Dynamics of He atoms adsorbed on a carbon nanotube
The spectrum of a helium atom adsorbed on an armchair single-walled carbon nanotube is calculated. It is shown that a helium adatom tunnels over the surface of the nanotube within a band 10 K (for ⁴He) to 14 K (for ³He) wide. The undersaturated system of helium adatoms on nanotubes can be used as an...
Gespeichert in:
| Datum: | 2011 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2011
|
| Schriftenreihe: | Физика низких температур |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/118591 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Dynamics of He atoms adsorbed on a carbon nanotube / M.A. Strzhemechny, I.V. Legchenkova // Физика низких температур. — 2011. — Т. 37, № 6. — С. 688–690. — Бібліогр.: 8 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-118591 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1185912017-05-31T03:08:18Z Dynamics of He atoms adsorbed on a carbon nanotube Strzhemechny, M.A. Legchenkova, I.V. Краткие сообщения The spectrum of a helium atom adsorbed on an armchair single-walled carbon nanotube is calculated. It is shown that a helium adatom tunnels over the surface of the nanotube within a band 10 K (for ⁴He) to 14 K (for ³He) wide. The undersaturated system of helium adatoms on nanotubes can be used as another system for studying quantum transport. It is also felt that this dynamics can be an important factor for the explanation of the large negative expansivities of nanotube bundles with adsorbed helium. 2011 Article Dynamics of He atoms adsorbed on a carbon nanotube / M.A. Strzhemechny, I.V. Legchenkova // Физика низких температур. — 2011. — Т. 37, № 6. — С. 688–690. — Бібліогр.: 8 назв. — англ. 0132-6414 PACS: 81.05.ub, 68.43.–h, 65.40.–b http://dspace.nbuv.gov.ua/handle/123456789/118591 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Краткие сообщения Краткие сообщения |
| spellingShingle |
Краткие сообщения Краткие сообщения Strzhemechny, M.A. Legchenkova, I.V. Dynamics of He atoms adsorbed on a carbon nanotube Физика низких температур |
| description |
The spectrum of a helium atom adsorbed on an armchair single-walled carbon nanotube is calculated. It is shown that a helium adatom tunnels over the surface of the nanotube within a band 10 K (for ⁴He) to 14 K (for ³He) wide. The undersaturated system of helium adatoms on nanotubes can be used as another system for studying quantum transport. It is also felt that this dynamics can be an important factor for the explanation of the large negative expansivities of nanotube bundles with adsorbed helium. |
| format |
Article |
| author |
Strzhemechny, M.A. Legchenkova, I.V. |
| author_facet |
Strzhemechny, M.A. Legchenkova, I.V. |
| author_sort |
Strzhemechny, M.A. |
| title |
Dynamics of He atoms adsorbed on a carbon nanotube |
| title_short |
Dynamics of He atoms adsorbed on a carbon nanotube |
| title_full |
Dynamics of He atoms adsorbed on a carbon nanotube |
| title_fullStr |
Dynamics of He atoms adsorbed on a carbon nanotube |
| title_full_unstemmed |
Dynamics of He atoms adsorbed on a carbon nanotube |
| title_sort |
dynamics of he atoms adsorbed on a carbon nanotube |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2011 |
| topic_facet |
Краткие сообщения |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118591 |
| citation_txt |
Dynamics of He atoms adsorbed on a carbon nanotube / M.A. Strzhemechny, I.V. Legchenkova // Физика низких температур. — 2011. — Т. 37, № 6. — С. 688–690. — Бібліогр.: 8 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT strzhemechnyma dynamicsofheatomsadsorbedonacarbonnanotube AT legchenkovaiv dynamicsofheatomsadsorbedonacarbonnanotube |
| first_indexed |
2025-07-08T14:17:20Z |
| last_indexed |
2025-07-08T14:17:20Z |
| _version_ |
1837088627781271552 |
| fulltext |
© Mikhail A. Strzhemechny and Irina V. Legchenkova, 2011
Fizika Nizkikh Temperatur, 2011, v. 37, No. 6, p. 688–690
Dynamics of He atoms adsorbed on a carbon nanotube
Mikhail A. Strzhemechny and Irina V. Legchenkova
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: strzhemechny@ilt.kharkov.ua
Received February 22, 2011
The spectrum of a helium atom adsorbed on an armchair single-walled carbon nanotube is calculated. It is
shown that a helium adatom tunnels over the surface of the nanotube within a band 10 K (for 4He) to 14 K
(for 3He) wide. The undersaturated system of helium adatoms on nanotubes can be used as another system
for studying quantum transport. It is also felt that this dynamics can be an important factor for the explanation of
the large negative expansivities of nanotube bundles with adsorbed helium.
PACS: 81.05.ub Nanotubes;
68.43.–h Chemisorption/physisorption: adsorbates on surfaces;
65.40.–b Thermal properties of crystalline solids.
Keywords: nanotubes, helium atoms, energy spectrum.
