ISO 25387:2026

Microbeam analysis — Analytical electron microscopy — Procedures for determining the point resolution of high-resolution transmission electron microscope

This document specifies a procedure for determining the point resolution, called Scherzer resolution, of high-resolution transmission electron microscopes (HREM), which can visualize sample structure with sub-nanometre fineness. This document also specifies the measurement procedure of the real spherical aberration coefficient of the objective lens used. The procedure specified in this document for measuring the spherical aberration coefficient uses the dark rings that appear in the fast Fourier transform (FFT) pattern of HREM images of amorphous thin films, in which, at least three dark rings need to be observable near the Scherzer focus. Therefore, this document is applicable to HRTEMs equipped with a cold field emission gun (CFEG), Schottky emission gun (SEG) or thermal field emission gun (TFEG), or HREMs equipped with a thermionic emission gun (TEG) in which three or more dark rings can be clearly observed in the FFT pattern. This document does not treat the information limits, lattice resolution and STEM resolution. In addition, this document is not applicable to Cs-corrected TEM.

Analyse par microfaisceaux — Microscopie électronique analytique — Modes opératoires de détermination de la résolution ponctuelle des microscopes électroniques à transmission à haute résolution

General Information

• Status: Published
• Publication Date: 08-Jun-2026
• ICS: 37.020 - Optical equipment
• Technical Committee: ISO/TC 202/SC 3 - Analytical electron microscopy
• Drafting Committee: ISO/TC 202/SC 3 - Analytical electron microscopy
• Current Stage: 6060 - International Standard published
• Start Date: 09-Jun-2026
• Due Date: 15-Nov-2027
• Completion Date: 09-Jun-2026

Overview

ISO 25387:2026 provides a standardized procedure for determining the point resolution-specifically, the Scherzer resolution-of high-resolution transmission electron microscopes (HREM) used in microbeam analytical electron microscopy. This international standard outlines how to assess the microscopic performance of HREM through precise measurement of point resolution and real spherical aberration coefficients of the objective lens. The methodology relies on analyzing dark rings in the fast Fourier transform (FFT) pattern of HREM images from amorphous thin films, offering reliable and consistent results across different transmission electron microscopes (TEMs).

ISO 25387:2026 is applicable to HREMs equipped with various electron gun types-including cold field emission gun (CFEG), Schottky emission gun (SEG), thermal field emission gun (TFEG), and thermionic emission gun (TEG)-where three or more dark rings can be distinctly observed in the FFT pattern. Notably, this standard does not cover Cs-corrected TEM, information limits, lattice resolution, or STEM resolution.

Key Topics

• Scherzer Resolution: Defines point resolution for HREM, determined by the first zero in the phase contrast transfer function (PCTF) under the Scherzer focus condition.
• Measurement of Spherical Aberration: Specifies how to accurately assess the real spherical aberration coefficient-critical for evaluating TEM performance-by observing the diameters of dark rings in FFT patterns.
• FFT Pattern Analysis: Utilizes FFT diffractograms of amorphous thin film images, requiring at least three observable dark rings near the Scherzer focus.
• Applicable Electron Guns: Addresses CFEG, SEG, TFEG, and TEG types used across modern HREMs.
• Exclusions: Does not treat information limit, lattice resolution, STEM resolution, or Cs-corrected TEM.
• Impact of Envelope Functions: Discusses how spatial and temporal coherency affect the appearance of diffractograms, but not the actual measured point resolution.

Applications

Implementing ISO 25387:2026 ensures reliable, repeatable, and internationally comparable determination of HREM point resolution. Its practical applications include:

• Quality Control in Materials Science: Enables laboratories to verify HREM performance by regularly measuring and documenting point resolution, helping maintain consistent high-resolution imaging standards.
• Instrument Benchmarking: Provides a universal methodology for comparing instrument capabilities across manufacturers and research facilities, fostering standardization in equipment procurement and performance evaluation.
• Method Validation: Laboratories developing or validating analytical techniques involving HREM can use the protocol to ensure the accuracy of sub-nanometre structural observations.
• Research & Development: Assists researchers in materials science, nanotechnology, and life sciences to accurately characterize and report on ultra-fine material structures, forming the basis for reproducible and credible scientific investigations.

Related Standards

When applying ISO 25387:2026 in analytical electron microscopy, these related international standards may also be relevant:

• ISO 15932: Terminology for electron microscopy and microbeam analysis-critical for consistent definitions used in ISO 25387:2026.
• ISO 20263: Imaging methods for scanning transmission electron microscopes (STEM).
• Other ISO TC 202 Standards: Various documents on sample preparation, calibration, and performance criteria for electron microscopes.
• IEC Electropedia & ISO Online Browsing Platform: Reference sources for standard terms and abbreviations used in the field.

