Chapter 20: Metallic Structures IV: Quasicrystals

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Welcome back to the Deep Dive.

Today we are strapping in for, well, one of the most intellectually jarring rides in material science.

Oh, for sure.

We're taking a deep dive into chapter 20 of our source material, metallic structures for EV quasicrystals.

And if you think you know how atoms arrange themselves in a solid?

Yeah.

Prepare for a complete revision of that.

This is such a monumental topic because it really represents a scientific revolution, one that was

highly controversial for years.

Right.

So our mission here is to synthesize the complexity of these quasicrystals.

We're talking about materials that have long range orientational order.

But and this is the key, they fundamentally lack 3D translational periodicity.

So they're ordered, but the pattern never ever precisely repeats.

It's like finding a perfect crystalline blueprint for chaos in a way.

That is the paradox we have to unravel.

And the cornerstone event really was this groundbreaking discovery in 1984 by Dan Sheckman and his team.

Yes, a legendary story.

They observed a sharply picked diffraction pattern in an al -14 % Nerder alloy.

And that pattern revealed something thought to be impossible.

Perfect icosahedral orientational order.

And why was that impossible?

Well, you tell us.

Because classical crystallography, I mean, the rules that had stood for a century dictated that crystals must belong to one of 32 crystallographic point groups.

And because of that requirement for translational symmetry, you know, the ability to shift the pattern and have it land on itself.

Rotations are severely restricted.

Two -fold, three -fold, four -fold, and six -fold axes.

That's it.

And icosahedral symmetry contains five -fold rotation axes.

Which are explicitly forbidden in a truly periodic lattice.

It was a structural impossibility and it was existing right there in the lab.

That's the classic crystallographic dogma.

Five -fold symmetry and periodicity cannot coexist.

So if the periodic model just fails completely, what mathematical framework do scientists even use to describe these things?

They had to invent a new one, basically.

They had to introduce mathematics that could capture these highly organized structures without relying on simple repetition.

A total paradigm shift.

A total paradigm shift.

It required defining order in a new way.

And three critical concepts emerged, which really form the bedrock of this entire chapter.

Okay, let's unpack those.

What's the first one?

First, they had to move away from strictly periodic functions for atomic density.

Instead, they started using what are called quasi -periodic functions.

And the differences?

Well, a periodic function repeats, you know, or x, or x plus a.

But a quasi -periodic function is defined by a sum of periodic functions, but their fundamental wave vectors have ratios that are irrational numbers.

Ah, the irrationality is the key.

That's the key.

It allows the pattern to be incredibly structured and ordered, but it guarantees it never perfectly overlaps itself when you translate it.

So the structure has rules, but those rules are based on irrational numbers, which guarantees non -repetition.

Okay, what's concept number two?

This is the one that's often the hardest to grasp.

It's the embedding of the non -crystallographic structure into a higher dimensional space.

Right.

This is where it gets a little sci -fi.

It sounds like it, but it's pure mathematics.

To maintain that non -crystallographic symmetry, like icosahedral, you treat the 3D structure we see as a projection, or you could think of it as a slice of a perfect periodic crystal that exists in, say, five, six, or even more dimensions.

Okay, let me make sure I've got this, because it's such a radical idea.

You're saying that up in 6D space, the crystal is perfectly periodic.

It obeys all the classical rules.

Yes, perfectly ordered.

But because we're seeing a 3D slice of it, and that slice is taken at an irrational angle to the 60 axes, what we see in our reality is this non -repeating quasi -periodic structure.

That's a perfect articulation.

The periodicity is just hidden in the higher dimension.

It's our perspective that creates the periodicity.

Okay, and the third tool?

Well, it flows directly from this.

We have to describe the atomic configurations and, more importantly, interpret the diffraction patterns using basis vectors defined in that higher dimensional space.

So no more just three miller indices?

No.

We need the indices that correspond to the full dimensionality of that embedding lattice.

That's the only way to account for all the diffraction spots they are seeing.

To understand the heart of this irrational projection, we have to start with the specific irrational number that governs the geometry of all this.

The golden mean?

You really can't talk about quasicrystals without it.

Think -Hum is just indispensable, especially for those structures with fivefold decagonal and icosahedral symmetries.

Numerically, it's about 1 .618.

0 .034, and so on.

But its real power in all of this comes from its algebraic identity.

Which is?

Satisfies the polynomial equation x sub by one equals zero.

Which means cap plus one.

Why is that identity so vital?

Because it's the key to the structure's ability to inflate without changing its nature.

Think about it.

When you multiply any length that's governed by by for itself?

You get two, which is just two plus one.

Exactly.

Scaling by relates the new size back to the original size plus a unit element.

So if you have a long length L and a short length S, and Ls is just scaled by relative to L.

It's a recursive property.

It's inherently recursive.

It allows the whole structure to be defined by these self -similar scaling operations, what we call inflation or deflation, just by using powers of hell.

The structure has to be similar.

So the non -repeating structure does repeat itself, but only at these specific irrational scales determined by powers of L.

And that recursive nature is what guarantees the long -range order, even without simple translation.

Precisely.

And the source makes this beautiful connection to history.

This whole idea is deeply linked to the Fibonacci sequence.

By Fibonacci's rabbit problem from what, 1202?

From Liber Abaci, yes.

The sequence Fk, Fk1 plus Fk2.

If you take the ratio of successive Fibonacci numbers, Fk plus one Fk, you generate continued fractions that get closer and closer to L.

It shows that the math for this has been hiding in plain sight for centuries.

Exactly.

And that recursion translates directly into the structural scaling properties.

Any coordinate in the quasicrystal, fifth V, has to be expressible as some linear combination involving X, V equals math plus P, where M and P are just integers.

So any inflation operation scaling the whole thing must use factors that are powers of death.

Or it falls apart.

The self -similarity has to be preserved.

Let's try to visualize this geometrically.

The source describes it in figure 20 .1, which makes the link between putt and fivefold symmetry really clear.

It's a great illustration.

You take a regular pentagon.

If you extend its sides out to create a star -stellating operation, the points of that star form a new, larger, regular pentagon.

And the ratio of the edge length of the new outer pentagon to the original inner one is?

It's precisely it.

That geometric relationship is just unavoidable in any system with pentagonal or icosahedral order.

And this isn't just a theoretical drawing.

It applies to the actual bricks used to build the 3D structures.

Yes.

The source points out that it defines these crucial dimensional ratios within the two types of rhombohedra that are the basis for many of these structures.

The long -to -short diagonal ratio of the faces.

Right.

And what's more, these same rhombohedra are found in what we call complex crystalline approximants, like the MG Alzin structure mentioned in the text.

So it shows a direct, quantitative link between the forbidden quasi -periodic structure and its closest periodic cousins.

The closer a structure gets to having that forbidden symmetry, the more its dimensions are governed by the golden mean.

It's a clear signature.

We've established the governing number.

Now let's move to the simplest physical model to see how this works in practice.

The one -dimensional quasi -periodic lattice.

The Fibonacci lattice.

It's a perfect place to start.

So instead of one unit cell repeating, you're using two different segments.

A short step S and a long step L.

And for the resulting chain of atoms to be quasi -periodic, the ratio of their lengths, LS,

must be the golden mean.

Because if it were a rational number, say 32.

You could just find a larger repeating unit and it would be periodic again.

Because, as irrational, the pattern of L and S segments never repeats, but it still follows this very strict rule.

And we generate the sequence using a recursion or substitution method, which mirrors the Fibonacci sequence itself.

Exactly.

We use that inherent recursion.

The substitution or inflation rules for this 1D lattice are actually elegantly simple.

You just define the rules for generating the next larger sequence.

Which are S goes to L and L goes to LS.

Right.

So you start with, say, L, you apply the rule, you get LS.

Apply it again to LS and you get LSL or LSL.

And again to LSL gives LSL LS, which is LSL LS.

And the length of the sequence, the period, increases by Fibonacci numbers.

One, two, three, five, eight.

And if you track the counts of the L and S segments, those counts also follow the Fibonacci sequence.

The whole process perfectly enforces that structural self -similarity we were talking about.

And there's a matrix for this, right?

There is an inflation matrix P.

It's a two by two matrix.

And when it operates on the vector of LS counts, it gives you the new counts.

The eigenvalues of this matrix are Nx1.

Which confirms that the only stable scaling factor is it all ties together.

The math is very tight.

So that's the substitution method.

But how do we link this back to the higher dimensional geometry?

We need the projection method.

The projection method is the really fundamental geometric picture.

To build this 1D quasi -periodic structure, we start with a perfectly periodic structure in a higher space.

A simple 2D square lattice.

A standard grid of points.

Exactly.

Then we draw a line, we'll call it L, through the origin of that grid.

This line represents our physical 1D space.

And critically, its slope must be the irrational number.

If the slope were rational, the projection would just give you a periodic 1D chain.

Correct.

The irrationality of the slope ensures the periodic result.

But you can't just project every point on the grid, or you'd just get a continuous line, a smear.

You'd get a smear.

So you introduce the projection strip.

This is a finite width band that runs parallel to our physical line L.

It exists in the 2D space.

And only the 2D lattice points that fall inside the strip get chosen.

And then they're projected down onto the 1D line L.

Orthognally projected, yes.

Why is the width of that strip so critical?

What happens if it's too wide or too narrow?

The width is mathematically defined.

It has to be just large enough to capture one and only one lattice point from every unit cell of the 2D lattice.

I see.

So if it were too narrow.

You'd miss points.

You'd get gaps, defects in your structure.

And too wide.

You might capture two points from a single cell, which would mess up the sequence, lead to overlaps or local periodicity you don't want.

So the strip's finite size and the line's irrational slope are the two essential ingredients to cut out this periodic sequence.

And the result of this precise cut is that alternating pattern of long L and short S segments on the line.

The L segments come from projections of points in one part of the strip's cross section and the S segments from another.

And the ratio of L to S segments over a long distance tends towards that.

Now we get to the proof.

Diffraction.

If this structure is quasi -periodic but highly ordered, what does its X -ray or electron diffraction pattern look like?

Is it diffused like a glass?

Or sharp like a crystal?

And that's the key finding.

The 1D Fibonacci lattice produces a set of discrete sharp Bragg peaks.

So it's not diffuse.

It has long range order.

It absolutely has long range orientational order.

But the spacing of those peaks must be unusual.

Highly unusual.

The diffraction pattern is described by the Fourier transform of the atomic density.

And what it reveals is that the intensity of any given Bragg peak is actually determined by the shape and density of that projection strip we just talked about.

It acts as a kind of form factor.

It modulates the strength of the peaks.

But here's the crucial non -crystallographic result.

How many indices do you need to describe the position of these sharp peaks?

For a 1D lattice, you'd think one.

You'd think one.

But you need two Miller indices.

Wait, two indices for a one -dimensional object?

That sounds absurd from a classical perspective.

It does.

It implies that the reciprocal space itself must be two -dimensional to contain all the possible reflection vectors.

And that's the mathematical signature of its higher dimensional origin.

It is.

The diffraction condition, QMN, shows that the wave vector for any reflection is indexed by a linear combination of two basis vectors, G1 and G2.

And their critical feature is that the ratio of the magnitudes of these two basis wave vectors is an irrational number.

So the reflections themselves are non -rational.

If you tried to index this using traditional 1D crystallography,

the indices would have to be irrational numbers.

Which is for bitter.

But by moving to 2D indices,

you can use simple integers, M and N, to define the position of a reflection that is physically located at an irrational position.

Precisely.

The equation for the peak positions has embedded right in it.

This is the first concrete demonstration that the law of rational indices, which is a cornerstone of crystallography, has to be generalized when you replace periodicity with quasi -periodicity.

We scale up to 2D and we get something visually stunning and historically significant.

The Penrose tiling.

Invented by Roger Penrose back in 1974, well before anyone had actually seen a physical quasicrystal.

And he designed these tilings to cover a plane completely, using only a small number of shapes or prototiles.

While preserving that forbidden five -fold rotational symmetry and making sure the pattern never repeats.

The source material focuses on the two rhombi, the acute one and the obtuse one.

What determines how often each one shows up?

Just like the L and S segments in 1D, the areas of the acute and obtuse rhombi when you tile them out to infinity have to appear in a ratio that's determined by...

So the ratio of the number of obtuse rhombi to acute rhombi is approximately no.

Yes.

The golden mean is dictating the composition of the structure again.

Let's talk about the inflation -deflation method here.

How does self -similarity work in 2D?

It's really remarkable if you take a Penrose tiling and you inflate it uniformly by a factor of two squared.

The resulting larger tiling is still composed entirely of the original small rhombi.

They're just rearranged.

So it proves the structural integrity in the self -similar order.

But you know, if you just gave someone a box of these tiles, they might accidentally create a little periodic patch.

How did Penrose stop that from happening?

The matching rules.

Right.

This seems philosophically important.

They are absolutely crucial.

There are local constraints.

You can think of them as little arrows drawn on the edges or colors on the corners that have to be obeyed when you join two tiles together.

So a local rule enforces a non -local property.

Exactly.

These local rules compel the tiling to be quasi -periodic across infinite space.

Without them, the natural tendency for the tiles to form little, stable periodic clusters would just take over.

So the matching rules are the mechanism that enforces the long -range orientational order and forbids the translational periodicity.

And what's the geometric result of these rules?

Well, because the rules are so strict, the local environment of any point in the infinite tiling is highly restricted.

The source describes the seven possible vertex configurations.

The star, sun, king, and queen, and so on.

Yes.

At every single vertex, the sum of the angles from the tiles meeting there must add up to 360 degrees to Paris.

It's the specific non -periodic arrangement of these seven configurations that creates the whole macroscopic order.

Now let's turn to the geometric construction.

We needed 2D space for the 1D Fibonacci lattice for a 2D Penrose tiling.

You need to go to even higher dimensions.

How high?

The 2D Penrose tiling is constructed by projecting points from a 5D space down onto the 2D plane.

Why five dimensions?

It comes down to the symmetry.

You need five independent basis vectors to properly describe the five -fold symmetry of the resulting 2D structure.

And what does that projection look like?

You start with a perfect hypercubic lattice in 5D.

You project it down onto a 2D physical space at an irrational angle.

And the basis vectors for the 5D space are defined in such a way that when they're projected onto our 2D plane, they line up with the vertices of a regular pentagon.

Which ensures the five -fold symmetry is mathematically baked into the projection process.

It's preserved.

And as a direct consequence, I'm guessing the diffraction pattern for this 2D tiling must be indexed by five Miller indices.

Exactly.

H1, H2, H3, H4, H5.

We keep adding indices as we increase the dimensionality of the embedding space.

Each index corresponds to one of those five basis vectors needed for the 5D reciprocal lattice.

And it's important to stress this method is general, right?

It's not just for five -fold symmetry.

Not at all.

The projection method is generalized to create other polygonal quasicrystals.

You can generate octagonal 8 -fold, decagonal 10 -fold, and dodecagonal 12 -fold structures this way.

So for 8 -fold and 12 -fold?

You need to project from a 4D space.

For 10 -fold, which is really the Penrose case, you need 5D.

The dimension you need is determined by the minimum number of basis vectors required to rationally index the reciprocal lattice that generates that particular rotational symmetry.

Let's just reflect on the magnitude of this challenge.

Quasicrystals forced us to confront the very core of crystallography.

What specifically does their existence do to the classical law of rational indices?

It delivers a direct conceptual strike.

I mean, the classical law relies on the fundamental assumption that translational periodicity must coexist with rotational symmetry.

It's all baked into the space group concept.

Right.

And that coexistence forces the restriction of rotation angles to only those that can tile space periodically.

Two, three, four, and six -fold.

But the core insight of quasicrystals is that orientational order and translational periodicity are separable.

They're separable.

If you let go of the requirement for translational periodicity, the rotational symmetries become free again.

So you can have these non -rational rotations like five -fold or eight -fold or 12 -fold fully expressed as long -range orientational order in the material.

And that fundamentally rewrites the rules for what constitutes an ordered solid.

But if these rotations are forbidden in our 3D space, how are they mathematically managed in the higher dimensional space to make sure the order actually persists after you project it?

Because the higher dimensional lattice is periodic,

and the symmetry operations you perform in that space have to be compatible with its periodicity.

So for the eight -fold and 12 -fold structures, these rotations are represented by 4D rotation matrices.

Okay.

Can you describe conceptually how a 4D matrix enforces a forbidden symmetry in 2D space?

Sure.

Imagine a 4D hypercube.

The matrix for eight -fold rotation, D8, is a 4x4 matrix made up of only integers 0, 1, and the megas 1.

Okay.

When this matrix acts on a vector in the 4D lattice, it rotates it.

If you apply the matrix eight times, the vector returns to its original position.

That's eight -fold periodicity, perfectly valid in the 4D space.

And when this whole operation is projected down to the 2D plane?

The effect we see is an eight -fold rotation.

The magic is that while the rotation is non -crystallographic in 2D, it's perfectly rational and compatible with the translations in the 4D embedding space.

So the symmetry is conserved by a rational rotation in the higher space, and it only appears irrational when we view the projected slice.

That's why the entire mathematical scaffolding is necessary.

We started with a 3D discovery of icosahedral symmetry.

Let's finish the structural discussion by looking at how we build and describe these 3D quasicrystals.

Okay.

So the 3D Penrose tiling is the structural model.

It replaces the 2D tiles with two specific lombohedral bricks.

The acute ramahedron and the obtuse ramahedron.

Right.

These are the two basic building blocks that fill 3D space, but they have to adhere to very strict matching rules to maintain that a periodic icosahedrally symmetric structure.

And just like the 2D case, I'm guessing the face angles and diagonal ratios of these 3D bricks are all determined by curse.

The geometric signature of terse is everywhere.

And to achieve icosahedral symmetry, which is the most complex of these forbidden symmetries, the 3D structure must be constructed via projection from a 6D space.

Okay.

Why 6D?

What does icosahedral symmetry demand that requires six dimensions?

An icosahedron has an incredible amount of symmetry.

Six five -fold axes, 10 three -fold axes, 15 two -fold axes.

The standard way to preserve all of that symmetry while keeping the embedding space periodic is to use six basis vectors that correspond to the six orthogonal two -fold rotation axes of the icosahedron.

And those six vectors span the 6D space needed to house the full symmetry.

Yes.

And their components, of course, involve off.

Which means the indexing of the diffraction pattern must reflect this 6D origin.

We're talking about six miller indices.

Precisely.

To index the position of any Bragg peak in a 3D icosahedral quasicrystal, you need six integer miller indices, hkl, hkl.

So if classical crystallography uses three indices, hkl, quasicrystography uses six, what do those extra three indices even represent physically?

They relate the physical scattering vector we measure to the basis vectors of the 6D reciprocal lattice.

The first three indices relate it to components in our physical 3D space, and the other three relate it to components in that perpendicular space, the hidden dimension.

It's the only way to define a reflection that would otherwise have an irrational spacing in 3D.

It is.

The six indices are required.

So we can measure the position of the reflections, calculate the Q values, and confirm that the pattern really requires this six index irrational scheme.

The map is validated by the experiment.

And we can go even further.

We can define a measure for this structure that's analogous to a conventional lattice constant.

Okay.

We call it the quasi -lattice constant, i -th force.

How is IR defined and measured?

It's defined relative to a scale factor that you get from the diffraction pattern.

If you plot the squared magnitude of the wave vector Q squared against an index scale factor, you get a straight line.

The slope of that line gives you a scale factor, which is then used to calculate R.

So it provides a crucial measurable dimension.

Yes.

For example, for a specific T.

nifessi quasicrystal, the constant is measured to be 0 .4784 nanometers.

That's a huge takeaway.

Despite being a periodic and requiring 6D math, you can still assign a physical measurable constant, AUR, that links the structure directly to the size of its bricks and its relationship to nearby periodic phases.

It's vital to remember the context of 1984, though.

This quasicrystal model was so revolutionary that it was met with intense skepticism.

Oh, yeah.

Including from Nobel laureates.

Famously so.

This led to alternative structural models being proposed to explain Sheckman's bizarre diffraction patterns.

