carbon nanotubes have been extensively studied because of   their interesting physical properties and potential applications . motivated by this success , scientists have been exploring nanotubes and nanostructures made of different materials . in particular ,
theoretical studies have also been carried out to investigate the electronic , optical and elastic properties of bc2n nanotubes using the first - principles and tight - binding methods , respectively [ 3 - 6 ] .
besides the elastic and electronic properties , theoretical and experimental research on phonon properties of bc2n nanotubes is also useful in understanding the properties of the nanotubes .
furthermore , symmetry properties of nanotubes have profound implications on their physical properties , such as photogalvanic effects in boron nitride nanotubes .
studies on the symmetry properties of carbon nanotubes predicted the raman- and infrared - active vibrations in the single - walled carbon nanotubes , which are consistent with the experimental data   and theoretical calculations .
a similar work was carried out by alon on boron nitride nanotubes , and the results were later confirmed by first - principles calculations . and the symmetry of bc2n nanotube was reported .
the purpose of this study is to extend the symmetry analysis to bc2n nanotubes and to determine their line groups .
the number of raman- and infrared ( ir)-active vibrations of the bc2n nanotubes is determined accordingly .
similar to carbon or boron nitride nanotubes , a single - walled bc2n nanotube can be completely specified by the chiral vector which is given in terms of a pair of integers ( nm ) .
however , compared to a carbon and boron nitride nanotubes , different bc2n nanotubes can be obtained by rolling up a bc2n sheet along different directions , as shown in fig . 1a , because of the anisotropic geometry of the bc2n sheet . if we follow the notations for carbon nanotubes , at least two types of zigzag bc2n nanotubes and two types of armchair nanotubes can be obtained . for convenience ,
we refer the two zigzag nanotubes obtained by rolling up the bc2n sheet along the a1 and the a2 directions as zz-1 and zz-2 , respectively , and two armchair nanotubes obtained by rolling up the bc2n sheet along the r1 and r2 directions as ac-1 and ac-2 , respectively .
the corresponding transactional lattice vectors along the tube axes are ta1 , ta2 , tr1 , and tr2 , respectively , as shown in fig .
it is noted that ta2 is parallel to r2 , tr1 to b1 , and tr2 to a2 .
we first consider the achiral carbon nanotubes with the rotation axis of order n , i.e. , zigzag ( n , 0 ) or armchair ( nn ) .
the nonsymmorphic line - group   describing such achiral carbon nanotubes can be decomposed in the following way : ( 1 ) where   is the 1d translation group with the primitive translation tz = |tz| , and e is the identity operation .
the screw axis   involves the smallest nonprimitive translation and rotation . the corresponding bc2n sheet of the zigzag ( n , 0 ) bc2n nanotubes ( zz-1 ) ( fig .
in this case , thednhanddndpoint groups reduce tocnvdue to the lack of horizontal symmetry axis / plane , ands2nvanishes for the lack of the screw axis .
2,(3 ) to determine the symmetries at the  point of the 12 n ( nis the number of unit cells in the tube andn = nfor zz-1 bc2n nanotubes ) of phonons inzz-1 bc2n nanotubes and the number of raman- or ir - active modes , we have to associate them with the irreducible representations ( irrep s ) ofcnv . here , two cases need to be considered .
2d projections of zigzag bc2n nanotubes ( zz-1).zis a glide plane nis odd ( orn = 2 m + 1 , m is an integer ) the character table of c(2m+1)v possesses m + 2 irrep s , i.e.,(4 ) the 12 n phonon modes transform according to the following irreps:(5 ) stands for the reducible representation of the atom positions inside the unit cell . the prefactor of 4 in   reflects the four equivalent and disjoint sublattices made by the four atoms in the zz-1 bc2n nanotubes .  
is the vector representation . of these modes , the ones that transform according to   ( the tensor representation ) or    are raman- or ir - active , respectively . out of the 12 n modes , four have vanishing frequencies , which transform as   and   corresponding to the three translational degrees of freedom giving rise to null vibrations of zero frequencies , and one rotational degree about the tube s own axis , respectively.(7)(8 ) nis even ( orn = 2m , mis an integer ) the character table of c2mv possesses m + 3 irrep s , i.e.,(9 ) the 12 n phonon modes transform according to the following irreps:(10 )   is the vector representation . of these modes , the ones that transform according to   ( the tensor representation ) or   are raman- or ir - active , respectively
. out of the 12 n modes , four ( which transform as   and ) have vanishing frequencies .(12)(13 ) the numbers of raman- and ir- active modes are 30 and 18 , respectively , for zz-1 bc2n nanotubes irrespective   n. the armchair ( n , n ) bc2n nanotubes ( ac-1 ) ( fig .
thednhanddndpoint groups reduce tocnhowing to the lack ofc2axes ands2nvanishes for the lack of the screw axis.(14 ) the point group of the line group is readily obtained from eq .
