Tensors for Physics

62 5 Symmetric Second Rank Tensors

A molecule with three different moments of inertia is referred to asasymmetric
topmolecule. On the level of the second rank tensor, it has the same symmetry as a
brick stone.

5.1 Exercise: Show that the Moment of Inertia Tensors for Regular Tetrahedra
and Octahedra are Isotropic

Hint: Use the coordinates( 1 , 1 , 1 ),(− 1 ,− 1 , 1 ),( 1 ,− 1 ,− 1 ),(− 1 , 1 ,− 1 )for the
four corners of the tetrahedron and( 1 , 0 , 0 ),(− 1 , 0 , 0 ),( 0 , 1 , 0 ),( 0 ,− 1 , 0 ),( 0 , 0 , 1 ),
( 0 , 0 ,− 1 ), for the six corners of the octahedron.

5.3.2 Radius of Gyration Tensor.

Consider a cloud ofNparticles orNmonomers of a polymer molecule located at
positions∑ r(^1 ),r(^2 ),...,r(N). The geometric center shall correspond tor=0, thus
N
i= 1 r

(i)=0. Theradius of gyration tensoris defined by

Gμν=

∑N

i= 1

rμ(i)rν(i). (5.22)

The trace of this tensor is the square of the average radiusRof the group of particles
considered:

R^2 =Gλλ=

∑N

i= 1

rλ(i)rλ(i), (5.23)

whereRis a measure for the size of the group of particles. The full tensor

Gμν=

1

3

Gλλδμν+Gμν,

characterizes the size and the shape of the group of particles under consideration. The
symmetric traceless part, in particular, is a measure for the deviation from a spherical
symmetry. Whenall particles consideredhavethesamemassm, themoment of inertia
tensorΘμνis related toGμνby

Θμν=m

(

Gλλδμν−Gμν

)

. (5.24)

In applications, the average of the right hand side of (5.22) is referred to as radius of
gyration tensor, viz.Gμν=〈

∑N

i= 1 r

(i)

μr

(i)

ν〉.