Scattering-Angle Basis in Physics

• Scattering-angle basis is a representation where the scattering angle is the primary variable, enabling symmetry exploitation and reduced integral dimensionality.
• Its formulation simplifies complex scattering problems by reducing 3D integrals to functions of momentum and angle, facilitating efficient numerical solutions.
• This framework is widely applied in quantum, nuclear, optical, and condensed matter physics, bridging theoretical models with measurable experimental observables.

A scattering-angle basis refers to a representation, formulation, or measurement paradigm in which the scattering angle—typically the angle between the incident and outgoing particle (or wave) momenta—serves as a core variable for describing and analyzing scattering phenomena. This basis is widely utilized across quantum mechanics, atomic and nuclear physics, optics, condensed matter, high-energy particle physics, astrophysics, and photonics, with formalizations tailored to the spin, dimensionality, and physical context of the scattering process. In advanced formulations, the scattering-angle basis enables reductions in problem dimensionality, encodes symmetries, facilitates efficient numerical solutions, and provides direct access to experimentally relevant observables such as differential cross sections, polarization, and dynamical structure factors.

1. Foundations and General Formulation

The scattering-angle basis is rooted in decomposing the scattering process in terms of the initial and final momentum vectors, often focusing on the polar angle ($\theta$) and azimuthal angle ($\phi$) between these vectors in the laboratory or center-of-mass frame.

In quantum mechanical frameworks for spin-carrying projectiles, an explicit momentum-spin product basis is constructed: $$|\mathbf{p} \lambda\rangle = |\mathbf{p}\rangle \otimes |z, \lambda\rangle$$ with $\mathbf{p}$ the momentum (parametrized as $(p, \theta, \phi)$) and $\lambda$ the spin projection. By aligning the incident momentum along the $z$-axis, the problem is simplified, yielding a factorization of azimuthal dependence and a reduction to functions of $p$ and the scattering angle $\theta'$. For example, for scattering of a spin-1/2 particle off a spin-0 target, the potential and the T-matrix elements assume the form

$$V_{\lambda'\lambda}(\mathbf{p}',\mathbf{p}=\mathbf{p} z) = e^{-i(\lambda' - \lambda)\phi'} V_{\lambda'\lambda}(p',p,\theta')$$

and similarly for $T_{\lambda'\lambda}$. The scattering angle $\theta'$ thus emerges as the principal variable defining the angular structure of the problem (Fachruddin et al., 2012).

The utility of adopting the scattering angle as a basis variable includes:

• Direct parameterization of experimental observables.
• Reduction of integral equations' dimensionality.
• Transparent encoding of symmetries and selection rules.

2. Integral Equations and Reduction via Angular Factorization

Exploiting the structure offered by the scattering-angle basis dramatically simplifies the formalism of scattering theory. For spin-dependent scattering, the Lippmann–Schwinger equation for the T-matrix becomes, after azimuthal reduction,

$$T_{\lambda'\lambda}(p',p,\theta') = V_{\lambda'\lambda}(p',p,\theta') + 2\mu \lim_{\epsilon\to 0} \sum_{\lambda''} \int dp''\, \frac{p''^2}{p^2 + i\epsilon - p''^2} \int d\cos\theta''\, V^{\lambda}_{\lambda'\lambda''}(p',p'',\theta',\theta'')T_{\lambda''\lambda}(p'',p,\theta'')$$

where the only angular variable is the (outgoing) scattering angle $\theta'$ (and analogously $\theta''$ in the kernel). The azimuthal dependence is fully encapsulated by simple phase factors, and the kernel $V^{\lambda}_{\lambda'\lambda''}$ is computed by integrating the potential over the intermediate azimuthal angle (Fachruddin et al., 2012).

This approach enables the following:

• Transition from a 3D integral equation to a coupled set of 2D equations in $p,\,\theta'$.
• Efficient numerical solution due to reduced computational complexity.
• Straightforward symmetry analysis.

