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PROFESSOR: All right, let's
resume our discussion.

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And we'll finish it for sure
by the end of today.

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00:00:14,850 --> 00:00:17,715
And as you say, you can tuck
this away for study on your

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00:00:17,715 --> 00:00:24,010
own or for future reference or
whatever you want to do.

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00:00:24,010 --> 00:00:30,480
So if we could assign sizes to
atoms and ions, we could do a

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lot towards predicting what sort
of structures they might

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00:00:35,320 --> 00:00:37,500
form if we knew the
composition.

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00:00:37,500 --> 00:00:38,680
The composition--

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00:00:38,680 --> 00:00:44,000
again, assuming there is only
one type of anion and one type

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00:00:44,000 --> 00:00:48,380
of cation, the composition is
determined by the valence.

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The composition fixes the ratio
of coordination numbers.

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And the structure will have
lowest energy when we have the

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maximum number of anions around
cation and vice-versa.

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So we're in great shape.

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And this is simple, idealized,
but very powerful stuff if we

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knew the sizes that were
assumed by ions.

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And that, really, is an absurd
notion, to talk about an atom

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as though it behaved like a
sphere with finite dimensions.

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If we plotted the density of the
electrons as a function of

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distance R from an atom, it
does something like this.

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At very small distances, that
defines a shell of a rather

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small volume, so there are
not many electrons.

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Rises up to a maximum, and
then tails off gradually.

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So what is the radius?

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The maximum in the distribution
of density.

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So the notion of radius is
absolutely nonsense in terms

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of the actual distribution
of electrons on the atom.

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However, one can get from your
fraction measurements a

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00:02:14,710 --> 00:02:16,590
determination of the
lattice constant.

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00:02:20,200 --> 00:02:25,680
And even with routine methods,
you can get the dimension of

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00:02:25,680 --> 00:02:28,530
the lattice to extraordinary
precision.

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00:02:28,530 --> 00:02:31,520
The typical lattice constant
is about 10 Angstroms.

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00:02:34,140 --> 00:02:38,270
You can get this through routine
methods to plus or

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minus 1 in the fourth
decimal point.

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So this is a precision of
1 part in 10 to the 5th.

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You can get atomic positions
when they are not fixed by

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00:03:02,840 --> 00:03:11,705
symmetry to get a coordinate to
plus or minus 0.yyyy plus

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or minus 1 in good cases.

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So that is 1 part in
10 to the 5th.

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Missing a factor or
10 somewhere.

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This is plus or minus 1.

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That's 1 in 10 to the 4th.

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00:03:30,090 --> 00:03:33,850
So you can get this information
very accurately.

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00:03:33,850 --> 00:03:37,620
One thing I always like to
point out here is that a

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thermal expansion coefficient
for most inorganic materials

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00:03:46,990 --> 00:03:52,080
is something that never gets
much lower than about 1 or 2

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00:03:52,080 --> 00:03:54,410
times 10 to the minus 6.

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00:03:54,410 --> 00:03:57,210
So if you're going to push the
measurement of a lattice

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00:03:57,210 --> 00:04:00,610
constant to the ultimate
accuracy, you'd better have a

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00:04:00,610 --> 00:04:04,030
thermometer on the top of the
X-ray generator, or your

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00:04:04,030 --> 00:04:05,880
measurement is meaningless.

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00:04:05,880 --> 00:04:09,190
The thermal expansion can be
more than the uncertainty in

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00:04:09,190 --> 00:04:10,420
your measurement.

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00:04:10,420 --> 00:04:13,360
So you should report, for very
precise measurements, the

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00:04:13,360 --> 00:04:15,218
temperature at which you
made your measurement.

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00:04:17,970 --> 00:04:22,220
OK, you can get interionic
distances with very high

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precisions.

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And if you look at a series
of compounds--

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00:04:28,100 --> 00:04:30,520
and these are the ones that are
discussed in the notes,

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00:04:30,520 --> 00:04:33,340
which have the rock salt
structure type.

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So here is the cation,
here is the anion.

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The anion-cation distance is
simply equal to 1/2 of A for

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the rock salt structure.

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00:04:49,560 --> 00:04:50,400
So that's easy.

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00:04:50,400 --> 00:04:52,700
You don't even have to know
the atomic positions.

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00:04:52,700 --> 00:04:54,670
Just measure the lattice
constants.

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00:04:54,670 --> 00:05:01,030
And then you would find, for
example, that R, NA plus R, CL

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00:05:01,030 --> 00:05:05,700
is equal to 1/2 of A for NaCl.

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Very precisely know the sum.

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00:05:11,370 --> 00:05:12,640
How do we split this up?

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00:05:12,640 --> 00:05:14,700
Well, let's look at
another compound.

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00:05:14,700 --> 00:05:20,780
Let's look at R, NA plus
R for bromine.

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00:05:20,780 --> 00:05:24,010
And that would be 1/2 of the
lattice constant of NaBr.

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Hm.

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00:05:27,810 --> 00:05:29,890
More data, but we haven't
solved anything.

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We have three radii, only
two measurements.

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00:05:34,220 --> 00:05:37,540
And you can see, if you try to
push this further, you can

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00:05:37,540 --> 00:05:43,330
never get enough information
to split up the interionic

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00:05:43,330 --> 00:05:47,050
distances into sums of
individual radii in any sort

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of unique fashion.

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So you have to start
somewhere.

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And there are different sets
of ionic radii that were

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00:05:56,120 --> 00:05:58,910
proposed early in the game.

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And they are discussed
in the notes.

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I give you an example of some
of the sums of radii for the

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00:06:13,910 --> 00:06:16,430
oxides and sulfides
and selenides of

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00:06:16,430 --> 00:06:18,120
the transition metals.

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You can see that as the anion
atomic number increases, the

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00:06:22,560 --> 00:06:24,620
size of the lattice
constant goes up.

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00:06:24,620 --> 00:06:28,560
So you could get, for example,
the difference in size between

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a sulfur and an oxygen and the
difference in radius between a

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selenium and a sulfur.

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00:06:34,920 --> 00:06:36,715
But you could never get
the absolute values.

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There are four early
sets of radii.

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And they are tabulated for you
part way through the notes.

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And there's a cautionary
lesson here.

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One the early series
of radii--

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see, even the old workers in
this field appreciated the

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role of ionic sizes in
determining structure.

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So they wanted to set up

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00:07:16,170 --> 00:07:19,030
self-consistent schemes of radii.

