Symmetry of solids/Topic 3

From WolfWikis

The roots of one

For all elements of a translation group (under closure for N) we have that

(Tⁱ)^N = (T^N)ⁱ= Eⁱ=E

Furthermore all irreps are one-dimensional, because all elements commute. This means that the element E is always represented by the one-dimensional unit matrix, aka the number one. (1). So we could write χ(E) =1 for all N irreps.

As representations (including irreducible ones) represent the behavior of our elements this means that χ(A)^N= χ(E) for all elements A and all irreps.

Thus χ(A) is always a root of the equation x^N=1!

These N-th roots of unity are well known to anyone familiar to the complex plane.

The complex plane

All complex numbers can be depicted in a 2D plane with a real axis and an imaginary one. Any point in the plane is a complex number Z. We can describe Z by its Cartesian coordinates:

Z= a + b.i

where a and b are the real and the imaginary part (coordinates)

Alternatively we can transform to polar coordinates (r,φ) by using:

r²=a²+b²

tan(φ)= b/a

We could call r the magnitude of the complex number and φ its phase angle. A special set of complex numbers is formed by those that have a magnitude of unity: r=1. They lie on what is known as the unit circle.

Interestingly, if we multiply two such numbers on the unit circle, the result is yet another complex number on that circle, with the phase angles added.

Euler notation

This is particularly easy to see in the Euler notation of a complex number:

Z= r.exp^iφ = r[cos(φ)+ i.sin(φ)]

We should note that φ=0 corresponds to the real number Z=1, likewise φ=π gives Z=-1.

For two numbers on the unit circle (r=1) we get:

Z₁.Z₂ = exp^iφ₁.exp^iφ₂=exp^{i[φ₁+iφ₂]}

This fact means that if we distribute N numbers, Z₁, Z₂, Z₃,...,Z_N evenly along the circle this gives us the solutions of the equation x^N=1, because Z_i taken to the power N will always be Z=1, because N times the phase angle of Z_i will always give φ=2π or a multiple thereof.

The first root Z₁ will have a phase angle φ=2π/N. Let us denote this particular complex number by ε. All the others will be powers of it:

Z₁, Z₂, Z₃,...,Z_N = ε¹, ε², ε³,...,ε^N=1

Collectively these numbers, all of magnitude r=1 (i.e. |Z_i|=1) are known as the roots of one

The irreps of a one-dimensional translation group

As we said above the elements of the abelian translation group should have irreps consisting of roots of one and so we can easily construct a character table for any such group using the complex numbers described above. We simply assign 1 to χ(E) and one of the roots of one to the first element T. Then we apply the multiplication rules of the group to find all others. We get:

The last irrep is more usually put at the head of the table because it is the totally symmetric one Γ_N