Trajectory aberrations & local magnification

Different phases need different electric fields to evaporate. When a precipitate sits at the apex of the needle, the surface quietly re-shapes itself until every phase is evaporating at once, and that re-shaped surface acts like a bad lens. Ions from the precipitate and ions from the matrix get steered into each other, and the interface you measure is a blend of both.

Field evaporation is a threshold process: an atom leaves when the local field reaches that phase's evaporation field. The field at a surface goes roughly as F ≈ V / (k_f R), so at a single applied voltage the only free parameter is the local radius of curvature. A phase that evaporates too easily gets shaved down until it is flatter: a bigger local radius, a weaker local field. A stubborn phase is left behind as a bump: a smaller local radius, a stronger local field. Within a few nanometres the apex settles into a lumpy, faceted shape that keeps every phase at its own threshold.

That lumpiness is the problem. Ions leave along the local surface normal. On a smooth hemisphere all the normals point back to one centre, which is exactly the point projection the reconstruction assumes. On a lumpy apex they do not, so the magnification is different from place to place, and near the phase boundary, trajectories from the two sides cross.

Aberration sandbox Hover the detector strip · drag the sliders

Matrix Precipitate Ion trajectory Un-perturbed hemisphere

Evaporation field of precipitate0.80×

Precipitate diameter6.0 nm

Tip radius 45 nm

Lateral point-spread σ0.40 nm

Trajectories drawn28

What the reconstruction reports

Measured solute (after reconstruction) True solute Reconstructed atomic density

Local radius

–

R_ppt / R_matrix

Apparent density

–

inside the image, × matrix

Apparent width

–

vs. true diameter

Peak solute

–

true value 25 at.%

Interface width

–

10–90 %, truly atomically sharp

–

The one thing to remember

A low-field precipitate is flattened, its ions are focused into too small a patch of detector, and it comes back too small and too dense. A high-field precipitate protrudes, its ions are sprayed over too much detector, and it comes back too big and too sparse. Either way the atoms that should have been at the interface end up somewhere else, which is why an atomically sharp interface never measures as atomically sharp.

Go deeper: the first-order model behind the sandbox

At steady state each phase evaporates when its own local field is reached. Since F = V/(k_fR) and the applied voltage V is common to the whole apex:

R_ppt / R_matrix = F_matrix / F_ppt = 1/φ φ = F_ppt/F_matrix, the evaporation-field ratio. k_f is the field factor, ≈ 3–5 for a real shanked needle (not the ≈ 1 of an isolated sphere).

The point-projection magnification onto a detector at flight distance L is M = L/(ξR), with ξ ≈ 1.5 the image compression factor. So to first order:

M_ppt / M_matrix = R_matrix / R_ppt = φ A precipitate that evaporates at 0.8× the matrix field is imaged at 0.8× the magnification: its image is 20 % too narrow, and about 1/0.8² ≈ 1.6× too dense.

The sandbox does not use those formulae directly. It builds the re-shaped profile, takes the true surface normal at every launch point, projects each ion along that normal to the detector, and then back-projects with the single magnification a reconstruction would assume. The magnification ratio, the density change and the trajectory crossing all fall out of the geometry, which is the point: none of this is added noise, it is a systematic consequence of assuming a sphere.

Why it hits small precipitates hardest. The aberration is set by how much the local curvature departs from the global curvature. A 2 nm precipitate on a 45 nm apex is almost pure aberration; a 20 nm precipitate carries enough surface to establish its own well-behaved local projection, and only its rim is badly imaged. This is why measured compositions of very small precipitates are systematically pulled toward the matrix, and why cluster-analysis parameters that work on 10 nm features misbehave on 2 nm ones.

Original treatment: Vurpillot, Bostel & Blavette, Appl. Phys. Lett. 76, 3127 (2000).

Go deeper: how to live with it

Report an integral, not a peak. Trajectory aberration moves atoms around but does not destroy them. Quantities that integrate over the interface (Gibbsian interfacial excess, total solute in a precipitate) survive far better than a peak concentration. (See proxigrams & interfacial excess.)
Watch the density map. Atomic density is reconstructed, not measured, so a density anomaly co-located with a composition anomaly is the fingerprint of local magnification rather than real chemistry. Most commercial software will draw a density iso-surface. Use it.
Match the field. Running at a lower base temperature or a different pulse fraction shifts the required evaporation field and can shrink φ. Laser energy does the same thing in laser mode, which is one reason measured compositions drift with laser energy.
Calibrate the shape independently. Imaging the same needle in TEM before the run gives you the real radius and shank angle to feed the reconstruction, instead of inferring them from the voltage curve. (See correlative APT + TEM.)

Depth vs. lateral resolution

APT routinely resolves individual atomic planes going into the specimen and almost never resolves them across it. Depth comes from an ordered sequence; lateral position comes from a projection. A sequence is nearly impossible to scramble. A projection loses information the moment it happens.

Depth is reconstructed from the order in which ions arrive. Field evaporation eats a crystal terrace by terrace, from the ledges inward, so atoms in one atomic plane are overwhelmingly removed before atoms in the next. For depth order, the reconstruction only has to count. The size of each depth sliver, though, comes from the reconstruction model (assumed atomic volume, analysed area, detector efficiency), so the ordering is sounder than the absolute scale. Nothing about the flight path can re-order the events, so depth resolution near a crystal pole reaches 0.05–0.3 nm: atomic planes, visibly.