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And the earliest one is by,
really, the founder of crystal

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chemistry, a man named
Goldschmidt.

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00:07:26,660 --> 00:07:31,460
Linus Pauling got into the
act very early on.

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If you look at this table that
I copied from these separate

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publications, four different
people--

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Goldschmidt, Pauling, Zacharias
and Ahrens.

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Some people left out the radii
of certain species.

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Other people left out different
radii depending on

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the compounds that
they looked at.

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If people looked at the same
ion, they very often got very,

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very different values,
depending on

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their starting point.

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Look at the size of barium,
for example.

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Zacharias said 1.29.

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Ahrens said 1.34.

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Pauling said 1.35.

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Goldschmidt said 1.43.

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Depends on the starting
point that they used

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for the initial radius.

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And once you've determined
that,

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everything falls into place.

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Another basis for determining
radii, Bragg determined a

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number of the structure of
important silicates.

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Why silicates?

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Because as you walk
along the ground,

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you're walking on silicates.

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These are part of the minerals
that form the Earth's crust,

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so there was a lot of
interest in those.

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Bragg, from his early work way
back in the late 1920s, found

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that the size and shape of the
Si04 tetrahedron forms

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tetrahedral coordination because
silicon is so small

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compared to oxygen.

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The interionic distances were
always about the same.

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The tetrahedron was
always regular.

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So what Bragg said was, a-ha,
this is a tough little nut in

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silicate structures.

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Doesn't deform.

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The size is always the same.

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It's always regular.

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The reason must be that the
oxygens are just in contact.

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So he used the dimensions of an
Si04 tetrahedron, took the

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radius of oxygen as
half the edge.

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Pauling used a different
starting point

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that has been followed.

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There's still another way
of determining radii.

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Is there any physical property
of a material which might also

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be shown to be proportional
to the size of the ion?

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And the answer to that,
which I'll pass

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over quickly, is yes.

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Quantity called the ionic
electrical polarizability,

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00:10:01,600 --> 00:10:05,850
which is the relation between an
applied electric field and

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00:10:05,850 --> 00:10:09,130
the dipole moment that is
induced on the atom when you

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separate the electron cloud
and the nucleus.

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And a simple model shows--

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and this is carried
out for you--

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that you can relate the ionic
electronic polarizability to

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the volume of the atom, and it
turns out to be beautifully

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simple, that the polarizability
is simply 4 pi

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00:10:37,010 --> 00:10:40,500
epsilon 0 times the radius
of the ion cubed.

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So therefore, you can split up
ionic radii in proportion to

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00:10:47,510 --> 00:10:49,910
the cube root of the
polarizability of the ions.

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Well, that trades one
problem for another.

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How do you get polarizabilities?

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It turns out that the index of
refraction of materials is

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00:11:00,230 --> 00:11:07,480
given by the sum of the number
of each species per unit

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00:11:07,480 --> 00:11:08,800
volume times the ionic

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00:11:08,800 --> 00:11:12,050
polarizability of that species.

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00:11:12,050 --> 00:11:16,530
And there is a famous
equation called the

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00:11:16,530 --> 00:11:18,850
Lorentz-Lorenz equation.

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00:11:18,850 --> 00:11:23,040
I can never resist calling
it not the Lorenz-Lorentz

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00:11:23,040 --> 00:11:26,250
equation but the Lorenz-Lorentz
equation

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00:11:26,250 --> 00:11:27,610
equation because it
sounds like you're

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00:11:27,610 --> 00:11:30,140
saying everything twice.

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If you note that the dielectric
constant of a

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00:11:37,130 --> 00:11:39,960
material is related to the
square of the index of

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00:11:39,960 --> 00:11:43,640
refraction, instead of n
squared, you can replace the n

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00:11:43,640 --> 00:11:47,970
squared by epsilon, the
dielectric constant, and then

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00:11:47,970 --> 00:11:49,860
the equations become
known as the

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00:11:49,860 --> 00:11:52,720
Clausius-Mossotti equations.

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00:11:52,720 --> 00:11:56,400
And I always take that as a nice
illustration of the fact

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that a very simple change of
variables can be all it takes

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00:12:00,760 --> 00:12:03,722
to become famous and immortal.

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00:12:03,722 --> 00:12:04,270
[INAUDIBLE]

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00:12:04,270 --> 00:12:07,570
what happened to Clausius
and Mossotti.

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00:12:07,570 --> 00:12:09,500
Where do you get
polarizabilities from?

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00:12:09,500 --> 00:12:12,440
This was looked at by
a number of people.

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00:12:12,440 --> 00:12:16,540
One of the most complete were
three people, Tessman, Kahn,

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00:12:16,540 --> 00:12:20,540
and Wild Bill Shockley
of transistor fame.

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00:12:20,540 --> 00:12:22,480
So you can find these numbers,
and you can get a

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00:12:22,480 --> 00:12:24,520
self-consistent set of radii.

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00:12:24,520 --> 00:12:27,740
But note in the table that
I give you here that the

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00:12:27,740 --> 00:12:30,490
individual numbers are
very different.

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00:12:30,490 --> 00:12:36,600
And two caveats that I have to
issue is you can't mix ionic

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00:12:36,600 --> 00:12:41,530
radii in different people's
tables because they have

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00:12:41,530 --> 00:12:45,060
different starting assumptions
for the standard radius.

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00:12:45,060 --> 00:12:48,000
If you do that, you're mixing
apples and oranges.

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00:12:48,000 --> 00:12:51,660
And I wish I had a cookie for
every half-baked book on

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00:12:51,660 --> 00:12:55,440
chemistry that said, hey, Ahrens
has got something for

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00:12:55,440 --> 00:12:57,060
americium and Pauling doesn't.

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00:12:57,060 --> 00:12:59,430
Let me put the value for
americium in there.

202
00:12:59,430 --> 00:13:02,580
Totally wrong that they
are established

203
00:13:02,580 --> 00:13:04,140
on different bases.

204
00:13:04,140 --> 00:13:06,730
So you can't mix apples
and oranges.

205
00:13:06,730 --> 00:13:09,780
You have to use a
self-consistent set of radii.

206
00:13:09,780 --> 00:13:13,590
The ones that were followed
until very recently are

207
00:13:13,590 --> 00:13:15,900
Pauling's radii.

