A review of methods for the accurate determination of the chiral indices of carbon nanotubes from electron diffraction patterns

C A R B O N 4 9 ( 2 0 1 1 ) 4 9 6 1 – 4 9 7 1

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Review

A review of methods for the accurate determinationof the chiral indices of carbon nanotubes from electrondiffraction patterns

C.S. Allen a,*, C. Zhang b, G. Burnell a, A.P. Brown c, J. Robertson b, B.J. Hickey a

a School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UKb Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UKc School of Process, Environmental and Materials Engineering, University of Leeds, Leeds LS2 9JT, UK

A R T I C L E I N F O

Article history:

Received 18 April 2011

Accepted 23 June 2011

Available online 4 August 2011

0008-6223/$ - see front matter � 2011 Elsevidoi:10.1016/j.carbon.2011.06.100

* Corresponding author.E-mail address: [email protected] (C.S

A B S T R A C T

The strong link between the precise molecular structure and electronic properties of single

wall carbon nanotubes is well known, however the experimental investigation of these

links remains a challenge. This is in part due to the difficulty in determining the structure,

or chirality of individual carbon nanotubes. Performing diffraction studies within a trans-

mission electron microscope has become arguably the most powerful technique for the

accurate determination of the chirality of, not only individual single wall, but also multi-

wall and small bundles of carbon nanotubes. In this paper we give an up to date review

of the analytical approaches to chirality assignment. For comparison we perform each of

the discussed techniques on the same diffraction pattern, obtained using standard electron

diffraction techniques, and discuss the ability of each to allow for the unambiguous deter-

mination of chirality within experimental uncertainties. The aim of this review is to give

guidance as to the most pragmatic approach to extracting carbon nanotube chirality from

electron diffraction patterns without the need for performing complicated and time con-

suming simulations.

� 2011 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4962

2. Description of analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4962

2.1. Axial measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4963

2.2. Radial measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4964

3. Analysis of an experimental diffraction pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4964

3.1. Chirality assignment from axial measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4965

3.2. Chirality assignment from radial measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4966

3.3. Fitting to the radial intensity distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4967

4. Multi-wall tubes and ropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4968

er Ltd. All rights reserved.. Allen).

4962 C A R B O N 4 9 ( 2 0 1 1 ) 4 9 6 1 – 4 9 7 1

5. Dealing with deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4968

6. Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4969

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4970

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4970

1. Introduction

The remarkable electronic properties of single wall carbon

nanotubes (SWCNTs) and the sensitivity of these properties

to atomic configuration [1] has driven the need for the devel-

opment of a reliable method for determining tube structures.

Techniques to determine the structure of carbon nano-

tubes fall into three broad categories: optical spectroscopy,

probe microscopy and electron microscopy.

Resonant Raman spectroscopy has emerged as the most

powerful of the optical techniques as a signal can be obtained

from an individual SWCNT due to the unusually large optical

response when the excitation energy of the laser is in reso-

nance with an electronic transition of the tube [2]. Interpreta-

tion of Raman spectra requires comparison with often

complex quantum mechanical models [3] and can be depen-

dent on environmental interactions. Furthermore in order to

study SWCNTs of a variety of structures a range of laser ener-

gies is required.

Recent work performing Raman and Rayleigh spectros-

copy on carbon nanotubes structurally characterised by elec-

tron diffraction techniques has confirmed the relationships

between optical properties and tube structure [4–8]. Evidence

for the modification of these properties due to tube–tube

interactions in SWCNT bundles has been found [6] and the

possibility of characterising SWCNTs with a single laser

wavelength demonstrated [7].

Scanning tunnelling microscopy (STM) can be used to im-

age the structure of a carbon nanotube on a surface allowing

for a direct measurement of the chiral angle combined with

electronic spectroscopic measurements [9,10]. This powerful

technique gave the first experimental verification of the link

between the electronic and structural properties of SWCNTs.

However, STM studies are limited by the fact that samples

have to be carefully prepared with the SWCNT laying on a

conductive substrate. Furthermore STM only allows for the

study of the outermost shell of a multi-wall carbon nanotube.

Early electron diffraction studies using standard selected

area electron diffraction (SAED) techniques were performed

on multi-wall [11–13], bundled [14,15] and individual single

wall tubes [16,17]. The diffraction pattern quality was typi-

cally poor and therefore analysis limited. In 1997 the use of

nano-beam electron diffraction, in which a coherent electron

beam of width a few tens of nanometres is incident on the

carbon nanotube, was first used to study ropes containing

multiple SWCNTs [14,18]. This technique significantly im-

proves the quality of electron diffraction patterns and allows

for a more analytical approach to chirality assignment.

Analysis of SWCNT electron diffraction patterns can be

performed by measuring the positions of the layer lines in

the direction parallel to the tube axis [19–22] or alternatively

by measuring the radial intensity distribution of the layer

lines [13,23–25]. Two major factors in the correct determina-

tion of the chiral indices of a SWCNT are the accurate calibra-

tion of the diffraction pattern and the errors due to a finite tilt

of the tube with respect to the incident electron beam.

