Semiclassical approach to the description of the basic properties of nanoobjects
Present paper is a review of results, obtained in the framework of semiclassical approach in nanophysics. Semiclassical description, based on Electrostatics and Thomas–Fermi model was applied to calculate dimensions of the electronic shell of a fullerene molecule and a carbon nanotube. This simpli...
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irk-123456789-1175642017-05-25T03:03:26Z Semiclassical approach to the description of the basic properties of nanoobjects Kornyushin, Y. Carbon nanotubes, quantum wires and Luttinger liquid Present paper is a review of results, obtained in the framework of semiclassical approach in nanophysics. Semiclassical description, based on Electrostatics and Thomas–Fermi model was applied to calculate dimensions of the electronic shell of a fullerene molecule and a carbon nanotube. This simplified approach yields surprisingly accurate results in some cases. Semiclassical approach provides rather good description of the dimensions of the electronic shell of a fullerene molecule. Two types of dipole oscillations in a fullerene molecule were considered and their frequencies were calculated. Similar calculations were performed for a carbon nanotube also. These results look rather reasonable. Three types of dipole oscillations in carbon nanotube were considered and their frequencies were calculated. Frequencies of the longitudinal collective oscillations of delocalized electrons in carbon peapod were calculated as well. Metallic cluster was modeled as a spherical ball. It was shown that metallic cluster is stable; its bulk modulus and the frequency of the dipole oscillation of the electronic shell relative to the ions were calculated. 2008 Article Semiclassical approach to the description of the basic properties of nanoobjects / Y. Kornyushin // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1063–1071. — Бібліогр.: 26 назв. — англ. 0132-6414 PACS: 73.63.–b http://dspace.nbuv.gov.ua/handle/123456789/117564 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid |
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Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid Kornyushin, Y. Semiclassical approach to the description of the basic properties of nanoobjects Физика низких температур |
| description |
Present paper is a review of results, obtained in the framework of semiclassical approach in nanophysics.
Semiclassical description, based on Electrostatics and Thomas–Fermi model was applied to calculate dimensions
of the electronic shell of a fullerene molecule and a carbon nanotube. This simplified approach
yields surprisingly accurate results in some cases. Semiclassical approach provides rather good description
of the dimensions of the electronic shell of a fullerene molecule. Two types of dipole oscillations in a
fullerene molecule were considered and their frequencies were calculated. Similar calculations were performed
for a carbon nanotube also. These results look rather reasonable. Three types of dipole oscillations in
carbon nanotube were considered and their frequencies were calculated. Frequencies of the longitudinal collective
oscillations of delocalized electrons in carbon peapod were calculated as well. Metallic cluster was
modeled as a spherical ball. It was shown that metallic cluster is stable; its bulk modulus and the frequency
of the dipole oscillation of the electronic shell relative to the ions were calculated. |
| format |
Article |
| author |
Kornyushin, Y. |
| author_facet |
Kornyushin, Y. |
| author_sort |
Kornyushin, Y. |
| title |
Semiclassical approach to the description of the basic properties of nanoobjects |
| title_short |
Semiclassical approach to the description of the basic properties of nanoobjects |
| title_full |
Semiclassical approach to the description of the basic properties of nanoobjects |
| title_fullStr |
Semiclassical approach to the description of the basic properties of nanoobjects |
| title_full_unstemmed |
Semiclassical approach to the description of the basic properties of nanoobjects |
| title_sort |
semiclassical approach to the description of the basic properties of nanoobjects |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Carbon nanotubes, quantum wires and Luttinger liquid |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117564 |
| citation_txt |
Semiclassical approach to the description of the basic
properties of nanoobjects
/ Y. Kornyushin // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1063–1071. — Бібліогр.: 26 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT kornyushiny semiclassicalapproachtothedescriptionofthebasicpropertiesofnanoobjects |
| first_indexed |
2025-07-08T12:28:27Z |
| last_indexed |
2025-07-08T12:28:27Z |
| _version_ |
1837081777833771008 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10, p. 1063–1071
Semiclassical approach to the description of the basic
properties of nanoobjects
Yuri Kornyushin
Maître Jean Brunschvig Research Unit, Chalet Shalva, Randogne CH-3975, Switzerland
E-mail: jacqie@bluewin.ch
Received November 22, 2007
Present paper is a review of results, obtained in the framework of semiclassical approach in nanophysics.
Semiclassical description, based on Electrostatics and Thomas–Fermi model was applied to calculate di-
mensions of the electronic shell of a fullerene molecule and a carbon nanotube. This simplified approach
yields surprisingly accurate results in some cases. Semiclassical approach provides rather good description
of the dimensions of the electronic shell of a fullerene molecule. Two types of dipole oscillations in a
fullerene molecule were considered and their frequencies were calculated. Similar calculations were per-
formed for a carbon nanotube also. These results look rather reasonable. Three types of dipole oscillations in
carbon nanotube were considered and their frequencies were calculated. Frequencies of the longitudinal col-
lective oscillations of delocalized electrons in carbon peapod were calculated as well. Metallic cluster was
modeled as a spherical ball. It was shown that metallic cluster is stable; its bulk modulus and the frequency
of the dipole oscillation of the electronic shell relative to the ions were calculated.
PACS: 73.63.–b Electronic transport in nanoscale materials and structures.
Keywords: fullerene molecule, carbon nanotube, metallic cluster.
1. Introduction
Semiclassical description, based on Electrostatics and
Thomas–Fermi model was applied rather successfully to
study atomic, molecular and nanoobjects problems [1–5].
Fullerene molecule was studied in [1,3–5]. Carbon nano-
tube was studied in [2,3–5]. Carbon peapod was studied
in [5].
