Download Chapter 9 Canonical ensemble

Chapter 9
Canonical ensemble
9.1
System in contact with a heat reservoir
We consider a small system A1 in thermal interaction with a heat reservoir A2 such that
A1 << A2 , i.e., A1 has fewer degrees of freedom than A2 (see Fig. 9.1).
Figure 9.1: Small system A1 compared to the reservoir A2 .
A1 is characterized by E1 , V1 , N1 and A2 by E2 , V2 , N2 with:
E2 >> E1 ,
N2 >> N1 ,
N1 = const,
N2 = const,
and E1 + E2 = E = const.
Both systems are in thermal equilibrium at temperature T . The wall between them
allows interchange of heat but not of particles. The system A1 may be any relatively
small macroscopic system such as, for instance, a bottle of water in a lake, while the lake
acts as the heat reservoir A2 .
The question we want to answer is the following:
⇒ under equilibrium conditions, what is the probability Pα of finding the small system
A1 in any one particular microstate α of energy Eα ? In other words, what is the
distribution function ρ of the system A1 ?
103
104
CHAPTER 9. CANONICAL ENSEMBLE
we assume that the combined system A = A1 + A2 is isolated and that A1 and A2 are
weakly interacting so that their energies are additive. The energy of A1 , E1 , is of course
not fixed. It is only the total energy E = E1 + E2 of the combined system A that has a
constant value in some range between E and E + ∆.
The Hamilton function of the combined system A is
H(q(1), p(1), q(2), p(2)) = H1 (q(1), p(1)) + H2 (q(2), p(2)).
In the following, we will use the notation:
q = (q(1), q(2)),
p = (p(1), p(2)).
Since the combined system A is isolated, the distribution function in the combined Γphase space is given by the microcanonical distribution function ρ(q, p); which we can
write for ∆ → 0 as:
where
ρ(q, p) = R
δ (E − H1 (q(1), p(1)) − H2 (q(2), p(2)))
,
dΓ δ (E − H1 (q(1), p(1)) − H2 (q(2), p(2)))
Z
dΓ δ(E − H) = Ω(E).
ρ(q, p) is the distribution function of the total system A, but it is the distribution function of the small system A1 , ρ1 (q(1), p(1)), that we are interested in. In order to find
ρ1 (q(1), p(1)), we proceed in the following way:
⇒ We integrate the total probability density ρ(q, p) over the phase space of the reservoir
A2 , i.e., we integrate out the A2 probabilities:
Z
ρ1 (q(1), p(1)) =
dΓ2 ρ(q(1), p(1), q(2), p(2))
R
dΓ2 δ(E − H1 − H2 )
Ω2 (E − H1 )
=
=
.
(9.1)
Ω(E)
Ω(E)
Now, we make use of the fact that A1 is a much smaller system than A2 and therefore
the energy E1 given by H1 is much smaller than the energy of the combined system:
E1 << E.
In this case, we can approximate equation (9.1) by expanding the slowly varying logarithm
of Ω2 (E2 ) = Ω2 (E − H1 ) around the value E2 = E as
∂ ln Ω2
ln Ω2 (E2 ) = ln Ω2 (E − H1 ) = ln Ω2 (E) −
H1 + . . .
∂E2 E2 =E
9.2. INTERPRETATION OF THE EXPONENTIAL:
BOLTZMANN FACTOR
105
and neglect the higher-order terms since H1 = E1 << E. Then,
Ω2 (E − H1 ) = eln Ω2 (E−H1 )
"
#
∂ ln Ω2 (E2 ) = exp ln Ω2 (E) −
· H1 + . . .
∂E2
E2 =E
= Ω2 (E)e
= Ω2 (E)e
h
∂S2
H1
B ∂E
− k1
H1
BT
−k
since
∂S2
1
=
∂E
T
i
or, equivalently,
∂ ln Ω2 (E2 ) 1
≡ β.
=
∂E2
kB T
E2 =E
T is the constant temperature parameter characterizing the heat reservoir A2 . Physically,
this means that the reservoir A2 is so large compared to A1 that its temperature remains
unaffected by whatever small amount of energy it gives to A1 .
