**3. Nonresonant Raman spectra: Model and method**

#### **3.1. Raman scattering**

Raman scattering from molecules and crystals was treated by many authors, among them Long [25] and Turell [26]. The main origin of the scattered light is considered to be an electric oscillating dipole, P, induced in the medium (molecules, amorphous materials, glasses, and crystals), by the electromagnetic incident field E. At first order, the induced dipole moment is given by

$$\mathbf{P} = \widetilde{\alpha} \,\mathbf{\overline{E}}\tag{5}$$

n(ω) is the Bose factor, the Raman shift *ω* is *ω* = *ω<sup>i</sup>*

*a*(*j*) = ∑*<sup>k</sup>*

the mass of the kth atom, *ω<sup>j</sup>*

They are obtained by expanding the atomic polarizability tensor π˘ <sup>k</sup>

*th* vibration mode.

the jth mode. The coefficients παγ,<sup>δ</sup>

is the frequency of the *j*

*π*,*<sup>δ</sup>*

**3.2. The bond polarizability model**

perpendicular to the bond, *α<sup>p</sup>*

*πij*

be written as follows:

*πij*,*<sup>k</sup>*

are given by:

*<sup>n</sup>* = ∑*<sup>m</sup>*

*<sup>l</sup>* <sup>=</sup> ( ∂ *α* \_\_\_*l* <sup>∂</sup>*<sup>r</sup>* )*<sup>r</sup>*=*r*<sup>0</sup> , *α* ′ *<sup>p</sup>* <sup>=</sup> ( ∂ *αp* \_\_\_ <sup>∂</sup>*<sup>r</sup>* )*<sup>r</sup>*=*r*<sup>0</sup>

\_\_1 <sup>3</sup>(*α* ′

*<sup>l</sup>* + 2*α* ′

*<sup>p</sup>*) *δij r<sup>k</sup>* + (*α* ′

and *r*<sup>0</sup>

is one parameter that determines the Raman intensities for SWCNT.

*<sup>l</sup>* − *α* ′

*πij*,*<sup>k</sup>*

where *α* ′

parameters (*α*′

butions, arising only from the polarizability *αij*

*b* (*r*) = \_\_1

where Mk

mentsu<sup>δ</sup> k , with  − *ω<sup>d</sup>* and

Structural and Vibrational Properties of C60 and C70 Fullerenes Encapsulating Carbon Nanotubes

∂*π k* \_\_\_\_\_ ∂*u<sup>δ</sup>* (*k*))<sup>0</sup>

*eδ*(*j*|*k*) (8)

http://dx.doi.org/10.5772/intechopen.71246

in terms of atom displace-

(9)

77

*th* mode, and *ω<sup>j</sup>*

, and a polarizability

<sup>3</sup> *δij*), (10)

*<sup>r</sup>* (*δik* **r***<sup>j</sup>* + *δjk r<sup>i</sup>* − 2 *r<sup>i</sup>* **r***<sup>j</sup>* **r***k*) (11)

(j| k) are the frequency and (δj) component of

of bonds *b* between nearest neighbor atoms. In

. Thus, the polarizability contribution of a particular bond *b* can

<sup>k</sup> connect the polarization fluctuations to the atomic motions.

*π*,*<sup>δ</sup> k* \_\_\_\_ √ \_\_\_ *Mk*

and e<sup>δ</sup>

*<sup>k</sup>* <sup>=</sup> <sup>∑</sup>*<sup>k</sup>*′ (

The Raman intensities can be calculated within the framework of the nonresonant bond polarizability model [27]. The basic assumption of the bond polarizability model is that the optical dielectric susceptibility of the material or molecule can be decomposed into individual contri-

*b*

<sup>3</sup>(*α<sup>l</sup>* <sup>+</sup> <sup>2</sup> *<sup>α</sup>p*) *<sup>δ</sup>ij* <sup>+</sup> (*α<sup>l</sup>* <sup>−</sup> *<sup>α</sup>p*)(*r<sup>i</sup>* **<sup>r</sup>***<sup>j</sup>* <sup>−</sup> \_\_1

where *i* and *j* are the Cartesian directions (x, y, z) and *r* is the unit vector along the bond *b* which connects the atom *n* and the atom *m* covalently bonded. Within this approach, one can assume that the bond polarizability parameters are functions of the bond lengths *r* only. The derivatives of Eq. (10) with respect to the atomic displamcent of the atom *n* in the direction *k*,

*<sup>n</sup>* , are linked to the Raman susceptibility of modes (see [27] for the detailed formalism) and

*<sup>p</sup>*)(*r<sup>i</sup>* **<sup>r</sup>***<sup>j</sup>* <sup>−</sup> \_\_1

the C60, the sum over bonds in Eq. 11 includes two types of bonds, single and double, so there

<sup>3</sup> *δij*) *r<sup>k</sup>* +

) are usually fitted with respect to the experiments. For example, in the case of

*<sup>α</sup><sup>l</sup>* <sup>−</sup> *<sup>α</sup>* \_\_\_\_\_*<sup>p</sup>*

is the equilibrium bond distance. The values of these

*aαγ*(*j*) is the *αγ* component of the so-called Raman polarizability tensor of the *j*

this model, each bond is characterized by a longitudinal polarizability, *α<sup>l</sup>*

where *α*˜ is the electronic polarizability tensor.

