Angular Correlations

The angle between two sequentially emitted gamma rays has a distribution that is dependent on the spins of the three states involved and the mixing ratio of the transitions. Measuring an angular correlation can assist in spin assignment or measurements of particular transition strengths.

To create an angular distribution for GRIFFIN two main steps are needed:

1. Creating 2D histograms of gamma-gamma energies for all possible angle combinations using grsiframe and the provided AngularCorrelationHelper.
2. Analysing a specific cascade using the AngularCorrelations program.

grsiframe and AngularCorrelationHelper

To create the 2D histograms we use the AngularCorrelationHelper and grsiframe like this: grsiframe <grsiframe options, i.e. --max-workers 2> $GRSISYS/examples/AngularCorrelationHelper.cxx <analysis file(s)> <parameter file with specfic settings> 2>&1 | tee <log-file name> where the parameter file (see $GRSISYS/examples/AngularCorrelation.par for an example) holds all the specific settings that determine what angles to use:

Parameter name	default	possible values	comments
GriffinDistance	145.	145. or 110.	distance of the detector front face from the center of the arrya, note the trailing .
Addback	true	true or false	whether to use addback or not when creating the spectra
Folding	false	true or false	whether to fold all angles around 90 degrees, i.e. map 180 degree to 0 degree, 135 degree to 45 degree and so on
Grouping	false	true or false	whether to group specific angles together which are close to each other, this also activates Folding unless custom grouping is used
CustomGrouping	empty	comma separated list of integers	list of groups for angles, needs to be as many entries as there are angles
ExcludeDetector	empty	comma separated list of integers (1-16)	list of all detectors that will be excluded in calculating the angles and creating the spectra
ExcludeCrystal	empty	comma separated list of integers (1-64)	list of all crystals that will be excluded in calculating the angles and creating the spectra
MaxPromptTime	200.	positive floating point number	maximum absolute time difference between two gammas for them to be considered to be in coincidence with each other
TimeRandom.Low	400.	positive floating point number	minimum of the absolute time difference between two gammas for them to be considered time random
TimeRandom.High	600.	positive floating point number	maximum of the absolute time difference between two gammas for them to be considered time random
NumberOfBins	3000	positive integer	number of bins in the x- and y-axis of the matrices
MinimumEnergy	0.	positive floating point number	minimum on the x- and y-axis of the matrices
MaximumEnergy	3000.	positive floating point number	maximum on the x- and y-axis of the matrices
NumberOfMixedEvents	10	positive integer	number of events to be used for event mixing

This helper will create prompt, time random, and event mixed histograms for each angle. "Prompt" means these are matrices for the energies of two gamma rays that are in coincidence with each other, as determined by the "MaxPromptTime" parameter. "Time random" means these matrices are for gamma rays that are not in coincidence with each other but are randomly within a certain time frame of each other, as determined by the "TimeRandom.Low" and "TimeRandom.High" parameters. These time random matrices can be used to correct for background from time random gamma rays within the coincidence window. "Mixed" means these matrices are generated using two gamma rays from two different events, ensuring that they are completely uncorrelated with each other. The number of previous event that are held in memory for this mixing is determined by the "NumberOfMixedEvents" (default is 10).

The names of the histograms are "AngularCorrelationN" for the prompt matrices, "AngularCorrelationBGN" for the time random matrices, and "AngularCorrelationMixedN" for the mixed event matrices, and N is an integer index starting at 0. The instances of TGriffinAngles and TUserSettings are also written to the output file, so one can see what settings have been used to create the root-file.

The difference in angles under which we consider the angles identical defaults to 0.001. This value can not be changed by a user setting, but could be changed by adding a line TGriffinAngles::Rounding(new value) to the constructor of the helper. This setting should only be changed if you are absolutely certain you know need it to be changed and know what you are doing.

If the statistic of the data set is limited, it can help to use folding and/or grouping to increase the statistics per histogram. Folding means that we take advantage of the symmetry of the angular distributions around 90 degree and map 180 degree to 0 degree, 135 degree to 45 degree, and so on. This will double the statistics in the histograms (apart from the 0 degree histogram since we only have the 180 degree data in there), and reduce the number of histograms from 51 to 26. If folding alone does not help, grouping can be enabled which groups angles that are close to each other together. This reduces the number of histograms further down to 11 (110 mm distance) or 12 (145 mm distance). Using grouping will reduce the information in the data (unlike folding), so the TGriffinAngles class enforces folding to be used if grouping is enabled.

It is also possible to use custom grouping by providing a list of the indices of the group each angle belongs to. Adding a line CustomGrouping: 0, 1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 7, 8, 8, 8, 8, 9, 9, 10, 11, 12, 13, 14, 15, 15 would create a total of 16 groups with 1 to 4 angles in each group. If custom grouping is provided folding does not have to be enabled, i.e. custom grouping allows to create groups for all 51 angles without mapping angles above 90 degree to those below 90 degree.

