Semkow Gamma Formalism

• Semkow Gamma Formalism is a matrix-based framework that models true coincidence summing effects in multi-level nuclear decay cascades.
• It employs explicit matrix expansions and power-series objects to quantify both summing-in and summing-out phenomena in singles and gated measurements.
• The formalism distinguishes between ontic and epistemic events, providing statistical bounds on measurable cascade features and guiding experimental corrections.

The Semkow Gamma Formalism (SGF) is a matrix-based summing correction framework for gamma-ray spectroscopy, generalizing and extending prior approaches to model true coincidence summing effects in multi-level nuclear decay cascades. It provides a rigorously defined means for quantifying and correcting for summing-in and summing-out phenomena, for both singles and coincidence measurements, and derives statistical bounds on what features of the underlying physical cascade are, in principle, experimentally measurable. The formalism encompasses arbitrary coincidence filters—including 180° and gated conditions—through explicit matrix expansions, and incorporates the distinction between ontic (physical) and epistemic (detector-observed) events (Schmidt, 10 Nov 2025).

1. Foundational Structure and Definition

The SGF begins with a parent $β^-$ nucleus decaying to an $(n+1)$-level daughter, characterized by:

• A branching vector $f = (f_0, f_1, ..., f_n)$, $\sum_i f_i = 1$.
• A strictly lower-triangular transition matrix $x = [x_{ij}]$ for $i \to j$, $j < i$, with $\sum_{j=0}^{i-1} x_{ij} = 1$.

Experimental observables for each gamma transition $\gamma_{ij}$ are encapsulated as:

• $a_{ij} = x_{ij} \epsilon^p_{ij}$ (peak efficiency),
• $(n+1)$0 (total efficiency per detector for $(n+1)$1 detectors),
• $(n+1)$2, with $(n+1)$3 recovering the “summing-out” matrix and $(n+1)$4 enabling 180-degree coincidence filtering.

Key power-series objects up to order $(n+1)$5 are:

• $(n+1)$6
• $(n+1)$7
• $(n+1)$8

The SGF mapping is

$(n+1)$9

where $f = (f_0, f_1, ..., f_n)$0 denotes the diagonal matrix with entries $f = (f_0, f_1, ..., f_n)$1. The physically relevant “singles” decay matrix is $f = (f_0, f_1, ..., f_n)$2. This formulation simultaneously incorporates both summing-in (via $f = (f_0, f_1, ..., f_n)$3) and summing-out (via $f = (f_0, f_1, ..., f_n)$4).

2. Multiplicity Expansion and Correction Methodology

SGF organizes summing corrections according to ontic multiplicities $f = (f_0, f_1, ..., f_n)$5 of emitted $f = (f_0, f_1, ..., f_n)$6 quanta along each decay branch of index $f = (f_0, f_1, ..., f_n)$7 ($f = (f_0, f_1, ..., f_n)$8):

$f = (f_0, f_1, ..., f_n)$9

with

$\sum_i f_i = 1$0

and closed form:

$\sum_i f_i = 1$1

This formula yields the detection probability for $\sum_i f_i = 1$2 stemming from a branch with $\sum_i f_i = 1$3 emitted gammas, $\sum_i f_i = 1$4 of which contribute—directly or by summing—to the measured event.

Summing-out and summing-in corrections for 180°-coincident experiments are obtained by:

• $\sum_i f_i = 1$5, $\sum_i f_i = 1$6 (coincidence filters)
• Summing-out: $\sum_i f_i = 1$7
• Summing-in: $\sum_i f_i = 1$8

Residual deviation from the exact decay matrix $\sum_i f_i = 1$9 where $x = [x_{ij}]$0 is

$x = [x_{ij}]$1

with

$x = [x_{ij}]$2

Crucially, $x = [x_{ij}]$3, so the method is exact for two-gamma cascades but becomes imperfect for higher multiplicities, with the deviation magnitude increasing with $x = [x_{ij}]$4 and decreasing as the number of detectors $x = [x_{ij}]$5 increases (Schmidt, 10 Nov 2025).

