E04 · Electron microscopy / foundations

Electron Resolution & Aberration Budget

Open or close the aperture and switch on spherical, chromatic, source, drift, vibration and astigmatism terms. Compare the pupil phase, Gaussian sample response and weak-phase contrast-transfer curve. Tune the aperture within ten percent of the best probe in this budget, then turn on correction and inspect the remaining floor.

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

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

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 $0\,\mathrm{nm}$

Largest contribution $\text{—}$

Experiment task $\text{—}$

Instrument settings

Electron Resolution & Aberration Budget

Spherical aberrationInclude this term in the displayed budget.Chromatic aberrationInclude this term in the displayed budget.Finite sourceInclude this term in the displayed budget.DriftInclude this term in the displayed budget.VibrationInclude this term in the displayed budget.AstigmatismInclude this term in the displayed budget.Spherical correctorRetain residual spherical and fifth-order terms.Accelerating voltage

V

200\,\mathrm{kV}

Convergence semi-angle

\alpha

10\,\mathrm{mrad}

Spherical coefficient

C_s

1\,\mathrm{mm}

Chromatic coefficient

C_c

1\,\mathrm{mm}

Energy width

\Delta E

0.7\,\mathrm{eV}

Source blur diameter

d_s

0.08\,\mathrm{nm}

Drift blur diameter

d_{\mathrm{drift}}

0.03\,\mathrm{nm}

Vibration blur diameter

d_v

0.02\,\mathrm{nm}

Astigmatic blur diameter

d_a

0.1\,\mathrm{nm}

CTF defocus

\Delta f

0\,\mathrm{nm}

Enable correction, then separately disable chromatic, drift and source terms.

Quadrature engineering blur budget, not a rigorous coherent PSF. Sample response is an elliptical Gaussian surrogate. Correction leaves one percent of the chosen spherical coefficient plus a fixed 10 mm fifth-order term. The weak-phase CTF is a separate third-order, radial model with source/drift/vibration envelope; it omits chromatic temporal coherence, fifth-order phase, astigmatic phase, sample structure, inelastic transport and detector response.

Physics tutorial

What limits the probe after spherical correction?

BackgroundOpen or close the aperture and switch on spherical, chromatic, source, drift, vibration and astigmatism terms. Compare the pupil phase, Gaussian sample response and weak-phase contrast-transfer curve. Tune the aperture within ten percent of the best probe in this budget, then turn on correction and inspect the remaining floor.

Why it mattersExplore how the electron source, column and specimen constrain an instrument before interpreting an image.

Start with the essentials

Focus question: What limits the probe after spherical correction?
One-sentence intuition: Correcting one aberration moves the optimum aperture and exposes the next largest contribution. A blur budget and a signed CTF answer different imaging questions.

Core mathematical model

Quadrature blur budget

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

The terms use characteristic diameter conventions; adding them in quadrature is an engineering approximation.

Competing aperture terms

d_{\mathrm{diff}}=0.61\lambda/\alpha,\quad d_{\mathrm{sph}}=C_s\alpha^3/2,\quad d_{\mathrm{chr}}=C_c\alpha\Delta E/E

Diffraction falls with aperture angle while the aberration terms rise.

Residual correction

C_{s,\mathrm{res}}=0.01C_s,\qquad d_5=C_5\alpha^5,\quad C_5=10\,\mathrm{mm}

These residual settings are illustrative, not a commercial corrector performance claim.

Separate weak-phase transfer

\mathrm{CTF}(q)=-\sin\!\left[\pi\left(\frac12C_s\lambda^3q^4-\Delta f\lambda q^2\right)\right]\exp(-2\pi^2\sigma^2q^2)

The Gaussian envelope includes source, drift and vibration. The zero-frequency transfer vanishes for a weak phase object.

Common difficulties

Keep the model boundary explicit

Typical misconceptionTurning on a corrector eliminates every resolution limit.

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

Run the experiment

01
Bracket the optimum
Compare narrow and wide presets, predict the best angle, then tune within ten percent of the best budget and Check target.
What to observe: An intermediate aperture balances diffraction and aberrations.
02
Expose the floor
Enable correction, then separately disable chromatic, drift and source terms.
What to observe: Residual terms can limit the corrected probe even when spherical blur is small.
03
Read contrast reversal
Keep the blur budget fixed and change CTF defocus through positive and negative values.
What to observe: The signed transfer changes; the independent diameter budget stays fixed.