Adsorption properties of bundles of carbon nanotubes,
especially with lighter species like hydrogen and helium
isotopes as adsorbates, attract interest due to the possibility
[1] of creating low-dimensional condensed systems with
unusual quantum behavior [2–4]. Low-temperature dilato-
metry of single-walled carbon nanotubes (SWNT) satu-
rated with 4He showed [5] a considerable (as compared to
pure nanotubes) increase of the negative thermal expan-
sivity below 4 K. Despite the lack of a specific physical
mechanism responsible for this increase, the authors tenta-
tively ascribed this effect to the tunneling nature of the
helium subsystem. The recent similar experiments of the
same group on SWNT samples with adsorbed 3He revealed
[6] a further huge increase of the negative expansivity at
low temperatures. When considered together, these two
results add weight to the assumption of the crucial role of
tunneling due to the quantum nature of He atoms. It should
be noted that the total content of helium in both cases men-
tioned above amounted to a helium-to-carbon atomic ratio
below 0.1, which makes the He–He interaction a less im-
portant factor and allows description of the helium subsys-
tem as a set of independent particles in the potential relief
of a nanotube. The aim of this communication is to consid-
er the quantum dynamics of helium atoms adsorbed on the
nanotube surface in an attempt to understand the above-
mentioned low-temperature phenomena.
From scratch we preferred to deal with a certain specif-
ic model of SWNT, which would provide a good general
understanding of the problem in question. To start with, we
had at our disposal the potential curves [7] of a He atom
interacting with the outer and inner surfaces of a (12,12)
armchair SWNT. We restrict our consideration to the outer
surface, the respective potential curves being shown in
Fig. 1.
We are to deal with a quantum-mechanical eigenvalue
problem of cylindric symmetry described by the Schrödin-
ger equation
Fig. 1. Interaction energy of an outside He atom with a nanotube
of 16.27 Å in diameter [7]. The distance is counted from the na-
notube center normal to the tube axis and going through: a hex-
agon center (solid), carbon atom (dashed), and center of C–C
bond (dotted). The thick horizontal line indicates the radial
ground state energy level. The inset: analytical approximation of
the first among the three potential curves.
0
–50
–100
–150
–200
E
ne
rg
y,
K
E
ne
rg
y,
K
11 12 13 14 15
R, Å
R, Å
0
–100
–200
0 2 4 6 8 10 12 14
Dynamics of He atoms adsorbed on a carbon nanotube
Fizika Nizkikh Temperatur, 2011, v. 37, No. 6 689
2 2 2
2 2 2
1 1 ( , , ) = .
2
R U R z E
m R R R R z
⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞− + + Ψ+ φ Ψ Ψ⎢ ⎥⎜ ⎟∂ ∂⎝ ⎠ ∂φ ∂⎢ ⎥⎣ ⎦
(1)
Here R is the distance from the outer nanotube surface,
z is the coordinate along the tube axis, and φ is the cir-
cumferential angle coordinate. The actual potential energy
( , , )U R z φ does not separate into a sum of terms each of
which depends only on one variable. However, there are
reasonable arguments which allow us to approximate
( , , )U R z φ as
( , , ) = ( ) ( , ) ,U R z V R u zφ + φ (2)
where ( , )u z φ is a periodic function in two dimensions,
which will be treated in more detail below. Indeed, since
all slopes of the three potential curves in Fig. 1 are steep,
the uniformly continued function ( , )u z φ will add almost
nothing to the wave function outside the potential well and
will distort but slightly the real shapes of both the true
( , , )U R z φ and the wave function. But as soon as the po-
tential surface is represented as in Eq. (2), the variables
separate and Eq. (1) can be easily solved. In order to do
that analytically, we have to approximate both functions in
Eq. (2). We took ( )V R in the form
0
0
, = 11 Å,
( ) =
/ , > ,N
R R
V R
A R R R
⎧∞⎪
⎨
⎪⎩
(3)
where the fitting parameters are 10;N 12 107·10 K·ÅA −
(cf. the fit in the inset in Fig. 1). We used the variatio-
nal approach to evaluate the energy of the ground state for
the potential of (3) to find that the most appropriate trial
function has the form 2
0 0( ) exp ( ( ) )R R R R− −α − with
2.0404α Å 2− , which means that the wave function,
shown in Fig. 2 is centered at 11.52R Å with a charac-
teristic width of about 0.70 Å. The relevant energy level of
131.0− K is shown in Fig. 1 as a thick horizontal line.
Given this rather sharp shape of the radial wave function,
the 21/ R factor in front of the 2 2/∂ ∂φ operator can be
substituted by a constant, which will bring about an insig-
nificant shift to the ground state energy upon account of
the remaining periodic part of the Hamiltonian.
For the specific nanotube geometry considered here the
periodic potential energy can be represented in the form
0
2 3( , ) = cos cos .z z Ru z u
L L L
⎡ ⎤⎛ ⎞π π π φ
φ + +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
(4)
Here 0 7.5u K, = 2.05L Å is the natural period, name-
ly, the distance between the centers of opposite C–C bonds
in a hexagon, and 11.52R Å, which is slightly larger
than 0R and thus this difference can be disregarded. It
should be also noted that 2 3 / 2R NLπ where = 12N
is the number of hexagon strips along our specific armchair
nanotube.
