Peak overlaps & isotope deconvolution
An atom probe measures mass-to-charge ratio, not identity. Nature supplies several isotopes per element, the instrument supplies several charge states per isotope, and the specimen supplies molecular ions on top. Sooner or later two different things land on the same number, and no amount of better hardware will separate them. Only arithmetic will.
Time of flight gives mass-to-charge through m/n = 2eV t² / L². Two species with the same
m/n are the same peak. ⁵⁶Fe²⁺ sits at 27.967 and ²⁸Si⁺ at 27.976: nine thousandths of a
dalton apart, needing a mass resolving power near 3,000 to separate. Real instruments deliver roughly
500–2,000. So in any steel with silicon in it, the 28 Da peak is a mixture, and the silicon
concentration you report depends entirely on how you split it.
Untangling one overlap
The way out is that isotopes come in fixed ratios. If an element has one peak nobody else is sitting on, that peak tells you how much of the element is present, and therefore how much of it is hiding inside the contested peak. Subtract, and what remains belongs to the other species.
| Method | Result | What it assumes |
|---|
Subtracting a big estimated number from a big measured number leaves a small number with a large uncertainty. The counting error on the corrected peak carries the error of the sister peak multiplied by the abundance ratio: for ⁵⁶Fe/⁵⁷Fe that multiplier is 43. A silicon measurement built on an iron subtraction can easily be ten times noisier than the raw count suggests.
Go deeper: the overlaps you will actually meet
| m/z | Species | Δ(m/z) | MRP needed | Way out |
|---|---|---|---|---|
| 27.0 | ⁵⁴Fe²⁺ / ²⁷Al⁺ | 0.012 | ≈ 2 300 | ⁵⁷Fe²⁺ subtraction |
| 28.0 | ⁵⁶Fe²⁺ / ²⁸Si⁺ | 0.009 | ≈ 3 000 | ⁵⁷Fe²⁺ subtraction |
| 29.0 | ⁵⁸Fe²⁺ / ⁵⁸Ni²⁺ | 0.001 | ≈ 28 000 | Isobaric: abundance only |
| 25.0 | ⁵⁰Ti²⁺ / ⁵⁰V²⁺ / ⁵⁰Cr²⁺ | < 0.002 | > 20 000 | Isobaric: abundance only |
| 16.0 | ¹⁶O⁺ / ¹⁶O₂²⁺ | 0.000 | ∞ | Impossible: model the O₂⁺/O₂²⁺ ratio |
| 12.0 | ²⁴Mg²⁺ / ¹²C⁺ | 0.006 | ≈ 2 100 | ²⁶Mg²⁺ subtraction |
| 28.0 | ¹⁴N₂⁺ / ²⁸Si⁺ / ¹²C¹⁶O⁺ | 0.011–0.029 | 1 000–2 500 | High MRP, or ¹⁵N¹⁴N⁺ at 29 |
Two patterns are worth internalising. Isobaric overlaps (⁵⁸Fe/⁵⁸Ni, the mass-50 triplet) differ by the nuclear binding energy of a few thousandths of a dalton and will never be separated by any atom probe; they can only be apportioned using other isotopes of the same elements. Molecular overlaps are worse, because the ratio of molecular to atomic ions is not a constant of nature: it depends on field, temperature and laser energy, so it has to be measured on your own specimen, not looked up.
Go deeper: ranging, background and the things that quietly bias a composition
- Where you put the range matters. Peaks are not Gaussian; they have tails, especially in laser mode. Clipping the tail loses counts from the major element (which has the most tail) and biases the minor element upward. A common convention is to range down to a fixed fraction of the local background.
- Background is not flat. It comes from ions that evaporated between pulses, so it scales with the standing DC field and rises under the big peaks. Subtracting a single flat level over-corrects the quiet regions and under-corrects the busy ones.
- Multi-hits. When two ions land within the detector's dead time and dead space, one may be lost. Multi-hit loss is species-dependent (light elements and molecular fragments arrive in correlated pairs), so it can bias composition, not just reduce counts.
