Elements of Modern Physics
170
E 0 = –
2
2
0 1
'
4πε
e Z
a
(5.88)
The calculation of the expectation value of λV is straight forward, and gives
V
λ
= –
2
2
0 1
0 1
(
)
5
2
3 2
− ′ ′
′
+
πε
πε
Z Z Z e
e Z
a
a
(5.89)
From the condition in Eq. (5.86,)
Z' = Z –
5
16
(5.90)
which gives an estimation of the screening of the nuclear charge. With this
value for Z', the ground state energy is
E = –
2
2
0 1
5
4
1 6
−
πε
e
Z
a
(5.91)
On substracting this expression from the energy of the ionized state, the
ionization energy is
E = –
2
2
2
0 1
5
1
4
1 6
2
−
−
πε
e
Z
Z
a
(5.92)
For the helium atom Z = 2, so that
I ≈ 23.1 eV
(5.93)
which is in very good agreement with the experimental value of 24.6 eV. For the
singly ionized lithium, Li
+
, the ionization energy from Eq. (5.92) with Z = 3,
comes out to be 74.1 eV which may be compared with the experimental value
of 75.6 eV.
Example 6
The highest-energy characteristic x-ray lines are obtained from U
92
. The energies
of K-lines are estimated from
E ≈ 13.6 Z
2
eV
(5.94)
≈ 115 keV
with a corresponding wavelength of about 0.11 Å.
Example 7
A knowledge of the molecular dissociation energy enables the estimation of its
repulsive energy.
For a KCl molecule, the dissociation energy is 4.42 eV (E = – 8.76 eV), so
that from Eq. (5.59),
170
E 0 = –
2
2
0 1
'
4πε
e Z
a
(5.88)
The calculation of the expectation value of λV is straight forward, and gives
V
λ
= –
2
2
0 1
0 1
(
)
5
2
3 2
− ′ ′
′
+
πε
πε
Z Z Z e
e Z
a
a
(5.89)
From the condition in Eq. (5.86,)
Z' = Z –
5
16
(5.90)
which gives an estimation of the screening of the nuclear charge. With this
value for Z', the ground state energy is
E = –
2
2
0 1
5
4
1 6
−
πε
e
Z
a
(5.91)
On substracting this expression from the energy of the ionized state, the
ionization energy is
E = –
2
2
2
0 1
5
1
4
1 6
2
−
−
πε
e
Z
Z
a
(5.92)
For the helium atom Z = 2, so that
I ≈ 23.1 eV
(5.93)
which is in very good agreement with the experimental value of 24.6 eV. For the
singly ionized lithium, Li
+
, the ionization energy from Eq. (5.92) with Z = 3,
comes out to be 74.1 eV which may be compared with the experimental value
of 75.6 eV.
Example 6
The highest-energy characteristic x-ray lines are obtained from U
92
. The energies
of K-lines are estimated from
E ≈ 13.6 Z
2
eV
(5.94)
≈ 115 keV
with a corresponding wavelength of about 0.11 Å.
Example 7
A knowledge of the molecular dissociation energy enables the estimation of its
repulsive energy.
For a KCl molecule, the dissociation energy is 4.42 eV (E = – 8.76 eV), so
that from Eq. (5.59),