arXiv:cond-mat/0412161v1 [cond-mat.other] 7 Dec 2004
On the diffraction pattern of C60peapods.
J. Cambedouzou1, V. Pichot2, S. Rols1, P. Launois2, P.
Petit3, R. Klement3,4, H. Kataura5and R. Almairac1
1Groupe de Dynamique des Phases Condens´ ees (UMR CNRS 5581),
Universit´ e Montpellier II, 34095 Montpellier Cedex 5, France
2Laboratoire de Physique des Solides (UMR CNRS 8502),
Universit´ e Paris Sud, 91405 Orsay
3Institut Charles Sadron, 67000 Strasbourg, France
4Department of Physical Chemistry,
Faculty of Chemical and Food Technology,
Slovak Technical University, Radlinsk´ eho 9,
812 32 Bratislava, Slovak Republic and
5Nanotechnology Research Institute,
National Institute of Advanced Industrial Science and Technology (AIST) Central 4,
Higashi 1-1-1, Tsukuba, Ibaraki 305-8562, Japan
(Dated: February 2, 2008)
We present detailed calculations of the diffraction pattern of a powder of bundles of C60peapods.
The influence of all pertinent structural parameters of the bundles on the diffraction diagram is
discussed, which should lead to a better interpretation of X-ray and neutron diffraction diagrams.
We illustrate our formalism for X-ray scattering experiments performed on peapod samples syn-
thesized from 2 different technics, which present different structural parameters. We propose and
test different criteria to solve the difficult problem of the filling rate determination.
PACS numbers: 61.46.+w,61.10.Dp,61.10.Nz
Since their discovery in 1991 , carbon nanotubes have been the purpose of a large
number of studies, dealing both with their mechanical and electronic properties.In
particular, it has been shown that the intercalation of electron donors or acceptors [2, 3, 4]
into single wall carbon nanotubes (SWNT) bundles could dramatically modify the electronic
properties of these objects. Rather large molecules are expected to be inserted into the
hollow core of a nanotube that has been shown to present very stable adsorption sites
. C60 is one of the molecules that have successfully been inserted into SWNT, and a
lot of studies have recently been achieved on the so-called ”peapods”. Those systems
consist of SWNT in which C60 fullerene molecules are inserted . Their study stands
within the fascinating field of systems in a confined geometry [7, 8, 9, 10]. Peapods
structural analysis can be performed on a small number of tubes (and even on a single
tube), using transmission electron microscopy (TEM) [6, 11] or electron diffraction [12, 13],
or on macroscopic assemblies, using Raman spectroscopy [11, 14, 15] or X-ray scattering
[15, 16, 17].
Theoretical and experimental studies have already been published on the diffraction dia-
gram of powder of SWNT bundles [18, 19]. In particular, the importance of the distribution
of tube diameter on the position of the (10) Bragg peak in the X-ray and neutron diffrac-
tion patterns was pointed out in ref. [19, 20]. The complete study of the influence of all
structural parameters of the bundles was performed using a simple numerical model. It was
shown that modeling is essential for a correct determination of the structural parameters for
such inhomogeneous samples of fairly crystallized nano-objects. Intercalated SWNT bundles
form even more complex nano-crystalline systems. The adsorption sites can be separated
into 3 main locations: inside the tubes, on the outer surface of the bundles (including the
so-called ”grooves”) and into the interstitial channels of the 2D triangular lattice. Ad-
sorption of a molecule into a SWNT bundle can involve modifications of the 2D triangular
lattice (symmetry and/or lattice parameter expansion). These modifications lead to site-
dependent diffraction diagram governed by the structure of the host (the nanotube bundle),
the structure of the adsorbed species inside the bundles and by crossed interferences between
the host and the molecules adsorbed. The modifications of the diagrams are also found to be
radiation dependent . Therefore, an efficient and correct interpretation of the diffraction
diagram from such complex systems requires the use of simulation.
The study of the structure of C60(and C70) peapods by X-ray diffraction has recently been
achieved by Kataura, Maniwa and co-workers [15, 16]. The authors could estimate the C60
(and C70) filling rates from the analysis of their measurements. Also, the 1D lattice con-
stant of the C60chains inside the tubes was found to be smaller than those for 3D crystals
of C60. More strikingly, the temperature dependence of the corresponding feature in the
diffraction diagram shows no dependence, indicating a nearly zero thermal expansion of the
C60chain inside the tubes, raising questions about possible polymerization of the C60chains.
X-ray diffraction is found to be a very powerful tool to probe the structure of C60inside
the tubes. However, further experimental work is needed to understand the properties of
C60peapods. Such experimental work should include diffraction investigations on samples
showing peapods having various structural characteristics:
1. different tube diameters
2. different filling rates
3. bundles with various sizes e.g. various numbers of tubes
and particularly, to allow variable degrees of freedom for the fullerene molecules. Therefore,
in this paper, we propose to give a detailed ”step by step” and complete study of the diffrac-
tion patterns of peapods and we discuss the characteristic signatures linked to the insertion
of C60inside the SWNTs. We also show how the variation of different structural parameters,
such as tube diameter, filling rate of SWNTs by C60, and C60adsorption in the outer groove
sites at the surface of the bundles can change the shape of the diffraction pattern. The
reader has to keep in mind that this report is an attempt to give experimentalists the nec-
essary tools to characterize their samples by X-ray and/or neutron diffraction. Very often
it appears that unexpected impurity phases are present in SWNT samples, as revealed by a
comparison between X-ray and neutron diffraction patterns measured on the same powder.
Therefore both techniques are complementary. In the first part of this paper, we present the
theoretical model used to achieve the simulations. We consider a powder of uncorrelated
tubes filled with C60in the second part, and the third part deals with powder of bundles of
peapods. Polymerization of the C60chains is also considered. We finally compare the results
of our calculations with experimental diffraction patterns. A large part of the discussion is
concerned with the determination of the filling rate in the investigated samples.
