Unit 1.9 – Crystal = Lattice + Motif


In this unit I want to come back
to the term crystal structure. This was already mentioned but
not explained in the last units. There is a simple formula –
a simple short form to memorize of what a crystal structure is. A crystal structure is a lattice plus the motif. The motif is sometimes
also called the base. So, we first have to look what lattice means, then what a motif is and in which
way it is linked to our crystals. A lattice is something mathematical
– it is defined as follows: A lattice is an infinite arrangement of points in space or in the plane or on a line, in which all points have the same surroundings. That means, a lattice in the plane looks like that: A genuine lattice would be infinite,
but that wouldn’t fit onto this slide. So just imagine that it is almost infinite. And now, we see that every single point
of the lattice has the same surroundings. That is true for this point with this neighbouring points in the
first, second, third sphere and so on. And this point has also the same surroundings. Now it should be also clear, why
a lattice has to be infinite according to its definition: Namely, because a point at the border does not
have the same surroundings as a point at the centre!
Therefore, there must be no border. Of course, in reality crystals have borders. And this leads to different physical or
chemical properties at their surface compared to the bulk phase. Now, we need the connection
to our crystal structure. What does such a lattice point represent? You can think of it as a connection
point between neighboring unit cells. This means, place a lattice point
at every corner of every unit cell. Here we have a piece of a crystal
with repeated unit cells that were joined together in
all three space directions. And now we place a lattice point at
every corner of these unit cells. And if we now look at the points only – for simplicity let’s take only two of the three
dimensions, namely the y,z-plane – then we see this lattice. And because our crystal consists
of an almost infinite number of unit cells our lattice can be
approximated as a real infinite lattice. A lattice is characterized by its lattice vectors. These are translational vectors; it is also sometimes said that
these vectors span the unit cell. And the lattice points can be transferred
into each other by these vectors, along the b-direction, along the c-direction, and of course the third dimension as well. Now only the motif is left. Our unit cells are not empty. They are filled with something –
and this is called the motif. The motif consists of the arrangement
of the building blocks of the unit cell. So this means normally some atoms, or a molecule. But, in principle, it could be
everything – like this car for instance. And the motif is represented by a lattice point. And if we apply this translational principle to this car we get this arrangement of cars, like a parking place. This parking place can be regarded as
a kind of a 2-dimensional crystal, if the cars are regularly arranged, and
if the cars are all of the same type. It is important that you understand that
the lattice is only a virtual construct, which describes the distance and the
direction from one motif to another. Okay, usually crystals are not made
by cars but by atoms or molecules. Let’s take, for instance, this three-atomic molecule – and let’s apply the translational principle – then we get this crystal. We see that the orange atoms are
translated by this orange lattice. And this must be also the case for the blue atoms. All building blocks of a crystal structure are subject to the same translational principle. Therefore also the green atoms
build the same lattice or are transferred to each
other by the same lattice. So, one can conclude that all lattices, which are built by the different atoms must be congruent, they are superimposable.

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