# Symmetry elements and reciprocal lattices.

by Urmi Roy
Tags: elements, lattices, reciprocal, symmetry
 P: 692 Hi, There are some points I really want to clear up in this topic....I promise to finish my chain of doubts as quicly as possible! I'll put in my first questions.... 1. Rotoinversion is a combination of inversion and rotation-- often it ends up as having the same effect on the crystal as reflection or just inversion--why,then is it recognized as a separate symmetry element? I could also put my question as 'what is the 'physical significance of rotoinversion,' since I read that all the symmetry elements have important roles in determining the chemical or physical properties of a crystal'-what is it in case of rotoinversion? 2. Inversion through a centre is also a symmetry element -- it is said that an object having one centre of inversion has one 'roto-inversion axis'--what is the relation between the centre of symmetry and 'rotoinversion'--why are they defined in terms of eachother? Can an object have more than one centre of symmetry? 3. Why is it not possible for a crystal to have a 5- or 8- fold rotational symmetry axis? 4. Again,translational symmetry elements must have some significance in determining the properties of crystals--firstly,why are they of only two types,screw and glide axes--couldn't we produce any other kind-like rotation followed by inversion? (I read that "A dislocation may glide only in the plane which contains both its line and its Burgers vector" - Is this the significance of 'glide' in crystal dislocations?) Thanks in advance for your help!
 P: 416 1. Any spacegroup will have some small set (usually 2 or 3) of generator operations that can be combined to produce the whole set of symmetry operations for a crystal. Symmetry operations can be represented by a rotation matrix and a translation vector, and the set of symmetry operations for a crystal are all the operations that can be generated that are unique. The matrix representing a rotoinversion will be different than the rotation matrices, since a rotation matrix always has determinant = 1, and the rotoinversion will have a determinant -1. Symmetries show up in many properties of a crystal. The most basic is the Bragg reflections that will be observed are very closely tied to the symmetries that exist in a crystal. There are some other specific examples; a crystal with a rotation axis can't have any magnetic or electric dipole perpendicular to that axis, and a crystal with inversion symmetry can't have any magnetic or electric dipoles at all. A thorough answer to this question is probably something only a hardcore crystallographer could give. 2. Part of the answer to this question I gave above; the generator matrices for a particular spacegroup might include a rotation about some axis and an inversion, and then the set of symmetry operations in that spacegroup will include all unique ways to multiply those matrices together. A crystal can definitely have more than one center of inversion although I think one center of inversion is as good as many. Anyway, a crystal will have a center of inversion in every unit cell since it is periodic so I don't see how it would make a difference. 3. In order to have a rotation axis be a symmetry in a periodic crystal, that means that in a 2D plane perpendicular to that axis, you must have a shape with the same rotation symmetry that fills space when translated. The basic geometrical figures with 5 or 8 fold rotation symmetry are the regular pentagon and octagon, but it's impossible to fill a 2D plane with those shapes. (Try it, draw a pentagon and then try to fill space with other pentagons, without throwing in any 180 degree rotations.) 4. I'm not sure I understand your example; a rotation followed by inversion can certainly be done but it does not have anything to do with the translation. Or if you mean a rotation followed by inversion then translation then that's probably the same as choosing a different rotation + translation. I suspect the answer to your question is any more complex symmetry operation that includes a translation can be reduced to screw and glide axes, because there is a great deal of freedom in choosing a translation vector.
P: 692

## Symmetry elements and reciprocal lattices.

In regard to your response to my first question,I gather that if we can recognise 2-3 basic symmetry operations in a particular crystal, those basic operations imply some other,perhaps more compex, symmetry operations in the crystal--we then have the point group of the crystal.

In case of rotoinversion,though incidentally it might end up with the same effects on a crystal as rotation,or reflection,it has certain requirements of its own (like in regard to the electic dipole directions,as you mentioned)--is that right?

The thing with rotoinversion is that its a very peculiar kind of operation--it seems to have sprung out of nowhere! If we can combine rotation and inversion to form an indepedant symmetry operation, could we not have other basic operations combine to form totally independant operations?

However,as you said,a crystal may have both a rotoinversion and a centre of symmetry operation,but any one of them does not necessitate the other.....does that mean we can have a centre of symmetry without a rotoinversion axis?

I should have been a little more clear about the second part of the second question--I meant to ask if there could be more than one centre of inversion in a [I]unit cell [/I]of a crystal,not the entire lattice.

For the last question,again I should have been a little more clear--I meant that if it was possible to have a symmetry element with 'rotation followed by translation' or 'inversion followed by translation.' I felt a little surprised that there were only two kinds of translational symmetry operations.
From your answer,I gather that all such operations,(like the ones I suggested) that include translation can be reduced to screw or glide axis-is that right?

Please forgive me for asking such naive questions,I'm very new to all this.
 P: 692 Perhaps someone can tell me a few things about the reciprocal lattice instead.... 1. What is the physical significance of the reciprocal lattice? (I read that it's got something to do with diffraction,but I don't see the link.Please explain.) 2. I found on a webpage that "All the reciprocal lattice points can be described by a linear combination of multiples of two basis vectors a* and b*."--does that mean that a* and b* are basically the edges of the cube formed by 8 lattice points in the reciprocal lattice? 3.Does the proportionality factor relating the original lattice vectors to the reciprocal lattice vectors always have to be 2(pi)?---does that hold any special meaning or significance? 4. Why is the reciprocal lattice vector called the momentum vector? 5.Is it safe to say that the reciprocal lattice vector provides us a scaled down image of the original vector,and thus the reciprocal lattice gives us a scaled down picture of the original lattice? 5.Why are the planes in the reciprocal lattice constructed perpendicular to the original lattice vector? 6.How are Miller indices related to reciprocal lattice vectors?
P: 416

 Quote by Urmi Roy In regard to your response to my first question,I gather that if we can recognise 2-3 basic symmetry operations in a particular crystal, those basic operations imply some other,perhaps more compex, symmetry operations in the crystal--we then have the point group of the crystal.
Yes.. if those symmetry operations include translations then we have the spacegroup as well as the point group.

