Introduction to Quantum Ideas
47
where λ ∞ ~ 3646 Å is the limiting value shown in Fig. 2.5. A simple but important
step in the analysis of atomic spectra was taken by Rydberg (1890) who pointed
out that the sequence in Eq. (2.39) could be represented in a more suggestive
form in terms of the reciprocal of the wavelength called the wave number,
related to the frequency:
2
2
1
1
1 ,
3, 4, ...
2
n
R
n
n
=
−
=
λ
(2.40)
R = 1.0972 × 10
7
m
-1
where R is called the Rydberg constant. The spectral lines represented by
Eq. (2.40) from what is known as the Balmer series. Further investigations
showed that the hydrogen spectrum has other series, in the ultraviolet and infrared
regions. They are represented by formulae similar to Eq. (2.40), e.g. Lyman
series (1906) in the ultraviolet region by
2
2
1
1 1 ,
2,3,...
1
n
R
n
n
=
−
=
λ
(2.41)
Paschen series (1908) in the infrared region by
2
2
1
1 1 ,
4, 5, ...
3
n
R
n
n
=
−
=
λ
(2.42)
Brackett series (1922) in the infrared region by
2
2
1
1
1 ,
5, 6, ...
4
n
R
n
n
=
−
=
λ
(2.43)
The frequencies of the lines in the hydrogen spectrum can be obtained from
a single formula
1
2
2
,
1
1
1 ,
m n
T
m n
m
n
=
−
<
λ
(2.44)
m = 1 giving the Lyman series, m = 2 giving the Balmer series, etc.
The spectra of other atoms also show some order, and their frequencies
can be represented by
1
2
,
1
( )
( ),
m n
T m T n m n
=
−
<
λ
(2.45)
However, the form of T(n) is generally more complicated than for the
hydrogen atom, one of the most useful being R(n–d)
–2
where δ is a constant
known as the quantum defect, δ << n.
47
where λ ∞ ~ 3646 Å is the limiting value shown in Fig. 2.5. A simple but important
step in the analysis of atomic spectra was taken by Rydberg (1890) who pointed
out that the sequence in Eq. (2.39) could be represented in a more suggestive
form in terms of the reciprocal of the wavelength called the wave number,
related to the frequency:
2
2
1
1
1 ,
3, 4, ...
2
n
R
n
n
=
−
=
λ
(2.40)
R = 1.0972 × 10
7
m
-1
where R is called the Rydberg constant. The spectral lines represented by
Eq. (2.40) from what is known as the Balmer series. Further investigations
showed that the hydrogen spectrum has other series, in the ultraviolet and infrared
regions. They are represented by formulae similar to Eq. (2.40), e.g. Lyman
series (1906) in the ultraviolet region by
2
2
1
1 1 ,
2,3,...
1
n
R
n
n
=
−
=
λ
(2.41)
Paschen series (1908) in the infrared region by
2
2
1
1 1 ,
4, 5, ...
3
n
R
n
n
=
−
=
λ
(2.42)
Brackett series (1922) in the infrared region by
2
2
1
1
1 ,
5, 6, ...
4
n
R
n
n
=
−
=
λ
(2.43)
The frequencies of the lines in the hydrogen spectrum can be obtained from
a single formula
1
2
2
,
1
1
1 ,
m n
T
m n
m
n
=
−
<
λ
(2.44)
m = 1 giving the Lyman series, m = 2 giving the Balmer series, etc.
The spectra of other atoms also show some order, and their frequencies
can be represented by
1
2
,
1
( )
( ),
m n
T m T n m n
=
−
<
λ
(2.45)
However, the form of T(n) is generally more complicated than for the
hydrogen atom, one of the most useful being R(n–d)
–2
where δ is a constant
known as the quantum defect, δ << n.