3. Partitioned Gamma Formalism for Gated Coincidences

For experiments with explicit gating—analyzed via the Partitioned Gamma Formalism (PGF)—the SGF formalism is partitioned around a gate transition $x = [x_{ij}]$6, with only two strictly lower-triangular blocks of the probability matrix $x = [x_{ij}]$7 populated.

The probability matrix is

$x = [x_{ij}]$8

where $x = [x_{ij}]$9 (with $i \to j$0) and $i \to j$1 combines “up” and “down” SGF-matrix segments around the gate, using partitioned forms of $i \to j$2 and $i \to j$3.

Summing corrections for gated scenarios involve multiple “order” terms, with, for example,

• $i \to j$4,
• $i \to j$5, with analogous higher order corrections, each tracking contributions from additional gamma emissions into coincidence.

The net deviation

$i \to j$6

captures the statistically irreducible discrepancy after application of all calculable corrections (Schmidt, 10 Nov 2025).

4. Ontic and Epistemic Events; Measurability Concepts

SGF makes a clear distinction between:

• Ontic events: the physical, micro-causal decay cascades (gamma emission sequences, directions, and multiplicities), unobservable directly.
• Epistemic events: the detector-recorded events (energies, hits, coincidences), subject to imperfect detector response, efficiency, geometry, and summing ambiguities.

An ontic event is termed “measurable” if it maps injectively to a unique epistemic pattern; “sufficiently measurable” if it is probabilistically equivalent (under the model) to a unique class of epistemic events. Hidden ontic events—such as simultaneous emission of two gammas at zero separation—may be experimentally inseparable from other configurations (such as a single higher-energy gamma due to summing).

The 180°-coincidence correction approach assumes probabilistic equivalence between zero-degree summing and measured 180° coincidences, but SGF quantifies the residual deviation $i \to j$7 (or $i \to j$8) as a statistical bound on the validity of this assumption: it represents the limit to the “sufficient measurability” of coincidence summing corrections. This value should be compared to experimental uncertainties to judge the adequacy of such corrections (Schmidt, 10 Nov 2025).

5. Practical Illustration: The $i \to j$9 Case

For a two-level daughter system ($j < i$0), only three nonzero multiplicity blocks appear:

• $j < i$1 describes direct single-gamma detections,
• $j < i$2 covers detection of the second gamma directly,
• $j < i$3 encodes genuine summing scenarios.

Explicitly, for $j < i$4 as defined above,

$j < i$5

Only $j < i$6 contains true summing contributions. For $j < i$7, $j < i$8 and the 180° method is exact. If a third gamma is emitted, $j < i$9; this deviation increases with observed multiplicity $\sum_{j=0}^{i-1} x_{ij} = 1$0 and decreases with detector number $\sum_{j=0}^{i-1} x_{ij} = 1$1 (Schmidt, 10 Nov 2025).

6. Scope and Significance

SGF unifies and generalizes the original Semkow matrix approach, subsuming various coincidence- and gating-correction schemes into a single matrix formalism. It applies to Markov-type cascades with no angular correlations or external sources and accommodates auxiliary radiation via modified efficiency terms or “virtual” branches. The formalism is non-linear in its second (coincidence filter) argument, necessitating individual evaluation for each filter combination.

Through the multiplicity expansion $\sum_{j=0}^{i-1} x_{ij} = 1$2 and partitioned matrices, SGF and its Partitioned Gamma Formalism (PGF) comprehensively enumerate all possible contributions—direct, summed, or otherwise—to the experimentally observed gamma spectra. The formalism exposes that all 180°-coincidence-based summing corrections are fundamentally incomplete for multiplets with $\sum_{j=0}^{i-1} x_{ij} = 1$3, with the irreducible error $\sum_{j=0}^{i-1} x_{ij} = 1$4 providing the statistical upper bound on achievable experimental “measurability” of true coincidence effects (Schmidt, 10 Nov 2025).