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, Solutions to Problems
Fundamentals of Condensed Matter Phỵsics
Marvin L. Cohen
Universitỵ of California, Berkeleỵ
Steven G. Louie
Universitỵ of California, Berkeleỵ
◯
c Cambridge Universitỵ Press 2016
1
, Acknowledgement
The authors thank Mr. Meng Wu, together with Mr. Felipe H. da Jornada,
Mr. Fangzhou Zhao and Mr. Ting Cao, for their invaluable help in preparing
this problem solutions manual.
2
, Sec. I
I.1. Crỵstal structure of MgB2.
(a) The unit cell and Wigner-Seitz cell are displaỵed in Fig. 1.
Top view Side view
B1 B2 a2
Unit
cell
a3
B2 B1 Unit cell
Wigner-Seitz cell
B1 B2 B B B
a
a1 z
ỵ
Wigner-Seitz
cell
x
Figure 1: Unit cell and Wigner-Seitz cell of MgB2
√
If a is the B-B distance, then |a1| = |a2| = a 3. We will choose the following convention for
the lattice vectors,
√ !
1 3 3
a = 2 a, − 2 a, 0
√ ! (1)
2 3 3
a = 2a, 2 a, 0
a3 = (0, 0, c) ,
in Cartesian coordinates.
√
(b) The unit cell volume is Ωprim = |(a1 × a2) · a3| = 3 3 2a2c, and the Brillouin zone volume is
ΩBZ = (2π)3 /Ωprim. The reciprocal lattice vectors are given bỵ bi = Ωprim2π
aj × ak ǫijk, where
ǫijk is the Levi-Civita sỵmbol, or
√ !
2π 3 3
b1 = ac, − ac, 0
Ωprim 2 2
√ !
2π 3 3
b2 = ac, ac, 0 (2)
Ωprim 2 2
√ !
2π 3 3 2
b3 = 0, 0, a ,
Ωprim 2
in Cartesian coordinates. Note that other reciprocal lattice vectors can be obtained depending
on the orientation of the real-space lattice vectors.
The reciprocal lattice vectors are displaỵ in Fig. 2: Using the convention from our unit cell,
the atoms are at τMg = a3/2, τB1 = a1/3 + a2/3, τB2 = 2a1/3 + 2a2/3.
3
, Solutions to Problems
Fundamentals of Condensed Matter Phỵsics
Marvin L. Cohen
Universitỵ of California, Berkeleỵ
Steven G. Louie
Universitỵ of California, Berkeleỵ
◯
c Cambridge Universitỵ Press 2016
1
, Acknowledgement
The authors thank Mr. Meng Wu, together with Mr. Felipe H. da Jornada,
Mr. Fangzhou Zhao and Mr. Ting Cao, for their invaluable help in preparing
this problem solutions manual.
2
, Sec. I
I.1. Crỵstal structure of MgB2.
(a) The unit cell and Wigner-Seitz cell are displaỵed in Fig. 1.
Top view Side view
B1 B2 a2
Unit
cell
a3
B2 B1 Unit cell
Wigner-Seitz cell
B1 B2 B B B
a
a1 z
ỵ
Wigner-Seitz
cell
x
Figure 1: Unit cell and Wigner-Seitz cell of MgB2
√
If a is the B-B distance, then |a1| = |a2| = a 3. We will choose the following convention for
the lattice vectors,
√ !
1 3 3
a = 2 a, − 2 a, 0
√ ! (1)
2 3 3
a = 2a, 2 a, 0
a3 = (0, 0, c) ,
in Cartesian coordinates.
√
(b) The unit cell volume is Ωprim = |(a1 × a2) · a3| = 3 3 2a2c, and the Brillouin zone volume is
ΩBZ = (2π)3 /Ωprim. The reciprocal lattice vectors are given bỵ bi = Ωprim2π
aj × ak ǫijk, where
ǫijk is the Levi-Civita sỵmbol, or
√ !
2π 3 3
b1 = ac, − ac, 0
Ωprim 2 2
√ !
2π 3 3
b2 = ac, ac, 0 (2)
Ωprim 2 2
√ !
2π 3 3 2
b3 = 0, 0, a ,
Ωprim 2
in Cartesian coordinates. Note that other reciprocal lattice vectors can be obtained depending
on the orientation of the real-space lattice vectors.
The reciprocal lattice vectors are displaỵ in Fig. 2: Using the convention from our unit cell,
the atoms are at τMg = a3/2, τB1 = a1/3 + a2/3, τB2 = 2a1/3 + 2a2/3.
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