In a Raman experiment (**Figure 4**), a visible, or near infrared light, of frequencyω<sup>1</sup> , wave vector k1 , polarization unit vector e1 , is incident in an isotropic medium, (ε<sup>0</sup> is the dielectric constant of vacuum, c is the speed of light in vacuum, and η<sup>1</sup> is the indice of refraction of the medium at the laser frequency).

The time-averaged power flux of the Raman-scattered light in a given direction, with a frequency between ω<sup>f</sup> and ω<sup>f</sup>  + dω<sup>f</sup> in a solid angledΩ, is related to the differential scattering cross section,

$$\frac{\mathbf{d}^2 \sigma}{\mathbf{d} \Omega \mathbf{d} \,\omega\_d} = \frac{\hbar \omega\_d^2}{8\pi^2 \mathbf{c}^4} \,\omega\_1 \omega\_d^3 [\mathbf{n}(\omega) + 1] \sum\_{\alpha \flat \gamma \lambda} \mathbf{e}\_{2\alpha} \mathbf{e}\_{2\flat} \mathbf{I}\_{\alpha \flat \beta \lambda}(\omega) \,\mathbf{e}\_{1\gamma} \,\mathbf{e}\_{1\nu} \tag{6}$$

where

$$I\_{\rm q;\mu i}(\omega) = \sum\_{\langle \ }a\_{\mu}^\* \langle \mathbf{j} \rangle a\_{\mu} \langle \mathbf{j} \rangle \frac{1}{2} \frac{1}{\omega\_{\rangle}} (\delta(\omega - \omega\_{\rangle}) - \delta(\omega + \omega\_{\rangle})) \tag{7}$$

**Figure 4.** Sketch of a Raman scattering experiment.

n(ω) is the Bose factor, the Raman shift *ω* is *ω* = *ω<sup>i</sup>*  − *ω<sup>d</sup>* and

$$a\_{\alpha\gamma}(j) = \sum\_{k\delta} \frac{\pi\_{\alpha\beta}^{\mathbb{A}}}{\sqrt{M\_k}} e\_{\delta}(j \mid k) \tag{8}$$

where Mk the mass of the kth atom, *ω<sup>j</sup>* and e<sup>δ</sup> (j| k) are the frequency and (δj) component of the jth mode. The coefficients παγ,<sup>δ</sup> <sup>k</sup> connect the polarization fluctuations to the atomic motions. They are obtained by expanding the atomic polarizability tensor π˘ <sup>k</sup> in terms of atom displacementsu<sup>δ</sup> k , with

$$
\pi\_{\text{eq},b}^k = \sum\_{k'} \left( \frac{\partial \pi\_{\text{eq}}^k}{\partial u\_\circ(k)} \right)\_0 \tag{9}
$$

*aαγ*(*j*) is the *αγ* component of the so-called Raman polarizability tensor of the *j th* mode, and *ω<sup>j</sup>* is the frequency of the *j th* vibration mode.

#### **3.2. The bond polarizability model**

**3. Nonresonant Raman spectra: Model and method**

where *α*˜ is the electronic polarizability tensor.

76 Fullerenes and Relative Materials - Properties and Applications

and ω<sup>f</sup>

dΩd ω<sup>d</sup>

**Figure 4.** Sketch of a Raman scattering experiment.

stant of vacuum, c is the speed of light in vacuum, and η<sup>1</sup>

 + dω<sup>f</sup>

<sup>=</sup> ħω<sup>d</sup> 2 \_\_\_\_\_ <sup>8</sup> <sup>π</sup><sup>2</sup> c4 <sup>ω</sup><sup>i</sup> <sup>ω</sup><sup>d</sup>

(*ω*) = ∑*<sup>j</sup> a*

, polarization unit vector e1

medium at the laser frequency).

quency between ω<sup>f</sup>

d2 \_\_\_\_\_\_ <sup>σ</sup>

*I*

cross section,

where

Raman scattering from molecules and crystals was treated by many authors, among them Long [25] and Turell [26]. The main origin of the scattered light is considered to be an electric oscillating dipole, P, induced in the medium (molecules, amorphous materials, glasses, and crystals), by the electromagnetic incident field E. At first order, the induced dipole moment is

**P** = ˜α **E** (5)

The time-averaged power flux of the Raman-scattered light in a given direction, with a fre-

<sup>∗</sup> (*j*) *<sup>a</sup>* (*j*) \_\_\_1 2 *ω<sup>j</sup>*

, is incident in an isotropic medium, (ε<sup>0</sup>

in a solid angledΩ, is related to the differential scattering

3[n(ω) + 1] ∑αβγλ e2αe2βIαγβλ(ω) e1<sup>γ</sup> e1λ, (6)