208
00:13:15,900 --> 00:13:20,820
And Pauling's radii are based
on a standard radius for

209
00:13:20,820 --> 00:13:26,210
oxygen, which is 1.40
Angstroms for

210
00:13:26,210 --> 00:13:27,460
the radius of oxygen.

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00:13:31,660 --> 00:13:34,670
The other thing that I have to
issue as a caveat, which I

212
00:13:34,670 --> 00:13:39,710
hope you won't forget, is that
radius ratios do a pretty good

213
00:13:39,710 --> 00:13:44,850
job of predicting the expected
coordination number.

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00:13:44,850 --> 00:13:49,070
But if you are very, very close
to a limiting critical

215
00:13:49,070 --> 00:13:52,980
radius ratio, all
bets are off.

216
00:13:52,980 --> 00:13:56,430
Because then you're really
asking the question, if I have

217
00:13:56,430 --> 00:14:02,500
this polyhedron of ions, if I
squish them a little bit so

218
00:14:02,500 --> 00:14:06,130
that I increase the coordination
number, does that

219
00:14:06,130 --> 00:14:10,250
give me a lower energy in return
for the price that I

220
00:14:10,250 --> 00:14:14,060
have to pay to deform the ions
that I'm packing together?

221
00:14:14,060 --> 00:14:18,560
And that is not possible to
answer on the basis of a rigid

222
00:14:18,560 --> 00:14:19,840
sphere model.

223
00:14:19,840 --> 00:14:24,710
So when you're very close to a
critical radius ratio, you

224
00:14:24,710 --> 00:14:28,900
really can't say for sure
whether the structure will go

225
00:14:28,900 --> 00:14:31,110
to the lower coordination
number or the higher

226
00:14:31,110 --> 00:14:32,140
coordination number.

227
00:14:32,140 --> 00:14:35,700
So you have to be
careful of that.

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00:14:35,700 --> 00:14:37,670
Finally, people have
continued on.

229
00:14:37,670 --> 00:14:42,190
And stuck in the margin of
these older radii is a

230
00:14:42,190 --> 00:14:48,670
reference to a paper by two
people, Bob Shannon, who was a

231
00:14:48,670 --> 00:14:53,380
crystallographer at DuPont,
and Charlie Prewitt.

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00:14:53,380 --> 00:14:57,260
I'm very proud that he became so
notable, because he was my

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00:14:57,260 --> 00:14:58,840
office mate in graduate
school.

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00:14:58,840 --> 00:15:02,030
So I like to promote his
radii whenever I can.

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00:15:02,030 --> 00:15:07,080
And in Acta Crystallographica in
1969, they published a set

236
00:15:07,080 --> 00:15:15,620
of radii, which is in the nice
typeset version that's Xeroxed

237
00:15:15,620 --> 00:15:19,470
directly from their paper in
Acta Crystallographica.

238
00:15:19,470 --> 00:15:24,060
And the headings need
explanation.

239
00:15:24,060 --> 00:15:27,890
There's the ion and its valence,
then the electron

240
00:15:27,890 --> 00:15:29,810
configuration.

241
00:15:29,810 --> 00:15:33,600
And what's interesting is that
some of the transition metals,

242
00:15:33,600 --> 00:15:37,870
for example iron, can be in
a high-spin or a low-spin

243
00:15:37,870 --> 00:15:40,410
configuration depending on
whether the moments on the

244
00:15:40,410 --> 00:15:43,660
electrons are all parallel
or anti-parallel.

245
00:15:43,660 --> 00:15:46,810
High-spin and low-spin
configurations give ions with

246
00:15:46,810 --> 00:15:48,910
very, very distinct radii.

247
00:15:54,850 --> 00:15:57,880
So the second column, EC,
stands for Electron

248
00:15:57,880 --> 00:15:59,030
Configuration.

249
00:15:59,030 --> 00:16:01,310
Then comes coordination
number.

250
00:16:01,310 --> 00:16:05,900
And here is another feature
that we really should not

251
00:16:05,900 --> 00:16:14,280
overlook, that as even
rigid spheres--

252
00:16:14,280 --> 00:16:17,280
here's an A, and here's a B--

253
00:16:17,280 --> 00:16:22,880
acquire progressively higher
coordination numbers--

254
00:16:22,880 --> 00:16:25,580
let's look at just coordination
number three and

255
00:16:25,580 --> 00:16:28,105
coordination number four,
because they're easy to draw.

256
00:16:30,990 --> 00:16:33,980
In addition to the attractive
force between anion and

257
00:16:33,980 --> 00:16:36,470
cation, there is also
a repulsive

258
00:16:36,470 --> 00:16:41,850
force between the anions.

259
00:16:41,850 --> 00:16:46,790
So here's the repulsive force
between this B and its

260
00:16:46,790 --> 00:16:49,750
neighboring B and this B and
its neighboring B. And the

261
00:16:49,750 --> 00:16:54,900
resultant is a force that
tends to push apart

262
00:16:54,900 --> 00:16:57,720
the A and the B ion.

263
00:16:57,720 --> 00:17:00,770
For four-coordination, remember
first of all, the Bs

264
00:17:00,770 --> 00:17:03,650
are closer, so the repulsive
force is stronger.

265
00:17:03,650 --> 00:17:05,410
But there are four
of them, as well.

266
00:17:05,410 --> 00:17:09,280
So this turns out to be a
larger repulsive force.

267
00:17:09,280 --> 00:17:13,730
So the conclusion is that the
radius of A would be expected

268
00:17:13,730 --> 00:17:24,790
to go up slightly as the
coordination number increased,

269
00:17:24,790 --> 00:17:27,800
simply because there is a
repulsive interaction between

270
00:17:27,800 --> 00:17:29,240
the coordinating Bs.

271
00:17:29,240 --> 00:17:32,380
And that will tend to push them
apart more strongly as

272
00:17:32,380 --> 00:17:33,970
the coordination goes up.

273
00:17:33,970 --> 00:17:37,930
So the third column in the
Shannon and Prewitt tables is

274
00:17:37,930 --> 00:17:39,770
coordination number.

275
00:17:39,770 --> 00:17:46,200
And if you look at several
ions, they're--

276
00:17:46,200 --> 00:17:48,740
for example, barium has
6-coordination,

277
00:17:48,740 --> 00:17:49,840
7-coordination.

278
00:17:49,840 --> 00:17:52,690
8, 9, 10, and even 12.