II.PRINCIPLE OF THE CALCULATIONS
In our attempt to reproduce the diffraction pattern of peapods, we developed a model
based on the general equations for x-ray and neutron diffraction in the kinematical approx-
imation [22, 23] for which the diffracted intensity is proportional to the squared modulus of
the scattering amplitude. The latter is defined as the Fourier transform of a configuration
of atoms in the scattering volume. The diffracted intensity thus writes as follows:
fsρ(? r)ei?Q·? rd3? r
where?Q is the scattering vector, ρ(? r) is the density of scatterer at position ? r in the
sample, and fsdepends both on the scattering element (e.g the atomic species) and on the
nature of the incident radiation (fsis a function of the wave-vector modulus Q for X-rays
while it is constant for neutrons). Since we consider the case of a powder experiment, the
diffraction is to be averaged over all the directions of space or, equivalently, over all the
orientations of the scattering vector in the reciprocal space. The mean diffracted intensity
is thus given by:
where the integration is performed over the sphere of radius Q and d2S(?Q) is a surface
element of this sphere. Therefore, the model consists in choosing a convenient -and physically
correct- mathematical form of the density of scatterer ρ(? r). A powerful approach consists
in replacing the discrete carbon atoms by uniformly charged surfaces for both nanotubes
and C60molecules, with a surface atomic density σc∼ 0.37 atom/˚ A2. This value is a little
underestimated for C60molecules (∼ 0.39 atom/˚ A2) but we take it as equal for the simplicity.
This assumption results in a loss of information concerning the atomic arrangement at the
surface of the objects forming the sample. It limits the reliability of our results to Q-values
lower than 2˚ A−1. Below this value, the diffraction pattern is indeed insensitive to the
detailed atomic order. It is however very much affected in this Q range by the medium
range organization e.g. the 2D hexagonal nanotubes bundles and the 1D C60packing. In
the following, we will therefore be concerned with the two latter levels of organization in the
framework of the homogenous approximation.
The detailed parameters of the model and the definition of the different variables used
thorough this work are presented in Figure 1:
• The upper part represents a single tube filled with a linear chain of C60s. Each C60-
filled nanotube will be considered as a linear superposition of 1D unit cell, each cell
consisting of a cylinder of length L and of a single C60molecule located at its center.
• The lower part represents the peapods organized into bundles e.g. on a 2D hexagonal
lattice, the parameter of which is a function of the tubes diameter forming it. We
introduced a random shift Tzbetween the position of the C60molecules on one tube
with respect to the corresponding position on the central tube to avoid unrealistic
correlations between the positions of the C60s inside the different nanotubes of the
III.ISOLATED TUBES FILLED WITH A CHAIN OF C60 MOLECULES: ISO-
A. Complete filling of the nanotubes
In this part, we discuss the main features of the diffraction pattern calculated for a powder
of 380˚ A long and 13.6˚ A large peapods that is obtained by stacking 40 cylinders of length
9.5˚ A, each cylinder containing a 3.5˚ A radius sphere at its center (see Figure 1). This value
of 9.5˚ A is not deduced from measurements. It must be considered as a parameter of the
model. We present in figure 2 the calculated intensity for this object (bottom), for the tube
alone (top) and for a C60chain alone (middle).
Let Itbe the intensity diffracted by a powder of empty nanotubes. It is a pseudo-periodic
oscillating function which is proportional to the squared modulus of the zero order cylindrical
Bessel function J0, as it appears in the following expression of It:
with Atthe amplitude scattered by the empty nanotube:
At(?Q) = 2πLrhfsσcJ0(Qrhsin(u))sin(QL
and where u, L and rhare defined in figure 1 and where n labels the 1D unit cells (cylinders
of length L).
Let Icbe the intensity diffracted by a powder of linear chains of C60molecules:
where Acis the amplitude scattered by the linear chain of C60molecules:
Ac(?Q) = 4πr2
and where rC60is the radius of a C60molecule (3.5˚ A). This intensity is also a pseudo-periodic
oscillating function, but with a pseudo-period that is nearly twice as long as the one of the
nanotubes.Indeed, the radius of a C60 is nearly half the radial dimension of the tube.
Another interesting feature of the latter curve is the asymmetric peak at 0.68˚ A−1, which
results from the periodic positions of C60s along the linear chain .
The first minimum of the intensity diffracted by the whole peapod (pointed by an arrow in
fig.2) is located at Q=0.44˚ A−1, whereas it was located at 0.38˚ A−1in the empty SWNT
diffraction pattern. Following the work on gas adsorption inside SWNT by Maniwa et al.
, H. Kataura et al. gave a simple explanation for this feature : the reason for the
presence of this minimum at this precise value of Q is due to the sum of the structure factors
of the tube and of the C60which equals zero for Q=0.44˚ A−1. We will discuss this effect in
slightly different term. Let us write the intensity Ip(Q) diffracted by a peapod as follows:
? ????At(?Q) + Ac(?Q)
According to the fact that Atand Acmay be complex, relation ( 7) can be developed as:
Ip(Q) ∝ It(Q) + Ic(Q) + 2
Re(At(?Q))Re(Ac(?Q)) + Im(At(?Q))Im(Ac(?Q))
where Re(At) and Im(At) stand for the real and imaginary parts of At, respectively.
Relation (8) can be rewritten as:
Ip∝ It+ Ic+ ICI
If attention is given to figure 2c, it is clear that the profile of Ipis quite different from (It+Ic).
This latter quantity is indeed the signature of a sample containing a tube decorrelated from
a linear chain of C60. In a peapod sample, there is a strong correlation in the relative
positions of the tube and the chain of C60, since the C60s are located inside the tube.
This correlation is revealed by the presence of the crossed interference term ICIwhich can
be positive or negative. At Q=0.44˚ A−1, the crossed term compensates the (It+Ic) term,
lowering Ipto zero or nearly zero . This value for Q is accidently the same as that of the
(10) peak position for a bundle made of empty SWNTs stacked into a 2D hexagonal lattice.
This coincidence will cause dramatic changes in the diffraction pattern relative to peapods
bundles (see part IV).
The increase of the tube diameter induces a decrease of the oscillation period as it is expected
when considering larger diffracting object (see figure 3). The C60periodicity characteristic
peak at 0.68˚ A−1, as well as its first harmonic at 1.36˚ A−1, are observable in all the calculated
diffraction profiles. For peapods of radius r=5.42˚ A (for example C60@(8,8) peapods) the first
peak is not separated from the first oscillation, but is clearly visible. The same behavior is
observed for r=8.1˚ A peapods (for example C60@(12,12) peapods) on the second oscillation,
but the peak amplitude is weaker. In fact, the larger the tube diameter and the weaker this
peak. The latter effect can be easily understood: an increase of the tube diameter implies
an increase of the tube surface which leads to a preponderance of the response of the tubes
over the response of the C60chains, in which the number of scatterers remains unchanged.
