Gamma-ray intensities in multi-gated spectra
GGamma-ray intensities in multi-gated spectra
Camille Ducoin , Guillaume Maquart , Olivier St´ezowski Univ Lyon, Universit´e Lyon 1, CNRS/IN2P3, IPN-Lyon, F-69622, Villeurbanne, France
The level structure of nuclei offers a large amount and variety of information to improve ourknowledge of the strong interaction and of mesoscopic quantum systems. Gamma spectroscopy isa powerful tool to perform such studies: modern gamma multi-detectors present increasing perfor-mances in terms of sensitivity and efficiency, allowing to extend ever more our ability to observeand characterize abundant nuclear states. For instance, the high-spin part of level schemes oftenreflects intriguing nuclear shape phenomena: this behaviour is unveiled by high-fold experimen-tal data analysed through multi-coincidence spectra, in which long deexcitation cascades becomeobservable. Determining the intensity of newly discovered transitions is important to characterizethe nuclear structure and formation mechanism of the emitting levels. However, it is not trivialto relate the apparent intensity observed in multi-gated spectra to the actual transition intensity.In this work, we introduce the basis of a formalism affiliated with graph theory: we have obtainedanalytic expressions from which data-analysis methods can eventually be derived to recover this linkin a rigorous way.
I. INTRODUCTION
Gamma spectroscopy is one of the most important experimental techniques allowing to characterize the quantumstructure of atomic nuclei. Gamma spectra produced under selection criteria that impose coincidence relations betweenthe photon emissions are of particular importance. First, setting different coincidence conditions and observing theresulting presence or absence of gamma rays allows to construct the level scheme of the nucleus. Furthermore,coincidence conditions have a selective role that is crucial if the studied nucleus is only one of the possible exit channelsof the production reaction (e.g. fusion-evaporation, fission...), and if we want to observe low-intensity gamma rays.This consideration is especially relevant in now-a-days experiments aiming at the knowledge of exotic nuclei (e.g.neutron-rich nuclei), and exotic states in nuclei (e.g. high-spin states), both having low production cross-section. Inthis context, progress in gamma-detector resolution and efficiency has been very important in the last decades, and isthe leading criterion for the development of new detector arrays such as AGATA [1] and GRETA [2]. Working withhigh-fold data, multi-gated spectra allow to observe nuclear structure regions that would be otherwise concealed bythe background.With increasing amount and complexity of experimental data, efforts have been dedicated to establish automatedprocedures to construct level schemes on the basis of coincidence data (see e.g. [3–7]). Although important stepshave been taken, these works generally conclude that human intervention is still crucial to obtain valid level schemeswhen realistic data are employed. Among the cited papers, of specific importance for the present study is the workof Demand et al. [7], where the relation between nuclear level scheme and graph theory is explored. Although ourgoal is different, since we are mainly focused on characterizing a new transition appearing in a previously known levelscheme, the framework of graph theory has proven very useful in the treatment of our problem. Many textbooks existon this mathematical formalism that has wide-spread applications; we only indicate here one of the classic references,by Bondy and Murty [8].Besides level-scheme solving, many other works have been dedicated to the improvement of gamma-ray data analysis.The most practical ones concern software developments that offer to the user an optimized environment to obtain andanalyze gated spectra, in relation with nucleus level scheme, such as the famous toolkit Radware [9]. Concerning theissue of intensity measurements, we can cite works on γ − γ coincidence matrices [10], a method focused on the effectof angular correlations [11], and studies to quantify coincidence-summing effects, especially the analytic approachpresented in [12]. The work of Beausang et al. [13] calls attention on the bias on intensity measurements due to thespiking effect in gated spectra, which is particularly relevant for the present study.The present work is focused on the issue of relating gamma-ray intensities observed in multi-gated spectra to thecorresponding absolute emission probabilities. This relation is non-trivial, especially in the case of gating conditionswhere different combinations of photons in coincidence are allowed to select an event. It is important to establishthis relation in a rigorous way in order to obtain accurate values of the emission probabilities, which contain valuableinformation to characterize the nuclear structure and also to study reaction mechanism in the light of level feeding.Starting from the graph-theory-inspired framework established by Demand et al. [7], we have developed a formalismand analytic expressions to establish this relation on a well-controlled basis. The present article is dedicated to thepresentation of this formalism: for clarity, this is done using simplifying assumptions that place this study in a veryidealized framework (in substance, every transition yields a photon that is fully detected). However, this first version a r X i v : . [ phy s i c s . d a t a - a n ] F e b a) Level scheme l l l l l E n e r g y t t t t t t b) Level-space graph l l t t t t t t l l l l t t t t t t l c) Transition-space graph l l FIG. 1: (Color online) A simple illustration of nuclear structure representations: (a) usual level-scheme representation, (b) rep-resentation of a level-space graph, (c) representation of a transition-space graph. Levels are denoted by l i (ordered by increasingenergy), transitions are denoted by t i (ordered by decreasing energy of the emitting level). In the graph representations, verticesare represented as dots and directed edges as arrows. will be used as a sound basis for further developments, and we plan next to make this formalism applicable to theanalysis of real experimental data.The present article is organized as follows. Section II presents the level-space and transition-space treatmentof nuclear structure, in the framework of graph theory. Section III is dedicated to the detailed description of theformalism we have derived. It includes the demonstration of the analytical relations we have obtained to express thegated intensities in terms of the absolute emission probabilities and of the transition probability matrix deduced frombranching ratios. An example of application to a schematic level scheme is presented in Section IV. A summary andplan for future developments are given in Section V. II. NUCLEUS DEEXCITATION AS AN APPLICATION OF GRAPH THEORY
The quantum states of an atomic nucleus are linked by a network of possible transitions, whose probabilities aredetermined by the properties of the physical interaction and the quantum numbers associated with the differentstates. This is one of the numerous situations that can be modelled by a mathematical object called ”graph”. Agraph G is defined as a triple ( V , E , Ψ), where V is a set of vertices, E a set of edges (links between vertices), and Ψa relationship associating each edge with a pair of vertices. The usual representation of nuclear structure is a levelscheme: it that can be seen as the representation of a graph for which the elements of V are the quantum states, andthe elements of E are the existing transitions. In addition to identifying the levels associated with a given transition,the relationship Ψ can carry some information about the probability of this transition: in this case, G is called aweighted graph. If transitions occur in response to an excitation, they can go towards either higher or lower energystates, and their probabilities depend on the properties of the excitation source. However, we will focus on the studyof nuclear deexcitation cascades following the formation of an excited nucleus: in this case, transitions occur onlytowards lower energy levels and their probabilities are determined by the branching ratios. Nuclear branching ratiosonly depend on the structure of the nucleus under study, and they are abundantly documented in databases such asENSDF [14]. Each transition can happen only in one direction (from higher to lower energy), which means that wehave a directed graph (also called digraph). To summarize, the level scheme that describes the different energy statesof a nucleus and the transitions that can occur during its deexcitation is modelled mathematically by a weighteddigraph.Concerning the deexcitation cascades, each one defines a directed path, which is a sequence of distinct verticeslinked by specific directed edges. In the present work, we will consider only simple graphs: namely, there is no morethan one edge between two vertices. In physical terms, this means that we only consider the existence of a transitionfrom one energy level to another, and we do not distinguish different kinds of transitions between these two levels. Infuture work, we will also consider different kinds of transitions in order to distinguish for instance between gammatransitions and electronic conversions: in this case, several edges can link two vertices, and the graph is no longer asimple graph. A. Level space
Level schemes are the usual representation of the structure of a nucleus, and they are focused on the descriptionof nuclear state properties. This point of view is called the level-space representation [7]. As stated above, in termsof graph theory, nuclear levels are vertices and transitions are edges: this is illustrated by Figure 1. We can noticethat usually, each excited level is able to decay following a cascade down to the ground state (GS), although nucleusdisintegration may also occur before reaching the GS. Assuming that there is always a decay branch that reaches theGS, we obtain a connected graph: it cannot be separated in two non-communicating sets of vertices.Let us specify that, in the present work, the level space is limited to the discrete part of the spectrum: the continuumis not explicitly treated. In this approach, in order to describe the deexcitation of a nucleus formed in a given reaction,the following information is needed: • List of possibly involved nuclear levels: vector l = { l , ..., l D l } . The number of levels D l gives the dimension ofthe level space. Note that this list can vary depending on the way the excited nucleus is produced. • Primary feeding: vector F (1) = { F (1)1 , ..., F (1) D l } . Each component F (1) i gives the probability that level l i is thefirst discrete level to be populated in the decay cascade. This quantity, again, highly depends on the way thenucleus is produced. It corresponds to either a direct feeding at the time of nucleus formation, or a decay fromthe continuum part of the spectrum. • Branching ratios: matrix B of dimension D l × D l . One element B ij gives the probability that level l i decaysdirectly to level l j . Since B ij does not depend on the way level l i was formed, these elements only depend onthe nucleus itself, and can be found in nuclear databases. Note that, in terms of graph theory, the branchingmatrix is the so-called adjacency matrix describing the connexions between the vertices of a graph.In this approach, the probability of a given transition t x = l i → l j is given by: P x = F i B ij , where F i is the totalfeeding of level l i , i.e. the probability that level l i is populated during the deexcitation. This can be expressed as afunction of the primary feeding vector F (1) and the branching matrix B . Indeed, the branching matrix B can be usedto determine a secondary feeding matrix F , where the element F ij gives the probability that level l j is populated iflevel l i has been populated before, with an arbitrary number of steps inbetween. This corresponds to the followingrelation, adapted from the derivation presented by Demand et al. [7]: F ij = (cid:32) ∞ (cid:88) n =1 B n (cid:33) ij = (cid:0) [ I − B ] − − I (cid:1) ij F = ∞ (cid:88) n =1 B n = [ I − B ] − − I where I is the identity matrix. For the element F ij , each term of the summation over n expresses the probability that l j is reached n steps after l i . For instance, B ij = (cid:80) k B ik B kj is the probability that l i decays to l j in two steps, andso on. The second part of the equality just results from the well-known Taylor development of [ I − B ] − . From thesecondary feeding matrix F , we obtain the secondary feeding vector F (2) , which gives the probability for each levelto be populated by the decay of any other discrete level. For each level l i , we have: F (2) i = (cid:88) k F (1) k F ki and the total feeding is simply given by F i = F (1) i + F (2) i . We can thus obtain the occurrence probability P x of eachtransition t x using input on level-space quantities F (1) (primary feeding) and B (branching-ratio matrix). However, forour purpose, we also need to express the probability of a transition under the condition that other specific transitions(gates) occur in the same deexcitation cascade. To treat this problem, it is more straightforward to adopt a differentpoint of view, the transition-centered description of the deexcitation. B. Transition space
