So et al. Applied Microscopy(2021) 51:10
https://doi.org/10.1186/s42649-021-00060-z

Applied Microscopy

RESEARCH

Open AccessSensitivity of quantitative symmetry
measurement algorithms for convergent
beam electron diffraction techniqueHyeongsub So1,2, Ro Woon Lee1, Sung Taek Hong1 and Kyou-Hyun Kim1*

Abstract
We investigate the sensitivity of symmetry quantification algorithms based on the profile R-factor (Rp) and thenormalized cross-correlation (NCC) coefficient (γ). A DM (Digital Micrograph©) script embedded in the Gatan digital
microscopy software is used to develop the symmetry quantification program. Using the Bloch method, a variety of
CBED patterns are simulated and used to investigate the sensitivity of symmetry quantification algorithms. The
quantification results show that two symmetry quantification coefficients are significantly sensitive to structuralchanges even for small strain values of < 1%.
Keywords: Convergence beam electron diffraction, Symmetry, Transmission electron microscopy

Introduction
Convergent beam electron diffraction (CBED) withtransmission electron microscopy (TEM) is a powerfultool to extract submicron information (Zuo et al. 1999;
Zuo et al. 1989; Zuo 1998). Especially, local symmetry
determination with the CBED technique is of importance in terms of understanding fine structures as regionsof interest scale down to submicron scale. The generalprocedure for symmetry determination with the CBED
technique follows from investigation of the zero-order
Laue zone (ZOLZ) details in the zone axis pattern (ZAP)(Buxton et al. 1976). Symmetry in the ZOLZ CBED pattern is classified into ten two-dimensional (2D) point
groups (Buxton, et al. 1976; Loretto 1994). The symmetry elements of the rotational or mirror develop point
groups by generating regularly repeated constituentsfrom a reference motif. The generated motifs have a specific pattern, which is called rocking curve information(or intensity profile). The 2D point group determinationin the ZOLZ CBED pattern is based on the regularly* Correspondence: khkim1308@kitech.re.kr
1
Korea-Russia Innovation Center, Korea Institute of Industrial Technology,
Incheon 22004, Republic of Korea
Full list of author information is available at the end of the article

repeated constituents. The obtained 2D point group isthen used to specify the projection diffraction groups
and possible diffraction groups which, in turn, determine
the point group (Buxton, et al. 1976; Loretto 1994).
The symmetry recorded in the CBED patterns is in
general determined using direct visual inspection. In
many practical applications, however, experimental
CBED patterns often contain uncertainty of rocking
curve information. The uncertainty may stem from either experimental error or the original structure. The
deviation can be ignored or taken into account for symmetry determination. In consequence, the symmetry recorded in the CBED pattern can be interpreted indifferent ways.
Recently, Kim et al. proposed a symmetry quantification method for CBED patterns using the profile R-factor (Rp) (Jansen et al. 1994; Toby 2006) and thenormalized cross-correlation coefficient (γ) (Lewis 1995)
to revel symmetry breaks and nanodomain structures inpiezoelectric material (Jeon and Kim 2020). Rp and γ
have been widely used to numerically calculate thedegree-of-agreement between two objects. First, the profile R-factor (Rp) has been used in Rietveld refinement toquantify the correlation between an experimental and© The Author(s). 2021 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate ifchanges were made. The images or other third party material in this article are included in the article's Creative Commonslicence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commonslicence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain
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So et al. Applied Microscopy(2021) 51:10