The cases of C60@(8,8) and C60@(12,12) peapods are discussed here as extreme cases for the
influence of the tube diameter. However, it must be pointed out that the insertion of C60
in a (8,8) peapod is unlikely due to the too small diameter of the tube, and C60s inside a
(12,12) peapod would possibly lead to zigzag or helical chains in place of linear chains, since
the center of the tube is no longer the most favorable location for C60regarding to van der
The bottom part in figure 3 reveals the consequences of a change in the inter-C60length L
inside the linear chain. A downshift of the characteristic peak of C60s periodicity is evidently
observed when L increases, but we also note a reinforcement of its intensity. One remarks a
striking effect: although the relative density of C60increases, the characteristic peak relative
to the C60 periodicity strongly decreases. This is due to the multiplication factor arising
from the intensity diffracted by a C60molecule, which reaches its first zero at 0.9˚ A−1, as
shown in figure 2. The closer the peak position to this value, the weaker the peak. One
must be careful with the fact that the respective proportion between tube and C60s is no
longer the pertinent parameter to account for the explanation of the changes in the peak
intensities. Thus, it may prove dangerous to focus only on the relative intensity of this peak,
and it seems necessary to consider the whole diffraction pattern to derive reliable structural
In several cases, the use of an analytical formula can prove more comfortable than the method
presented above. For infinite tubes, numerical calculations are performed after the limit of
infinite tube length was taken in the above equations, leading to the following equation
(demonstrated in appendix 1 eq. 16), where the first term comes from the Fourier transform
of the structure projected in a plane perpendicular to the tube axis, while the second one
comes from the periodicity of the C60chains:
(2πLrhσcJ0(Qrh) + 4πr2
In this expression Int(QL
2π) is the integer part of (QL
B.Partial filling of the nanotubes
In the case of a real sample, it seems reasonable to consider that all tubes may not be
fully filled with C60s, despite efforts made to obtain filling rates as high as possible .
Two different hypotheses are discussed here.
The first hypothesis is to consider a random filling of the tubes without assuming clustering
effects inside the tubes : the molecules are randomly positioned within a tube, the only
constraint being a minimum distance of L between them (one may note that it implies
that in the limiting case of 100% filling, the molecules necessarily form ordered chains).
Calculations are performed within the finite tube length model, for a given filling ratio of
the nanotubes. The effects of a random incomplete filling of the nanotubes are presented in
the upper part of figure 4. One observes the vanishing of the C60-C60characteristic peak with
decreasing filling rates; it completely disappears for filling rates below 85%. If we compare
the shapes of the 85% and the 100% diffraction profiles, one finds that the minimum at
0.44˚ A−1for the 85% filled sample no longer lowers to zero. The additional intensity can
be attributed to the effect of the disorder induced by the random filling of the nanotubes
by the C60molecules. Moreover, the first minimum in intensity shifts to lower Q values for
decreasing rate of fullerenes, which can be explained as in figure 2 by compensation effects
between tube, fullerene and interference terms.
As a second hypothesis, one can consider the partial filling of the tubes with long (’quasi-
infinite’) chains of C60. We assume here that the molecules tend to cluster within nanotubes.
Indeed, this should correspond to a low energy configuration of the system, the energy being
lowered by the attractive C60-C60interactions. This hypothesis is supported by observations
reported in ref. . Calculated diffraction patterns for different filling rates (50%, 75% and
100%) are presented in the lower part of figure 4. Here we use the infinite tube length model
(detailed calculations are given in appendix 2). As it was already mentioned in the first
hypothesis, we observe the vanishing of the C60-C60characteristic peak at low filling rates,
but it interestingly disappears here for a filling rate of 50% which is much lower than the 85%
observed in the first part. This is due to the long chains assumption : the C60-C60distance
is preserved for the different filling rates. Another feature of these diffraction patterns is
that the first minimum goes to zero for all filling rates, contrarily to what was found within
the first hypothesis, which is due to lower disorder.
If one takes into account the differences pointed out between the 2 hypothesis discussed
here, one should in principle be able to determine the way a sample of isolated peapods is
filled. Unfortunately, the Q range around 0.44˚ A−1is usually perturbed by parasitic signals
(intense scattering at small wave-vectors ), so it is very difficult to use the value of the
first minimum of intensity as a clue to determine the filling mode. As a result, the C60-C60
characteristic peaks remain the only observable features for the estimation of the filling rate
and of the filling mode. An important result from our calculations is that C60molecules in
isolated peapods with a random filling rate below 85% and peapods with a long chain filling
rate under 50% are undetectable by diffraction.
C.Polymerized C60molecules inside nanotubes
One of the remarkable properties of the C60 molecule is its ability to form covalent
bonds through quite different routes. In crystalline C60, polymers can be obtained (i) by
photopolymerization, (ii) under pressure and at high temperature , (iii) in doped samples
[27, 28]. Photo-induced [11, 15] and charge transfert induced  polymerization have also
been demonstrated in C60peapods, using Raman spectroscopy. It is interesting to consider
the effect of polymerization on diffraction patterns. Let us consider peapods where the chains
can be formed of n-polymers (n=1: monomers, n=2: dimers, n=3: trimers...). For the sake
of simplicity, we deal with the case of completely filled nanotubes. The distance Lbbetween
bonded C60molecules is taken to be 9.2˚ A as in crystalline polymerized samples, and the
distance L between unbonded molecules is fixed at 9.5˚ A as above. The distance between
monomers is smaller in peapods than in crystalline C60, where it is equal to about 10˚ A,
possibly because of interactions with the nanotubes. However, there is no reason to take a
smaller value for the distance between bonded molecules because it is mainly determined by
the covalent bonding between them. It is shown in appendix 3 that the scattered intensity
of a powder of peapods filled with chains of n-polymers writes:
Q((2πrh(L + (n − 1)Lb)σcJ0(Qrh) + 4πr2
+2(1 − δM,0)
sin(kπnLb/(L + (n − 1)Lb))
sin(kπLb/(L + (n − 1)Lb)))2)(11)
where the second term appears only for Q values such that M -the integer part of
each Q = k2π/(L+(n−1)Lb) value (with k integer), asymmetric peaks characteristic of the
) - is not zero. Calculated patterns for n=1, 2 and 3 are drawn in fig.5. At
chain periodicity can be observed. Despite the change of the period -equal to (L+(n-1)Lb)
- with n, the spectra look quite similar. It can easily be explained from eq. 11 : the asym-
for k=n, 2n,... The first intense peak of the n-polymer diffraction pattern is thus located
metric peak intensity is multiplied by
, which is close to zero except
at Q=n2π/(L + (n − 1)Lb) : Q=0.672˚ A−1for dimers, 0.676˚ A−1for trimers, to be com-
pared with 0.661˚ A−1for monomers. The upper value, corresponding to infinite polymers, is
Q=2π/9.2 =0.683˚ A−1.
