350 Solutions
Replacing LS → L
′
and L → S
′
in the last line giv es the configuration of the second
but last line (self-similarity or fractality). For the Fourier transform see [
35].
1.3: Given two vectors a
1
, a
2
,with|a
1
| = a
1
, |a
2
| = a
2
,anda
1
· a
2
= a
1
a
2
cos α,
spanning a plane. The following five cases can be distinguished:
a
1
= a
2
,α= π/2 square
α = π/3 triangular or hexagonal
α = π/2,π/3
a
1
= a
2
,α= π/2 rectangular
α = π/2
1.4: T
R
n
is the translation operator. It acts on a function according to
T
R
n
φ(r)=φ(r + R
n
)
and commutes with the system Hamiltonian, [T
R
n
,H] = 0. Therefore, there exist
simultaneous eigenfunctions of H and T
R
n
with the property
T
R
n
φ
k
(r)=e
ik·R
n
φ
k
(r)
i.e. the wave eigenfunctions in different Wigner–Seitz cells differ only by a phase
factor with wave vector k from the first Brillouin zone.
1.5: Count nearest neighbors (n.n.) and spheres per cube:
sc 6 n.n., 1 sphere →
4π
3
a
2
3
=
a
3
=
π
6
=0.52,
bcc 8 n.n., 2 spheres → 2
4π
3
√
3a
4
!
3
>
a
3
=
√
3π
8
=0.68,
fcc 12 n.n., 4 spheres → 4
4π
3
a
2
√
2
!
3
>
a
3
=
π
3
√
2
=0.74,
diamond 4 n.n., 8 spheres → 8
4π
3
√
3a
8
!
3
>
a
3
=
√
3π
16
=0.34.
1.6: A mass density n(r)=δ(r −r
i
) and using δ(r −r
i
)=
q
exp (iq · (r − r
i
))/V
gives for the scattering amplitude
F (k, k
′
)=F (k − k
′
)=
i,q
e
iq·r
i
1
V
V
e
i(k−k
′
+q)·r
=
i,q
e
iq·r
i
δ
q,k−k
′
and with r
i
→ R
n
+ τ for a crystalline solid
F (q)=
τ
e
iq·τ
n
e
iq·R
n
,
where the last sum vanishes except for q = G and
n
e
iq·R
n
= Nδ
q,G
.Thusthe
scattering amplitude, which equals the static structure factor (up to a factor N)has
peaks for the reciprocal lattice vectors.