Fullerene molecule (C60) forms a spherical ball. Let us
denote the radius of the sphere, on which the carbon at-
oms are situated R f . Chemical bonds determine the value
of this radius. The electronic configuration of the constit-
uent carbon atom is 1 2 22 2 2s s p . It was assumed [1,3,4]
that in a fullerene molecule two 1s electrons of each atom
belong to the core (forming the ion itself), two 2s elec-
trons form molecular bonds, and two 2 p electrons are
delocalized, or free. Hence it was assumed in [1,3,4] that
the total number of the delocalized electrons in a fullerene
molecule is 120. Another assumption was made, that the
delocalized electrons couldn’t penetrate inside the sphere
of the ions as they repel each other.
The latest experimental data [6] show that R f � 0.354
nm, the total number of the delocalized electrons, N �
= 240, the equilibrium internal radius of the electron
shell, Rie � 0.279 nm (that is the delocalized electrons do
penetrate the sphere of the ions), and the equilibrium ex-
ternal radius of the electronic shell, Re � 0.429 nm.
In the model considered here it is assumed like in
[1,3–5] that the positive charge of the ions, � �eN e( 0 is
the electron charge), is distributed homogeneously on the
surface of a sphere r R rf� ( is the distance from the cen-
ter of a fullerene molecule). The charge of the delocalized
electrons, eN , is assumed to be distributed homogene-
ously in a spherical layer R r Ri � � (see Fig. 1).
Now let us calculate the electrostatic energy of a
fullerene molecule [5].
2. Calculation of the electrostatic energy
of a fullerene molecule
The electrostatic potential �( )r , arising from electric
charge, can be obtained as a solution of the Poisson’s
equation [7]
�� ��( ) ( )r r� �4 , (1)
where �( )r is the charge density.
© Yuri Kornyushin, 2008
The uniformly distributed charge density in the spheri-
cal layer of the electronic shell of a fullerene molecule is
� �( ) ( )r � �3 4 3 3eN/ R Ri . (2)
The charge of the delocalized electrons produces the
following electrostatic potential inside the spherical
layer, where the delocalized electrons are found:
� e i ieN/ R R R / r / R /r� � � �[ ( )][( ) ( ) ( )3 3 2 2 33 2 2 . (3)
The charge of the ions creates the following potential
on the sphere r R f� and outside of it:
� i r eN/r( ) � � . (4)
At r R f� the positive charge of the sphere creates elec-
trostatic potential equal to �eN/R f . According to Eqs. (3),
(4), the total electrostatic potential, � � �( ) ( ) ( )r r re i� , is
zero at r R� . Hence for r R
it is zero also. Inside the neu-
tral fullerene molecule, when 0 � �r Ri , the total electro-
static potential is as follows:
� in i i i feN R R R RR R eN/R� �[ ( ) / ( )]3 2 2 2 . (5)
As on the external surface of a neutral fullerene mole-
cule, at r R� , the total electrostatic potential is zero, it
should be zero on the internal surface, at r Ri� , also.
Otherwise the electrons will move to the locations, where
their potential energy is smaller. In a state of equilibrium
� in � 0. This yields:
R R R /i f f� � �0 5 32 2 1 2. { [ ( )] } ,/� � (6)
where � � �R Ri is the thickness of the electronic shell.
The latest experimental data [6] are: R f � 0 354. nm and
� � 0 15. nm. For these values Eq. (6) yields Ri � 0.274 nm.
The experimental value is 0.279 nm. The calculated value
is in very good agreement with the observed one.
The total electrostatic energy of a system, U , consists
of the electrostatic energy of the ions, electrostatic energy
of the delocalized electrons, and electrostatic energy of
the interaction between them. It is described by the fol-
lowing expression:
U e N / R e N / Rf� �( ) ( )2 2 2 22 2
� � � � [ ( )]( )e N / R R R R R R Rf i f i f
2 2 3 3 2 3 32 3 2
� � �0 1 9 5 53 3 2 5 5 6 3 2. [ ( )] [ ( ) ]eN/ R R R R R /R R Ri i i i . (7)
Here the first term describes the electrostatic energy of
the ions. The third term describes the interaction energy.
The second and the fourth term describe the electrostatic
energy of the delocalized electrons. The second term rep-
resents the energy of the electrostatic field outside the
charged spherical layer. The fourth term represents the
energy of the field inside the spherical layer. This term
was calculated as an integral over the layer of the square
of the gradient of � e r( ), divided by 8�.
3. Calculation of the kinetic and total energies
of a fullerene molecule
The value of the kinetic energy, T , of a gas of delo-
calized electrons, confined in a volume of regarded spher-
ical layer 4 33 3�( )R R /i� , and calculated in accord with
the ideas of Thomas–Fermi model, is [1]
T /m N / R Ri
/� �1105 2 5 3 3 3 2 3. ( ) ( )/
� . (8)
The total energy, W T U� , is a function of a single
variational parameter, �. Indeed, when Eq. (6) is substi-
tuted in Eqs. (7), (8) together with R Ri� � �� 0 5.
� �{ [ ( )] }/R R /f f� �2 2 1 23 , we see that the total energy
is a function of a single variational parameter, �. In a state
of equilibrium the total energy reaches a minimum value
Wmin � 9.422 keV. The minimum (equilibrium) value of
the total energy is reached when � �� �e 0.1374 nm. At
that R Ri ie� � 0.2808 nm, and R Re� � 0.4182 nm.
The minimum (equilibrium) energy of a positive ion of
a fullerene molecule (with N = 239) was calculated to be
10.327 keV. At that in this case Rie = 0.2808 nm and Re �
= 0.4182. The difference, appearing in further digits, is
very small.