279
00:17:52,690 --> 00:17:58,250
And as you go down that list
of radii, it gets larger as

280
00:17:58,250 --> 00:18:00,020
the coordination goes up.

281
00:18:00,020 --> 00:18:04,240
And it's a relatively small
effect that's on the order of

282
00:18:04,240 --> 00:18:06,300
5 or so percent.

283
00:18:06,300 --> 00:18:10,560
But still, when trying to
calculate an interionic

284
00:18:10,560 --> 00:18:13,950
distance, you should use the
radius that is appropriate for

285
00:18:13,950 --> 00:18:17,090
the coordination numbers.

286
00:18:17,090 --> 00:18:20,930
The fourth column is spin.

287
00:18:20,930 --> 00:18:24,630
And the spin stands for
high spin or low spin.

288
00:18:24,630 --> 00:18:28,700
And you will find that for the
transition metal atoms--

289
00:18:28,700 --> 00:18:32,990
for example, CO2 plus low spin
or high spin, and note the

290
00:18:32,990 --> 00:18:36,640
rather different radii for these
two configurations of

291
00:18:36,640 --> 00:18:37,410
the electron moments.

292
00:18:37,410 --> 00:18:37,878
Yeah.

293
00:18:37,878 --> 00:18:40,686
AUDIENCE: Under coordination
number, what's the difference

294
00:18:40,686 --> 00:18:44,785
between 4SQ and just Q?

295
00:18:44,785 --> 00:18:50,140
PROFESSOR: You would guess
that the lowest energy

296
00:18:50,140 --> 00:18:52,290
configuration for
four-coordination would be

297
00:18:52,290 --> 00:18:54,560
tetrahedral, because
that keeps the

298
00:18:54,560 --> 00:18:56,390
anions most widely separated.

299
00:18:56,390 --> 00:19:00,020
There are cases where the
coordination is plainer.

300
00:19:00,020 --> 00:19:04,350
So the "SQ" stands for square,
and that's unusual.

301
00:19:07,360 --> 00:19:11,590
And you'll notice that that
appears for things like

302
00:19:11,590 --> 00:19:18,040
silver, which has a considerable
covalent

303
00:19:18,040 --> 00:19:18,800
character to it.

304
00:19:18,800 --> 00:19:20,110
For gold, 3-plus.

305
00:19:20,110 --> 00:19:22,270
That's one of the noble metals,
again, that has a

306
00:19:22,270 --> 00:19:25,590
considerable covalent
character to it.

307
00:19:25,590 --> 00:19:28,240
So you'll notice that as the
coordination number goes up,

308
00:19:28,240 --> 00:19:29,490
you get a different radius.

309
00:19:34,320 --> 00:19:44,020
Now, there are then a set of
numbers, and these are numbers

310
00:19:44,020 --> 00:19:47,460
that are based on different
starting points.

311
00:19:52,190 --> 00:19:57,240
The furthest column to the
right, quote "IR" quote, this

312
00:19:57,240 --> 00:20:00,430
is based on Pauling's
starting point.

313
00:20:00,430 --> 00:20:03,230
And Pauling's starting point,
if you go way down the list

314
00:20:03,230 --> 00:20:09,090
onto the second page, is that
for 02 minus for oxygen in

315
00:20:09,090 --> 00:20:13,950
six-coordination, the standard
radius from which everything

316
00:20:13,950 --> 00:20:18,010
else hinges is 1.40.

317
00:20:18,010 --> 00:20:21,650
The Pauling radii had been so
commonly used that Shannon and

318
00:20:21,650 --> 00:20:26,810
Prewitt decided to use 1.40,
Pauling's radii.

319
00:20:26,810 --> 00:20:30,060
So all of their numbers are
compatible with the early

320
00:20:30,060 --> 00:20:34,320
Pauling radii, except they're
based on thousands more

321
00:20:34,320 --> 00:20:35,570
crystal structure
determinations.

322
00:20:38,620 --> 00:20:41,930
There is another column,
second from the

323
00:20:41,930 --> 00:20:43,900
right, called CR.

324
00:20:43,900 --> 00:20:46,750
And that stands for
Crystal Radii.

325
00:20:46,750 --> 00:20:52,810
And there were two people that
proposed that Pauling's radii

326
00:20:52,810 --> 00:20:57,880
really didn't give a feel for
the proper sizes of the ions.

327
00:20:57,880 --> 00:21:00,070
And they went on at great
length to explain what

328
00:21:00,070 --> 00:21:01,750
"proper sizes" is.

329
00:21:01,750 --> 00:21:05,560
And the only difference for the
sizes that are based on,

330
00:21:05,560 --> 00:21:09,190
quote, "crystal radii" is that
if you go all the way down to

331
00:21:09,190 --> 00:21:13,540
oxygen, the size of the oxygen
ion in six-coordination is

332
00:21:13,540 --> 00:21:18,020
given by 1.26.

333
00:21:18,020 --> 00:21:23,240
So all of the anions in the
crystal radius column are

334
00:21:23,240 --> 00:21:29,080
smaller by 1.40 minus 1.26.

335
00:21:29,080 --> 00:21:30,290
I can do that in my head.

336
00:21:30,290 --> 00:21:33,990
That's 0.14 Angstroms.

337
00:21:33,990 --> 00:21:34,790
Run down the list.

338
00:21:34,790 --> 00:21:38,180
They're 1.4 Angstroms smaller
for the anions.

339
00:21:38,180 --> 00:21:41,340
And the cations are 0.14
Angstroms larger.

340
00:21:41,340 --> 00:21:45,130
So there are two alternative
sets of radii, one based on

341
00:21:45,130 --> 00:21:51,860
the traditional Pauling radii,
1.40 for oxygen in

342
00:21:51,860 --> 00:21:58,600
six-coordination, and the
Fumi-Tosi radii, 1.26, which

343
00:21:58,600 --> 00:22:03,650
maybe give a better picture
of what the sphere sizes

344
00:22:03,650 --> 00:22:05,660
actually, quote, "look
like," unquote.

345
00:22:09,170 --> 00:22:12,040
But note the tendencies.

346
00:22:12,040 --> 00:22:15,590
There is a progressive
change--

347
00:22:15,590 --> 00:22:18,420
whoever's starting point you
use, there's a progressive

348
00:22:18,420 --> 00:22:21,250
change in size with coordination
number.

349
00:22:21,250 --> 00:22:24,500
And let me give you, if I may,
at this point, give you--

350
00:22:24,500 --> 00:22:29,010
again, a little knowledge
is a dangerous thing.

