UNC Nanotube Diffraction Simulator
http://www.physics.unc.edu/project/lcqin/www1/qin.htm
Geometry and Simulation of Electron Diffraction Pattern of Carbon Nanotube
On the graphene net, if the crystallographic basis vectors a1 and a2 are chosen to have an
inter-angle of 60 , the diameter d and helicity  of nanotube [u,v] are
d  a0 u 2  v 2  uv 
(1)
where a0  0.2461 nm is the lattice constant of graphene and
 3v 


  tan 1 

 2u  v  .
2
2
2
u

v

uv





  cos 1 
2u  v
(2)
On the diffraction plane, the principal reflections of {100} type form three principal layer lines,
they are labeled as L1, L2, L3, L0, L-3, L-2, and L-1 located at vertical position Z1, Z2, Z3, Z0, Z-3, Z-2,
and Z-1. The intensity distribution Ij(R) vs. R on the horizontal layer line lj are given in the unit
of 1/a (a = 0.2461 nm is the lattice constant of grapheme) by
2u  v
2
I1 ( R)  J v ( R u 2  v 2  uv ) at Z1 
3(u 2  v 2  uv
u  2v
2
I 2 ( R)  J u ( R u 2  v 2  uv ) at Z 2 
3(u 2  v 2  uv
2
I 3 ( R)  J ( u v ) ( R u 2  v 2  uv ) at Z 3 
uv
3(u 2  v 2  uv
(3)
(4)
(5)
2
I 0 ( R)  J 0 ( R u 2  v 2  uv ) at Z 0  0
2
I 3 ( R)  J u v ( R u 2  v 2  uv ) at Z 3  
2
I 2 ( R)  J u ( R u 2  v 2  uv ) at Z 2  
2
I 1 ( R)  J v ( R u 2  v 2  uv ) at Z 1  
(6)
u v
3(u  v 2  uv
2
u  2v
3(u 2  v 2  uv
2u  v
3(u 2  v 2  uv
(7)
(8)
(9)
There are two additional reflections of {110} type are of significant intensity when the nanotube
has indices u  v .
The Java Applet for this site is designed by Tao Wan of University of Bristol, UK.
UNC Nanotube Diffraction Simulator
http://www.physics.unc.edu/project/lcqin/www1/qin.htm
2
I 4 ( R)  J u v ( R u 2  v 2  uv ) at Z 4 
3(u  v)
3(u 2  v 2  uv
2
I  4 ( R)  J u v ( R u 2  v 2  uv ) at Z  4  
3(u  v)
3(u 2  v 2  uv
(10)
(11)
In addition, the following equations are useful
v 2 Z 2  Z1

u 2 Z1  Z 2
(12)
Z 4  2 Z1  Z 3
(13)
Online calculations can be done at the Nanotube Diffraction Simulator.
References
1.
L.-C. Qin, J. Mater. Res. 9, 2450 (1994).
2.
L.-C. Qin, Chem. Phys. Lett. 298, 23 (1998).
3.
Z. Liu and L.-C. Qin, Chem. Phys. Lett. 400, 430 (2004).
4.
Z. Liu and L.-C. Qin, Chem. Phys. Lett. 402, 202 (2005).
5.
Z. Liu and L.-C. Qin, Chem. Phys. Lett. (2005) in press.
6.
Z. Liu and L.-C. Qin, Chem. Phys. Lett. (2005) in press.
7.
L.-C. Qin, Rep. Prog. Phys. (2005).
The Java Applet for this site is designed by Tao Wan of University of Bristol, UK.