- Charge-state ratio drifts. The 2+/1+ ratio depends on field, so it changes with tip radius through a run and between phases within one dataset. Any quantification that assumes a fixed charge-state split is assuming something that is measurably untrue.
- Report the ranging. Two people ranging the same dataset can differ by a factor of two on a trace element. Peak assignments, range limits and background treatment belong in the methods section, not in the analyst's head.
Proxigrams & iso-concentration surfaces
A reconstruction is a bag of labelled dots. To say anything about an interface you first have to decide where the interface is, and the standard answer is to build a smooth concentration field, draw a surface at some threshold, and then measure every atom's distance to that surface. The result is only as good as the threshold you picked.
The recipe has three steps. Voxelise: divide the volume into small cells and count what is in each. Delocalise: smear each atom over neighbouring voxels with a Gaussian kernel, so the concentration field is continuous instead of shot noise. Threshold: draw the surface where the field crosses a chosen concentration. Then the proximity histogram, the proxigram, bins every atom by its signed distance to that surface and reports composition in each bin.
Composition against distance
A straight cylinder can only be normal to a curved interface in one place. Everywhere else it cuts
the interface at an angle and the profile it returns is broadened by 1/cos θ: pure
geometry, nothing to do with the material. The proxigram measures distance to the actual surface, so
curvature costs it nothing, and it averages over the whole surface instead of one narrow tube, which
buys it an enormous amount of counting statistics.
Go deeper: thresholds, delocalisation, and the number you should report instead
The threshold moves the answer. Nothing physical happens at 10 at.%. Raising the
threshold shrinks the surface inward, which shifts the whole proxigram along the distance axis and
changes the apparent composition at d = 0. Convention is to place the threshold near
the inflection of the composition profile, halfway between matrix and precipitate, and then to
check that the conclusion survives a sweep of the threshold. If it does not, it was not a
conclusion.
Delocalisation is a lie you need. Typical defaults are of order 3 nm laterally and 1.5 nm in depth, chosen to match the anisotropic resolution from Part 1. Too little and the surface is a noise-driven mess; too much and you have manufactured a gradient that is not there. It is a smoothing parameter, and it must be reported like one.
Report the excess, not the peak. For a segregated interface the robust quantity is the Gibbsian interfacial excess:
Γi = Niexcess / A = (1/A) ∫ ρ [ Ci(z) − Cibulk ] dz Atoms per unit interface area. The numerator is an integral, so it is invariant under any broadening that conserves atoms: delocalisation, trajectory aberration, a badly chosen bin width. The peak concentration is not. The denominator, A, is another matter entirely.In practice Γ is read off a ladder diagram: plot the cumulative count of solute atoms against the cumulative count of all atoms across the interface, and the height of the step is the excess. No thresholds, no bin widths, no smoothing on the numerator, which is exactly why it is the number to quote.
The half of that sentence people drop. Γ is a ratio, and only the top of it is threshold-free. The bottom is an area, and on a real dataset that area comes from an iso-concentration surface you chose the threshold for. Sweeping the demo above across delocalisation and threshold, the excess count stays within about 15% of the truth throughout, while the recovered interface length runs from 1.0× the true area down to 0.6× as the threshold rises. Divide the same honest count by the true area instead of the recovered one and the bias falls from 1.16× to 1.02× with roughly half the spread. So Γ is robust to everything that happens to the profile, and inherits everything that happens to the surface. Two laboratories quoting Γ for the same boundary at different iso-surface thresholds are not quoting the same quantity, and the difference does not appear anywhere in the profile they publish alongside it.
Proximity histograms: O. C. Hellman et al., Microsc. Microanal. 6, 437 (2000).
Counting statistics in a small volume
Atom probe compositions are quoted to two decimal places from volumes containing a few thousand atoms. A 3 nm precipitate analysed at 50 % efficiency contains fewer detected atoms than a small lecture theatre holds people, and the error bar that follows from that is usually bigger than the effect people are arguing about.
