Marder M.P. Condensed Matter Physics
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Fullerenes and nanotubes 143
been found.
Three-dimensional quasicrystals can be described using theoretical tools very
similar to those employed in two dimensions, and they are discussed by Janot
(1992).
The essential idea of how fivefold symmetry is realized in nature is ex-
plained by the two-dimensional case, but three-dimensional quasicrystallography,
discussed by Mermin (1992), is needed to obtain detailed comparison with most
experimental systems.
5.11 Fullerenes and nanotubes
A new class of carbon compounds is the fullerenes, whose physical properties are
reviewed by Smalley (1997) and by Weaver and Poirier (1994). The prototype
for these compounds is
COO,
where 60 carbon atoms are arranged at the vertices
of a molecule resembling a soccer ball, or one of the structures designed by the
architect Buckminster Fuller.
Ij ima (1991) discovered carbon nanotubes in 1991 while examining waste prod-
ucts created during synthesis of fullerenes. Single-walled carbon nanotubes con-
sist in single atomic layers of carbon in a honeycomb structure (graphene) rolled
around an axis into a tube (Problem 1.3). When more than a single atomic layer
of carbon rolls up into a tube, the result is a multi-walled
carbon
nanotube.. Great
effort has been spent investigating their properties, as documented by Dresselhaus
et al. (2001) and Ebbesen (1996). Nanotubes are extremely stiff and strong, and
can grow to lengths of microns.
Novoselov et al. (2004) found that by attaching tape to graphite crystals and
pulling it off, it was possible to create micron-sized crystals of graphene that in
places were literally one atom thick (Figure 1.3). Thus the study of graphene now
encompasses small clusters, tubes, and infinite sheets.
Problems
1.
Coordination numbers:
(a) Find the coordination number z for fee and bec lattices
(b) Find the coordination number z for hep lattices, depending on c/a.
(c) Find the average coordination number z for sodium chloride and perovskite.
(d) Find the average coordination number z for a continuous random network of
Si0
2
.
2.
Verlet method:
(a) Derive Eq. (5.38) and show that errors are only of order dt
4
after each time
step by considering the Taylor expansion of the function r(t + dt).
(b) Find an expression for the velocity r" in terms of r"
+1
and r"~
l
that has
corrections at order dt
2
.
144 Chapter 5. Beyond Crystals
3.
Phase separation:
(a) Given
a
specific function
5(c),
the endpoints
of
phase separation
in
Figure
5.5 can be determined by two algebraic equations. Find these equations.
(b) Consider the specific case of
1
S"(c)
=
-(c-0.2)
2
(c-0.8r+ c/2.
(5.133)
Show that the conditions
for
phases separation are satisfied
for c
a
=
0.2 and
c
b
=
0.8.
(c) Again considering the free energy (5.133), what is the final equilibrium state
of a system that initially is homogeneous and has
c =
0.6?
4.
Peritectic phase diagram:
Figure 5.32. Phase diagram with peritectic.
A peritectic
is
a point in a phase diagram above which phase
ß
simultaneously
melts and phase separates. Use
a
series
of
sketches analogous
to
Figure
5.6
to show how
a
such a phase diagram can arise from the competition between
liquid and solid free energies.
5.
Monte Carlo: Consider 20 particles living in one dimension whose potential
energy
is
υ
=
Σφ{χ
ι
-χ
ι
,)
ί
φ{χ) =
φ
0
(χ-
η
-χ-
6
)
(5.134)
Kl'
and which
are
constrained
to sit in the
interval [0, L], with
L =
30. Write
a Monte Carlo program
to
calculate
g(x) =
ri2{x)/n
2
at
two temperatures,
ß
= \/k
B
T =
10/00 and
ß =
100/φ
0
.
(a) Create
an
array
of
length
20 to
hold particle locations,
and
also create
an
array of length NBINS=300 to hold g(x).
Problems
145
(b) Space the particles evenly initially, at distance
1
from each other.