Experiments linking the transport properties of SWCNTs

to their structure, as determined from electron diffraction

patterns, remain relatively few [26–30] due in no small part

to the technical difficulties in achieving this. Recently, struc-

tural changes induced by the application of tensile stress to

a SWCNT have been observed from diffraction pattern analy-

sis [31].

The development of aberration correction in transmission

electron microscopes (TEM) lenses has allowed for high reso-

lution imaging of the carbon atoms in SWCNTs [32–34]. This

has enabled the location of defects in the tube wall, and the

study of the evolution of these defects at high temperatures.

In this paper we present a review of the numerous tech-

niques which have been proposed for the determination of

carbon nanotube structure from the analysis of electron dif-

fraction patterns. For comparison of the techniques described

we perform detailed analysis of electron diffraction patterns

from a single wall and a double wall carbon nanotube using

standard SAED techniques at a beam energy of 80 keV. When

careful consideration is given to the propagation of experi-

mental uncertainties throughout the calculations we show

that the use of certain analyses in isolation can lead to ambi-

guity in chirality determination. However, when multiple

analyses are used in conjunction with one another the chiral

indices can be determined from electron diffraction patterns

with a high degree of confidence.

2. Description of analyses

The structure of a SWCNT is typically described in terms of

the chiral indices (n,m) which define the vector that maps

around the circumference of the tube in terms of the unit vec-

tor of the graphene lattice. The chiral indices give a complete

description of the structure of the tube and define both the

diameter

D0 ¼a0

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þm2 þ nm

pð1Þ

and the chiral angle

h ¼ tan�1

ffiffiffi3p

m2nþm

!ð2Þ

of the tube (where a0 = 0.246 nm, the length of the basis vec-

tors of the graphene lattice) [35].

A kinematical description of diffraction from a SWCNT

was first introduced by Qin in 1994 [13], and subsequently

developed by Lambin, Bruyninckx and Lucas [36,37]. Here

we are primarily interested in the application of the theory

to experimental data so will not describe the theoretical back-

ground, rather discuss the proposed analyses derived from it

Fig. 1 – (a) A schematic depiction of the diffraction pattern obtained from a SWCNT. The hexagons mark the first order

reflections from the graphene lattice. The first three layer lines are indicated (L1–L3) as is the equatorial oscillation (Eq) and its

oscillatory period, d. The chiral angle h is shown and the axial distances of each of the layer lines (d1 to d3) are marked. (b) The

theoretical form of the radial intensity distribution of one of the layer lin es. The separation of the first and second order

peaks (P1 and P2) are shown as is the separation of the first order minima (M1) and the oscillatory period (D12).

C A R B O N 4 9 ( 2 0 1 1 ) 4 9 6 1 – 4 9 7 1 4963

(the interested reader is directed towards a number of excel-

lent review articles in the literature, for example [21,38]).

The diffraction pattern from a SWCNT is most simply

understood by considering diffraction from two parallel

sheets of graphene. An electron beam incident normal to

the tube axis passes sequentially through the two graphene

sheets (corresponding to the ‘top’ and ‘bottom’ wall of the

carbon nanotube). The resultant diffraction pattern is the

sum of that from each graphene sheet. If the graphene sheets

are rotated with respect to one another about the axis normal

to the plane of the sheets, the two diffraction patterns will

also be rotated with respect to one another by the same angle

(see Fig. 1). Due to radial curvature the sharp diffraction spots

appear smeared in the radial direction, forming a series of

lines known as layer lines [37].

More precisely, the diffraction pattern from a SWCNT is

the projection of the points in reciprocal space where the

Ewald sphere intersects with the reciprocal lattice of the crys-

tal. Due to the high energy electrons used in electron diffrac-

tion experiments the Ewald sphere can be approximated to a

plane [24]. The reciprocal space representation of a carbon

nanotube consists of a series of concentric discs oriented per-

pendicular to the tube axis with the layer lines in Fig. 1 being

the points at which the Ewald plane slices through these

discs. The axial positions of the layer lines are determined

by the axial periodicity of the SWCNT and their radial form

by the cylindrical curvature of the tube [21].

2.1. Axial measurements

In principle the chiral angle of the SWCNT, h can be measured

directly from the diffraction pattern. Combined with a mea-

surement of the diameter of the tube and using Eqs. (1) and

(2) the chiral indices can be determined. However, due to the ra-

dial curvature of the tube the positions of the diffraction spots

do not lie exactly on the points of the reciprocal lattice of graph-

ene [39,40]. The structure of a SWCNT cannot therefore be

accurately determined from a direct measurement of h.