computed intensity profile (Jansen, et al. 1994; Toby
2006). The normalized cross-correlation coefficient (γ) is
another numerical measure to quantify the correlation
between two image templates. The γ value is 1 (or100%) when two image plates (symmetric CBED discs)
are identical. In contrast, the γ value gets close to − 1(or − 100%) as the amount of symmetry difference increases. The CBED disc consists of an intensity profile,
so Rp can be used to compare the degree-of-agreement
between two motifs. The whole disc image can also be
considered as an image template. In this perspective, γ
can be used to measure the amount of degree-ofagreement between the selected discs. The use of thesymmetry quantification method, therefore, provides amore precise way to determine the symmetry in a CBED
pattern. It was also proposed that the symmetry quantification method should be combined with a scanningelectron diffraction technique for symmetry mapping(Tao et al. 2009; Zuo and Tao 2011).
In this study, we investigate the sensitivity of symmetry quantification methods for Rp and γ (NCC). Eachalgorithm is applied to several simulated structures suchas strained Si and a perovskite structure to investigate
the sensitivity of mathematical methods for symmetry
quantification. This is because the experimental CBED
patterns have unavoidable background noise generated
by a CCD camera or inelastic scattering induced bysomething such as phonon vibration. For the simulationof CBED patterns, this study uses a Bloch wave method
based on atomic scattering factors of Doyle and Turner(Doyle and Turner 1968) and the absorption parametersof Bird and King (Bird and King 1990).

Methods of quantitative symmetry measurement
The symmetry quantification algorithms are embedded
as DM (Digital Micrograph©) script. As proposed byKim et al. (Kim and Zuo 2013), the developed algorithms consist of (1) disc selection, (2) alignment, (3) application of symmetry operation, and (4) symmetry
quantification. Details on the imaging process for symmetry quantification have been reported elsewhere (Huet al. 2000).
For the symmetry quantification of a CBED pattern, Rp
is modified to quantify the similarity between two selected CBED discs, as in the following,
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
sP
fIB ðx; y Þ−IA ðx; y Þg2
Rp ¼
ð1Þ
P
IA ðx; y Þ2
where IA(x, y) and IB(x, y) are the intensities of the selected CBED discs A and B at the address of (x, y) in
pixels, respectively. Because the two selected templates
are similar, the intensity difference sum between selected
CBED discs A and B approaches zero, so that the

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smaller Rp value provides the better match. The two selected CBED discs will be referred to as image templates
A and B. The DM script is then expressed as follows to
calculate the Rp value.
Number RProfileFactor(Image ImgA, Image ImgB){
Number Numerator, Denominator
Numerator = sum((ImgB-ImgA)**2)
Denominator = sum(ImgA**2)
return sqrt(Numerator/Denominator)}
The normalized cross-correlation (NCC) coefficient, γ,
basically follows from the sum of the multiplication ofdifferences between the image template and the mean of
the image template for two templates. The crosscorrelation coefficient is then normalized as follows,P
x;y f½IA ðx; y Þ−IA   ½IB ðx; y Þ−IB g
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
r
y¼ n
o ð2Þ
P
P
2
2
½
I
ð
x;
y
Þ−I


½
I
ð
x;
y
Þ−I

A
A
B
B
x;y
x;y
where I A and I B are the mean values of two templates(Lewis 1995). In the Eq. (2), the numerator and denominator have exactly the same values if the two templates
are absolutely identical. In contrast to Rp, the crosscorrelation coefficient is close to 1 when the two templates are identical. For a symmetry quantification algorithm based on γ, the DM script can be written as
follows.
Number CrossCorrelation(Image ImgA, Image ImgB).
{
Number Numerator, Denominator.
Numerator = sum((ImgA-mean(ImgA))*(ImgBmean(ImgB))).
Denominator = sqrt(sum((ImgA-Mean(ImgA))**2)*sum((ImgB-mean(ImgB))**2)).
return Numerator/Denominator.
}

Applications and discussion
Strained Si

Strained Si is attractive as a potential structure for advanced complementary
metal-oxide-semiconductor(CMOS) technology or electro-optic devices (Erdtmann
and Langdo 2006; Jacobsen et al. 2006). The amount ofstrain rate is very small, at a few percentage points, andis difficult to detect for local areas using X-ray diffraction technique due to the relatively large beam probe
size of X-ray (Kim and Zuo 2014). An Si single crystal
has a space group of Fd3m with lattice parameters ofa = b = c = 5.4309 Å and with atomic coordinates of Si (0,0, 0). The original structure of Si is then artificiallystrained along the [100]C, [010]C, and [001]C directions
by (0.5%, 0.5%, − 0.25%), (1%, 1%, − 0.5%), and (2%, 2%,

