196 11 Isotropic Tensors
All these triple products of three tensors are invariant under a rotation of the coordi-
nate system.
Now let the tensorsa,b,cbe constructed from the components of unit vectorsa,
bandc,viz.
aμ 1 μ 2 ·μ 1 =aμ 1 aμ 1 ···aμ 1 ,bν 1 ν 2 ·ν 2 =bν 1 bν 1 ···bν 2 ,cκ 1 κ 2 ·κ=cκ 1 cκ 2 ···cκ.Then the scalar quantity
P(^1 ,^2 ,)(a,b,c), = 1 + 2 , 1 + 2 − 2 ,...,| 1 − 2 |, (11.60)constructed in the manner just described, defines ageneralized Legendre polynomial,
apart from a numerical factor. For 1 =, 2 =0, one hasP(,^0 ,)∼P, whereP
is a Legendre polynomial, cf. Sect.9.3. For the cases 1 = 0 , 2 =, and 1 = 2 ,
=0, the generalization also reduces to ordinary Legendre polynomials. An explicit
example for= 1 + 2 is
P(^1 ,^2 ,^1 +^2 )(a,b,c)=aμ 1 ..μ 1 bν 1 ..ν 2 cμ 1 ..μ 1 ν 1 ..ν 2
=aμ 1 ..aμ 1 bν 1 ..bν 2 cμ 1 ..cμ 1 cν 1 ..cν 2. (11.61)The special case 1 = 2 ==2is
aμaλbλbνcνcμ.The interaction potential between non-spherical particles, see e.g. [34], involves
scalar invariants of the kind discussed here.
11.7.2 Interaction Potential for Uniaxial Particles
Consider two uniaxial non-spherical particles whose symmetry axes are specified
by the unit vectorsu 1 andu 2. These particles can be rod-like or disc-like. In many
cases, weakly biaxial particles can be treated as effectively uniaxial. The centers of
mass of the two particles are at the positionsr 1 andr 2 ,r=r 1 −r 2 is their relative
position vector. The interaction potentialbetween the particles depends on their
orientations, viz. onu 1 andu 2 , and onr=r̂r. The angle dependence ofcan be
described by the functionsP(^1 ,^2 ,)defined above. Thus the potential function is
expressed as the expansion