Gamma electron Angular Correlations
The Pentagonal Array for Conversion Electron Spectroscopy (PACES) at TRIUMF is an ancillary detection system used for in-vacuum high-resolution conversion electron spectroscopy. PACES is comprised of 5 lithium-drifted silicon (Si(Li)) detectors.
The installation of PACES necessitates the removal of one GRIFFIN HPGe clover (position 13 in the upstream lampshade) such that the maximum number of coincidence angles between GRIFFIN and PACES detectors is 300 (60 HPGe crystals x 5 Si(Li)). In the GRIFFIN coordinate system, the PACES Theta and Phi coordinates are as follows, with a diagram of PACES included for clarity (beam travels out of the page in this view).
In the analysis of gamma-electron angular correlations, it is essential that both detectors share the same coordinate system. Newer versions of GRSISort contain the coordinates of all HPGe crystals and PACES detectors.
The angle between GRIFFIN and PACES detectors can now be found simply:
double angle = TGriffin::GetPosition(gd, gc, distance).Angle(TPaces::GetPosition(p))*180./TMath::Pi();,
where gd is the GRIFFIN HPGe clover position (1-16), gc is the HPGe crystal number (0-3) and p is the PACES detector number (0-4). The distance of the HPGe clovers from the source, distance is 110 or 145 mm (the latter used for full Compton suppression). A histogram of calculated angles is included here: the number of angles per bin is 4.
GRSISort contains coordinates of all HPGe crystals in TGriffin.cxx so that the relative angles can be extracted via TVector.
Similarly the PACES coordinates can be defined in TPaces.cxx.
Note that GRSISort will need to recompiled after this modification.
Examples are provided on how to incorporate the GRIFFIN-PACES angles into a data processing script. Here, GRSIProof is used to process the AnalysisTree which includes time-correlated GRIFFIN and PACES events. The procedure is outlined as follows:
- A map of coincidence angles is constructed and grouped into a given number of bins.
- Relative angle of GRIFFIN-PACES detector hits is found event-by-event.
- Depending on the angle, a different electron-gamma matrix is filled.
- Final electron-gamma matrices are processed to extract the angular correlation.
In practice, calculating the angle for each correlation is not necessary. Instead each detector pair is assigned an index based on the following formula:
double index = (64*p)+(((gd-1)*4))+gc,
where index takes the values 0-319. Due to the missing HPGe clover (position 13), index values from 48-51, 112-115, 176-179, 240-243 and 304-307 are not used. Indexes may be grouped together on the basis of their corresponding angle, forming angle bins. This is typically 20-30 bins but may vary depending on the analysis. Grouping is useful as it results in a more managable number of coincidence spectra to analyze, as well as summing together statistics. Consider the coincidence angles available for a single PACES detector (e.g. #2 at phi=94 degrees). The index map (grouped into bins of 2-3 angles) is as follows:
double group[nbin][w]={{90,85,-1},{86,118,91},{119,84,88},{87,94,-1},{117,81,116},{95,65,93},{125,80,66},{82,92,70},{124,64,126},{83,122,67},{98,71,121},{69,127,109},{99,123,97},{68,108,120},{110,77,96},{111,102,105},{101,78,106},{79,103,75},{104,74,-1},{100,72,107}};,
where nbin=20, w=3 and -1 is a dummy variable/placeholder. More complex maps can be constructed if including more PACES detectors. In general, the number of angles per bin (the weight) should be managed so that the difference in weight between each bin is minimised. This helps reduce the scatter observed in the final angular correlation. However, the relative geometry of the GRIFFIN-PACES detector systems results in an angular coverage strongly peaked around 90 degrees and this enhances the variation.
The angle map can be included in the header file (.h) of the GRSIProof Selector. For example,
std::vector<std::pair<double, double>> PACESAngleCombinations(){
double group[nbin][w]={...};
std::vector<std::pair<double, double>> result;
for(int i=0;i<int(nbin);++i){
for(int j=0;j<int(w);++j){
if(group[i][j]<0) continue;
result.push_back(std::make_pair(group[i][j], double(i)));
}
}
return result; }
In ..public TGRSISelector{ define:
std::vector<std::pair<double, double>> fPACESAngleCombinations;
std::map<double, double> fPACESAngleMap;
int maxanglebin;
and in the nested loop EventSelector(TTree * /*tree*/ =0) : TGRSISelector(), fGrif(0), fPaces(0) {, define:
for(int i = 0; i < static_cast<int>(fPACESAngleCombinations.size()); ++i) {
if(fPACESAngleCombinations[i].second>0) maxanglebin = fPACESAngleCombinations[i].second;
fPACESAngleMap.insert(std::make_pair(fPACESAngleCombinations[i].first, fPACESAngleCombinations[i].second));
}
Tip: remember to define the PACESAngleCombinations vector at the beginning of the header file.
In the main PROOF program (.C) we can now refer to PACESAngleMap to determine the angle bin to which each detector pair belongs. In FillHistograms() {,
double index = (64*p)+(((gd-1)*4)+gc);
auto angle_bin = fPACESAngleMap.lower_bound(index-0.5);
int whichindex = int(angle_bin->first);
int whichbin = int(angle_bin->second);
The PACESAngleMap key (by default the key is the first column of the array) is queried to find the bin number corresponding to the index. In this example, the map using a single PACES detector would return bin 2 for indexes 84, 88 and 119. The variables whichindex and whichbin can be used as diagnostics to check the map is functioning properly. Examples are shown here using only PACES detector #2 with no grouping applied.
As a minimum requirement, two versions of the electron-gamma matrix should be produced by PROOF for each angle bin corresponding to time-correlated and time-random coincidence events, defined by appropriate cuts on the electron-gamma time-difference spectrum. The time-random matrix is scaled by a factor equal to the ratio of the time-correlated and time-random cuts, then subtracted from the time-correlated matrix.
A projection of the (time-random background-subtracted) time-correlated matrix is produced by gating on the ICE or gamma-peak of interest. A second projection gated on the Compton background near the peak must also be subtracted. This subtraction and (importantly) the scaling of the Compton background are handled automatically by the jROOT and TBGSubtraction() environments available via GitHub (J.Smallcombe) and GRSISort respectively. It is recommended to use either of these applications when performing this step. The peak areas extracted from the appropriately-subtracted projections form the basis of the angular correlation measurement.
Peak areas must be normalized according to the weight of each angle bin. This is achieved by using either
- the combinatorial weight of the bin (i.e. integer representing number of detector pairs) or
- the event-mixed weight determined from uncorrelated physics events.
The event-mixed weights are preferred as these account for variations in the detector-pair efficiency. The event-mixed weights correspond to the coincidence-gated electron (or gamma-ray) peak areas previously described; however the electron-gamma matrices are constructed from hits in separate, uncorrelated physics events. For an example of how to include event-mixed matrices in a PROOF script, see the link below for gamma-gamma angular correlations:
The process of extracting peak areas from correlated or uncorrelated electron-gamma matrices is identical with the exception of the time-random background subtraction which is unnecessary in the latter case.