So et al. Applied Microscopy(2021) 51:10− 1%). Hereinafter, the degree of applied strain will bereferred to as ε0, εI, εII, and εIII for (0%, 0%, 0%), (0.5%,
0.5%, − 0.25%), (1%, 1%, − 0.5%), and (2%, 2%, − 1%), respectively. The strain rates were determined based on
the usual amount observed in the CMOS device. In thestrained Si structure, a zone axis is properly selected toobserve the effect of strain on the symmetry breaking in
the recorded pattern symmetry. For example, the symmetry element along [100]C is more sensitive to the applied strain than is that of [111]C. This is because thesymmetry element is projected onto a two-dimensional
CBED pattern along the observing direction. Thus,
CBED patterns are simulated for zone axes of [100]Cand [111]C to quantify the pattern symmetry based on
the proposed symmetry quantification algorithms.
Figure 1a shows the simulated CBED patterns of Si for thezone axis of [100]C at the thickness of 60 nm. The patternsymmetry of Si has 4 mm at the zone axis of [100]C, as
shown in Fig. 1a. Figures 1a-d show the simulated CBED
patterns for the strains of εI, εII, and εIII. Overall features ofsimulated CBED patterns are very similar for different
amounts of strain while the amount of strain increases from
εI to εIII. Differences in the rocking curve information are
only observable in the magnified discs images. Figure 1f, forexample, shows magnified CBED discs of (004) and (040)m,
where ‘m’ indicates that the mirror symmetry is applied tothe (040) disc. In the magnified image with εI, the two CBEDdiscs have considerably small differences in the rocking
curve information, as indicated by the dotted circle and therectangle. Similarly, other simulated CBED patterns (Figs. 1g
and h) for the larger strain values also induce very smallchanges in the rocking curve information; these changes arevery hard to determine only by visual inspection.

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Symmetry breaking induced by strain is then measured byquantitative methods based on the R-factor and the γ coefficient, as shown in Fig. 2. Symmetry measurements were performed for the 1st order reflections and for the 2nd orderreflections. For mirror symmetry, the (022)/(0 2 2) and (0 22)/(02 2) discs along m1 are selected for 1st order reflections
and the (004)/(040) and (00 4)/(0 4 0) discs are selected for2nd order reflections. The same discs sets are selected to
calculate the 4-fold rotational symmetry. Figures 2a and b
show the variations of mirror symmetry calculated by γm
and Rp(m) for the 1st and the 2nd order CBED discs. For the
1st order CBED discs, the quantification values indicate thatmirror symmetry is almost maintained while the amount ofstrain increases from ε0 to εIII. In contrast, it can be observedthat mirror symmetry is obviously broken in the 2nd orderreflections. The γm value decreases from 100% to 92.5% as
the amount of strain increases. Also, the Rp(m) value increases from 0.028 to 0.346 as the amount of strain increases. The sensitivities to the structural change, however,are different between the applied algorithms. From the symmetry quantification results, the perfect mirror symmetry ofγm (=100%) slightly decreases to 99.2%, 95.9%, and 92.5% forεI, εII, and εIII, respectively. In contrast, the Rp(m) value has
0.028 for the perfect Si structure and dramatically increases
to 0.104, 1.245, and 0.346. By comparting the two algorithms, the Rp value shows an almost 15 times difference inthe calculation results between εI and εII, while changes of
only a few percentage points are only observed for the γvalue. The γ value shows very similar results for the 4-foldrotational symmetry, as shown in Fig. 2c. The 4-fold rotational symmetry is almost maintained for the 1st order,
while the quantification results for the 2nd order are affected

Fig. 1 Calculated CBED patterns for the perfect Si struct. and (b-d) the strained Si structures at the zone axis of [100]C. The amount of strain isdenoted in each simulated CBED patterns

