Term Symbols

• Great video reference:

  • Term Symbols Explanation

• Term symbols define the energy levels of a system, including for degenerate states. These arise from the coupling of angular momentum of spins and orbitals.

• The general notation involves capital letters for spin and orbital angular momentum:

  • $L=l_1+l_2,l_1+l_2-1,...,|l_1-l_2|$
  • $S=s_1+s_2,s_1+s_2-1,...,|s_1-s_2|$
  • $J=L+S,L+S-1,...,|L-S|$
  • Degeneracy $ = 2J + 1$

• Hund's Rules (for organizing energy levels):

  1. The largest $S$ corresponds to the most stable configuration.
  2. For a given $S$, the largest $L$ corresponds to the most stable configuration.
  3. For a given $S$ and $L$:
     • If the shell is less than half-filled, the smallest $J$ is the most stable.
     • If the shell is more than half-filled, the largest $J$ is the most stable.

• Particle-Hole Symmetry:

  • Instead of focusing on the particles, angular momentum coupling can also apply to the holes. This can simplify the problem in multi-electron systems.

• Procedure for Finding Term Symbols:

  1. Determine total electron arrangements:

     • For each $m_l$ value, there can be an up or down electron. Use the binomial coefficient $(\genfrac{}{}{0ex}{}{n}{x})$, where $n=2(\mathrm{\#}{m}_l)$ and $x=\mathrm{\#}{e}^{-}$.
     • Total possibilities: $\frac{n!}{x!(n-x)!}$.

  2. List all electron configurations and their $M_L$, $M_S$ values:

     • Write out the electron arrangements for each $m_l$ and their associated $M_L$ and $M_S$.

  3. Assign possible $L$ and $S$ values:

     • $L=l_1+l_2,l_1+l_2-1,...,|l_1-l_2|$
     • $S=s_1+s_2,s_1+s_2-1,...,|s_1-s_2|$

  4. Check if the symbols are valid:

     • Assign symbols to configurations and check if they cover all possible $M_L$ and $M_S$ values. If a configuration is used, check it off.

  5. Find $J$ values:

     • Calculate $J=L+S,L+S-1,...,|L-S|$ for each symbol.

  6. Determine degeneracy:

     • Degeneracy $ = 2J + 1$. The sum of all degeneracies should match the total number of microstates.

  7. Organize symbols by energy:

     • Use Hund's rules to order the symbols by increasing energy.

Short Questions:

• Number of Microstates:

  • Microstates = $(2L+1)(2S+1)$

• Possible $J$ Value:

  • Check if $J$ is in the series from $|L-S|$ to $L+S$.

• Allowed Transitions:

  • Selection rules for transitions:
    • $\mathrm{\Delta }l=0,\pm 1$ except when $L=0‡L^{\prime }=0$.
    • $\mathrm{\Delta }S=0$
    • $\mathrm{\Delta }J=0,\pm 1$ except when $J=0‡J^{\prime }=0$.
  • $‡$ indicates non-combinable states.

Letter Notations for $L$:

• $L=0$ corresponds to $S$ (sharp)
• $L=1$ corresponds to $P$ (principal)
• $L=2$ corresponds to $D$ (diffuse)
• $L=3$ corresponds to $F$ (fundamental)
• $L=4$ corresponds to $G$
• $L=5$ corresponds to $H$
• $L=6$ corresponds to $I$