In summary, scattering analysis of polymerization of C60 molecules in peapods samples
should be based on a careful study of the position Q0of the first intense asymmetric peak,
and on the search for lower intensity peaks at kQ0/n (n=2,3...; k=1 to (n-1)) to identify
IV. BUNDLES OF PEAPODS.
A. Complete filling
Peapods are packed into bundles where they are maintained together by van der Waals
inter-tubes interactions. This organization is clearly visible on TEM pictures [11, 15, 16]. In
this part, we calculate and discuss the diffraction pattern for such objects. Calculations are
performed both for peapods of finite length and for peapods of infinite length (as detailed
in appendices 1 and 2). The relative positions of the C60chains along the tube axes (Tzin
fig.1) are assumed to present no correlation from one tube to another. Indeed, a C60chain
interacts the most with the nanotube in which it is located, and nanotubes within a bundle
present different helicities .
Figure 6 shows the comparison between the diffraction profiles calculated for a powder of
bundles of 12 empty nanotubes and that of bundles of 12 peapods, with different tube radii.
In all cases, we consider 380˚ A long nanotubes organized on a 2D hexagonal lattice with a 3.2
˚ A van der Waals length between 2 adjacent nanotubes. Let us first consider the upper part
of figure 6, which deals with (10,10) tubes and peapods, of radius r=6.8˚ A. For peapods,
the additional peaks characteristic of the 1D periodicity of the C60 chains are indicated
by an arrow. We remark that most of the characteristic peaks observed in the diffraction
pattern of the empty SWNT bundles show up in the diffraction pattern of the peapod
bundles, except those located in the low Q range where a lack of intensity is obtained for the
peapods. An important difference between the pattern of the peapod bundles and the empty
nanotube bundles is thus the disappearance of the (10) Bragg peak at 0.44˚ A−1. This peak
(of finite width because of the small bundle size) is replaced by a minimum delimited by two
smaller peaks. This can be explained on the basis of part III results, since we observed that
the intensity diffracted by a single peapod lowers to zero at 0.44˚ A−1, which is around the
position of the (10) lattice peak of the bundles. This important extinction process in the case
of SWNT bundles has already been found and discussed for different kinds of intercalated
molecules like gas molecules [15, 31] or in iodine doped nanotubes samples . Middle
and lower parts of figure 6 show that the extinction phenomenon is modified when the tube
diameter is changed. For example, we consider the case of bundles of 12 empty nanotubes
and 12 peapods of radius r=5.42˚ A ((8,8) tubes, middle part of figure 6) and r=8.1˚ A ((12,12)
tubes, lower part of figure 6). One can see that the (10) peak is not split into two parts for
the bundle of (8,8) peapods, but appears shifted to lower Q values. By contrast, this peak
completely disappears in the case of the bundle of (12,12) peapods. When tube is thinner
or larger in diameter, the progressive loss of the accidental adequate conditions implies the
extinction phenomenon to be lost.
It is important to be aware that the extinction can occur for values of Q that are slightly
different from the (10) peak position. In that case the (10) peak does not appear split, but
seems shifted because only one side of the peak is lowered. Therefore, a direct interpretation
of such apparent shift in terms of a change of the lattice parameter is inappropriate, and
direct conclusions about structural changes based on the observation of the (10) peak alone,
prove very hazardous. The analysis of peapods diffraction patterns is consequently not
straightforward and should be based on comparison between measurements and calculations.
B. Partial filling
As for isolated tubes, all tubes may not be fully filled with C60s. In this part, 4 different
filling modes are discussed.
The first case (case a) consists in a random filling, where each tube of the bundle is filled as
described in part III.B. Calculations are performed within the finite tube length model. In
the 3 other cases, which are treated within the infinite tube assumption (detailed calculations
are given in appendix 2), the C60s are all stacked into long chains, but the way these
chains are distributed into the bundles changes with the case. One can indeed consider an
homogeneous filling (case b), where the tubes are all filled with the same number of C60
molecules, or an inhomogeneous filling (case c), where the filling rate of each tube of the
bundle is slightly different. The last case to be discussed (case d) consists in a mix of fully
filled bundles and empty bundles (see the right part of figure 7).
We present the results for filling rates of 85% and 50% for bundles of 12 nanotubes of radius
r=6.8˚ A in the left part of figure 7. The main difference between all diffraction patterns
calculated for a 85% filling rate and all those calculated for a 50% filling rate is visible in
the low Q range. The (10) Bragg peak is indeed still splitted and almost invisible at the
85% filling rate, while it clearly reappears at the 50% filling rate. This observation can be
explained by the fact that the accidental conditions allowing the extinction of the (10) Bragg
peak are progressively lost when the proportion of C60decreases in the sample.
If more attention is given to the a), b), c) and d) diffraction profiles for a given filling rate,
other features linked to the different filling modes can be extracted from the figure. One
can first consider the Q range below 0.6˚ A−1. The intensity in this region is the least for
the b) configuration, which is the least disordered configuration. If we compare with the
inhomogeneous filling (mode c), and then with the random filling (mode a), we observe a
progressive increase of the intensity in the low Q range, corresponding to the progressive
increase of the disorder in the system, as already mentionned in part III.B. The d) case must
be considered separately from the other cases because the intensity in the low Q range is here
the sum of the intensities of 2 decorrelated systems ( full peapods and empty nanotubes).
One can also focus on the C60-C60characteristic peak. This asymmetric peak is clearly visible
on the diffraction profiles of the b), c) and d) filling modes, even for the 50% filling rate,
whereas it never appears at this rate for isolated peapods. In addition, this peak is invisible
at both filling rates in the a) filling mode. Thus, the observation of such a feature in an
experimental diffraction pattern should stand as the signature of a long chain organization
of the C60molecules inside the tubes.