As pointed out in the work of Demand et al. [7], although level space offers the most natural representation ofnuclear properties, it can be more useful in the framework of experimental data analysis to switch to a transition-centered representation. Indeed, transitions are the experimental observables from which the level scheme has tobe deduced. The set of observed transitions is then the natural starting point in the search for an automated levelscheme construction procedure. Our purpose is different, since it aims at adding further knowledge to a partiallyknown level scheme. However, also in our case, quantities associated with the transition space are the relevant inputneeded to determine what we want: an expression of the gamma-ray intensities measured in multi-gated spectra. Incontrast with the approach of Demand et al. where the transition space contains only the observed transitions, weconsider here that it contains all the transitions occurring during the deexcitation process: we make the simplifyingassumption that every transition is detected, leaving for later work further refinement of the formalism.In the transition-space approach, in terms of graph theory, transitions are vertices. The edges are links betweentransitions: namely, a level (in our approach), or possibly a group of levels linked by unobserved transitions (in theobservable-based approach developed by Demand et al.). Note however that, even in our case, the situation is notsymmetrical to the one in the level space: there is no one-to-one correspondance between the set of edges and the setof levels, since one level can be associated with several edges (one for each of its decay modes). This can be seen inFigure 1, where we can also notice that the transition-space graph is not necessarily connected. In order to describethe deexcitation process, the following information is needed: • List of possibly involved transitions (dependent on the nucleus formation mechanism): transition vector t = { t , ..., t D t } , where D t is the dimension of the considered transition space. • Transition probabilities: vector P = { P , ..., P D t } , giving the probability of each transition to occur during adeexcitation cascade. • Adjacency matrix A , where the element A ij gives the probability that the transition t i is immediately followedby the transition t j .Note that the above transition-space quantities can be easily deduced from level-space input (level vector l , primaryfeeding vector F (1) , branching matrix B ): • t is obtained by listing all possible transitions from one level to the other, using l and B ; • as shown in the previous subsection, P is deduced from F (1) and B ; • A is closely related to B . Let us call l x,e the emitting level of a transition t x and l x,r its receiving level: anelement A ij is non-zero only if l i,r = l j,e , and in this case it is equal the branching ratio of the decay mode from l j,e to l j,r .We can remark that conversely, the level-space fundamental quantities ( l , F (1) , and B ) could be deduced from thetransition-space ones ( t , P , and A ), if each transition t x is associated with identified emitting and receiving levels; ifnot, we have to face the difficulties of level-scheme reconstruction. We will not address this subject.Let us now introduce a transition-space quantity that occupies a central place in the formalism we are developing:the transition probability matrix P . The relation between A and P is analogous to the one obtained in level spacebetween B (branching matrix) and F (secondary feeding matrix). Namely, an element P ij gives the probability thattransition t j occurs if transition t i has occurred before, with an arbitrary number of steps inbetween. This correspondsto the relation presented in Ref. [7]: P = ∞ (cid:88) n =1 A n = [ I − A ] − − I (1) III. FORMALISM TO CALCULATE GAMMA-RAY INTENSITIES IN GATED SPECTRA
A gate condition selects events for which a given set of gamma rays are emitted in coincidence. The chosengate condition has a direct impact on the presence and intensity of each gamma ray in the resulting spectrum. Inthis section, we develop a formalism that allows one to calculate the apparent intensity of any ray emitted during adeexcitation cascade, depending on the kind of gate condition that has been applied. We will distinguish the relativelysimple case of a gate condition of type ”and” from the more complex situation occurring when a gate condition oftype ”or” is applied.
A. Simplifying hypotheses and external input
In order to introduce the formalism, we assume some simplifying hypotheses (keeping for later work the general-ization to more realistic cases): • The nucleus emits pure gamma cascades until the ground state (no electron conversion, no decay of excitedstates to another nucleus by nucleon emission or β disintegration) • We ignore the problem of degeneracy, which has to be considered if transitions taking place in different parts ofthe level scheme lead to similar gamma emissions • The gamma detection is ideally performed, with 100% absolute photopeak efficiency (every emitted gamma rayis fully detected).Furthermore, the feeding of the entry states (primary feeding, which depends on the reaction mechanism) is givenas an input.
B. Selected definitions
To formalize the situation, we will need to use some specific terminology and notations. We introduce here themost fundamental ones in order to settle the frame of the following discussion.
1. Gate conditions
A gate condition is based on the detection of specific gamma rays, called gates . For each event, a gate is said to be open when the corresponding ray is detected, and closed if it is not. The list of N gates involved in the expressionof a given condition will be written: L = { g , ..., g N } , where g k identifies an individual gamma ray used as a gate.Depending on the way these gates are involved, we can distinguish different kinds of conditions. For the present study,we need to define three kinds: • Positive explicit gate conditions (type ”and”): all the gates of the list L have to be open. Such condition willbe denoted by G = { g · ... · g N } , called a positive explicit condition of order N . • Exclusive explicit gate conditions : each gate of the list L is specified to be either open or closed. This conceptis particularly useful for the treatment of gate conditions of type ”or”, as we will see later. The term ”explicit”means that each gate of L has a specified status (open or closed), ”exclusive” means that some of them arerequired to be closed. An exclusive explicit condition of order ( n, ¯ n ) involves n open gates and ¯ n closed gates,with n + ¯ n = N . The list L is then decomposed in two sublists: L ( o ) = { g ( o )1 , ..., g ( o ) n } containing the open gates,and L ( c ) = { g ( c )1 , ..., g ( c )¯ n } containing the closed gates. As we will see in Section III D 2, exclusive conditionscan be treated by performing developments in terms of positive conditions, with terms involving gates from thesublist L ( c ) whose status changes from ”closed” to ”open”. For this reason, to express an exclusive condition,we specify both the gate identification (position within the sublist L ( o ) or L ( c ) ) and its status (open: g , orclosed: ¯ g ). This leads to the notation: G = { g ( o )1 · ... · g ( o ) n · ¯ g ( c )1 · ... · ¯ g ( c )¯ n } . Although this notation may now lookredundant, it will be useful in future developments: see for instance Eq. (6). • Optional gate conditions (type ”or”): a minimal number of gates from the list L have to be open. If m is thisminimal number, any event for which at least m gates of L are open is selected (whatever the status of theremaining gates). Such condition is denoted by G = { g + ... + g N } m .Optional gate conditions will be studied in detail in the following. For the treatment of this case, it will be useful toconsider the various explicit gate conditions that can be defined using sublists of L = { g , ..., g N } : • G α ( n, L ) denotes a positive explicit condition of order n ≤ N : it involves a list L ( α ) = { g ( α )1 , ..., g ( α ) n } that isa sublist of L . For a given order n , the number of possible combinations of n gates picked from the list L isgiven by the well-known binomial coefficient C Nn = N ! / [ n !( N − n )!]. The α index, which identifies the differentcombinations, then takes the values 1 ≤ α ≤ C Nn . = \ e e e e e e e e e e ∩e ∩e e ∩e e ∩e ∩e FIG. 2: Schematic illustration of the basic relation used to express exclusive elementary sets in terms of positive elementarysets. • G β ( n, ¯ n, L ) denotes an exclusive explicit condition of order ( n, ¯ n ), with n + ¯ n ≤ N : it involves the lists L ( β,o ) = { g ( β,o )1 , ..., g ( β,o ) n } and L ( β,c ) = { g ( β,c )1 , ..., g ( β,c )¯ n } that are sublists of L . For a given order ( n, ¯ n ), there are C Nn +¯ n combinations of n + ¯ n specified gates picked from the list L , and C n +¯ nn combinations of n open gates pickedfrom the n + ¯ n specified gates. The β index then takes the values 1 ≤ β ≤ C Nn +¯ n × C n +¯ nn .
2. Associated sets of events
Experimentally, an event corresponds to the formation of an excited nucleus and the following deexcitation cascade.In the dataset, it is identified as a list of energy deposits localized at different places in the detection system. Eventreconstruction from individual deposits is a first step for data analysis, which is not addressed here. In gammaspectroscopy, each emitted photon may remain unobserved or partially dectected (Compton effect); furthermore, sometransitions can be non-radiative (e.g. occurring by electronic conversion). In our simplified scheme, the deexcitationprocess is purely radiative, and all the photons are fully detected: each event is then simply characterized by the listof transitions that occurred in the corresponding deexcitation cascade. For a given event, a specific gate is open if thecorresponding transition is present in the list, and closed otherwise. A gate condition yields a set of selected events.Let us first remind some conventional symbols and properties of set algebra: • Set union: E = E ∪ E contains all events that belong to E and all events that belong to E . The union of aseries of sets reads: E ∪ ... ∪ E n = n (cid:91) i =1 E i • Set intersection: E = E ∩ E contains all events that belong to both E and E . The intersection of a seriesof sets reads: E ∩ ... ∩ E n = n (cid:92) i =1 E i • Set difference: E \ E contains all events that belong to E but not to E . • Set complement: ¯ E contains all events that do not belong to E ; it can be written ¯ E = U \ E , where U is theuniverse of events (i.e. the set that contains all of them). • Intersection of a set E with a set complement ¯ E : E ∩ ¯ E = E ∩ [ U \ E ] = [ E ∩ U ] \ [ E ∩ E ] = E \ [ E ∩ E ]This relation will be particularly useful for the treatment of optional gate conditions. It is illustrated by agraphical example in Figure 2.Let us now present the different kinds of event sets we will deal with. • A single set e is associated with single-gate conditions G = { g } . e e e a) E({g }) = e e e e e e e e e e e e e e e e b) E({g .g }) = e ∩e c) E({g .g .g }) = e ∩e ∩e d) E({g }) = e e) E({g .g }) = e ∩e f) E({g .g .g }) = e ∩e ∩e FIG. 3: Elementary sets: basic examples. (a) Positive elementary set of order 1 : E ( G = { g } ). (b) Positive elementary set oforder 2 : E ( G = { g · g } ). (c) Positive elementary set of order 3 : E ( G = { g · g · g } ). (d) Exclusive elementary set of order(0 ,
1) : E ( G = { ¯ g } ). (e) Exclusive elementary set of order (1 ,
1) : E ( G = { g · ¯ g } ). (f) Exclusive elementary set of order (2 , E ( G = { g · g · ¯ g } ). • A positive elementary set E ( G ) of order N is associated with a positive explicit gate condition G = { g · ... · g N } :in short-hand notation, it is denoted by E g ...g N . It corresponds to the intersection of single sets e i = E ( { g i } ): E g ...g N = e ∩ ... ∩ e N • An exclusive elementary set E ( G ) of order ( n, ¯ n ) is associated with an exclusive explicit gate condition G = { g ( o )1 · ... · g ( o ) n · ¯ g ( c )1 · ... · ¯ g ( c )¯ n } : in short-hand notation, it is denoted by E g ( o )1 ...g ( o ) n ¯ g ( c )1 ... ¯ g ( c )¯ n . It corresponds to theintersection of single sets e ( o ) i = E ( { g ( o ) i } ) and single set complements ¯ e ( c ) i = ¯ E ( { g ( c ) i } ): E g ( o )1 ...g ( o ) n ¯ g ( c )1 ... ¯ g ( c )¯ n = e ( o )1 ∩ ... ∩ e ( o ) n ∩ ¯ e ( c )1 ∩ ... ∩ ¯ e ( c )¯ n = e ( o )1 ∩ ... ∩ e ( o ) n ∩ ( U \ e ( c )1 ) ∩ ... ∩ ( U \ e ( c )¯ n ) • A combined set E ( G ) is associated with an optional gate condition G = { g + ... + g N } m . It corresponds to theunion of several elementary sets. This case will be detailed later.Basic examples of elementary and combined sets are illustrated in Figures 3 and 4.
