Allowed Atomic Transitions

Atomic transitions between energy levels are governed by selection rules. For electric dipole transitions, which are the most common type, the following rules apply under LS coupling:

• Total Angular Momentum (J): The change in total angular momentum quantum number $\Delta J$ must be $0, \pm 1$. However, a transition from $J=0$ to $J=0$ is forbidden.
• Orbital Angular Momentum (L): The change in orbital angular momentum quantum number $\Delta L$ must be $0, \pm 1$.
• Spin Angular Momentum (S): The change in spin angular momentum quantum number $\Delta S$ must be $0$. This is known as the spin selection rule.
• Parity: The parity of the initial and final states must be different. Parity is determined by $(-1)^L$. This means a transition is allowed only if $L$ changes from even to odd, or odd to even (i.e., $\Delta L$ must be odd if $\Delta L \ne 0$, or if $\Delta L = 0$, then the states must still have different parity which is not possible under $(-1)^L$). More precisely, the parity of the state with orbital quantum number $L$ is $(-1)^L$. The Laporte rule states that transitions between states of the same parity are forbidden. Thus, $\Delta L$ must be odd for allowed transitions (since if $L \rightarrow L'$, $(-1)^L \neq (-1)^{L'}$ implies $L$ and $L'$ have different parity, meaning $L-L'$ is odd, so $\Delta L$ is odd). If $\Delta L=0$, the parity rule $(-1)^L \neq (-1)^{L'}$ requires $L$ to change parity, which contradicts $\Delta L=0$. Hence, for electric dipole transitions, $\Delta L = \pm 1$ or $\Delta L=0$ with parity change. Combining $\Delta L$ rule and parity rule, the effective rule becomes $\Delta L = \pm 1$. The $\Delta L=0$ transition is allowed only if the states have different parity, which happens for higher order transitions (like magnetic dipole), but for electric dipole, $\Delta L=0$ means same parity unless combined with another change. So for electric dipole, $\Delta L = \pm 1$ and parity must change.

Let's analyze the given transition from the initial state ³F₄.

For the state ³F₄:

• Spin Multiplicity $2S+1 = 3 \implies 2S = 2 \implies S = 1$.
• Orbital Angular Momentum $L$: F corresponds to $L=3$.
• Total Angular Momentum $J = 4$.
• Parity is $(-1)^L = (-1)^3 = -1$ (odd).

Now, let's check each option for the transition $^{3}$F$_{4} \rightarrow$ Final State.

Transition Option 1: $^{3}$F$_{4} \rightarrow ^{3}$D$_{3}$

Initial State: $S=1, L=3, J=4$, Parity = odd.

Final State: $^{3}$D$_{3}$

• $2S'+1 = 3 \implies S' = 1$.
• $L'$: D corresponds to $L'=2$.
• $J' = 3$.
• Parity is $(-1)^{L'} = (-1)^2 = +1$ (even).

Checking Selection Rules:

• $\Delta J = |J' - J| = |3 - 4| = 1$. Allowed ($\Delta J = \pm 1$).
• $\Delta L = |L' - L| = |2 - 3| = 1$. Allowed ($\Delta L = \pm 1$).
• $\Delta S = |S' - S| = |1 - 1| = 0$. Allowed ($\Delta S = 0$).
• Parity changes from odd to even. Allowed (Parity must change).

All selection rules are satisfied for this transition.

Transition Option 2: $^{3}$F$_{4} \rightarrow ^{1}$D$_{3}$

Initial State: $S=1, L=3, J=4$, Parity = odd.

Final State: $^{1}$D$_{3}$

• $2S'+1 = 1 \implies S' = 0$.
• $L'$: D corresponds to $L'=2$.
• $J' = 3$.
• Parity is $(-1)^{L'} = (-1)^2 = +1$ (even).

Checking Selection Rules:

• $\Delta S = |S' - S| = |0 - 1| = 1$. Not allowed ($\Delta S \ne 0$).

This transition violates the spin selection rule.

Transition Option 3: $^{3}$F$_{4} \rightarrow ^{3}$P$_{4}$

Initial State: $S=1, L=3, J=4$, Parity = odd.

Final State: $^{3}$P$_{4}$

• $2S'+1 = 3 \implies S' = 1$.
• $L'$: P corresponds to $L'=1$.
• $J' = 4$.
• Parity is $(-1)^{L'} = (-1)^1 = -1$ (odd).

Checking Selection Rules:

• $\Delta L = |L' - L| = |1 - 3| = 2$. Not allowed ($\Delta L \ne 0, \pm 1$).
• Parity changes from odd to odd. Not allowed (Parity must change).

This transition violates the $\Delta L$ rule and the parity rule.

Transition Option 4: $^{3}$F$_{4} \rightarrow ^{3}$D$_{2}$

Initial State: $S=1, L=3, J=4$, Parity = odd.

Final State: $^{3}$D$_{2}$

• $2S'+1 = 3 \implies S' = 1$.
• $L'$: D corresponds to $L'=2$.
• $J' = 2$.
• Parity is $(-1)^{L'} = (-1)^2 = +1$ (even).

Checking Selection Rules:

• $\Delta J = |J' - J| = |2 - 4| = 2$. Not allowed ($\Delta J \ne 0, \pm 1$).

This transition violates the $\Delta J$ rule.

Based on the analysis of selection rules, only the transition $^{3}$F$_{4} \rightarrow ^{3}$D$_{3}$ is allowed for electric dipole transitions in an atomic system.