Calibration and Quantification

From peak area to concentration

A LIBS calibration converts measured peak area (or intensity ratio) to elemental concentration using a set of matrix-matched standards.

cal_ds <- ls_example_data("calibration")
cal_ds
conc <- cal_ds$sample_info$concentration
unique(conc)
#> [1]     0   100   250   500  1000  2500  5000  7500 10000

cal <- ls_calibrate(cal_ds, element = "Ca", line_nm = 393.37,
                    concentrations = conc, method = "univariate",
                    verbose = FALSE)
cal

ls_plot_calibration(cal)

ls_plot_residuals(cal)

wl_is <- c(seq(392, 395, by = 0.02), seq(587, 591, by = 0.02))
spectra_is <- lapply(seq_along(conc), function(i) {
  response <- 5 + 0.1 * conc[i] * exp(-0.5 * ((wl_is - 393.37) / 0.12)^2) +
    20 * exp(-0.5 * ((wl_is - 589.00) / 0.12)^2) +
    0.1 * sin(3 * wl_is + i)
  ls_spectrum(wl_is, response,
              metadata = list(sample_id = paste0("ratio_standard_", i)))
})
is_ds <- ls_dataset(spectra_is)
cal_is <- ls_calibrate(is_ds, "Ca", 393.37, concentrations = conc,
                       method = "internal_std",
                       internal_std_nm = 589.00,
                       verbose = FALSE)
cal_is
ls_quantify(cal_is, is_ds)
#> # A tibble: 27 × 6
#>    sample_id         element concentration unit  below_lod below_loq
#>    <chr>             <chr>           <dbl> <chr> <lgl>     <lgl>    
#>  1 ratio_standard_1  Ca              -3.28 ppm   TRUE      TRUE     
#>  2 ratio_standard_2  Ca              -1.83 ppm   TRUE      TRUE     
#>  3 ratio_standard_3  Ca              -1.46 ppm   TRUE      TRUE     
#>  4 ratio_standard_4  Ca              98.2  ppm   TRUE      TRUE     
#>  5 ratio_standard_5  Ca              95.6  ppm   TRUE      TRUE     
#>  6 ratio_standard_6  Ca              94.2  ppm   TRUE      TRUE     
#>  7 ratio_standard_7  Ca             244.   ppm   FALSE     TRUE     
#>  8 ratio_standard_8  Ca             248.   ppm   FALSE     TRUE     
#>  9 ratio_standard_9  Ca             251.   ppm   FALSE     TRUE     
#> 10 ratio_standard_10 Ca             503.   ppm   FALSE     FALSE    
#> # ℹ 17 more rows

cal_pls <- ls_calibrate(cal_ds, "Ca", 393.37, concentrations = conc,
                        method = "pls", n_components = 5,
                        verbose = FALSE)
cal_pls

Quantifying unknowns

The calibration retains its exact wavelength axis, ordered preprocessing, integration window, and internal-standard line or PLS feature columns. Apply the same processing to new measurements. ls_quantify() reuses the saved recipe and rejects a different supplied window, changed channel labels or missing legacy contracts. Refit models created before the contract was introduced.

ls_quantify(cal, cal_ds)
#> # A tibble: 27 × 6
#>    sample_id    element concentration unit  below_lod below_loq
#>    <chr>        <chr>           <dbl> <chr> <lgl>     <lgl>    
#>  1 std_00000_r1 Ca             1708.  ppm   TRUE      TRUE     
#>  2 std_00000_r2 Ca               27.1 ppm   TRUE      TRUE     
#>  3 std_00000_r3 Ca              679.  ppm   TRUE      TRUE     
#>  4 std_00100_r1 Ca               27.1 ppm   TRUE      TRUE     
#>  5 std_00100_r2 Ca               27.1 ppm   TRUE      TRUE     
#>  6 std_00100_r3 Ca               27.1 ppm   TRUE      TRUE     
#>  7 std_00250_r1 Ca               27.1 ppm   TRUE      TRUE     
#>  8 std_00250_r2 Ca             1057.  ppm   TRUE      TRUE     
#>  9 std_00250_r3 Ca               27.1 ppm   TRUE      TRUE     
#> 10 std_00500_r1 Ca             1858.  ppm   TRUE      TRUE     
#> # ℹ 17 more rows

LOD and LOQ

For univariate and internal-standard fits, the estimates are 3 * sigma / abs(slope) and 10 * sigma / abs(slope), where the slope and residual standard deviation are from an auxiliary response-on-concentration regression. These estimates have the calibration concentration unit and remain invariant under instrument gain. The inverse concentration-on-response model still supplies predictions.

