From WolfWikis

Symmetry of solids
Topic 1
Topic 2
Topic 3
Topic 4
Topic 5
Topic 6
Topic 7
Topic 8
Topic 9
Topic 10
Topic 11
Topic 12
Topic 13
Topic 14
Topic 15

Contents 1 3D structures and rotation symmetry 1.1 Translations in higher dimensions: 2D and 3D 1.2 Combinations of rotation and translations 1.3 Other than triclinic groups 1.3.1 Primitive case 1.3.2 Centered case 1.3.3 Non-symmorphic elements 1.3.4 Space groups 1.3.5 References

3D structures and rotation symmetry

So far we have tried to ignore the fact that we live in a 3D, or if we include time 4D world, although we have made a few generalizing remarks about translations being possible in 3D. Another point we have only touched upon is rotation symmetry. As all crystalline solids are three-dimensional and most also have rotational elements like twofold axes or inversion centers this leaves us with a number of important questions:

1. How do we generalize our translation group to 2D, 3D etc.?
2. How do we combine rotational and translational elements into a group?
3. Are there any new types of elements possible by combination?
4. What do the irreps look like?
5. Can we do anything useful with the irreps?

Translations in higher dimensions: 2D and 3D

Although 3D is far more important in science translations in 2D are easier to visualize.

The figure shows a 2D pattern of objects that can be generated form one such object by translating them according to two translation vectors t_a and t_b repeated endlessly to fill the 2D plane. Interestingly the two basic vectors make an arbitrary angle with each other. In addition the can describe the pattern with more than one set of such vectors and the point of departure of the vectors is arbitrary as well. In one set I have chosen the trunk of the elephant as the origin, in the other the ear, but in reality the vectors are active in any point and the choice of origin is entirely arbitrary.

In three dimensions we have a similar situation: we now have three non-coplanar unit vectors. When applied to an arbitrary origin, they form a parallellopiped that has three different sides (a,b,c) and three different angles between them α is the angle between b and c, β between a and c and γ between a and b. This oddly shaped box is what is known as the unit cell and its six parameters as the cell or lattice parameters.

Because there are three different angles to the box it is known as a triclinic unit cell, -clinic meaning leaning in Greek.

Combinations of rotation and translations

The easiest rotation element to combine is the inversion center i. It takes anything at point (x,y,z) (say elephant tail) to point (-x,-y,-z).

We can combine this element with our 3D translation group as follows: we will write {i|t} when we mean: apply the inversion followed by a translation by a vector t'. This means that any pure translation is written as {E|t} (do nothing follow by shift t) and the identity element is {E|0}

To specify the shifts it is easiest to express them as multiples of the unit vectors a,b and c. In that case we can write e.g {i|(103)} for the operation of inverting followed by one shift in the a' direction and three in the 'c direction.

If we take all the elements {i|t} and {E|t} together we have a new group, known as the space group P1^bar.

Notational problem

Unfortunately this wiki does not allow me to write a bar above the one, so instead I will write ^bar. This is all the more unfortunate because in crystallography it is customary to write negative coordinates not as -x but as x^bar, i.e. with the minus sign written over the coordinate.

If we only take the elements {E|t} we have our old friend the 3D translation group, aka P1 (without the bar).

Other than triclinic groups

P1^bar and P1 are actually the only triclinic space groups. The reason for that is that any other rotational element introduced into the translational world will impose its requirements on the shape of the unit cell. For example if we introduce a mirror plane m' two of the angles of the box must become equal to 90^o.

Primitive case

The first two translation vectors -that we will take to be a and c for reasons of tradition- we can conveniently choose to lie in the mirror plane. The angle β between them is free to be any size. But the third axis b is a problem. We can either choose it perpendicular to the mirror plane. That leads to a perfectly good box albeit with α=γ= 90 . This case is known as a primitive monoclinic cell. Monoclinic because there is only one 'leaning' (as opposed to perpendicular) angle. The space group we get is known as Pm in short notation or as P 1 m 1 in long notation. The last notation specifies that the mirror plane m is perpendicular to the second (b) axis. This is traditionally the most common choice, but P m 1 1 or P 1 1 m are also possible. They are essentially the same group with the same multiplication table but with a different choice of axes.

