144
3 Ensembles: Systems of Particles
p
B
0 (T ) =
1
1 + 3e −359.7/T , p
B
1 (T ) =
3e −359.7/T
1 + 3e −359.7/T .
For temperature T = 450 ◦ C, for example, the fractional population of the excited
state of system A will be
p
A
1 =
0.4870
1.4870
= 0.328 ,
while that for system B will be
p
B
1 =
1.4611
2.4611
= 0.594 .
Thus, we see that the triple degeneracy associated with 1 in system B has increased
the fractional population of the excited state for system B relative to that for system
A by just over 80% at a temperature of 450 ◦ C (or 723.15 K).
Example 3.2 Silicon and germanium atoms.
To illustrate in a more concrete manner the role played by energy degeneracy, let
us compare Si and Ge atoms, whose three lowest energy levels have 1, 3, 5 states,
respectively, with the pseudo-atoms Si* and Ge* for which the three lowest energy
levels all have 1 state only.
If we represent the (filled shell) electronic configurations of Ne and Ar by
[Ne] and [Ar], then the electronic configuration of Si can be written in the form
[Ne](3s) 2 (3p) 2 , while that of Ge is given as [Ar](4s) 2 (3d) 10 (4p) 2 . All atoms
possess a hierarchy of energy levels, beginning with the lowest, or ground, energy
level, which is equivalent to 0 from the previous example. Atomic energy levels
have been systematically characterized using an agreed-upon convention expressed
in the form of atomic term symbols (see Appendix D for an explanation of their
origin and meanings). The ground electronic term for both Si and Ge is designated
by 3 P , made up of three distinct levels, denoted in turn by 3 P 2 , 3 P 1 , and 3 P 0 .
The ground level for the Si atom is the 3 P 0 level, with the 3 P 1 , 3 P 2 levels lying
[3] 77.115 cm −1 , 223.16 cm −1 , respectively, above it; the ground level for the Ge
atom is also a 3 P 0 level, with 3 P 1 , 3 P 2 levels lying [3] 557.13 cm −1 , 1409.96 cm −1
above it. The next electronic term for each of these atoms is a 1 D 2 term lying
6298.85 cm −1 , respectively, 7125.30 cm −1 above the 3 P 0 levels of the Si and
Ge atoms. For atomic levels, the degeneracy is given by 2J + 1, with J being
the subscript on the atomic term symbol: thus, the 3 P 2 , 3 P 1 , 3 P 0 levels have
degeneracies 5, 3, 1, respectively, and each 1 D 2 term has degeneracy 5.
We shall compare the actual fractional populations for the three electronic levels
of the ground 3 P terms of Si and Ge with the fractional populations that would have
been obtained were all three 3 P levels nondegenerate.
As in our first example for systems A and B, we shall work with a temperature
T = 723.15 K (or 450 ◦ C) to keep matters simple. The relevant partition functions
are
3 Ensembles: Systems of Particles
p
B
0 (T ) =
1
1 + 3e −359.7/T , p
B
1 (T ) =
3e −359.7/T
1 + 3e −359.7/T .
For temperature T = 450 ◦ C, for example, the fractional population of the excited
state of system A will be
p
A
1 =
0.4870
1.4870
= 0.328 ,
while that for system B will be
p
B
1 =
1.4611
2.4611
= 0.594 .
Thus, we see that the triple degeneracy associated with 1 in system B has increased
the fractional population of the excited state for system B relative to that for system
A by just over 80% at a temperature of 450 ◦ C (or 723.15 K).
Example 3.2 Silicon and germanium atoms.
To illustrate in a more concrete manner the role played by energy degeneracy, let
us compare Si and Ge atoms, whose three lowest energy levels have 1, 3, 5 states,
respectively, with the pseudo-atoms Si* and Ge* for which the three lowest energy
levels all have 1 state only.
If we represent the (filled shell) electronic configurations of Ne and Ar by
[Ne] and [Ar], then the electronic configuration of Si can be written in the form
[Ne](3s) 2 (3p) 2 , while that of Ge is given as [Ar](4s) 2 (3d) 10 (4p) 2 . All atoms
possess a hierarchy of energy levels, beginning with the lowest, or ground, energy
level, which is equivalent to 0 from the previous example. Atomic energy levels
have been systematically characterized using an agreed-upon convention expressed
in the form of atomic term symbols (see Appendix D for an explanation of their
origin and meanings). The ground electronic term for both Si and Ge is designated
by 3 P , made up of three distinct levels, denoted in turn by 3 P 2 , 3 P 1 , and 3 P 0 .
The ground level for the Si atom is the 3 P 0 level, with the 3 P 1 , 3 P 2 levels lying
[3] 77.115 cm −1 , 223.16 cm −1 , respectively, above it; the ground level for the Ge
atom is also a 3 P 0 level, with 3 P 1 , 3 P 2 levels lying [3] 557.13 cm −1 , 1409.96 cm −1
above it. The next electronic term for each of these atoms is a 1 D 2 term lying
6298.85 cm −1 , respectively, 7125.30 cm −1 above the 3 P 0 levels of the Si and
Ge atoms. For atomic levels, the degeneracy is given by 2J + 1, with J being
the subscript on the atomic term symbol: thus, the 3 P 2 , 3 P 1 , 3 P 0 levels have
degeneracies 5, 3, 1, respectively, and each 1 D 2 term has degeneracy 5.
We shall compare the actual fractional populations for the three electronic levels
of the ground 3 P terms of Si and Ge with the fractional populations that would have
been obtained were all three 3 P levels nondegenerate.
As in our first example for systems A and B, we shall work with a temperature
T = 723.15 K (or 450 ◦ C) to keep matters simple. The relevant partition functions
are