Hard X-radiation emitted by a charged particle moving in a carbon nanotube

a.__

p &I

I8 November

1996

PHYSICS

LETTERS

A

Physics Letters A 222 (1996) 424-428

EL.SEl’IER

Hard X-radiation emitted by a charged particle moving in a carbon nanotube V.V. Klimov a,b*c*‘,VS. Letokhov a,b ofSpectroscopy, Russian Academy of Sciences, 142092 Troitsk. Moscow Region. Russian Federation h Optical Sciences Center; University of Arizona, Tucson, USA ’ P.N. Lebedev Physical Institute, Russian Academy ofSciences, 53 Leninsky Prospekt, I I7294 Moscow, Russian Federation a Institute

Received 28 August 1996; accepted for publication Communicated

4 September

1996

by L.J. Sham

Abstract We consider the radiation emitted by a charged particle propagating inside a single-layer carbon nanotube and interacting without delay with the screened charges of the nuclei of carbon atoms. It is shown, for example, that a beam of positrons with an energy of around 1 GeV and a divergence of some 10-j rad is captured by the nanotube and emits hard X-rays with an energy of about 0.3 MeV in the direction of its propagation. The theoretical results obtained allow one to consider

carbon nanotubes as a new source of hard X-radiation.

Considerable advances have been made in recent years in the synthesis of so-called carbon nanotubes important objects for use in nanotechnologies [ l-41. These wonderful objects, a mere nanometer in diameter, may have a quite macroscopic length of up to a few centimeters. Being hollow, nanotubes seem to be natural candidates for transporting both neutral and charged particles in various nanodevices. On the other hand, charged particles propagating in nanotubes interact with their walls and can thus generate a coherent electromagnetic radiation which can be of special interest in nanodevices. The radiation due to charged particles channeling in crystals has been well studied. Such a radiation has a whole range of properties making it of practical use [5]. The aim of the present paper is to investigate the electromagnetic effects taking place in the course of propagation of positively charged particles in nanotubes. From our point of view the most important features of radiating nanotubes are due to their large (in comparison with channels in common crystals) diameter. On the other hand, in a system consisting of a number of parallel nanotubes [ I], the ratio of the cross-sectional area of the walls to the total cross-sectional area is small, so that the system can effectively capture sufficiently wide beams and hence emit radiation as effectively. The geometry of the problem is shown in Fig. 1. The specific electromagnetic effects are due entirely to the structure of the nanotube. So let us consider this structure in more detail. Most developed to date are the methods for synthesizing nanotubes from carbon atoms [l-4]. d nanotube in this case can be visualized as a hexagonal graphite layer rolled into a tube with a diameter of the order of 1 nm. ’ E-mail: [email protected]. 0375-%01/96/$12.00 Copyright PII SO375-9601(96)00674-3

0 1996 Elsevier Science B.V. All rights reserved.

V.V. Klimov, VS. Letokhnv / Physics Letters A 222 (19961424-428

425

Nanotube

Fig.

1, Motion trajectory of a charged particle in a carbon nanotube

The radius of the single-layer tube thus obtained with the conservation of the carbon bond length,

can easily be found from the elementary

relation

associated

(1) where N is the number of elementary periods present along the tube perimeter and a = 1.42 A is the carbon bond length. Since the nanotube diameter is usually around 11 A, it is not very difficult to estimate the number of elementary cells present along the perimeter of the tube: N = 14. The main type of interaction is the Coulomb interaction between a particle and the charge of a nucleus, partially screened by the electron shell of the atom. If delay is disregarded, the interaction potential between an atom with the nuclear charge Zze located at the point 11 and a particle with the charge Zte located at the point r may be approximated by the expression [6] 1 u=e2ZtZ2

d& (

Jh

-

r-212

-t

(a*&)2

>

’

a*=T

213

3%-

QB

-

(

4(fl+a)

>

’

(2)

where a* is the screening parameter and aa = /i2/me2 is the Bohr radius. The total particle-nanotube interaction energy can be obtained by summing (2) over all the atoms of the nanotube. Insofar as the nanotube radius is large enough in comparison with the interatomic distance, the expression for the interaction potential can in a first approximation be averaged over the periodic coordinates (z, VP) to obtain the expression Z1Z2e2