In order to get insight into the problem of the variables
z and φ , we solve an auxiliary problem. Ignoring for the
time being the circumferential degree of freedom, we ap-
proximate the periodic potential along the line that unites
neighbor carbon hexagon centers and goes through the
center of the C–C bond common for these two neighbor
hexagons:
0
2( ) = cos .zu z u
L
π (5)
Now we transform Eq. (1) to dimensionless variables,
making sure that the Schrödinger equation for the periodic
function ( )u z is in the traditional dimensionless form of
the Mathieu equation [8]
2
2
( ) ( 2 cos 2 ) ( ) = 0 .d u p q u
d
ζ
+ − ζ ζ
ζ
(6)
Here 0= / aζ ξ with 0 = / 0.652a L π Å; 0= /p E ε is
the dimensionless energy eigenvalue with
2
0 2
0
= 14.26 K;
2ma
ε (7)
and 0 0= / 2 0.263q u ε is the dimensionless amplitude
of the spatial variation of the potential relief. The lowest
eigenstate of Eq. (6) is described by the Mathieu function
0 ( , )ce qζ . For sufficiently small q this periodic function
can be represented [8] as an expansion in powers of q
2
0
1 cos 4 1( , ) = 1 cos 2 .
2 32 162
qce q q⎡ ζ ⎤⎛ ⎞ζ − ζ + − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
… (8)
The third term can be omitted since for the actual values of
the parameters involved the ratio of the third term to a sum
of the first two varies in a periodic fashion between
0.0065− and –0.0019. The respective eigenvalue 0 ( )a q ,
which is the energy shift, can also be estimated from the
expression [8] 2 2
0 ( ) = / 2 7 /128 0.0343a q q q− + − −… ,
which to a good accuracy amounts to 0.49− K.
Fig. 2. The normalized partial wave function ( )RΨ .
4
3
2
1
0
11 12 13
R, Å
�
Mikhail A. Strzhemechny and Irina V. Legchenkova
690 Fizika Nizkikh Temperatur, 2011, v. 37, No. 6
The simple cosine form of the wave function in Eq. (6)
is indicative of the fact that the actual approximation for
the lowest band is a tight-binding one. This conclusion
simplifies the task of calculating the dispersion law of the
4He atom tunneling over the periodically corrugated cylin-
drical surface of a SWNT. Then the effective tunneling
Hamiltonian tH can be cast in the form
< , >
= ,
2t i j
i j
JH a a+∑ (9)
where summation runs over nearest neighbors and the
jump frequency J in absolute values is 5 K for 4He and
7 K for 3He, respectively. The spectrum ( )n kε , which is
easily calculable, is quasi-continuum along z and quantiz-
ed for lateral motion:
( ) = [cos 2 2cos cos / ]n k J kL kL n Nε π + π π . (10)
For the specific type of SWNT, = 6N and 1 6n≥ ≥ , and
the spectrum consists of three pairs of degenerate bands.
The width of the main band is 10.1 K for 4He atoms and
14.3 K for 4He atoms.
In conclusion, we point out that the experiments which
initiated this short note clearly suggest that a system of
undersaturated helium atoms on carbon nanotubes can be
viewed as a new quantum tunneling system enabling stu-
dies of quantum transport of solitary adatoms.
The authors thank V.G. Manzhelii and A.V. Dolbin for
valuable discussions and K.A. Chishko for critical reading
of the manuscript.
1. M.M. Calbi, F. Toigo, and W.W. Cole, Phys. Rev. Lett. 86,
5062 (2001).
2. S.O. Diallo, B. Fåk, M.A. Adams, O.E. Vilches, M.R. John-
son, H. Schober, and H.R. Glyde, EPL 88, 56005 (2009).
3. T.N. Antsygina, I.I. Poltavsky, K.A. Chishko, T.A. Wilson,
O.E. Vilches, and O.E. Vilches, Fiz. Nizk. Temp. 31, 1328
(2006) [Low Temp. Phys. 31, 1007 (2006)].
4. T.N. Antsygina, I.I. Poltavsky, and K.A. Chishko, J. Low
Temp. Phys. 138, 223 (2005).
5. A.V. Dolbin, V.B. Esel'son, V.G. Gavrilko, V.G. Manzhelii,
N.A. Vinnikov, S.N. Popov, and B. Sundqvist, Fiz. Nizk.
Temp. 36, 797 (2010) [Low Temp. Phys. 36, 635 (2010)].
6. A.V. Dolbin, V.B. Esel'son, V.G. Gavrilko, V.G. Manzhelii,
N.A. Vinnikov, S.N. Popov, and B. Sundqvist, Fiz. Nizk.
Temp. 37, 685 (2011) [Low Temp. Phys. 37, No. 6 (2011)].
7. L. Firlej and B. Kuchta, Colloids and Surfaces A: Physico-
chem. Eng. Aspects 241, 149 (2004).
8. G. Blanch, Mathieu Functions in: Handbook of Mathema-
tical Functions, M. Abramowitz and I.E. Stegun (eds.), NBS
Appl. Math. Ser. 55 (1964).
|