Multipole Expansion in Shell Model

Nuclear Physics

The multipole expansion of two-body interactions is a fundamental technique in nuclear shell model calculations. By exploiting rotational invariance, we can decompose the interaction into angular momentum channels, dramatically simplifying the computation of two-body matrix elements.

Rotational Invariance

The key assumption is that the two-body interaction depends only on the distance between particles:

V (1, 2) = V (∣ r_{1} - r_{2} ∣)

Because the interaction is a scalar under rotations, it can be expanded into irreducible spherical tensors. This property is crucial for the shell model, where angular momentum is a good quantum number.

Multipole Expansion

Using the addition theorem for spherical harmonics, the interaction can be expanded as:

V (∣ r_{1} - r_{2} ∣) = k = 0 \sum \infty V_{k} (r_{1}, r_{2}) m = - k \sum k Y_{km} (\overset{r}{^}_{1}) Y_{km}^{*} (\overset{r}{^}_{2})

Each value of $k$ labels an angular-momentum transfer channel. The expansion separates the radial dependence $V_{k} (r_{1}, r_{2})$ from the angular dependence encoded in the spherical harmonics.

Two-Body Matrix Elements

For shell-model states $∣ n l jm ⟩$ , the two-body matrix element takes the form:

⟨ ab ∣ V ∣ c d ⟩ = k \sum (Clebsch-Gordan / 6j / 9j algebra) \times F^{k} (ab, c d)

All angular dependence is handled algebraically through Clebsch-Gordan coefficients and 6j/9j symbols. This is a major advantage: instead of computing multi-dimensional angular integrals, we use well-established angular momentum coupling rules.

Slater Integrals (Radial Part)

The radial physics is encoded in the Slater integrals:

F^{k} (ab, c d) = \int d r_{1} r_{1}^{2} \int d r_{2} r_{2}^{2} R_{a} (r_{1}) R_{c} (r_{1}) V_{k} (r_{1}, r_{2}) R_{b} (r_{2}) R_{d} (r_{2})

where $R_{n l} (r)$ are the radial wavefunctions.

Properties of Slater Integrals

Encodes all radial physics: The entire radial dependence of the interaction is contained in these integrals
Independent of magnetic quantum numbers: The Slater integrals depend only on the principal and orbital quantum numbers, not on $m$ or $j$
Reusable: The same Slater integrals appear in many different matrix elements, making calculations efficient

Benefits of Multipole Expansion

Algebraic simplification: Converts multi-angle integrals into exact angular momentum algebra
Selection rules: Exposes angular momentum selection rules automatically
Separation of concerns: Separates geometry (angular momentum) from dynamics (radial interaction)
Parameterization: Allows effective interactions to be parameterized by a small number of $k$ -channels

Physical Interpretation

Each multipole channel $k$ has a distinct physical meaning:

$k$	Physical Meaning
0	Monopole (average interaction, shell evolution)
2	Quadrupole (collectivity, deformation)
4	Hexadecapole (higher-order correlations)
Higher	Higher-order multipole correlations

Core Idea

Rotational invariance forces the two-body interaction to decompose into multipole ( $k$ ) channels, turning shell-model matrix elements into angular-momentum algebra times reusable radial Slater integrals.

Computational Advantages

This decomposition provides several computational benefits:

Reduced dimensionality: Instead of computing 6D integrals, we compute 2D radial integrals and use algebraic formulas
Selection rules: Many matrix elements vanish due to angular momentum conservation, reducing computational cost
Efficiency: Slater integrals can be precomputed and reused across many matrix elements
Systematic truncation: The multipole expansion can be truncated at a finite $k_{m a x}$ , with higher-order terms typically being smaller

Connection to Effective Interactions

In practice, effective interactions for the shell model are often parameterized in terms of a few multipole channels. This allows:

Fitting to experimental data (binding energies, spectra)
Systematic improvement by including more channels
Physical interpretation of interaction strengths