Finally, we have that
Ω2 (E − H1 ) = Ω2 (E)e
and
ρ1 (q(1), p(1)) =
H1
BT
−k
Ω2 (E) − kH1T
e B .
Ω(E)
(9.2)
2 (E)
is a constant independent of H1 , we define the normalization of
Since in eq. (9.2) ΩΩ(E)
ρ1 by integrating over the phase space Γ1 :
ρ1 (q(1), p(1)) = R
e
−
H1 (q(1),p(1))
kB T
dΓ1 e
−H1 (q(1),p(1))
kB T
.
(9.3)
⇒ ρ1 is the distribution function in the canonical ensemble.
9.2
Interpretation of the exponential:
Boltzmann factor
Let us go back to the probability Pα of finding the system A1 (which is in thermal
equilibrium with the heat reservoir A2 ) in any one particular microstate α of energy Eα .
Following the procedure given in the previous section, one expresses Pα as
Pα = c e−βEα ,
(9.4)
where c is some constant of proportionality independent of α. By making use of the
normalization condition for probabilities,
X
Pα = 1,
α
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CHAPTER 9. CANONICAL ENSEMBLE
and expressing the constant c as
c−1 =
X
e−βEα ,
α
one can write equation (9.3) in the form:
e−βEα
Pα = P −βEα ,
αe
(9.5)
which describes the probability function of the microstate α with energy Eα .
If A1 is known to be in a definite one of its microstates, the reservoir A2 can be in any one
of the larger number Ω2 (E − H1 ) of states accessible to it. The number of states Ω2 (E2 )
accessible to the reservoir is a very rapidly increasing function of its energy. Thus, if A1 is
in a state α where its energy Eα is higher, the conservation of energy for the total system
implies that the energy of the reservoir is correspondingly lower so that the number of
states accessible to the reservoir is reduced. The probability of encountering this situation
in the ensemble is accordingly very much less. The exponential dependence of Pα on Eα
in equation (9.5) basically expresses this fact in mathematical terms.
Example:
Suppose that the number of states accessible to A1 and A2 for various values of their
respective energies is given by the graphs in Fig. 9.2.
Figure 9.2: Density of states diagrams for A1 and A2 .
Let us assume that the total energy of the combined system is 1007. First, we
consider the situation where A1 is in the state α with energy 6. In this case, the
energy of the reservoir A2 must be 1001 so that it can be in any of the 300,000
accessible states. In an ensemble consisting of many copies of the system A, there
will be 300,000 different possible examples of systems for which A1 is in the state α.
9.3. CANONICAL PARTITION FUNCTION
107
Now, if A1 is in the state γ with energy 7, the reservoir must have energy 1000 so
that only 100,000 states are accessible to it. In this case, the ensemble contains only
100,000 different possible examples of systems for which A1 is in the state γ.
The exponential factor e−βEα is called the Boltzmann factor and the probability distribution, eq. (9.5), is the canonical distribution. An ensemble of systems all of which are
in contact with a heat reservoir at temperature T is called a canonical ensemble.
The probability P (E) that A1 has energy in a small range between E and E + ∆ is
obtained by adding the probabilities for all states whose energies lie in this range:
X
P (E) =
Pα ,
α
with E < Eα < E + ∆. All these states are equally probable (principle of equal a priori
probability) and are characterized by the same exponential factor e−βE so that
P (E) = c Ω1 (E) e−βE ,
(9.6)
where Ω1 (E) is the number of states in the energy range between E and E + ∆ (density
of states). Since A1 is a large system, although very much smaller than A2 , Ω1 (E) is a
rapidly increasing function of E. Due to the presence of the rapidly decreasing factor
e−βE , the product Ω1 (E)e−βE has a maximum. The larger A1 is, the sharper is this
maximum in P (E).
We will put equation (9.6) in a more rigorous context in section 9.6.
9.3
Canonical partition function
In section 9.1 we obtained that the distribution function in the canonical ensemble is
given by
e−βH(q,p)
ρ(q, p) = R
,
dΓ e−βH(q,p)
where we dropped all the indices ”1” for simplicity, though in fact we are still describing
the properties of a ”small” system (which is nevertheless macroscopically big) in thermal
equilibrium with a heat reservoir.
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CHAPTER 9. CANONICAL ENSEMBLE
The average value of an observable O in the canonical ensemble is given as
Z
Z
3N
hOi = d q d3N p ρ(q, p) O(q, p)
and its probability distribution as
ρ(o) = hδ(o − O)i =
Z
with
ρ(q, p) = R
d3N q d3N p ρ(q, p) δ(o − O(q, p)),
e−βH(q,p)
.
d3N q d3N p e−βH(q,p)
We define the following function ZN :
ZN =
Z
d3N q d3N p −βH(q,p)
e
,
h3N N !
(9.7)
i.e., the sum over states (”kanonische Zustandssumme”) with the appropriate h3N N ! norm
factors, as the canonical partition function. ZN is analogous to the microcanonical space
volume Γ(E) in units of Γ0 :
Z
1
Γ(E)
d3N q d3N p
= 3N
Γ0
h N ! E<H(q,p)<E+E+∆
Z 3N 3N
d qd p
=
(Θ(E + ∆ − H) − Θ(E − H)) .
h3N N !
We will show that it is possible to obtain all thermodynamical observables by differentiating the partition function ZN . In particular,
ZN = e−βF (T,V,N )
⇒ F (T, V, N ) = −kB T ln ZN (T, N ) ,
where F (T, V, N ) is the Helmholtz free energy.
Proof :
We perform the differentiation:
R dΓ −βH
∂
e
∂
1 ∂ZN
∂β
h3N N !
ln ZN =
= R dΓ −βH
∂β
ZN ∂β
e
h3N N !
R
−βH
dΓ(−H)e
R
=
=−<H>.
dΓe−βH
Since < H >= E = U is the internal energy, we have that
U =−
∂
∂
ln ZN = kB T 2
ln ZN .
∂β
∂T
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9.3. CANONICAL PARTITION FUNCTION
Now we use the thermodynamic relations:
F
∂
∂F
2 ∂
(βF )
= −T
=
U = F + TS = F − T
∂T V,N
∂T T
∂β
⇒
−
∂
∂
ln ZN =
(βF )
∂β
∂β
and get
ln ZN = −βF
⇒
which is what we wanted to prove.
ZN = e−βF ,
This is correct up to a constant cV,N . In order to make sure that the constant in not
dependent on V and N and therefore we can set it to zero, we will perform in the following
a more complete derivation of the Helmholtz energy.
Let us consider the expression of F as a Legendre transformation:
F (T, V, N ) = U (T, V, N ) − T S(T, V, N )
dF = dU − d(T S)
(dU = T dS − P dV )
1
1
− 2 dT
dβ =
kB
T
dF = −SdT − P dV
d(−βF )
= −βdF − F dβ
F
dT
= −β [−SdT − P dV ] +
kB T 2
F
= β S+
dT + βP dV
T
U
dT + P dV
= β
T
(9.8)
1) we note that:
U =−
eq. (9.9) =
∂
ln ZN
∂β
V,N
∂
kB T
ln ZN
∂T
2
(9.9)
V,N
2) in class we showed that
H = H(p, q, x) x: external parameter, f. i., V
∂H
∂H
= dV
δW = dx
∂x
∂V
| {z }
−P
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CHAPTER 9. CANONICAL ENSEMBLE
P =−
∂H
∂V
T,N
The integral
1
3N
h N!
Z
dΓ
∂H
∂V
e
R
dΓ
=− R
−βH
∂H
∂V
e−βH
1
3N
h N!
dΓ e−βH
1 ∂
=−
β ∂V
−βH
dΓ ∂H
e
∂V
=−
3N
h N !ZN
R
Z
dΓ e
−βH
=−
1 ∂
ZN
β ∂V
Then
1
1 1 ∂
ZN =
P =−
β ZN ∂V
β
∂
ln ZN
∂V
(9.10)
T,N
Eq. (9.9) and (9.10) into eq. (9.8):
1
d(−βF ) =
kB T
∂
ln ZN
kB T
∂T
1
dT +
(kB T )
kB T
V,N
∂ ln ZN
∂V
dV
T,N
This implies:
−βF = ln ZN + c
c is independent of V and N and is set to zero.
