Chemistry Reference
In-Depth Information
p
k
cos
1
«
a
2
b
1
¼
ð
3
:
68
Þ
N
þ
where
1
denotes the inverse function.
Hence, remembering from elementary analysis that:
d cos
1
u
d u
Þ
1
=
2
u
2
¼ð
1
ð
3
:
69
Þ
we have:
þ
cos
1
«
a
2
b
N
1
k
¼
ð
3
:
70
Þ
p
<
2
4
3
5
2
=
1
2
@
k
@«
¼
1
2
b
N
þ
1
«
a
2
b
1
p
ð
3
:
71
Þ
!
1
=
2
:
1
2
b
N
þ
1
p
k
1
2
b
N
þ
1
p
k
sin
2
¼
¼
cosec
p
N
þ
1
p
N
þ
1
the same result as before.
To introduce further details of the theory of solids in an elementaryway,
we can resort to the results given in Section 3.2 for the closed chainwithN
atoms.We have shown there that the general solution in complex form for
the N-atom closed chain with N
¼
odd is:
!
<
:
2
p
k
N
1
x
k
¼
2 cos
k
¼
0
;
1
;
2
; ;
N
þ
1
2
!
ð
3
:
72
Þ
i
mk
N
c
mk
/
exp 2
p
where i is the imaginary unit. Apart from the ground state
ð
k
¼
0
Þ
,
roots (3.72) occur in pairs, each level being hence doubly degenerate.
Let consider as an example the cases N
15. We have the
numerical results of Table 3.2 which are plotted in Figure 3.4.
For solids, the quantum number k is replaced by the wave vector
¼
5 and N
¼
:
k
2
k
N
;
p
;
a
N
; ;
a
2
1
N
a
¼
k
¼
0
1
ð
3
:
73
Þ
k
where a is the lattice spacing.
