*2.3. Quadrupole Sum Rules*

The nuclear shape can be inferred indirectly from transition probabilities or spectroscopic quadrupole moments, but this approach is not always unambiguous and generally depends on comparisons with models. An alternative model-independent approach, proposed by K. Kumar [1] and D. Cline [7], exploits the specific properties of the electromagnetic multipole operators. As these operators are spherical tensors, their zero-coupled products are rotationally invariant. The expectation values of these products are observables, and they are strictly related to the parameters describing the shape of the charge distribution.

The electric quadrupole operator in the principal axis system can be represented using the variables *Q* and *δ*, whose expectation values are equivalent to the Hill–Wheeler parameters (*β*2, *γ*) describing the quadrupole shape [1,7]. The simplest invariants read:

$$\{E2 \times E2\}^0 = \frac{1}{\sqrt{5}} Q^2 \, , \tag{3}$$

$$\left\{ \left[ E2 \times E2 \right]^2 \times E2 \right\}^0 = -\sqrt{\frac{2}{35}} Q^3 \cos 3\delta. \tag{4}$$

The expectation values of these invariants for a state *In* can be expressed through *E*2 matrix elements defined in the laboratory system. For instance:

$$
\langle I\_{\rm li} \vert Q^2 \vert I\_{\rm li} \rangle = \frac{\sqrt{5}(-1)^{2I\_{\pi}}}{\sqrt{2I\_{\pi} + 1}} \sum\_{m} M\_{mm} M\_{mm} \begin{Bmatrix} 2 & 2 & 0 \\ I\_{\rm n} & I\_{\rm n} & I\_{\rm m} \end{Bmatrix} \tag{5}
$$

$$
\langle I\_n | Q^3 \cos 3\delta | I\_n \rangle = -\sqrt{\frac{35}{2}} \frac{(-1)^{2I\_n}}{2I\_n + 1} \sum\_{ml} M\_{nl} M\_{lm} M\_{mn} \begin{Bmatrix} 2 & 2 & 2 \\ I\_n & I\_m & I\_l \end{Bmatrix} \tag{6}
$$

where *Mab* ≡ *Ia*||*E*2||*Ib* and the expression in curly brackets is a 6*j* coefficient. Higherorder invariants can be defined, such as *Q*4, which can be linked to the dispersion in *Q*2 via

$$
\sigma(Q^2) = \sqrt{\langle Q^4 \rangle - \left( \langle Q^2 \rangle \right)^2}. \tag{7}
$$

A similar definition applies to *σ*(*Q*<sup>3</sup> cos 3*δ*). In principle, this approach can be extended to more complex, non-quadrupole shapes.

The invariants obtained from quadrupole sum rules provide a model-independent description of the nuclear shape in the intrinsic reference system. However, the experimental determination of such invariants requires numerous matrix elements to be known. While for the lowest-order shape invariant, *Q*2, all matrix elements enter the sum in squares, this is not true for most higher-order invariants. In particular, the *Q*<sup>3</sup> cos <sup>3</sup>*δ* invariant is constructed from triple products of *E*2 matrix elements, *In*||*E*2||*IlIl*||*E*2||*ImIm*||*E*2||*In*, where |*In* is the state in question and |*Il* and |*Im* are the intermediate states. The diagonal matrix elements (i.e., |*Il* = |*Im* ) and their signs are necessary to extract this invariant, as well as the relative signs of all relevant transitional matrix elements.

While the sums in Equations (5) and (6) formally run over all intermediate states that can be reached from the state in question via a single *E*2 transition, usually only a few key states contribute to the invariant. In particular, for the ground state of an even– even nucleus, the contributions to *Q*2 are dominated by the coupling to the 2<sup>+</sup> <sup>1</sup> state, which typically amounts to well over 90% of the total. Similarly, the largest contributions to *Q*<sup>3</sup> cos <sup>3</sup>*δ* for the ground state come from the 0<sup>+</sup> <sup>1</sup> ||*E*2||2<sup>+</sup> <sup>1</sup> 2<sup>+</sup> <sup>1</sup> ||*E*2||2<sup>+</sup> <sup>1</sup> 2<sup>+</sup> <sup>1</sup> ||*E*2||0<sup>+</sup> 1 and 0<sup>+</sup> <sup>1</sup> ||*E*2||2<sup>+</sup> <sup>1</sup> 2<sup>+</sup> <sup>1</sup> ||*E*2||2<sup>+</sup> <sup>2</sup> 2<sup>+</sup> <sup>2</sup> ||*E*2||0<sup>+</sup> <sup>1</sup> products. The situation becomes much more complicated for excited states, and the number of intermediate states that need to be included in the sum rules varies from one case to another. While theoretical approaches can, in principle, provide a complete set of electromagnetic matrix elements, this is not always true for experiments. Systematic studies employing the Shell Model addressed this convergence issue [13–15]. The contributions of individual products of matrix elements to the experimentally determined invariants have also been analysed in some cases [14,16–18].
