E26 · Electron imaging / spectroscopy

TEM Mass–Thickness Contrast

Why does a denser inclusion look dark in bright-field TEM? It scatters more electrons to large angles, and the objective aperture blocks those electrons from the bright-field image. Open a wider aperture and some of the scattered electrons get back in, weakening this mass-thickness contrast.

Pass a parallel beam through an amorphous wedge with denser inclusions. Open the objective aperture, select zero-loss electrons and defocus the image. Follow the direct, scattered and excluded electrons all the way to the recorded image.

Recorded intensity $0\,\mathrm{\%}$

Dense-inclusion contrast $0\,\mathrm{\%}$

Geometric blur $0\,\mathrm{nm}$

Counts per 4 nm pixel $0$

More measurements and target

Aperture rejected $0\,\mathrm{\%}$

Filter rejected $0\,\mathrm{\%}$

No elastic scattering $0\,\mathrm{\%}$

Zero-loss fraction $0\,\mathrm{\%}$

Plural inelastic scattering $0\,\mathrm{\%}$

Experiment task $\text{—}$

Simulated examples; each plot identifies its data source.

Zero-loss filterAfter angular collection, also reject electrons that lost energy.Reference thickness

t

120\,\mathrm{nm}

Aperture semiangle

\alpha

20\,\mathrm{mrad}

Defocus

\Delta f

0\,\mathrm{nm}

Advanced settings

Beam energy

E_0

120\,\mathrm{keV}

Background density

\rho

2.33\,\mathrm{g\,cm^{-3}}

Atomic number

Z

14

Incident areal dose

D

30\,\mathrm{e^{-}\,nm^{-2}}

Learning goal and model limits

Instrument settings

TEM Mass–Thickness Contrast

Keep recorded intensity above 15 percent, inclusion contrast at least 15 percent, plural inelastic scattering below 30 percent, and geometric blur below 4 nm.

Model assumptions

Contrast comes from an incoherent amplitude model. The elastic and inelastic mean free paths are assumed, and the angular distribution is a direct beam plus a Gaussian spread. The zero-loss filter removes every electron that has lost energy. Defocus only blurs the image geometrically; Fresnel phase contrast, diffraction, multislice and dynamical scattering are not modeled. The mean free paths are illustrative, not tabulated material data.

Start with the essentials

Focus question: Why does a denser inclusion look dark in bright-field TEM?
One-sentence intuition: The objective aperture turns a change in scattering angle into a change in brightness. A wide aperture lets scattered electrons back in and weakens mass contrast. Zero-loss electrons and unscattered electrons are not the same group.

Core mathematical model

Direct and zero-loss fractions

P_{\mathrm{dir}}=e^{-t/\lambda_{\mathrm{el}}},\quad P_0=e^{-t/\lambda_{\mathrm{in}}}

Elastic and inelastic events are independent in this teaching model.

Fraction passing the aperture

T_{\alpha}=P_{\mathrm{dir}}+(1-P_{\mathrm{dir}})\left(1-e^{-\alpha^2/(2\sigma_\theta^2)}\right)

The width of the diffuse part is chosen to keep the assumed mean-square scattering angle.

Filtered intensity

T_{\mathrm{filtered}}=T_{\alpha}P_0,\quad P_{\ge2}=1-(1+t/\lambda_{\mathrm{in}})P_0

The rejected intensity is kept as its own fraction; no absorption is calculated.

Common difficulties

Missing electrons and absorption

Typical misconceptionEvery electron missing from a bright-field image was absorbed by the specimen.

Better mental modelThe model splits the incident electrons into three parts: recorded, stopped by the aperture and removed by the filter. Scattering electrons to other angles can darken the image on its own, without any absorption term.

Run the experiment

01
Predict the aperture effect
Predict whether opening the objective aperture makes the dense inclusion brighter or darker. Compare Contrast target and Open aperture at the same specimen thickness.
What to observe: The objective aperture turns a change in scattering angle into a change in brightness. Opening it lets scattered electrons back in, which weakens the mass contrast. Zero-loss and unscattered electrons are different groups.
02
Meet the contrast target
Keep recorded intensity above 15 percent, inclusion contrast at least 15 percent, plural inelastic scattering below 30 percent, and geometric blur below 4 nm.
What to observe: Check the numbers against the target, then compare the instrument view with the data it produces.
03
Separate the losses
Compare the zero-loss filtered and unfiltered cases. Then thicken the specimen and defocus it. Explain lost counts, plural scattering and blur separately.
What to observe: Contrast comes from an incoherent amplitude model. The elastic and inelastic mean free paths are assumed, and the angular distribution is a direct beam plus a Gaussian spread. The zero-loss filter removes every electron that has lost energy. Defocus only blurs the image geometrically; Fresnel phase contrast, diffraction, multislice and dynamical scattering are not modeled. The mean free paths are illustrative, not tabulated material data.