It is worthwhile noting that the total energy of a ful-
lerene molecule also contains large negative terms, corre-
sponding to the energy of the ionic cores and the energy
of the chemical bonds. These terms were not calculated
here.
1064 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
Yuri Kornyushin
Fig. 1. Model, applied to a fullerene molecule and a carbon
nanotube [5]. The charge of the ions is distributed homoge-
neously on the surface of a sphere of a radius R f , or cylinder
of a radius Rn for a carbon nanotube. The delocalized electrons
are distributed uniformly in a spherical (or cylindrical for a
carbon nanotube) layer R r Ri � � .
4. Collective dipole oscillation of delocalized
electrons in a fullerene molecule
Linear-response theory was used by G.F. Bertsch et al.
to calculate the frequency of a giant collective resonance
in a fullerene molecule [8]. The value, calculated in [8]
was of about 20 eV. This value, predicted in [8], was mea-
sured experimentally and reported in [9].
Two peaks of dipole collective oscillations of
delocalized electrons in C60 positive ions, observed ex-
perimentally by S.W.J. Scully et al., were reported in
[10]. The authors associated the lower peak near 20 eV
with a surface plasmon, its frequency being 3 times
smaller than the Langmuir frequency [2]. The other peak
(about 40 eV) is associated by the authors with some vol-
ume plasmon of unknown origin [10].
Mie Oscillation. Mie oscillation is a collective delo-
calized electron oscillation in a thin surface layer of a
conducting sphere [11,12], where positive and negative
charges are not separated in space. So it does not look
suitable to regard such an oscillation in the thin surface
layer of the electronic shell of a fullerene molecule. Any-
way let us estimate possible frequency of such oscillation.
Let us consider a thin surface layer of a thickness d, situ-
ated on the external surface of the electronic shell of a
fullerene molecule. The volume of this layer, 4 2�R de , con-
tains electric charge 4 2�Re end (here n is the number of
delocalized electrons per unit volume of the electronic
shell). Let us shift the layer as a whole by the distance s
along the z-axis. This shift creates a dipole moment
P R ensde� 4 2� and restoring electric field (in a spherically
symmetric object) E P/ R ensd/r e� �4 3 4 32� �( ) corre-
spondingly. Restoring force at that is ( )4 32�enR sd/e . The
mass of the regarded thin surface layer is 4 2�R nmde (here
m denotes the mass of the delocalized electron). Restoring
force leads to the dipole oscillation. From Newton equa-
tion follows that the frequency of the oscillation consid-
ered is
M � ( ) /4 32 1 2�e n/ m . This frequency, as was men-
tioned by S.W.J. Scully et al. [10], is 3 times smaller
than Langmuir frequency. In the model described above
n N/ R Re ie� �3 4 3 3�( ). From two last equations written
above follows that
M e iee N/m R R� �[ ( )] /2 3 3 1 2. This equa-
tion yields �
M � 21.45 eV when the experimental val-
ues of the parameters are used. S.W.J. Scully et al. re-
ported the value of one of the two measured peaks, �
M ,
near 20 eV [10]. The agreement is quite reasonable. Using
theoretical values of Re = 0.4182 nm and Rie = 0.2808
nm, calculated here, one can obtain the value of the peak
22.72 eV. This result is not too bad either.
Simple Dipole Oscillation. Let us consider a dipole
type collective quantum oscillation of the electronic shell
as a whole (having mass mN ) relative to the ion skeleton.
Such an oscillation looks more plausible in a fullerene
molecule than Mie oscillation. Regarded system consists
of two spherically symmetric objects, the ion skeleton
and the electronic shell. Hence this oscillation causes no
electric field in the blank interior of a fullerene molecule
around its center (cavity). The value of the electrostatic
potential inside the electronic shell is � e eeN/ R� �[ ( 3
� � �R R / r / R /rie e ie
3 2 2 33 2 2)][( ) ( ) ( )] [see Eq. (3)]. Let us
consider the shift of the electronic shell by a small dis-
tance, s, relative to the ion skeleton and calculate the
change in the electrostatic energy of a fullerene molecule
(in the framework of the accepted model). The first term
in Eq. (3) does not contribute to the change in the electro-
static energy, as it is constant. The contribution of the
third term is also zero because the third term depends on
the distance like 1/r, a potential produced by a point
charge. Interaction of the ion skeleton with the point
charge, situated near the center of the fullerene molecule,
does not depend on their relative position, because the
electrostatic potential is constant for r R f� . The change
in the electrostatic energy does not depend on the direc-
tion of the shift (s vector) because of the spherical sym-
metry of the problem. This means that the change cannot
contain term proportional to s. Let us choose the x-axis
along the s vector. Then after the shift we shall have in the
third term of Eq. (3) [( ) ]x s y z� 2 2 2 instead of r 2 �
� ( )x y z2 2 2 . Taking into account that the term, pro-
portional to s in the change of the electrostatic energy is
zero, we find that the third term (the only one, which con-
tributes) yields the following change in the electrostatic
energy of a fullerene molecule due to the shift regarded:
U s eNs / R Re ie( ) ( ) ( )� �2 3 32 [5]. This energy, U s( ), is a po-
tential of a 3-d harmonic oscillator [13]. For a 3-d har-
monic oscillator with mass m NU s mN s /s( ) �
2 2 2 [5,13].
As follows from the last two equations, the frequency,
s ,
is equal to Mie frequency,
M . The distance between the
quantum levels is �
M . Its calculated value is 21.45 eV.
Its experimental value is about 20 eV. Quantum excitation
to the first excited level manifests itself as 20 eV peak, to
the second one as 40 eV peak. Conception of some vol-
ume plasmon of unknown origin [10] is not needed.
