Welcome to Unit 5.7!
And welcome to the sequel of the thrilling story
about the discovery of quasicrystals!
We still write the year 1982.
Dan Shechtman carried out an
electron diffraction experiment
with a new kind of alloys and
perceived such a diffraction pattern:
a diffraction pattern with
ten-fold rotational symmetry!
He was convinced that he
discovered something new!
But in the first two years after his
discovery, he was the only one!
And probably this is not surprising,
because for someone to say
“I have discovered crystals with
forbidden symmetry” is somewhat
similar to “I found a planet which is a flat slice!”
Many of his research colleagues
believed that the experiments
were not carried out with sufficient care,
and that the patterns he observed are
some kind of artefacts.
One possibility he explicitly excluded
was that this pattern could result
from the investigations of crystals,
which look like single chunks,
but are in fact composed
of two or more domains.
This actually occurs rather often and is called twinning.
One mineral which relatively often shows
an unusual twinning is pentagonite,
a rare silicon containing mineral.
And this mineral was named according to the circumstanc
that the specific twinning 
occurring in pentagonite leads to an
apparent five-fold, a pentagonal
symmetry.
So it looks like something with five-fold symmetry
but is in fact
only composed of several single-crystals,
which as such show only
ordinary symmetry.
But again, Shechtman was a good
experimentalist, he trusted his research results
and he knew: I investigated
single-crystals - with “forbidden” symmetry!
Two years later, finally, he was able
to publish his results in the journal
Physical Review Letters, with the
title: Metallic phase with long-range
orientational order and no translational symmetry!
Bit by bit, and in particular with
help from the theoretical side -
in the person of Paul Steinhardt -, the
discovery of Shechtman was very slowly,
but surely accepted in the end.
And three years ago, in 2011,
he was awarded with the Nobel prize in
chemistry for the discovery of the quasicrystals.
Interestingly, there was one famous exception.
Not all believed in this new state of
matter: No one less than Linus Pauling,
one of the most esteemed chemists
having received the Nobel prize twice,
disagreed with Shechtman’s beliefs -
and this until his dying day in 1994!
I just briefly mentioned Paul
Steinhardt and his work with Shechtman,
and this leads us to the famous Penrose tilings.
We saw in the last unit that it
is not possible to fill the floor
or to cover the plane
completely with pentagons.
But in 1974, Sir Roger Penrose
could show that it is possible,
by choosing two special rhombi,
a fat and a thin one to cover
the plane completely and which
is (a) non-periodic and
(b) that this pattern shows
five-fold rotational symmetry!
It is very interesting that this
pattern has no strict periodicity.
For a long time it was
believed that is not possible
to cover a plane with something non-periodic.
But this is true for this pattern.
It is not strictly regular;
it is not possible to define a
certain area as the unit cell
and to complete the pattern by pure
translations along two directions.
If we look for instance at the center
of this tiling, we see a pattern,
which I marked here in a different color.
And one might think that we could see
this pattern again in other areas,
for instance here, but
this is not the case:
it is very similar, that’s
true, but it is not identical!
Yeah, this is I think a
very strange pattern!
Back to Steinhardt: he was also
working on the question if a
quasiregular state of matter could
give rise to diffraction patterns
with icosahedral, meaning
with five-fold symmetry -
and he was aware of the Penrose tilings!
And when he saw the work of
Shechtman, actually a preprint
of his publication, he
was very fascinated.
He realized that Shechtman’s
quasicrystals show
exactly the features
that he was looking for!
Later it could be shown that it is
indeed possible to obtain a 3D analogon
of the 2D Penrose tilings, in which not
rhombi but rhombohedra are joined together.
This must resemble in a way the atomic
arrangement of the Shechtman’s alloys.
It is not possible to describe
quasicrystals as normal crystals in
3D space - however, 3D
quasicrystals can be described as
regular formations in the
six-dimensional space -
which will lead us to crystallography
in higher dimensions.
Did I say us? Well, not me!
Don’t ask me anything about
crystallography in higher dimensions.
I find crystallography in three
dimensions already tough enough!
So, one question remains: are
quasicrystals now crystals or not?
Well, they lack one particular feature
that is characteristic for crystals:
long-range translational order!
How could they be crystals? On the other hand:
quasicrystals do give defined diffraction
patterns in diffraction experiments.
And only crystals give such patterns!
The International Union of Crystallography
joined on the latter aspect and
redefined the term crystal as:
"A crystal is a state of
matter, which causes defined
Bragg peaks (diffraction spots)
in an diffraction experiment."
So, this was the last unit of chapter 5!
And this was actually my last
proper unit in this course.
Now I want to hand over to Michael, who is
responsible for the remaining two chapters!
We consider, for instance, the
question: is it possible to combine
crystalline features with
empty space, namely pores?
Stay tuned - and be curious
about metal-organic frameworks
and their description as nets.
Thank you!
