# Counting body centre as lattice point when the basis consists of 2 atoms

An article in my book gives the following informatiom :-

I understood part a) and b) but faced problems with c).

It says that here the basis consists of 2 atoms which doesnt seem to be very convincing. It would have been clear if it said something like 'a molecule' or 'an ion'. How can it contain 2 particles at once?

Then it says that the smaller particle is at the body centre but we won't count it as a lattice point thus making it a simple cubic rather than a body centered cubic. Can't a particle be only placed at a lattice point ? If yes, why we hadn't we counted it here?

• Please cite the chapters, helps others as well. It's NCERT, no? Commented Apr 1, 2018 at 2:10
• It's a private institution book ,coaching/tution book to be precise. Citing the source won't do any good since it is only available for the students part of it. Commented Apr 1, 2018 at 3:39
• The book is rather sloppily written, I'd say. To begin with, a basis can't consist of two, or one, or any number of atoms. Basis is a mathematical concept. It doesn't consist of atoms; instead, it consists of vectors. Then again, what's that with particles placed outside of lattice points? Of course it may happen; you saw an example as early as (b), where big atoms are placed in lattice points, but the small ones are not. Commented Apr 1, 2018 at 6:06
• In maths, yes, a basis is a set of vectors. However in crystallography a basis usually means the set of atoms that are associated with each lattice point and their positions. See e.g. ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/… Commented Apr 1, 2018 at 8:17

I'm not an expert here, hopefully somebody who really does know their stuff can answer this definitively. But anyway here goes ...

From http://www.physics-in-a-nutshell.com/article/4/lattice-basis-and-crystal "A lattice is in general defined as a discrete but infinite regular arrangement of points (lattice sites) in a vector space". Notice how this says nothing about atoms, molecules or ions; it is purely a regular layout of points in space. We define the atoms and their positions via the basis . This is simply a list of species types and their positions relative to an arbitrary origin. We then associate the basis with the lattice points, that is we use the lattice points as the origin for the basis, and as we have a crystal (and so translational symmetry) every lattice point is associated with the same basis. Thus to define a crystal structure

1. We first define a lattice - a set of regularly laid out points
2. We then define the basis - a list of atom types and positions and offsets
3. We then associate an identical basis with every lattice point

Thus for a given lattice we can have any number of bases, and each basis can contain any number of atoms.

Now we can simplify things a little bit by noting that we can always shift the origin of the basis so that one of the atoms is at the origin, and thus in the crystal a lattice point coincides with one of the atoms in the basis. This is what has happened in your case c). Now note that for us here atoms are always point particles, so rather than large or small I'm going to use blue and red - I think giving the atom a size may be confusing here as here atoms only have a position and a type. So

1. We have a simple cubic lattice, so the lattice points (which have nothing to do with atoms) are on a cubic layout throughout all space
2. For the basis (i.e. the atoms) we have chosen to place a blue one at the origin, and a red one offset along the diagonal
3. We then associate the basis with the lattice to make the crystal, the result being a blue atom at every lattice point (because the basis has a large atom at its origin), and a red atom offset diagonally from the lattice point, coincidentally in this case half way to the next lattice point due solely to the choice of basis, not the lattice

Also note that the basis may indeed be the positions and types of atoms in a molecule. We would then form a crystal of that molecule. But bonding is not important in this context, all we are interested in is the layout of the atoms in space, so for the moment we can forget all about chemical bonding and think purely in terms of atoms.

• I understood your answer and I think it's very much correct but what you said raises another doubt in my mind. Could you please check out chemistry.stackexchange.com/questions/94456/… Commented Apr 2, 2018 at 5:45

How can it contain 2 particles at once?

[ we are discussing crystal structures here - - of course crystals can contain two different atom - e.g. sodium and chlorine - common salt ]

Then it says that the smaller particle is at the body centre but we won't count it as a lattice point thus making it a simple cubic rather than a body centered cubic. Can't a particle be only placed at a lattice point ? If yes, why we hadn't we counted it here?

[- a particle can be placed anywhere in space - and there can be more than one sort of particle-it is up to us to categorize it mathematically and topologically ... and the text is trying to explain the three different catagories

what they are saying is that in a crystal of two different atoms , if the second atom is at the centre of the cube being repeated in the crystal - then you can ignore it in terms of geometrical symmetry and crystal lattice maths categorization . ]

• What I feel is that the basis consists of the two particles(blue and red) and they are placed in such a manner that the blue one is present on the lattice point(corner one) and the orange point happens to be on the body centre position. And then just like case b) where we took lattice point on the center of the blue one, we will do the same here therefore its still a simple cubic but the way the basis was placed at the lattice point, it makes it appear like a body centered cubic. Commented Apr 1, 2018 at 12:12