Marder M.P. Condensed Matter Physics

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Quasicrystals

133

A and B in Eq. (5.98) gives

|£o|2

' = ^5ΠΓΤ

= \A*o<Ar2(

))/(3D/*). (5.103)

In a backscattering geometry, light is collected instead at x = 0 and the opposite

sign of the derivative must be used in Eq. (5.102). In the limit L

—>

oo the result is

(E*(0)E(t)) ,

|£o|2

' = e~™\

= ^/ck

(Ar^t))/(3Dl*). (5.104)

Now suppose that the particles in the fluid are diffusing with particle diffusion

constant

so that according to Eq. (5.16) the square distance a particle moves in

time t is is Δ^(ί) =

6T>

Then one has

^^6tk

(5.105)

Weitz and Pine (1993) recommend taking xo/l* « 2.1 in order to compare with

experiment. As shown in Figure 5.21, the log of (E*(0)E(t)) is a linear function

y/T>

The slope of such curves can be used to find T)

and hence as shown in

Problem 10 can be used to estimate particle sizes in suspensions that are not dilute.

5.10 Quasicrystals

First Observation. Shechtman et al. (1984) were not setting out to challenge

crystallography, but were preparing melt-spun ribbons of AlgöMn^. The alloys

were made to cool at rates of 10

K s~' by pouring molten metal onto a rapidly

spinning wheel, cooling rates that could be expected to produce a metallic glass.

After examination with an electron microscope, the samples were placed under X-

ray diffraction. The startling result was a set of diffraction patterns indicating axes

of threefold and fivefold symmetry, shown in Figure 5.22. Fivefold symmetry was

completely unexpected, since as proved in Problem 4 of Chapter 1, a fivefold axis

is crystallographically impossible. It is impossible to build a lattice with a fivefold

axis.

Nonetheless, the data unambiguously indicated that the system had this sym-

metry. In fact, the scattering patterns exhibited the symmetry of an icosahedron,

giving tenfold, sixfold, and fivefold symmetric diffraction patterns when tilted at

appropriate angles. A picture of a small crystal exhibiting such symmetry appears

in Figure 5.23.

The claim that nature had found a way to realize true fivefold symmetries met

with some resistance. A collection of five crystals might conceivably bond together

at five different orientations to produce an apparent fivefold axis. They would not

fit together perfectly, but the scattering pattern would not easily reveal this fact,

and such a point of view was vigorously advanced by Pauling (1985). In some

materials it turned out to be correct. However, the most interesting proposal, and

the one that seems to have carried the day for the original AIMn alloy as well as

134 Chapter 5. Beyond Crystals

79.2°

Figure 5.22. X-ray scattering patterns from single grain of AlgóMn^. Angles refer to

various orientations of the grain. Note clear fivefold symmetry axis in two of the scattering

patterns. [Source: Shechtman et al. (1984), p. 1952.]

many other materials, holds that the explanation of the diffraction pattern lies in

a quasi-periodic filling of space which has a true icosahedral symmetry. Levine

and Steinhardt (1984) christened this type of lattice a quasicrystal. Such ordering

generalizes the idea of the crystal, and the simplest explanation of how it works

begins in one dimension.

5.10.1 One-Dimensional Quasicrystal

A one-dimensional quasicrystal is a collection of points lying on a line, spaced

quasiperiodically. As an example, and to demonstrate the meaning of quasiperi-

odic,

consider the Fibonacci sequence:

71+ (r — 1)ϊηί(η/τ). "intOO" is the function that finds the largest (5.106)

integer smaller than x.

Quasicrystals 135

Figure 5.23. Quasicrystal of AlLiCu:

[Source: Kortan (1996).]

This sequence of points is composed of long and short intervals: The distance

between successive points is either r, the golden mean,

r=l + -= „

=1.618. . . , (5.107)

T 2

or 1. The sequence has a deflation rule. If one writes out the sequence of differ-

ences between successive points (a sequences of l's and r's), then replaces every r

with the little sequence r, 1, and also replaces every 1 with a r, the same sequence

returns.

Example. Start with the sequence

rlrrl (5.108)

Using the deflation rule r

—>

rl and 1

—>

r gives

rlrrlrlr (5.109)

Another Construction. One can use this procedure to generate longer and longer

portions of the Fibonacci sequence, starting from a small subsection. One can also

generate the sequence by the construction

Xn+i =-X/i-Xn-i, (5.110)

which means that one builds the sequence in this way:

X-i

=T;X

= T1;X\ =

1τ; Χ

= τΐττΐ . . . (5.111)

{

= rlrrlrlr. (5.112)

These properties are meant to indicate that the series has many regularities without

being periodic. However, it is quasiperiodic, which means that it is periodic in a

higher-dimensional space. In this case, the function is periodic in two dimensions,

but has been projected into one.

The Fibonacci sequence can be interpreted as a shadow cast in one dimension

by a slice through a periodic lattice in two dimensions, as shown in Figure 5.24.

To prove this assertion, proceed as follows. Rewrite Eq. (5.106) as

= m + y ηθ(η — m/τ +

)θ(ΐη/τ — η)/τ Here θ(χ) is the Heaviside step function, which

^—' equals one if its argument is larger than 0, and

" vanishes otherwise. A factor of (τ

— 1 )

has been

replaced by l/τ, using Eq. (5.107).

(5.113)

136

Chapter 5. Beyond Crystals

Figure 5.24. Construction showing how the one-dimensional quasicrystal results from a

projection down from two dimensions, and how one obtains the deflation rule. The shaded

bar has slope 1/r, and vertical height 1. The sequence x

defined by Eq. (5.106) results

from projection of the points lying in this bar onto the x axis. The bar enclosed by the

dotted line has slope l/τ and width 1, and the sequence

X„

defined by Eq. (5.118) results

from projection of the points inside onto the x axis.

Now consider the points that fall inside the shaded strip in Figure 5.24. The strip

has vertical thickness of

and has slope 1/r, and points lie inside it if

x/r>y>x/r-l.

(5.114)

So for every vertical column of the lattice, indexed by integer m, there is precisely

one point inside the strip at location

[m,

ηθ(ηΐ/τ

—

η)θ(η—

[m/r— 1])].

This

js a Cartesian vector, whose first compo- (5.115)

To find where the shadow of this point falls on the

axis along the direction

(1,

—τ), one adds an appropriate multiple of (1, —r) to (5.115) so that the y co-

ordinate vanishes, and then one checks where one lands along x. The answer is

precisely Eq. (5.113). Therefore the geometrical construction of Figure 5.24 is the

same as Eq. (5.106).

This geometrical construction can be used in order to prove the deflation rule.

Consider the collection of points that fall within the smaller strip of Figure 5.24,

which is surrounded by a dotted line and has unit width rather than unit height.

Points lie in this smaller strip if

(JC+1)/T-1

>y>x/r-

(5.116)

Quasicrystals 137

Because this strip has unit width, one lattice point must fall within it from every

horizontal row, and one finds the lattice points satisfying Eq. (5.116) to be

[J2 m9((m+ l)/r -

-η)θ(η - [m/r - 1]), n}. (5.117)

Projecting this set of points onto the x axis as before results in the sequence

n+]

=^m0((m+l)/T-/i-l)0(n-/n/r + l)+/i/i"· (5.118)

This location is defined to be

rather than X

so that

the lower left point in Figure 5.24 will correspond to X\

as well as to x\.

The set of distances X

is depicted in Figure 5.24 as the hollow circles.

It is not hard to show that the sequence X

is almost exactly the same as the

sequence x

. After a few simple changes of variables (Problem 11), one finds

= -\/T + Tx

. (5.119)

Thus X

is nothing but the sequence x

expanded by a factor of r and slightly

displaced; in particular, the sequences of long and short segments of X

and x

are exactly the same. It is evident from Figure 5.24 that X

is obtained from x

by sliding the upper dotted line up slightly, replacing all long intervals by a long

followed by a short, and leaving all short intervals alone. This argument shows the

origin of the deflation rule.

Scattering from a One-Dimensional Quasicrystal. The advantage of viewing the

quasicrystal as a projection down from higher dimensions is that it makes it possi-

ble to compute its Fourier transform and to understand how scattering from such a

structure leads to sharp peaks. What will emerge from the analysis is that the math-

ematical structure of scattering from a quasicrystal differs in some striking ways

from that from a crystal. The quasicrystal scattering peaks are countably infinite

and dense; any finite strip of

space contains an infinite number of

peaks,

but most

of them have amplitude too small to be seen. This type of scattering spectrum is

called singular continuous.

The idea behind the calculation is that, as in Figure 5.24, the points contained in

a one-dimensional quasicrystal can be expressed as the product of two functions.

The first is the two-dimensional square lattice, and the second is a function that

equals one within the shaded strip of Figure 5.24 but is zero outside of it. So,

roughly speaking, one expects the Fourier transform of the quasicrystal to be a

convolution of

the

Fourier transform of

the

square lattice with the Fourier transform

of the shaded strip.

The sum to be computed is

E, = Ve

i?I

" ^

■ , (5.120)

v z

—' The

functions are only

nonzero when m and n have

= J2e

^^e(n-m/r+l)9(m/r-n) ^j^^^ (5.121)

n,m contribution to the sum.

138

Chapter 5. Beyond Crystals

= / dxdy e

iH*,y)

δ(χ —

m)S(y

—

where q = (q, q/τ).

0(y -χ/τ + 1)θ(χ/τ -

yt>5A22)

(5.123)

Equation (5.122) is a two dimensional Fourier transform that is to be evaluated

along a particular line in q space. To carry out the integral one can use the convo-

lution theorem, Eq. (A.42), to carry out the Fourier transforms of the two halves of

the integrand and then integrate them together in the end. The first piece is

A{q) = j dxdy Σ δ{χ-πι)δ{

-η)ε^

m,n

^ψ~ Σ S{q

-2irn')6(q

-2irm!).

(5.124)

(5.125)

Eq. (5.124) describes the scattering from a two-dimensional square

lattice, which, according to the two-dimensional analog of

Eq.

(3.19)

produces Bragg peaks at the reciprocal lattice vectors 2π(η', m

The second piece is

B{q) = [ dx f

x+ici

= ί dx e^

J Jx/τ-Ι J

) —

'1y(x/T-\)

(5.126)

Convoluting Eqs. (5.125) and (5.126) with q given by Eq. (5.123) gives

E^oc

ö(q-q'

-2Kn')

dxdq'

dq'

η',ηι' K '^V

// ' Hy

,ί(η/τ—2πιη')(χ/τ) _ J(q/r—2nm')(x/T—l)

^Σ

,m'

x5(q/r

—

q'y

—

2πηι')

■e

ίφ/τ) _Aq[,(x/T-\)

iq/r

—

2nim'

i(q-2irn')x

-i(q/T—2nm')

271"

> ; —δ(\2πιη' ]/τ + 2πη'

—

q).

f^, iar-l-Kim'

The peaks of (5.129) are at

2π(ηι'

/τ + n')

r-2 + 1

q All integers m' and n' are allowed.

and their square amplitude is proportional to

sin 7Γ

m'r

—

T +

j (q/τ

—

2nm'

e'^(5.127)

(5.128)

(5.129)

(5.130)

(5.131)

Therefore, the Fourier spectrum is made up of countably many sharp peaks, as

shown in Figure 5.25; in contrast to those produced by a lattice, these are dense

and form a singular continuous spectrum. However most of them are sufficiently

weak that they are practically invisible. The important ones are those produced by

and m' whose ratio is near the golden mean—that is, when these integers are

neighboring Fibonacci numbers.

Quasicrystals

139

Figure 5.25. Plot

the scattering amplitude

predicted

by Eq.

(5.131). This singular con-

tinuous scattering spectrum

has

sharp peaks

■§,

that fill

the k

axis

in a

dense fashion,

yet are 3

zero almost everywhere.

Wave number

5.10.2 Two-Dimensional Quasicrystals—Penrose Tiles

Figure 5.26. Penrose tiles. These are only allowed

join when the arrows match together.

The smaller angle

the

fat

rhombus

2π/5, and the smaller angle

the skinny rhombus

is 2π/10. Taking

the

lengths

all

the

sides

to be 1, the

long diagonal

the

fat

rhombus

(dashed line)

has

length

r = (\/5 +

l)/2,

and the

short diagonal

of the

skinny rhombus

(dashed line) has length

1/r.

Figure 5.27.

this decoration

engraved

each tile

a Penrose lattice,

the

result will

Penrose tiling with more tiles

on a

smaller scale.

Penrose introduced two-dimensional quasicrystals as a playful mathematical

problem in filling the plane nonperiodically; the results were first published for

practical purposes by Gardner (1977). The Penrose lattice is a tiling of the plane

140 Chapter 5. Beyond Crystals

Figure 5.28. The

first

three applications of

the

deflation rule to two fat rhombuses and one

skinny rhombus are shown here on the left, while on the right is a large circular section of

quasicrystal, produced by the method of Problem 13.

by a collection of two tiles, shown in Figure 5.26. These are decorated by arrow-

heads,

and the tiles must be placed together so that matching arrows are always

adjacent. It is possible to tile the plane with these objects, and although the result

is not periodic, every finite area segment of the lattice repeats infinitely often else-

where in the lattice. One can partly verify these claims by noticing that the Penrose

lattice has a deflation rule. If one decorates each tile as shown in Figure 5.27 with

smaller tiles, then a set of big tiles which obeys the matching rules generates a set

of many more smaller tiles also obeying the matching rules. Proceeding this way

long enough generates a lattice with an arbitrarily large number of

tiles

in it, shown

in Figure 5.28.

