delta L = \pm1$ (total orbital angular momentum must change by 1)
  • $\Delta J = 0, \pm1$ (total angular momentum must change by 0 or 1, but $J = 0$ to $J = 0$ is forbidden)
  • Parity must change (odd to even or even to odd)
• The Laporte selection rule states that transitions between states with the same parity (even to even or odd to odd) are forbidden in centrosymmetric systems
  • Example: In a centrosymmetric atom, the transition from an $s$ orbital ($l=0$, even parity) to another $s$ orbital is forbidden

Forbidden Transitions

• Forbidden transitions can still occur, but they are much less likely than allowed transitions and have longer lifetimes
• Forbidden transitions can become weakly allowed due to perturbations, such as vibronic coupling or magnetic dipole interactions
  • Example: The transition from the $^1S_0$ ground state to the $^3P_1$ excited state in helium is forbidden by the spin selection rule ($\Delta S = 0$), but it can occur weakly due to spin-orbit coupling
• Forbidden transitions often result in metastable states, which have longer lifetimes than states that can undergo allowed transitions
  • Example: The $^1S_0$ ground state of helium is metastable because transitions to the $^3P$ states are forbidden by the spin selection rule

Analyzing Atomic Spectra

Spectral Lines and Energy Levels

• Atomic spectra consist of emission or absorption lines corresponding to transitions between different electronic states of an atom
• The wavelength of a spectral line is related to the energy difference between the initial and final states of the transition according to the equation $\Delta E = hc/\lambda$, where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength
  • Example: The Lyman series in the hydrogen spectrum corresponds to transitions from various excited states to the $n=1$ ground state
• The relative intensities of spectral lines depend on the transition probabilities and the population of the initial states
  • Stronger lines correspond to more probable transitions and higher population of the initial state
  • Example: In the hydrogen spectrum, the Balmer series lines are more intense than the Lyman series lines because the $n=2$ state has a higher population than the $n=1$ state at typical temperatures

Fine Structure and Rydberg's Formula

• The fine structure of spectral lines arises from the coupling of the orbital and spin angular momenta (spin-orbit coupling)
• The splitting of spectral lines into multiple components can be used to determine the $J$ values of the electronic states involved
  • Example: The sodium D lines (589.0 nm and 589.6 nm) arise from the fine structure splitting of the $^2P$ state into the $^2P_{1/2}$ and $^2P_{3/2}$ levels
• Rydberg's formula can be used to calculate the wavelengths of spectral lines in hydrogen-like atoms: $1/\lambda = R(1/n_1^2 - 1/n_2^2)$, where $R$ is the Rydberg constant, and $n_1$ and $n_2$ are the principal quantum numbers of the initial and final states, respectively
  • Example: The wavelength of the Balmer-alpha line in the hydrogen spectrum ($n=2$ to $n=3$ transition) can be calculated using Rydberg's formula

Deducing Electronic Configurations and Term Symbols

• The electronic configuration and term symbols of an atom can be deduced by comparing the observed spectral lines with the predicted transitions based on selection rules and energy level diagrams
• The presence or absence of certain spectral lines can provide information about the electronic configuration and the allowed transitions between states
  • Example: The absence of certain transitions in the helium spectrum indicates that the ground state has a $^1S_0$ term symbol and a $1s^2$ electronic configuration
• The relative intensities and fine structure of spectral lines can help determine the term symbols of the electronic states involved
  • Example: The strong sodium D lines and their fine structure splitting suggest that the excited state has a $^2P$ term symbol and a $3p$ electron configuration