The positions of the layer lines in the axial direction are

unaffected by the radial curvature of the tube. The use of

the axial positions of the layer lines to determine SWCNT

structure was first explicitly performed by Gao et al. in 2003

[19]. In this work, using simple geometrical arguments the

chiral angle was determined from the axial distances of the

layer lines from the equatorial line using the expression

h ¼ tan�1 1ffiffiffi3p d2 � d1

d3

� �¼ tan�1 1ffiffiffi

3p 2d2 � d3

d3

� �ð3Þ

with di the distance of the ith layer line from the equatorial

line. This analysis was further expanded by Liu et al. in 2005

[20] who introduced a technique to allow for the determina-

tion of the ratio of the chiral indices from measurements of

di [20,21].

mn¼ d2 � d1

d2 þ 2d1ð4Þ

mn¼ d3 � 2d1

d3 þ d1ð5Þ

mn¼ 2d2 � d3

2d3 � d2ð6Þ

As well as being relatively simple to perform, the use of the

ratio of the di distances enables the calibration-independent

analysis of the diffraction pattern. Furthermore, the use of

di ratios also ensures these analyses are robust against any tilt

of the axial plane of the SWCNTaway from normal to the inci-

dent electron beam.

Approximating the Ewald sphere to a plane (due to the

high energy of the electrons used in diffraction experiments),

simple geometric considerations show that tilting the tube

away from normal to the incident electron beam increases

the measured di distances by a factor of 1cos s, with s being

the axial tilt angle of the tube [22]. For very large tilt angles

(s > 45�) the planar approximation of the Ewald sphere will

introduce a systematic error approaching the order of magni-

tude of experimental uncertainties. However, for realistic val-

ues of s this approximation is sufficient.

An alternative technique to determine the chiral indices

from measurements of the di distances was introduced in

2007 by Jiang et al. [22]. The di distances are divided by the per-

iod of the equatorial oscillation, d to give the dimensionless

(and therefore calibration independent) parameters ni. The

chiral indices can be directly calculated using pairs of ni

parameters:

4964 C A R B O N 4 9 ( 2 0 1 1 ) 4 9 6 1 – 4 9 7 1

n12 ¼p

3ffiffiffi3p :ðn2 þ 2n1Þ; m12 ¼

p

3ffiffiffi3p :ðn2 � n1Þ

n13 ¼p

3ffiffiffi3p :ðn3 þ n1Þ; m13 ¼

p

3ffiffiffi3p :ðn3 � 2n1Þ

n23 ¼pffiffiffi3p :ð2n3 � n2Þ; m23 ¼

pffiffiffi3p :ð2n2 � n3Þ ð7Þ

Here the subscripts correspond to the pairs of di distances

used in the calculation with n12 = n13 = n23 = n and m12 = m13 =

m23 = m. The chiral indices calculated in this way, while being

independent of the calibration of the diffraction pattern, are

not independent of any tilt of the axial plane of the SWCNT

with respect to the incident electron beam. The measured

values of ni are therefore a factor of 1cos s greater than the real

values. Thus axial tilt will result in the calculation of non-

integer values for ni,j and mi,j.

2.2. Radial measurements

A more complete description of electron diffraction through

an individual SWCNT, first introduced by Qin in 1994 [13], is

obtained by considering the tube as a series of molecular heli-

ces [21]. Within this description the radial intensity distribu-

tion of each layer line is described by the square of a Bessel

function, the order of which is directly related to the chiral

indices of the tube:

I1ðXÞ / jJnþmðpD0XÞj2

I2ðXÞ / jJnðpD0XÞj2

I3ðXÞ / jJmðpD0XÞj2

I4ðXÞ / jJn�mðpD0XÞj2 ð8Þ

where Ii(X) is the radial intensity distribution of the ith layer

line, D0 the diameter of the SWCNT and X the reciprocal space

distance in the radial direction. Within this molecular helix

model the radial intensity of the equatorial oscillation, IEq

can be described by the square of a zero order Bessel function:

IEq / jJ0ðpD0XÞj2 ð9Þ

A zero order Bessel function J0(x) can, for x� 0, be approx-

imated to [41]:

J0ðxÞ ¼ffiffiffiffiffiffi2px

rcos x� x

4

� �ð10Þ

Eq. (10) is equal to zero at values of x ¼ ðj� 14Þp where j is an

integer. As in this case x = pD0X, the diameter of the SWCNT

can therefore be approximated from the period of the equato-

rial oscillations, d:

D0 ¼1d

ð11Þ

The oscillatory period of a Bessel function of a particular

order is unique. The ratio of peak separations is defined as:

RP ¼P1

P2ð12Þ

with P1 and P2 the separation of the first and second order

peaks, respectively (see Fig. 1). First proposed by Liu and Qin

in 2005, the order of the dominant Bessel function of a partic-

ular layer line can be determined from measurement of P1

and P2, calculation of RP and subsequent comparison with

theoretical values [23]. As the Bessel function order increases,

the difference between theoretical RP values decreases,

resulting in the need for very accurate measurement of P1

and P2 to assign high order Bessel functions.

This limitation was addressed in the subsequent work of

Jiang and co-workers in 2006 [24]. They suggest that high or-

der Bessel functions can be more accurately determined by

the use of the ratio of the first peak (P1) or minimum separa-

tion (M1) to the distance between the first two minima (D12,

see Fig. 1) [24]:

RPD ¼P1

D12ð13Þ

RMD ¼M1

D12ð14Þ

Any tilt of the axial plane of a SWCNT away from normal to

the incident electron beam results in a change in the posi-

tions of the peaks in the radial intensity distribution [24,25].

