E04 · Electron microscopy / foundations

Electron Resolution & Aberration Budget

Once spherical aberration is corrected, what limits how small the probe can be? Whatever is now the biggest remaining contribution, such as chromatic aberration, the source size, drift or vibration. Removing one aberration also moves the best aperture size, and the next limit comes into view. Go through the remaining contributions one by one to find the new floor.

Open or close the aperture and switch the spherical, chromatic, source, drift, vibration and astigmatism terms on and off. Compare three views: the radial phase, which the transfer curve also uses; a Gaussian estimate of the combined blur; and the weak-phase transfer cut off by the aperture. Pick a spatial frequency, then vary the energy width to tell chromatic damping apart from the aperture cutoff. Tune the aperture to within ten percent of the smallest probe these blur terms allow, then turn on correction and see what limit remains.

Total probe scale $0\,\mathrm{nm}$

Diffraction contribution $0\,\mathrm{nm}$

Largest contribution $\text{—}$

Signed transfer at test frequency $\text{—}$

More measurements and target

Spherical contribution $0\,\mathrm{nm}$

Chromatic contribution $0\,\mathrm{nm}$

Source and mechanical floor $0\,\mathrm{nm}$

Fifth-order residual $0\,\mathrm{nm}$

Best semi-angle in scan $0\,\mathrm{mrad}$

Best probe in scan $0\,\mathrm{nm}$

Pupil frequency cutoff $0\,\mathrm{nm^{-1}}$

Chromatic focal standard deviation $0\,\mathrm{nm}$

Envelope at test frequency $\text{—}$

Test frequency acceptance $\text{—}$

Experiment task $\text{—}$

Simulated examples; each plot identifies its data source.

Spherical correctorThe corrector leaves residual spherical and fifth-order terms.Convergence semi-angle

\alpha

10\,\mathrm{mrad}

Spherical coefficient

C_s

1\,\mathrm{mm}

Energy width: FWHM

\Delta E

0.7\,\mathrm{eV}

CTF defocus: underfocus positive

\Delta f

0\,\mathrm{nm}

Advanced settings

Spherical aberrationInclude this term in the probe-size estimate.Chromatic aberrationInclude this term in the probe-size estimate.Finite sourceInclude this term in the probe-size estimate.DriftInclude this term in the probe-size estimate.VibrationInclude this term in the probe-size estimate.AstigmatismInclude this term in the probe-size estimate.Accelerating voltage

V

200\,\mathrm{kV}

Chromatic coefficient

C_c

1\,\mathrm{mm}

Source blur: Gaussian FWHM

d_s

0.08\,\mathrm{nm}

Drift blur: Gaussian FWHM

d_{\mathrm{drift}}

0.03\,\mathrm{nm}

Vibration blur: Gaussian FWHM

d_v

0.02\,\mathrm{nm}

Astigmatic blur diameter

d_a

0.1\,\mathrm{nm}

Test spatial frequency

q

3\,\mathrm{nm^{-1}}

Learning goal and model limits

Instrument settings

Electron Resolution & Aberration Budget

Compare the pupil cutoff with the test frequency. Switch chromatic aberration off to isolate the damping from its temporal envelope.

Model assumptions

The probe size comes from an engineering blur budget, drawn as a peak-normalized Gaussian; it is not a coherent point spread function. Correction leaves one percent of the spherical aberration plus a fixed 10 mm fifth-order term. A separate on-axis weak-phase reference uses the same semi-angle as a hard pupil cutoff. Its phase includes defocus, counted positive for underfocus, and third- and fifth-order spherical terms. Energy width is a Gaussian FWHM and sets the chromatic focal spread. Source size, drift and vibration enter as Gaussian image blurs given as FWHM. There is no acquired image, angular source coherence, astigmatic phase, lens instability, specimen or detector model. Ray errors and hardware are schematic, and the phase colors wrap around and are sampled for display.

Physics tutorial

Once spherical aberration is corrected, what limits how small the probe can be?

