Weird Universe. In another universe, the electron is a spin -(3)/(2) rather than a spin -(1)/(2)

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Weird Universe. In another universe, the electron is a spin...

$\textbf{Weird Universe.}$ In another universe, the electron is a spin -$\frac{3}{2}$ rather than a spin -$\frac{1}{2}$ particle, but all other physics are the same as in our universe. In this universe, (a) what are the atomic numbers of the lightest two inert gases? (b) What is the groundstate electron configuration of sodium?

Key Concepts

Pauli Exclusion Principle

The Pauli Exclusion Principle states that no two electrons can occupy the same quantum state simultaneously. This rule applies to both the spatial and spin components of the electron’s state. With an increased spin multiplicity, each spatial orbital can hold more electrons before the orbital is fully occupied, yet the principle still restricts electrons to unique combinations of quantum numbers.

Electron Configuration and Energy Ordering

Electron configuration involves the distribution of electrons into various orbitals based on their energy levels, following an order that minimizes the overall energy of the atom. The energy ordering is determined by quantum numbers and influences both the filling sequence of orbitals and the eventual closure of shells. Modifications in the intrinsic properties, such as an increase in spin multiplicity, affect the maximum number of electrons in each orbital and thus alter the electron configurations of atoms.

Electron Spin and Multiplicity

Electron spin is an intrinsic form of angular momentum carried by electrons and is quantified by a spin quantum number. The spin quantum number determines the number of available spin states through its multiplicity (given by 2s + 1). Changing the spin value alters this multiplicity and, consequently, the number of electrons that can share the same spatial orbital. This concept is fundamental to understanding how modifications in spin, such as from 1/2 to 3/2, affect the filling of atomic orbitals.

Quantum Shells and Orbital Capacities

Atomic electrons are arranged into shells characterized by the principal quantum number (n). Each shell can accommodate a number of electrons determined by both the orbital degeneracy (related to the magnetic quantum number, with a typical 2l+1 count for a given orbital angular momentum quantum number l) and the spin multiplicity. When the spin multiplicity is altered, the electron capacity of each shell changes proportionally, affecting the overall structure and closure of electron shells.

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