For high order Bessel functions this can be accounted for by

scaling the radial distance along each layer line using the

expression

X0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

i sin2 sþ X2q

Lkð15Þ

where di is the distance of the layer line in the axial direction,

s the axial tilt of the tube, k the wavelength of the incident

electrons and L the distance between the sample and the

imaging plane (the product Lk is generally referred to as the

camera constant of the microscope). However, as noted by

Jiang et al., for low order Bessel functions a tube tilt of as little

as 6� can cause the strongest peaks in the radial intensity dis-

tribution to merge, thus causing the Bessel function assign-

ment to fail [24].

In the literature the chiral indices of individual SWCNTs

have been robustly determined using each of the techniques

described above, as well as through comparison with simula-

tion [42–44] which we will not discuss here.

In the next section we will apply each of the analyses to

the same SWCNT diffraction pattern to enable direct

comparison.

3. Analysis of an experimental diffractionpattern

Carbon nanotubes were grown by chemical vapour deposition

(CVD) across a perforated silicon nitride TEM grid (SPI Supplies)

using a method described elsewhere [30]. Real space imaging

and SAED of individual, isolated carbon nanotubes lying across

holes in the TEM grid were performed in a Philips/FEI CM200

field emission gun (FEG) TEM fitted with a Gatan imaging filter

(GIF 200). The TEM was operated at 80 keV to avoid damaging

the SWCNTs as the threshold for knock-on damage is 86 keV

[45]. Magnification calibration of the image was performed

using graphite (0002) lattice fringes and of the diffraction pat-

tern using the gold and graphite reflections from a gold on car-

bon combined test specimen (Agar Scientific Ltd.) prior to

insertion of the carbon nanotube sample.

Fig. 2 shows a diffraction pattern and real space image of

an individual suspended SWCNT. The four principal layer

lines (L1–L4) are marked as is the equatorial line (Eq).

The tube diameter, as measured directly from the real

space GIF–CCD image, is 2.4 ± 0.5 nm. The large uncertainty

Fig. 2 – Selected area diffraction pattern from an individual

SWCNT and real space image (insert).

C A R B O N 4 9 ( 2 0 1 1 ) 4 9 6 1 – 4 9 7 1 4965

ascribed to this value is due to the under-estimation of tube

diameter caused by the contribution of phase contrast to

the image [46], coupled with the errors from imperfections

in the microscope, calibration and experimental factors such

as vibration and sample drift.

An alternative measure of the diameter of the SWCNT can

be found from analysis of the diffraction pattern. The period

of the equatorial oscillation is related to the tube diameter by

Eq. (11). As previously noted Eq. (11) is only approximately true,

however for a zero order Bessel function and for the values of

the argument, pD0X of interest (that is X in the range 0.2–

0.8 A�1 and D0 of the order of a few nanometres) the resultant

error introduced by using this approximation is less than

0.1%. This is, as we shall see, over an order of magnitude lower

than the experimental uncertainty in the measurement of d.

To extract the period of the equatorial oscillation an area

which completely encompassed all of the signal was selected

and the grey scale value averaged in the axial direction. The

positions of eight minima (four either side of the central sat-

uration) were then determined by manual selection followed

by fitting of a third order polynomial to the data in the vicinity

of the chosen minima. The minima of the fitted polynomial

were taken to be the position of the data minima. The period

of the equatorial oscillation was found to take a value of

d = 101 ± 2 pixels. Using a calibration of 1 pixel = 3.294 ·10�4 A�1 for the diffraction pattern, as determined from a

standard sample, the diameter of the SWCNT was found to

be D0 = 3.01 ± 0.06 nm. This value does not agree with that ob-

tained from direct measurement of the real space image.

However, both the direct diameter measurement from the

real space image and the diameter determined from the equa-

torial oscillation depend on the accurate calibration of the

real space image and the diffraction image, respectively. Both

of these calibrations are strongly dependent on the precise

imaging conditions used [47,48]. As such these measurements

could only be considered accurate if the imaging conditions

were identical to those used when calibrating with a standard

sample, a requirement which can be met (see for example

Ref. [49]) but is in general difficult to ensure in most experi-

mental situations.

3.1. Chirality assignment from axial measurements

Measurement of the di distances was performed by projecting

the intensity of the entire diffraction pattern onto the kz axis

(see Fig. 1). The peak positions were then determined by

manually picking then fitting each peak with a third order

polynomial as described previously. The distances (in pixels)

were determined to be d1 = 183 ± 3 px, d2 = 1009 ± 3 px and

d3 = 1195 ± 3 px.

From simple geometrical considerations the chiral angle

can be calculated using Eq. (3), and is found to be

h = 21.7� ± 0.2�. Combined with the diameter determined from

the period of the equatorial oscillation the tube is identified as

having the chiral indices (27,16).

