Coincidence Algebra Bundle for Decay Quivers: An Algebraic Approach to Gamma-ray Spectroscopy
Liam Schmidt
TL;DR
This work introduces a general algebraic framework for gamma-ray coincidence spectroscopy by modeling decay schemes as quivers and employing the path algebra $K\mathcal{Q}$, extended to a tensor decay space to incorporate non-consecutive transitions. It then defines a coincidence algebra bundle with fiber algebras $\mathcal{A}_d\cong\mathcal{C}$, enabling fibrewise multiplication and detection-mapped probabilities $\Gamma(d)$ and $\Gamma_C(d)$ that account for summing in and out. A coincidence tensor space $\mathcal{T}=\bigoplus_{i=0}^n K\mathcal{D}^{\otimes^i}$ and a physically realized coincidence set $\mathcal{O}$ underpin a fibrewise algebra that captures non-connected path contributions and branching structure. The framework extends the classic transition-matrix approach, offering a more granular, separable treatment of coincidence terms and potential avenues for incorporating angular, energy, and spin structure within a geometric-nuclear-theory context.
Abstract
Motivated by the need for a more comprehensive algebraic structure to calculate coincidence probabilities of a general decay scheme for gamma ray spectroscopy, we model the decay scheme, rather naturally, as a quiver through which we define a decay quiver. The path algebra of quivers is the underlying, more general, algebra for transition matrices that is typically used in modeling decay schemes. The path algebra allows for concatenation of transitions which affords the calculation of cascade probabilities. We extend the path algebra to allow for the multiplication of non-composable paths, i.e., transition that don't directly share a level connecting them. We define the coincidence algebra as the algebra that allows for such an extension and realize it as the fibres for a coincidence algebra bundle, the base space of which is the path algebra where decay schemes live. A given decay schemes coincidence probabilities are calculated on its fibre. \textit{Detection maps} are defined as linear maps on the base space that map transition probabilities to detected probabilities.