3. Symmetry Relations and Reduction of Coupled Equations

Physical symmetries constrain the potential and T-matrix matrix elements. In the context of spin-1/2 projectiles on spin-0 targets, time-reversal and other invariances yield

$$V_{\lambda'\lambda}(p',p,\theta') = (-1)^{\lambda'-\lambda} V_{-\lambda',-\lambda}(p',p,\theta')$$

and, at the level of the azimuthally-integrated kernel,

$$V^{\lambda}_{\lambda'\lambda''}(p',p'',\theta',\theta'') = (-1)^{\lambda'-\lambda''} V^{-\lambda}_{-\lambda',-\lambda''}(p',p'',\theta',\theta'')$$

These relations, propagated to $T_{\lambda'\lambda}$, show

$$T_{\lambda'\lambda}(p',p,\theta') = (-1)^{\lambda'-\lambda} T_{-\lambda',-\lambda}(p',p,\theta')$$

In practical terms, only two independent $T$-matrix elements must be computed, and all others are determined by these symmetry relations, halving the computational burden (Fachruddin et al., 2012).

4. Physical Observables and Experimental Consequences

Once the T-matrix elements are solved as functions of the scattering angle, all physical spin observables can be constructed. For the spin-1/2 + spin-0 case, the differential cross section, polarization, analyzing power, and depolarization tensor are given by

$$I_0 = \langle \frac{d\sigma}{d\Omega} \rangle = (4\pi^2 \mu)^2 \left[ |T_{1/2,1/2}(p,p,\theta')|^2 + |T_{-1/2,1/2}(p,p,\theta')|^2 \right]$$

$$P_y = A_y = \frac{2}{I_0} (4\pi^2 \mu)^2 \Im \left\{ T^*_{1/2,1/2} T_{-1/2,1/2} \right\}$$

$$D_{x'x} = D_{z'z} = \frac{1}{I_0}(4\pi^2 \mu)^2 \left[ \left(|T_{1/2,1/2}|^2 - |T_{-1/2,1/2}|^2\right)\cos\theta_{\text{lab}} + 2\Re(T^*_{1/2,1/2} T_{-1/2,1/2})\sin\theta_{\text{lab}} \right]$$

and analogous expressions for other spin-transfer coefficients, with the laboratory angle connected to the scattering angle by an analytic relation (Fachruddin et al., 2012).

This mapping means that the entire suite of measurable quantities in such scattering experiments is encoded in a small set of angularly-resolved T-matrix elements, highlighting the operational significance of the scattering-angle basis.

5. Numerical and Schematic Implementation

Adopting the scattering-angle basis facilitates discretization strategies and efficient matrix methods:

• The reduced integral equations—functions of $p$ and $\theta'$—can be discretized on a two-dimensional grid, and the symmetries reduce the dimensionality of the linear algebraic system to be solved.
• Observables are often output as angular distributions, directly accessible from the calculated $T_{\lambda'\lambda}(p,p,\theta')$ on the grid.
• For practical calculations, the azimuthal phase factors are trivially included, and the numerically intensive part is confined to the physical scattering angle and momentum variables.

The equations and formalism are suitable for both potential models and more general operator structures, provided appropriate azimuthal factorizability exists.

6. Broader Context and Generalizations

The concept of a scattering-angle basis extends beyond nonrelativistic quantum scattering. It underpins:

• Formulations in nuclear and hadronic physics involving partial-wave decompositions, where the scattering angle or its cosine, $\cos\theta$, is the argument of Legendre polynomials.
• Light and neutron scattering in condensed matter, where the momentum transfer $q$ is tied to the scattering angle by $q = (4\pi n/\lambda) \sin(\theta/2)$, and data are presented either as $I(\theta)$ or $I(q)$ distributions (Pauw, 2013, Honecker et al., 2022).
• Modern amplitude-based approaches, including the use of scattering equations and H-bases, which relate variables directly to particle kinematics and, for $n$-body scattering, generalize the notion of a scattering-angle basis (Bosma et al., 2016).
• The application in classical mechanics and semiclassical analyses, where the angular deflection function determines forbidden “shadow” regions for particle trajectories, as with repulsive potentials (Curtright et al., 11 Apr 2024).

7. Impact and Limitations

The value of the scattering-angle basis lies in computational economy, direct connection to physical observables, and the systematic exploitation of symmetry. Nevertheless, it presupposes the ability to factorize azimuthal dependence (which may fail for certain non-central or spin-dependent interactions), and for more complex systems (e.g., higher-spin targets), the bookkeeping of matrix elements may become nontrivial.

In summary, the scattering-angle basis provides a mathematically rigorous, physically transparent, and computationally efficient framework for both theoretical analysis and the quantitative comparison of predictions with experimental results in a wide array of scattering phenomena. It serves as a central tool that bridges rigorous formalism with operational definitions of observables across quantum and classical scattering theory.