C. Chains of C60molecules in the outer groove sites of the bundles
We finally discuss the effects induced by the filling of the channels located at the bundle
surface with linear chains of C60s , for bundles of various size (figure 8). All tubes have
a radius of 6.8˚ A. When we consider the diffraction pattern of a 6 peapods bundle where
the external channels have been completely filled, we note 2 main differences with regards
to a naked peapod bundle. First, the peak at 0.68˚ A−1is significantly reinforced due to
the additional C60 molecules in the external channels. Secondly, the intensity in the low
Q range as well as between 1 and 1.5˚ A−1appears strengthened. As it is shown in figure
2, the C60response is high in these Q ranges, so this effect is just due to the increase of
the C60proportion in the sample. It is evidently clear that the higher the number of tubes
in the bundle, the weaker the effect described just above. The reason for that lies in the
simple fact that volume grows faster than surface when the size of the bundle increases.
According to these results, we are able to conclude that an experimental diffraction pattern
with a high intensity in the 1 to 1.5˚ A−1Q range may be relative to a sample made of small
sized bundles, saturated with C60s in both the inner space of the tubes and in the external
D.Determination of the filling rate
In this section we try to define a method to determine the filling rate from experimental
data. Two criteria are proposed which are based on the analysis of the results obtained
from the model. As it was explained previously, the (10) Q range is strongly perturbed, so
this Q range is disregarded. The first criterion consists to remark that the peak centered
around 0.7˚ A−1contains two contributions (fig. 7): i) the response of the C60 lattice at
0.663˚ A−1and ii) the (20) reflection of the bundle lattice centered at 0.706˚ A−1. The
ratio of the two respective amplitudes is expected to vary as a function of the filling rate.
However two problems arise. The first one concerns the type of filling which is unknown.
The second concerns the background determination. We assume that the background can
be represented by a straight line between 0.6˚ A−1and 0.8˚ A−1as is shown in the inset of
fig. 9a. In fig. 9a the amplitude ratio is drawn as a function of the filling rate and for the
two extrem types of filling, i.e. random and homogeneous (a) and b) cases, respectively).
Cases c) and d) have intermediate behaviors, so they are not presented in this figure. Thus
for a given experimental amplitude ratio (on vertical axis) one gets a range for the filling
The second criterion consists to use: i) the amplitude of the C601-D lattice peak around
1.3˚ A−1as this peak arises in a Q range where the C60contribution is strong, and ii) the
amplitude of the (21) reflection of the bundles around 1.1˚ A−1as the C60contribution is
small in this Q range. The calculated diagrams of fig. 7 were analyzed with a 3 peaks model
from which we extracted the two amplitudes (fig. 9b). Again the background estimation
has a large incidence on the results.
The second criterion seems to be more reliable as it presents a smaller dependence on the
type of filling. It is of easier use. However it has been established for SWNT bundles
corresponding to diameters occuring in the case of electric arc or laser synthesis, i.e.
centered around 1.4 nm and with a small dispersion in diameter distribution (FWHM ≤ 2
˚ A ). In the next section these criteria are applied to the estimation of the filling rate in two
V.COMPARISON BETWEENCALCULATEDAND EXPERIMENTAL
Diffraction patterns were performed using a powder diffractometer equiped with a curved
position sensitive detector INEL CPS120 allowing one to measure simultaneously a range
of 2θ angles from 2 to 120◦. A wavelength of 1.542˚ A was used. We measured the diffrac-
tion pattern of two different peapod samples (powder sample or numerous small pieces of
bucky paper in glass capillaries). Sample A (figure 10a) was synthesized as follows : the
SWNT material was first soaked in concentrated nitric acid (HNO365%) and sonicated for
approximatively 30 min, then treated in boiling HNO3 at 140-150◦C for 4 hours. After
sedimentation of the acid treated SWNT material in distilled water, SWNTs were separated
from the acidic supernatant by centrifugation. The acid treated material was washed several
times with distilled water (several centrifuge-washing-decantation cycles) then washed twice
with ethanol. Finally, the SWNT material was dried under low pressure during overnight.
The dried acid treated and washed SWNTs were subsequently heated in air at 420-430◦C
for 30-40 min. After oxidation, the material was mixed with an excess of C60 powder in
a glass tube. The glass tube was sealed under high vacuum, then heated at temperatures
550-600◦C for 72 hours. After heating, the ampoule was cooled down to room temperature
and opened. The excess of C60was removed by washing the SWNT material with toluene.
Finally, the material was washed once with ethanol and dried under reduced pressure for
overnight. The preparation method of sample B (figure 10b) has been described previously
The diffraction patterns of these two samples look quite different. We note the presence of a
broad underlying structure from 1 to 2.2˚ A−1in the diffraction patterns of sample A (dotted
line in fig. 10a). Applying the two criteria defined in part IV.D to the determination of the
filling rate (after substracting the dotted line for sample A), one obtains 75 to 95% (using
the first method) compared to 85% (second method) for sample A, and 77 to 96% (first)
compared to 95% (second) for sample B (see the horizontal lines in fig. 9). However, as is
shown below, if the criteria can be applied to sample B where characteristic C60peaks are
clearly observed, they only give an estimation of a upper value of the filling rate for sample
Diffraction profiles were fitted by optimized calculations. Calculated diffraction patterns
were convoluted with a convenient resolution function in order to be compared with the
experimental profiles. We also introduced a distribution of tube diameter in the calculation.
The latter was considered to be of Gaussian shape [19, 20].