351
00:22:29,010 --> 00:22:32,890
One of the structures which I
worked on many years ago was a

352
00:22:32,890 --> 00:22:36,810
puzzle, and this was the
prototype fast ion conductor

353
00:22:36,810 --> 00:22:38,280
silver iodide.

354
00:22:38,280 --> 00:22:40,880
And nobody had been able to
determine the structure for

355
00:22:40,880 --> 00:22:41,590
several reasons.

356
00:22:41,590 --> 00:22:44,760
Nobody had ever made a single
crystal of it, and all the

357
00:22:44,760 --> 00:22:47,110
diffraction data you had
available for the structure

358
00:22:47,110 --> 00:22:49,830
were five or six intensities.

359
00:22:49,830 --> 00:22:53,040
And there were about seven
parameters necessary to fully

360
00:22:53,040 --> 00:22:55,890
describe the structure, so
you were out of luck.

361
00:22:55,890 --> 00:22:58,740
We learned how to make
single crystals.

362
00:22:58,740 --> 00:22:59,990
But before that--

363
00:23:03,770 --> 00:23:05,500
no, he will go nameless.

364
00:23:05,500 --> 00:23:08,125
He said, OK, let's see
what's going on here.

365
00:23:08,125 --> 00:23:12,670
Oh, we'll get the radius
of silver.

366
00:23:12,670 --> 00:23:17,280
We know that in silver iodide,
the iodines are in

367
00:23:17,280 --> 00:23:19,800
body-centered cubic packing.

368
00:23:19,800 --> 00:23:22,530
And we know what the lattice
constant is, because we got

369
00:23:22,530 --> 00:23:24,640
that from powder diffraction
data.

370
00:23:24,640 --> 00:23:29,690
Huh, the silver ion is too
big to fit in either the

371
00:23:29,690 --> 00:23:33,580
tetrahedral interstitial site or
the octahedral interstitial

372
00:23:33,580 --> 00:23:37,160
site in silver iodide.

373
00:23:37,160 --> 00:23:41,600
So he said, therefore, the
structure has to distort, and

374
00:23:41,600 --> 00:23:44,410
it's really tetragonal.

375
00:23:44,410 --> 00:23:45,770
But there are going
to be domains.

376
00:23:45,770 --> 00:23:49,400
a tetragonal domain in one
orientation here, a tetragonal

377
00:23:49,400 --> 00:23:51,580
domain in another orientation
here.

378
00:23:51,580 --> 00:23:54,370
So he gets himself a random
number generator and takes

379
00:23:54,370 --> 00:23:58,000
numbers one, two, and three and
puts it on a cubic lattice

380
00:23:58,000 --> 00:24:00,080
to determine what the
distortion will be.

381
00:24:00,080 --> 00:24:03,130
Said, hey, I got a bunch of
ones here and a bunch of

382
00:24:03,130 --> 00:24:04,360
threes here.

383
00:24:04,360 --> 00:24:07,100
This crystal is built
of domains.

384
00:24:07,100 --> 00:24:10,540
And the reason the silver ions
migrate rapidly is that a

385
00:24:10,540 --> 00:24:15,940
domain can pop from one mode of
distortion to another, and

386
00:24:15,940 --> 00:24:18,360
that's essentially going to
transport the silver ion from

387
00:24:18,360 --> 00:24:19,760
one end of the domain
to another.

388
00:24:19,760 --> 00:24:21,800
This guy got two papers
out of it.

389
00:24:21,800 --> 00:24:26,370
What he had done was he looked
up the radius of silver in

390
00:24:26,370 --> 00:24:27,630
Pauling's tables.

391
00:24:27,630 --> 00:24:31,870
And he didn't realize that all
of Pauling's original radii

392
00:24:31,870 --> 00:24:39,140
were normalized to octahedral
six-hole coordination, which

393
00:24:39,140 --> 00:24:40,760
makes the radius larger.

394
00:24:40,760 --> 00:24:43,420
And that's why the silvers
didn't fit in any of the

395
00:24:43,420 --> 00:24:44,490
interstices.

396
00:24:44,490 --> 00:24:47,660
If you take the tetrahedral
radius for silver, which is

397
00:24:47,660 --> 00:24:51,040
where it actually sits in the
structure, it fits as snugly

398
00:24:51,040 --> 00:24:54,210
and nicely as you please in the
tetrahedral interstitial

399
00:24:54,210 --> 00:24:56,660
site in the body-centered
cubic arrangement.

400
00:24:56,660 --> 00:24:58,980
So this guy made an idiot of
himself because he didn't

401
00:24:58,980 --> 00:25:02,160
realize the basis on which
these tables had been

402
00:25:02,160 --> 00:25:02,630
established.

403
00:25:02,630 --> 00:25:05,050
But he's nameless, so I
can call him names.

404
00:25:05,050 --> 00:25:07,120
So anyway, again, a little
knowledge can

405
00:25:07,120 --> 00:25:10,420
be a dangerous thing.

406
00:25:10,420 --> 00:25:14,560
Moving on, in the few minutes
that we have left to devote to

407
00:25:14,560 --> 00:25:15,810
crystal chemistry.

408
00:25:18,090 --> 00:25:22,140
Pauling's radii, and even
better still now, the

409
00:25:22,140 --> 00:25:25,280
Shannon-Prewitt radii, let you
predict nearest-neighbor

410
00:25:25,280 --> 00:25:26,690
configurations--

411
00:25:26,690 --> 00:25:34,680
coordination numbers with fair
degree of confidence.

412
00:25:34,680 --> 00:25:35,370
Pauling--

413
00:25:35,370 --> 00:25:38,890
and this was one of the things
that he received the Nobel

414
00:25:38,890 --> 00:25:40,860
Prize for--

415
00:25:40,860 --> 00:25:44,830
proposed a set of five rules
that are called Pauling's

416
00:25:44,830 --> 00:25:47,050
Rules for ionic structure.

417
00:25:47,050 --> 00:25:50,490
The first thing he said was
something that we've already

418
00:25:50,490 --> 00:25:53,480
stated, a coordination polyhedra
of anions is formed

419
00:25:53,480 --> 00:25:57,320
about each cation, and the
cation-anion distance is

420
00:25:57,320 --> 00:26:00,570
determined by the sum of the
radii, and the coordination

421
00:26:00,570 --> 00:26:03,490
number is determined
by radius ratio.

