The spin of fundamental particles determines if they are bosons or fermions. The atoms also have bosonic or fermionic behavior, for example $\require{mhchem}\ce{^4He}$ has bosonic and $\ce{^3He}$ has fermionic statistics. Which quantity of atom determines its statistics?
Answer
A composite particle consisting of an even (odd) number of fermions behaves like a boson (fermion).
Swapping $\require{mhchem}\ce{^3He}$ atoms involves swapping an odd number of fermions (3 nucleons and 2 electrons): an odd number of sign changes of the wave function corresponds to no sign change or bosonic symmetry. Swapping $\ce{^4He}$ atoms involves swapping an even number of fermions (4 nucleons and 2 electrons): an even number of sign changes of the wave function corresponds to a sign change or fermionic behavior.
The same applies to other atoms: $\ce H$ = 1 proton and 1 electron = 2 (even) fermions: bosonic, $\ce{D}$ = 2 nucleons and 1 electron = 3 (odd) fermions: fermionic, etc.
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