Physics 481: Condensed Matter Physics  Test prep Homework 7
due date: Friday, March 5, 2011
Problem 1: Tetragonal lattice
(15 points)
A hypothetical monoatomic substance crystallizes in a centered tetragonal structure. The con
ventional unit cell can described by primitive vectors (
a,
0
,
0)
,
(0
,a,
0)
,
(0
,
0
,c
) with
c
= 3
a/
2 and
a basis consisting of two atoms at positions (0
,
0
,
0) and (
a/
2
,a/
2
,c/
2). The lattice constant is
a
= 4
.
2
˚
A.
a) Calculate the maximum space ﬁlling for this lattice.
b) Find the primitive vectors of the reciprocal lattice.
c) A powder specimen of the substance is analyzed by Xray diﬀraction using the DebyeScherrer
method. The wavelength of the Xrays is 1.5
˚
A. Calculate the angles of the ﬁrst four diﬀraction
rings.
Problem 2: Phonon dispersion in a chain with longrange interactions
(AshcroftMermin
22.1 , 15 points)
Consider a onedimensional chain of identical atoms of mass
M
. The springs are not only between
nearest neighbors but between all pairs of atoms. Thus, the elastic energy reads
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 Spring '11
 ThomasVojta
 Physics, Crystallography, Work, Diffraction, Wavelength, Fundamental physics concepts, Condensed matter physics, Phonon

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