Theory of nuclear resonant inelastic x-ray scattering from ${}^{57}\mathrm{Fe}$ in a single-walled carbon nanotube

A computationally efficient phonon Green’s-function method is described for calculation of frequency spectra of single-walled carbon nanotubes containing point defects. The method is generally applicable to defects in other nanostructures. The phonon Green’s function is used to calculate line shapes of one-phonon lines in the nuclear resonant inelastic x-ray scattering from ${}^{57}\mathrm{Fe}$ embedded in a single-walled carbon nanotube. The line shapes are anisotropic and have some unusual features that can provide insight into the physical processes in nanomaterials at atomistic scales and can be used to characterize them. In particular, it may be possible to use the line shapes in certain directions to determine the chirality of a nanotube.

Figure 1

The frame of reference showing the origin and the $X$ and $Y$ axes for a cylindrical SWNT. The $Z$ axis is along the axis of the cylinder, which is normal to the plane of the paper. All the atoms are located at the surface of the cylinder. The figure shows the locations of atoms in two adjacent unit cells, $L=0$ and $L=1$, each containing two atoms, $\kappa =0$ and $\kappa =1$, and also the angular separation between atoms 00 and 10.

Figure 2

(Color online) Frequency spectra $g\left(\omega \right)$ of perfect SWNTs of similar chiralities but different diameters as function of frequency. The chiral indices and diameters are the following: top panel, (10,10), 34.6; middle panel, (8,8), 27.7; and bottom panel, (6,6), 20.8. The diameters are in units of $a∕\pi$, where $2a$ is the lattice constant of the SWNT lattice. Functions $g\left(\omega \right)$ are normalized such that their integral over the entire frequency range is unity. The frequencies are normalized with respect to the maximum phonon frequency for the SWNT.

Figure 3

(Color online) Same as in Fig. 2 for SWNTs of different chiralities but almost equal diameters. The chiral indices and diameters are the following: Top panel, (8,8), 27.7; middle panel, (10,6), 28; and bottom panel, (14,0), 28. The units and normalization are the same as in Fig. 2.

Figure 4

(Color online) Change in the frequency spectra $\Delta g\left(\omega \right)$ due to substitutional ${}^{57}\mathrm{Fe}$ defects in SWNTs of similar chiralities but different diameters as function of frequency. Functions $\Delta g\left(\omega \right)$ are normalized such that their integral over the entire frequency range is unity. Other notations are the same as in Fig. 2.

Figure 5

(Color online) Same as in Fig. 4 for SWNTs of different chiralities but almost equal diameters. Chiral indices and diameters are the same as in Fig. 3.

Figure 6

(Color online) Dependence of the line shape $U_X$ of the one-phonon NRIXS line in the radial direction ($X$ direction in Fig. 1) on the diameter of SWNTs with similar chiralities. Line shapes are normalized such that their integral over the entire frequency range is unity. Other notations are the same as in Fig. 2.

Figure 7

(Color online) Dependence of the line shape $U_Y$ of the one-phonon NRIXS line in the $Y$ direction on the chirality of SWNTs with almost equal diameters. Chiral indices and diameters, same as in Fig. 3. Normalization as described in Fig. 6.

Figure 8

(Color online) Same as Fig. 7 but in the axial direction ($Z$ direction) of the SWNTs.