So, the frequency of the dipole oscillation in a ful-
lerene molecule is as follows:
f e iee N/m R R� �[ ( )] /2 3 3 1 2 . (9)
Mie oscillation is also a 3-d (quantum) harmonic oscil-
lator. So the results are the same for both types of oscilla-
tions. Only Mie oscillation in a fullerene molecule looks
less plausible.
The frequency of the simple dipole oscillation is equal
to Mie frequency in spherically symmetric objects [1]. So
it is rather difficult to determine which type of the oscilla-
tion occurs in the object under investigation.
Semiclassical approach to the description of the basic properties of nanoobjects
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1065
5. Calculation of the electrostatic energy
of a carbon nanotube
Let us consider a long (comparative to its diameter)
carbon nanotube. Let us denote N n the number of delo-
calized electrons in a carbon nanotube per unit length. We
denote the radius of the cylinder, on which the ions of a
carbon nanotube are situated, Rn .
The Gauss theorem [7] allows calculating the electro-
static field inside the electronic shell of a long carbon
nanotube (at R r Ri � � ):
E r eN /r r R / R Re n i i( ) ( )( ) ( )� � �2 2 2 2 2 . (10)
The electrostatic field, produced by the ions at R r Rn � �
can be calculated also [7]:
E r eN /ri n( ) � �2 . (11)
At r Rn� the electrostatic field, produced by the ions,
is zero. The electrostatic potential difference between the
internal and external surfaces of the electronic shell in a
carbon nanotube should be zero in a state of equilibrium.
Otherwise the delocalized electrons will move to the loca-
tions, where their potential energy is smaller. The electro-
static potential difference is an integral of minus electro-
static field between the surfaces mentioned. It is zero
when
R R /Ri n� 2 0 5exp . . (12)
Eq. (12) yields for the thickness of the electronic shell,
� � �R Ri , the following result:
� � �( ) (exp . )/R R Ri n i
1 2 0 25 . (13)
As � � �R Ri , it follows from Eq. (13) that
R R R Ri i n� �� ( ) exp ./1 2 0 25 . (14)
The total electrostatic energy of a system is equal to
the integral of the square of the electrostatic field, divided
by 8�. The value of it per unit length of a nanotube is de-
scribed by the following expression:
U e N / R R R R/R R R /Rn i n i n i� � [ ( ) ][ ln ( ) ln ( )]2 2 2 2 2 4 4
� � �[ ( )]( . . )e N / R R R R Rn i n i
2 2 2 2 2 2 20 75 0 75 (15)
Here [see Eq. (14)], R R Ri n
/� ( ) exp .1 2 0 25. From this fol-
lows that the electrostatic energy of the system is a func-
tion of only one variational parameter, Ri .
6. Calculation of the kinetic and total energies
of a carbon nanotube
The kinetic energy per unit length, T , of a gas of
delocalized electrons, confined in a cylindrical layer of a
volume per unit length of a nanotube, � ( )R Ri
2 2� , and
calculated in accord with the ideas of the Thomas–Fermi
model, is as follows [2,3]:
T /m N /R R Rn i n i� �1 338 0 52 5 3 2 3 2 3. ( ) [ (exp . ) ]/ / /
� . (16)
Eq. (14) was used here to express the kinetic energy as a
function of Rn and one variational parameter, Ri . Thence,
the total energy, W T U� , is a function of a single varia-
tional parameter, Ri . In a state of equilibrium the total
energy reaches a minimum value Wmin � 71.51 keV/nm at
Rn = 0.7 nm and N n = 670 nm–1. Parameter Rn = 0.7 nm
was taken from [4]. It was assumed here also that the total
number of delocalized electrons in a carbon nanotube is
four times larger than the number of carbon atoms. For the
same values of Rn = 0.7 nm and N n = 670 nm–1 the mini-
mum (equilibrium) value of total energy is reached when
Ri = Rnie = 0.577 nm. At that R Rne� � 0.816 nm and � e �
= 0.239 nm.
It is worthwhile noting that the total energy of a carbon
nanotube also contains large negative terms, correspond-
ing to the energy of the ionic cores and the energy of the
chemical bonds. These terms were not calculated here.
7. Collective dipole oscillations of delocalized
electrons in a carbon nanotube
Let us consider first a longitudinal surface dipole col-
lective oscillation of delocalized electrons in a long carbon
nanotube [2]. The electric current can exist only in the
layer occupied by collective electrons. In this layer the cur-
rent carriers can move in an oscillatory fashion. We assume
that the collective electrons move as a whole without de-
formation along the axis of a carbon nanotube relative to
the ions. We shall call such an oscillation longitudinal. Let
us shift the current carriers in the layer by a distance h
along the axis of a cylinder. As a result of the shift we have
opposite charges on the bases of a cylinder, q eN hn� � .
The two bases are situated rather far away from each other,
so they should be treated as two separate charged disks. A
disk is a limiting case of a flattened ellipsoid of revolution
with half-axes a b c� � . So the capacitance of a disk is that
of an ellipsoid with a � 0 and b c Rne� � , that is the capa-
citance, C R /ne� 2 � [14]. The electrostatic energy of the
two disks is U q /Cd � � �2 0 5 2. ( / )� e N R hn ne
2 2 22 . The
force, acting on all the electrons as a whole, is a derivative
� �dU /dsd �� e N /R hn ne
2 2( ) . Thus we come to the Newton
equation for the lN n collective electrons as a whole:
mlN d h dr e N /R hn n ne( / ) ( )2 2 2 2 0 �� . (17)
Assuming h h tl� 0 sin
, and using Eq. (17), we come
immediately to the expression:
�l n nee N /mlR� ( ) /1 2 . (18)
1066 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
Yuri Kornyushin
The angular frequency of the longitudinal oscillation
is inversely proportional to the square root of the length
of the carbon nanotube [2]. At N n = 670 nm–1, l = 30 nm,
Rne = 0.816 nm we have �
l = 3.065 eV.
