E30 · Electron wave imaging

HRTEM Phase Contrast & CTF

Focus a transmission electron microscope on a thin crystal with a missing column. Change defocus and spherical aberration, watch the signed contrast transfer, and reveal the known column positions to test whether bright spots really locate atoms.

Electron wavelength $0\,\mathrm{pm}$

Scherzer defocus $0\,\mathrm{nm}$

Transfer at column spacing $\text{—}$

Coherence at column spacing $\text{—}$

Image RMS contrast $0\,\mathrm{\%}$

Maximum specimen phase $0\,\mathrm{rad}$

Experiment target $\text{—}$

Change the lens, keep the specimen

Turn dark columns bright

Reveal known column positionsGold markers belong to the ideal specimen. They do not change the detector or infer positions from the image.Accelerating voltage

V

200\,\mathrm{kV}

Defocus · negative is underfocus

\Delta f

-42\,\mathrm{nm}

Spherical aberration

C_s

0.5\,\mathrm{mm}

Weak specimen thickness

t

4\,\mathrm{nm}

Focus spread · standard deviation

\sigma_f

2\,\mathrm{nm}

Illumination angle · standard deviation

\sigma_\alpha

0.1\,\mathrm{mrad}

Objective frequency cutoff

q_{\max}

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

Predict whether underfocus or overfocus makes columns bright. Set transfer at the 0.32 nm spacing above 0.85, reveal the known positions and check the target. Compare with Dark columns and In focus.

First-order weak-phase model of a synthetic projected potential, not a named material. The 3.84 nm periodic field has 0.32 nm column spacing and a maximum phase below 0.24 rad over the control range. Thickness scales phase; voltage also changes the relativistic interaction constant; tilt, absorption, multiple scattering, detector noise and atomic relaxation are omitted. Negative defocus means underfocus. Focus spread and illumination spread are Gaussian standard deviations; the spatial envelope uses a first-order angular expansion. Images use a fixed linear 0.7–1.3 display range. The gold position overlay is known specimen truth, not a reconstruction. Ideal relay rays are schematic with angles enlarged 12 times; phase transfer, not geometric rays, calculates the image.

Start with the essentials

Focus question: Can the same atoms turn from dark to bright?
One-sentence intuition: The signed transfer weights each spatial frequency. Opposite signs reverse that component of the image; zeros erase it. A sharp-looking image does not guarantee a direct map of atom positions.

Core mathematical model

The weak exit wave

\psi_{\mathrm{exit}}(\mathbf r)\simeq1+i\phi(\mathbf r),\quad \phi(\mathbf r)=\sigma V_{\mathrm{proj}}(\mathbf r)

The phase is small and absorption is neglected. Here a synthetic Gaussian-column potential defines phase directly; no material-specific interaction constant is fitted.

Lens phase and signed transfer

\begin{aligned}\chi(q)&=\pi\lambda\Delta f q^2+\frac{\pi}{2}C_s\lambda^3q^4\\H(q)&=e^{-i\chi(q)}\\T(q)&=A(q)E(q)\sin\chi(q)\end{aligned}

Spatial frequency is in cycles per nanometre. All lengths in the calculation are converted to nanometres. Negative defocus means underfocus; the aperture passes frequencies up to its cutoff.

The image is computed from the same phase

I(\mathbf r)\simeq1+2\mathcal F^{-1}\!\left[T(q)\mathcal F\{\phi(\mathbf r)\}\right]

This is the first-order intensity, not the squared modulus of a full multislice wave. The DC component remains one and the image has no automatic contrast normalization.

Coherence limits transfer

\begin{aligned}E_t&=e^{-\frac12(\pi\lambda q^2\sigma_f)^2}\\E_s&=e^{-\frac12[2\pi\sigma_\alpha(\Delta f q+C_s\lambda^2q^3)]^2}\\E&=E_tE_s\end{aligned}

Gaussian focus and angle spreads average phase differences. The spatial expression linearizes the lens phase in illumination angle; it is an envelope approximation.

A useful conventional focus

\Delta f_{\mathrm{Sch}}\simeq-1.2\sqrt{C_s\lambda}

For positive spherical aberration this conventional underfocus broadens the first useful passband. At zero spherical aberration it gives zero, which does not create first-order phase contrast at exact focus.

Common difficulties

Bright points are not universally atoms

Typical misconceptionEvery bright dot locates an atom independently of focus.

Better mental modelIntensity is interference after a frequency-dependent lens phase. The overlay is separate specimen truth and cannot certify an unknown experimental structure.

Thickness has a model boundary

Typical misconceptionIncreasing thickness here predicts a thick crystal quantitatively.

Better mental modelThickness only scales the weak phase. Dynamical scattering in thick crystals requires a different wave-propagation model.

Run the experiment

01
Predict, then reverse
Choose Dark columns, reveal known positions, then choose Contrast reversal and reveal positions again.
What to observe: The specimen phase is identical. The signed transfer and simulated intensity reverse when the corrected lens changes defocus sign.
02
Make the crystal disappear
Choose In focus · invisible. Compare the phase panel, image RMS contrast and detector.
What to observe: The phase remains nonzero, but an ideal corrected lens at exact focus has zero first-order phase contrast. This does not mean the specimen vanished.
03
Lose information without moving atoms
Choose Scherzer focus, then Poor coherence. Lower the objective cutoff below the column frequency.
What to observe: The envelope damps high frequencies, and an aperture removes them. The position overlay still knows the specimen; the measured image alone does not.
04
Complete the target
Tune defocus until the transfer at column spacing exceeds 0.85, turn on the known positions and check the target.
What to observe: You verified the sign of one frequency band. Other bands can have different signs, so the complete image is more complicated than a binary bright-or-dark rule.