Table 1 shows the possible chiral indices calculated from

Eqs. (4)–(6) which agree (within experimental uncertainties)

with the diameter measurement determined from either the

real space or diffraction pattern measurement. The two pos-

sible tube structures which appear in all three cases are

(20,12) and (25,15). Neither of these agrees with the (27,16)

structure determined from the combination of chiral angle

and tube diameter. Furthermore, the theoretical diameters

of both a (20,12) and a (25,15) SWCNT lie outside the error

bounds of that calculated from the equatorial oscillations

(D0 = 3.01 ± 0.06 nm). However the theoretical diameters

shown in Table 1 are calculated assuming a value for the car-

bon–carbon bond length of 0.142 nm. In reality this value is

not precisely known and is reported to lie between 0.142

and 0.144 nm [48]. Furthermore it has been shown that for

very small diameter carbon nanotubes the carbon–carbon

bond length can be stretched [50].

Analysis of the diffraction pattern in terms of only the di ra-

tios and calculated tube diameter is, in this case, insufficient to

unambiguously determine the chiral indices of this SWCNT.

We now proceed with an alternative analysis using a combina-

tion of di and the period of the equatorial oscillation d.

The possible chiral indices determined from Eq. (7) are dis-

played in Table 2. The data in Table 2 suggests that if this

analysis technique is considered in isolation, due to experi-

mental uncertainties, the chiral indices are not uniquely

defined.

The effect of any tilt of the axial plane of the SWCNT away

from the normal to the incident electron beam is to increase

the calculated mi,j and ni,j away from the expected integer val-

ues. Calculating the tilt angles for the possible combinations

of n and m from each ni reveals that only (n,m) = (25,15) gives

a consistent tube tilt for all ni. The calculated tilt of the SWCNT

away from the normal to the incident electron beam is s < 9�.For this particular diffraction pattern, when tilt of the

SWCNT is taken into account and the condition of consistent

tube tilt is applied the possible chiral indices are confined to a

single set.

3.2. Chirality assignment from radial measurements

The distances P1, P2, M1 and D12 were measured for each layer

line both above and below the equatorial line (marked in

Table 1 – The ratio of chiral indices calculated from the ratios of sums and/or differences of di, the axialpositions of the layer lines (Eqs. (4)–(6)). The corresponding possible chiral indices and theoretical tubediameters and chiral angles are also displayed. Only chiral indices which agree with the diameter determinedfrom either the real space or diffraction pattern measurements are included.

Layer lines mn range Possible Theoretical

n m D0 (nm) h (deg.) mn

d1, d2 0.601 ± 0.007 20 12 2.2 21.8 0.60025 15 2.7 21.8 0.60028 17 3.1 22.0 0.607

d1, d3 0.601 ± 0.008 20 12 2.2 21.8 0.60023 14 2.5 21.4 0.60925 15 2.7 21.8 0.60028 17 3.1 22.0 0.607

d2, d3 0.597 ± 0.004 20 12 2.2 21.8 0.60025 15 2.7 21.8 0.60027 16 2.95 21.6 0.593

Table 2 – Chiral indices calculated from the dimensionless ni

parameters (Eq. (7)).

Layer lines mi,j m ni,j n

n1, n2 14.8 ± 0.3 14 or 15 24.7 ± 0.3 24 or 25n1, n3 14.9 ± 0.3 14 or 15 24.7 ± 0.3 24 or 25n2, n3 14.8 ± 0.5 14 or 15 24.8 ± 0.5 24 or 25

4966 C A R B O N 4 9 ( 2 0 1 1 ) 4 9 6 1 – 4 9 7 1

Table 3 with subscripts a and b, respectively). The peak pick-

ing and fitting procedure outlined earlier was again used.

The calculated ratios RP, RPD and RMD (Eqs. (12)–(14)) are dis-

played in Table 3. The range of possible Bessel function or-

ders, OP, OPD and OMD are also displayed. These were

selected from comparison of the measured RP, RPD and RMD

with matching values from calculated Bessel functions.

For high order Bessel functions the theoretical RP values

converge, that is, the difference between RP of neighbouring

Bessel function order becomes very small. For Bessel orders

above 20 the difference between subsequent values of RP is

less the 1%. As such the assignment of chiral indices using

this technique is sensitive to experimental uncertainties.

The ratios RPD and RMD are more robust than RP at higher

Bessel order with a difference of �3% between subsequent

Table 3 – The ratio of peak positions, RP, RPD and RMD (as dof Bessel function orders, OP,OPD and OMD, from the radialLayer lines marked a lie above the equatorial oscillationthe beam blanker.

Layer line RP OP

1a 1.157 ± 0.006 37–40 161b 1.156 ± 0.006 38–41 15

2a 1.229 ± 0.009 22–23 11

3a 1.28 ± 0.02 16–17 7.43b 1.28 ± 0.02 16–18 7.3

4a 1.42 ± 0.02 9–10 5.84b 1.41 ± 0.02 9–10 5.7

values for a Bessel order of 20. However, the value of D12 is

typically more than 10 times smaller than P1, P2 or M1. The

greater relative uncertainty of the measured D1,2 compared

to that of P1, P2 or M1 negates the benefits of the larger spacing

between subsequent values of RPD and RMD over RP. As such all

three values give similar ranges of possible chiral indices,

with RP being the more accurate at larger Bessel function or-

ders despite the convergence of theoretical values.

