194
7 Quadrupole Contributions from Interface and Bulk
The formulas (i)-(vi) are derived as follows. Here we employ the Einstein’s
convention of contraction.
(i)
(A × B) · (C × D) = ε ij k A j B k ε ilm C l D m
=
δ jl δ km − δ jm δ kl
A j B k C l D m = A j B k C j D k − A j B k C k D j
= (A · C) (B · D) − (A · D) (B · C)
(ii)
[∇ × (∇ × A)] i = ε ij k ∂ j [∇ × A] k = ε ij k ∂ j (ε klm ∂ l A m )
= ε ij k ε lmk ∂ j ∂ l A m =
δ il δ jm − δ im δ jl
∂ j ∂ l A m = ∂ j ∂ i A j − ∂ j ∂ j A i
= ∂ i (∇ · A) − ∇
2 A i
(iii)
∇ · (∇ × A) = ∂ i
ε ij k ∂ j A k
= ε ij k ∂ i ∂ j A k = 0
∵ ε ij k ∂ i ∂ j A k = −ε jik ∂ i ∂ j A k = 0
(iv)
[∇ × (∇φ)] i = ε ij k ∂ j (∂ k φ) = ε ij k ∂ j ∂ k φ = 0
(v)
∇ · (A × B) = ∂ i
ε ij k A j B k
= ε ij k (∂ i A j )B k + ε ij k A j (∂ i B k ) = B k ε kij ∂ i A j − A j ε jik ∂ i B k
= B · (∇ × A) − A · (∇ × B)
(vi)
[∇ × (A × B)] i = ε ij k ∂ j [A × B] k = ε ij k ∂ j (ε klm A l B m )
= (δ il δ jm − δ im δ jl )∂ j A l B m = ∂ j A i B j − ∂ j A j B i
= A i (∇ · B) + (B · ∇) A − B i (∇ · A) − (A · ∇) B i
Appendix
A.1 Physical Meaning of χ IQB
Among the four terms of nonlinear susceptibility summarized in Fig. 7.5, χ IQB
may be difficult to understand intuitively. It arises from the integral of nonlinear
polarization over the interface region in Sect. 7.2.3, though it is a pure bulk property.
The χ IQB term is known to have significant contribution in some SFG spectra
[12, 13]. Here we explain its origin and physical meaning in an illustrative manner.
As we discussed in Sect. 7.2.3, the χ IQB term in Eq. (7.30) originates from the
integral of quadrupole over the interface. Accordingly, we consider the situation in
Fig. 7.6 that the induced quadrupole is distributed over the whole interface system,
and evaluate the net polarization of interface P I in Eq. (7.26) by integrating the
dipole polarization over the interface region from z = z b to ∞. The lower bound z =
z b is arbitrarily chosen in the bulk region so that the integral encompasses the whole
interface region. In this integral of polarization, the distributed quadrupole comes
7 Quadrupole Contributions from Interface and Bulk
The formulas (i)-(vi) are derived as follows. Here we employ the Einstein’s
convention of contraction.
(i)
(A × B) · (C × D) = ε ij k A j B k ε ilm C l D m
=
δ jl δ km − δ jm δ kl
A j B k C l D m = A j B k C j D k − A j B k C k D j
= (A · C) (B · D) − (A · D) (B · C)
(ii)
[∇ × (∇ × A)] i = ε ij k ∂ j [∇ × A] k = ε ij k ∂ j (ε klm ∂ l A m )
= ε ij k ε lmk ∂ j ∂ l A m =
δ il δ jm − δ im δ jl
∂ j ∂ l A m = ∂ j ∂ i A j − ∂ j ∂ j A i
= ∂ i (∇ · A) − ∇
2 A i
(iii)
∇ · (∇ × A) = ∂ i
ε ij k ∂ j A k
= ε ij k ∂ i ∂ j A k = 0
∵ ε ij k ∂ i ∂ j A k = −ε jik ∂ i ∂ j A k = 0
(iv)
[∇ × (∇φ)] i = ε ij k ∂ j (∂ k φ) = ε ij k ∂ j ∂ k φ = 0
(v)
∇ · (A × B) = ∂ i
ε ij k A j B k
= ε ij k (∂ i A j )B k + ε ij k A j (∂ i B k ) = B k ε kij ∂ i A j − A j ε jik ∂ i B k
= B · (∇ × A) − A · (∇ × B)
(vi)
[∇ × (A × B)] i = ε ij k ∂ j [A × B] k = ε ij k ∂ j (ε klm A l B m )
= (δ il δ jm − δ im δ jl )∂ j A l B m = ∂ j A i B j − ∂ j A j B i
= A i (∇ · B) + (B · ∇) A − B i (∇ · A) − (A · ∇) B i
Appendix
A.1 Physical Meaning of χ IQB
Among the four terms of nonlinear susceptibility summarized in Fig. 7.5, χ IQB
may be difficult to understand intuitively. It arises from the integral of nonlinear
polarization over the interface region in Sect. 7.2.3, though it is a pure bulk property.
The χ IQB term is known to have significant contribution in some SFG spectra
[12, 13]. Here we explain its origin and physical meaning in an illustrative manner.
As we discussed in Sect. 7.2.3, the χ IQB term in Eq. (7.30) originates from the
integral of quadrupole over the interface. Accordingly, we consider the situation in
Fig. 7.6 that the induced quadrupole is distributed over the whole interface system,
and evaluate the net polarization of interface P I in Eq. (7.26) by integrating the
dipole polarization over the interface region from z = z b to ∞. The lower bound z =
z b is arbitrarily chosen in the bulk region so that the integral encompasses the whole
interface region. In this integral of polarization, the distributed quadrupole comes