The package retains its stated three-sigma convention. The IUPAC definition makes the factor depend on the chosen certainty, while ICH Q2(R2), section 3.2.3.3 uses 3.3 for the response-standard-deviation method and requires suitable verification near the proposed limits. These functions do not establish compliance with a validation protocol.

A supplied blank needs at least two replicate shots with the fitted wavelength and processing contract. Its noise is the variation of the integrated response (or internal-standard ratio) across shots, not variation across channels. Confirm performance independently near any proposed instrument limit. PLS returns NA for LOD/LOQ and reports training RMSE separately; missing limits produce missing below-limit flags rather than a claim that a result exceeds a known limit.

ls_lod(cal)
#> [1] 3361.061
ls_loq(cal)
#> [1] 11203.54

Relative elemental amounts under a two-stage LTE model

This calculation requires independently justified electron density, first ionization energies, partition functions, resolved transitions and instrument-response factors. The line wavelengths must use the same air or vacuum convention as the measured axis. A good fit does not establish LTE, optical thinness or stoichiometric ablation.

The example below uses fictional X/Y transitions and intensities calculated by an independent high-precision forward model. The ordinary spectrum simulator is a Gaussian display model and does not simulate Boltzmann or Saha populations.

source(system.file("examples", "saha-boltzmann-synthetic.R", package = "libscanR"))
inputs <- saha_synthetic_example()
result <- do.call(ls_saha_boltzmann, inputs)
result
#> # A tibble: 2 × 7
#>   element temperature_k concentration_rel electron_density n_lines r_squared
#>   <chr>           <dbl>             <dbl>            <dbl>   <int>     <dbl>
#> 1 X              10000.             0.700             1e17       3         1
#> 2 Y              10000.             0.300             1e17       3         1
#> # ℹ 1 more variable: log_population_sd <dbl>
attr(result, "line_diagnostics")
#> # A tibble: 6 × 19
#>   element ionization wavelength_nm wavelength_medium wavelength_vacuum_nm    A_s
#>   <chr>        <int>         <int> <chr>                            <int>  <dbl>
#> 1 X                1           520 vacuum                             520    1e7
#> 2 X                1           530 vacuum                             530    2e7
#> 3 X                1           540 vacuum                             540    5e7
#> 4 Y                2           560 vacuum                             560    3e7
#> 5 Y                2           570 vacuum                             570    1e7
#> 6 Y                2           580 vacuum                             580    4e7
#> # ℹ 13 more variables: g_upper <int>, energy_upper_ev <dbl>, response <dbl>,
#> #   source <chr>, requested_nm <dbl>, integrated_intensity <dbl>,
#> #   effective_window_nm <dbl>, log_line_strength <dbl>,
#> #   log_partition_function <dbl>, log_stage_fraction <dbl>,
#> #   log_total_population <dbl>, fitted_log_line_strength <dbl>, residual <dbl>

The expected temperature is 10000 K and the relative elemental number fractions are X = 0.7 and Y = 0.3. Fractions are normalized over the supplied elements only; they are not mass fractions or absolute concentrations. The model includes neutral and singly ionized stages, excluding molecules and higher charge states. No real instrument or tissue data were used to qualify this example.

For experimental work, supply verified atomic records and retain their references. ls_element_db() reads user-supplied atomic reference fields, but missing rates or vacuum wavelengths must be resolved from primary sources before those rows can be used. Partition functions and response factors are not provided by the line database.

See the NIST line and Saha/LTE documentation for the conventions. The previous incomplete implementation omitted required physics; its outputs cannot serve as quantitative references and must be recalculated.