16
H. Kaur et al.
where, α
is the linear polarizability and E is the electric field component of incident
IR radiation. The polarization vector (P) for the incident electric field E can be given
as [27, 32]:
P = ε 0 χ
(1) E
(20)
Here, ε 0 is the vacuum permittivity and χ
(1) is defined as the linear electric susceptibility. Thus P gives the average number of dipoles induced per unit volume and it
varies linearly with the dielectric susceptibility of the sample and the incident electric field. The light-matter interaction process based on Eq. (20) which is a linear
optical process can be studied by applying ATR-IR spectroscopy [26, 27, 32, 33].
The wavelength-dependent propagation of IR radiation through the sample induces
the response in the form of dispersion and the wavelength-dependent transfer of IR
energy to the sample yields the absorption. Thus χ
(1) in Eq. (20) is a complex tensor
quantity and can be defined as a function of the complex refractive index of the
sample [22, 27, 31, 33]:
χ
(1)
= real
χ
(1)
+ iIm
χ
(1)
=
n
2
− 1
+ i[κ]
2
(21)
It is evident from Eq. (21) in collaboration with Eqs. 4 and 5 that the absorption
of IR radiation by the sample completely relies on the imaginary part of linear
electronic susceptibility tensor i.e., Im
χ
(1)
and the spectral profile of Im
χ
(1)
in the IR absorption spectrum depends on the resonant vibrational modes of the
molecules present in the sample and the number of molecules present in the region
[32, 33]. The Im
χ
(1)
is directly proportional to the IR absorbance by ATR-FTIR
spectroscopy (A AT R ) and it is defined as [27, 32, 33]:
Im
χ
(1)
= c AT R A AT R
(22)
c AT R ∝
2n 1 cos θ
N R d p ω I R
(23)
where c AT R represents the ATR correction factor. The oscillator strength of the
absorption band can be quantified by data fitting of the observed spectral profile
in Lorentzian, Gaussian, or Voigt functions. The fit of the Voigt function profile
contains the contribution of both Lorentzian and Gaussian profiles and defined as the
convolution of Lorentzian and Gaussian functions [33]:
Im
χ
(1)
=
q
A q
exp
− (ωIR−ωq)
2
(q)
2
1 + (ωIR−ωq)
2
(q)
2
(24)
H. Kaur et al.
where, α
is the linear polarizability and E is the electric field component of incident
IR radiation. The polarization vector (P) for the incident electric field E can be given
as [27, 32]:
P = ε 0 χ
(1) E
(20)
Here, ε 0 is the vacuum permittivity and χ
(1) is defined as the linear electric susceptibility. Thus P gives the average number of dipoles induced per unit volume and it
varies linearly with the dielectric susceptibility of the sample and the incident electric field. The light-matter interaction process based on Eq. (20) which is a linear
optical process can be studied by applying ATR-IR spectroscopy [26, 27, 32, 33].
The wavelength-dependent propagation of IR radiation through the sample induces
the response in the form of dispersion and the wavelength-dependent transfer of IR
energy to the sample yields the absorption. Thus χ
(1) in Eq. (20) is a complex tensor
quantity and can be defined as a function of the complex refractive index of the
sample [22, 27, 31, 33]:
χ
(1)
= real
χ
(1)
+ iIm
χ
(1)
=
n
2
− 1
+ i[κ]
2
(21)
It is evident from Eq. (21) in collaboration with Eqs. 4 and 5 that the absorption
of IR radiation by the sample completely relies on the imaginary part of linear
electronic susceptibility tensor i.e., Im
χ
(1)
and the spectral profile of Im
χ
(1)
in the IR absorption spectrum depends on the resonant vibrational modes of the
molecules present in the sample and the number of molecules present in the region
[32, 33]. The Im
χ
(1)
is directly proportional to the IR absorbance by ATR-FTIR
spectroscopy (A AT R ) and it is defined as [27, 32, 33]:
Im
χ
(1)
= c AT R A AT R
(22)
c AT R ∝
2n 1 cos θ
N R d p ω I R
(23)
where c AT R represents the ATR correction factor. The oscillator strength of the
absorption band can be quantified by data fitting of the observed spectral profile
in Lorentzian, Gaussian, or Voigt functions. The fit of the Voigt function profile
contains the contribution of both Lorentzian and Gaussian profiles and defined as the
convolution of Lorentzian and Gaussian functions [33]:
Im
χ
(1)
=
q
A q
exp
− (ωIR−ωq)
2
(q)
2
1 + (ωIR−ωq)
2
(q)
2
(24)