3. Associated spectra
A spectrum is a histogram representation of events, displaying the number of photons counted in each energyinterval. For each gate condition G , there is a set-spectrum S ( E ( G )) representing the associated event set E ( G ):it gives the actual counting of photons emitted during the selected events. For simplicity, S ( E ( G )) can be directlydenoted by S ( G ).Note however that a spectrum does not necessarily provide a one-to-one representation of an event set: otherkinds of spectra can be obtained by combining set-spectra. Let us consider for instance several set-spectra S ( E i )representing event sets E i . A new spectrum S can be obtained by performing a linear combination of S ( E i ) such as: S = (cid:88) i c i S ( E i ) e e e e e e e e e e e e e e e e e e ∩e e ∩e e ∩e U U e e e b) E ( G ={g +g +g } ) = a) E ( G ={g +g } )e e e = U FIG. 4: Combined sets: basic examples. (a) Combined set E ( G = { g + g } ). (b) Combined set E ( G = { g + g + g } ). where the photon numbers of S ( E i ) are counted c i times (or subtracted | c i | times if c i < spiking effect (artificial enhancement of some peaks). We will call sum-spectrum a spectrum of this kind.For the present work, we need to define the following kinds of set-spectra: • Positive elementary spectrum S ( G ) of order N , representing the event set E ( G ) associated with a positiveexplicit condition G = { g · ... · g N } . In short-hand notation, it is denoted by S g ...g N . • Exclusive elementary spectrum S ( G ) of order ( n, ¯ n ), representing the event set E ( G ) associated with an exclusiveexplicit condition G = { g ( o )1 · ... · g ( o ) n · ¯ g ( c )1 · ... · ¯ g ( c )¯ n } . In short-hand notation, it is denoted by: S g ( o )1 ...g ( o ) n ¯ g ( c )1 ... ¯ g ( c )¯ n .Building experimentally such a spectrum would require combining gating and anti-gating methods as developedin Ref. [15]. However, in the present work, exclusive spectra are used as an intermediate step and eventuallyexpressed in terms of positive spectra, as detailed in Section III D 2. • Combined spectrum S ( G ), representing a combined set E ( G ) associated with the optional condition G = { g + ... + g N } m . We will see later how to express it as a combination of elementary spectra.It is also useful to introduce a dedicated notation for specific kinds of sum-spectra, that will appear in laterexpressions: • Positive sum-spectrum of order n , denoted by σ ( n, L ), defined as the sum of spectra associated with all positiveexplicit conditions of order n that can be defined by picking n gates in a given list L = { g , ..., g N } . It reads : σ ( n, L ) = C Nn (cid:88) α =1 S ( G α ( n, L )) • Exclusive sum-spectrum of order ( n, ¯ n ), denoted by σ ( n, ¯ n, L ), defined as the sum of spectra associated with allexclusive explicit conditions of order ( n, ¯ n ) that can be defined by picking n open gates and ¯ n closed gates in agiven list L = { g , ..., g N } . It reads : σ ( n, ¯ n, L ) = C Nn +¯ n × C n +¯ nn (cid:88) β =1 S ( G β ( n, ¯ n, L )) • Spiked spectrum : it is a usual kind of sum-spectrum, which is employed in practice as the simplest way torepresent the events of a combined set. Obtained by summing set-spectra of overlapping event sets, it involvessome multi-counting, hence the ”spiked” qualification. The case of spiked spectra will be addressed in a dedicatedsubsection.
4. Gamma-ray intensity and relative intensity
Let us consider a given gamma ray emitted during the transition t i , occurring with the probability P i in thedeexcitation cascade following nucleus formation: P i = N i N tot , where N tot is the total number of events (i.e. the number of nucleus formations followed by deexcitation) and N i is thenumber of transitions t i that occur. Experimentally, a typical goal when a new transition t i is observed is to quantifythe probability P i by measuring the corresponding peak size in a gamma-emission spectrum. Usually, the studiedspectra are subject to gate conditions that make this peak more visible by reducing the background and the numberof alternative cascades. The purpose here is then to relate the peak size associated with t i in a gated spectrum tothe emission probability P i . Let us define the following quantities, for a given set of events E associated with a gatecondition G : • Gated transition probability P { G,i } : probability for an event to verify condition G and to contain transition t i .It corresponds to the ratio: P { G,i } = N { G,i } N tot where N { G,i } is the number of events of E ( G ) that involve t i . This quantity will be expressed later as a functionof the transition probability vector P and matrix P . • Gated intensity I i ( G ): fraction of counts in the gated spectrum that belong to the peak of t i . It corresponds tothe ratio: I i ( G ) = N { G,i } N { G,γ } where N { G,γ } is the total number of gamma rays emitted during the events E ( G ). This number is directly givenby the spectrum integral, but it is not determined by the formalism we are developing here. Indeed, N { G,γ } includes a number of emissions from the continuum of the level scheme, which is not treated by our formalism(in its present version). • Relative gated intensity I ( r ) i ( G ): ratio between the peak sizes associated with t i and with a reference transition t ref . It corresponds to: I ( r ) i ( G ) = N { G,i } N { G,ref } = P { G,i } P { G,ref } where N { G,i } and N { G,ref } can be directly measured in the gated spectrum while P { G,i } and P { G,ref } can beexpressed in terms of the transition probabilities involved in vector P and matrix P .Let us finally define the relative intensity I ( r ) i , which compares the occurrence of t i and t ref in the total set of events: I ( r ) i = N i N ref = P i P ref This quantity is often given in the litterature to characterize the strength of a transition t i observed in an experiment.Let us note that it is in principle different from any gated relative intensity, although the measurement of I ( r ) i ( G ) isusually assumed to give an approximation of I ( r ) i . Since the validity of such an approximation depends on the detailsof the gate condition and on the cascade structure, it is important to establish a quantitative relation between gatedand ungated relative intensities, which is the aim of this work. C. Positive explicit gate condition (”and”)
As defined above, a positive explicit gate condition consists in a list of gates that are all required to be open. It isdenoted by G = { g · ... · g N } , and gives rise to a positive elementary spectrum S ( G ) that represents the set of selected0events E ( G ). We also specify that the gate list is ordered in such a way that the gates of lower indices correspondto transitions occurring earlier in the cascade, i.e. emitted by a higher energy level. This will be symbolized by therelation: g > ... > g N . Our purpose is now to express the gated probability P { G,i } of a transition t i as a functionof the transition probability vector P and matrix P . We remind that each element P k of the transition probabilityvector gives the probability that transition t k occurs during the deexcitation process while each element P ij = P t i → t j of the transition probability matrix P gives the probability that, once transition t i has occurred, it is followed bytransition t j after an arbitrary number of steps. For the homogeneity of some expressions, we will also use the notation P t k = P k .Let us start with examples for restricted numbers of gates N . The shortest list is of course the single gate : G = { g } . A transition t i that occurs in coincidence with g can take place either ”above” or ”below” g in thedeexcitation cascade. Namely, ”above g ” means earlier in the cascade, and is denoted by t i > g ; ”below g ” meanslater in the cascade, and is denoted by t i < g . Depending on each case, the gated intensity I i ( G ) is expresseddifferently as a function of the transition probability vector P and matrix P : • If t i > g : P { G,i } = P t i × P t i → g • If g > t i : P { G,i } = P g × P g → t i Globally, we can write: P { G,i } = P t i × P t i → g + P g × P g → t i since if g > t i we have P t i → g = 0, and if t i > g we have P g → t i = 0.Let us now consider a double gate G = { g · g } (ordered as g > g ): • If t i > g > g : P { G,i } = P t i × P t i → g × P g → g • If g > t i > g : P { G,i } = P g × P g → t i × P t i → g • If g > g > t i : P { G,i } = P g × P g → g × P g → t i which corresponds to the global expression, where only one term is non-zero: P { G,i } = P t i × P t i → g × P g → g + P g × P g → t i × P t i → g + P g × P g → g × P g → t i In order to generalize the expression of P { G,i } , let us introduce the transition cascade vectors T h ( G ): h indicatesthe position of t i among the gates g x . In the following, the dependence of T h on G will be implicit. For G = { g · g } ,there are three possible cascade vectors: • If t i > g > g : T = ( t i , g , g ) • If g > t i > g : T = ( g , t i , g ) • If g > g > t i : T = ( g , g , t i )We will denote by T hk the transition associated with the component k of the cascade vector T h (with the conventionthat k starts from zero). Now we can use T h to write P { G,i } : P { G,i } = P T × P T → T × P T → T + P T × P T → T × P T → T + P T × P T → T × P T → T = (cid:88) h =0 P T h × (cid:89) j =1 P T hj − → T hj This last expression can be easily generalized to a positive explicit gate condition G = { g · g · ... · g N } implying anynumber N of gates : P { G,i } = N (cid:88) h =0 P T h × N (cid:89) j =1 P T hj − → T hj (2)Note that, although the h summation offers an elegant mathematical expression that is independent from thetransition location in the cascade, it will be more efficient in numerical calculation to determine for each consideredtransition t i the corresponding position h ( t i ) before performing the product (since all other h terms are zero).1 D. Optional gate conditions (”or”)
With optional gate conditions, the list L = { g , ..., g N } is a list of optional gates: a minimal number m of themis required to be open. Such a condition is denoted by G = { g + ... + g N } m = L m/N , and is fulfilled every time acombination of at least m gates among L are open. The set of events E ( G ) that are selected by this condition includesdifferent elementary sets. Indeed, for any list L (cid:48) = { g (cid:48) , ..., g (cid:48) n } that is a sublist of L with n ≥ m , the elementary set E ( { g (cid:48) · ... · g (cid:48) n } ) is included in E ( G ). Several such elementary sets have to be united in order to obtain E ( G ), hencethe denomination of combined set .The most simple way to obtain E ( G ) is to unite all the elementary sets corresponding to the minimal requirement of m open gates. Each such set is associated with a positive explicit gate condition of order m , G α ( m, L ) = { g ( α )1 · ... · g ( α ) m } ,where L ( α ) = { g ( α )1 , ..., g ( α ) m } is a sublist of L . Note however that the spectrum S ( G ) that represents the combinedset E ( G ) does not correspond to the sum of elementary spectra (cid:80) α S ( G α ). Indeed, the sets E ( G α ) are overlapping,which means that one event of E ( G ) can belong to several sets E ( G α ), giving rise to an artificial enhancement of peaksizes (spiking effect).Let us illustrate this with a specific combined gate condition G = { g + g + g } . Here, an event is selected ifat least 2 gates are open, among a list of 3. Namely, it has to fulfill at least one of the explicit gate conditions { g · g } , or { g · g } , or { g · g } , which define the elementary sets E ( { g · g } ) = E g g , E ( { g · g } ) = E g g and E ( { g · g } ) = E g g , respectively. We can easily realize that these sets are overlapping: indeed, every event for whichthe three optional gates are open belongs to all elementary sets E g g , E g g and E g g . As a result, such events arecounted three times in the sum-spectrum S g g + S g g + S g g .Simple examples of optional conditions are detailed in Appendix A, which can be consulted in parallel with thepresent subsection.