( at the  point ) of the 12 n ( n = n ) phonons in ac-1 bc2n nanotubes and the number of raman- or ir - active modes , two cases need consideration , by associating them with the irrep s of cnh .
2d projections of armchair bc2n nanotubes ( ac-1).zis a glide plane nis odd ( n = 2 m + 1 ) the character table of c(2m+1)h possesses 4 m + 2 irrep s , i.e.,(16 ) the 12 n phonon modes transform according to the following irreps:(17 ) and   is the vector representation
. of these modes , the ones that transform according to   ( the tensor representation ) or   are raman- or ir - active , respectively . out of the 12 n modes , four ( which transform as   and ) have vanishing frequencies .(19)(20 ) the character table of c2mh possesses 4 m irrep s , i.e.,(21 ) the 12
n phonon modes transform according to the following irreps:(22 )   is the vector representation . of these modes , the ones that transform according to   ( the tensor representation ) or  
are raman- or ir - active , respectively . out of the 12 n modes , four ( which transform as   and ) have vanishing frequencies .(24)(25 ) the numbers of raman- and ir- active modes are 19 and 10 , respectively , for ac-1 bc2n nanotubes in irrespective of   n. the numbers of raman- and ir- active phonon modes for zz-1 bc2n nanotubes are almost twice as for ac-1 bc2n nanotubes , which is similar to boron nitride nanotubes .
the nonsymmorphic line group describing the ( ) -chiral carbon nanotubes can be decomposed as follows:(26 ) where ; wheredris the greatest common divisor of   and ; dis the greatest common divisor of   and ; sn / dand snare the screw - axis operations with the orders ofn / dandn , respectively .
26,(27 ) where   and   are the rotations embedded in sn / dand sn , respectively .
for chiral ( nm ) bc2n nanotubes , the point group dn reduces to cn due for the lack of c2 axes . here ,
, where dr is the greatest common divisor of    and ; d is the greatest common divisor of   and .
4a and b , respectively , for zz-2 and ac-2 bc2n nanotubes , n = 4n .
the point group corresponding to the two models is expressed as:(28 ) the character table of cn has n irrep s , i.e.,(29 ) the 12 n phonon modes transform according to the following irreps:(30 ) where   and . of these modes , the ones that transform according to   and/or    are raman- and/or ir- active , respectively .
out of the 24 n modes , four ( which transform as   and ) have vanishing frequencies .(31)(32 ) experimentally , only several raman / ir - active modes can be observed .
the e2 g mode around 1580 cm is related to the stretching mode of c  c bond .
the experimental raman spectra between 100 and 300 cm should be attributed to e1 g and a1 g modes .
nis odd ( orn = 2 m + 1 , m is an integer ) the character table of c(2m+1)v possesses m + 2 irrep s , i.e.,(4 ) the 12 n phonon modes transform according to the following irreps:(5 ) stands for the reducible representation of the atom positions inside the unit cell . the prefactor of 4 in   reflects the four equivalent and disjoint sublattices made by the four atoms in the zz-1 bc2n nanotubes .  
is the vector representation . of these modes , the ones that transform according to   ( the tensor representation ) or    are raman- or ir - active , respectively . out of the 12 n modes , four have vanishing frequencies , which transform as   and   corresponding to the three translational degrees of freedom giving rise to null vibrations of zero frequencies , and one rotational degree about the tube s own axis , respectively.(7)(8 )
nis even ( orn = 2m , mis an integer ) the character table of c2mv possesses m + 3 irrep s , i.e.,(9 ) the 12 n phonon modes transform according to the following irreps:(10 )   is the vector representation . of these modes , the ones that transform according to   ( the tensor representation ) or   are raman- or ir - active , respectively
. out of the 12 n modes , four ( which transform as   and ) have vanishing frequencies .(12)(13 ) the numbers of raman- and ir- active modes are 30 and 18 , respectively , for zz-1 bc2n nanotubes irrespective   n. the armchair ( n , n ) bc2n nanotubes ( ac-1 ) ( fig .
thednhanddndpoint groups reduce tocnhowing to the lack ofc2axes ands2nvanishes for the lack of the screw axis.(14 ) the point group of the line group is readily obtained from eq .