Surface dipole collective oscillation of delocalized
electrons in a conductive sphere is called, as was men-
tioned above, Mie oscillation [11,12]. The frequency of
Mie oscillation is square root from 3 times smaller than
Langmuir frequency [2],
�L e n/m� ( ) /4 2 1 2 (here n is the
number of the delocalized electrons per unit volume of
the electronic shell). In general case [12] the frequency of
Mie oscillation,
�M De n/m� ( ) /4 2 1 2 (here D is the depo-
larization factor [12]). Since for a sphere [7] D = 1/3, we
have for the frequency of Mie oscillation in a sphere
�M e n/ m� ( ) /4 32 1 2, as was mentioned above. For a cyl-
inder [7] D = 1/2 and n N / R Rn ne nie� ��( )2 2 . Thence we
have for the frequency of the transverse surface (Mie) os-
cillation in carbon nanotube following result:
M n ne niee N /m R R� �[ ( )]2 2 2 2 . (19)
For N n = 670 nm–1, Rne = 0.816 nm, and Rnie = 0.577
nm we have �
M = 21 eV.
Let us consider now a simple dipole oscillation of the
electronic shell as a whole relative to the ion skeleton per-
pendicular to the carbon nanotube axis. Regarded system
consists of two cylindrically symmetric objects, the ion
skeleton and the electronic shell. Hence this oscillation
causes no electric field in the blank interior of a carbon
nanotube around its center (cavity).
Like for a fullerene molecule, when the electronic
shell of a carbon nanotube is shifted as a whole by a small
distance, h, relative to the ion skeleton, the electrostatic
energy of a carbon nanotube is changed byW h( ), which is
a potential of a 2 � d harmonic oscillator. Let us calculate
this change. The electrostatic potential is equal to the in-
tegral of minus electric field [7]. Inside the electronic
shell of a carbon nanotube, as follows from Eq. (10), the
electrostatic potential is as follows:
� e en ne nieeN r / R R� � � �[ ( )]2 2 2
� �2 2 2 2
0eN R R R / r/rn nie ne nie[ / ( )] ln( ) , (20)
where r0 is some constant. The second term in Eq. (20) co-
incides with the potential produced by some linear
charge. This term does not contribute to the change in the
electrostatic energy of a carbon nanotube, because the in-
teraction between the ion skeleton and a linear charge is
zero. This is so because the electrostatic potential is con-
stant for r Rn� . The change in the electrostatic energy
does not contain a term, proportional to h because of the
radial symmetry of the problem. So the only contribution
comes from the first term of Eq. (20). Let us choose the
x-axis along the h vector. Then after the shift we shall
have in the first term of Eq. (20) [( ) ]x h y� 2 2 instead of
r x y2 2 2� ( ). Taking into account that the term, propor-
tional to h in the change of the electrostatic energy is zero,
we find that the first term (the only one, which cont-
ributes) yields the following value of the change in the
electrostatic energy of a carbon nanotube (per unit length)
due to the shift regarded: U h eN h R Rn ne nie( ) ( ) / ( )� �2 2 2
[5]. This energy, U h( ), is a potential of a 2 � d harmonic
oscillator [5,13]. For a 2 � d harmonic oscillator [13]
(mass per unit length of a carbon nanotube mN n )U s( ) �
� mN s /n s
2 2 2. As follows from the last two equations,
the frequency,
s , is equal to Mie frequency,
M . The dis-
tance between the levels is �
M . Its calculated value is 21
eV. So, according to the calculations performed, the quan-
tum transition to the first excited level should manifest it-
self as 21 eV peak, and to the second one as 42 eV peak.
Mie oscillation in a carbon nanotube is also a 2 � d har-
monic oscillator. So the results are the same for both types
of the oscillations. Only Mie oscillation in a carbon
nanotube looks less plausible.
Obtained results show that a very simple semiclassical
concept of Thomas–Fermi model and Electrostatics often
gives rather good agreement between experimental and
theoretical results. In particular, this model often gives
rather reasonable description of nanoobjects.
8. Carbon peapod
As in [15] we assume that the fullerene molecules are
encapsulated in a carbon nanotube, and there is an inter-
action between collective electrons of a carbon nanotube
and fullerene molecules.
Let us consider the longitudinal oscillations [4]. We re-
mind that Rne denotes equilibrium external radius of the
electronic shell of a carbon nanotube and Rnie denotes equi-
librium internal radius of a carbon nanotube. Let us calcu-
late the part of the electrostatic energy (per unit length),
which depends on the shift of the electronic shell of a
fullerene molecule s and that of a carbon nanotube h. It con-
sists of the energy of the charges arising on the ends of a car-
bon nanotube, 2 0 5 22 2 2 2� �. ( / )q /lC e N lR hn ne� , the energy
of a fullerene molecule, U s eNs / R Re ie( ) ( ) ( )� �2 3 32 , multi-
plied by the number of fullerene molecules in a unit length
of a carbon peapod n f , the energy of the interaction of the
dipole moments of the fullerene molecules en Nsf with
electric field inside a carbon nanotube E q/lCn � �( )2
� �e N /lR hn ne( ) [this interaction energy is equal to
�e n NN /lR shf n ne
2( ) ], and the energy of a dipole–dipole in-
teraction between the dipoles of fullerene molecules. The
last term was calculated for a linear array of fullerene mole-
cules with the assumption that the size of the dipoles is
smaller than the distance between them. The part of the total
electrostatic energy, depending on the shifts s and h is as
follows:
Semiclassical approach to the description of the basic properties of nanoobjects
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1067
U s h e n N / R Rf e ie( , ) {[ . ( )]� � �2 2 3 30 5
� �2 2
2 2 2 2 2( ) ( )} ( )n /l f n l s e N / lR hf f n ne�
� �e n NN /lR shf n ne
2( ) ,
where f k k /i /i
i
k
i
k
( ) ( ) ( )� �
�
�
�
�
� �1 13
1
1
2
1
1
. (21)
The term �2 2 2 3 2e N n /l f n l sf f( ) ( ) in Eq. (21) repre-
sents the energy of the dipole–dipole interaction in a peri-
odic array of fullerene molecules, calculated in the ap-
proximation which neglects the real size of a dipole. This
approximation somewhat underestimates the effect of
dipole–dipole interaction.
