Abstract Submitted to the ; NANOTUBE'05 Conference:

The usual description of atomic positions of the atoms forming a carbon nanotube is based on the high symmetry of these tubes. Each chiral nanotube is a single orbit of its symmetry group G={ (C_q^r|na/q)^tC_n^sU^u | t=0,1,-1,2,-2,..., s=0,1,...,n-1, u=0,1 } with trivial stabilizer. There is a one-to-one correspondence between the set of all the atoms of the nanotube and the set S=Zx{0,1,...,n-1}x{0,1}, where Z={...,-2,-1,0,1,2,...} is the set of all integers. The use of S as a mathematical model offers some advantages, but the description of nearest neighbours of an atom is rather complicated and the scalar product has an unusual expression.
Our aim is to present an alternate rather different description which can be more advantageous in certain cases. Let e₀=(1,0), e₁= (cos 2p/3,sin 2p/3), e₂= (cos 4p/3,sin 4p/3). The points x₀e₀+x₁e₁+ x₂e₂ corresponding to all the elements of the subset L={ (x₀,x₁,x₂) | x₀+x₁+x₂=0,1 } of Z³ are distinct and form a honeycomb lattice. This allows us to use the set L as a mathematical model for honeycomb lattice. In this description the nearest neighbours of x=(x₀,x₁,x₂) are (x₀+s(x),x₁,x₂), (x₀,x₁+s(x),x₂) and (x₀,x₁,x₂ +s(x)), where s(x) =(-1)^{x₀+x₁+x₂}. The symmetry transformations and the geometric invariants have a very simple form. The subset T={ (x₀,x₁,x₂) | x₀+x₁+x₂=0 } of L corresponds to the translation subgroup of G.
By starting from a fixed element c of T we define in L the following equivalence relation: x~y if x-y is an integer multiple of c. The coset [x]={ y | y~x } corresponding to x is [x]=x+Zc. We use the factor space L=L /~ formed by all the cosets [x] as a mathematical model for the carbon nanotube with the chiral vector c. There is a one-to-one correspondence between the set of all the atoms of the nanotube and the set L. Each rational number is a class of equivalent fractions, called its representatives. In computation, the rational numbers are replaced by representatives but the result does not depend on the representatives we choose. In a very similar way, in our approach [ http://xxx.lanl.gov;, math-ph/0403011 ] each atom of the nanotube corresponds to a coset [x] and in computation we use representatives of these cosets. A mathematical object defined on L can have geometric or physical meaning only if it is independent on the representatives we choose. In our approach the scalar product has the usual form, and generally, the equations and dispersion relations occuring in the usual theories become simpler and more symmetric [ http://fpcm5.fizica.unibuc.ro/~ncotfas; ].

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