E30 · Electron wave imaging

HRTEM Phase Contrast & CTF

Can the same atoms look dark in one high-resolution TEM image and bright in another? Yes, just by changing defocus. Defocus and spherical aberration decide how each spatial frequency, each scale of detail, passes into the image and with which sign. Where the sign flips, that detail reverses contrast. Until you have checked this contrast transfer function, a sharp image is not a direct map of atom positions.

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

Lens transfer · horizontal columns $\text{—}$

Specimen factor · horizontal columns $\text{—}$

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

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

More measurements and target

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

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

Coherence at column spacing $\text{—}$

Beam path in specimen $0\,\mathrm{nm}$

Projected column span $0\,\mathrm{nm}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Defocus · negative is underfocus

\Delta f

-42\,\mathrm{nm}

Spherical aberration

C_s

0.5\,\mathrm{mm}

Weak specimen thickness

t

4\,\mathrm{nm}

Specimen tilt · about image vertical

\theta

0\,\mathrm{mrad}

Focus spread · standard deviation

\sigma_f

2\,\mathrm{nm}

Advanced settings

Reveal known column positionsGold circles mark the known column midpoints; lines show their projected spans under tilt. The detector image is unchanged. These markers are specified references.Accelerating voltage

V

200\,\mathrm{kV}

Illumination angle · standard deviation

\sigma_\alpha

0.1\,\mathrm{mrad}

Objective frequency cutoff

q_{\max}

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

Learning goal and model limits

Focus and specimen alignment

Turn dark columns bright

Raise the lens transfer above 0.85, reveal known positions and check the target. Drag the specimen to tilt it, then compare the projected phase with the image.

Model assumptions

This is a first-order weak-phase model of a synthetic projected potential, not of a named material. The periodic field is 3.84 nm across with 0.32 nm column spacing, and the maximum phase stays below 0.24 rad over the control range. The finite slab contains Gaussian columns extruded uniformly through its depth. Tilt integrates this same potential along the beam, broadening columns horizontally and shortening their projected spacing; opposite tilt signs give the same phase for this symmetric slab. Thickness is measured along the slab normal, and voltage changes the relativistic interaction constant. This projection neglects propagation inside the slab, absorption, multiple scattering, detector noise and atomic relaxation. Negative defocus means underfocus. Focus spread and illumination spread are Gaussian standard deviations, and the spatial envelope uses a first-order expansion in angle. Images use a fixed linear display range from 0.7 to 1.3. Gold circles show known column midpoints, and horizontal segments show their projected spans. These are specimen references, not an image reconstruction. The relay uses schematic rays with angles enlarged 12 times. Slab depth uses a separate schematic scale; the specimen tilt angle is drawn at its actual value. The image is calculated from Fourier phase transfer.

Start with the essentials

Focus question: Can the same atoms look dark in one high-resolution TEM image and bright in another?
One-sentence intuition: The lens multiplies each spatial frequency by a signed transfer: a negative value reverses that part of the image and a zero erases it, so even a sharp image need not map where the atoms are.

Core mathematical model

The weak exit wave

\begin{aligned}\psi_{\mathrm{exit}}(\mathbf r)&\simeq1+i\phi(\mathbf r)\\\phi(\mathbf r)&=\sigma V_{\mathrm{proj}}(\mathbf r)\end{aligned}

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

\begin{aligned}G(\mathbf q)&=T(q)\mathcal F\{\phi(\mathbf r)\}\\I(\mathbf r)&\simeq1+2\mathcal F^{-1}\{G(\mathbf q)\}\end{aligned}

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

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

Tilt changes the projected specimen

\begin{aligned}L_X&=L\cos\theta\\q_X&=q_x/\cos\theta\\\Delta X&=t\sin\theta\\M_{\rm cont}(q_x)&=\frac{\operatorname{sinc}(q_xt\tan\theta)}{\cos\theta}\\\operatorname{sinc}(s)&=\frac{\sin(\pi s)}{\pi s}\end{aligned}

Tilt is about the image vertical. The slab is uniform through its normal thickness t. Integrating along the beam spreads a column by the signed span shown here. The continuum projection multiplies each normalized Fourier coefficient by the factor shown here. The normalized sinc function is one at zero. The calculation integrates a periodic linear interpolation of the sampled potential; the specimen-factor readout uses that discrete integral. The lens uses the foreshortened horizontal frequency; it keeps the same radial transfer law.

Scherzer 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 dots are not always atoms

Typical misconceptionEvery bright dot marks an atom, whatever the focus.

Better mental modelThe intensity comes from interference after the lens adds a phase that depends on spatial frequency. The overlay is the known specimen, shown separately; it cannot vouch for an unknown structure in a real experiment.

What the thickness control can predict

Typical misconceptionIncreasing thickness here predicts a thick crystal quantitatively.

Better mental modelThickness scales the normal weak phase and sets the projection spread under tilt. The columns are uniformly extruded through the depth; thick-crystal dynamical scattering and discrete layering need a different 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 the aperture removes them. The position overlay still shows the true specimen; the recorded image by itself does not.
04
Tilt a finite slab
Choose Tilt spreads the columns, then set specimen tilt to zero without changing thickness or focus. Compare phase, intensity and the specimen factor.
What to observe: Tilt spreads the potential across neighboring columns and weakens the horizontal lattice component. The vertical component stays sharp. Opposite tilt signs give the same projection for this depth-symmetric specimen, so the image cannot determine the sign.
05
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.

HRTEM Phase Contrast & CTF

A bright spot is an interference result

Core mathematical model

The weak exit wave

Lens phase and signed transfer

The image is computed from the same phase

Coherence limits transfer

Tilt changes the projected specimen

Scherzer focus

Common difficulties

Bright dots are not always atoms

What the thickness control can predict

Run the experiment

Predict, then reverse

Make the crystal disappear

Lose information without moving atoms

Tilt a finite slab

Complete the target

Related experiments and articles

Electron probes: reading the evidence