Elements of Modern Physics
136
E n
(0)
=
2
2
2
0
1
2 4
m Ze
n
−
πε
(5.20)
and the total energy is the sum of the energies of the N electrons,
E
(0)
=
(0)
1
( )
N
n
i
E
i
=
∑
(5.21)
However, the states that can be occupied by the electrons are constrained
by Pauli’s exclusion principle. The ground-state energy is therefore obtained by
placing successive electrons in the lowest-energy, unocoupied states. It may be
noted (see Eq. (4.27)) that for each value of the principal quantum number n,
there are 2n
2
states (including the factor of 2 due to the states) with the same
energy. Thus, the first two electrons are to be placed in the n = 1 states, the next
8 electrons in the n = 2 states, the next 18 electrons in the n = 3 state, etc.
Electrons with the same value of n form what are known as shells which are
designated by the letters K for n = 1, L for n = 2, M for n = 3, etc.
It may be recollected (Sec. 4.1) that the degeneracy of the different l states
(with l ≤ n – 1) for a given value of the principal quantum number n, is a special
property of the 1/r potential. The average potential V(r i ), arising from the
interaction with the other electrons will remove this degeneracy and states with
different l value but the same n value, will have different energies, Since V (r i )
is positive and becomes more important as r i increases, it may be expected that
the states with larger l values will be raised more than those with smaller l
values. Explicit perturbative calculations can be made for the first two terms of
the potential V(r i ) given in Eq. (5.19). From Eq. (3.125),
E n, l ≈ E n
(0)
+
2
2
1
2
0
(
1)
1
(3
( 1))
4
4
a
Z
e
n l l
b
b Z
−
−
−
+
πε
(5.22)
where Eq. (4.29) has been used for 〈 r 〉, a 1 being the radius of the first Bohr
orbit with Z = 1. It is seen here that the screening effects due to other electrons
remove the l-degeneracy, the energies now increasing as l increases. This implies
that each shell is made up of subshells that have the same n value but different
l values, the subshells with larger l values having higher energy. Indeed, it so
happens that the energy of a subshell with sufficiently large l may be higher
than that of another with larger n but a lower l. The relative positions of the
various energy levels which follow from detailed calculations, and also from
experimental observations, are shown in Fig. (5.1) and form the basis of the
shell structure of the atoms.
136
E n
(0)
=
2
2
2
0
1
2 4
m Ze
n
−
πε
(5.20)
and the total energy is the sum of the energies of the N electrons,
E
(0)
=
(0)
1
( )
N
n
i
E
i
=
∑
(5.21)
However, the states that can be occupied by the electrons are constrained
by Pauli’s exclusion principle. The ground-state energy is therefore obtained by
placing successive electrons in the lowest-energy, unocoupied states. It may be
noted (see Eq. (4.27)) that for each value of the principal quantum number n,
there are 2n
2
states (including the factor of 2 due to the states) with the same
energy. Thus, the first two electrons are to be placed in the n = 1 states, the next
8 electrons in the n = 2 states, the next 18 electrons in the n = 3 state, etc.
Electrons with the same value of n form what are known as shells which are
designated by the letters K for n = 1, L for n = 2, M for n = 3, etc.
It may be recollected (Sec. 4.1) that the degeneracy of the different l states
(with l ≤ n – 1) for a given value of the principal quantum number n, is a special
property of the 1/r potential. The average potential V(r i ), arising from the
interaction with the other electrons will remove this degeneracy and states with
different l value but the same n value, will have different energies, Since V (r i )
is positive and becomes more important as r i increases, it may be expected that
the states with larger l values will be raised more than those with smaller l
values. Explicit perturbative calculations can be made for the first two terms of
the potential V(r i ) given in Eq. (5.19). From Eq. (3.125),
E n, l ≈ E n
(0)
+
2
2
1
2
0
(
1)
1
(3
( 1))
4
4
a
Z
e
n l l
b
b Z
−
−
−
+
πε
(5.22)
where Eq. (4.29) has been used for 〈 r 〉, a 1 being the radius of the first Bohr
orbit with Z = 1. It is seen here that the screening effects due to other electrons
remove the l-degeneracy, the energies now increasing as l increases. This implies
that each shell is made up of subshells that have the same n value but different
l values, the subshells with larger l values having higher energy. Indeed, it so
happens that the energy of a subshell with sufficiently large l may be higher
than that of another with larger n but a lower l. The relative positions of the
various energy levels which follow from detailed calculations, and also from
experimental observations, are shown in Fig. (5.1) and form the basis of the
shell structure of the atoms.