(*δ*(*ω* − *ωj*) − *δ*(*ω* + *ωj*)), (7)

, wave vec-

is the dielectric con-

is the indice of refraction of the

In a Raman experiment (**Figure 4**), a visible, or near infrared light, of frequencyω<sup>1</sup>

**3.1. Raman scattering**

given by

tor k1

The Raman intensities can be calculated within the framework of the nonresonant bond polarizability model [27]. The basic assumption of the bond polarizability model is that the optical dielectric susceptibility of the material or molecule can be decomposed into individual contributions, arising only from the polarizability *αij b* of bonds *b* between nearest neighbor atoms. In this model, each bond is characterized by a longitudinal polarizability, *α<sup>l</sup>* , and a polarizability perpendicular to the bond, *α<sup>p</sup>* . Thus, the polarizability contribution of a particular bond *b* can be written as follows:

$$
\pi\_{\vec{\eta}}^{b}(\mathbf{r}) = \frac{1}{3} (\boldsymbol{\alpha}\_{l} + \boldsymbol{\Omega} \, \boldsymbol{\alpha}\_{p}) \, \boldsymbol{\delta}\_{\vec{\eta}} + (\boldsymbol{\alpha}\_{l} - \boldsymbol{\alpha}\_{p}) \Big(\mathbf{r}\_{i} \, \mathbf{r}\_{j} - \frac{1}{3} \, \boldsymbol{\delta}\_{\vec{\eta}}\Big), \tag{10}
$$

where *i* and *j* are the Cartesian directions (x, y, z) and *r* is the unit vector along the bond *b* which connects the atom *n* and the atom *m* covalently bonded. Within this approach, one can assume that the bond polarizability parameters are functions of the bond lengths *r* only. The derivatives of Eq. (10) with respect to the atomic displamcent of the atom *n* in the direction *k*, *πij*,*<sup>k</sup> <sup>n</sup>* , are linked to the Raman susceptibility of modes (see [27] for the detailed formalism) and are given by:

$$\boldsymbol{\pi}\_{\boldsymbol{\upbeta},k}^{\*} = \sum\_{\boldsymbol{\upalpha}} \frac{1}{3} (\boldsymbol{\alpha}^{\prime}{}\_{\boldsymbol{l}} + 2\boldsymbol{\alpha}^{\prime}{}\_{\boldsymbol{p}}) \, \boldsymbol{\delta}\_{\boldsymbol{\upbeta}} \mathbf{r}\_{\boldsymbol{k}} + \left(\boldsymbol{\alpha}^{\prime}{}\_{\boldsymbol{l}} - \boldsymbol{\alpha}^{\prime}{}\_{\boldsymbol{p}}\right) \left(\mathbf{r}\_{\boldsymbol{l}} \mathbf{r}\_{\boldsymbol{j}} - \frac{1}{3}\boldsymbol{\delta}\_{\boldsymbol{\upbeta}}\right) \mathbf{r}\_{\boldsymbol{k}} + \frac{a\_{\boldsymbol{l}} - a\_{\boldsymbol{r}}}{r} (\boldsymbol{\delta}\_{\boldsymbol{\upalpha}} \mathbf{r}\_{\boldsymbol{j}} + \boldsymbol{\delta}\_{\boldsymbol{\upmu}} \mathbf{r}\_{\boldsymbol{i}} - 2\mathbf{r}\_{\boldsymbol{r}} \mathbf{r}\_{\boldsymbol{j}} \mathbf{r}\_{\boldsymbol{k}}) \tag{11}$$

where *α* ′ *<sup>l</sup>* <sup>=</sup> ( ∂ *α* \_\_\_*l* <sup>∂</sup>*<sup>r</sup>* )*<sup>r</sup>*=*r*<sup>0</sup> , *α* ′ *<sup>p</sup>* <sup>=</sup> ( ∂ *αp* \_\_\_ <sup>∂</sup>*<sup>r</sup>* )*<sup>r</sup>*=*r*<sup>0</sup> and *r*<sup>0</sup> is the equilibrium bond distance. The values of these parameters (*α*′ ) are usually fitted with respect to the experiments. For example, in the case of the C60, the sum over bonds in Eq. 11 includes two types of bonds, single and double, so there is one parameter that determines the Raman intensities for SWCNT.

#### **3.3. The spectral moment's method**

The direct method to calculate the Raman spectrum requires, besides the polarization parameters, direct diagonalization of the dynamical matrix to obtain the eigenvalues and the eigenvectors of the system. The diagonalization fails or requires long computing time when the system contains a large number of atoms, as for a long C60 and C70 chains inside nanotubes. By contrast, the spectral moment's method allows computing directly the Raman responses of harmonic systems without any diagonalization of the dynamical matrix [28, 29]. In the case of C60 and C70 peapods, calculations of the Raman spectra of peapods showed that approximately 500 moments are sufficient to obtain good results for larger samples (~25,000 degrees of freedom).