So et al. Applied Microscopy(2021) 51:10

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Fig. 2 Symmetry quantification for mirror and 4-fold rotational symmetry recorded in the simulated CBED patterns for ε0(0%, 0%, 0%), εI(0.5%, 0.5%, −0.25%), εII(1%, 1%, − 0.5%), εIII(2%, 2%, − 1%). The symmetry elements are respectively quantified based on γ and Rp

by the strain. Unlike the results of the γ values, however, theRp values vary from 0 ~ 0.061 for the 1st order. Nevertheless,Rp of εII has a smaller value than εI and εIII, even thoughRp(4R) values are expected to increase gradually as theamount of strain increases. This suggests that the γ coefficient is more correlated with the symmetry change than isRp, while the Rp is more sensitive to the symmetry change.
In addition, the above results indicate that symmetry broken
by structural distortion has a greater effect on symmetrical
relation in the 2nd or higher order reflections than in the
1st order reflections.
Figure 3 shows simulated CBED patterns for Si withdifferent strain values. The CBED patterns were calculated for the sample thickness of 60 nm and for thestrain values used in Figs. 2 and 3. Like the simulated[100]C CBED patterns, the broken symmetry is hardto observe in the simulated CBED patterns by visualinspection alone, as shown in Figs. 3a-d. Differencesin the rocking curve information are compared forthe mirror symmetry using the magnified second
order reflections of (2 4 2)/(22 4)m, as shown in Figs.

3f-h. Only small differences are observed even in themagnified CBED discs, as well as in the results fromthe zone axis of [100]C. The quantitative symmetry
measurements then follow using Rp and the γcoefficient.
Figure 4 shows the effect of strain on the deviation of
the symmetry element along the zone axis of [111]C.
The recorded symmetry is quantified for the mirror andfor the 6-fold rotational symmetry based on the 1storder and 2nd order reflections, respectively. Similar to
the results of [100]C, the quantification results indicatethat the degree of broken symmetry gradually increases
as the amount of applied strain increases. Nevertheless,the symmetry quantification values are not strictly related to the amount of applied strain. For example, theγm values for the second order reflections decrease from~ 99.9% (ε0) to 99.3% (εI) → 99.04 (εII) → 97.7 (εIII). The
calculated γm value abruptly decreases at εIII, while theapplied strain values uniformly increase by two times for
εI → εII → εIII. The RP(m) values also dramatically increase at εIII as well.

So et al. Applied Microscopy(2021) 51:10

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Fig. 3 Calculated CBED patterns for the perfect Si structure and (b-d) the strained Si structures at the zone axis of [111]C. The amount of strain isdenoted in each simulated CBED patterns

Fig. 4 Symmetry quantification for mirror and 6-fold rotational symmetry recorded in the simulated CBED patterns for ε0(0%, 0%, 0%), εI(0.5%, 0.5%, −0.25%), εII(1%, 1%, − 0.5%), εIII(2%, 2%, − 1%). The symmetry elements are respectively quantified based on γ and Rp

So et al. Applied Microscopy(2021) 51:10

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Table 1 Crystallographic information of PMN-31PT for cubic, monoclinic, rhombohedral, and tetragonal
Crystal system

Cubic

Monoclinic

Rhombohedral

Tetragonal

Space group

Pm3m

Cm

Pm

R3m

P4mm

Lattice parametersa = b = c = 4.0191
α = β = γ = 90oa = 5.6951
b = 5.6813
c = 4.0138
α = γ = 90oβ = 90.136o

a = 4.0183
b = 4.0046
c = 4.0276
α = γ = 90oβ = 90.146oa = b = c = 4.0364
α = β = γ = 89.8826o

a = b = 3.9920
c = 4.0516
α = β = γ = 90o

Atomic coordinates Pb(0, 0, 0)
Pb(0, 0, 0)
TiNb/Mg(0.5, 0.5, 0.5) TiNb/Mg(0.5250, 0, 0.498)
O(0.5, 0.5, 0)O1(0.54, 0, −0.01)
O2(0.317, 0.267, 0.48)