Due to experimental uncertainties, this analysis technique

clearly does not allow for the unambiguous determination of

the structure of the SWCNT.

3.3. Fitting to the radial intensity distribution

Bessel functions (encompassing the entire range of possible

chiral indices determined from the previous analyses) were

fit to the normalised radial intensity distribution of each layer

line. A Gaussian background was subtracted from each exper-

imental data set to account for the zero order beam. The

SWCNT diameter D0 is extracted from these fits. An exponen-

tial damping term is included in the fitting function to ac-

count for the fast fall-off in intensity of the experimental

data with increasing kx. The results of the fits to the layer

lines are shown in Fig. 3.

efined by Eqs. (12)–(14)) and the corresponding rangeintensity distribution of the four principal layer lines.and b below. The data for layer line 2b is obscured by

RPD OPD RMD OMD

.5 ± 1.0 39–45 18.1 ± 1.1 39–46

.9 ± 0.9 37–43 17.5 ± 1.0 37–44

.4 ± 0.8 23–27 13.1 ± 0.9 24–28

± 0.5 13–15 8.7 ± 0.6 12–15± 0.5 13–14 8.6 ± 0.6 12–14

± 0.5 9–10 7.3 ± 0.6 9–11± 0.5 9–10 7.2 ± 0.6 9–10

0.4 0.45 0.5 0.55 0.60

0.2

0.4

0.6

0.8

1

kx (Å−1)

Nor

mal

ised

inte

nsity

L1aO=39O=40O=41

0.25 0.3 0.35 0.4 0.450

0.2

0.4

0.6

0.8

1

kx (Å−1)

Nor

mal

ised

inte

nsity

L2aO=24O=25O=26

0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

kx (Å−1)

Nor

mal

ised

inte

nsity

L3aO=14O=15O=16

0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

kx (Å−1)

Nor

mal

ised

inte

nsity

L4aO=9O=10O=11

(a) (b)

(c) (d)

Fig. 3 – Fits of Bessel functions to the experimental radial intensity distribution of the four principal layer lines. The order of

the Bessel function (O) which describes L1 corresponds to the sum of the chiral indices n + m, that which describes L2 to the

chiral index n, L3 to m and L4 to n �m. Only positive kx is displayed for clarity and only the ‘a’ layer lines for brevity.

C A R B O N 4 9 ( 2 0 1 1 ) 4 9 6 1 – 4 9 7 1 4967

Inspection of the positions of the first two peaks of the fits

for the four layer lines show they are almost indistinguish-

able. This illustrates why the calculations of RPD, RMD and RP

(which rely on the accurate determination of the positions

of the first two peaks and/or minima) are so sensitive to

experimental uncertainties.

In the fits of Figs. 3c and d the separation between the peak

position of subsequent order Bessel functions becomes great-

er at larger peak numbers (higher kx values). In Fig. 3c only the

fit for Bessel function of order 15 remains in phase with the

experimental data for values of kx J 0.3, with the Bessel

functions of order 14 and 16 becoming clearly out of phase.

Similarly for Fig. 3d only the Bessel function of order 10 re-

mains in phase at larger kx. This is not so evident in the fits

to layer lines L1a and L2a (Fig. 3a and b) due to the few resolv-

able peaks. The same conclusions are drawn from analysis of

the ‘b’ layer lines (not shown here).

Referring to Eq. (8), the order of the Bessel function which

describes L3 is equal to the chiral index m and that which de-

scribes L4 equal to n �m. Therefore the analysis of these two

lines alone completely describes the tube structure as a

(25,15) SWCNT. The fits to L2 and L1 (the Bessel function or-

ders of which are equal to n and n + m, respectively) support

this analysis, although it is not possible to conclusively assign

unique Bessel orders to these lines.

Table 4 shows the diameters extracted from the fits to the

various layer lines. The consistency of the diameters for

L1 = 40, L2 = 25, L3 = 15 and L4 = 10 supports the chiral index

assignment. The theoretical diameter for a (25,15) SWCNT

calculated using Eq. (1) is 2.74 nm and the theoretical chiral

angle equal to h = 21.8�. The discrepancy of the diameters

returned from the fitting procedure compared to the theoret-

ical diameter is due to either the incorrect calibration of the

diffraction pattern or due to tilt of the tube with respect to

the incident electron beam (see Eq. (15)).

As noted by Kociak et al. in 2003 [48] once the chiral indices

have been determined the diffraction pattern can be accu-

rately self-calibrated. Here we re-calibrate the diffraction pat-

tern so that the inverse of the period of the equatorial

oscillation is equal to 2.74 nm, the theoretical diameter of a

(25,15) SWCNT. Using the re-calibration, the fits for the Bessel

functions corresponding to a (25,15) tube give diameters in

the range 2.72–2.81 nm.

The chiral angle of 21.7 ± 0.2�calculated from the di dis-

tances agrees with the theoretical value for a (25,15) SWCNT

within experimental uncertainties.

