<!doctype html public "-//w3c//dtd html 4.0 transitional//en">
<html>
Philip,
<p>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.
<p>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.
<p>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).
<p>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
<<font color="#3333FF"><A HREF="http://www.rdmag.com/archives/basics/10microscopy.htm">http://www.rdmag.com/archives/basics/10microscopy.htm</A></font>>.
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.
<p>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).
<p>Mike O'Keefe
<br>
<p>Philip Koeck wrote:
<blockquote TYPE=CITE>------------------------------------------------------------------------
<br>The Microscopy ListServer -- Sponsor: The Microscopy Society of America
<br>To Subscribe/Unsubscribe -- Send Email to ListServer@MSA.Microscopy.Com
<br>On-Line Help <a href="http://www.msa.microscopy.com/MicroscopyListserver/FAQ.html">http://www.msa.microscopy.com/MicroscopyListserver/FAQ.html</a>
<br>-----------------------------------------------------------------------.
<p>Hallo everybody,
<p>I've started wondering about what actually the point resolution of
<br>a TEM in phase-contrast imaging is.
<p>>From Raleigh's criterion I would conclude that for a typical 120 kV
<br>TEM with a numerical aperture of about 20 mrad distances larger than
<br>about 2 Angstrom or 'density waves' with a wavelength longer
than
<br>2 Angstrom simply cannot be resolved.
<p>In Scherzer defocus the Phase Contrast Transfer Function (PCTF) has
its
<br>first zero at about the same resolution but there seems to be
<br>information
<br>at higher frequencies (even if it is inverted in contrast).
<p>Which is the true resolution limit?
<p>Yours sincerely,
<p>Philip
<p>--
<br>Philip Koeck
<br>Karolinska Institutet
<br>Dept. of Bioscience
<br>Novum
<br>S-14157 Huddinge
<br>Sweden
<br>Tel.: +46-8-608 91 86
<br>Fax.: +46-8-608 92 90
<br>Email: Philip.Koeck@csb.ki.se
<br><a href="http://www.csb.ki.se/em">http://www.csb.ki.se/em</a></blockquote>
</html>