Pb(0, 0, 0)
Pb(0, 0, 0)
TiNb/Mg(0.509, 0.50, 0.5479) TiNb/Mg(0.534, 0.534, 0.534)
O1(0.47, 0, 0.57)
O(0.541, 0.541, 0.03)
O2(0.417, 0.5, 0.509)
O3(−0.02, 0.5, 0.57)

By considering the results from the zone axis of[100]C, quantitative symmetry measurement along thezone axis of [111]C shows that the amount of broken
symmetry is smaller than for the results along the zone
axis of [100]C. The degree of symmetry broken by applied strain becomes small along the higher index of thezone axis because the pattern symmetry of CBED resultsfrom the projected atomic coordinates. Considering this,strain induced broken symmetry is more obviously observable along the zone axis of [100]C than the zone axisof [111]C.
Symmetry change in relaxor-based ferroelectric materials

Relaxor-based ferroelectric ceramics such as (1x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-xPT) have attracted
much research interest due to the strong dependence of
their polarization on the applied electric field. A single crystal of PMN-xPT with low contents of PT (x < ~30%) has
rhombohedral symmetry at room temperature, in which
PS, a spontaneous polarization, is constrained to the [111]direction. In a rhombohedral composition near the morphotropic phase boundary (MPB) (Choi et al. 1989; Kimet al. 2012), PMN-PT achieves ultrahigh piezoelectric responses along the non-polar direction of [001], which leads
to electric field-induced phase transition from R (rhombohedral, R3m) to T (tetragonal, P4mm) symmetry (Noheda
2002). Indeed, R and T symmetry are not allowed for directPb(0, 0, 0)
TiNb/Mg(0.5, 0.5, 0.532)
O1(0.5, 0.5, 0.054)
O2(0.5, 0, 0.601)

phase transition. The polarization vectors then rotate
within the mirror plane (Fu and Cohen 2000) common to
the R and T symmetry. The R symmetry undergoes a structural change into T symmetry via the paths of ‘R → M(monoclinic, Pm or Cm) → T (Noheda 2002). Nevertheless,
it is still unclear whether the giant piezoelectric properties
stem from the observed monoclinic phases (Viehland 2000;
Kisi et al. 2003). On the other hand, structural similarities
among the known R, M, and T have been a key issue in determining the exact crystallographic information in PMNxPT single crystals. The high temperature phase of cubic in
PMN-xPT is transformed into R, M, or T with very small
structural distortions, as listed in Table 1. In consequence,the conventional electron diffraction and X-ray diffraction
techniques are hard to apply to identify the phase of PMNxPT. The CBED patterns of PMN-PT with x = 31% are simulated for the reported crystallographic information and thepattern symmetry is quantified using γ and Rp.
Figure 5 shows the calculated CBED patterns of theC, M (Cm, Pm), R, and T phases at the zone axis of[001]C with respect to the pseudo-cubic axes at the
sample thickness of 80 nm. For the pseudo-cubic axisof [001]C, the corresponding zone axes of M (Cm,Pm), R, and T phases are respectively [001]M(Cm),[100]M(Pm), [001]R and [001]T. As shown in the simulated patterns, the pattern symmetries can be only
ambiguously distinguished by visual inspection, while

Fig. 5 Simulated CBED patterns for (a) the zone axis of [001]C (Pm3 m), (b) the zone axis of [001]M (Cm), (c) the zone axis of [100]M (Pm), (d) thezone axis of [001]R (R3m), and (e) the zone axis of [001]T (P4mm)

