Atomic Structure Theory

5.1 Hyperfine Structure 139
If we let Qij represent the nuclear quadrupole moment tensor, then the
scalar potential is given by
φ(r) = 1
4πɛ0
ij
xixj
2r 5 Qij .
The quadrupole tensor Qij is a traceless symmetric tensor of rank 2; it therefore
has five independent components. For a classical charge distribution ρ(r),
the Cartesian components of the quadrupole tensor are given by
Qij = d 3 r (3xixj − r 2 δij)ρ(r) .
The components of this tensor can be transformed to a spherical basis and
expressed in terms of the five components of the second-rank spherical tensor
Qλ defined by
Qλ = d 3 rr 2 C 2 λ(ˆr)ρ(r) ,
where C 2 λ (ˆr) is a normalized spherical tensor of rank 2. In particular, Q33 =
2Q0. The potential due to the quadrupole, expressed in a spherical basis, is
φ(r) = 1
4πɛ0
λ
(−1) λ C2 λ (ˆr)
r 3
Q−λ .
Here, Qλ is an irreducible tensor operator of rank 2 acting in the space of
nucleon coordinates and spins. The c-number quadrupole moment of the nucleus
Q is given in terms of the expectation value of the operator Q0 in the
extended state:
Q def
=2〈II|Q0|II〉 . (5.4)
The nuclear quadrupole moment Q is dimensionally a charge times a length
squared. It is commonly written in units of |e|×barn.
The hyperfine interaction Hamiltonian for a relativistic electron with the
nuclear magnetic-dipole and electric-quadrupole moments becomes
hhfs(r) = e
(−1)
4πɛ0 λ
λ i√2[α · C (0)
1λ (ˆr)]
cr2 µ−λ +
(−1)
λ
λ C2 λ (ˆr)
Q−λ .
r3 (5.5)
Both the electric and magnetic interactions are thereby expressed in terms of
tensor operators and the hyperfine interaction Hamiltonian takes the form
hhfs(r) =
(−1) λ t k λ(ˆr) T k −λ ,
kλ
where t k q (r) is an irreducible tensor operator of rank k that acts on electron
coordinates and spin, and T k q is a rank k irreducible tensor operator that