D(p) = ___ 3a

u*mt

where the dimensionless U*(p) In expression

= 4Nln

part of the potential

is described

by the expression

82+~2+~*+J(a2+~2+~*)2-(2R~)* 2R2

(4)) use is made of the dimensionless

>. quantities

(4)

426

V.V. Klimou. V.S. Letokhou / Physics Letrers A 222 f1996) 424-428

WA-,

- p

d=

P=g’

-

a*

a&

and it is assumed that the particle is inside the nanotube, i.e., p < R. Consider now the motion of a charged particle along a nanotube in potential (3). In this case, one should first of all bear in mind the fact that it is only sufficiently fast particles that can be described by the classical equations of motion. Channeling near the surface of the nanotube is possible for negatively charged particles. In the case of positively charged particles, the situation is more advantageous for their propagation inside the nanotube, because a particle coming close to the surface of the nanotube is acted upon by a repulsive force due to the incomplete screening of the positively charged atomic nuciei. Considering what has been said above, we investigate in this work a sufficiently fast (relativistic) positively charged particle entering a nanotube in a direction parallel to its axis and not too close to its surface. The motion trajectory will be flat, and the motion along the nanotube axis will be uniform, while oscillations will develop in the transverse direction with their amplitude equal to the radius of the point of entry of the particle into the nanotube. The equation of motion in the transverse plane may be written down in the usual form,

$(rmVJ =-g,

(5)

is a relativistic factor wherein the radial oscillation velocity is taken to be negligible where y = l/VW in comparison with the motion velocity along the x-axis and m is the mass of the particle at rest. In the case where the particle enters the nanotube close to its axis, one may use for the potential the harmonic approximation 0 = Un $ U2p2,

6.0 =

When using the above harmonic be solved,

02 =

4NZt Z2e2 (3a)3

approximation,

the equation

of motion

z2 (R2+B

(6)

2)2'

in the radial direction

(5) can easily

p = pocos(nt>,

(7)

where po is the radius of the point of entry of the particle frequency given by n2

=

2 = 8NZl Z2e2 (@ my my(3a)3

In the case of a typical nanotube a=

2.8 x 10”

E

into the nanotube

and R is the radial oscillation

2

+.q

(8)

2)2’

with a diameter of 11 A, we have, instead of (8),

[s_,I (9)

Ji; which for positrons with an energy of 1 GeV gives fin/27r = 1013 Hz. The total radiation power P of a particle moving with an acceleration by the well-known formula (see, for example, Ref. [7 J)

where R' is the instantaneous radius of curvature of the motion inverse square of the radius of curvature is given by the relation

along a curvilinear

trajectory.

trajectory

is defined

In our case, the period-averaged

V.V. Klimov, V.S. Lerokhou/Physics

1 -=-R’2 Substituting

1 fb; 2 c4

421

Letters A 222 (1996) 424-428

(11)

’

this relation

P = 3c3m2Cl,Y 4de2 -2

into ( lo), we obtain the final expression

2 = _2

64e6( NZt Z2) 2

PO

3~3d(3a)4

E4 (82

+ d 2)4

Y2Y.

for the radiation

power of the particle,

2

(12)

where DO= pa/3a is the dimensionless radius of the point of entry of the particle into the nanotube. Accordingly, for the loss per unit length, I, we have E4

64e6( NZt 2~)~ 3c4mq

3a)4

(P

In the case of typical nanotube unit length will be as follows, I z &210-‘4

+ E 2)JY

2 (13)

.

with N = 14 ( 11 8, in diameter),

the expression

for the loss by radiation

[erg/cm].

per

(14)

For positrons with an energy of E, = 1 GeV, y = 2000, we have, at PO = 1 (pa z 4.3 A), I = 4 x 10e8 erg/cm = 25 keV/cm. In that case, the transverse oscillation energy and velocity are Ep M 10 eV and V, M 4.2 x lo6 cm/s, so that the maximum angle of the deviation of the particle from the nanotube axis is around low4 rad. Correspondingly, if the incoming beam of positrons has a divergence of the same order of magnitude, the overwhelming proportion of the particles will be captured by the nanotube and will effectively emit radiation. Our analysis of the spatial-temporal structure of the radiation shows that the radiation maximum is located at w(O) =

n (15)

1 -&cos8

where 0 is the angle between the nanotube axis and the observation direction, which is mainly governed by the Doppler effect. It can be seen from (15) that with the angle 0 fixed, radiation is emitted at a perfectly definite frequency. Radiation at the maximum frequency wmnx is emitted forward, i.e., at 0 = 0. This frequency is expressed as R

4e

-~--

w max =

1 -

P

E

(16)

3aP+E2

i.e., it rises in a nonlinear fashion as the energy of the particle increases. In the case of a typical nanotube 11 8, in diameter at pa M 4.3 A, we obtain the following

expression

instead

of (16), WmaxM 5.6 x 10’5y3/2 [s-l].