AngularCorrelations program

To analyze a specific gamma cascade we use the AngularCorrelations program. This program takes the matrices created by the helper in the previous step, projects an energy range around the first peak on the y-axis onto the x-axis, subtracts projections of background areas, and fits these with the second peaks. The areas of these peaks are then used to calculate the angular distributions by dividing the areas in the projections of the time random corrected prompt matrices by the areas in the projections of the event mixed matrices. This division ensures that any effects from the different number of combinations of crystals that can contribute to a specific angle, or the different efficiencies of the detectors are accounted for. Parameters for this part of the program are:

Parameter name	possible values	comments
Projection.Low	positive floating point number	the low edge of the range of values on the y-axis that will be projected onto the x-axis, can be overwritten by the projection-low flag
Projection.High	positive floating point number	the high edge of the range of values on the y-axis that will be projected onto the x-axis, can be overwritten by the projection-high flag
TimeRandomNormalization	positive floating point number	the ratio between the time window of the prompt gamma-gamma spectra, and the time window of the time random gamma-gamma spectra, can be overwritten by time-random-normalization flag
Peak.Position	positive floating point number	the position of the peak to be fitted in the projections, can be overwritten by the peak-pos flag
Peak.Position.Low	positive floating point number	the low edge of the range of the peak fit in the projections, can be overwritten by the peak-low flag
Peak.Position.High	positive floating point number	the high edge of the range of the peak fit in the projections, can be overwritten by the peak-high flag
Background.N.Low	positive floating point number	the low edge of the Nth background projection to be subtracted from the projection of the first gamma peak, n is give by N which is an integer starting at 0
Background.N.High	positive floating point number	the high edge of the Nth background projection to be subtracted from the projection of the first gamma peak, n is give by N which is an integer starting at 0
BackgroundPeak.Position.N	positive floating point number	positions of background peaks that are included in the fit of the projection but whose areas are ignored, N is the index starting at 0
Peak.Parameter.N	floating point number	initial value for Nth parameter of the peak, this value is only used if the limits are also set
Peak.Parameter.N.Low	floating point number	lower limit for Nth parameter of the peak, if this is the same as the upper limit, the parameter is fixed to this value
Peak.Parameter.N.Hihg	floating point number	upper limit for Nth parameter of the peak, if this is the same as the lower limit, the parameter is fixed to this value
BackgroundPeak.Parameter.N	floating point number	initial value for Nth parameter of the background peak(s), this value is only used if the limits are also set
BackgroundPeak.Parameter.N.Low	floating point number	lower limit for Nth parameter of the background peak(s), if this is the same as the upper limit, the parameter is fixed to this value
BackgroundPeak.Parameter.N.High	floating point number	upper limit for Nth parameter of the background peak(s), if this is the same as the lower limit, the parameter is fixed to this value
Background.Offset(.Low/.High)	floating point number	value, lower, and upper limit for the constant offset in the global background of the fit
Background.Linear(.Low/.High)	floating point number	value, lower, and upper limit for the linear factor in the global background of the fit
Background.Quadratic(.Low/.High)	floating point number	value, lower, and upper limit for the quadratic factor in the global background of the fit
Background.Xoffset(.Low/.High)	floating point number	value, lower, and upper limit for the offset in x used in the global background of the fit

The created angular distributions can be compared to simulated angular distributions. To do so, the user has to provided the path to a .root file via the theory flag. That .root file is expected to contain three TGraphErrors called "graph000", "graph010", and "graph100", which are the simulated "Z₀", "Z₂", and "Z₄" distributions as described in method 2 of the paper by Smith et al. If these are provided, the following parameters are being used:

Parameter name	possible values	comments
TwoJ.Low	comma separated integers	two times J for the lowest state involved in the cascade, this has to be a vector, even if it has only one entry that entry must be followed by a comma
TwoJ.Middle	comma separated integers	two times J for the middle state involved in the cascade, this has to be a vector, even if it has only one entry that entry must be followed by a comma
TwoJ.High	comma separated integers	two times J for the highest state involved in the cascade, this has to be a vector, even if it has only one entry that entry must be followed by a comma
ConfidenceLevel	positive floating point number	the value for the 99% confidence level that the reduced chi2 is smaller than this (depends on the degrees of freedom)

Of the three vectors only one can have more than one entry, and the program will produce for each entry a graph showing the reduced chi² vs. the mixing angle. These graphs are all saved to the output file, and also plotted on a canvas that is also saved to the output file under the name "MixingCanvas".

Finally the program also does a free fit of a₂ and a₄, with the results saved to the output file and plotted on a canvas called "A2a4Canvas".

The parameters for the best fit for each spin as well as the parameters of the free fit are also saved to the output file as ParametersN (N being an index starting with 0), or ParametersA2a4Fit. These parameters can be retrieved from the output file by first declaring a pointer to a vector of doubles, and then using TFile::GetObject to read the desired parameter set:

std::vector<double>* par; _file0->GetObject("ParametersSpin0", par);

Further information about angular correlations

The angular correlation NIM paper is an excellent resource, as well as these example scripts and the README within. You may need simulations, in which case the GitHub repos here will be helpful. If your data have poor enough statistics that you don't necessarily need simulations, you'll at least need some attenuation factors, which can be found for the 16-clover setup here.