(c) Write a subroutine to compute the sum Eq. (5.134). The sum can be limited
to near neighbors; five are recommended, as in
for i=0 through n
{
for j= (maximum of i-5 and 0) through i
{
e+=phi(x(i)-x(j));
}
}
(d) Wait 100,000 time steps, and then at every 20th time step compute a new
contribution to g; the relevant line might look like
for i=0 through n-1
{
for j=0 through i-1
{
k=(int) (abs (x(i)-x(j)
)/L*NBINS)
g(k)=g(k)+l
}
}
Find the correct normalization of g(x). In a liquid of infinite extent, g
—>
1 as x
—>
oo. If case of a discrepancy of two, think about the effect of
fabs(x[i]-x[j] ).
(e) Carry out a total of 500,000 Monte Carlo time steps for each of the two
temperatures. Do not worry about the quality of the random number generator.
(f) Landau and Lifshitz (1980), p. 537, prove that long-range order is impossi-
ble in one dimension at nonzero temperatures for particles with short-range
forces. Please comment.
6. Von Neumann's law: Consider a two-dimensional array of polycrystalline
grains (see Figure 5.33). Assume four facts about the grain boundaries.
Figure 5.33. (A) An array of two-dimensional soap cells, which obey laws very similar to
those of metal grains. (B) Geometry of von Neumann's law.
146 Chapter 5. Beyond Crystals
(a) When grain boundaries intersect, there are always exactly three arcs depart-
ing from the intersection point.
(b) The sides of the grains are all arcs of circles.
(c) Where the arcs meet, the angles between them are all exactly 120°. To see
why, consider three ropes meeting at a point, all being pulled with the same
force. Only with 120° angles can they be in equilibrium.
(d) For every arc with radius of curvature R and length /, the corresponding grain
grows at rate —kl/R. Outward-bowing segments cause the grain to shrink;
inward bowing segments cause it to grow, because the curved grain boundary
exerts pressure like the skin of a balloon. Here k is a constant.
Use these four facts to relate the growth rate of a grain to its number of sides.
The result is called von Neumann's law, and it was derived by von Neumann
while he sat in
a
conference listening to
a
talk on grain growth. A recent
review of the topic is by Weaire and McMurry (1997).
7.
Polymer integrals:
Consider the probability
5>i
(/?) that a single polymer segment reaches from 0
to R, and its Fourier transform 7\ (k) defined in Section
5.8.1.
Show that
0
>
i(ik
=
O)
= l.
(5.135)
(a) Show that in a Taylor expansion to cubic order in components of k, the only
terms allowed are of the form
?
x
{k)^\-
C
-k
2
^e-
ck2
l
2
(5.136)
Assume '?\(—R) =
7\(R),
and that T(/?) depends upon the magnitude of R
but not its direction.
8. Polymer stiffness: Suppose one has
a
polymer composed of a sequence of
rigid rods, of length a, and suppose that the rods are connected by springs so
that
if
the angle between rod
/
and rod /
+
1
is θι, the energy of the joint is
K9J
(assume low temperatures so that βκ^> 1). Show that for long enough
polymers the chain executes an ideal random walk.
(a) Write down the probability
"Ρ(θ\
. .
.
0jy)of having some set of angles θ\
. .
.
0#
for a polymer at temperature
T.
(b) Confine the polymer to two dimensions, and ignore the possibility of
self-
intersections. Find the x coordinate of the Mh bead for some particular set of
angles θ\
. . .
ΘΝ,
given that bead
1
is at the origin.
(c) Find the expectation value (xjj) of the end of the polymer. Take
NkßT/κ
to
be much greater than
1.