BackgroundOpen or close the aperture and switch the spherical, chromatic, source, drift, vibration and astigmatism terms on and off. Compare the radial phase, the Gaussian reference for the blur budget and the weak-phase transfer cut off by the aperture. Tune the aperture to within ten percent of the best probe this budget allows, then turn on correction and see what limit remains.

Why it mattersBefore you trust an electron image, you need to know what the source, the lens column and the specimen itself allow.

Start with the essentials

Focus question: Once spherical aberration is corrected, what limits how small the probe can be?
One-sentence intuition: Correcting one aberration moves the best aperture and exposes the next largest blur term. The blur budget estimates probe size; the signed CTF describes how each spatial frequency gets through. They answer different imaging questions.

Core mathematical model

Adding blur terms in quadrature

\begin{aligned}d^2={}&d_{\mathrm{diff}}^2+d_{\mathrm{sph}}^2+d_{\mathrm{chr}}^2\\&+d_s^2+d_{\mathrm{drift}}^2+d_v^2+d_a^2+d_5^2\end{aligned}

Each term is a characteristic diameter. Adding them in quadrature is an engineering approximation.

How aperture angle affects each term

\begin{aligned}d_{\mathrm{diff}}&=0.61\lambda/\alpha\\d_{\mathrm{sph}}&=C_s\alpha^3/2\\d_{\mathrm{chr}}&=C_c\alpha\Delta E/E\end{aligned}

A wider aperture angle shrinks the diffraction term and makes the aberration terms grow.

What correction leaves behind

\begin{aligned}C_{s,\mathrm{res}}&=0.01C_s\\d_5&=C_5\alpha^5\\C_5&=10\,\mathrm{mm}\end{aligned}

These residual values are for illustration; they do not describe the performance of any commercial corrector.

Weak-phase transfer function

\begin{aligned}\chi(q)={}&-\pi\Delta f\lambda q^2\\&+\frac{\pi C_s\lambda^3q^4}{2}\\&+\frac{\pi C_5\lambda^5q^6}{3}\\T(q)&=-A(q)E_t(q)E_b(q)\sin\chi(q)\end{aligned}

Underfocus counts as positive defocus. The pupil view shows the same radial phase. The hard pupil passes only frequencies inside the selected semi-angle. The fifth-order term appears only when spherical correction is on. This signed weak-phase reference is computed separately from the probe diameter budget and leaves out astigmatic phase.

Gaussian coherence envelopes

\begin{aligned}\sigma_f&=\frac{C_c\Delta E}{E\sqrt{8\ln2}}\\E_t(q)&=\exp[-(\pi\lambda\sigma_f q^2)^2/2]\\E_b(q)&=\exp[-2\pi^2\sigma_b^2q^2]\end{aligned}

The energy width is given as FWHM and converted to a standard deviation of focus. A separate image-blur standard deviation combines the source, drift and vibration FWHM values in quadrature. Neither envelope is a measured information limit.

Common difficulties

What a corrector does not remove

Typical misconceptionTurning on a corrector eliminates every resolution limit.

Better mental modelSource size, energy spread, drift, vibration and higher-order residuals remain as limits.

Run the experiment

01
Find the best aperture
Compare the Narrow aperture and Wide aperture presets and predict the best angle. Then tune to within ten percent of the best probe size and press Check target.
What to observe: The best aperture lies in between, where diffraction and aberrations balance.
02
Find the remaining limit
Turn on correction, then switch off the chromatic, drift and source terms one at a time.
What to observe: Residual terms can limit the corrected probe even when spherical blur is small.
03
See contrast reversal
Sweep defocus through positive and negative values. Choose a test frequency, close the aperture until that frequency is blocked, then reopen it and increase the energy width.
What to observe: Defocus changes the phase but not the diameter budget. A frequency outside the aperture is transferred with exactly zero amplitude; chromatic damping lowers the envelope even inside the pupil.