Conclusion

By adopting ISO 25387:2026, laboratories and researchers in analytical electron microscopy can confidently determine and report the point resolution of their HREM instruments. This standard delivers transparency, comparability, and precision for advanced imaging applications vital to material science, nanotechnology, and related fields, supporting global best practices in ultra-high-resolution electron microscopy.

Standards Content (Sample)

International
Standard
ISO 25387
First edition
Microbeam analysis — Analytical
2026-06
electron microscopy — Procedures
for determining the point resolution
of high-resolution transmission
electron microscope
Analyse par microfaisceaux — Microscopie électronique
analytique — Modes opératoires de détermination de la
résolution ponctuelle des microscopes électroniques à
transmission à haute résolution
Reference number
© ISO 2026
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on
the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below
or ISO’s member body in the country of the requester.
ISO copyright office
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Published in Switzerland
ii
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and abbreviated terms. 5
5 Definition of point resolution based on information theory . 7
5.1 General .7
5.2 Definition of Scherzer condition .7
5.3 Definition of the theoretical point resolution .9
6 Diffractogram .11
7 Effect of the envelope function .13
7.1 General . 13
7.2 Envelope function for temporal coherency . 13
7.3 Envelope function for spatial coherency .14
7.4 Effect of the total envelope function on the contrast distribution of diffractogram. 15
8 Sample for point resolution measurement .18
8.1 General .18
8.2 Sample for point resolution measurement .18
9 Measurement procedures for determining experimental point resolution of HREM .18
9.1 General .18
9.2 Calculation of provisional Scherzer focus value . 20
9.3 Sample preparation . 20
9.4 Set up the TEM . 20
9.5 Defocus setting and image recording . 22
9.6 Obtain diffractogram series . 23
9.6.1 Set a ROI in the image . 23
9.6.2 Obtain a series of diffractogram .24
9.7 Obtain two different line-profiles from two lines drawn on a diffractogram . 25
9.8 Measure distance between two spots reflecting Au lattice plane spacing .27
st nd rd
9.9 Measure diameter of dark rings reflecting 1 , 2 and 3 zero position in square of
realistic PCTF function . 28
9.10 Calibrate the spatial frequency axis . 29
9.11 Calculate three defocus values from the diameter of three dark rings . 30
9.11.1 Relationship between defocus value and diameter of dark ring . 30
9.11.2 Calculate average defocus value of the image . 30
9.12 Determine real spherical aberration coefficient . 30
9.13 Determine the experimental point resolution . 33
9.13.1 Relationship between square of spatial frequency and defocus . 33
9.13.2 Procedure for determining experimental point resolution . 33
10 Uncertainty in point resolution measurement .34
Annex A (informative) Crystal lattice spacing of gold .36
Annex B (informative) Example of measurement procedure for experimental point resolution
determination .37
Bibliography .52

iii
Foreword
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Analytical electron microscopy.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.

iv
Introduction
High-resolution transmission electron microscopes (HREM) that can observe ultra-fine structures with
sub-nanometre resolution have been applied in various fields such as medical, biological, materials science,
classical and innovative materials, and others.
One of the key factors in knowing the limitations of the microscopic performance of HREM is the point
resolution. Although there are various categories of TEM resolution definitions, point resolution for HREM is
generally defined by the Scherzer resolution.
Scherzer resolution is measured from the high-resolution image of a weak phase object observed under a
specific defocusing condition (called the Scherzer focus or the Scherzer condition) derived from the spherical
aberration coefficient of the objective lens and the electron wavelength. However, different definitions have
been proposed for the Scherzer condition, which is an important factor in the resolution measurement
of TEM, depending on the purpose of the observation and the targeting materials. To determine the TEM
resolution, this document adopts the Scherzer condition, which is derived from information theory.
Furthermore, although the spherical aberration coefficient is usually provided by the TEM manufacturer,
it is important to know the real spherical aberration coefficient to properly evaluate the instrumental
performance of the TEM. This document specifies the procedure for measuring real spherical aberration
coefficient of objective lens used and the procedure for determining the point resolution (Scherzer
resolution) of HREM using the measured real spherical aberration coefficient.
Recently, ultra-high-resolution electron microscopes equipped with a spherical aberration corrector, called
“C -corrected TEMs”, have been developed and widely used. These state-of-the-art TEMs can reduce spherical
s
aberration to as close to zero as possible, contributing to a dramatic improvement in TEM resolution beyond
the Scherzer resolution. In general, the resolution of these state-of-the-art TEMs is defined by the critical
structure size (called the information limit) at which the phase contrast carried by the phase contrast
transfer function vanishes by the envelope function. For this reason, the resolution for C -corrected TEMs
s
should be treated as a separate category from the Scherzer resolution for HREMs.