An additional curious feature is that one can decorate the tiles with lines as

shown in Figure 5.29; then when the tiles are laid on the plane, all the lines drawn

on the tiles march across the plane. These lines are called the Ammanti lines. The

spacing between them takes two values, whose ratio is the golden mean. When

the deflation rule shown in Figure 5.28 is applied to the Penrose lattice, one can

ask what happens to the lattice of Ammann lines. Say the large spacing between

lines is r. After deflation, the new lattice will replace this large spaced pair of lines

with three lines, along the same direction; the first is spaced by 1, and the second is

spaced by

/r. All the lines separated by the smaller distance, 1, are not changed.

This deflation rule is precisely the deflation rule for the one-dimensional Fi-

bonacci sequence, showing that the spacings between the Ammann lines are given

by Eq. (5.106) Therefore, the Penrose tiling is a a two-dimensional generalization

of the quasicrystal.

One can build it by choosing five unit vectors, ê

, which point at angles of 2π/5

with respect to one another, along the sides of a pentagon. Then the quasicrystal

Quasicrystals 141

Figure 5.29. In (A) is shown a set of decorative lines drawn upon the fat and skinny tiles.

They have the curious property, shown in

(B),

that when the tiles cover the plane, the lines

join

and form

five

grids of

parallel

lines, with the spacings of

lines

in any given direction

described by the one-dimensional Fibonacci sequence.

lattice points are the set of points r such that

r-ê

, r-eß=x

nß

, (5.132)

where x

is a member of the Fibonacci sequence. In other words, one places down

a set of

lines

in the plane. The lines travel along the five angles given by the sides of

a pentagon. The spacing between groups of parallel lines proceeds as the Fibonacci

sequence. The points in the lattice occur at the intersections of two lines. A more

general way to create two-dimensional quasicrystals is the subject of Problem 13.

One immediate question is why one has to vary the spacing between the lines

in such a complicated way. Because one has created a fivefold axis by brute force,

why not choose all the spacings between the lines to be constant? The answer is

that if one does this, the vertices where lines cross can come arbitrarily close to

one another, and atoms that built such a structure would have to do the same. This

difficulty is at the heart of the proof that fivefold symmetry is crystallographically

impossible. The Penrose lattice avoids this difficulty, and because it is constructed

by assembling together simple geometrical objects, it is possible that molecules

might naturally pack together so as to form this particular structure. In fact, dozens

of materials are now known where such packing occurs.

5.10.3

Experimental Observations

To understand why naturally forming crystals might have a tendency to create qua-

sicrystals, it is helpful to return to the problem of packing hard spheres. In close-

packing arrangements such as fee or hep lattices, every sphere has 12 neighbors,

but the neighbors are not arrayed about a central sphere with perfect symmetry.

142

Chapter 5. Beyond Crystals

Suppose one takes a sphere and places 12 identical spheres around it in the most

symmetrical arrangement. Connecting the centers of the neighbors with lines, one

finds that they form an icosahedron, as shown in Figure 5.30. Because of the five-

fold symmetry evident in this structure, one knows immediately that it cannot fill

space uniformly. Approximate icosahedral symmetry is the basis for a complicated

arrangement of atoms first proposed by Frank and Kasper (1958), and reviewed

more recently by Nelson and Spaepen (1989). In the Frank-Kasper

phases,

a unit

cell is packed with several approximate icosahedra; they must be distorted to fit

together, and the unit cell is very complicated. Quasicrystalline phases can be un-

derstood as another way for materials to realize icosahedral symmetry.

Figure 5.30. Twelve neighbors symmetri-

cally arranged about a sphere form an icosa-

hedron.

The first quasicrystals were only metastable, and were filled with defects, but

alloys such as AI6L13CU have since been found for which the quasicrystalline phase

is truly a free energy minimum. The alloy system shown in Figure 5.31 has the

remarkable property of forming a two-dimensional quasicrystal, arranged in layers.

Figure

5.31.

(A) Scanning tunneling microscope image of

the

two-dimensional quasicrys-

tal AI65CU15C020, viewed from an oblique angle. The three-dimensional crystal consists

of stacks of identical two-dimensional quasicrystals. (B) Top view shows how to decorate

the atomic locations with Penrose tiles. [Source: Kortan (1996).]

The majority of quasicrystals found to date has the icosahedral symmetry illus-

trated in Figure 5.22. However, octagonal and decagonal quasicrystals have also