The last term in Eq. (21) can be positive or negative
depending on the direction of the shifts s and h. It is posi-
tive when the shifts are parallel, and negative when they
are antiparallel.
Let us consider now the transverse oscillations. In ac-
cepted here model the shift of the electronic shell (as a
whole) relative to the ion skeleton does not produce elec-
trostatic field inside the fullerene molecule or carbon
nanotube. This is because the ion skeleton and the elec-
tronic shell do not lose their symmetry as a result of the
shift. So there is no interaction between transverse oscil-
lations of a fullerene molecule and carbon nanotube. This
means that the two transverse oscillations are not changed
in a carbon peapod; they remain the same as they were in a
free fullerene molecule and a free carbon nanotube.
Let us write the energy of the longitudinal oscillation
U s h( , ) in the following form:
U s h As Bh Gsh( , ) . .� �0 5 0 52 2 . (22)
The specific values of the factors A, B, and G are given
in Eq. (21).
Now let us write Newton equations for all the collec-
tive electrons per unit length of a carbon nanotube, and
for all the collective electrons of all the fullerene mole-
cules per unit length of a carbon nanotube, for the case of
the longitudinal oscillations:
mN h/ t Bh Gsn ( )� �2 2 0 � � ,
mn N s/ t As Ghf f ( )� �2 2 0 � � .
(23)
Two Eqs. (23) yield the following equations for the
amplitudes of the oscillations h0 and s0:
( ) ,B mN h Gsn� � �
2
0 0 0
� � �Gh A mn N sf0
2
0 0( )
. (24)
Eqs. (24) yield the frequencies of the oscillations (the same
frequencies for both parallel and antiparallel modes):
1 2
2 2 2 2
0 5 2, . {( )� � �f a l
� � � [( ) ( ) ] } ,/
�
f a l f ne f ln R /lR
2 2 2 2 3 2 2 1 22 4
(25)
a f f fe n N f n l /lm
2 2 2
2� ( ) . (26)
Here
f is the angular frequency of the plasmon in an
isolated C60 molecule [Eq. (9)],
l is the angular fre-
quency of the longitudinal oscillations in an isolated car-
bon nanotube [Eq. (18)]; ( ) /
f a
2 2 1 22� is the angular
frequency of the dipole mode of the plasma longitudinal
oscillations of a linear periodic array of fullerene mole-
cules. We find that the dipole–dipole interaction between
the fullerene molecules reduces the plasmon energy for
longitudinal oscillations. We also find in the accepted
model that in the limit of infinite length carbon peapod,
the coupling between the carbon nanotube and C60 mole-
cules vanishes.
Eqs. (25), (26) do not take into account spatial disper-
sion (dependence on the wave vector q). In [4] Eqs. (25),
(26) were generalized, taking into account spatial disper-
sion for small wave vectors. Figure 2 shows the coupled
longitudinal modes in a finite length carbon peapod (Fig.
1 in [4]), where the dotted line presents
l . For small
wave vector q, the coupling between the carbon nanotube
and C60 molecules has a negligible effect. In the vicinity
1068 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
Yuri Kornyushin
18
16
14
12
10
8
6
4
2
0
1 2 3 4 5 6
q, nm–1
E
n
er
g
y,
eV
Fullerene branch
Isolated nanotube
Nanotube branch
Fig. 2. Plasmon energy as a function of wavevector q along the
carbon nanotube direction, for a carbon peapod of finite length
(as calculated in [4]). The dotted curve is the plasmon disper-
sion of an isolated carbon nanotube, from Fig. 2 of [16]. The
lower (higher) solid curve is the carbon nanotube (fullerene)
branch, when coupling between the carbon nanotube and C60
molecules is taken into account. Here the following parameters
values were used [4]: �
f � 20 eV, N = 120, n f � 1.03 nm
–1
,
Nn � 333 nm
–1
, Re � 0.538 nm, R Rf ie� � 0.353 nm, Rne �
0.885 nm, R Rn nie� � 0.7 nm, and l � 29.1 nm (n lf � 30). With
these values, we have f n lf( ) � 34.4, �
a � 9.07 eV,
4
3�( )n R /lRf e ne � 0.0783, and �( ) /
f a
2 2 1 22� = 15.3 eV.
of
l f a� �( ) /2 2 1 22 , the coupling lowers the carbon
nanotube branch while it raises the fullerene branch,
forming an anticrossing.
9. Metallic cluster
A double-jelly model (delocalized electrons jelly and
ions jelly) was applied for the description of metallic
clusters [1]. This model does not take into account micro-
scopic electrostatic field around the ions. Here we shall
take into account this microscopic field. We shall analyze
here the degenerate delocalized electrons.
A spherical ball models the shape of a cluster. We shall
model the conditions as adiabatic ones under given pres-
sure. We shall restrict our consideration here by given en-
tropy and given pressure condition. In this case the ther-
modynamic potential, having a minimum in the state of
equilibrium, is the enthalpy (it is a function of thermody-
namic variables, entropy S and pressure P: H S P( , ) [17]).