The physics is a coin flip. Each detected atom either is or is not the species of interest, so the
count follows a binomial distribution and the standard error on a measured concentration
c from N detected atoms is σ = √(c(1−c)/N). Nothing about the
instrument changes that; only more atoms do.
Two consequences fall straight out of the square root. First, halving the error bar costs four
times the atoms, and you cannot always get four times the atoms: the precipitate is the size
it is. Second, detection efficiency enters as a straight multiplier on N: going from a
37 % reflectron instrument to an 80 % straight-flight-path instrument is worth a factor of
√2.2 ≈ 1.5 on every error bar in the dataset, which is often a better
investment than any amount of post-processing.
Counting statistics is the smallest uncertainty your measurement can have. Real error bars also carry peak-overlap corrections, background subtraction, ranging choices, trajectory aberrations and the fact that the reconstruction placed some of those atoms in the wrong bin. A paper that quotes only √N has quoted the best case and called it the answer.
Go deeper: the arithmetic, and how many atoms you actually have
A worked case. A 4 nm spherical precipitate has a volume of 33.5 nm³. In iron that is about
2,800 atoms, of which a 52 % efficient instrument records roughly 1,480. If the true solute content
is 5 at.%, you have about 74 solute counts, and
σ = √(0.05 × 0.95 / 1480) = 0.57 at.%. The honest quotation is
5.0 ± 1.1 at.% (2σ): a ±23 % relative uncertainty on a number people routinely
report as "5.0".
Detection limits. With a flat background of b counts per dalton under
a peak of width w, the smallest concentration distinguishable from nothing at 3σ is
roughly 3√(2bw)/Ntotal. In a clean 10 M-ion dataset that lands in the tens
of atomic parts per million: genuinely excellent, and one of the few things APT does that nothing
else does at this spatial scale.
Clusters are worse than they look. Cluster-search algorithms select the atoms they count, so the counting error on a "cluster composition" is not binomial around the true value: the search itself biases toward solute-rich configurations. Small clusters found by maximum-separation methods are systematically enriched. Always test the algorithm against a randomised copy of your own dataset before believing a cluster composition.
Sources & further reading
- O. C. Hellman, J. A. Vandenbroucke, J. Rüsing, D. Isheim, D. N. Seidman, “Analysis of three-dimensional atom-probe data by the proximity histogram,” Microsc. Microanal. 6, 437 (2000).
- B. Gault, M. P. Moody, J. M. Cairney, S. P. Ringer, Atom Probe Microscopy, Springer (2012): mass spectrometry, ranging and data-analysis chapters.
- D. J. Larson, T. J. Prosa, R. M. Ulfig, B. P. Geiser, T. F. Kelly, Local Electrode Atom Probe Tomography: A User's Guide, Springer (2013).
- D. J. Barton, B. C. Hornbuckle, K. A. Darling, G. B. Thompson, “The influence of isoconcentration surface selection in quantitative outputs from proximity histograms,” Microsc. Microanal. 25, 401 (2019). The threshold you pick for the iso-surface changes the numbers that come off the proxigram, which is the effect the builder above lets you measure directly.
- B. M. Jenkins, F. Danoix, M. Gouné, P. A. J. Bagot, Z. Peng, M. P. Moody, B. Gault, “Reflections on the analysis of interfaces and grain boundaries by atom probe tomography,” Microsc. Microanal. 26, 247 (2020). Asks plainly whether Gibbsian interfacial excess is still fit for purpose given how many of its inputs the operator chooses, and argues for publishing the composition profile beside the excess rather than the excess alone.
- Isotopic masses and natural abundances: IUPAC / NIST atomic weights and isotopic compositions, NIST.
- D. N. Seidman, “Three-dimensional atom-probe tomography: advances and applications,” Annu. Rev. Mater. Res. 37, 127 (2007).