v
International Standard ISO 25387:2026(en)
Microbeam analysis — Analytical electron microscopy —
Procedures for determining the point resolution of high-
resolution transmission electron microscope
1 Scope
This document specifies a procedure for determining the point resolution, called Scherzer resolution, of
high-resolution transmission electron microscopes (HREM), which can visualize sample structure with
sub-nanometre fineness. This document also specifies the measurement procedure of the real spherical
aberration coefficient of the objective lens used.
The procedure specified in this document for measuring the spherical aberration coefficient uses the dark
rings that appear in the fast Fourier transform (FFT) pattern of HREM images of amorphous thin films, in
which, at least three dark rings need to be observable near the Scherzer focus. Therefore, this document
is applicable to HRTEMs equipped with a cold field emission gun (CFEG), Schottky emission gun (SEG) or
thermal field emission gun (TFEG), or HREMs equipped with a thermionic emission gun (TEG) in which
three or more dark rings can be clearly observed in the FFT pattern.
This document does not treat the information limits, lattice resolution and STEM resolution. In addition, this
document is not applicable to C -corrected TEM.
s
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
3.1
chromatic aberration
lens defect which arises because electrons from the same point in the specimen, but of slightly different
energies, will be focused at different positions in the image plane
[SOURCE: ISO15932: 2013, 3.2.2.1]
3.2
cold field emission gun
CFEG
electron gun (3.5) employing cold field emission
[SOURCE: ISO15932: 2013, 3.1.2.1]
3.3
defocus spread
fluctuation of the defocus caused by the energy spread (3.6) of the incident electron beam, and fluctuation in
the power supply for accelerating voltage and the objective lens exciting current

