Elements of Modern Physics
48
The discreteness of the spectral lines is a property of the absorption spectrum
as well. When radiation passes through a vapour, the vapour absorbs radiation
of discrete frequencies which correspond to the frequencies of the emission
spectrum, given by Eq. (2.45). This results in the appearance of dark lines
corresponding to the frequencies of the radiation absorbed, which coincide with
the positions of the spectral lines in the emission spectrum. In fact, helium was
first discovered by its absorption lines in the solar spectrum before it could be
identified terrestrially.
A photon associated with radiation of frequency ν carries in energy hν.
Therefore, since energy is conserved, it is plausible to conclude that the discrete
frequencies of the emission and absorption spectra, given by Eq. (2.45),
correspond to transitions of the atom between states characterized by the integers
m, n, which have discrete energies
E n = – hcT (n)
(2.46)
In particular, the energy levels of the hydrogen atom are given by
2
( )
n
hcR
E H
n
= −
(2.47)
The numerical value of hcR is 13.6 eV, and is called the ionization potential
of the hydrogen atom.
One of the first attempts to have a model of an atom, which can explain the
discrete spectrum of the atom, was due to J.J. Thomson (1903). It had been
recognized that the negatively charged electron is one of the fundamental
constituents of the atom. Since the atom as a whole is neutral, it should also
contain a positively-charged part, called positive ion to balance the negative
charge of the electron. It was also known that a great majority of the mass is
associated with the positive ion. This led Thomson to propose a model of the
atom in the form of a sphere of uniform positive charge, in which the small,
negatively charged electrons are embedded. The electrons perform simple
harmonic motion about their positions of equilibrium, which results in the emission
of radiation of characteristic frequencies. Quantitatively, the potential energy of
the electron inside the atom is
2
2
2
0
3
0 0
( )
(
3 )
8
Ze
V r
r
R
R
=
−
πε
(2.48)
where R 0 is the radius of the atom and Z is the atomic number. This potential will
cause the electron to oscillate with a frequency
1/ 2
2
3
0
0
1
2 4
Ze
mR
ν =
π πε
(2.49)
48
The discreteness of the spectral lines is a property of the absorption spectrum
as well. When radiation passes through a vapour, the vapour absorbs radiation
of discrete frequencies which correspond to the frequencies of the emission
spectrum, given by Eq. (2.45). This results in the appearance of dark lines
corresponding to the frequencies of the radiation absorbed, which coincide with
the positions of the spectral lines in the emission spectrum. In fact, helium was
first discovered by its absorption lines in the solar spectrum before it could be
identified terrestrially.
A photon associated with radiation of frequency ν carries in energy hν.
Therefore, since energy is conserved, it is plausible to conclude that the discrete
frequencies of the emission and absorption spectra, given by Eq. (2.45),
correspond to transitions of the atom between states characterized by the integers
m, n, which have discrete energies
E n = – hcT (n)
(2.46)
In particular, the energy levels of the hydrogen atom are given by
2
( )
n
hcR
E H
n
= −
(2.47)
The numerical value of hcR is 13.6 eV, and is called the ionization potential
of the hydrogen atom.
One of the first attempts to have a model of an atom, which can explain the
discrete spectrum of the atom, was due to J.J. Thomson (1903). It had been
recognized that the negatively charged electron is one of the fundamental
constituents of the atom. Since the atom as a whole is neutral, it should also
contain a positively-charged part, called positive ion to balance the negative
charge of the electron. It was also known that a great majority of the mass is
associated with the positive ion. This led Thomson to propose a model of the
atom in the form of a sphere of uniform positive charge, in which the small,
negatively charged electrons are embedded. The electrons perform simple
harmonic motion about their positions of equilibrium, which results in the emission
of radiation of characteristic frequencies. Quantitatively, the potential energy of
the electron inside the atom is
2
2
2
0
3
0 0
( )
(
3 )
8
Ze
V r
r
R
R
=
−
πε
(2.48)
where R 0 is the radius of the atom and Z is the atomic number. This potential will
cause the electron to oscillate with a frequency
1/ 2
2
3
0
0
1
2 4
Ze
mR
ν =
π πε
(2.49)