Tensors for Physics

9.4 Cartesian and Spherical Tensors 161

Y 2 (±^2 )=c( 2 )

1

2

√

2

r−^2 (x±iy)^2 =c( 2 )

1

2

√

2

sin^2 θexp(± 2 iφ). (9.28)

For allone has
Y(m)∼exp(miφ). (9.29)

Furthermore, theY(^0 ), which do not depend onφ, are proportional to the Legendre

polynomialsP(cosθ),viz.Y(^0 )=

√

( 2 + 1 )/ 4 πP. The spherical harmonicsY(^0 ),

as well asY(m)+Y(m)∗andi(Y(m)−Y(m)∗)are real functions.

Notice: the namespherical harmonicsdoes not indicate the symmetry of these func-
tions, but rather their dependence on the polar angles of spherical coordinates. As
far as symmetry is concerned, a preferential axis, usually chosen as thez-axis, is
linked with the spherical harmonics. In Quantum Mechanics, this reference axis is
also referred to asquantization axis.

9.5 Cubic Harmonics

9.5.1 Cubic Tensors

Irreducible Cartesian tensors, which reflect the symmetry of cubic crystals are tensors
of ranks= 4 , 6 ,....Lete(i), withi= 1 , 2 ,3 be unit vectors parallel to the axes of
a cubic crystal. The first of these tensors with full cubic symmetry, as used in [25],
are

Hμνλκ(^4 ) ≡

∑^3

i= 1

e(μi)e(νi)e(λi)e(κi)=

∑^3

i= 1

e(μi)e(νi)e(λi)e(κi)−

1

5

(δμνδλκ+δμλδνκ+δμκδνλ),

(9.30)

Hμνλκσ τ(^6 ) ≡ e(μ^1 )e(ν^1 )e(λ^2 )e(κ^2 )eσ(^3 )e(τ^3 ), Hμ(^81 )···μ 8 ≡

∑^3

i= 1

e(μi 1 )···eμ(i) 8. (9.31)

These tensors are invariant against the exchange of the cubic axes.

9.5.2 Cubic Harmonics with Full Cubic Symmetry

Multiplication ofHμνλκ(^4 ) with the irreducible fourth rank tensor r̂μ̂rνr̂λr̂κ, con-

structed from the components of the unit vector̂r=r−^1 r, yields