3.4
diffractogram
image showing how the intensity of Fourier transform pattern of the high-resolution TEM image distributes
with the structure size
3.5
electron gun
component that produces an electron beam with a well-defined kinetic energy
[SOURCE: ISO15932: 2013, 3.1]
3.6
energy spread
diversity of energy of electrons in the incident beam
[SOURCE: ISO15932: 2013, 2.1.1.1]
3.7
envelope function
function that expresses the damping of the phase contrast transfer function (3.17), caused by energy spread
(3.6) of the incident electron beam, fluctuation in the power supply for accelerating voltage and the objective
lens exciting current, angular aperture of an incident electron beam, and environmental instability
3.8
fast Fourier transform
FFT
efficient algorithm to compute the discrete Fourier transform
[SOURCE: ISO15932: 2013, 5.4.1.1]
3.9
first zero
position where the phase contrast transfer function (3.17) first intersects the spatial frequency axis, and the
reciprocal of the wave number indicated by this position corresponds to point resolution (3.19) of HREM
3.10
illumination semi-angle
divergence semi-angle of the incident electron beam to a specimen
3.11
information limit
theoretical limit of the resolution of HREM, established by the observation of an amorphous specimen and
used as a performance index of a TEM
[SOURCE: ISO15932: 2013, 7.3]
3.12
lattice spacing
line spacing of periodic structure formed by interference between a scattered wave from a set of lattice
planes of the crystal and transmitted wave or between two scattered waves.
3.13
lattice resolution
resolution under imaging, which corresponds to the periodic lattice structure of the specimen
[SOURCE: ISO15932: 2013, 7.6]
3.14
live fast Fourier transform
live FFT
fast Fourier transform (3.8) processing technology which follows changes in processing target images in real
time
3.15
Miller index
notation system in crystallography for planes and directions in crystal lattices, in which a family of lattice
planes or directions is determined by three integers h, k, l
[SOURCE: ISO15932: 2013, 8.4]
3.16
phase contrast
image contrast due to the interference of transmitted and phase-shifted diffracted waves, which provides
high-resolution TEM image
[SOURCE: ISO15932: 2013, 6.2, modified — “interference of transmitted and” has been added]
3.17
phase contrast transfer function
PCTF
function which provides image contrast modulation caused by the phase shift produced by combination of
given spherical aberration and defocus
3.18
phase-distortion function
function representing the equiphase surface distribution of the electron wave passing through the lens,
which depends on the defocus amount, spherical aberration coefficient, spatial frequency, and wavelength
of electrons
3.19
point resolution
resolution under Scherzer focus (3.24), which is defined as the reciprocal of the spatial frequency where the
phase contrast transfer function (3.17) crosses the abscissa for the first time
[SOURCE: ISO15932: 2013, 7.6]
3.20
projected potential
electrostatic potential of a crystal projected along a low-index zone axis
3.21
region of interest
ROI
sub-dataset picked out from the entire dataset for a specific purpose
[SOURCE: ISO20263: 2017, 3.1.24]
3.22
Scherzer condition
state of observation set to Scherzer focus
3.23
Scherzer defocus
state in which the focus is shifted to Scherzer focus (3.24)
3.24
Scherzer focus
∆∆f
Scherzer
defocusing condition in which high-resolution images of the weak phase object (3.35) are formed by shifting
the phase of the scattered beam by -0,63 π (phase delay) which maximizes the spatial frequency of the first
zero of the contrast transfer function
Note 1 to entry: ∆ f is given by the Formula:
Scherzer
fC11, 2 
Scherzer s
where C is the spherical aberration coefficient and λ is the electron wavelength.
s
Note 2 to entry: ∆ f > 0 indicates the condition of under focus.
Scherzer
[SOURCE: ISO15932: 2013, 5.9.3, modified — “the diffracted beam by 0,5 π” has been replaced by “the
scattered beam by -0,63 π (phase delay)”, “fC11, 2  ” has been replaced by “
Scherzer s
fC11, 2  ”, and “Note 2 to entry” has been added. ]
Scherzer s
3.25
Scherzer resolution
d
Scherzer
point resolution (3.19) of HREM defined by the structural size corresponding to the first zero position of the
phase contrast transfer function (3.17) at Scherzer focus (3.24)
Note 1 to entry: d is given by the Formula:
Scherzer
1 3
4 4
dC0,668 
Scherzer s
where C is the spherical aberration coefficient and λ is the electron wavelength.
s
3.26
Schottky emission
thermionic electron emission that takes place under an electric field that enhances emission by lowering the
surface barrier
[SOURCE: ISO 15932: 2013, 3.1.1]
3.27
Schottky emission gun
SEG
electron gun (3.5) employing Schottky emission (3.26)
3.28
spatial frequency
reciprocal of structure size
3.29
spherical aberration
lens defect arising from the varying strength of an electromagnetic lens with distance from the optic axis,
which causes rays further from the optic axis to be focused more strongly than those nearer the optic axis
[SOURCE: ISO 15932: 2013, 3.2.2.2]
3.30
temporal coherency
correlations between waves observed at different points in time
3.31
thermal field emission gun
TFEG
electron gun (3.5) employing thermal field emission
[SOURCE: ISO 15932: 2013, 3.1.2.3]

3.32
thermionic emission gun
TEG
electron gun (3.5) employing thermionic emission
[SOURCE: ISO 15932: 2013, 3.1.3]
3.33
through-focus technique
image recording method that records images sequentially while varying the focal length at defined intervals
3.34
under focus
focusing condition of the objective lens in which its excitation is adjusted to slightly decreased rather than
focusing on a specimen
3.35
weak phase object
WPO
very thin TEM sample to which weak phase object approximation (3.36) can be applied
3.36
weak phase object approximation
WPOA
approximation assuming that the weak phase object (3.35) does not change the amplitude of the incident
wave but shifts the phase slightly in proportion to the projected potential (3.20)-
[SOURCE: ISO15932: 2013, 6.2.1, modified — “in which the specimen” has been replaced by “assuming that
the weak phase object (3.35)” and “slightly shifts the phase” has been replaced by “shifts the phase slightly in
proportion to the projected potential (3.20)”.]
4 Symbols and abbreviated terms
C chromatic aberration coefficient
c
CFEG cold field emission gun
C spherical aberration coefficient
s
the spherical aberration coefficient provided by TEM manufacturer
C
sprvd
the real spherical aberration coefficient of the objective lens used
C
sr

lattice spacing indicated by the (hkl) plane of the colloidal gold nano crystal
d
hklAu
theoretical point resolution of HREM
d
Scherzer
d provisional Scherzer resolution
Scherzer prvs

d reciprocal of the u value, representing measured point resolution of the TEM used
Scherzer r Scherzer r
 
D diameter (in pixels) of the n-th dark ring of the diffractogram for j-th image
n j