Entropy is a function of a radius of a cluster R. So R could
be taken as thermodynamic variable instead of entropy.
Under different condition the contribution of the entropy
term to the Gibbs free energy is negligible.
We shall consider here only atmospheric pressure,
which can be neglected. So really, we will take into ac-
count the energy of a cluster only.
9.1. Electrostatic energy of a separate ion
Let us consider a ball of a metallic cluster of a volume
V R /� 4 33� (here R is the radius of a cluster), consisting
of Nc delocalized electrons. We consider here the ions as
point charges, and the delocalized electrons like a nega-
tively charged gas. In metallic clusters the delocalized
electrons density is so high, that they are degenerate over
all the temperature range of the existence of a metallic
cluster. Degenerate delocalized electrons screen a long-
range electrostatic field of point charges. The screening
Thomas–Fermi radius is as follows [18]:
1 6 0 47142 1 2 3 2 1 2/g VE / N e R E /N /eF c F c� � �( ) . ( )/ / /� ,
(27)
where E N /V / mF c� ( ) ( )/3 22 2 3 2� � is the Fermi energy
[18].
Fermi energy is proportional to 1 2/R . Thomas–Fermi
radius is proportional to R.
The electrostatic field around a separate positive ion,
submerged into the gas of degenerate delocalized elec-
trons, is as follows [18]:
� � � �( ) exp( )ze/r gr , (28)
where z is the number of delocalized electrons per one
atom, an r is the distance from the center of the ion.
The electrostatic energy of this field is the integral
over the ball volume of its gradient in a second power, di-
vided by 8�. The lower limit of the integral on r should be
taken as r0 (the radius of the ion), a very small value. Oth-
erwise the integral diverges. Calculation yields the fol-
lowing expression for the electrostatic energy of a sepa-
rate ion:
U z e r g gr0
2 2
0
1
00 5 0 5 2� �
�
. ( . ) exp . (29)
For 2 0gr essentially smaller than unity Eq. (29) yields:
U z e / r z e g0
2 2
0
2 22 0 75� �( ) . . (30)
The first term in the right-hand part of Eq. (30),
z e / r2 2
02 , represents the electrostatic energy of the bare ion.
It is worthwhile to note that the expansion of a cluster
leads to the decrease in the delocalized electron density.
From this follows the increase in the screening radius [see
Eq. (27)]. The electrostatic energy of a separate ion in-
creases concomitantly. One can see it, regarding Eq. (30).
9.2. Total energy of a cluster
We regard the ions of a cluster as randomly distrib-
uted. It is well known since 1967, that the electrostatic
energy of N /zc randomly distributed ions is just
U N /z Uc� ( ) 0 [19].
The kinetic energy, T , of a gas of delocalized electrons,
confined in a volume ( )4 3 3/ R� , and calculated in accord
with the ideas of the Thomas–Fermi model, is as follows [1]:
T N /mRc�1105
5 3 2 2. ( )
/
� . (31)
So the total energy of a cluster, W R( ), is as follows:
W R N /mR ze N / r ze N gc c c( ) . ( ) ( ) . .
/
� �1105 2 0 75
5 3 2 2 2
0
2
�
(32)
As a function of R,W R( ) has a minimum. So, the cluster is
stable. The minimum value of W R( ) is
W ze N / r z N me /e c c� �( ) . ( )/2
0
4 3 4 22 0 565 � . (33)
This minimum occurs when R Re� :
R N /z /mee c� 2 422 2 1 3 2 2. ( ) ( )/
� . (34)
The equilibrium volume per one ion is
v zR / N /me /ze e c� �4 3 59 532
3 2 2 3� . ( )� . (35)
For z ve� � � �1 8 821 10 24. cm3. It is about 3 times
smaller than expected. For this value of ve we have
average interatomic distance 2.564�10–8 cm and 2 /g �
� � �5 27 10 9. cm, which is 2.43 times smaller than average
interatomic distance. This means that electrostatic fields
of adjacent ions [Eq. (28)] do not overlap. So the electro-
static energy of N /zc ions is U N /z Uc� ( ) 0 anyway, are
they distributed randomly or not. Overlapping takes place
Semiclassical approach to the description of the basic properties of nanoobjects
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1069
when v /me /ze � 0 2871 2 2 3. ( )� only. According to Eq. (35)
this is not so at ambient pressure.
9.3. Bulk modulus of a cluster
When condensed matter subject is expanded, the in-
crease in its elastic energy is as follows [20]:
� � �W K V /V KV R/Re e e� �0 5 4 52 2. ( ) . ( ) , (36)
where K is the bulk modulus, Ve is the initial equilibrium
volume, and Re is the initial equilibrium radius. Here a
well-known relation, � �V/V R/Re e� 3 , was used. It is valid
when �R is essentially smaller than Re .
The change in the total energy of a cluster [Eq. (32)] is
� � �W W/ R dRR Re
� �0 5 2 2 2. ( ) ( ) . (37)
Equations (36), (37) yield:
K / R W/ Re R Re
� �( ) )( )1 12 2 2� � � . (38)
Equations (32), 34), and (38) yield the following ex-
pression for the bulk modulus:
K z m e //� 0 00105 10 3 4 10 8. ( )� . (39)
For z � 1 Eq. (13) yields K � �3 102 1011. erg/cm3. This
value is a very reasonable one indeed.
