LQ Cut

LQ (late charge) is a pulse-shape parameter that measures the amount of charge collected late in a detector pulse. Events with an increased LQ, for example interactions close to the detector surface, are rejected with a one-sided cut. LegendSpecFits.jl corrects the drift-time dependence of LQ with the double escape peak (DEP) of the 2614.5 keV Th-228 line, normalizes LQ to the width of the DEP distribution, and measures the survival fractions of the cut. The LQ value itself is computed in the signal-processing stage and is an input here.

Inputs and Pre-Normalization

The functions use the energy-corrected LQ (lq / e in the dataflow), an effective drift time dt_eff (qdrift / e), the calibrated energies e_cal, and the mean dep_μ and standard deviation dep_σ of the DEP peak in energy, taken from the energy calibration. Before the correction, the dataflow scales LQ so that the main population is near 1, with cut_single_peak and its field max:

cuts = cut_single_peak(lq_e_corr, 0.0, quantile(filter(isfinite, lq_e_corr), 0.99); n_bins = -1)
lq_e_corr ./= cuts.max

Drift-Time Correction

LQ of single-site events depends on the drift time. lq_ctc_correction fits this dependence with DEP events and subtracts it:

Select events with energy within ctc_dep_edgesigma (default 3.0) standard deviations of the DEP mean.
Fit a truncated Gaussian to their LQ distribution (fit_single_trunc_gauss); the fit window is found with cut_single_peak and ctc_lq_precut_relative_cut (default 0.25). Keep events with LQ within lq_outlier_sigma (default 2.0) standard deviations of the fitted mean.
Restrict the drift time to a range selected by ctc_driftime_cutoff_method: :percentile (default) uses the quantiles ctc_dt_eff_low_quantile (0.15) and ctc_dt_eff_high_quantile (0.95); :gaussian uses dt_eff_outlier_sigma (2.0) standard deviations around a Gaussian fit and fails for detectors with two drift-time populations; :double_gaussian uses the range in which a double-Gaussian fit exceeds 10% of its maximum. Other values throw an ArgumentError.
Fit a polynomial of order pol_fit_order (default 1) of LQ as a function of drift time to the remaining events.

The result contains func, an expression string that subtracts the polynomial from LQ, the fit result fit_result, and the selection limits box_constraints. The keywords lq_e_corr_expression and dt_eff_expression set the names that appear in func:

( lq ) - p0 * (dt)^0 - p1 * (dt)^1

The report contains the fits and the selection limits for plotting with lplot from LegendMakie. The keyword uncertainty controls whether the fits compute parameter uncertainties (default false).

LQ Normalization

lq_norm scales the corrected LQ so that the DEP distribution has mean 0 and standard deviation 1. It builds the LQ histogram of events within dep_sideband_sigma (default 4.5) standard deviations of the DEP mean and subtracts the histograms of two sidebands, which together have the same energy width. If dep_σ is below 2 keV, the sidebands lie on both sides of the peak; otherwise both lie below the peak, to stay clear of the neighboring 1620.5 keV Bi-212 line. Negative bins after the subtraction are set to zero. A binned truncated Gaussian fit (fit_binned_trunc_gauss) to the subtracted histogram, truncated at cut_truncation_sigma (default 3.5) standard deviations around the estimated peak position, gives the mean $\mu_\mathrm{LQ}$ and standard deviation $\sigma_\mathrm{LQ}$. The result field func is the expression

\[\mathrm{LQ}_\mathrm{norm} = \frac{\mathrm{LQ}_\mathrm{ctc} - \mu_\mathrm{LQ}}{\sigma_\mathrm{LQ}}\,,\]

where $\mathrm{LQ}_\mathrm{ctc}$ is the drift-time-corrected LQ given by lq_class_expression. The report contains the histograms of the DEP and the sidebands, the fit report, and the sideband energy limits.

The following example applies both steps to synthetic data with a drift-time dependence and a flat energy background:

using LegendSpecFits, Random, Unitful

Random.seed!(3)
N, Nb = 20_000, 6_000
dep_μ, dep_σ = 1592.5u"keV", 1.5u"keV"
e_cal = [dep_μ .+ dep_σ .* randn(N); (1550 .+ 90 .* rand(Nb))u"keV"]
dt_eff = 0.5 .+ 0.1 .* randn(N + Nb)
lq = [0.02 .* randn(N); 0.2 .* rand(Nb) .- 0.05] .+ 0.01 .* dt_eff .+ 0.05

result_ctc, report_ctc = lq_ctc_correction(lq, dt_eff, e_cal, dep_μ, dep_σ;
    lq_e_corr_expression = "lq", dt_eff_expression = "dt")
lq_ctc = lq .- report_ctc.drift_time_func.(dt_eff)

result_norm, report_norm = lq_norm(dep_μ, dep_σ, e_cal, lq_ctc; lq_class_expression = result_ctc.func)
println(result_norm.func)
result_norm.fit_result.σ

\[0.01991 \pm 0.0001\]

Cut and Survival Fractions

The LQ cut is a configured upper limit on the normalized LQ, in units of the DEP standard deviation (high_cut_sigma in the dataflow); it is not fitted. The survival fractions use the same functions as the A/E cut, with the normalized LQ as the cut parameter and only an upper bound:

result_peaks, report_peaks = get_peaks_survival_fractions(
    lq_norm_values, e_cal, peaks, peak_names, windows_left, windows_right;
    high_cut = high_cut_sigma, fit_funcs = fit_funcs,
)
result_qbb, report_qbb = get_continuum_survival_fraction(
    lq_norm_values, e_cal, qbb, qbb_window; high_cut = high_cut_sigma,
)

get_peaks_survival_fractions fits each peak with and without the cut and returns the signal survival fraction sf in percent; get_continuum_survival_fraction counts events around $Q_{\beta\beta}$. Both are described in Survival Fractions. The A/E and LQ cuts are combined in the PSD efficiency step, which passes a Boolean selection built from both cuts as survival_flag to the same functions.

Order of the LQ Steps

For each LQ definition, the dataflow runs:

Pre-normalization with cut_single_peak.
lq_ctc_correction with the DEP parameters from the energy calibration; its func becomes the LQ expression.
lq_norm on the corrected LQ.
Survival fractions with the upper cut.

Compton-Band Fits

fit_lq_compton fits the LQ distribution in Compton bands with the same peak shape as the A/E bands (aoe_compton_peakshape). It takes the band histograms peakhists, the start values peakstats, and the band energies, in the format returned by generate_aoe_compton_bands, and returns an ordered dictionary of results and reports keyed by the band energy. The dataflow does not use it.