Dfrg( j) diffractogram obtained by FFT processing to the ROI set in the Img( j)
envelope function for temporal coherency based on the chromatic aberration
Eu

c
envelope function for spatial coherency based on the effective electron source size
Eu

s
F objective lens focal length at in-focus position
FFT fast Fourier transform/transformation
realistic phase contrast transfer function
F
PCTF r

HREM high resolution transmission electron microscope
Img( j) j-th image in the through-focus image series
objective lens exciting current
I
obj
L1( j) line connecting the spots reflected Au lattice spacing appeared in the Dfrg( j)
LP1( j) line-profile along L1( j)
L2( j) line passing through the centre of Dfrg( j)
LP2( j) line-profile along L2( j)
in the diffractogram for j-th image, distance (in pixels) between two peaks corresponding to
N
hklj

bright spots reflecting the lattice spacing of the (hkl) plane of the colloidal gold nano crystal
PCTF phase contrast transfer function
ROI region of interest
phase contrast transfer function (PCTF)
sin  u

SEG Schottky emission gun
STEM scanning transmission electron microscope/microscopy
TEG thermionic emission gun
TEM transmission electron microscope/microscopy
TFEG thermal field emission gun
u spatial frequency
spatial frequency corresponding to the n-th dark ring of the diffractogram for j-th image
u
nj

-1
in the diffractogram for j-th image, spatial frequency (in nm ) corresponding to one pixel
u
unit j

spatial frequency for the 1st zero of PCTF under Scherzer condition
u
Scherzer
st
spatial frequency corresponding to the 1 dark ring of Dfrg( j) obtained by applying the
u
Scherzer r

 f to the approximate quadratic equation
Scherzer r

accelerating voltage
V
acc
WPO weak phase object
WPOA weak phase object approximation
α
illumination semi-angle of incident electron beam
∆E
energy spread of electrons
∆ f
amount of defocus
 f average of three defocus values ( f (n=1 to 3)) for the j-th image
ave j n j
 
amount of Scherzer focus
∆ f
Scherzer
provisional Scherzer focus
 f
Scherzer prvs

real Scherzer focus
 f
Scherzer r

defocus value calculated from diameter of n-th dark ring of the diffractogram for j-th image
f
nj

λ
wavelength of electron
phase-distortion function
 u

5 Definition of point resolution based on information theory
5.1 General
It is widely recognised that the point resolution of a TEM is determined by the Scherzer resolution. However,
the coefficients included in the formula defining the Scherzer condition and the Scherzer resolution have
[2]
slightly different values depending on the target and purpose of observation. Therefore, there is a
possibility that different measurers can have different point resolution values.
In order to eliminate this issue, in this document, the point resolution is defined by applying a value
[3]
determined on the basis of information theory to the coefficients.
5.2 Definition of Scherzer condition
The observation condition (or defocus condition) that produces a phase contrast image reflecting the
projection potential of a weak phase object is called the Scherzer condition (or Scherzer focus; represented
[4]
by ∆ f ), and the Scherzer resolution is obtained under this condition .
Scherzer
[5]
The Scherzer condition is derived from the phase-distortion function ( u ), which describes the phase

difference between the electrons scattered by a weak phased object and the unscattered electrons, when
they pass through the objective lens and combine at the image plane. The phase-distortion function,
expressed as a composite of two phase-shifts induced by spherical aberration and defocusing of the objective
[6]
lens, is described by Formula (1) .
34 2
uC05, uf u (1)

Cf s
s
where
 u is the phase-distortion function;

δ is the phase shift caused by spherical aberration of objective lens;
C
s
 is the phase shift caused by defocus of objective lens;
 f
C is the spherical aberration coefficient of objective lens;
s
λ is the wavelength of the electron;
u is the spatial frequency;
NOTE u is the reciprocal of the structure size (1/d).
∆ f is defocus of objective lens.
NOTE  f  0 is under focus.
A graph of the phase-distortion function for  0,00251nm and C =1mm is shown in Figure 1. The
s
parameter is defocus values ( f 40nm,,60nm andn80 m ). The variation of the phase contrast with

spatial frequency can be derived from the phase-distortion function. Namely, the phase contrast is zero
when the phase-distortion function is an even multiple of π/2, and the contrast shows maximum when the
[7]
phase-distortion function is equal to an odd multiple . The Scherzer condition is a condition for obtaining
high phase contrast over a wide spatial frequency range, which is achieved by a defocus condition such that

the minimum value (Figure 1, Key 5) of the phase-distortion function is approximately equal to 