9.4. Collective oscillations
Frequency of the dipole oscillation of delocalized
electrons,
d , in any conductor of a spherical shape is 3
times smaller than Langmuir plasma frequency [12]. For
spherical cluster regarded here we have
d z me /� 0 2653 1 2 4 3. /
� . (40)
For z = 1 we have �
d = 7.219 eV. This value is close
to the classical surface-plasmon frequency in the case of
Na [21]. For Na instead of the factor 0.2653 in Eq. (40)
there is a factor 0.2497 [21]. So for Na we have �
d =
= 6.795 eV. This value is only 5.87% less then Eq. (40)
yields. Nobody could expect that some simple approach
like that used in this paper could compete with modern
techniques of Theoretical Physics used in [21,22]. Semi-
classical approach works only for simple basic problems.
So far it works fine for the problems discussed here. But it
cannot describe like in [22] a wide range of photon ener-
gies. It was shown in [22] that for lower energies collec-
tive effects are essential and for higher energies (higher
than 15 eV) a single-particle picture is relevant.
We are not discussing here the quadrupole and breath-
ing modes, but it is worthwhile mentioning that both of
them have an atomic branch of slow oscillations and an
electronic one of fast oscillations [1].
10. Discussion
Today many authors use the model of the electronic
shell accepted here. This model allows understanding ba-
sic experimental results. As a matter of fact this concept is
present in papers [6,10]. Two peaks of dipole oscillations
of the delocalized electrons in C60 ions, observed experi-
mentally by S.W.J. Scully et al., were reported in [10].
The authors ascribe these peaks to the surface (Mie) oscil-
lation [11,12] and some bulk dipole mode. The energy
values of the quanta of the two peaks were measured to be
about 20 and 40 eV. Using experimental data on N , R and
Ri and equation for the Mie frequency [2,10], one could
calculate the energy of the quantum of the Mie oscilla-
tion, 21.45 eV.
Simple dipole oscillation of the electronic shell as a
whole has the same frequency as Mie oscillation. Oscilla-
tion of the electronic shell as a whole relative to the skele-
ton of the ions is an oscillation of a 3 � d (for a fullerene
molecule) and 2-d (for a carbon nanotube) harmonic os-
cillator. The same is for Mie oscillation. As the frequen-
cies of the two types of oscillation are the same, cited ex-
perimental data could be explained both ways.
Concerning carbon nanotubes more sophisticated ob-
jects are now under investigation. In [23] collective exci-
tations in a linear periodic array of parallel cylindrical
carbon nanotubes, consisting of coaxial cylindrical tu-
bules, was studied. Multiwall carbon nanostructures were
studies in [24].
In [25] dynamic screening of an atom, confined within
a finite width fullerene molecule is a subject of research.
The authors used a classical dielectric approach. Now let
us consider a problem of possibility of the atom confine-
ment inside a fullerene molecule. There is no classical
reason to find neutral atom confined in the cavity of a
fullerene molecule or a neutral carbon nanotube. There is
no electric field there and electrostatic potential there is
zero like at the infinite distance from a fullerene molecule
or a carbon nanotube. It was discussed in Sections 4 and 7
of the present paper.
In this problem we have two objects, a fullerene mole-
cule and an atom. Let us denote the ionization energy of
one object as I and the affinity of the other as �A. Let us
transfer one electron with charge e from one object to an-
other one. The extra charge �e is situated on the external
surface of the electronic shell of a fullerene molecule,
r Re� . Atom (ion) is charged with the extra charge of the
same value and opposite sign. The electrostatic potential
on the external surface of the electronic shell of a ful-
lerene molecule is equal to that in the cavity. It was dis-
cussed in Sec. 2 of this paper. Both of them are equal to
� e/Re . The electrostatic energy of the interaction of the
two charges, one in the cavity and the other one on the
surface of the electronic shell, is �e /Re
2 . The total energy
is changed by I A e /Re� � 2 as a result of a charge transfer.
1070 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
Yuri Kornyushin
When this quantity is negative and minimal, the confine-
ment discussed is possible. Either positive or negative ion
could be found inside a fullerene molecule. It depends for
which ion the total change in the energy is minimal.
Using Eq. (3) it is possible to see that in case of a posi-
tive ion its energy in the cavity around the center of a
fullerene molecule is smaller than that inside the elec-
tronic shell. So a positive ion occupies the cavity of a
fullerene molecule. Positive ion is not localized in the
center of a fullerene molecule, like it was assumed in
[25]. It is expected to be found in any place in the cavity.
It is opposite for a negative ion. As follows from Eq.
(3), the minimal value of its energy is achieved when a
negative ion is situated on the surface, where the positive
ions of a fullerene molecule are located, r R f� . This en-
ergy is smaller than I A e /Re� � 2 , because the positive ion
skelton attracts a negative ion. This attraction is not taken
into account in the expression I A e /Re� � 2 .
In [26] a model of non-homogeneous oscillation of the
shell of a fullerene molecule and a carbon nanotube was
proposed. This model also can explain experimental re-
sults, obtained by S.W.J. Scully et al. [10]. But this model
does not look inherent to the topic. That is why it was not
discussed here.
We modeled the shape of metallic cluster here as a
spherical ball. The delocalized electrons in metallic clus-
ter are degenerate. Their kinetic energy is calculated in
the spirit of Thomas–Fermi model. The delocalized elec-
trons screen the electrostatic field of the ions. This field
was calculated; it was shown that it contributes essen-
tially to the energy of a cluster and its stability.
The equilibrium values of the energy of a cluster and
its volume were calculated. The bulk modulus of a cluster
and simple collective dipole oscillation of the electronic
shell as a whole relative to the ion skeleton were calcu-
lated also.
Obtained results show that a very simple semiclassical
concept, based on Electrostatics and Thomas–Fermi mo-
del often gives rather good agreement between experi-
mental and theoretical results. In particular, this model
gives rather reasonable description of nanoobjects.
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Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1071
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