 In case of rotoinversion,though incidentally it might end up with the same effects on a crystal as rotation,or reflection,it has certain requirements of its own (like in regard to the electic dipole directions,as you mentioned)--is that right?
Yes, although the requirements will (usually) be the intersection of the requirements for the elementary symmetry operations, which may end up being trivial, or completely forbidding electric dipole.

 The thing with rotoinversion is that its a very peculiar kind of operation--it seems to have sprung out of nowhere! If we can combine rotation and inversion to form an indepedant symmetry operation, could we not have other basic operations combine to form totally independant operations?
Yes.. a simple example is if you have a crystal with 6 fold rotation symmetry and no other symmetries. Then the generator is a rotation by 60 degrees, which can be applied multiple times to generate the whole symmetry group, which includes rotations by 120, 180, etc. degrees.

 However,as you said,a crystal may have both a rotoinversion and a centre of symmetry operation,but any one of them does not necessitate the other.....does that mean we can have a centre of symmetry without a rotoinversion axis?
The only way to have inversion symmetry without rotoinversion is to not have any rotation symmetries.

 I should have been a little more clear about the second part of the second question--I meant to ask if there could be more than one centre of inversion in a [I]unit cell [/I]of a crystal,not the entire lattice.
So just to clarify, the answer is yes.

 For the last question,again I should have been a little more clear--I meant that if it was possible to have a symmetry element with 'rotation followed by translation' or 'inversion followed by translation.' I felt a little surprised that there were only two kinds of translational symmetry operations. From your answer,I gather that all such operations,(like the ones I suggested) that include translation can be reduced to screw or glide axis-is that right?
Yes, I believe so, but I have not checked.

On the reciprocal lattice, it's maybe more illustrative to at a derivation of Bloch's theorem. The wavefunction can be separated into $$\psi_{nk}(r) = u_{nk}(r) e^{i k \cdot r}$$. where $$u_{nk}(r)$$ is periodic in the unit cell. k is called the pseudomomentum, it lives in the reciprocal space and the wavefunction is periodic in k, that is $$\psi_{nk} = \psi_{n,k+G}$$ where G is a reciprocal lattice vector. If you want to expand u_nk in planewaves, the planewaves would be $$\exp(i G \cdot r)$$ where G is a reciprocal lattice vector. So the reciprocal lattice is the set of all wave vectors for planewaves which are periodic in the unit cell.

1. The reciprocal lattice shows up just about everywhere. Diffraction is just one example.

2. I don't understand the context of this. What webpage?

3. Yes, it always has to be 2 pi, because of the periodicity of sin and cos functions.

4. The vector k in many ways plays the role of the momentum in a solid. It is conserved modulo a reciprocal lattice vector in interactions. If one block diagonalizes a periodic Hamiltonian with Bloch's theorem the momentum operator is transformed to (p + k)^2/2m.

5. No, notice that reciprocal lattice vectors have dimensions of 1/length, instead of the dimension of length that a lattice vector has. The reciprocal lattice is not just a scaled version of the direct lattice. There are many examples of a reciprocal lattice that is rather different from the direct lattice, for instance the reciprocal lattice of a face centered cubic crystal is body centered cubic.

6. Wikipedia has a detailed description: http://en.wikipedia.org/wiki/Miller_index
P: 692
 Quote by kanato 4. The vector k in many ways plays the role of the momentum in a solid. It is conserved modulo a reciprocal lattice vector in interactions. If one block diagonalizes a periodic Hamiltonian with Bloch's theorem the momentum operator is transformed to (p + k)^2/2m.
I really don't have the requisite background to completely understand this, it would be great if you could please interpret in simpler terms.

 Quote by kanato Yes.. a simple example is if you have a crystal with 6 fold rotation symmetry and no other symmetries. Then the generator is a rotation by 60 degrees, which can be applied multiple times to generate the whole symmetry group, which includes rotations by 120, 180, etc. degrees.
I could still say that there could be another independant symmetry operation called roto-reflection,just by combining rotation and reflection--but the fact is there isn't -that's what makes me wonder what's so special about roto-inversion that it get's recognised as an independant symmetry operation.

Also,in regard to this,let me go back to what you said,
 Quote by kanato Yes, although the requirements will (usually) be the intersection of the requirements for the elementary symmetry operations....
Let me clarify - rotoinversion is not only a consequence of the inersection of the reqirements of rotation and inversion -- it has its own completely different set of requirements,doesn't it?

By the way, you mentioned something such as plane waves - I read briefly about these on wikipedia,from where I gathered that these are basically a geometrical representation of ordinary waves --right? (It didn't say very much more on the physical significance of plane waves,and I didn't understand most of the mathematical part.)

 Quote by kanato 2. I don't understand the context of this. What webpage?.
I can't find the website for now..for the time being please ignore this point.

 Quote by kanato 6. Wikipedia has a detailed description: http://en.wikipedia.org/wiki/Miller_index
Thanks for this link.I read through it twice,but it'll take some more reading to finally seep through .
 P: 692 Please somebody,come back to my thread!! Kanato whatever happened to you?

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