 Intrinsic nuclear quadrupole moment

ID: intrinsic-nuclear-quadrupole-moment

The axial electric quadrupole moment of an intrinsic nuclear charge distribution, with the origin at its center and the

z

axis along its symmetry axis:

Q_{0} = \int ρ_{ch} (r) (2 z^{2} - x^{2} - y^{2}) d^{3} r .

This convention includes electric charge in

Q_{0}

; a tabulated moment divided by the elementary charge

e

instead has units of area. For the uniform spheroid, mapping a uniform unit ball to

(a x, a y, cz)

gives

⟨ x^{2} ⟩ = a^{2} /5

and

⟨ z^{2} ⟩ = c^{2} /5

, so

Q_{0} = \frac{2}{5} Z e (c^{2} - a^{2}), \frac{Q _{0}}{Z e R ^{2}} = \frac{2}{5} (η^{4/3} - η^{- 2/3}) .

The formula explains the signs in the figure. It measures an anisotropy of the nuclear charge distribution, rather than counting individual nucleons on a surface. An illustrative choice

η = 8

gives

a = R / 2

c = 2 R

and

Q_{0} = 7 Z e R^{2} /5

, while preserving

a^{2} c = R^{3}

exactly.

James Rainwater's lecture contrasts measured laboratory electric quadrupole moments with intrinsic moments inferred through Coulomb excitation, including even-even nuclei. Extracting intrinsic shape therefore uses a nuclear model, rather than equating every measured moment to

Q_{0}

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