The group consists of all elements

{E|t} and {m|t} where t is a vector in integer multiples of a,b and c, say (219) or (113^bar)

Of course the negative elements can also be written as positive, because we are still applying closure. So (113^bar) is equivalent to (1,1,N-3)

Centered case

We can also refuse to make the third unit vector perpendicular to the ac plane, but then we get what is known as a centered cell.

Once we start combining the mirror element with the b translation element we automatically generate translations perpendicular to the mirror (shown in red). It then becomes convenient to use the red box as a unit cell, once again only with one angle unequal to 90. Some a cell is known as centered monoclinic. The cell is centered because described in the red vectors the element {E|½½0} is also an element of the group. That means that we can still form a space group, known as Cm (short) or C 1 m 1 (long). The C indicates that the short translation {E|½½0} lies in the place opposite the c-axis (the ab plane). Other choices like Am etc. are also possible.

The elements of the group are still

{E|t} and {m|t} but t is now either a vector in integer multiples of a,b and c, say (219) or (113^bar) or it can contain the centering shift, so that elements like {m|2½,3½,16} are also part of the group.

This is an example of C-centering. For higher symmetry than monoclinic there are other forms of centering:

• A,B or C- centering: one extra shift in one of the planes {E|0½½} or {E|½0½}, or {E|½½0}
• F -centering an extra shift in each plane {E|0½½},{E|½0½} and {E|½½0}
• I -centering: body centering: an extra element {E|½½½}
• R-centering 2 extra elements involving 1/3 and 2/3 shifts in one description (hexagonal) or a cell with α=β=γ unequal 90 in the other

Non-symmorphic elements

So far all elements can be seen as a combination of a pure rotation and a pure translation and both these components occur separately in the group. E.g. {m|2½,3½,16} is composed of {m|000} and {E|2½,3½,16} a pure mirror and a pure translation both part of Cm.

However it is possible to form groups that have combined elements, the components of which do not occur in the group. A good example is a space group like Pc or P 1 c 1 in long notation. Apart from all the translation of a primitive cell it has an element {m|0,0,½} : mirror and shift by half the translation vector in the c-direction. The group does not contain the elements {m|0,0,0} or {E|0,0,½}, only the combined element. One may wonder how this can form a group, i.e. a closed system of elements. The reason is that if the element is combined with itself: {m|0,0,½}{m|0,0,½} = {E|0,0,1} is a whole translation and part of the translation lattice.

This is an example of a non-symmorphic element and in this case is known as a glide plane. There are a,b,c,n and d glides depending on the direction of the half-integer shift

Other typical and common non-symmorphic elements are the screw axes 2₁, 3₁,3₂,4₁,4₂,4₃,6₁,6₂,6₃,6₄,6₅.

Space groups

The way that all these elements can be combined has been thoroughly worked out and is described in exquisite detail in the International Tables. These details are not the topic of this course. Besides the Tables there are excellent books on the use of space groups in determining crystal structures ^[1]. But let us make a few observations. First of all, the presence of rotational elements puts limitations on the cell parameters of the cell. This is already for mirrors or twofold axes. More extensive rotational symmetry leads to fewer independent parameters:

Lattice	rotational elements	cell	# parameters	Bravais type
Triclinic	1bar	a,b,c,α,β,γ	6	P
Monoclinic	2, m , 2/m	a,b,c,β α=γ=90	4	P,C
Orthorhombic	222, mmm	a,b,c β=α=γ=90	3	P,C,F,I
Trigonal	3 3/m	a=b,c β=α=90 γ=120	2	P,R
Hexagonal	6 6/m	a=b,c β=α=90 γ=120	2	P
Tetragonal	4 4/mmm	a=b,c γ=β=α=90	2	P, I
Cubic	23 - m3m	a=b=c γ=β=α=90	1	P,F,I

Attentive observers may have noticed that the elephant lattice has its flaws. The trunk is not always quite touching the chosen origin. In practice this is quite common. Translation symmetry is an idealization and real crystals always have their flaws. Nevertheless crystallographers often proudly announce to have 'solved their structure'. This is a fib. What they mean is that they have solved the totally symmetric component of the structure, i.e. its temporal and spatial average', because they base their analysis on the Bragg reflections in the reciprocal space they have made visible through scattering. As we said before, these points represent all the aliases of the totally symmetric irrep. If one elephant in a mole looks they other way that will hardly affect the average, no matter how much that would affect other properties.

References

1. ↑ X-ray Structure determination, George H. Stout, Lyle H. Jensen, Wiley & Sons 1989 ISBN 0-471-60711-8