TEM resolution

Mon Dec 4 10:03:52 PST 2000

Philip,

In the field of high-resolution transmission electron microscopy, the
"resolution" is defined as the first zero in the phase contrast transfer
function (PCTF) at Scherzer (or optimum) defocus as you point out.  This
definition is chosen because any higher-frequency information is phase
inverted (as you also point out).  However, it results in defining the
resolution in terms of the spatial frequency at which transfer goes to zero,
rather than in terms of the highest transferred spatial frequency.

I have suggested a definition in which the resolution is defined at the 70%
pass limit within the Scherzer passband, giving d = 0.67 Cs^1/4 lambda^3/4
at an "extended" optimum defocus of sqrt(1.5Cs.lambda) instead of Scherzer's
d = 0.707 Cs^1/4 lambda^3/4 at an optimum defocus of sqrt(Cs.lambda).
Current practice seems to use the extended defocus with the crossover
(zero-transfer) frequency to get d = 0.64 Cs^1/4 lambda^3/4 at the
"extended" optimum defocus.

The information beyond the Scherzer cross-over (zero-transfer) frequency can
be utilized to provide structural information.  Because of the misphasings,
this information is difficult to interpret directly.  Originally, the way to
get at this higher-frequency (smaller-spacing) information was to compare
focal series of experimental images with images simulated from possible
models.  In this way, we "saw" the small tunnels in Nb12O29 (with a
2.5Angstrom spacing) with a JEOL 100B (Scherzer resolution of 3.5A) in a
focal series obtained by Sumio Iijima (see "Resolution-limiting effects in
electron microscope images", G.R. Anstis and M.A. O'Keefe, In 34th Ann.
Proc. EMSA, Miami Beach (1976) 480-481).  I used HRTEM image simulation
programs such as my SHRLI (simulated high-resolution lattice image) code, or
my later interactive TEMPaS (TEM processing and simulation) code (ported to
the Mac as MacTempas) with great success, but only when the model was known
or limited to several possible ones (for example "Inversion domains in GaN
grown on sapphire", L.T. Romano, J.E. Northrup and M.A. O'Keefe, Appl. Phys.
Lett. 69 (1996) 2394-2396).  Currently, post-Scherzer information can be
accessed by image reconstruction algorithms from focal series, or tilt
series, or holography.  I use the Philips/Brite-Euram software for
focal-series reconstruction by Coene and Thust in my one-Angstrom microscope
(OAM) project to extend the resolution of a modified Philips CM300FEG/UT
from its native resolution of 1.7Angstrom to a super-resolution of
0.89Angstrom that can image the "dumbbells" in [110] diamond (see “The NCEM
One-Ångstrom Microscope project reaches 0.89Å resolution”, M. A. O'Keefe in
58th Ann. Proc. MSA, Philadelphia, Pennsylvania (2000) 1192-1193).

For the equations that belong with a discussion of resolution, see
"Resolution in high-resolution electron microscopy", M.A. O'Keefe,
Ultramicroscopy 47 (1992) 282-297.  For a (simple) explanation of the use of
focal-series to obtain super-resolution see "Push TEM limits with
super-resolution", Michael A. O'Keefe (1999), R&D Magazine October 1999, p79
or  <http://www.rdmag.com/archives/basics/10microscopy.htm>.  For details of
the Philips/Brite-Euram software for focal-series reconstruction by Coene
and Thust, see W.M.J. Coene, A. Thust, M. Op de Beeck and D. Van Dyck,
Ultramicroscopy 64 (1996) 109-135 and A. Thust, W.M.J. Coene, M. Op de Beeck
and D. Van Dyck, Ultramicroscopy 64 (1996) 211-230.

In the above, I have dealt only with linear terms in the transfer of
information from the specimen (more properly, the exit-surface wave from the
specimen) into the HREM image.  Non-linear terms can give rise to
even-higher spatial frequencies in the image.  However, the question of
whether these can be termed "resolution" is doubtful (see
"Resolution-damping functions in non-linear images", M.A. O'Keefe, in 37th
Ann. Proc. EMSA, San Antonio, Texas (1979) 556-557).

Mike O'Keefe

Philip Koeck wrote:

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> Hallo everybody,
>
> I've started wondering about what actually the point resolution of
> a TEM in phase-contrast imaging is.
>
> >From Raleigh's criterion I would conclude that for a typical 120 kV
> TEM with a numerical aperture of about 20 mrad distances larger than
> about  2 Angstrom or 'density waves' with a wavelength longer than
> 2 Angstrom simply cannot be resolved.
>
> In Scherzer defocus the Phase Contrast Transfer Function (PCTF) has its
> first zero at about the same resolution but there seems to be
> information
> at higher frequencies (even if it is inverted in contrast).
>
> Which is the true resolution limit?
>
> Yours sincerely,
>
> Philip
>
> --
> Philip Koeck
> Karolinska Institutet
> Dept. of Bioscience
> Novum
> S-14157 Huddinge
> Sweden
> Tel.: +46-8-608 91 86
> Fax.: +46-8-608 92 90
> Email: Philip.Koeck at csb.ki.se
> http://www.csb.ki.se/em
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