1. Tiling of the combined set
In order to avoid the spiking effect and obtain the combined spectrum S ( G ) as an exact representation of the eventset E ( G ), we have to express E ( G ) as the union of elementary sets E ( G β ) that are not overlapping. In other words, theelementary sets E ( G β ) that are considered have to constitute a tiling of the combined set E ( G ). The non-overlappingcriterion means that the gate conditions G β have to exclude each other: this is possible only if the status of all thegates of L is specified by each explicit condition G β , using exclusive conditions in the cases where less than N gatesare required to be open.We have seen that exclusive explicit conditions of order ( n, ¯ n ) can be defined from the list L = { g , ...g N } ; theyare denoted by G β ( n, ¯ n, L ) = { g ( β,o )1 · ... · g ( β,o ) n · ¯ g ( β,c )1 · ... · ¯ g ( β,c )¯ n } , where L ( β,o ) = { g ( β,o )1 , ...g ( β,o ) n } and L ( β,c ) = { g ( β,c )1 , ...g ( β,c )¯ n } are sublists of L . For the tiling of E ( G ) by elementary sets, we need to consider all the conditions G β ( n, ¯ n, L ) such that n ≥ m and n + ¯ n = N . For a given value of n , the number of β combinations is then given by C NN × C Nn = C Nn . This leads to the tiling relation : E ( G = L m/N ) = N (cid:91) n = m C Nn (cid:91) β =1 E ( G β ( n, N − n, L )) (3)The tiling relation allows to obtain the combined spectrum representing E ( G ) as: S ( G ) = N (cid:88) n = m C Nn (cid:88) β =1 S ( G β ( n, N − n, L )) = N (cid:88) n = m σ ( n, N − n, L ) (4)where the second part of the equation is obtained by replacing, for each value of n , the β summation on elementaryspectra by the corresponding sum-spectrum σ .Let us consider for instance the optional condition G = { g + g + g } . The tiling relation reads in this case: E ( G ) = g g ¯ g ∪ g g ¯ g ∪ g g ¯ g ∪ g g g The combined spectrum is then given by: S ( G ) = S g g ¯ g + S g g ¯ g + S g g ¯ g + S g g g E ( G ) is counted once and only once. It involves the exclusive sum-spectra of order ( n, ¯ n ) suchthat n ≥ n + ¯ n = 3: σ (2 , , { g , g , g } ) = S g g ¯ g + S g g ¯ g + S g g ¯ g σ (3 , , { g , g , g } ) = σ (3 , { g , g , g } ) = S g g g in terms of which we can express the combined spectrum: S ( G = { g + g + g } ) = σ (2 , , { g , g , g } ) + σ (3 , , { g , g , g } )
2. Development in positive elementary spectra
We have seen that the tiling relation (3) allows to obtain a combined spectrum S ( G ) as a sum of elementary spectra S ( G β ). However, this summation involves exclusive elementary spectra, associated with conditions that impose gateclosures. In order to apply directly Eq. (2) to establish the gamma-ray intensities in S ( G ), we need to expressthe combined spectrum as a combination of positive elementary spectra. From the experimentalist point of view,data analysis could involve anti-gating (namely, gates imposed to be closed); however, one often prefers to constructspectra based on positive gating conditions. Thus, the present approach will also allow a more classical correspondancebetween modelling and construction from an experimental data set.Let us first remind the expression of a set intersection with a complementary set: E = E ∩ ¯ E E ∩ [ U \ E ] = E \ [ E ∩ E ]. The corresponding spectrum is expressed as a subtraction: S ( E = E ∩ ¯ E
2) = S ( E ) − S ( E ∩ E )This principle can be applied to spectra associated with exclusive explicit conditions. The exclusive elementary setassociated with a condition G β ( n, ¯ n, L ) is: E g ( β,o )1 ...g ( β,o ) n ¯ g ( β,c )1 ... ¯ g ( β,c )¯ n = e ( β,o )1 ∩ ... ∩ e ( β,o ) n ∩ ¯ e ( β,c )1 ∩ ... ∩ ¯ e ( β,c )¯ n − ∩ ¯ e ( β,c )¯ n = (cid:104) e ( β,o )1 ∩ ... ∩ e ( β,o ) n ∩ ¯ e ( β,c )1 ∩ ... ∩ ¯ e ( β,c )¯ n − (cid:105) ∩ (cid:104) U \ e ( β,c )¯ n (cid:105) = (cid:104) e ( β,o )1 ∩ ... ∩ e ( β,o ) n ∩ ¯ e ( β,c )1 ∩ ... ∩ ¯ e ( β,c )¯ n − (cid:105) \ (cid:104) e ( β,o )1 ∩ ... ∩ e ( β,o ) n ∩ ¯ e ( β,c )1 ∩ ... ∩ ¯ e ( β,c )¯ n − ∩ e ( β,c )¯ n (cid:105) so that the spectrum S ( G β ( n, ¯ n, L )) can be decomposed as: S g ( β,o )1 ...g ( β,o ) n ¯ g ( β,c )1 ... ¯ g ( β,c )¯ n = S g ( β,o )1 ...g ( β,o ) n ¯ g ( β,c )1 ... ¯ g ( β,c )¯ n − − S g ( β,o )1 ...g ( β,o ) n ¯ g ( β,c )1 ... ¯ g ( β,c )¯ n − g ( β,c )¯ n (5)As a result, any exclusive elementary spectrum of order ( n, ¯ n ) can be expressed as the combination of exclusiveelementary spectra of order ( n, ¯ n −
1) and ( n + 1 , ¯ n − S ( G β ) can bedeveloped as a combination of positive elementary spectra of order p , with n ≤ p ≤ n + ¯ n . To represent the process,it is convenient to use for an elementary spectrum the notation S ( G β ) = g ( β,o )1 ...g ( β,o ) n ¯ g ( β,c )1 ... ¯ g ( β,c )¯ n , and to symbolizethe relation (5) by a factorisation: S ( G β ( n, ¯ n, L )) = g ( β,o )1 ...g ( β,o ) n ¯ g ( β,c )1 ... ¯ g ( β,c )¯ n − (1 − g ( β,c )¯ n )Along the recursive steps, each closed gate factor ¯ g x is eventually replaced by the factor (1 − g x ), leading to thefollowing expression of S ( G β ) in terms of positive elementary spectra: S ( G β ( n, ¯ n, L )) = g ( β,o )1 ...g ( β,o ) n (1 − g ( β,c )1 ) ... (1 − g ( β,c )¯ n ) (6)Note that this expression makes use of the notation g ( β,c ) i (with no bar on g ) indicating that the gate i of the list L ( β,c ) has changed its original status from ”closed” to ”open”. Hence the distinction that was introduced in Section III B 1for the definition of exclusive conditions: the lists L ( β,o ) and L ( β,c ) identify the gates involved in the condition, whilethe condition G β = { g ( β,o )1 · ... · g ( β,o ) n · ¯ g ( β,c )1 · ... · ¯ g ( β,c )¯ n } specifies their status, that vary when S ( G β ) is developed interms of positive conditions. Now, the development (6) can be expressed as: S ( G β ( n, ¯ n, L )) = n +¯ n (cid:88) p = n ( − p − n C Np (cid:88) α =1 k α,β S ( G α ( p, L )) (7)3where the α index identifies the different gate combinations L ( α ) defining the positive explicit conditions G α ( p, L ).We have introduced the coefficient k α,β , which is 1 if the α combination appears in the development of S ( G β ), and 0otherwise. The condition to have k α,β = 1 is then: • L ( α ) ⊆ ( L ( β,o ) + L ( β,c ) ) (all gates open in G α are specified in G β ) • L ( β,o ) ⊆ L ( α ) (all gates open in G β are open in G α )A given term S ( G α ( p, L )) appears in the development of several spectra S ( G β ( n, ¯ n, L )). Indeed, to determine acondition G β such that k α,β (cid:54) = 0: • There are C pn choices to pick the n gates of L ( α ) that belong to L ( β,o ) • The remaining p − n gates of L ( α ) necessarily belong to L ( β,c ) • The list L ( β,c ) contains ¯ n gates, of which p − n belong to L ( α ) . The remaining ¯ n − ( n − p ) closed gates of G β have to be picked from the N − p gates of L that are not specified by G α : there are C N − p ¯ n − ( p − n ) = C N − p ¯ n + n − p possiblecombinations.Thus, for fixed values of n , ¯ n , p and α , there are C pn × C N − p ¯ n + n − p values of β such that k α,β = 1. As a result, byperforming a summation of Eq. (7) over the β index (for fixed values of n and ¯ n ), we obtain: (cid:88) β S ( G β ( n, ¯ n, L )) = (cid:88) β n +¯ n (cid:88) p = n ( − p − n (cid:88) α k α,β S ( G α ( p, L ))= n +¯ n (cid:88) p = n ( − p − n (cid:88) α S ( G α ( p, L )) (cid:88) β k α,β (cid:88) β S ( G β ( n, ¯ n, L )) = n +¯ n (cid:88) p = n ( − p − n ( C pn × C N − p ¯ n + n − p ) (cid:88) α S ( G α ( p, L ))where the β summation runs over the C Nn +¯ n × C n +¯ nn possible combinations to define G β ( n, ¯ n, L ); the α summation runsover the C Np possible combinations to define G α ( p, L ); and for given values of p and α , we have (cid:80) β k α,β = C pn × C N − p ¯ n + n − p .For a given gate list L = { g , ..., g N } , the above equation gives the sum of all exclusive elementary spectra of order( n, ¯ n ) in terms of positive elementary spectra of order p with n ≤ p ≤ N . Replacing the β and α summations bysum-spectra σ , this expression reads: σ ( n, ¯ n, L ) = n +¯ n (cid:88) p = n ( − p − n ( C pn × C N − p ¯ n + n − p ) σ ( p, L ) (8)Namely, for a given gate list L = { g , ..., g N } , the exclusive sum-spectrum of order ( n, ¯ n ) can be expressed by acombination of positive sum-spectra of order p with n ≤ p ≤ n + ¯ n .This can be directly applied to the expression of the combined spectrum given by Eq.