(Figure 1, Key 4). For example, a defocus value of 60 nm (Key 2) in Figure 1 is close to the Scherzer condition.
The relationship between the minimum value ( u ) of phase-distortion function and defocus (∆ f ) is

min
expressed by Formula (2).
 
 f
 u    (2)

min
 
2 C 
 s 
[3]
According to information theory, the maximum information of the sample is transferred to the image
[8]
when the value of  u is equal to 0,63 π. In this document, the defocus value expressed in Formula (3)

min
derived from this condition is defined as the Scherzer condition (∆ f ).
Scherzer
fC12,,61 12 C 2 (3)

Scherzer ss
For example, under the TEM conditions used to draw Figure 1, ∆ f is calculated as +56 nm (under
Scherzer
focus).
Key
-1
X spatial frequency ( u nm )
Y phase-distortion ( u ) of scattered wave against transmitted wave

1 graph for defocus at 40 nm.
2 graph for defocus at 60 nm.
3 graph for defocus at 80 nm.

4 dashed-line indicating  u  .

5 black dots indicating the minimum position of each  u curve.

Figure 1 — Phase-distortion function ( u ).

TEM conditions; Accelerating voltage is 200 kV ( 0,00251nm ), and C is 1,0 mm.
s
5.3 Definition of the theoretical point resolution
The change in phase contrast as a function of spatial frequency ( u ) can be expressed as a sine function of
 u , represented by Formula (4).

34 2
sinsuCin 05, uf u (4)


s
The sin  u is known as the phase contrast transfer function (PCTF). Examples of PCTF series due to

differences in defocus are shown in Figure 2. The TEM conditions for obtaining these graphs are the same as
for Figure 1. The appearance of the graph depends on the defocus value, but their common rule is to oscillate
st
as u increases after a graph crosses the u -axis first (so called 1 zero).
Figure 2b shows the PCTF under the defocus condition of ∆ f . In this graph, it is clear that the sin  u

Scherzer
is close to -1, i.e., high contrast, over a wide range of u . In contrast, Figures 2a and 2c show the PCTFs when
both defocus conditions are away from the ∆ f , indicating a narrower range of high contrast.
Scherzer
a) ∆ f=+40nm
b)  f 56 nm
Scherzer
c) ∆ f=+70nm
Key
-1
X spatial frequency ( u nm )
Y phase contrast (sin  u )

st
1 Dot indicates 1 zero position under the Scherzer condition.
Figure 2 — Series of PCTF curves (sin  u ) calculated for different defocus values (∆∆f ).

TEM conditions; Accelerating voltage is 200 kV ( 0,00251nm ) and C is 1,0 mm.
s
st
In the lower spatial frequency of u than the 1 zero [Figure 2b, Key 1] of PCTF under the Scherzer condition,
st
the image represents projected potential of the sample quite faithfully. But, beyond the 1 zero, the PCTF
oscillates and the information reflecting the projection potential cannot be reproduced on the image.

Therefore, the theoretical point resolution ( d ) of high-resolution transmission electron microscope
Scherzer
st
(HREM), known as Scherzer resolution, is defined by the structure size corresponding to the 1 zero position
( u ) of u-axis represented as Formula (5). Therefore, Formula (6) is the expression for the theoretical
Scherzer
point resolution ( d ).
Scherzer
2f
22, 4
Scherzer
u  (5)
Scherzer
21 3
 
C 
s
 
2 2
C 
s
 
 
 
1 3
1 4 4
d 0,668C  (6)
Scherzer s
u
Scherzer
where
d is the theoretical point resolution;
Scherzer
NOTE 1 d is sometimes referred to as Scherzer resolution.
Scherzer
st
u is the spatial frequency for the 1 zero of PCTF under Scherzer condition;
Scherzer
C is the spherical aberration coefficient of objective lens;
s
λ is the wavelength of electron used.
NOTE 2 "Information limit" is often used as the resolution of state-of-the-art TEM (C -corrected
s
TEM) equipped with "C corrector". The "information limit" indicates the limit of contrast
s
transmission for the image components, and is represented by the structural size corresponding
to the point where the PCTF disappears due to the envelope function. This is classified in a
different category from the "Scherzer resolution", which indicates the limit of transmission for
the projection potential of the sample.
6 Diffractogram
The 2D pattern generated by FFT processing of the TEM image is called diffractogram. Figure 3 shows an
example of the ideal diffractogram of the Scherzer focused image of an amorphous thin film (WPO). Since
the radial intensity distribution of the diffractogram reflects the square of PCTF curve (sin  u ) shown as

