The purpose of this article is to show how to detect, diagnose, and correct calibration curvature in ICP-OES so analysts can restore linearity and improve quantitative accuracy.
1. Recognize curvature quickly.
Plot residuals versus concentration for each wavelength. Look for systematic positive or negative arcs. Check back-calculated concentrations against known values. Confirm the linear dynamic range for each line. Use low, mid, and high standards plus a carryover blank. Document % relative error at each point.
| Symptom | Likely cause | Fast check |
|---|---|---|
| High-end flattening | Self-absorption or detector saturation | Dilute top standards 2–10× and re-plot. |
| Low-end uplift | Background overcorrection or contamination | Measure reagent blank series and adjust background points. |
| Mid-range S-curve | Matrix mismatch or ionization effects | Matrix-match acid and ionic strength. Add ionization buffer. |
| All points biased | Nebulizer flow or plasma power drift | Optimize gas flows and RF power. Re-stabilize baseline. |
| Element-specific curvature | Wrong line selection | Switch to a less sensitive or interference-free line. |
2. Stabilize the instrument first.
Warm up until plasma and detector stabilize. Verify coolant, peristaltic pump rate, and sample uptake tubing ID. Optimize nebulizer gas flow for maximum S/N at a mid standard. Set RF power per method. Confirm alignment and viewing mode. Flush the sample path with acidified rinse until baseline is flat.
3. Match the chemistry.
Matrix-match acid type and molarity between samples, blanks, and standards. Match total dissolved solids within a factor of two. Add ionization buffer for easily ionized elements. Use 1000–2000 mg/L K or Cs when sodium or potassium is high. Viscosity-match with small amounts of methanol or surfactant only if the method allows it.
4. Choose the right lines.
Select wavelengths within the linear dynamic range. Prefer alternate lines with lower sensitivity when high concentrations are required. Avoid lines with strong nearby backgrounds or known molecular bands. Validate each line with a five to seven point range check.
5. Use better calibration models.
Start with ordinary least squares and inspect residuals. If curvature persists, apply one of the following approaches.
5.1 Weighted least squares (WLS).
Use 1/y or 1/y² weights when variance grows with signal. Recompute residual plots. Confirm improved % relative error at the high end.
# WLS concept For points (C_i, S_i): Minimize Σ w_i · (S_i − (a + b·C_i))² Choose w_i = 1 / S_i² when heteroscedastic. Report slope b, intercept a, and RMSE_w. 5.2 Quadratic fit with guardrails.
Use a second-order model only within a validated range. Lock the range with bracket standards in every batch. Reject runs where the vertex falls inside the working range.
# Quadratic calibration S = a + b·C + c·C² Back-calc concentration via: C = [-b + sqrt(b² - 4·c·(a − S))] / (2·c) Use only if c is stable and residuals random. 5.3 Segmented linear calibration.
Fit separate linear segments for low and high ranges. Place the breakpoint at the mid standard. Report the applied segment for each sample.
5.4 Standard addition for difficult matrices.
Spike samples at two or three levels. Extrapolate to the x-intercept to cancel matrix effects. Use this when matrix matching fails.
# Standard addition slope-intercept method Measure S_0, S_1, S_2 at added ΔC levels. Fit S = a + b·(C_sample + ΔC). Then C_sample = (S_intercept − a) / b, where S_intercept at ΔC = 0. 6. Control the physical contributors.
Reduce concentration of upper standards to avoid self-absorption. Verify the detector is below 70–80% of full scale. Shorten integration time if needed. Use radial view for high TDS or high concentration to extend linearity. Keep sample uptake rate stable. Replace worn pump tubing. Clean or replace the nebulizer and injector tube if baseline noise rises.
7. Strengthen background correction.
Place background points on both sides when possible. Avoid structured background regions. Increase replicate count at low concentrations to stabilize subtraction. Re-verify blanks after any line or background change.
8. Validate with statistics.
Run lack-of-fit testing against pure error from replicates. Apply Mandel’s fitting test to compare linear and quadratic fits. Track RMSE, %RE at each level, and residual randomness.
# Mandel test outline Compute SSE_linear and SSE_quadratic. F = [(SSE_linear − SSE_quadratic)/(p_q − p_l)] / [SSE_quadratic/(n − p_q)]. If F > F_crit, prefer quadratic within validated range. n = number of points, p_l=2, p_q=3. 9. Implement a robust daily workflow.
- Prepare fresh multi-element standards with matched acid and ionic strength.
- Run blank, low, mid, and high standards, then a carryover blank.
- Optimize gas flow and confirm detector headroom on the high standard.
- Build linear, then evaluate WLS, segmented, or quadratic if curvature remains.
- Plot residuals and store back-calculated results.
- Verify with an independent check standard and a fortified sample.
- Release data only if %RE limits and residual criteria pass.
10. Acceptance criteria checklist.
| Item | Target | Action if failed |
|---|---|---|
| Linearity R² | ≥ 0.999 for primary lines | Switch line or range. Consider WLS. |
| %RE per level | ≤ 5% low and mid, ≤ 10% high | Dilute top points. Refit. |
| Residual pattern | Random about zero | Change model or segment range. |
| Check standard | Within ±5% | Recalibrate or apply dilution. |
| Blank signal | Stable and near zero | Re-clean path. Adjust background. |
11. Example SOP block.
# ICP-OES curvature correction SOP 1. Start with five-point linear calibration per line. 2. Inspect residuals and %RE. If arc present, proceed. 3. Dilute high standard until detector < 75% FS. Re-run. 4. If curvature persists, apply WLS with w_i=1/S_i². Re-assess. 5. If still curved, split at mid standard and refit segments. 6. For hard matrices, switch to standard addition for affected elements. 7. Re-verify with check standard and fortified sample. 8. Document model, range, and verification results in the run log. FAQ
When should I switch wavelengths?
Switch when the high standard exceeds 75–80% full scale or residuals show persistent high-end flattening. A less sensitive line often restores linearity without exotic models.
Is quadratic calibration acceptable for compliance data?
Yes if the method allows it, the range is validated, and lack-of-fit testing supports the choice. Always include bracket standards and report the applicable range.
How do ionization buffers help?
They stabilize the plasma electron population for easily ionized elements. This reduces matrix-driven curvature, especially in high sodium or potassium samples.
What if only the low end is biased?
Rebuild blanks. Reposition background points. Increase replicates and integration time at low concentration. Confirm there is no carryover.