( at the  point ) of the 12 n ( n = n ) phonons in ac-1 bc2n nanotubes and the number of raman- or ir - active modes , two cases need consideration , by associating them with the irrep s of cnh .
nis odd ( n = 2 m + 1 ) the character table of c(2m+1)h possesses 4 m + 2 irrep s , i.e.,(16 ) the 12 n phonon modes transform according to the following irreps:(17 ) and   is the vector representation . of these modes , the ones that transform according to   ( the tensor representation ) or   are raman- or ir - active , respectively . out of the 12 n modes , four ( which transform as   and ) have vanishing frequencies .(19)(20 )
the character table of c2mh possesses 4 m irrep s , i.e.,(21 ) the 12 n phonon modes transform according to the following irreps:(22 )   is the vector representation . of these modes , the ones that transform according to   ( the tensor representation ) or   are raman- or ir - active , respectively
. out of the 12 n modes , four ( which transform as   and ) have vanishing frequencies .(24)(25 ) the numbers of raman- and ir- active modes are 19 and 10 , respectively , for ac-1 bc2n nanotubes in irrespective of   n. the numbers of raman- and ir- active phonon modes for zz-1 bc2n nanotubes are almost twice as for ac-1 bc2n nanotubes , which is similar to boron nitride nanotubes .
the nonsymmorphic line group describing the ( ) -chiral carbon nanotubes can be decomposed as follows:(26 ) where ; wheredris the greatest common divisor of   and ; dis the greatest common divisor of   and ; sn / dand snare the screw - axis operations with the orders ofn / dandn , respectively .
26,(27 ) where   and   are the rotations embedded in sn / dand sn , respectively .
for chiral ( nm ) bc2n nanotubes , the point group dn reduces to cn due for the lack of c2 axes . here ,
, where dr is the greatest common divisor of    and ; d is the greatest common divisor of   and .
4a and b , respectively , for zz-2 and ac-2 bc2n nanotubes , n = 4n .
the point group corresponding to the two models is expressed as:(28 ) the character table of cn has n irrep s , i.e.,(29 ) the 12 n phonon modes transform according to the following irreps:(30 ) where   and . of these modes , the ones that transform according to   and/or    are raman- and/or ir- active , respectively . out of
the 24 n modes , four ( which transform as   and ) have vanishing frequencies .(31)(32 ) experimentally , only several raman / ir - active modes can be observed .
the e2 g mode around 1580 cm is related to the stretching mode of c  c bond .
the experimental raman spectra between 100 and 300 cm should be attributed to e1 g and a1 g modes .
in summary , the symmetry properties of bc2n nanotubes were discussed based on line group .
all bc2n nanotubes possess nonsymmorphic line groups , just like carbon nanotubes   and boron nitride nanotubes .
contrary to carbon and boron nitride nanotubes , armchair and zigzag bc2n nanotubes belong to different line groups , depending on the index n ( even or odd ) and the vector chosen . by utilizing the symmetries of the factor groups of the line groups , it was found that all zz-1 bc2n nanotubes have 30 raman- and 18 ir- active phonon modes ; all ac-1 bc2n nanotubes have 19 raman- and 10 ir - active phonon modes ; all zz-2 , ac-2 , and other chiral bc2n nanotubes have 33 raman- and 21 ir - active phonon modes .
it is noticed that the numbers of raman- and ir- active phonon modes in zz-1 bc2n nanotubes are almost twice as in ac-1 bc2n nanotubes , but which is almost the same as those in chiral , zz-2 , and ac-2 bc2n nanotubes .
the situation in bc2n nanotubes is different from that in carbon or boron nitride nanotubes .
 the symmetry properties of the single - walled bc2n nanotubes were investigated . 
 all the bc2n nanotubes possess nonsymmorphic line groups . 
 in contrast with the carbon and boron nitride nanotubes , armchair and zigzag bc2n nanotubes belong to different line groups , depending on the index n ( even or odd ) and the vector chosen . 
 the number of raman- active phonon modes is almost twice that of the infrared - active phonon modes for all kinds of bc2n nanotubes . 
Introduction
Structures of BC
Symmetry of BC
Case 1
Case 2
Case 1
Case 2
Conclusions