422
00:26:03,490 --> 00:26:06,350
So there are three principles
in here.

423
00:26:06,350 --> 00:26:12,380
The two most important ones is
that expressing faith in a set

424
00:26:12,380 --> 00:26:15,660
of radii to determine interionic
distances and the

425
00:26:15,660 --> 00:26:18,490
fact that if there are radii
that can be assigned to the

426
00:26:18,490 --> 00:26:21,140
atoms, you can predict a
coordination number.

427
00:26:21,140 --> 00:26:22,770
So that's Pauling's
first rule.

428
00:26:22,770 --> 00:26:25,550
Actually, Pauling's first
rule was first stated by

429
00:26:25,550 --> 00:26:26,740
Goldschmidt.

430
00:26:26,740 --> 00:26:29,730
But to have a complete listing
of guiding principles, he

431
00:26:29,730 --> 00:26:33,000
included that there and didn't
claim it was his own.

432
00:26:33,000 --> 00:26:39,160
But then he does something that
is rather interesting.

433
00:26:39,160 --> 00:26:43,790
This is the only one of the
rules that is quantitative.

434
00:26:43,790 --> 00:26:47,470
If you had an ionic structure,
and you could determine its

435
00:26:47,470 --> 00:26:53,360
unit cell, the structure should
be electrically neutral

436
00:26:53,360 --> 00:26:56,520
if it's composed of ions
that are charged.

437
00:26:56,520 --> 00:26:59,570
So therefore, you would not make
a structure in which all

438
00:26:59,570 --> 00:27:03,380
the cations were up in one
corner of the cell, and all

439
00:27:03,380 --> 00:27:05,830
the anions were down in the
opposite end of the cell.

440
00:27:05,830 --> 00:27:08,140
That would be something that
might be electrically neutral,

441
00:27:08,140 --> 00:27:11,660
but it would be a high-energy
configuration.

442
00:27:11,660 --> 00:27:17,940
So Pauling's second rule is a
very cute way of expressing

443
00:27:17,940 --> 00:27:21,880
the fact that a structure
should be atomistically

444
00:27:21,880 --> 00:27:25,690
electrically neutral, as well
as microscopically.

445
00:27:25,690 --> 00:27:29,570
And he did that in the
following way.

446
00:27:29,570 --> 00:27:37,970
He said, let us look at the
coordination number the

447
00:27:37,970 --> 00:27:42,790
cation, A, and look at the
coordination polyhedron.

448
00:27:42,790 --> 00:27:54,850
And let's define as the bond
strength, S, as the ratio of

449
00:27:54,850 --> 00:28:01,020
the cation charge plus Q divided
by the coordination

450
00:28:01,020 --> 00:28:04,720
number of A.

451
00:28:04,720 --> 00:28:08,130
And then he said that in a
stable structure, if we look

452
00:28:08,130 --> 00:28:14,810
at the anion, the sum of all
the bonds donated to--

453
00:28:14,810 --> 00:28:19,230
he visualize the charge being
donated to the anion.

454
00:28:19,230 --> 00:28:24,980
The sum of all of the bonds from
the same cations or from

455
00:28:24,980 --> 00:28:27,870
different cations, if that's the
nature of the structure,

456
00:28:27,870 --> 00:28:32,525
should be equal to the
charge of the anion.

457
00:28:36,120 --> 00:28:37,680
So let me say "cation" here.

458
00:28:41,590 --> 00:28:45,070
So it's a cute way of
expressing local

459
00:28:45,070 --> 00:28:46,320
electroneutrality.

460
00:28:52,830 --> 00:28:56,930
Actually, that's been superseded
by an extension of

461
00:28:56,930 --> 00:28:58,510
this general principle.

462
00:28:58,510 --> 00:29:02,340
And I may not have time to
discuss it today, but I'll

463
00:29:02,340 --> 00:29:03,730
give you a handout next time.

464
00:29:03,730 --> 00:29:04,900
It's not in the notes, either.

465
00:29:04,900 --> 00:29:08,800
This is a relatively recent tool
that most people are not

466
00:29:08,800 --> 00:29:14,210
familiar with, so I'll return
to it briefly next time.

467
00:29:14,210 --> 00:29:21,840
The next rule says, OK, we
know how to determine the

468
00:29:21,840 --> 00:29:23,090
coordination numbers.

469
00:29:25,940 --> 00:29:32,140
To describe the structure, we
should be able to say how

470
00:29:32,140 --> 00:29:35,840
these coordination polyhedra
fit together.

471
00:29:35,840 --> 00:29:38,730
So looking at the coordination
numbers--

472
00:29:38,730 --> 00:29:40,490
well, let me do a
three-dimensional case,

473
00:29:40,490 --> 00:29:42,890
because that's the realistic
situation.

474
00:29:42,890 --> 00:29:46,920
Let's suppose we have
coordination number eight,

475
00:29:46,920 --> 00:29:48,040
which we often do.

476
00:29:48,040 --> 00:29:49,990
And besides, that's
easy to draw.

477
00:29:49,990 --> 00:29:54,130
So he says that in a stable
structure, the coordination

478
00:29:54,130 --> 00:29:59,760
polyhedra tend to
share corners.

479
00:30:05,870 --> 00:30:09,780
And I might say also that they
tend to have the line between

480
00:30:09,780 --> 00:30:14,970
cation, anion, cation to be a
straight line, as opposed to

481
00:30:14,970 --> 00:30:16,220
sharing edges.

482
00:30:22,940 --> 00:30:28,890
So for eight-coordination, this
might be a pair of cubic

483
00:30:28,890 --> 00:30:30,850
coordination polyhedra
sharing edges.

484
00:30:33,700 --> 00:30:39,430
And especially corner sharing
is more favorable

485
00:30:39,430 --> 00:30:40,680
than sharing faces.

486
00:30:44,870 --> 00:30:48,380
It's a very specific statement
on how the coordination groups

487
00:30:48,380 --> 00:30:52,380
whose geometry is determined by
size should fit together in

488
00:30:52,380 --> 00:30:53,210
a structure.

489
00:30:53,210 --> 00:30:55,290
And it has a very
simple basis, an

490
00:30:55,290 --> 00:30:57,410
almost trivial basis.