(4), which results from thetiling relation. Some simplifications occur due to the condition n + ¯ n = N (the exclusive explicit conditions involvedin the tiling relation have to specify the status of every gate of L ). We have in this case C N − p ¯ n + n − p = C N − pN − p = 1, so theexclusive sum-spectra appearing in Eq.(4) are given by: σ ( n, N − n, L ) = N (cid:88) p = n ( − p − n C pn σ ( p, L ) = N (cid:88) p = n a n,p σ ( p, L ) (9)where we have introduced the coefficients a n,p = ( − p − n C pn . The combined spectrum associated with the condition G = L m/N is expressed as the following combination: S ( L m/N ) = N (cid:88) n = m σ ( n, N − n, L )= N (cid:88) n = m N (cid:88) p = n ( − p − n C pn σ ( p, L ) = N (cid:88) p = m p (cid:88) n = m ( − p − n C pn σ ( p, L ) = N (cid:88) p = m p (cid:88) n = m a n,p σ ( p, L )4 [p=1] [p=2] [p=3] [p=4] [p=5] [p=6] [p=7] [p=8] [p=9] [p=10]1 -2 3 -4 5 -6 7 -8 9 -10 [n=1]1 -3 6 -10 15 -21 28 -36 45 [n=2]1 -4 10 -20 35 -56 84 -120 [n=3]1 -5 15 -35 70 -126 210 [n=4]1 -6 21 -56 126 -252 [n=5]1 -7 28 -84 210 [n=6]1 -8 36 -120 [n=7]1 -9 45 [n=8]1 -10 [n=9]1 [n=10]TABLE I: Coefficients a n,p that allow to express any combined spectrum S ( L m/N ) with gate list size N ≤ The sum inversion is performed thanks to the relation verified by any function f ( n, p ): N (cid:88) n, p = mn ≤ p f ( n, p ) = N (cid:88) n = m N (cid:88) p = n f ( n, p ) = N (cid:88) p = m p (cid:88) n = m f ( n, p )Introducing the coefficients c p ( m ) = (cid:80) pn = m a n,p to express the linear combination, the combined spectrum develop-ment in terms of positive elementary spectra reads: S ( L m/N ) = N (cid:88) p = m c p ( m ) σ ( p, L ) = N (cid:88) p = m c p ( m ) C Np (cid:88) α =1 S ( G α ( p, L )) (10)with c p ( m ) = p (cid:88) n = m a n,p = p (cid:88) n = m ( − p − n C pn The coefficients c p ( m ) involved in the development of S ( L m/N ) can be easily obtained in practice by representing thecoefficients a n,p = ( − p − n C pn in a universal table, where n refers to the line number and p to the column number.The binomial coefficients can even be recovered by hand, applying the Pascal relation C p +1 n +1 = C pn + C pn +1 that allowsto construst the well-known Pascal triangle. The a n,p coefficients are shown in Table I up to p = 10. The coefficients c p ( m ) are obtained by summing the elements of column p , starting at line m . A schematic representation showingthe principle of this development is shown in Figure 5.
3. Gated intensity in a combined spectrum
We consider here the combined spectrum S ( G ) representing the set of events selected by the optional condition G = L m/N = { g + ... + g N } m . Following Eq. (10), this combined spectrum is given by a linear combination of positiveelementary spectra. As a consequence, the number of events of E ( G ) for which the transition t i has occurred is givenby a similar combination: N {G ,i } = N (cid:88) p = m c p ( m ) C Np (cid:88) α =1 N { G α ( p,L ) ,i } as well as the gated probability of transition t i : P {G ,i } = N {G ,i } N tot = N (cid:88) p = m c p ( m ) C Np (cid:88) α =1 P { G α ( p,L ) ,i } Each gated probability P { G α ( p,L ) ,i } associated with a positive explicit condition G α ( p, L ) can be expressed in termsof the transition probabilities according to Eq.(2), where the cascade vectors T h are determined by the list L α of5 n n n,p Nma) s (n ,n ,L)as a function of s (p,L) b) Combined spectrum S (L m/N ) as a function of s (p,L)n n n +n
00 N-m n,p
FIG. 5: (Color online) Schematic illustrations of the development in positive spectra. (a) Recursive development allowing toexpress an exclusive elementary spectrum of order ( n , ¯ n ) in terms of positive elementary spectra of order p, with n ≤ p ≤ n + ¯ n . The horizontal axis ( n or p ) is the number of open gates; the vertical axis (¯ n ) is the number of closed gates. One squareof the grid represents an exclusive elementary spectrum of order ( n, ¯ n ); if ¯ n = 0, it is a positive elementary spectrum or order p .The two arrows from each square ( n, ¯ n ) symbolize its development in two elementary spectra of order ( n, ¯ n −
1) and ( n +1 , ¯ n − p involved in the development of the original elementary spectrum of order ( n , ¯ n ). For correspondance with Eq. (8), we can alsoidentify each square ( n, ¯ n ) with the exclusive sum-spectrum σ ( n, ¯ n, L ), and each square ( p,
0) with the positive sum-spectrum σ ( p, L ). (b) Similar illustration, applied to the combined spectrum S ( L m/N ) composed of exclusive sum-spectra σ ( n, N − n, L )(red squares) according to the tiling relation S ( L m/N ) = (cid:80) Nn = m σ ( n, N − n, L ) given by Eq. (4). The arrows showing thesteps of the recursive developments are not shown here. Each horizontal arrow on the lower line illustrates the expressionof an exclusive sum-spectrum σ ( n, N − n, L ) in terms of positive sum-spectra σ ( p, L ), corresponding to the development σ ( n, N − n, L ) = (cid:80) Np = n a n,p σ ( p, L ) given by Eq. (9). Each square σ ( p, L ) is involved in the development of all the squares σ ( n, N − n, L ) with n ≤ p , hence the coefficients c m ( p ) = (cid:80) pn = m a n,p in the final development S ( L m/N ) = (cid:80) Np = m c p ( m ) σ ( p, L )given by Eq. (10). the corresponding gate combination. We remind that the h exponent gives the position of transition t i among thesequence of gate transitions. For G α = { g ( α )1 · ... · g ( α ) p } , we have for instance T ( G α ) = ( t i , g ( α )1 , ..., g ( α ) p ). Note that,for fixed values of p and h , the transition cascade vectors T h ( G α ) corresponding to the different α combinations canbe viewed as the different lines of a transition cascade matrix T h ( p, L ). An element T hα,j of this matrix correspondsto the component j of the cascade vector T h ( G α ). The matrix T h ( p, L ) has C Np lines and p + 1 columns. In thefollowing, the dependence of each matrix T h on p and L will be implicit. The gated probability P {G ,i } is then givenby: P {G ,i } = N (cid:88) p = m c p ( m ) C Np (cid:88) α =1 p (cid:88) h =0 P T hα, × p (cid:89) j =1 P T hα,j − →T hα,j (11)The same formula applies to express the gated probability of a reference transition t ref , so that the relative gatedintensity I ( r ) i ( G ) given by the ratio: I ( r ) i ( G ) = N {G ,i } N {G ,ref } = P {G ,i } P {G ,ref } can be obtained either by measuring the peak areas N {G ,i } and N {G ,ref } in the combined spectrum S ( G ), or byimplementing Eq. (11) to calculate the gated probabilities P {G ,i } and P {G ,ref } .
