Department
of
Physics
Savitribai Phule Pune University
PHY-C-205
(2018 Pattern): Solid State Physics
Tutorial
# 4 July 12- July
17,
2021
I.
The
dispersion
relation for a one-dimensional
crystal
with
two
atoms
per
cell
1s
-2C
+
2C2
(1-
cos
Ka)
= 0;
where
a
is
the
length
of
the
cell and
Mi,
MMa
M2
are
the
masses
of
the
two
atoms.
Determine
the
roots
to
the
dispersion
relation
at
the
Brillouin
zone
center
and
at
the
zone
boundary.
Also
plot
the relation.
2.
Suppose
that
we
allow the
two
masses
Mi
and
M2
in a
one-dimensional
diatomic
lattice
to
become
equal.
What
happens
with
the
frequency
gap?
3.
Prove
that in a
one-dimensional
diatomic
lattice,
both
acoustic
and
optical
branches
in
dispersion
curve
meet
the
zone
boundary normally.
4.
Prove
that the
gradient
of
optical
branch
of
a dispersion
curve
at
the
maximum
frequency
1s
zero.
5.
Consider
a
one-dimensional
chain
of
identical
atoms.
The
springs
between
them
alternate
in
strength
between
values Ki and
K2.
a.
Find
the
vibrational
frequencies
as
a
fiunction
of
wave
number
q.
Study
the
low
q
limit
and
find
the
sound
velocity.
b.
Discuss
the
physical
meaning
of
the
two
branches.
Sketch
the
way
the
atoms
move
in
both
cases!
c.
Discuss
the dispersion
and
the
normal
modes
for
K1>>
K2.
d.
Discuss
the
limit
Ki
*
K2
and
compare
with
the
homogeneous
chain
where all
springs
are
identical.
6.
The
speed
of
sound
in a
certain
linear
monoatomic
chain
is
1.08x
10* m/s.
Ifthe
mass
of
each
atom
is
6.81
x 10-40 kg
and
the
atomic
separation
at
equilibrium
is
4.85Ä
find
a)
the force
constant
and
b)
the
maximum
normal
mode
angular
frequency.
Assume
one
nearest-neighbor
atoms
interact.
