The One-Electron Atom
119
than that of the proton. Therefore, the hyperfine interaction is much larger for
the positronium. In particular, the separation between the F = 1 and F = 0 levels
corresponds to a frequency of v = 2.034 × 10
11
s
–1
. The electron and the positron
in the positronium annihilate each other, emitting two photons in the F = 0 state
and three photons in the F = 1 state. The lifetimes of the positronium in these
two states are different: 1.25 × 10
–10
s for the F = 0 state and 1.4 × 10
–7
s for the
F = 1 state.
Muonium
Muonium is a bound state of an electron and a µ
+
meson (µ meson or muon
and µ
+
meson are similar to an electron and a positron respectively, except that
they are heavier, their mass being about 206.84 m e ). They are produced when a
beam of µ
+
is stopped by a gas. Their energy levels are similar to those of the
hydrogen atom except for a small difference due to the difference in the reduced
mass. The hyperfine energy levels of the muonium are of importance since they
can be calculated precisely, and serve as a test of the theory.
Muonic Helium
Muonic helium is formed by replacing one of the electrons in a helium atom
by a muon. Since the Bohr radius of the muon is smaller by a factor of about 207
than that of the electron, the electron essentially sees a nucleus of charge 2|e|
with a muon moving around close to the nucleus. Therefore, the energy levels
are similar to those of the hydrogen atom. However, the hyperfine splitting is
due to the electron magnetic moment interacting with the muon magnetic moment.
Using Eq. (4.76) as a first approximation but taking g = 1 and replacing m N by
m µ , it is found that the hyperfine splitting for muonic helium corresponds to a
frequency of v = 4.515 × 10
9
s
–1
, close to the experimental value of
4.465 × 10
9
s
–1
.
Muonic atoms are very useful for probing the structure of nuclei since the
Bohr radius of the muon is quite small, and therefore the probability of finding
the muon inside the nucleus may be quite substantial.
Rydberg Atoms
When an electron in an atom is in a state with a sufficiently large principal
quantum number n, it is influenced mainly by the net positive charge of the ionic
core and not by its distribution. These excited states of atoms are similar to
those of a hydrogen atom. They are termed Rydberg states and the atoms are
called Rydberg atoms. It is the advent of tunable lasers (see Sec. 6.5) that has
helped to excite and investigate the Rydberg states. They are of interest for the
following reasons:
119
than that of the proton. Therefore, the hyperfine interaction is much larger for
the positronium. In particular, the separation between the F = 1 and F = 0 levels
corresponds to a frequency of v = 2.034 × 10
11
s
–1
. The electron and the positron
in the positronium annihilate each other, emitting two photons in the F = 0 state
and three photons in the F = 1 state. The lifetimes of the positronium in these
two states are different: 1.25 × 10
–10
s for the F = 0 state and 1.4 × 10
–7
s for the
F = 1 state.
Muonium
Muonium is a bound state of an electron and a µ
+
meson (µ meson or muon
and µ
+
meson are similar to an electron and a positron respectively, except that
they are heavier, their mass being about 206.84 m e ). They are produced when a
beam of µ
+
is stopped by a gas. Their energy levels are similar to those of the
hydrogen atom except for a small difference due to the difference in the reduced
mass. The hyperfine energy levels of the muonium are of importance since they
can be calculated precisely, and serve as a test of the theory.
Muonic Helium
Muonic helium is formed by replacing one of the electrons in a helium atom
by a muon. Since the Bohr radius of the muon is smaller by a factor of about 207
than that of the electron, the electron essentially sees a nucleus of charge 2|e|
with a muon moving around close to the nucleus. Therefore, the energy levels
are similar to those of the hydrogen atom. However, the hyperfine splitting is
due to the electron magnetic moment interacting with the muon magnetic moment.
Using Eq. (4.76) as a first approximation but taking g = 1 and replacing m N by
m µ , it is found that the hyperfine splitting for muonic helium corresponds to a
frequency of v = 4.515 × 10
9
s
–1
, close to the experimental value of
4.465 × 10
9
s
–1
.
Muonic atoms are very useful for probing the structure of nuclei since the
Bohr radius of the muon is quite small, and therefore the probability of finding
the muon inside the nucleus may be quite substantial.
Rydberg Atoms
When an electron in an atom is in a state with a sufficiently large principal
quantum number n, it is influenced mainly by the net positive charge of the ionic
core and not by its distribution. These excited states of atoms are similar to
those of a hydrogen atom. They are termed Rydberg states and the atoms are
called Rydberg atoms. It is the advent of tunable lasers (see Sec. 6.5) that has
helped to excite and investigate the Rydberg states. They are of interest for the
following reasons: