We can start.

Yes.

So it doesn't matter if I have little
gaps.

It doesn't matter at all.

They should edit out quite easily.

Yes, quite easily.

Yes.

Okay.

Yes.

Now, please.

Two cameras are now rolling.

Well, this talk is about metal bonding.

The bonding between atoms that hold a
piece of metal together at the atomic level.

It's A simplified version, which I wrote
for my family and friends.

I mean, the full paper is, I don't know,
15 pages of theoretical physics.

But this is just about 3 pages of simple
explanation of the paper.

The full reference to the paper is

It's published, it was published at the
beginning of last year, that's 2024 last

year, yes, and by myself and Siyu Chen,
a graduate student who was helping me

with things.

In Journal of Physics, Condensed Matter,
Volume 36, Paper 353002, published in 2024.

There are two strange things about metal
bonding.

The first is that if you have a grain of
sand or sugar or a crystal of salt and

you just hit it with a hammer, it'll
crush into a powder.

But a metal doesn't do that.

the metal just flattens, and that's
different, it's unusual.

The bond between neighboring atoms is a
covalent bond, the same as in hydrogen

bond, H2, or oxygen, O2, or water, H2O,
or indeed all the molecules that make up

our bodies.

So it's not that they're different, it's
the

how they work together, that is
different.

Somehow the atoms of a metal hold
strongly together.

But they care far less about how they
are arranged in their crystal structure

or whatever with respect to their
neighbors.

That's why if you hit it, the atoms
shift around, but they're still close

together because the metal doesn't
expand or contract.

It still has the same volume as before.

It's just that it's changed.

In chemistry, we learn about how two
neighboring atoms join together to form

a covalent bond between them, such as
the oxygen we breathe and the water that

we've just mentioned.

The other strange thing about metals

is that if we make a picture of how we
think the bonds are between atoms, sort

of a classical kind of picture, we would
have all these atoms and place the

electrons between them making the bonds.

But what that would be strange is that
we only have three electrons in

aluminium and 12 neighbors.

And in sodium as a more extreme example,
we only have one electron per atom and

12 or 14 neighbors.

How can one electron per atom form
enough bonds, bonding to hold all this together?

That is the second thing that is strange
about metals.

A short piece of mathematics in the main
paper shows that in quantum mechanics,

that in quantum mechanics it's
different.

This physical picture that I described
of the electrons sort of between the
55
00:05:38,805 --> 00:05:40,6
atoms, that's correct.

But in classical physics it would be
that in one instant and then another

instant the electrons would have moved
and so they would be moving about and on

average it would be everywhere.

everywhere, but it would be moving.

The electrons are entities and they
would be moving like that.

In quantum mechanics that is different
because the quantum mechanical thing is

the electron is a wave function that is
spread out.

And so the electron can be, is indeed
everywhere

all the time.

And it's the ground state, as we call
it, the basic state of the system is

then a mixture of all possible pictures
that you can draw at the same time all together.

And it was pointed out by Pauling
already in 1938 that if you look between

two atoms,

The covalent bond looks just like any
other covalent bond, but it's this

totality that is different with only one
electron in sodium or three electrons

per atom in aluminum that somehow spread
themselves around in a quantum kind of

way that then holds everything together.

short-term piece of mathematics applied
to this, the quantum mechanics of this,

shows that the bonding of an atom is not
to its neighbor individually, but as a

covalent cluster bond to its cluster of
covalent surrounding atoms as a whole.

That's different from this, that, and
the other.

It's one

cluster bond to all its neighbors
simultaneously.

And its bond strength, the bond strength
is proportional to the square root of

the number of neighbors for each such
bonding electron.

As c is the total coordination number,
we call it, of neighbors around the atom

that we are discussing.

It is totally an inherent quantum effect
that all of these pictures are included

in the ground state.

Ground state, meaning lowest energy
state in quantum mechanics.

They're not sort of alternatives in
time.

There's no classical counterpart to this
in classical physics.

There's underlined, emphasized in a
paper by Schrodinger.

1935, discussing this.

A lot of things immediately follow.

First of all, they explain why metals
have close-packed crystal structures.

I mean, you can have crystal structures
with an atom having four neighbors or

six or something, but in metals, they
all have 12 or 14

neighbors, and that is because they form
these cluster bonds, and the cluster

bonds, they like to have as big a
cluster as possible, and so they have a

shell around the atom enclosing it.

That's one of the basic things.

Secondly, we note what a huge
multiplicative effect is enhancement by

the square root of the coordination
number.

I mentioned that the energy of the bond
is proportional to the square root of

the coordination number for each
electron.

So each covalent bond

has this square root of the coordination
number as a multiplicative factor.

And that's a lot.

I mean, 12 neighbors, or 14, is the norm
for metals.

And the square root of 12 is 3.46.

And so you're multiplying the strength
of the bond that you think the bond

might have by 3.46

to make it this cluster bond and bond to
the cluster of neighbors as a whole.

The next thing that it explains is the
easy structural phase transitions.

I mean that

Usually we think of phasic transitions
classically that, well, they have one

structure and at another temperature or
pressure they may have another

structure, but it's a big disruption to
change from one structure to another.

In metals, this is not so.

I mean, the important thing is the
cluster.

And the cluster is still there.

It may be just atoms have moved
slightly, but even that may not be so.

I mean, phase-centered cubic and
hexagonal close-packed are made-up out

of the same clusters, but stacked
differently.

And so the phase transitions between one
and the other as a function of

temperature or pressure is very common
amongst metals.

Fourth effect that I've noted is there
was a couple of metallurgists called Daw

and Basques in 1983.

Now this is a long time later, I mean.

They found that just they were modelers.

putting in atomic models to understand
dislocations and things in metals,

complicated effects in metals, because
at that time...

People were studying nuclear power, and
you have all these nuclear transition,

nuclear particles being expelled with
big force, and they cause damage in the

shells that hold the plutonium or
uranium together.

And so there's a lot of radiation
damage, and it's very important to

understand that.

Another problem that had a puzzle that
had really cleared up was the energy of

a vacancy.

I mean, a vacancy is just as an atom is
missing.

And if you think of 12 neighbors, well,

You've broken 12 bonds if you take an
atom out.

Then when you put it on the surface, you
can't destroy atoms.

So what you do is the way physics works.

is you take the atom out, then you add
it just as an extra atom on the surface.

Well, on the average, with 12 neighbors,
every atom has six.

You're making six new bonds each time
you add an atom.

So you've taken out 12, and you've got
six back.

So that you might expect the, and so
you've lost six.

Well, when you just build up a solid
from atoms, you're adding six and six

and six to build up the hole.

And so you might expect the energy of a
vacancy to equal the opposite of the

binding energy of the material.

So in other words, yes, the vacancy is
the same as taking away an atom.

Well, it isn't, it's half that.

And the half is, well, I have to do some
bit of complicated algebra.

It's not really terribly complicated,
but it comes from the square root.

The square root means to the power of
one half.

And so it's from that that this factor
of a half comes in.

The paper, it was published at the
beginning of 2024, and also has a couple

of interesting applications to
chemistry.

The first of these is catalysis.

Now, my simple picture doesn't

is not discussing any particular
catalytic reaction, but it does discuss

something that contributes.

It's a grossly oversimplified system.

Let us consider a monovalent metal, just
one electron per atom, and let us

consider a

pair molecule of that same atom element.

And then we can consider the element for
the molecule, the pair, first as a

molecule in the vacuum above the
surface, and then we think of it on the

surface and we're concerned with the
energy of the molecule dividing.

And, well, in the vapor phase, it's
obvious it just divides.

We have one electron from each atom in
our simple picture, so it's two bonding electrons.

On the surface, it's different.

because the two atoms, the molecule,
sits on the surface and it's attached to

the surface, but they're now moving
apart but still attached to the surface.

And it's that difference that makes a
big difference to the energetics.

If the coordination of the

the material that it's on.

It's the coordination, the extra
coordination at the surface.

So you have an atom on the surface
sitting on a, if we think of just a

simple cubic structure, it would, the
atom that it's sitting on is enhanced
180
00:17:07,93 --> 00:17:07,894
from having

four neighbors to now having five
neighbors to now having its total six,

which is what the simple cubic structure
would have.

And so it's a change in the bonding of
the surface atoms of the material from

five to six.

Well, that gives it a better, that is,
extra bonding from the point of view of the

metal that it's sitting on.

But that is retained when the two atoms
on the surface, the molecule, move apart.

And so instead of the atoms on the
surface, instead of having lost their

total bonding, they've reduced it from

from 1 to something like the difference
between 5 and 6.

If you take a look at square roots, the
difference between 6 and 5 is quite small.

The difference between 1 and 0 of the
molecule in the vacuum in the gaseous

phase is big.

one born broken, well it's two up and
down spin, and the same is applied on

the surface, but it's one compared with
the difference between six and five, and

the difference in fact between the
square root of six and the square root

of five.

And you can work that out, and that's of
course quite a small quantity.

And so the presence of sitting on a
surface makes the

breaking open of the bond of the
molecule from the vapor is much reduced.

And I mean, chemists may be surprised
that one can make a sort of a general

theory of catalysis, because I think
chemists tend to think of each chemical

reaction as different.

But they're of course correct.

But running through any such calculation
that they might make, there would be

this theme, there would be this fact of
quantum mechanics, that the energy for

pulling them apart on the surface is a
lot less than pulling them apart in the

gas phase.

And that will remain.

This is where a physicist thinks in
terms of these simple models.

But I mean, chemistry was my first love
at school.

So I keep that orientation.

Now we come to a second application to
chemistry.

And that is the benzene molecule.

This is benzene, Z-E-N-E, not benzene,
Z-I-N-E, that you put into your America.

And this has a structure of six atoms of
which the atoms in the plane of the molecule

They're all filled and they give the
structure of the six hexagon.

But there is one electron per atom, a pi
electron, p2pz, perpendicular to the

plane of the molecule that is left over.

And that travels around the ring in a
band of, well, it could have 12

electrons because it has six atoms and

up and down spin, but we only have 6
electrons, and so it is a half-filled

band, like an alkaline metal.

And that is what gives it its great
stability.

Also, we

neighboring atom, I mean a benzene ring,
exists in molecules and coffee, all

sorts of biological biochemistry, it's
all over biochemistry.

And so we can think of a benzene with an
extra little group added.

And that will of course disturb the six
electrons and cause a ripple in the

electron density of that band.

And that oscillates up and down exactly
the same way as Jacques Friedel

discussed for metals in the 1950s and
60s.

If you put an impurity or any change in
a metal, it sets up ripples in the

electron density with a wavelength, wave
number of twice kf.

relating to the Fermi surface and that's
spread out.

Well, it's the same around the ring if
we do something like that to a benzene ring.
237
00:22:42,896 --> 00:22:51,4
And these are called the para, meta, and
ortho positions around the ring.

This part I hadn't really prepared
enough.

The answer to the puzzles about metallic
bonding that I mentioned at the

beginning were already

implied, but not really thought about
and discussed, somewhat hidden in the

1929 by Felix Bloch, the first paper to
think of quantum mechanics applied to metals.

And it was that paper that created what
we now call functions that span the

whole metal.

And the point about it was it had to
assume, in order for the mathematics to

work, that it was a perfect crystal
extending in all directions to infinity,

just pure that metal atom everywhere.

Well of course that's not the reality
and we showed that I talked about

surfaces of metals and hitting them and
changing the structure and so on.

So this approach that I was using was a
more atomic approach which was started by

Professor Jacques Friedel in Paris in
the 1960s, where I visited.

And it was one of his graduate students,
Francois Lachmann, who was doing that work.

And it was just, yes, it was doing
quantum mechanics differently, at least

applying it differently to metals.

And this was

And I was interested in this, but they
didn't, they made some use of it, but

not very much.

And not in the way that I started to
think with the square root of the

coordination number and so on.

That then came in the, I think the first
time in a review article around 19, or

end of the 19th.

Yes, 1980, a review article about local
methods.

That's, and then it's developed
gradually since then in my thinking to

what was contained in the paper
published at the beginning of 2024.

Yes.

The end.