From trigonometric considerations the theoretical di

distances can be simply calculated given h and the reciprocal

lattice vector of graphene. We calculate d1 = 0.67 nm�1,

d2 = 3.69 nm�1, and d3 = 4.36 nm�1. Measured directly from

the experimental diffraction pattern using the re-calibration

the di distances were d1 = 0.66 ± 0.02 nm�1, d2 = 3.64 ±

0.07 nm�1 and d3 = 4.32 ± 0.08 nm�1, in good agreement with

the theoretical values.

4. Multi-wall tubes and ropes

The diffraction pattern obtained from a multi-wall carbon

nanotube (MWCNT) or a rope of tubes can be considered as

the superposition of the diffraction pattern through multiple

SWCNTs. Provided the layer lines from the component dif-

fraction patterns do not coincide the analysis can proceed

Table 4 – Tube diameter determined from fitting of the layer line radial intensity distribution to Bessel functions. The data forL2b is obscured by the beam blocker.

Layer Order D0 (nm) Layer Order D0 (nm)

L1a 39 2.95 L1b 39 2.9240 3.02 40 2.9841 3.08 41 3.05

L2a 24 2.91 L2b – –25 3.01 – –26 3.12 – –

L3a 14 2.84 L3b 14 2.8215 3.01 15 3.0116 3.18 16 3.16

L4a 9 2.82 L4b 9 2.8410 3.06 10 3.0811 3.30 11 3.32

4968 C A R B O N 4 9 ( 2 0 1 1 ) 4 9 6 1 – 4 9 7 1

as for the single wall case. The layer lines can be easily sepa-

rated into those coming from each tube since the simple rela-

tionships [21]

d3 ¼ d1 þ d2

2d3 ¼ d2 þ d4 ð16Þ

must be satisfied.

Analysis of the equatorial oscillation, however, deviates

from the single wall case as there will always be a contribu-

tion from all walls. To illustrate this we will consider the

equatorial oscillation of a diffraction pattern taken from a

double wall carbon nanotube (DWCNT) shown in Fig. 4. The

intensity along the equatorial line oscillates with a period

equal to e ¼ 1=D within an envelope period E = 1/dD, where D

is the mean diameter of the two tubes and dD the interlayer

distance [26].

The DWCNT show in Fig. 4 is found to be a (30,17) tube

nested inside a (31,26) tube with an inter-wall spacing of

0.36 nm. In general DWCNTs have been found to have an in-

ter-wall spacing slightly higher than the interlayer spacing in

bulk graphite of 0.335 nm [42,48].

The procedures described in Sections 2.1 and 2.2, com-

bined with the modified equatorial oscillation analysis, have

been successfully applied to the study of MWCNTs [20,51,52]

Nor

milis

ed In

tens

ity

Fig. 4 – Left, selected area diffraction pattern from an individual

profile of the right half of the equatorial line.

and ropes containing multiple tubes [53,54]. In general the

use of several complimentary analyses is used for increased

confidence in chirality determination [52] as we have also

illustrated here. Interestingly the study of inner tubes of

MWCNTs has shown the length of the carbon–carbon bond

in the radial direction is stretched from that of the graphene

lattice, while in the axial direction it remains as expected

[52,50].

5. Dealing with deformation

Up to now we have considered only the case in which the

SWCNT is isolated and not subject to any external forces

which could lead to deformation of the tube. Any deformation

would in turn alter the diffraction pattern and lead to possible

mis-assignment of the chiral indices. Three types of deforma-

tion have been considered in the literature: elliptical deforma-

tion, asymmetric deformation and torsional twisting.

The effect of elliptical deformation was first tackled by Liu

and Qin in 2005 [55]. Elliptical deformation causes breaking of

the mirror symmetry across the tube axis when diffraction

patterns are taken at a non-zero tilt angle [55,56]. The amount

of deformation, the eccentricity, can be extracted from com-

parisons of diffraction patterns with simulations. As elliptical

−0.8 −0.6 −0.4 −0.2 00

0.2

0.4

0.6

0.8

1

kx (Å−1)

2Ee

DWCNT and real space image (insert). Right, radial intensity

C A R B O N 4 9 ( 2 0 1 1 ) 4 9 6 1 – 4 9 7 1 4969

deformation does not alter the structure of the SWCNT in the

axial direction analyses based on measurements along this

axis should still be valid. Indeed elliptical deformation has a

negligible effect on the order of the dominant Bessel function

which describes each layer line, but does alter the value for

the diameter of the SWCNT (D0 in Eq. (8)). In Ref. [55] analysis

of a diffraction pattern taken from a portion of a SWCNT on

which amorphous carbon has been adsorbed is performed

and shows the expected symmetry breaking consistent with

elliptical deformation.

Following a procedure introduced by Zhang and Zuo, we

evaluate the axial asymmetry of the diffraction pattern shown

in Fig. 2 in the following way [52]. A Gaussian is fitted to each of

the peaks of the experimental radial intensity distribution of a

layer line in order to obtain a radial peak position, XG. The posi-

tions of the peaks from the Bessel function fit to the layer lines

XB is also determined. The average deviation between the

experimental and Bessel fit peak positions (XG � XB) on the left

(dL) and right (dR) of the layer line is calculated. The difference

between these average deviations, dL � dR, is taken as a mea-

sure of the axial asymmetry of the layer line.