Lateral position is reconstructed from where the ion hit, run backwards through an assumed projection. Every physical uncertainty on the way out (the atom's thermal motion at the instant it ionises, the roll-up over a ledge, faceting on the apex, the aberrations from the previous section) lands directly in x and y. Lateral resolution therefore sits around 0.3–1 nm at best, strongly specimen- and trajectory-dependent, and it is worst away from the poles.

Resolution explorer Two blurs, one lattice

Where on the tip?

Material & plane

Reconstructed atom position Spatial distribution along that axis

Depth blur σ_z0.040 nm

Lateral blur σ_xy0.22 nm

Detection efficiency52 %

Plane spacing

–

d_hkl, nm

Plane contrast in depth

–

from σ_z

Plane contrast laterally

–

from σ_xy

–

Drag the depth blur up and watch the horizontal bands survive far longer than the vertical ones ever could. That asymmetry (sharp in z, soft in x and y) is the single most characteristic feature of an atom-probe reconstruction, and it is why crystallographic information in APT is extracted from depth histograms (spatial distribution maps) rather than from looking at the point cloud.

It is not the detector's fault

A common first guess is that the delay-line detector is what limits lateral resolution. It is not, and the numbers are worth doing once. Set the flight path, tip radius and compression factor below and see what a detector pixel is actually worth back at the specimen.

Projection arithmetic Where the resolution really goes

Tip radius R45 nm

Flight path L90 mm

Image compression ξ1.50

Detector resolution100 µm

Magnification

–

M = L / ξR

Detector limit

–

back at the specimen

Field of view

–

80 mm detector

Ions per nm of depth

–

Fe, 52 % efficiency

–

⚠ The quiet consequence

Because M ∝ 1/R and the radius grows throughout a run, the magnification falls, the field of view widens, and the effective lateral resolution gets slowly worse as you go deeper. The end of a dataset is not the same instrument as the beginning of it.

Go deeper: the reconstruction equations

Lateral. A hit at detector coordinate X maps back through the point projection:

x = X / M , M = L / (ξ R) ξ ≈ 1.5 is the image compression factor: the real projection is not a pure point projection, and ξ absorbs the difference. R itself is usually inferred from the applied voltage, R = V / (k_f F_evap), so an error in k_f is an error in every x and y.

Depth. Each detected ion represents 1/η atoms of volume Ω, spread over the analysed area, which is the detector area divided by M²:

Δz = Ω M² / (η A_det) Ω = atomic volume (≈ 0.0118 nm³ for Fe, 0.0166 nm³ for Al); η = detection efficiency; A_det = active detector area. Note there is no flight-path term: depth never passes through the projection, which is exactly why it survives.

Why depth still is not perfect. Retained atoms (η < 1) punch random holes in the sequence; atoms that roll along a terrace before leaving arrive out of order; and where the surface is not locally flat, "one plane" is not a plane. Off-pole, the terraces are so small that the layer-by-layer ordering washes out and depth resolution degrades toward the lateral value.

Detection efficiency is set by the detector's open area: the fraction of the microchannel plate that is hole rather than wall. Reflectron-fitted instruments trade counts for mass resolution and land around 37–52 %; straight-flight-path instruments reach up to 80 %. There is no such thing as a complete atom probe dataset.

Go deeper: how the resolution is actually measured

You do not measure APT resolution with a ruler; you measure it with a spatial distribution map (SDM). Take every pair of atoms within a small neighbourhood, histogram the vector between them along one axis, and repeat over millions of pairs. If the reconstruction has preserved the lattice, the histogram shows peaks at multiples of the plane spacing; the width of those peaks is the resolution along that axis.

The histograms drawn under the lattice above are a simplified one-dimensional version of the same idea. On a pole in a well-behaved metal the depth SDM shows clean peaks out to several plane spacings; the lateral SDM usually shows one broad hump and nothing else.

SDMs are also the practical way to calibrate a reconstruction: adjust ξ and k_f until the known plane spacing comes out right, and the rest of the volume is far more trustworthy. Getting the spacing right in z and the pole positions right in x, y is the closest thing APT has to an internal standard.

Next · Part 2

Data & analysis

Peak overlaps and isotope deconvolution, proxigrams and iso-concentration surfaces, and why a few million atoms is a small number.

Part 3

Instrument, prep & applications

Voltage vs. laser pulsing, FIB lift-out and annular milling, grain-boundary segregation, and correlative APT + TEM.

Sources & further reading

F. Vurpillot, A. Bostel, D. Blavette, “Trajectory overlaps and local magnification in three-dimensional atom probe,” Appl. Phys. Lett. 76, 3127 (2000): abstract.
B. Gault, M. P. Moody, J. M. Cairney, S. P. Ringer, Atom Probe Microscopy, Springer (2012): chapters on reconstruction and spatial resolution.
M. K. Miller, R. G. Forbes, Atom-Probe Tomography: The Local Electrode Atom Probe, Springer (2014).
D. J. Larson et al., “Advances in the reconstruction of atom probe tomography data,” Ultramicroscopy 111, 506 (2011).