Concerning sample A, the comparison with the pristine SWNT powder allows us to put for-
ward the following main characteristics of the peapod diffraction profile: a weak (10) peak,
an enlargement of the second peak at its low Q-side (0.68˚ A−1) and additional intensity
around 1.3˚ A−1. We obtained the best fit of the diffraction pattern considering a structure
made of small bundles of 6 peapods, with a large tube radius distribution centered around
6.8˚ A and chains of C60s in the external channels of the bundles. Considering the 4 filling
modes described above, a very high filling rate implies a sharp peak at 0.68˚ A−1, standing
for the C60-C60periodicity. Such a sharp peak is not observed in the diffraction pattern, so
we introduced a distribution of C60-C60lengths from 9 to 10˚ A into the model, leading to
an improvement of the fit. This distribution could testify to the presence of mixed dimer,
trimer or n-polymers chains of C60into the tubes in this sample. However, if by-products
of the chemical treatment were present inside nanotubes with the C60molecules, this would
explain the decrease of the (1,0) peak intensity together with an enlargement of the C60
periodicity peak since periodicity would be perturbated. We should thus note here that on
the basis of the present results, the filling rate values determined here, above 75%, might be
Concerning sample B, the general shape of the diffraction profile is better fitted. Different
kinds of filling modes have been studied in order to give a reasonnable range for the filling
rate. If a d) filling mode is chosen, we obtain the plain line of figure 10b, and a filling rate
of about 75%. The model includes 30 tubes per bundle with a very narrow distribution of
tube radii centered at 6.76˚ A (same radius as in ref. ), and an inter-C60length of 9.8˚ A.
The good agreement between the calculation and the rest of the diffraction pattern is the
proof of a reliable characterization of the sample and of a high quality process of synthesis.
The diffraction patterns measured for both samples show up a clear difference in their respec-
tive structures. Clear features in the diffraction pattern of sample B allow an unambiguous
characterization and a reliable estimation of its filling rate. Furthermore, our results are in
good agreement with what was previously determined for this sample in other x-ray diffrac-
tion and transmission electonic microscopy studies. On the other hand, one must be
more careful with the determination of the filling rate of sample A, as noted above.
We presented in detail the formalism to calculate the diffraction diagram of peapods at
different level of organization: isolated and organized into bundles. This formalism allows
to numerically investigate the main characteristics of the diffraction patterns of peapod
samples, and to discuss those driving to a pertinent characterization of real samples. In
particular, the isolated peapod study shows how the concentration of C60s inside the tubes
can shift the positions of the diffracted intensity zeros, and consequently lead to an accidental
extinction of the (10) Bragg peak in the diffraction profile of peapod bundles, whereas there
is no change in the bundle arrangement. These results show that one must be extremely
careful about the correct interpretation of the changes in position and intensity of the (10)
Bragg peak in experimental data concerning peapods and also all the inserted samples of
SWNTs. Furthermore, we saw that the diffraction pattern of peapods has to be considered
in its entire 0 to 2˚ A−1Q range in order to derive reliable characterization of the sample.
In particular, much attention has to be paid to both the intensity and shape of the C60-C60
Those features are discussed in the measured diffraction patterns of two different samples.
It is our pleasure to acknowledge J.-L. Sauvajol and P.A. Albouy for fruitful discussions.
H. Kataura acknowledges for a support by Industrial Technology Research Grant Program
in ’03 from New Energy and Industrial Technology Development Organization (NEDO) of
The calculation of the scattering pattern from a powder of peapods is detailed in this
appendix for the case of nanotubes of infinite length.
For a given wavevector?Q the intensity coming from an isolated peapod is:
I(?Q) = F(?Q)F∗(?Q)
The form factor of a peapod of length L, containing one C60 molecule at half height,
F(?Q) = fs(2πrhLσcJ0(Q//rh)sin(QzL/2)
In this expression− →
Q//is the projection of the wave-vector?Q perpendicularly to the nanotube
axis and Qzis its projection along the axis.
For a peapod of length NcL, it becomes
F(?Q) = fs(2πrhLσcJ0(Q//rh)sin(QzL/2)
The intensity per length unit thus writes
I(?Q) = f2
Now we consider an infinite tube: Nc→ ∞. Using the relation
where δ(x) is the Dirac distribution and where k is an integer, it follows that the intensity
per unit length scattered by an isolated peapod of infinite length writes
I(?Q) = f2
which can be written as:
I(?Q) = f2
The δ(Qz) dependent term is the Fourier transform of the structure projected on a plane
perpendicular to the nanotube axis, while the δ(Qz− 2πk/L) dependent term comes from
the periodicity of the C60chain.
Now we use eq. 2 to calculate powder average. If g(Q//,Qz) represents the factor multi-
plicating δ(Qz) in the previous expression, the integration of the δ(Qz) dependent term over
the angles u and ϕ gives
where Qz= Qcos(u). The integration of the δ(Qz− 2πk/L) dependent term reduces to
δ(Qcos(u) − 2πk/L)sin(u)du
which is equal to [22π
QInt(QL/2π)] . Here Int(QL/2π) is the integer part of (QL/2π) : the
asymmetric shape of the peaks characteristic of the C601D periodicity can be understood
through this formula (the sawtooth line shape).
It follows that the intensity scattered by a powder of isolated peapods is given by
[(2πrhLσcJ0(Qrh) + 4πr2
Let us now calculate the intensity scattered by a powder of peapod bundles. The expres-
sion of the form factor of a bundle of peapods of length NcL is :
F(?Q) = fs
Riis the tube i position and Tz(i) is a random number between 0 and L (see fig. 1).
Using the above procedure, one finds that the intensity scattered by a powder of peapod
bundles is given by
[(2πrhLσcJ0(Qrh) + 4πr2
The lattice term is (?
Rj|); NT is the number of tubes per bundle. If one considers a distribution of tube
i,jJ0(QRij)) where Rijis the distance between tubes i and j (Rij=
|− →Ri−− →
diameters inside bundles, or distribution of bundle sizes, one can extrapolate the average
procedures presented in ref. .
IX. APPENDIX 2
This appendix details some intensity calculations in the case of a powder of nanotube
bundles, for incomplete filling of the nanotubes by ’long’ (quasi-infinite) chains of fullerenes.
The chains within each tube are assumed to be sufficiently long to allow one to describe
their scattered intensity with Dirac distributions. This assumption implies that the formula
derived below cannot be used for too small filling rates.
Giving the site (n) of a molecule in tube (i), we define the filling factor of the site (i,n) by a
function f(i,n) which is 1 if the site is occupied and zero otherwise. To calculate the intensity
Ip(Q) in the case of partial filling, one multiplies the C60term in eq. 17 in appendix 1 by
this filling factor.
One follows the orientational average procedure given in appendix 1 but with I(?Q) replaced
by its average over occupancies f(i,n). One finds
[(2πrhLσcJ0(Qrh))2+ < p(i)p(j) > (4πr2
+2 < p(i) > (2πrhLσcJ0(Qrh))(4πr2
Int(QL/2π) < p(i)p(i) > (4πr2
During the course of the calculation terms containing < p(i)p(j) > arise, where p(i) is
the filling factor of tube i : p(i) =
n=1f(i,n) where Nc(i) is the number of C60sites
in tube i.