So et al. Applied Microscopy(2021) 51:10

the CBED patterns are simulated with different crystal
structures. For the zone axis of [001]C, the C, M (Pm,
Cm), R, and T phases have pattern symmetries of4mm, m, m, m, and 4mm, respectively, as indicated inFig. 5. The mirror symmetry element in the CBED
patterns is then selected for the symmetry quantification because the polarization vector, PS, in PMN-PT
lies on the mirror plane of each phase. To see theamount of mirror elements, mirror symmetry is quantified along the directions of [100]C, [010]C, and[110]C for each CBED pattern in order to see thePage 7 of 9variations of mirror symmetry elements subjected tostructural distortions, i.e., phase transformation.
Figure 6 shows profiles of mirror symmetry quantification results for the C, R, M (Cm), M (Pm), and T phases
along the different directions of [100]C, [010]C, and[110]C. As shown in the quantification results, the γmvalues of the 1st order reflections are almost maintained
for all directions. This indicates that, among the reported crystallographic information, the structural distortions induced by phase transformation are very small.
Similarly, by considering the result of strained Si, the Rp

Fig. 6 Quantified symmetry element for simulated CBED patterns of PMN-31%PT

So et al. Applied Microscopy(2021) 51:10

values of the 1st order reflections are found to vary bysmall amounts. In contrast, broken symmetry is obviously observed in the 2nd order reflections. From thequantification results, the amount of mirror symmetry
has the lowest value for the M phase of Cm. Then, theamount of mirror symmetry gradually increases. This ismainly due to polarization rotation. In a previous study,
it was found that the polarization vector rotates from R([111]C) to T ([001]C) via the monoclinic phases of Cmand Pm. On the pseudo-cubic axes, the polarization vector rotates along (100)C of M (Cm) → (010)C of M (Pm).
It is well known that phase transformation occurs via
C → R → M (Cm) → M (Pm) → T (P4mm). Based onthat previous report, the phase transformation from R to
M (Cm) is subjected to the first order transition, whileother transformation steps follow by second order transition. From a structural aspect, the rhombohedral crystal structure requires severe distortion to transform into
monoclinic axes, as listed in Table 1. In consequence,the drastic γ changes between R and M (Cm) are considered as the degree of amount of structural change forthe phase transition between R → M.

Conclusion
We investigated the sensitivity of two symmetry quantification coefficients of the profile R-factor (Rp) and the normalized
cross-correlation
coefficient(γ).
Thequantification results show that the two different coefficients are significantly sensitive to deviation of the symmetry element. Only the few strain values of < 1% are
obviously detected by Rp and the γ coefficient. In addition,
small structural distortions can also be evidently differentiated using the two coefficients. Nevertheless, the two coefficients Rp and γ show differences in application tosymmetry quantification. Compared to γ, Rp shows huge
differences for small structural changes, so it is more applicable to visualizing small differences in a crystal structure. Because it is not normalized, however, Rp cannot be
used to compare CBED patterns recorded from different
samples. Moreover, in some cases, Rp only uncertainlyshows the amount of symmetrical change, as demonstrated in Fig. 2d. In comparison, because it is normalized,
the γ value can be directly applied to quantify the symmetry elements recorded in different CBED patterns. Dueto its normalization, the γ value is not affected by different
operation conditions. Also, the γ value exactly agrees withthe structural change in all cases.
Acknowledgements
None.
Authors’ contributions
Hyeongsub So: Investigation, Data curation, Writing – original draft. Ro Woon
Lee: Investigation. Sung-Taek Hong: Investigation. Kyou-Hyun Kim: Resources,
Advisor, Supervision, Project administration, Writing – review & editing. The
authors read and approved the final manuscript.

Page 8 of 9

Funding
This work was supported by a research program funded by the Korea
Institute of Industrial Technology (KITECH).

Availability of data and materials
Not applicable.

Declarations
Competing interests
The authors declare that they have no competing interests.
Author details
1
Korea-Russia Innovation Center, Korea Institute of Industrial Technology,
Incheon 22004, Republic of Korea. 2Department of Materials and Science
Engineering, Korea University, Seoul 02841, Republic of Korea.
Received: 4 June 2021 Accepted: 23 June 2021

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