Elements of Modern Physics
216
E = Σq i ε i
=
3/2
1
1
2
1
3 exp ( )
2
2
q
m
f
V
βε
π β
β
(7.31)
It then follows that average energy is
ε =
3
2β
(7.32)
Since ε is equal to (3/2) kT for the classical particles where T is the absolute
temperature it follows that
β =
1
kT
(7.33)
In terms of T, the various distributions can be written as
q = f i exp (– α 1 – ε i /kT) Maxwell-Boltzmann,
(7.34)
r i =
2
Bose-Einstein,
exp (
/ ) 1
i
i
g
kT
α + ε
−
(7.35)
s i =
3
Fermi- Dirac
exp (
/ ) 1
i
i
h
a
kT
+ ε
+
(7.36)
These are known as the Maxwell-Boltzmann (for distinguishable particles),
Bose-Einstein (for bosons), and Fermi-Dirac (for fermions) distributions,
respectively. The constant α 1 , α 2 and α 3 are determined from the conditions in
Eq. (7.12) for the number of particles.
The following properties of the distributions may be noted:
1. For large ε i /kT when exp (α 2,3 + ε i /kT) >> 1, all the three distributions
have the some form, i.e. that of the Maxwell-Boltzmann distribution.
2. For distinguishable particles, the quantity q i /f i called the particle index,
satisfies the relation
/
/
i i
i j
q f
q f
= exp [ – (ε i = ε j )/kT]
(7.37)
which implies that there is always a greater tendency for the particle to
occupy a lower energy state than a higher energy state. This tendency is
observed for bosons and fermions as well [see Eqs. (7.35) and (7.36)].
Equation (7.37) is valid for bosons and fermions also, if the 1 in the
denominator of Eqs. (7.35) and (7.36) can be neglected.
3. It is noted that when the total number of bosons is unrestricted, α 2 = 0
and the distribution of the bosons is given by Eq. (7.27). In this case, it is
216
E = Σq i ε i
=
3/2
1
1
2
1
3 exp ( )
2
2
q
m
f
V
βε
π β
β
(7.31)
It then follows that average energy is
ε =
3
2β
(7.32)
Since ε is equal to (3/2) kT for the classical particles where T is the absolute
temperature it follows that
β =
1
kT
(7.33)
In terms of T, the various distributions can be written as
q = f i exp (– α 1 – ε i /kT) Maxwell-Boltzmann,
(7.34)
r i =
2
Bose-Einstein,
exp (
/ ) 1
i
i
g
kT
α + ε
−
(7.35)
s i =
3
Fermi- Dirac
exp (
/ ) 1
i
i
h
a
kT
+ ε
+
(7.36)
These are known as the Maxwell-Boltzmann (for distinguishable particles),
Bose-Einstein (for bosons), and Fermi-Dirac (for fermions) distributions,
respectively. The constant α 1 , α 2 and α 3 are determined from the conditions in
Eq. (7.12) for the number of particles.
The following properties of the distributions may be noted:
1. For large ε i /kT when exp (α 2,3 + ε i /kT) >> 1, all the three distributions
have the some form, i.e. that of the Maxwell-Boltzmann distribution.
2. For distinguishable particles, the quantity q i /f i called the particle index,
satisfies the relation
/
/
i i
i j
q f
q f
= exp [ – (ε i = ε j )/kT]
(7.37)
which implies that there is always a greater tendency for the particle to
occupy a lower energy state than a higher energy state. This tendency is
observed for bosons and fermions as well [see Eqs. (7.35) and (7.36)].
Equation (7.37) is valid for bosons and fermions also, if the 1 in the
denominator of Eqs. (7.35) and (7.36) can be neglected.
3. It is noted that when the total number of bosons is unrestricted, α 2 = 0
and the distribution of the bosons is given by Eq. (7.27). In this case, it is