2.1 Basic Concepts and Formulae
89
where μ is the reduced mass given by
μ =
m e m p
m e + m p
(2.17)
and Z = 1 for H-atom.
The radius of the smallest orbit, called the Bohr radius,
a 0 = r 1 =
ε 0 h
2
πme 2 = 0.529 ˚
A
(2.18)
Orbital velocity in the nth orbit
v n =
ze
2
2ε o nh
(2.19)
Fine structure constant
α =
v
c
=
e
2
2ε ohc
=
1
137
(2.20)
α is a measure of the electromagnetic interaction
Kinetic Energy of electron
K n =
1
2
m v
2
n =
ze
2
8πε o r n
(2.21)
Potential energy of electron
U n = −
ze
2
4πε o r n
(2.22)
Total Energy of electron
E n = K n + U n = −
ze
2
8πε o r n
= −
mz
2 e
4
8ε 2
o h 2 n 2 =
−13.6
n 2 (ev)
(2.23)
˜
ν if =
1
λ if
= R
1
n
2
f
−
1
n
2
i
(2.24)
Mesic atom
A negatively charged muon or pion when captured by the nucleus forms a bound
system called mesic atom before absorption or decay system.
i. r n ∝ 1/μ; therefore radii are shrunk by a factor of ∼ 200 for muonic atom
compared to H-atom.
ii. E n ∝ μ; therefore energy levels are spaced 200 times greater than those of
H-atom. X-rays are emitted instead of visible light, when the muon cascades
down.