Figure 4, the radius of the first dark ring [Figure 3, Key 1] indicates the point resolution of the TEM [Figure 4,
Key 1].
Key
-1
X spatial frequency ( u nm )
1 radius of the first dark ring in the diffractogram
Figure 3 — Ideal diffractogram of the Scherzer focused image of WPO.
TEM conditions; accelerating voltage is 200 kV ( 0,00251nm ), and C is 1,0 mm.
s
Key
-1
X spatial frequency ( u nm )
Y intensity (:Square of PCTF (sin  u ) )

1 first zero position corresponding to the first dark ring in the diffractogram
Figure 4 — Square of PCTF curve (sin  u ) calculated under the Scherzer condition

(∆∆f =+56 nm). TEM conditions; Accelerating voltage is 200 kV ( 0,00251nm ) and C is 1,0 mm.
s
7 Effect of the envelope function
7.1 General
The ideal diffractogram with high contrast dark rings up to the higher spatial frequency region, as shown
in Figure 3, can only be produced from the image obtained with a fully coherent electron beam and ideal
observation conditions. In real situation, however, the actual diffractogram looks slightly different due to
contrast damping in the higher spatial frequency region caused by the reduced coherency of the electron
beam. There are two main factors contributing to the reduction in the coherency of the electron beam. One
is the reduction in spatial coherency based on the size of the effective electron source and the other is the
[9]
reduction in temporal coherency based on the degree of chromatic aberration. The effect of these two
factors on the contrast of the diffractogram is expressed as the result of multiplying their envelope functions
to the PCTF. In this clause, the effect of these two envelope functions on the contrast of the diffractogram is
presented.
NOTE The envelope function only affects the contrast of the diffractogram and does not change the position of
the dark rings. Therefore, the envelope function affects the information limit but not the point resolution.
7.2 Envelope function for temporal coherency
A factor that affects temporal coherency is chromatic aberration formed by the temporal defocus fluctuation
introduced by electron energy spread and instabilities of acceleration voltage and objective lens current.
[10]
The envelope function (Eu ) for temporal coherency is expressed by Formulae (7) and (8) .

c
 
 

u
   
Eu exp (7)
  
c  
 
 
 
 
 22 
 I 
 V   
E
obj
 acc 
 C  4  (8)
   
0 c
    
 
V V I
   
accacc obj
 
 
 
where
u is the spatial frequency;
λ is the wavelength of electron;
∆ is the defocus spread;
C is the chromatic aberration coefficient of objective lens;
c
V is the accelerating voltage;
acc
∆V
acc
is the instability in the accelerating voltage;
V
acc
∆E is energy spread of electrons;
NOTE In general, ∆E is around 1,5 eV for thermionic electron gun (TEG; LaB source), 0,7 eV
[11]
for Schottky electron gun (SEG), and 0,3 eV for cold field emission gun (CFEG) .
I is the objective lens exciting current;
obj
∆I
obj
is the instability of the objective lens current.
I
obj
Figure 5 shows Eu curves for each of the 200 kV TEM ( C = 1,4 mm) with three different electron guns.

c c
This shows that TEMs with TEG (LaB source) [Figure 5, Key 1] are more affected by Eu than TEMs with

c
SEG [Figure 5, Key 2] or CFEG [Figure 5, Key 3].

Key
-1
X spatial frequency ( u nm )
Y envelope function for temporal coherency (Eu )

c
1 E 15, eV
2 E 07, eV
3 E 03, eV
Figure 5 — Three envelope functions (Eu ) for temporal coherency with ∆∆E as parameter.

c
TEM condition; Accelerating voltage is 200 kV, λ is 0,002 51 nm, and C is 1,4 mm.
c
7.3 Envelope function for spatial coherency
The parallelism of the beam irradiating the sample depends on the electron source size. Then, the finiteness
of the electron source size creates tilting of the irradiated electron beam, which affects spatial coherency.
The envelope function (Eu ) due to the degradation of spatial coherency caused by the tilted beam having

s
[12]
illumination semi-angle of α is expressed by Formula (9). An example of Eu is shown in Figure 6.

s
 
 
 
