Five Fold Rotational Symmetry of Crystal - A Different Approach of Discussion

                Symmetry is a very vast and important part for not only crystallography but also for many other parts of physics. we will discuss the symmetry of a crystal more specifically about rotation symmetry of the crystal.

When a crystal can be divided into two mirror images about a plane or about an axis or about a point that is called crystal symmetry.

why symmetry is so important for crystallography ??

                Among many importance of symmetry three basic importance are  (i) Identification of materials, (ii) Prediction of atomic structure and (iii) Relation to physical properties ( optical, mechanical, electrical and magnetic etc.)

In crystallography, symmetry is used to characterize crystals, identify repeating parts of molecules and simplify both data collection and nearly all calculations. Also, the symmetry of physical properties of a crystal such as thermal conductivity and optical activity must include the symmetry of the crystal. Thus, a thorough knowledge of symmetry is essential to a crystallographer. 

Symmetry can be various types like translation symmetry, rotational symmetry etc. Here I will talk about rotational symmetry only. Rotational symmetry means rotation of the crystal by a certain angle about an axis (rotational axis) makes no difference to the crystal i.e crystal will remain exactly same as before after that rotation.

If we can rotate the crystal n times by an angle α where n and α satisfy nα=360 degrees then this kind of rotational symmetry is called n-fold symmetry. That means for n-fold symmetry n times rotation by angle α  make a total of 360 degrees rotation of the crystal. Let take an example of a square of side a. If we rotate the square by 90 degrees about the axis perpendicular to its center it will remain same as before and we can rotate the square 4 times to make a total rotation of 360 degrees. So the square has 4-fold rotational symmetry. If we take an equilateral triangle it a 3-fold rotational symmetry.

Coming back to the crystal, now we are going to check how many possible folds of symmetry a crystal can possess.
 Crystal can have 1,2,3,4,6-fold symmetry but can not have 5 or 7 fold symmetry. Now I have reached to the main topic of this discussion that is '5-fold symmetry can not exist in crystal'.

This can be shown by many different methods but I am going to discuss three methods of showing that 5-fold symmetry does not exist for crystal.

1.First one is very easy and you may find it in any standard introductory textbook of solid sate physics*.

                                                                               

Let the lattice have n-fold rotational symmetry about the axis passing through the lattice point and perpendicular to the plane of the paper. By performing a rotation of angle α= 2π/n we can get back the same lattice. So in the figure A' and B' are also lattice points as A and B thus we can write A'B' = m*AB where m is an integer.

                                                                 A'B' = m*AB

                                                Hence,   A'X + XY+ YB' = ma

                                                                acosα +a+ acosα = ma

                                                                cosα = (m-1)/2 ...................(1)

As  -1 ≤ cosα ≤ 1  so m can not be any random integer. m must have some specific values shown in the below table.

               

m	cosα	Α        (degrees)	Symmetry
0	-0.5	120	3-fold
1	0	90	4-fold
2	0.5	60	6-fold
3	1	360	1-fold (identity)
-1	-1	180	2-fold

So from this table, we can see 1,2,3,4,6-fold symmetry exist for crystal but 5-fold symmetry does not exist.

2. The second approach of showing that 5-fold symmetry doesn't exist**.

This approach is much more physical than the first one. Here we only use some simple geometry to reach our goal. Before that, we need to know that in the arrangement of lattice the most important thing is periodicity. The basic structure should repeat again and again to form an infinite lattice. There should not be any gap between these repetition.

Area filled with a connected array of square.

so if we use either a square or an equilateral triangle or a hexagon to cover up some place by repeating them again and again, we can arrange them such a way that there will be no gap between them. That is why there are no problems with these kind of symmetries.

Area filled with a connected array of a uniform triangle.

Now if we take a uniform pentagon and try to cover some plane area by repeating the pentagon many times it is not possible to find an arrangement where there will be no gaps (try it yourself). So it is impossible to fill all the area of a plane with a connected array of pentagons. This is the problem with five-fold symmetry of a crystal.

3.Now the third method.

The most simple method where we consider 5-fold symmetry is existed and then prove that something is wrong in this assumption.

Let’s begin by assuming 5-fold symmetry is possible. Set the shortest lattice vector

along the x-axis: a₀ = a₀(1, 0). Then any other lattice vector can be derived from this as:

                                                                a_n = a₀ [cos(2πn/5), sin(2πn/5)]

we know that some of two lattice vector is also a lattice vector but no lattice vector can be shorter than a₀ . let sum a₁ and a₄

                                                T= a₁ + a₄ =a₀ [{cos(2π/5)+cos(8π/5)}, {sin(2π/5)+sin(8π/5)}]

                                                Hence, T=a₀(0.62,0)

Here we found that the magnitude of the lattice vector T is shorter than a₀  but as per our consideration a₀ is the shortest lattice vector. This is a contradictory situation so something is wrong in our assumption that 5-fold symmetry does exist.

*Reference for the first method: book-1, book-2

** Reference for the second method: Introduction To Solid State Physics 8 Edition -Charles Kittel  or Introduction to Solid State Physics (Paperback) by Charles Kittel

Some Important Things:

First I would like to say that the statement  'five-fold symmetry does not exist'  is not completely true. In the above discussion all the things I mentioned is correct for periodic crystals (in general, all the crystals are periodic that is why we can use the statement of nonexistence of five-fold symmetry for the crystal in general ). There exist a different type of crystal called quasicrystal, which has a structure that is ordered but not periodic. Quasicrystals have long-range order but they are not periodic. Quasiperiodicity leads to unconventional symmetries (5, 8, 10, 12 -fold).

Dan Shechtman,  the Philip Tobias Professor of Materials Science, is the man behind the discovery of Quasicrystal and received the Nobel prize in chemistry in 2011.