4. Case of ”spiked” spectra
The optional gate condition G = { g + ... + g N } m involves the list of gates L = { g , ..., g N } . We have seen abovethat the simplest way to express the combined set E ( G ) of events selected by G is to perform the union of all positive6elementary sets E ( G α ), where each condition G α specifies a sublist of m gates chosen among L : E ( G ) = E ( L m/N ) = C Nm (cid:91) α =1 E ( G α ( m, L )) (12)On the other hand, since the sets E ( G α ) are overlapping, multi-counting of events occurs if we want to represent E ( G )by the sum of elementary spectra S ( G α ). This is why the resulting spectrum is called a spiked spectrum: S s ( G ) = C Nm (cid:88) α =1 S ( G α ( m, L ))Note that the spiked spectrum is nothing but the sum-spectrum of order m : S s ( G ) = σ ( m, L ) (13)In the spiked spectrum, the (distorted) counting of the t i transition is given by: N i ( S s ( G )) = C Nm (cid:88) α =1 N { G α ( m,L ) ,i } = C Nm (cid:88) α =1 N tot × P { G α ( m,L ) ,i } where, as previously, the gated probabilities are given in terms of the transition probability vector P and matrix P .The involved elements are now indicated by the transition cascade matrices T h ( m, L ): P { G α ( m,L ) ,i } = m (cid:88) h =0 P T hα, × m (cid:89) j =1 P T hα,j − →T hα,j The same formula applies to the reference transition t ref , so that the spiked relative intensity is obtained as: I ( r ) i ( S s ( G )) = N i ( S s ( G )) N ref ( S s ( G )) = (cid:80) α P { G α ( m,L ) ,i } (cid:80) α P { G α ( m,L ) ,ref } We could conclude that, although the spiked spectrum gives a distorted representation of the events selected by G ,it is also linked to the transition probability vector P and matrix P in a well-defined way. So it can also be usedas an analysis tool if the goal is, for instance, to obtain information on the transition probabilities by measuring thepeak ratio N i ( S s ( G )) /N ref ( S s ( G )). Note however that, if one of the gates is a doubled transition (namely, there isanother possible transition with the same energy), even if we intend to apply an explicit condition, the filtered eventsobey an effective condition that is combined: no elementary spectrum can be isolated. In this case the analysis has totake into account the combinatory effects associated woth ”or”-type gate conditions. The application of the presentformalism to the case of doubled (or even multi-degenerate) transitions will be addressed in a future work. IV. APPLICATION TO A SCHEMATIC LEVEL SCHEME
In this section, we illustrate the predictive power of the analytic formula we have derived, both in the case ofelementary and combined spectra. We have chosen to work on a schematic level scheme for the sake of clarityconcerning the points we want to illustrate, namely the difference between explicit and optional conditions, and therole of band communication. This idealized approach allows to avoid obstacles such as the presence of degeneratetransitions (which are not yet treated by the formalism). Furthermore, we can ignore the details of nucleus-formationmechanism and transition physical properties, thus ignoring constraints on the expected values of primary feedingand branching ratios (this simplifies the choice of the input, but has no impact on the future applicability of themethod to realistic cases). We then remain with the problem of determining gated intensities for a list of transitionsorganized in a level scheme, with given emission probability and adjacency matrix. This will be done following theformalism presented in the previous sections.The schematic level scheme and corresponding transition scheme that are studied in this section are shown inFigure 6. The level scheme is composed of two interacting structures, named ”ground-state band” and ”excitedband”. The transitions can link two successive levels in a given structure (intra-band transitions) or two neighboringlevels of each structure (inter-band transitions). This situation is quite usual in the structure of nuclei presenting for7 (B1) (B1) (B1) (B1) (B1) (B1) (B1) (B1) (B2) (B2) (B2) (B2) (B2) t t t t t t t t t t t t t t t t t t t t t FIG. 6: (Color online) Schematic level scheme used to illustrate the predictive power of the formalism. instance different kinds of deformation. However, let us remind that no hypothesis is made here on the nature of thebands and transitions: the role of electron conversion is ignored, and branching ratios are chosen arbitrarily. Mostimportantly for our purpose, with this kind of level scheme, band communication allows different possibilities to passfrom one transition to the other. As a result, if we apply an optional gate condition using for instance the lowertransitions of the ground-state band, all the terms of the combination of sum-spectra given by Eq. (10) contribute tothe gated intensity: this is what we need to check the full validity of this formula.Let us remind the two possible approaches to describe the deexcitation process: characterization of the level space(list of levels with associated primary feeding and branching matrix), or characterization of the transition space (listof transitions with associated emission probabilities and adjacency matrix). As stated in Section II, transition-spaceinformation can be deduced from level-space information. For convenience, in our code, the original input concernslevel-space information: • List of levels, with corresponding level energy. • Branching matrix. In the case of a real application, this can be readily obtained from usual databases; in ourcode, it was chosen by hand. • Primary feeding. For a given nucleus, this is not universal since it depends on the reaction mechanism: it shouldbe determined for each studied experiment. In our code, we have used an arbitrary function to distribute theprimary feeding among the different levels.The resulting level-space characterization is given by Table II and the branching matrix presented in Table III.The formalism that we use is based on a transition-space approach, where the useful input is the transition proba-bility vector P and the adjacency matrix A . Both can be deduced from the level-space input specified in Tables II andIII. The elements of P are given in Table IV, together with other transition properties; the matrix A is represented by8 Level Energy Identification Primary Feeding Total Feeding0 ( B .
089 1790 ( B .
048 0 . B .
097 0 . B .
049 0 . B .
099 0 . B .
050 0 . B .
099 0 . B .
049 0 . B .
096 0 . B .
048 0 . B .
094 0 . B .
092 0 . B .
089 0 . B ( B ( B ( B ( B ( B ( B ( B ( B ( B ( B ( B ( B ( B B B .
714 0 .
286 0 0 0 0 0 0 0 0 0 0 0( B .
231 0 .
769 0 0 0 0 0 0 0 0 0 0( B .
667 0 .
333 0 0 0 0 0 0 0 0 0( B .
714 0 .
286 0 0 0 0 0 0 0 0( B .
625 0 .
375 0 0 0 0 0 0 0( B .
741 0 .
259 0 0 0 0 0 0( B .
667 0 .
333 0 0 0 0 0( B .
800 0 .
200 0 0 0 0( B .
714 0 .
286 0 0 0( B B B . Each element B ij gives the probability that level i decaysdirectly to level j . Table V. This is all the data needed to characterize the transition space and predict the profile of any kind of gatedspectrum obtained from the corresponding set of events.
A. Comparison of analytical and numerical gated spectra
We now operate the presented formalism to obtain different gated spectra. The first step is to obtain the probabilitymatrix P by applying Eq. (1): the result is presented in Table VI. Next, we have to specify a gate condition G andrun an algorithm that yields the gated probability P { G,i } of each transition t i . In the case of a positive explicit gatecondition G = { g · ... · g N } , these numbers are given by a straightforward application of Eq. (2). Note that transitionsused as gates also have a gated probability attributed: it corresponds to the probability that an event belongs to theselected set, given by P { G } = P g × N (cid:89) j =2 P g j − → g j We then obtain an elementary spectrum such as those represented on the two upper panels of Figure 7. In the caseof an optional gate condition G = { g + ... + g N } m = L m/N , we have to apply Eq. (11), which requires several steps.Starting from an empty combined spectrum, for each given value of p such that m ≤ p ≤ N , we have to:1. Determine the C Np combinations of gates that will define the positive explicit conditions G α ( p, L ) (where α identifies each combination). In practice, we calculate a combination matrix where each line α gives a sub-listof p gates, identified by their position in the list L .9 Identification Emitting Level Receiving Level Transition Energy Emission Probability t ( B ( B
354 0 . t ( B ( B
276 0 . t ( B ( B
238 0 . t ( B ( B
312 0 . t ( B ( B
74 0 . t ( B ( B
617 0 . t ( B ( B
543 0 . t ( B ( B
843 0 . t ( B ( B
300 0 . t ( B ( B
606 0 . t ( B ( B
306 0 . t ( B ( B
638 0 . t ( B ( B
332 0 . t ( B ( B
549 0 . t ( B ( B
217 0 . t ( B ( B
466 0 . t ( B ( B
249 0 . t ( B ( B
570 0 . t ( B ( B
321 0 . t ( B ( B . t ( B ( B
790 0 . t t t t t t t t t t t t t t t t t t t t t t t .
286 0 .
714 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t .
200 0 .
800 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t .
333 0 .
667 0 0 0 0 0 0 0 0 0 0 0 0 0 t .
333 0 .
667 0 0 0 0 0 0 0 0 0 0 0 0 0 t .
259 0 .
741 0 0 0 0 0 0 0 0 0 0 0 t .
259 0 .
741 0 0 0 0 0 0 0 0 0 0 0 t .
375 0 .
625 0 0 0 0 0 0 0 0 0 t .
375 0 .
625 0 0 0 0 0 0 0 0 0 t .
286 0 .
714 0 0 0 0 0 0 0 t .
286 0 .
714 0 0 0 0 0 0 0 t .
333 0 .
667 0 0 0 0 0 t .
333 0 .
667 0 0 0 0 0 t .
769 0 .
231 0 0 0 t .
769 0 .
231 0 0 0 t .
286 0 .
714 0 t .
286 0 .
714 0 t t t t A . Each element A ij gives the probability that transition i is immediately followed by transition j . These numbers can be considered either as an input, or as deduced from level-spaceinformation about branching ratios.
2. Sum the C Np elementary spectra obtained by application of Eq. (2) with the sets of gates G α ( p, L ) given by eachline of the combination matrix. This gives the sum-spectrum σ ( p, L ).3. Calculate the coefficient c p ( m ) = (cid:80) pn = m ( − ( p − n ) C pn .4. Add to the combined spectrum the sum-spectrum σ ( p, L ) affected by the factor c p ( m ).We then obtain a combined spectrum such as those represented on the two lower panels of Figure 7.0 t t t t t t t t t t t t t t t t t t t t t t .
286 0 .
714 0 .
057 0 .
229 0 .
257 0 .
514 0 .
126 0 .
360 0 .
240 0 .
400 0 .
171 0 .
428 0 .
191 0 .
381 0 .
476 0 .
143 0 .
245 0 .
612 0 . t .
286 0 .
714 0 .
057 0 .
229 0 .
257 0 .
514 0 .
126 0 .
360 0 .
240 0 .
400 0 .
171 0 .
428 0 .
191 0 .
381 0 .
476 0 .
143 0 .
245 0 .
612 0 . t .
200 0 .
800 0 .
067 0 .
133 0 .
225 0 .
642 0 .
134 0 .
224 0 .
222 0 .
554 0 .
149 0 .
297 0 .
541 0 .
162 0 .
239 0 .
598 0 . t .
333 0 .
667 0 .
086 0 .
247 0 .
282 0 .
471 0 .
151 0 .
378 0 .
207 0 .
415 0 .
450 0 .
135 0 .
247 0 .
618 0 . t .
333 0 .
667 0 .
086 0 .
247 0 .
282 0 .
471 0 .
151 0 .
378 0 .
207 0 .
415 0 .
450 0 .
135 0 .
247 0 .
618 0 . t .
259 0 .
741 0 .
097 0 .
162 0 .
239 0 .
599 0 .
134 0 .
268 0 .
563 0 .
169 0 .
237 0 .
594 0 . t .
259 0 .
741 0 .
097 0 .
