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GATE 2022 PH – Question 14

Atomic and Molecular Physics · spin-orbit interaction: L-S and j-j coupling schemes · 1 mark · Multiple choice

An excited Ca configuration is $[\mathrm{Mg}]3p^54s^23d^1$. Which orbital spectroscopic terms occur?

  1. S, P, and D
  2. P, D, and F
  3. P and D
  4. S and P

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Correct answer: (B) P, D, and F

Explanation

The configuration $[\text{Mg}]\,3p^5\,4s^2\,3d^1$ has the closed $4s^2$ shell (which contributes $L=0$, $S=0$), an unfilled $3p^5$ shell and one $3d$ electron.

**The $3p^5$ shell** is equivalent to a single hole in a filled $p$ shell, so it behaves like one $p$ electron: $l_1=1$.

**The $3d$ electron:** $l_2=2$.

**Allowed total orbital angular momentum** for $l_1=1$ and $l_2=2$:
$$L=|l_1-l_2|,\dots,l_1+l_2=1,\ 2,\ 3 .$$

These correspond to the terms **P, D and F** (option B).