Evaluating layer line 4a (Fig. 3) in this way gives

dL � dR = 0.00 ± 0.01 A�1, indicating that, within experimental

uncertainties, the SWCNT analysed in Section 3 has not

undergone any elliptical deformation.

In Ref. [57] Jiang et al. tackle the problem of asymmetric

deformation of the tube cross-section, that is flattening of

one side of a SWCNT as opposed to elliptical deformation.

By comparison of simulated diffraction patterns from de-

formed and non-deformed SWCNTs they show that asym-

metric radial deformation leads to alteration of the radial

intensity distribution resulting in the possibility of incorrect

chirality assignment when using Bessel function based tech-

niques. As stated previously, the axial position of the layer

lines is not affected by radial deformation. The authors apply

their model to an experimental diffraction pattern taken of

two interacting SWCNTs and show that the intertube van

der Waals forces act to flatten the contact side of both tubes.

Recently Kim and co-workers have performed combined

electronic transport and diffraction experiments on small

bundles of SWCNTs indicating a change in electronic struc-

ture due to intertube van der Waals deformation [29].

The form of an electron diffraction pattern from a SWCNT

under torsional strain has been tackled by Liu and Qin [58].

Deformation of a SWCNT by torsional twisting results in a

modification to the spacing of the principal layer lines in

the axial direction. The Bessel function analysis is not sensi-

tive to torsional twist of the tube. In fact using this analysis

the handedness of a carbon nanotube can be determined by

the direction of the shifts in the axial positions of the layer

lines. The twisting angle Dh can be written as

Dh2p¼ f0 � f

f� nm

ð17Þ

where f0 = (2n + m)/(n + 2m) and f = d3/d2.

For the experimental diffraction pattern analysed in Sec-

tion 3 (Fig. 2) we would not expect any torsional twisting as

the SWCNT should not be under any external stresses. In-

deed, the good agreement between the theoretical di and

the experimental di demonstrated at the end of Section 3

rules out the possibility of any torsional twisting of the tube.

Using a nano-manipulator situated inside a TEM, Hirahara

et al. have recently shown that the application of tensile

stress can cause the decrease in the chiral indices of a carbon

nanotube [31]. By passing a current through the tube the acti-

vation energy for plastic deformation is overcome and, using

electron diffraction analysis, the authors demonstrate a de-

crease in the chiral indices of both the inner and outer wall

of a DWCNT by (1,1). They report similar changes in single

wall and triple wall carbon nanotubes.

6. Concluding remarks

In this article we have outlined the state of the art in diffrac-

tion pattern analysis. By applying each technique to the same

experimental data we have performed a direct comparison

and shown that many of the analyses, when used in isolation,

are insufficient to ensure unambiguous determination of

SWCNT chirality.

The chiral indices of the SWCNT were uniquely deter-

mined by just two of the analyses. Firstly, based on measure-

ments of the axial positions of the layer lines the technique

proposed by Jiang et al. [22] (in which the dimensionless ni

parameters are calculated, see Eq. (7)) was able to determine

the chiral indices of the SWCNT uniquely, provided that tilt

of the tube with respect to the incident beam was considered.

Secondly, as the experimental data was of sufficient quality to

resolve numerous oscillations, the extraction of Bessel func-

tion order from fits to the radial intensity distribution of the

layer lines was also able to uniquely determine the chiral

indices of the SWCNT.

The determination of the same unique chiral indices from

two analyses based on independent measurements combined

with the agreement, within experimental uncertainties, of all

the other analyses allows us to be confident of the assign-

ment of SWCNT chirality.

The most auspicious approach to chirality characterisa-

tion is clearly to perform both axial and radial measure-

ments on the diffraction pattern and ensure consistency in

the chiral indices determined over all techniques as demon-

strated here.

The effect of deformation of SWCNTs on the form of the

diffraction pattern has recently been addressed both theoret-

ically and experimentally. The deformation of SWCNTs can

lead to incorrect chirality assignment, particularly if a single

analysis is relied on. For the experimental data presented in

this review we have shown that the isolated SWCNT is not af-

fected by either elliptical deformation or torsional strain and

therefore the chirality assignment can be considered robust.

When analysing electron diffraction patterns of ropes con-

taining multiple tubes deformation has been shown to occur

and affect chirality assignment [29,57].

Advances in the incorporation of carbon nanotubes into

nano-electromechanical devices [59,60] and the investigation

of the changes in of their electrical properties when under

strain [29] can now be correlated to changes in the atomic

structure of tubes as determined by careful analysis of elec-

tron diffraction patterns.

4970 C A R B O N 4 9 ( 2 0 1 1 ) 4 9 6 1 – 4 9 7 1

Acknowledgements

The authors would like to thank EPSRC for financial support.

C.S.A. acknowledges funding from the QIP IRC and is grateful

to Dr. M.B. Ward for invaluable help with the operation of the

TEM.

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