3.3 Properties of χ (2)
63
interface. The hyperpolarizability of an individual molecule α (2) and the nonlinear
susceptibility of the interface χ (2) are treated on the same formulation. The relation
between χ (2) and α (2) is discussed as follows. In the following formulas about
molecular orientation, the suffixes p, q, r stand for the space-fixed coordinates
(x, y, z), while p , q , r for the molecule-fixed coordinates (ξ, η, ζ ).
Two Factors: Density and Orientation
Suppose that the interface system consists of molecules, the second-order susceptibility of the interface χ (2) is approximated by the sum of the second-order
susceptibility of constituent molecules α (2) ,
χ
(2)
pqr ≈
N
l=1
α
(2)
l,pqr = N · α
(2)
pqr ,
(3.41)
where the suffix l stands for the constituent N molecules, and α
(2)
pqr =
1
N
N
l=1
α
(2)
l,pqr
is the average of α (2) in the space-fixed coordinate. Equation (3.41) indicates that
χ
(2)
pqr is proportional to the number of molecules N and the average of hyperpolarizability α
(2)
pqr . α
(2)
pqr is the average over molecules with various orientations, and
thus an index to the orientational order of the molecules. The two factors, number
density and orientational order, are regarded to govern the intensity of SFG signal
in a qualitative sense.
We note that Eq. (3.41) expresses the nonlinear polarization of the whole
system simply as the assembly of the induced polarizations of molecules. This
χ (2) expression of Eq. (3.41) is widely utilized in qualitative analysis of SFG
amplitude. However, this expression neglects dielectric coupling among molecular
polarizations, and the underlying approximation can break down in Sect. 9.3 [6, 7].
More accurate treatment of polarization will be provided in Chap. 5.
Rotational Matrix
Equation (3.41) relates the tensor elements χ
(2)
pqr and α
(2)
pqr on the common spacefixed coordinates. However, the molecular property of α
(2)
pqr is conveniently presented on the molecule-fixed coordinates, while χ
(2)
pqr is usually given on the
space-fixed coordinates of the experimental geometry. Accordingly, we define
the transformation of the hyperpolarizability tensor α (2) from the molecule-fixed
coordinates (namely ξ, η, ζ ) to the space-fixed ones (x, y, z).
We denote the unit vectors along the molecule-fixed axes ξ, η, ζ by e ξ , e η , e ζ ,
respectively, and the unit vectors along the spaced-fixed axes x, y, z by e x , e y , e z .
63
interface. The hyperpolarizability of an individual molecule α (2) and the nonlinear
susceptibility of the interface χ (2) are treated on the same formulation. The relation
between χ (2) and α (2) is discussed as follows. In the following formulas about
molecular orientation, the suffixes p, q, r stand for the space-fixed coordinates
(x, y, z), while p , q , r for the molecule-fixed coordinates (ξ, η, ζ ).
Two Factors: Density and Orientation
Suppose that the interface system consists of molecules, the second-order susceptibility of the interface χ (2) is approximated by the sum of the second-order
susceptibility of constituent molecules α (2) ,
χ
(2)
pqr ≈
N
l=1
α
(2)
l,pqr = N · α
(2)
pqr ,
(3.41)
where the suffix l stands for the constituent N molecules, and α
(2)
pqr =
1
N
N
l=1
α
(2)
l,pqr
is the average of α (2) in the space-fixed coordinate. Equation (3.41) indicates that
χ
(2)
pqr is proportional to the number of molecules N and the average of hyperpolarizability α
(2)
pqr . α
(2)
pqr is the average over molecules with various orientations, and
thus an index to the orientational order of the molecules. The two factors, number
density and orientational order, are regarded to govern the intensity of SFG signal
in a qualitative sense.
We note that Eq. (3.41) expresses the nonlinear polarization of the whole
system simply as the assembly of the induced polarizations of molecules. This
χ (2) expression of Eq. (3.41) is widely utilized in qualitative analysis of SFG
amplitude. However, this expression neglects dielectric coupling among molecular
polarizations, and the underlying approximation can break down in Sect. 9.3 [6, 7].
More accurate treatment of polarization will be provided in Chap. 5.
Rotational Matrix
Equation (3.41) relates the tensor elements χ
(2)
pqr and α
(2)
pqr on the common spacefixed coordinates. However, the molecular property of α
(2)
pqr is conveniently presented on the molecule-fixed coordinates, while χ
(2)
pqr is usually given on the
space-fixed coordinates of the experimental geometry. Accordingly, we define
the transformation of the hyperpolarizability tensor α (2) from the molecule-fixed
coordinates (namely ξ, η, ζ ) to the space-fixed ones (x, y, z).
We denote the unit vectors along the molecule-fixed axes ξ, η, ζ by e ξ , e η , e ζ ,
respectively, and the unit vectors along the spaced-fixed axes x, y, z by e x , e y , e z .