(17)

For positrons with an energy of Ez = 1 GeV, y = 2000, the energy of the maximum-frequency quanta can easily be found from ( 17) to be liw,,, z 0.33 MeV, i.e., hard X-quanta are emitted. The radiation linewidth is mainly governed by the number of oscillations the particle executes while passing through the nanotube. Fig. 2 shows the radiation energy emitted into a unit solid angle in a unit frequency interval as a function of the radiation frequency at B = 0, while Fig. 3 shows the total radiation energy emitted into a unit solid angle as a function of the angle 8. The analysis of these figures clearly shows the concentration of hard X-radiation in the beam propagation direction within the limits of angles t9 w l/y. When the particle moves close enough to the surface of the nanotube, the radiation power increases, but our estimates in this region may prove incorrect, for here one must take into account the effects associated with the

V.V. Klimou.

428

VS. Letokhoo/

Physics

Lrrrers

A 222 (1996)

424-428

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Fig. 2. Radiation energy emitted into a unit solid angle in a unit frequency interval at t9 = 0 by a positron with an energy of 1 GeV propagating in a nanotube 11 A in diameter and 1 mm long at a distance of 4.3 8, from its axis as a function of the radiation frequency w (normalization fo e2/47r2c( 1 - p: ) ‘). Fig. 3. Total radiation energy emitted into a unit solid angle by a positron with an energy of 1 GeV propagating in a nanotube I1 8, in diameter and 1 mm long at a distance of 4.3 A from its axis as a function of the angle 0 (normalization to ~*w,,,~~/471*c( 1 - pZ )?),

discrete character of the interaction between the particle and the incompletely screened nuclei of the nanotube. Nevertheless, our estimates seem quite correct for the overwhelming volume inside the nanotube. Thus, considered in this work is the radiation emitted by a positively charged particle propagating inside a carbon nanotube and interacting without delay with the screened charges of the nanotube nuclei. Analytical expressions are obtained for the total radiation power and the relationship between the radiation frequency and the entry angle of the particle. It is shown that even in the case of positrons with an energy of 1 GeV, the channeling of a beam with a divergence of the order of lop4 rad is possible. In that case, in the region of small entry angles there occurs emission of hard X-quanta with an energy of around 0.33 MeV and a width of some 0.01 MeV (for a 0.1 cm long nanotube). Though this work deals with the spontaneous emission of radiation by a charged particle, stimulated emission and amplification are possible in this case as well (as is the case with the free-electron laser), because the emission and transition lines here are far from one another (for a long nanotube) as a result of the recoil effect. These possibilities will be examined in an separate publication. The authors thank the Russian Basic Research Foundation (Grant 96-02-19753) Defense (through the intermediary of the University of Arizona) for their financial

and the US Department support of the work.

of

References [ 1 I R.E. Smalley, From balls to tubes to ropes: new materials from carbon, presentation, American Institute of Chemical Engineers, Texas Section, January Meeting in Houston, 1996; T.W. Ebbesen, Phys. Today (June 1996) 26. 21 S. Iijima and T. Ichihashi, Nature 363 (1993) 603. 31 D.S. Bethune, C.H. Klang, MS. de Vries. G. Gorman, R. Savoy, J. Vazquez and R. Beyers, Nature 363 ( 1993) 605 i 41 T. Guo, I? Nikolaev, A. Thess, D.T. Colbert and R.E. Smalley, Chem. Phys. L&t. 243 ( 1995) 49. 51 M.A. Kumakhov and F.F. Komarov, Radiation of charged particles in solids (Minsk Univ. Press, Minsk, 1985). 61 J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34, No. 14 ( 1965). (Wiley, New York, 1962). I 7 ] J. Jackson, Classical electrodynamics

South