491
00:30:57,410 --> 00:31:03,200
As the polyhedra of the given
size share corners and then

492
00:31:03,200 --> 00:31:07,780
edges and then faces, the
cations inside of these

493
00:31:07,780 --> 00:31:10,600
polyhedra are progressively
getting

494
00:31:10,600 --> 00:31:12,720
closer and closer together.

495
00:31:12,720 --> 00:31:15,890
And clearly, that increases
the repulsive energy.

496
00:31:15,890 --> 00:31:21,850
So the cations are most widely
separated if not faces, not

497
00:31:21,850 --> 00:31:24,360
edges, but corners tend
to be shared.

498
00:31:24,360 --> 00:31:25,660
Puckered a little bit.

499
00:31:25,660 --> 00:31:31,010
Maybe this is not exactly
a 180-degree bond angle.

500
00:31:31,010 --> 00:31:33,590
But corner sharing is
still favorable.

501
00:31:37,260 --> 00:31:41,370
And rule D is essentially an
appendix to that, and it says,

502
00:31:41,370 --> 00:31:44,680
and how, if the charge of the
cation is high and the

503
00:31:44,680 --> 00:31:48,640
coordination number is low, the
higher the charge of the

504
00:31:48,640 --> 00:31:52,520
cations, the stronger that
repulsive interaction.

505
00:31:52,520 --> 00:31:56,830
The smaller the coordination
number of the cation, the

506
00:31:56,830 --> 00:32:01,670
closer together they become in,
let's say, face sharing

507
00:32:01,670 --> 00:32:05,500
for tubes as opposed to face
sharing for tetrahedra.

508
00:32:05,500 --> 00:32:08,400
The lower the coordination
number, the closer together

509
00:32:08,400 --> 00:32:12,780
the cations come, regardless
of the precise shape of the

510
00:32:12,780 --> 00:32:14,370
coordination polyhedra.

511
00:32:14,370 --> 00:32:18,840
So rule D says, and how for
high charge and low

512
00:32:18,840 --> 00:32:22,090
coordination number.

513
00:32:22,090 --> 00:32:25,240
The last rule, sometimes
called the "rule of

514
00:32:25,240 --> 00:32:30,030
parsimony," the "rule of
stinginess," says that the

515
00:32:30,030 --> 00:32:33,200
number of different kinds
of atoms in a

516
00:32:33,200 --> 00:32:35,020
structure tends to be small.

517
00:32:38,360 --> 00:32:42,630
You can read in two different
interpretations of the term

518
00:32:42,630 --> 00:32:46,840
"different kinds of cations,"
and both of them are right.

519
00:32:46,840 --> 00:32:50,460
You could mean different
in terms of

520
00:32:50,460 --> 00:32:52,300
different chemical species.

521
00:32:52,300 --> 00:32:56,160
In other words, if you take
a pot and dump half of the

522
00:32:56,160 --> 00:33:01,480
elements in the periodic table
into the pot, melt it up, and

523
00:33:01,480 --> 00:33:04,400
then cool it down to crystallize
the phase, you're

524
00:33:04,400 --> 00:33:06,930
not going to incorporate
all of those ions

525
00:33:06,930 --> 00:33:09,200
into one crystal structure.

526
00:33:09,200 --> 00:33:10,520
They have different sizes.

527
00:33:10,520 --> 00:33:13,490
They have different coordination
polyhedra.

528
00:33:13,490 --> 00:33:15,270
The different coordination
polyhedra

529
00:33:15,270 --> 00:33:18,950
have different sizes.

530
00:33:18,950 --> 00:33:22,160
So to try to fit all of those
things together in one

531
00:33:22,160 --> 00:33:24,870
structure is just going to
cost you a lot of energy.

532
00:33:24,870 --> 00:33:27,450
It's going to be much more
efficient to form two or more

533
00:33:27,450 --> 00:33:32,100
phases, which have much lower
energy, and then pay the price

534
00:33:32,100 --> 00:33:34,360
of an interface between them.

535
00:33:34,360 --> 00:33:38,520
So have a chemically complex
melt or solution, you're going

536
00:33:38,520 --> 00:33:39,710
to get different phases.

537
00:33:39,710 --> 00:33:43,210
You want to incorporate
everything into one structure.

538
00:33:43,210 --> 00:33:46,230
The other interpretation of
different is in terms of their

539
00:33:46,230 --> 00:33:48,645
coordination number and their
coordination polyhedra.

540
00:33:51,750 --> 00:33:55,680
If the size is such that a
particular cation wants to

541
00:33:55,680 --> 00:33:59,590
have tetrahedral coordination,
you would expect them all to

542
00:33:59,590 --> 00:34:03,500
have tetrahedral coordination
and not have a mixture of

543
00:34:03,500 --> 00:34:05,750
cubes and tetrahedra
and octahedra.

544
00:34:05,750 --> 00:34:08,989
If it works for one coordination
through the

545
00:34:08,989 --> 00:34:12,330
chemical species, it'll
work for them all.

546
00:34:12,330 --> 00:34:20,960
That rule tends to have some,
but not terribly many,

547
00:34:20,960 --> 00:34:21,850
exceptions.

548
00:34:21,850 --> 00:34:25,360
But there are cases where
you find two different

549
00:34:25,360 --> 00:34:27,389
coordination numbers.

550
00:34:27,389 --> 00:34:33,340
But very often, that's because
of a covalent character.

551
00:34:33,340 --> 00:34:35,929
So there's some discussion
and elaboration of

552
00:34:35,929 --> 00:34:37,870
each of those rules.

553
00:34:37,870 --> 00:34:40,400
I'm just about out of
time, and therefore

554
00:34:40,400 --> 00:34:42,420
out of crystal chemistry.

555
00:34:42,420 --> 00:34:48,320
Another entirely different
approach to structure is to

556
00:34:48,320 --> 00:34:56,389
say that the anions are
the big things.

557
00:34:56,389 --> 00:35:03,390
And if the energy of the
structure is lowered by taking

558
00:35:03,390 --> 00:35:09,660
ions of like charge and getting
them as close as

559
00:35:09,660 --> 00:35:14,790
possible to the larger ions,
say the anions of opposite

560
00:35:14,790 --> 00:35:19,040
charge, there is a strategy for
making the arrangement.

561
00:35:19,040 --> 00:35:22,720
And I think it's the same
strategy that I use when I'm

562
00:35:22,720 --> 00:35:25,350
packing a suitcase for a trip.