162 0 .
239 0 .
599 0 .
134 0 .
268 0 .
563 0 .
169 0 .
237 0 .
594 0 . t .
375 0 .
625 0 .
107 0 .
268 0 .
244 0 .
488 0 .
394 0 .
118 0 .
252 0 .
630 0 . t .
375 0 .
625 0 .
107 0 .
268 0 .
244 0 .
488 0 .
394 0 .
118 0 .
252 0 .
630 0 . t .
286 0 .
714 0 .
095 0 .
190 0 .
623 0 .
187 0 .
232 0 .
581 0 . t .
286 0 .
714 0 .
095 0 .
190 0 .
623 0 .
187 0 .
232 0 .
581 0 . t .
333 0 .
667 0 .
256 0 .
077 0 .
264 0 .
659 0 . t .
333 0 .
667 0 .
256 0 .
077 0 .
264 0 .
659 0 . t .
769 0 .
231 0 .
220 0 .
549 0 . t .
769 0 .
231 0 .
220 0 .
549 0 . t .
286 0 .
714 0 . t .
286 0 .
714 0 . t t t t P : each element P ij gives the probability that transition i is followed by transition j after anarbitrary number of steps. This matrix is deduced from the adjacency matrix (Table V) by application of Eq. (1). The gated spectra obtained by applying the analytical formula (2) and (11) can be compared with the resultsobtained with a purely numerical approach. We have performed the following procedure:1. List all possible cascades: starting from each transition, the different possible ways are determined thanks tothe adjacency matrix, following a recursive algorithm.2. Determine the probability associated with each cascade. For this, we first have to determine the probabilitythat the deexcitation process starts with the first transition of the cascade we are considering. This is done bycombining information contained in the emission probability vector and in the adjacency matrix. For the restof the cascade, each step is associated with a probability factor given by the corresponding adjacency-matrixelement.3. The gated probability P { G,i } for each transition t i is obtained by summing the probabilities of all cascades thatcontain both t i and the gates needed to pass the selection.The gated probability values we have obtained with the analytical and numerical approaches are strictly identical, forall kinds of gate conditions. This can be seen in Figure 7, where gated spectra are presented with four different gateconditions. The corresponding values of gated probabilities are listed in Table VII. B. Determination of the emission probability of a new transition
Finally, we can check that our approach allows to determine the emission probability of a newly observed transitionthat is added to the level scheme. This new transition is denoted by t x . We make the following suppositions: • t x is directly connected to the previously known level scheme, in the sense that at least its receiving level belongsto the known level list. In this situation, all the cascades below t x are already characterized. • Coincidence data allows to place correctly t x in the level scheme. • The emission probability has been previously determined for all other transitions present in the level scheme,or, at least, all transitions involved in the relative intensity of t x in the gated spectrum that is studied.Then, is it possible to determine the emission probability P x of the new transition. In practice, for the present study, westart from the nuclear structure described above (Figure 6, with the corresponding transition-space information given1 FIG. 7: (Color online) Gated spectra associated with the schematic level scheme of Figure 6, with transition-space propertiesgiven by Tables IV and V. Gate conditions are specified on each panel : positive explicit conditions G = { t · t } for panela), G = { t · t · t } for panel b); optional conditions G = { t + t + t + t } for panel c), G = { t + t + t + t } for panel d). The lines correspond to numerical results, and the markers to analytical results (dots: intra-band transitions ofthe ground-state band; squares: intra-band transitions of the excited band; stars: inter-band transitions). The vertical axiscorresponds to the gated probability of each transition. The percentage indicated on each panel is the probability that an eventbelongs to the selected set. Transitions used as gates are indicated by the letter ”g”.Transition t i P { G ,i } P { G ,i } P { G ,i } P { G ,i } t . . . . t . . . . t . . . . t . . . . t . . . . t . . t . . . t . . . . t . . . t . t . . t . . . . t . . t . . t . . . . t . . . . t . . . . t . . t . . t . . . . t . . G = { t · t } , G = { t · t · t } , G = { t + t + t + t } , G = { t + t + t + t } . t x . Information concerning t x is deleted fromthe emission probability vector P and adjacency matrix A . It has to be recovered by using the information remainingin P and A , together with the ”observation” of a gated spectrum (here, this spectrum is previously calculated usingthe complete P and A ). We are especially interested in recovering the value of P x .Let us first study the consequences of inserting a new transition t x in the level scheme. The transition-spacedimension increases: the element t x is added to the transition vector, and the transition probability vector P has tobe completed with the corresponding value of P x . The impact on the adjacency matrix A will be the following : • The line and column corresponding to t x have to be added. • For all transitions t i arriving on the emitting level of t x , the adjacency matrix elements A ij have to be re-normalized to take into account the new possible decay path: the new values depend on P x . We have: A ij = P j / (cid:80) j (cid:48) P j (cid:48) , where t x is now part of the list of transitions t j (cid:48) . • The line added for t x is similar to the lines corresponding to the transitions that have the same receiving levelas t x (it is determined by the branching properties of the receiving level, unaffected by P x ).The modification of A , in turn, has an impact on the probability matrix P , and consequently on the gated probabilitiesgiven by Eq. (11). (We will only refer to the gated probabilities in combined spectra, since this formula allows torecover as a specific case the gated probabilities for elementary spectra given by Eq. (2).) Note that the P x valuewill affect the gated probability P { G,i } of a transition t i if t x is between t i and one of the gates, or between one ofthe gates and t i , or between two gates. Let us also specify how the gated probability P { G,x } of t x depends on theemission probability P x . We notice that Eq. (11) can be split in two terms, one (that will be denoted by X ) treatingthe cases where t x is above the sub-list of gates, and one (that will be denoted by Y ) treating the cases where t x isbetween two gates of the sub-list: X ( P x ) = N (cid:88) p = m c p ( m ) C Np (cid:88) α =1 P x × P t x → g ( α )1 × p (cid:89) j =2 P g ( α ) j − → g ( α ) j Y ( P x ) = N (cid:88) p = m c p ( m ) C Np (cid:88) α =1 P g ( α )1 × p (cid:88) h =1 p (cid:89) j =1 P T hα,j − →T hα,j X and Y both depend on P x ; however, the dependence of X is explicit, since the probability matrix elements P ij that are involved do not depend on P x but only on the branching ratios of the levels below t x . We can then write X ( P x ) = P x × C x , where C x is a ”constant” that does not depend on P x , and can be calculated using only the reducedtransition-space information. On the other hand, Y is expressed in terms of probability matrix elements that dependon P x , through the modified adjacency values occuring within the cascade.Let us now turn to the study of the relative gated intensity I ( r ) x ( G ), that can be measured in a gated spectrum withrespect to a reference transition t ref : I ( r ) x ( G ) = N { G,x } N { G,ref } = P { G,x } P { G,ref } The gated probabilities P { G,x } and P { G,ref } are given by Eq. (11). In general, these values are affected by P x and therelative gated intensity has the following dependence: I ( r ) x ( G ) = P x × C x + Y ( P x ) P { G,ref } ( P x )This can be solved for P x using a numerical iterative procedure:1. Propose a value of P x in the emission probability vector P .2. Deduce the modifications that have to be done in the adjacency matrix.3. Re-calculate the probability matrix according to Eq. (1).4. Apply Eq. (11) to determine the new gated probabilities.5. Compare the ratio P { G,x } /P { G,ref } to the relative intensity I ( r ) x ( G ) measured in the gated spectrum.36. Modify the value of P x for a new iteration, until convergence is reached.We have applied this procedure to our example, using a simple dichotomy. Trantition t was used as the referencetransition, and the different gate conditions shown in Figure 7 have been applied. All the transitions different from t ref and from the gates have been treated in turn as being the new transition t x . In all cases, the correct value of P x was recovered, except in one case: the last transition, t . Indeed, for this last transition, changing the value of P x has no impact on the gated probabilities, since no adjacency element is affected, and t is never on top of a selectedcascade. Actually, changing P t only means changing the primary feeding of the emitting level ( B , which cannotbe reflected by a gated spectrum since t is a final transition. For such a transition, the emission probability has tobe determined without gate conditions. Let us note, however, that it is not a disadvantage for our purpose: indeed,our focus is on the transitions situated in the high region of the level scheme, where multi-gating is needed to makeobservations. Lack of knowledge about transition probabilities (such as P t ) that do not affect such spectra has, bydefinition, no consequence.Let us now turn to the case of transitions situated in the higher part of the level scheme. More specifically, weconsider that the new transition t x is situated above the set of gates, and above the reference transition t ref . In thiscase, P { G,ref } has no dependence on P x , while P { G,x } reduces to P x × C x . Since P { G,ref } and C x can be calculatedusing only the reduced transition-space information, we can determine P x directly once I ( r ) x ( G ) is measured in thegated spectrum: P x = I ( r ) x ( G ) × P { G,ref } C x We have applied this procedure to our example, still using trantition t as the reference transition, and applyingthe different gate conditions shown in Figure 7. The correct value of P x was obtained for all the transitions situatedabove the set of gates. V. SUMMARY AND OUTLOOK