The filling rate of the sample, called < p > or p, is the double average of f(i,n):
< p >=
where the number of C60sites has been taken to be the same for all tubes Nc(i) = Nc.
Three cases are considered.
(i)Homogeneous partial filling of the tubes with long chains of C60, all tubes having the same
mean filling rate p, independent of i : p(i) = p, independent of i.
In that case, equation 19 becomes :
[(2πrhLσcJ0(Qrh) + p4πr2
The C60form factor -4πr2
- is multiplied by the mean filling rate p. There is
no other modification of equation 18 because one assumes that the C60molecules agglomerate
within nanotubes to form long chains.
(ii)Partial filling of the tubes with long chains of C60molecules, with filling rates different
from one tube to another within each bundle. In that case, one has to consider that
< p(i)p(j) >=< p2> if i = j
< p(i)p(j) >=< p >2if i ?= j (no correlation from one tube to another).So that
< p2> − < p >2?= 0.
For instance, one can consider within each bundle average proportions p of fully filled
tubes and (1-p) of empty tubes (which can be due to the fact that p% of the tubes are
opened and (1-p)% are closed), which gives: < p(i) >=p and < p(i)p(i) >=p, then
< p(i)2> − < p(i) >2= p(1 − p) ?= 0.
The different fillings of nanotubes induce additional disorder in direct space, which corre-
sponds to diffuse scattering in reciprocal space. The additional scattering is the most intense
at small Q values as in the case of chemical disorder . This approach is detailed in ref.
for partial filling of zeolite channels with nanotubes. Eq. 18 becomes
[(2πrhLσcJ0(Qrh)+ < p > 4πr2
+NT(< p2> − < p >2)(4πr2
)2+2NTInt(QL/2π) < p2> (4πr2
(iii) Nanotubes in the same bundle are all filled or all empty, p% of the bundles corre-
sponding to filled tubes and (1-p)% to empty ones. The scattered intensity is the sum of the
intensities for p fully filled bundles and for (1-p) empty bundles. Equation 18 becomes :
Ip(Q) = p(2π)2f2
[(2πrhLσcJ0(Qrh) + 4πr2
This appendix details calculations in the case of polymerized C60 molecules inside the
tubes. The chains within the tubes are assumed to be formed of n-polymers of C60molecules.
The distance between bonded C60molecules within a n-polymer is Lb, which is smaller than
the distance L between C60neighbors belonging to different n-polymers. For instance, for
n=2, one considers a chain of dimers and for n=3 a chain of trimers.
The form factor of a n-polymer of C60molecules writes
Lb1 − eiQznLb
1 − eiQzLb
)Lbis the position of the C60molecule indexed by k within the n-polymer.
By replacing in eq. 15 the monomer form factor by that of the n-polymer and the period
L along the monomer chain by the one along the n-polymer chain, which is (L+(n-1)Lb),
δ(Qz− 2πk/(L + (n − 1)Lb))]
Powder average thus gives :
Ip(Q) ∝ f2
Q(2πrh(L + (n − 1)Lb)σcJ0(Qrh) + 4πr2
Q(1 − δN,0)
sin(πknLb/(L + (n − 1)Lb))
sin(πkLb/(L + (n − 1)Lb)))2)]
where M is the integer part of (Q(L+(n−1)Lb
); the term (1−δM,0) was introduced to avoid
the case M=0.
 : S. Iijima, Nature 354 (1991) 56
 : R.S. Lee, H.J. Kim, J.E. Fischer, A. Thess, R.E. Smalley, Nature 388 (1997) 255
 : A.M. Rao, P.C. Eklund, S. Bandow, A. Thess, R.E. Smalley, Nature 388 (1997) 257
 : N. Bendiab, A. Righi, E. Anglaret, J.L Sauvajol, L. Duclaux, F. Beguin, Phys. Rev. B 63
 : M.R. Johnson, S. Rols, P.Wass, M. Muris, M. Bienfait, P. Zeppenfeld, N. Dupont-Pavlovsky,
Chemical Physics 293 (2003) 217
 : B.W. Smith, M. Monthioux, D.E. Luzzi, Nature 396 (1998) 323
 : D.H. Kim, H.S. Sim and K.J. Chang, Phys. Rev. B 64 (2001) 115409
 : A. Rochefort, Phys. Rev. B 67 (2003) 115401
 : D.J. Hornbaker, S.J. Kahng, S. Misra, B.W. Smith, A.T. Johnson, E.J. Mele, D.E. Luzzi,
A. Yazdani, Science 295 (2002) 828
 : J. Vavro, M.C. Llaguno, B.C. Satishkumar, D.E. Luzzi and J.E. Fischer, Appl. Phys. Lett.
80 (2002) 1450
 : H. Kataura, Y. Maniwa, T. Kodama, K. Kikuchi, H. Hirahara, K. Suenaga, S. Iijima, S.
Suzuki, Y. Achiba and W. Kr¨ atschmer, Synthetic Metals 121 (2001) 1195
 : X. Liu, T. Pichler, M. Knupfer, M.S. Golden, J. Fink, H. Kataura, Y. Achiba, K. Hirahara
and S. Iijima, Phys. Rev. B 65 (2002) 045419
 : K. Hirahara, S. Bandow, K. Suenaga, H. Kato, T. Okazaki, H. Shinohara and S. Iijima,
Phys. Rev. B 64 (2001) 115420
 : H. Kuzmany, R. Pfeiffer, C. Kramberger, T. Pichler, X. Liu, M. Knupfer, J. Fink, H.
Kataura, Y. Achiba, B.W. Smith and D.E. Luzzi, Appl. Phys. A 76 (2003) 449
 : H. Kataura, Y. Maniwa, M. Abe, A. Fujiwara, T. Kodama, K. Kikuchi, H. Imahori, Y.
Misaki, S. Suzuki, Y. Achiba, Applied Physics A, 74 (2002) 349
 : Y. Maniwa, H. Kataura, M. Abe, A. Fujiwara, R. Fujiwara, H. Kira, H. Tou, S. Suzuki; Y.