563
00:35:25,350 --> 00:35:27,050
Take the big things--

564
00:35:27,050 --> 00:35:29,440
the books, the shoes--

565
00:35:29,440 --> 00:35:31,130
and arrange them
in the suitcase

566
00:35:31,130 --> 00:35:34,040
as densely as possible.

567
00:35:34,040 --> 00:35:36,140
Then I take the little things--
the toothbrush, the

568
00:35:36,140 --> 00:35:37,590
socks, and so on--

569
00:35:37,590 --> 00:35:41,040
and I stick them into the
interstices that are left.

570
00:35:41,040 --> 00:35:44,470
And this other view of examining
structures and

571
00:35:44,470 --> 00:35:48,020
interpreting says
the same thing.

572
00:35:48,020 --> 00:35:49,810
Take the big things--

573
00:35:49,810 --> 00:35:53,090
not the shoes but the anions--

574
00:35:53,090 --> 00:35:55,790
and arrange them as densely
as possible.

575
00:35:55,790 --> 00:35:57,320
And this gets into
the question of

576
00:35:57,320 --> 00:35:59,040
close packing of spheres.

577
00:35:59,040 --> 00:36:02,260
Then take the little guys, the
tiny little cations, and stuff

578
00:36:02,260 --> 00:36:05,120
them into the holes that
are left in the

579
00:36:05,120 --> 00:36:07,030
close-packed arrays.

580
00:36:07,030 --> 00:36:10,720
So another way of interpreting
structures and predicting

581
00:36:10,720 --> 00:36:14,640
structures is to look at the
sort of interstices that exist

582
00:36:14,640 --> 00:36:17,010
in a close-packed array.

583
00:36:17,010 --> 00:36:19,990
Regardless of the stacking
sequence of spheres, there are

584
00:36:19,990 --> 00:36:23,440
always two kinds of interstitial
sites, one

585
00:36:23,440 --> 00:36:26,910
tetrahedral, one octahedral.

586
00:36:26,910 --> 00:36:31,400
And there are two common, but
not exclusive, types of

587
00:36:31,400 --> 00:36:33,790
packing arrangements, which
you probably know.

588
00:36:33,790 --> 00:36:38,310
There is the cubic close
packed array.

589
00:36:38,310 --> 00:36:41,490
There's a little diagram of
close-packed sheets here.

590
00:36:41,490 --> 00:36:45,230
And there are two sorts of
interstices shown in the

591
00:36:45,230 --> 00:36:49,690
figure on the first page
of the notes on packing

592
00:36:49,690 --> 00:36:50,730
considerations.

593
00:36:50,730 --> 00:36:54,160
There's one set of triangles
that points in one orientation

594
00:36:54,160 --> 00:36:56,850
that forms a hollow in which
you can drop down a second

595
00:36:56,850 --> 00:37:00,210
layer of spheres, another
triangle that is the closest

596
00:37:00,210 --> 00:37:01,870
triangle to the first
one that points in

597
00:37:01,870 --> 00:37:03,900
the opposite direction.

598
00:37:03,900 --> 00:37:07,440
Each represents a potential
location for the next

599
00:37:07,440 --> 00:37:10,620
close-packed sheet, but those
two different locations are

600
00:37:10,620 --> 00:37:12,800
too close together
to both be used.

601
00:37:12,800 --> 00:37:15,790
So you have to pick
one or another.

602
00:37:15,790 --> 00:37:19,400
And if you label those three
different locations A, B, and

603
00:37:19,400 --> 00:37:25,830
C, any arrangement of spheres
that does not place two

604
00:37:25,830 --> 00:37:29,190
letters in succession will
give you a close-packed

605
00:37:29,190 --> 00:37:31,660
arrangement with exactly
the same density.

606
00:37:31,660 --> 00:37:39,280
So ABC AC, AC, ABC, AC, AC is
a packing of close-packed

607
00:37:39,280 --> 00:37:43,750
spheres that is exactly the same
density as AB, AB, which

608
00:37:43,750 --> 00:37:48,100
is hexagonal close packed, or
ABC, ABC, ABC, which could be

609
00:37:48,100 --> 00:37:51,490
shown to be the face-centered
cubic structure.

610
00:37:51,490 --> 00:37:54,890
So that is another way of
looking at structures.

611
00:37:54,890 --> 00:37:58,400
And specifying the space
filling in simple ionic

612
00:37:58,400 --> 00:38:00,910
structures is another way of
looking at structures.

613
00:38:00,910 --> 00:38:02,770
And at that point, I'll quit.

614
00:38:02,770 --> 00:38:05,460
There are some examples of
close-packed structures and

615
00:38:05,460 --> 00:38:08,890
derivatives thereof that
form the last pictures.

616
00:38:08,890 --> 00:38:12,640
And I'm going to stop and suck
in air and mention as one last

617
00:38:12,640 --> 00:38:15,740
thought that only five
of Pauling's

618
00:38:15,740 --> 00:38:17,190
rules are stated here.

619
00:38:17,190 --> 00:38:20,780
There was a sixth Pauling's
rule, and that's

620
00:38:20,780 --> 00:38:21,820
not commonly listed.

621
00:38:21,820 --> 00:38:22,540
And I'll tell you what it is.

622
00:38:22,540 --> 00:38:26,960
The sixth rule says, massive
doses of vitamin C can cure

623
00:38:26,960 --> 00:38:28,210
the common cold.

624
00:38:31,240 --> 00:38:33,970
Actually, that was something
that Pauling pushed.

625
00:38:33,970 --> 00:38:37,110
And interestingly, about three
years before he passed away,

626
00:38:37,110 --> 00:38:40,620
Pauling came to MIT and gave
a lecture in 6120.

627
00:38:40,620 --> 00:38:44,940
And as you might expect of MIT
undergraduates, after he

628
00:38:44,940 --> 00:38:47,410
finished his lecture, there
were a few questions.

629
00:38:47,410 --> 00:38:49,710
But there were about a dozen
students who came up with

630
00:38:49,710 --> 00:38:52,540
bottles of vitamin C and
asked Pauling to

631
00:38:52,540 --> 00:38:55,350
autograph them for them.

632
00:38:55,350 --> 00:38:59,800
And he did so with good humor
and thought he had perhaps won

633
00:38:59,800 --> 00:39:01,990
a few converts in the process.

634
00:39:01,990 --> 00:39:02,880
So that's it.

635
00:39:02,880 --> 00:39:05,230
Getting silly, so it's time
for all of us to go home.