In the present work, we have addressed the issue of recovering the absolute probability of a transition throughthe measurement of intensities appearing in multi-gated spectra, using different kinds of gate conditions (explicit oroptional). We have presented the base of a formalism that allows to treat this problem following an analytic approach,and we have demonstrated formulas linking the gated probability of a gamma ray with two objects that characterizethe transition space of the excited nucleus: the emission probability vector P , and the probability matrix P . Theformer is linked to the primary feeding of the levels, and branching ratios; the latter, whose elements P ij give theprobability that a transition t j occurs after a transition t i has taken place (whatever the number of steps inbetween),is deduced from the transition adjacency matrix A by the analytic formula presented by Demand et al. [7]. We havefound the graph-theory framework used in this reference to be very fructful and promizing for the type of problemsto be addressed in gamma spectroscopy. Although the intensity problem we address can in principle be treated in apurely numerical way, the analytic approach allows to gain more control on the complexity of the analysis, and offersboth a way to check the results and a powerful tool to extract emission probabilities in the case of new transitions ontop of the set of gates.Although the basic principles are soundly set down in this article, some developments are needed before the presentformalism can be applied to extract emission probabilities from real experimental data. We will address in futurework several generalizations, concerning both the physics of the deexciting nucleus and the characteristics of theexperimental setup. Concerning the nucleus, we should include the following possibilities: • different deexcitation modes (e.g. electronic conversion, eletron-positron pair emission); • cases where the deexcitation cascade is cut before reaching the ground-state: presence of isomeric states, ornuclear disintegration occurring from an excited state; • existence of degenerate transitions, which has an impact on the nature of the gating condition: namely, if agate energy corresponds to the energy of several transitions, an option (”or”) in the coincidence condition isintroduced.Note that, in order to treat the last point, the structure of the gate condition will have to be generalized beyond thetwo cases defined in this article (explicit/optional). It is anyway a useful development to consider gate conditionswith a mixed explicit and optional structure, depending on the part of the level scheme where the gates are situated.Concerning now the experimental setup, we should specify the following properties:4 • response of the detector system: detection efficiency, eventually including the role of angular correlations betweenthe emitted gamma rays; • possibility of having experimental data filtered by a multiplicity threshold; • treatment of the back-ground.These improvements will allow one to apply the analytic formalism introduced here in the case of real data. Theycould eventually lead to the development of a dedicated piece of software. This approach stresses the importance toobtain accurate (rather than approximately estimated) values of the emission probabilities, even when they concernweak gamma rays that can only be accessed via multi-gated spectra. Precise results can be used for instance as acriterion to check the tentative placement of a new transition in the level scheme: consistent values have to be obtainedwhen using different gate conditions. Most importantly, the emission probabilities contain fundamental informationthat should be used to improve our knowledge of nuclear structure and reactions. Appendix A: Examples of optional conditions and associated spectra
An optional condition is denoted by G{ g + ... + g N } m ; it involves a list of N optional gates L = { g , ..., g N } , and m isthe minimal number of open gates among this list. The combined set of events associated with this condition is E ( G );it is represented by the combined spectrum S ( G ). We want to develop S ( G ) in terms of positive elementary spectra S α = S ( G α = { g ( α )1 · ... · g ( α ) p } ). Each positive elementary condition G α involves a list of p gates L ( α ) = { g ( α )1 , ..., g ( α ) p } extracted from L , with m ≤ p ≤ N . In this appendix, we illustrate with specific examples the formulas that arederived in the text and summarized below: • Tiling relation (3) to express the combined set as non-overlapping exclusive elementary sets: E ( G = L m/N ) = N (cid:91) n = m C Nn (cid:91) β =1 E ( G β ( n, N − n, L )) • The resulting relation (4) between combined and exclusive elementary spectra: S ( G ) = N (cid:88) n = m C Nn (cid:88) β =1 S ( G β ( n, N − n, L )) = N (cid:88) n = m σ ( n, N − n, L ) • Expression of exclusive sum-spectra in terms of positive sum-spectra. We remind that a sum-spectrum is thesummation of all the elementary spectra of the same order, that can be defined from the same gate list L . Thisexpression is given by Eq. (9): σ ( n, N − n, L ) = N (cid:88) p = n a n,p σ ( p, L ) with a n , p = ( − p − n C pn This leads to the final expression (10) of the combined spectrum: S ( G ) = N (cid:88) p = m c p ( m ) σ ( p, L ) with c m (p) = p (cid:88) n=m a n , p = p (cid:88) n=m ( − p − n C pn • Concerning the spiked spectrum associated with condition G , we also give the expression of E ( G ) as the union ofpositive elementary sets, according to Eq. (12), and the corresponding biased sum-spectrum given by Eq. (13): E ( G ) = E ( L m/N ) = C Nm (cid:91) α =1 E ( G α ( m, L )) S s ( G ) = σ ( m, L )5
1. Optional gate condition G = { g + g + g } In this example, N = 3, m = 2, and L = { g , g , g } . • Combined set expressed by the tiling relation: E ( { g + g + g } ) = E g g ¯ g ∪ E g g ¯ g ∪ E g g ¯ g ∪ E g g g • Combined spectrum expressed as a sum of exclusive elementary spectra: S ( { g + g + g } ) = S g g ¯ g + S g g ¯ g + S g g ¯ g + S g g g = σ (2 , , L ) + σ (3 , , L )where the exclusive sum-spectrum of order (2 ,
1) is: σ (2 , , L ) = σ (2 , , { g , g , g } ) = S g g ¯ g + S g g ¯ g + S g g ¯ g and the exclusive sum-spectrum of order (3 , σ (3 , , L ) = σ (3 , , { g , g , g } ) = σ (3 , { g , g , g } ) = S g g g • Expression of the exclusive sum-spectrum σ (2 , , L ) in terms of positive sum-spectra, as given by Eq. (9): σ (2 , , L ) = a , σ (2 , L ) + a , σ (3 , L ) = σ (2 , L ) − σ (3 , L )with σ (2 , L ) = S g g + S g g + S g g σ (3 , L ) = S g g g As a result: S ( { g + g + g } ) = [ σ (2 , L ) − σ (3 , L )] + σ (3 , L ) = σ (2 , L ) − σ (3 , L )= S g g + S g g + S g g − S g g g Note that for such a reduced list of gates, the final result can easily be obtained in a pedestrian approach,applying intuitively the development of exclusive spectra involving one closed gate: S g g ¯ g = S g g − S g g g = g g (1 − g ) S g g ¯ g = S g g − S g g g = g g (1 − g ) S g g ¯ g = S g g − S g g g = g g (1 − g )which are specific examples of the general relation (6). We recover the final result: S ( { g + g + g } ) = S g g + S g g + S g g − S g g g • We finally consider the spiked spectrum. The combined set can be expressed as the union of all positiveelementary sets of order m = 2: E ( G ) = C (cid:91) α =1 E ( G α (2 , L )) = E g g ∪ E g g ∪ E g g The corresponding summation of elementary spectra (which involves multi-counting of events in the overlappingregions of the united sets) gives the spiked spectrum: S s ( G ) = σ (2 , L ) = C (cid:88) α =1 S ( G α (2 , L )) = S g g + S g g + S g g The relation between combined and spiked spectra is: S ( G ) = S s ( G ) − S g g g which, again, can be found intuitively in this simple example: one can see directly that the events of theoverlapping part E g g g are counted three times in the spiked spectrum, since they belong to all three sets E g g , E g g and E g g .6
2. Gate condition G = { g + g + g + g } In this example, N = 4, m = 2, and L = { g , g , g , g } . With only one more gate in the optional list, one findsthat the pedestrian approach to express the combined spectrum in terms of positive elementary spectra is alreadymuch more tedious, and the analytic expressions that have been derived are now helpful. • Combined set expressed by the tiling relation: E ( { g + g + g + g } ) = (cid:91) n =2 C n (cid:91) β =1 E ( G β ( n, − n, L ))= C (cid:91) β =1 E ( G β (2 , , L )) C (cid:91) β (cid:48) =1 E ( G β (cid:48) (3 , , L )) C (cid:91) β (cid:48)(cid:48) =1 E ( G β (cid:48)(cid:48) (4 , , L ))where C (cid:91) β =1 E ( G β (2 , , L )) = E g g ¯ g ¯ g ∪ E g g ¯ g ¯ g ∪ E g g ¯ g ¯ g ∪ E g g ¯ g ¯ g ∪ E g g ¯ g ¯ g ∪ E g g ¯ g ¯ g C (cid:91) β (cid:48) =1 E ( G β (cid:48) (3 , , L )) = E g g g ¯ g ∪ E g g g ¯ g ∪ E g g g ¯ g ∪ E g g g ¯ g C (cid:91) β (cid:48)(cid:48) =1 = E g g g g • Combined spectrum expressed as a sum of exclusive elementary spectra: S ( { g + g + g + g } ) = (cid:88) n =2 C n (cid:88) β =1 S ( G β ( n, − n, L ))= C (cid:88) β =1 S ( G β (2 , , L )) + C (cid:88) β (cid:48) =1 S ( G β (cid:48) (3 , , L )) + C (cid:88) β (cid:48)(cid:48) =1 S ( G β (cid:48)(cid:48) (4 , , L ))= σ (2 , , L ) + σ (3 , , L ) + σ (4 , , L )where the sum-spectra σ are: σ (2 , , L ) = S g g ¯ g ¯ g + S g g ¯ g ¯ g + S g g ¯ g ¯ g + S g g ¯ g ¯ g + S g g ¯ g ¯ g + S g g ¯ g ¯ g σ (3 , , L ) = S g g g ¯ g + S g g g ¯ g + S g g g ¯ g + S g g g ¯ g σ (4 , , L ) = S g g g g • Expression of the exclusive sum-spectra σ ( n, N − n, L ) in terms of positive sum-spectra σ ( p, L ). Following thepedestrian approach, each term can be developped recursively according to: S g g ¯ g ¯ g = S g g ¯ g − S g g g ¯ g = ( S g g − S g g g ) − ( S g g g − S g g g g )In the end, we recover the result expressed by the analytic formula: S ( G = { g + g + g + g } ) = (cid:88) p =2 c p ( m ) σ ( p, L )Term p = 2: c p ( m ) = p (cid:88) n =2 a n,p = p (cid:88) n =2 ( − p − n C pn = 1 σ (2 , L ) = S g g + S g g + S g g + S g g + S g g + S g g p = 3: c p ( m ) = p (cid:88) n =2 a n,p = p (cid:88) n =2 ( − p − n C pn = − − σ (3 , L ) = S g g g + S g g g + S g g g + S g g g Term p = 4: c p ( m ) = p (cid:88) n =2 a n,p = p (cid:88) n =2 ( − p − n C pn = 6 − σ (4 , L ) = S g g g g The combined spectrum is then expressed as: S ( G ) = σ (2 , L ) − × σ (3 , L ) + 3 × σ (4 , L )Note that the coefficients c p ( m ) = (cid:80) pn = m ( − p − n C pn can be obtained by column summation in a universal tablethat contains the coefficients a n,p (Table I). • Let us finally consider the spiked spectrum. The union of all positive elementary sets of order m = 2 is now: E ( G ) = C (cid:91) α =1 E ( G α (2 , L )) = E g g ∪ E g g ∪ E g g ∪ E g g ∪ E g g ∪ E g g The corresponding summation of elementary spectra (which involves multi-counting of events in the overlappingregion of the united sets) gives the spiked spectrum: S s ( G ) = σ (2 , L ) = C (cid:88) α =1 S ( G α (2 , L )) = S g g + S g g + S g g + S g g + S g g + S g g The relation between combined and spiked spectra is: S ( G ) = S s ( G ) − × σ (3 , L ) + 3 × σ (4 , L ))