Achiba, E. Nishibori, M. Takata, M. Sakata and H. Suematsu, J. of Phys. Soc. of Japan 72
 : W. Zhou, K. I. Winey, J. E. Fischer, T. V. Sreekumar, S. Kumar, and H. Kataura, Appl.
Phys. Lett. 84 (2004) 2172
 : A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, C. Xu, Y.H. Lee, S.G. Kim, D.T.
Colbert, G.E. Scuseria, D. Tomanek, J.E. Fischer, R.E. Smalley, Science 273 (1996) 483
 : S. Rols, R. Almairac, L. Henrard, E. Anglaret, J.L. Sauvajol, The European Physical Journal
B, 10 (1999) 263
 : E. Anglaret, S. Rols and J.-L. Sauvajol, Phys. Rev. Lett. 81 (1998) 4780
 : N. Bendiab, R. Almairac, S. Rols, R. Aznar, J-L. Sauvajol, I. Mirebeau, Phys. Rev. B 69
 : A. Guinier, X-ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies (Dover
 : S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter (Oxford Science 1984)
 The maximum of the asymmetric peak originating from the C60chains is situated at Q=2π/L
= 0.66˚ A−1for infinite chains. For finite chains, it becomes less abrupt and its maximum shifts
to higher Q values (namely, 0.68˚ A−1for chains of 40 molecules in figure 3).
 : Y. Maniwa, Y. Kumazawa, Y. Saito, H. Tou, H. Kataura, H. Ishii, S. Suzuki, Y. Achiba, A.
Fujiwara, H. Suematsu , Japanese Journal of Applied Physics 38 (1999) L668
 The compensation leads to a zero value of the intensity for peapods of infinite lengths, and to
values close to zero for finite lengths.
 : B. Sundqvist, Adv. Phys. 48 (1999) 1
 : L. Forr´ o and L. Mih´ aly, Rep. Prog. Phys. 64 (2001) 649
 : T. Pichler, H. Kuzmany, H. Kataura and Y. Achiba, Phys. Rev. Lett. 87 (2001) 267401
 : L. Henrard, A. Loiseau, C. Journet, and P. Bernier, Eur. Phys. J. B 13 (2000) 661
 : A. Fujiwara, K. Ishii, H. Suematsu, H. Kataura, Y. Maniwa, S. Suzuki and Y. Achiba,
Chem. Phys. Lett. 336 (2001) 205
 : P. Launois, R. Moret, D. Le Bolloc’h, P.A. Albouy, Z.K. Tang, G. Li and J. Chen, Solid
State Comm.116 (2000) 99
FIG. 1: Schematic representation of the system of coordinates and variables used in the calculations.
FIG. 2: Calculations of the intensity diffracted by several powders. a) nanotube of radius r=6.8
˚ A and of length 380˚ A. b) linear chain of 40 C60molecules. c) peapod (plain line), sum of the
intensities from a) and b) (dotted line), and crossed term (dashed line).
FIG. 3: Upper part of the figure : calculated powder diffraction patterns of peapods for different
tube radii : 5.42˚ A (8,8), 6.8˚ A (10,10), 8.1˚ A (12,12) , with an inter-C60distance L equal to 9.5
˚ A. Lower part of the figure : calculated diffraction patterns for different inter-C60distances L and
for a r=6.8˚ A tube.
FIG. 4: Calculated diffracted intensities of a powder of isolated nanotubes (rh=6.8˚ A). Up: for
random filling by the C60molecules (tube length is 380˚ A); down : for incomplete filling by long
(’infinite’) C60chains. Filling rates indicated in the figure are the same for all tubes in each case
FIG. 5: Component of the calculated diffracted intensities relative to periodicity effects (only the
second term of eq. 11 is drawn) of a powder of isolated nanotubes (rh=6.8˚ A), for C60monomers
(upper line), dimers (middle line) and trimers (lower line) chains inside the tubes.
FIG. 6: Comparison between the calculated intensities scattered by a powder of bundles of 12
empty nanotubes (upper curve of each part) and bundles of 12 peapods (lower curve of each part)
for (10,10) nanotubes (upper part) of radius r=6.8˚ A, (8,8) nanotubes (middle part) of radius
r=5.42˚ A and (12,12) nanotubes (lower part) of radius r=8.1˚ A. The tube length was fixed at 380
˚ A in all the calculations. The arrows in the upper part point toward assymetric peaks characteristic
of the 1D periodicity of C60chains.
FIG. 7: Left top : Diffraction patterns of 85% filled bundles of 12 peapods. Left bottom : Diffraction
patterns of 50% filled bundles of 12 peapods. All tubes have a radius of 6.8˚ A. Right part : Schematic
representations of the 4 different filling modes used in calculations : a) random positions for the
C60molecules within each tube, same filling for each tube, b) long C60chains, same filling for each
tube , c) long C60chains, inhomogeneous filling of the different tubes within a bundle, d) mix of
full and empty bundles.
FIG. 8: Effect on diffraction pattern of the filling of the external channels with C60s linear chains
for different sizes of bundles. All tubes have a radius of 6.8˚ A and are 380˚ A long. a) 4 tubes. b)
6 tubes compared with a 6 tubes naked bundle (dotted line). c) 12 tubes.
FIG. 9: Filling rate determination. a) First criterium giving the amplitude ratio
I(.663˚ A−1)/I(.706˚ A−1) as function of the filling rate p (see the inset), calculated for randomly
filled (empty circles) and homogeneously filled (full circles) bundles of peapods. b) Second cri-
terium: amplitude ratio of the peaks around 1.3˚ A−1and 1.1˚ A−1as function of p (see the inset),
calculated for randomly filled (empty circles), inhomogeneously filled (empty triangles) and homo-
geneously filled (full circles) bundes of peapods. The dashed and dotted horizontal lines represent
the experimental finding for the two investigated samples A and B.
FIG. 10: Experimental diffraction patterns of peapod samples. a) Peapod sample A (upper line),
corresponding MER pristine SWNT sample (lower line). The dotted line is the baseline used for
the fits. b) Peapods sample B (upper line) and a calculation of its diffraction profile (lower line).
Intensity (arb. units)
Intensity (arb. units)
Intensity (arb. units)
Intensity (arb. units)
Intensity (Arb. units)
Intensity (arb. units)
Intensity (arb. units)
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Amplitudes ratio ( * 2)
0.811.2 1.4 1.6 1.8