What was the main alternative before the projection method really took hold?

The first big one was the multiple twinning model.

This theory suggested that the material wasn't truly quasi -periodic at all.

Instead, it was just composed of extremely small, conventionally cubic crystals that were twinned together.

And twinned in such a regular way that they mimic the macroscopic icosahedral symmetry.

That was the idea.

So why did that fail to explain the data?

It struggled with the observed sharpness of the Bragg peaks.

In crystallography, the width of a diffraction peak is inversely related to the size of the coherent scattering domain.

Meaning bigger, more perfect crystals give sharper peaks?

Exactly.

If the structure was made of tiny micro -twin domains, as the model required, the diffraction peaks should have been significantly broadened.

But they weren't.

Sheckman observed peaks that were as sharp as those from perfect single crystals.

Which demanded long -range order that extended over many micrometers.

The twinning model simply could not account for that observed long -range coherence.

So the sharpness of the peaks was the initial executioner of that theory.

What about the other one, the icosahedral glass model?

The icosahedral glass models took a different approach.

They said, sure, the local arrangement of atoms might form these icosahedral clusters.

But those clusters are just randomly packed with no long -range correlation, like atoms in an amorphous glass.

That was the premise.

And if the arrangement was random, what would the diffraction pattern look like?

Well, a true glass gives you diffuse scattering.

You know, broad hazy rings.

No sharp Bragg reflections.

So while that model could explain the five -fold local symmetry.

It failed spectacularly to account for the sharp, discrete Bragg peaks they were actually seeing.

The existence of those sharp peaks proved the structure had long -range orientational order, which completely ruled out the random packing of a glass.

So the evidence that really cemented the quasicrystal model was purely observational.

The definitive proof, the smoking gun, came from selected area electron diffraction, or SAED,

studies on single quasicrystal grains.

And what did those confirm?

Two things at once.

One, sharp diffraction peaks proving the long -range order.

And two, the clear presence of five -fold, three -fold, and two -fold rotation axes confirming the icosahedral symmetry.

So when researchers rotated the crystal and looked down that five -fold zone axis.

They saw the unmistakable 10 -dot pattern that only perfect five -fold symmetry can produce.

It was undeniable.

And these observations, often on tiny crystals, maybe half a micrometer to two micrometers in size,

just permanently validated the whole quasi -periodic model.

We should also acknowledge the key historical figures here.

Dan Sheckman, of course, for the discovery and for facing enormous resistance for it.

Absolutely.

And John Werner Kahn, who was critical in the early analysis, applying the geometric principles and helping to establish that six -index notation.

Roger Penrose, who provided the fundamental mathematical geometry years before the physical material was even found.

It's an amazing example of abstract math predicting physical reality.

What an incredible journey into structural science.

We've covered ground that forced scientists to add an entirely new definition of order to their lexicon.

Let's provide you, the learner, with the absolute core takeaways to master this.

Okay, first.

Quasicrystals are ordered materials defined by non -crystallographic symmetries, like five -fold or eight -fold rotations.

And they violate the law of rational indices because their orientational order exists independent of translational periodicity.

Second,

the golden mean is the fundamental mathematical ratio governing their geometry.

Its unique algebraic property, total plus one, is what enables the system to be self -similar.

Third,

they are mathematically constructed using the cut and project method, where our 3D structure is generated by taking an irrational slice of a perfect periodic crystal living in a higher dimensional space.

5D for decagonal, 6D for icosahedral.

And fourth,

because of that higher dimensional origin, their diffraction patterns require more Miller indices, up to six for a 3D quasicrystal, to rationally index the reflection positions.

And the final provocative thought is this.

The discovery of quasicrystals forces us to realize that the definition of crystalline order is just so much broader than a simple repeating box.

It's a much richer concept.

It is.

It demonstrates that the most intricate, complex, and beautiful structures in the physical world are governed not by the simplicity of rational numbers, but by the infinite recursion and irrationality of concepts like the golden mean, hidden away in extra dimensions until experiment finally revealed them.

Absolutely fascinating.

Thank you for joining us on this deep dive into metallic structures and the geometry of the impossible.

Something to mull on until next time.