Eu exp Cu
fu (9)


 
ss


 
 
where
α is the illumination semi-angle;
NOTE In general, α is around 0,7 mrad for thermionic electron gun (TEG; LaB source)
[12]
[Figure 6, Key 1], and 0,1 mrad for cold field emission gun (CFEG) [Figure 6, Key 2] .
λ is the wavelength of the electron;
u is the spatial frequency;
C is the spherical aberration coefficient of objective lens;
s
∆ f is the defocus value.
Key
-1
X spatial frequency ( u nm )
Y envelope function for spatial coherency (Eu )

s
1  07, mrad
2  01, mrad
Figure 6 — Two envelope functions (E u ) for spatial coherency under Scherzer condition with αα

s
as parameter.
TEM condition; Accelerating voltage is 200 kV, λ is 0,002 51 nm, C is 1,0 mm, and defocus is 56 nm
s
(under focus).
7.4 Effect of the total envelope function on the contrast distribution of diffractogram
The total envelope function (Eu ) is expressed by the Formula (10).

t
Eu Eu Eu (10)
  
tc s
where
Eu is the envelope function for temporal coherency based on the chromatic aberration;

c
Eu is the envelope function for spatial coherency based on the effective electron source size.

s
Then, the realistic phase contrast transfer function ( F ) reflects the Eu is expressed by the

PCTF r t

Formula (11).
FE sinsuu uE in  uE u (11)
    
PCTF rt cs

Examples of F graphs considering the influence of the total enveloping function are shown in
 
PCTF r

Figure 7 for TEG (LaB source) and Figure 8 for CFEG. Figures 7(b) and 8(b) are realistic diffractograms
reflecting the contrast distribution of Figures 7(a) and 8(a) respectively. In both figures, the TEM conditions
are the same except for the electron source.

a) Square of realistic PCTF curve ( F ) under Scherzer condition (∆∆f =+56 nm).
 
PCTF r

b) Realistic diffractogram corresponding to (a).
Key
-1
X spatial frequency ( u nm )
Y intensity (Square of realistic PCTF; ( F )
 
PCTF r

1 total envelope function (Eu ) (thin solid line)

t
2 square of realistic PCTF function ( F ) (solid line)
 
PCTF r

Figure 7 — An example of (F ) distribution for TEM equipped with LaB source. Conditions;
PCTF r

Accelerating voltage is 200 kV, λ is 0,002 51 nm, C is 1,0 mm, C is 1,4 mm, ∆∆E is 1,5 eV and α is
s c
0,7 mrad.
a) Square of realistic PCTF curve (F ) under Scherzer condition (∆∆f =+56 nm).
PCTF r

b) Realistic diffractogram corresponding to a).
Key
-1
X spatial frequency ( u nm )
Y intensity (Square of realistic PCTF); F
 
PCTF r

1 total envelope function (Eu ) (thin solid line)

t
2 square of realistic PCTF ( F ) (solid line)
 
PCTF r

Figure 8 — An example of (F ) distribution for TEM equipped with CFEG.
PCTF r

TEM conditions; Accelerating voltage is 200 kV, λ is 0,002 51 nm, C is 1,0 mm, C is 1,4 mm, ∆∆E is
s c
0,3 eV and α is 0,1 mrad.
8 Sample for point resolution measurement
8.1 General
The theoretical treatment related to the point resolution of HREM is based on the assumption of weak phase
object approximation. Therefore, the sample used for the resolution measurement should be a weak phase
object.
8.2 Sample for point resolution measurement
A sample support film for TEM observation with fine granular structure characteristics is convenient for
measuring point resolution. Considering the condition that the sample should be a weak phase object, it is
necessary to use amorphous carbon film or amorphous germanium film with a thickness of 10 nm or less. In
addition, the fine particles having size calibration structure should be simultaneously recorded on the image
used for point resolution measurements. Colloidal gold nano-particles are a convenient internal calibration
material for this purpose. Figure 9 shows an example image of colloidal gold nano-particles [Figure 9, Key 1]
supported on amorphous carbon thin film (about 10 nm thick).
Key
1 All particles appeared black are colloidal gold nano-particles.
Figure 9 — Colloidal gold nano-particles supported on amorphous carbon thin film.
9 Measurement procedures for determining experimental point resolution of HREM
9.1 General
In this chapter, the measurement procedure for determining the experimental point resolution of HREM is
described. Figure 10 shows flow chart of the measurement procedure. For specific example of measurement
procedures, refer to Annex B.
Figure 10 — Flow chart of measurement procedure for determining point resolution of HREM

9.2 Calculation of provisional Scherzer focus value
To determine point resolution experimentally according to this document, it is necessary to prepare a set of
through-focus images with the Scherzer focus position approximately at its centre. To find out the Scherzer
focus, the appropriate spherical aberration coefficient must be applied to Formula (3). However, the real
value of the spherical aberration coefficient ( C ) of the objective lens used is unknown at this stage.
sr

Therefore, the spherical aberration coefficient ( C ) provided by the TEM manufacturer is applied t
...