[3dem] Cs question

[Ruben Diaz Avalos]

Hi David,

Nominal Cs is fundamentally a hardware property of the objective lens and
pole piece geometry, and manufacturers determine it from the
electromagnetic lens design. For a Krios, the stated value (~2.7 mm) is not
empirical tuning but a well-characterized design parameter. In principle,
it should not vary significantly between microscopes of the same
configuration.

What we refine in image processing, however, is not a direct physical
measurement of the lens geometry, but the coefficient of the k^4 term in
the CTF phase function. In Zernike language, this corresponds primarily to
the radially symmetric fourth-order component (the Z4^0 term of the wave
aberration). In practice, that fitted coefficient can absorb small modeling
imperfections — residual coma, higher-order aberrations not explicitly
modeled, envelope inaccuracies, or subtle systematic phase errors. So the
“refined Cs” should be interpreted as the best-fit fourth-order phase term
under the assumptions of the refinement model.

In one of our datasets (TMV, refined to 1.8 Å), the refined Cs converged to
2.7 mm when processing the full dataset. Importantly, pixel size is
extremely well calibrated in this case because TMV provides a very precise
internal ruler via its layer-line spacing and helical repeat, so I am
confident that pixel size error is not significantly contributing to the
k^4 term.

Out of curiosity, I split the dataset into three arbitrary subsets and
refined them independently. The refined Cs values were 2.75 mm, 2.69 mm,
and 2.8 mm. When I recombined all particles, the final reconstruction was
essentially identical to the original 1.8 Å map, with no meaningful change
in resolution or map features.

So empirically, at least in this regime, a ~2–3% variation in the refined
Cs did not have a noticeable impact on the outcome. That is not entirely
surprising, since the phase error introduced by a small fractional change
in Cs remains modest over the spatial frequency range that carries usable
signal, especially compared to the dominant defocus term.

It would certainly be interesting to test the robustness more aggressively
by starting from a deliberately incorrect Cs (for example 2.5 mm) and
seeing whether refinement converges back to the nominal value and whether
map quality degrades in the process. But based on this dataset, small
deviations from the manufacturer value appear to be well tolerated at ~1.8
Å.

Ruben.

Ruben Diaz Avalos,

La Jolla Institute for Immunology.


On Mon, Feb 16, 2026 at 1:07 PM Morgan, David Gene via 3dem <
3dem at ncmir.ucsd.edu> wrote:

> Hi,
>
> For those of you who have used any of the image processing programs to
> refine the Cs value for your microscope, how does the refined value compare
> to what the manufacturers claim?
>
> As I understand it, the manufacturers simply calculate a Cs value for each
> microscope/pole piece model.  I guess I have always assumed that those
> values are relatively accurate, but I don't really have any data to support
> that.  Nor am I certain what would constitute "relatively accurate."
>
> Any thoughts would be appreciated.  Thanks.
>
> --
>
> NOTE:  my email has changed to dagmorga at iu.edu.
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