212
8 Other Topics
These relations are readily confirmed from the fact that χ (2) for an isotropic material
should be invariant by arbitrary rotation around an arbitrary axis. For example, if we
rotate the system by 90 ◦ around the z axis, the coordinates should change as follows.
x → y, y → −x, z → z
(rotation around z).
Accordingly, χ
(2)
xyz = −χ
(2)
yxz is required in order that the rotation does not affect
the material properties. The similar argument holds to derive the other relations in
Eq. (8.20).
Equation (8.20) has one independent variable, χ
(2)
chiral . It is given by the average,
χ
(2)
chiral =
1
6
x∼z
p,q,r
ε pqr χ
(2)
pqr ,
(8.21)
where ε pqr is the Levi-Civita antisymmetric tensor (see Appendix A.2),
ε pqr =
⎧
⎨
⎩
1 (pqr = xyz, yzx, zxy),
−1 (pqr = yxz, zyx, xzy),
0 (otherwise).
(7.132)
Equation (8.21) clarifies the invariance of χ
(2)
chiral to arbitrary rotation, since the LeviCivita tensor is invariant as shown in Eq. (7.137) in Appendix A.2. Therefore, the
expression of Eq. (8.21) is valid for the molecule-fixed coordinates as well.
(ii) Interface Then, we consider the situation that the chiral material forms an
interface of C ∞ symmetry.
[Problem 8.2] Suppose a chiral interface system of azimuthal C ∞ symmetry. What
relations are required among the six χ (2) elements of chiral origin in Eq. (8.19)?
In a chiral system with interface, the nonvanishing χ (2) elements of this system
are attributed to either interface or chiral origin. We find from Sect. 3.3.3 and
Problem 8.2 that the χ
(2)
pqr elements of chiral origin are distinct from the χ
(2)
pqr
elements of interface origin. The nonvanishing elements of χ
(2)
pqr are summarized
below.
• χ (2) of interface origin (achiral)—
χ
(2)
xxz = χ
(2)
yyz ,
χ
(2)
xzx = χ
(2)
yzy ,
χ
(2)
zxx = χ
(2)
zyy ,
χ
(2)
zzz .
(3.48)
• χ (2) of chiral origin—
χ
(2)
xyz = −χ
(2)
yxz ,
χ
(2)
xzy = −χ
(2)
yzx ,
χ
(2)
zxy = −χ
(2)
zyx .
(8.22)
8 Other Topics
These relations are readily confirmed from the fact that χ (2) for an isotropic material
should be invariant by arbitrary rotation around an arbitrary axis. For example, if we
rotate the system by 90 ◦ around the z axis, the coordinates should change as follows.
x → y, y → −x, z → z
(rotation around z).
Accordingly, χ
(2)
xyz = −χ
(2)
yxz is required in order that the rotation does not affect
the material properties. The similar argument holds to derive the other relations in
Eq. (8.20).
Equation (8.20) has one independent variable, χ
(2)
chiral . It is given by the average,
χ
(2)
chiral =
1
6
x∼z
p,q,r
ε pqr χ
(2)
pqr ,
(8.21)
where ε pqr is the Levi-Civita antisymmetric tensor (see Appendix A.2),
ε pqr =
⎧
⎨
⎩
1 (pqr = xyz, yzx, zxy),
−1 (pqr = yxz, zyx, xzy),
0 (otherwise).
(7.132)
Equation (8.21) clarifies the invariance of χ
(2)
chiral to arbitrary rotation, since the LeviCivita tensor is invariant as shown in Eq. (7.137) in Appendix A.2. Therefore, the
expression of Eq. (8.21) is valid for the molecule-fixed coordinates as well.
(ii) Interface Then, we consider the situation that the chiral material forms an
interface of C ∞ symmetry.
[Problem 8.2] Suppose a chiral interface system of azimuthal C ∞ symmetry. What
relations are required among the six χ (2) elements of chiral origin in Eq. (8.19)?
In a chiral system with interface, the nonvanishing χ (2) elements of this system
are attributed to either interface or chiral origin. We find from Sect. 3.3.3 and
Problem 8.2 that the χ
(2)
pqr elements of chiral origin are distinct from the χ
(2)
pqr
elements of interface origin. The nonvanishing elements of χ
(2)
pqr are summarized
below.
• χ (2) of interface origin (achiral)—
χ
(2)
xxz = χ
(2)
yyz ,
χ
(2)
xzx = χ
(2)
yzy ,
χ
(2)
zxx = χ
(2)
zyy ,
χ
(2)
zzz .
(3.48)
• χ (2) of chiral origin—
χ
(2)
xyz = −χ
(2)
yxz ,
χ
(2)
xzy = −χ
(2)
yzx ,
χ
(2)
zxy = −χ
(2)
zyx .
(8.22)