Physics 481: Condensed Matter Physics  Midsemester test
Friday, March 4, 2011
Problem 1: Structure determination
(70 points)
DebyeScherrer Xray diffraction is used to study a powder specimen of a monoatomic substance
that is known to crystallize in a cubic Bravais lattice structure with primitive vectors
~a
1
=
(
a,
0
,
0)
, ~a
2
= (0
, a,
0) and
~a
3
= (0
,
0
, a
). The wavelength of the Xrays is 1.4
˚
A.
a) Find the primitive vectors of the reciprocal lattice. (15 points)
b) Find the four shortest possible lengths of reciprocal vectors. (20 points)
c) The first diffraction ring is at an angle of
ϑ
= 17
.
9
◦
from the incident direction. Determine the
lattice constant
a
. (20 points)
d) Find the angles of the next three diffraction rings. (15 points)
Problem 2: Onedimensional Morse solid
(80 points points)
Consider
N
identical atoms of mass
M
whose motion is restricted to the
x
axis. Nearest neighbor
atoms are coupled by the socalled Morse potential
V
M
(
r
) =
D
1

e

α
(
r

r
0
)
2

D
where
r
is the distance between them and
D
,
α
, and
r
0
are positive constants.
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 Spring '11
 ThomasVojta
 Physics, Crystallography, Diffraction, Condensed matter physics, cubic Bravais lattice, lattice constant a.

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