(d) The result has the same form as expected for an ideal random walk, but the
segment length a must be replaced by an effective length ä. What is al
Problems 147
9. Diffusing-wave spectroscopy: Demonstrate Eq. (5.83). In addition to the
definitions in Eq. (5.84), also define
Aki = ki(t)-ki(0) (5.137)
(a) Show that Eq. (5.82) can be rewritten as the sum of two terms, one of which
involves Δ£,·(ί) ·
[r,+ i
(ί)
—
?,·(?)]
and the other of which appears in Eq. (5.83).
Make use of the fact that Δ?, vanishes when / = 0 and i = N +
1
because the
starting and ending points of the light path are the laser and detector which do
not move.
(b) Argue that the sum involving Δ&,· should be much smaller than the sum in-
volving q, so long as particle motions over time / are small compared to the
distance between scattering sites, and by discarding the smaller sum, obtain
Eq. (5.83).
(c) Demonstrate Eq. (5.90).
(d) Complete the calculations to find Eqs. (5.103) and Eq. (5.104).
10.
Diffusion of spheres: Einstein (1905) first showed that the diffusion of spheres
in liquid can be used to deduce their size. To recover this result, start with the
Langevin equation, Eq. (5.40), with the external force F set to zero.
(a) Defining v(t) = r(t), show for a particle starting at rest that
v
a
(t)=
fdt'e-^'^Ut')
(5.138)
Jo
(b) Consider times t\ and
t2
that are large compared with \jb. Using Eq. (5.41),
show that
(vMvßfo)) = —δ
αβ
«H"-*^. (5.139)
m
By setting t\ =
?2
and invoking equipartition of
energy,
argue that the constants
in Eq. (5.41) were chosen correctly to reproduce thermal equilibrium.
(c) Again considering times t much larger than \/b, compute
(r
a
(t)r
0
(t)} (5.140)
and obtain the Einstein relation
D=
k
-f.
(5.141)
b
(d) Employing Stokes' relation for the drag resistance of fluid on a slowly mov-
ing sphere, Eq. (15.145), relate the diffusion constant D for a sphere in liquid
to its radius.
11.
One-dimensional quasicrystal: Verify that
Eq.
(5.119) follows from
Eq.
(5.118).
Begin by sending n
—>
m and m
—>
n, and then send n
—>
n +
m
+ 1.
148
Chapter 5. Beyond Crystals
12.
Quasicrystal scattering: Consider a two-dimensional quasicrystal that is cre-
ated in the following way. First take five unit vectors ê/, with / =
1
. . . 5 point-
ing along the angles
2π//5.
Construct an array of lines perpendicular to each
of these unit vectors, and whose spacing is given by the Fibonacci sequence
(5.106),
as described by Eq. (5.132).
(a) Find the two-dimensional Fourier transform of this crystal's density, thus
obtaining the diffraction scattering amplitude. Use Eq. (5.129) as a starting
point.
(b) Prepare a plot in two dimensions of some of the brighter scattering peaks.
13.
The generalized dual method: A method for generating two-dimensional
quasicrystals alternative to the one discussed in the text, and more general, is
the generalized dual method. It proceeds in the following steps:
Figure 5.34. Drawing upon which the generalized dual method for constructing quasicrys-
tals is based.
(a) Write down five unit vectors êj pointing along angles 27iy'/5, for j
—
0 . . .4.
(b) For each unit vector, draw a set of parallel equally spaced lines perpendicular
to the vector. Choose the starting points of the sets of lines so that no more
than two meet at an intersection point. Number the lines, and refer with
n-
s
to
the nth line along the y'th unit vector.
(c) Every open region between the lines will correspond to a vertex of the qua-
sicrystal. To obtain the location of the vertex, compute Σ)=ο
n
jêj where the
open region is after the nth line in the j direction and before the
(n
+
1
)st.
(d) Choose the intersection point of any two lines in Figure 5.34 and describe
what type of shape is generated in the quasicrystal as one travels 360° in a
small loop about this intersection point. What are the two basic possibilities
for what can happen?
(e) Write a program to construct the tiling corresponding to Figure 5.34; or,
better yet, starting from scratch.
References 149
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