F = −kB T ln ZN
9.3.1
Thermodynamic properties
We can now formulate the systematics of calculating the system’s properties in the canonical ensemble:
1 - calculate the partition function ZN ;
2 - calculate the free energy F (T, V, N ) = −kB T ln ZN (T, V, N );
3 - calculate the thermodynamic properties out of F as
∂F
P = −
,
∂V T,N
∂F
S = −
,
∂T V,N
∂F
;
µ =
∂N T,V
4 - through Legendre transformation, calculate the remaining thermodynamic properties.
9.4. COMPARISON BETWEEN CANONICALAND MICROCANONICAL ENSEMBLES111
9.4
Comparison between canonical
and microcanonical ensembles
We have seen that the calculations in the microcanonical and canonical ensembles reduce
to a phase space integration and a calculation of a thermodynamic potential:
Phase space
integration
Thermodynamic
potential
Microcanonical ensemble
Density of states:
R
ΩN (E) = d3N q d3N p δ(E − H)
S(E, V, N ) = kB ln
ΩN (E)∆
h3N N !
Canonical ensemble
Partition
Z 3N function:
d q d3N p −H(q,p)β
e
ZN (T ) =
h3N N !
F (T, V, N ) = −kB T ln ZN (T )
The relation between the density of states ΩN (E) and the partition function ZN (T ) can
be defined as a Laplace transformation in the following way. Multiplying both sides of
the equality
Z
∞
1=
0
dE δ(E − H)
by ZN and using the definition of ZN (eq. (9.7)), we have
ZN (T )
Z
∞
Z
d3N q d3N p −βH
e
dE δ(E − H)
h3N N !
0
Z 3N 3N
Z ∞
d qd p
−βE
δ(E − H)
dE e
=
h3N N !
0
Z ∞
ΩN (E)
dE e−βE 3N
.
=
h N!
0
=
We have thus shown that ZN (T ) is the Laplace transform of ΩN (E).
Advantages of the canonical ensemble with respect to the microcanonical ensemble:
In both ensembles we have to perform P
an integration which is usually difficult. When
the Hamilton function is additive, H = i Hi , the integration in the canonical ensemble
can be factorized, which is not the case for the microcanonical ensemble. Therefore, it is
usually easier to calculate in the canonical ensemble than in the microcanonical ensemble.
9.5
Additivity of F (T, V, N )
An important property of the free energy is that it has to be additive. Let us consider two
systems in thermal equilibrium. We neglect the interaction among the systems, therefore
the total Hamiltonian can be written as a sum of the Hamiltonians of each of the systems
112
CHAPTER 9. CANONICAL ENSEMBLE
H = H1 + H2
N = N1 + N2
The partition function of the total system is then:
Z
1
ZN (T, V ) = 3N
d3N q d3N p e−β(H1 +H2 )
h N1 !N2 !
Note that the exchange of a particle of system 1 with a particle of system 2 gives a new
microstate and therefore the factor in the denominator is N1 !N2 ! and NOT N !.
Then
Z
1
d3N1 q d3N1 p e−βH1 (q1 ,p1 )
ZN (T, V ) =
h3N1 N1 !
Z
1
× 3N2
d3N2 q d3N2 p e−βH2 (q2 ,p2 )
h N2 !
= ZN1 (T, V1 )ZN2 (T, V2 )
⇒
F (T, V, N ) = F1 (T, V1 , N1 ) + F2 (T, V2 , N2 )
By using the information that the partition function Z(T, V, N ) is the Laplace transform
of the density of microstates function Ω(E, V, N ) we can now obtain the relation between
density of microstates of two systems in thermal equilibrium
Ω̃Ni (E) =
1
h3Ni N
i!
ΩNi (E)
ZN = Z N 1 Z N 2
⇓
Z ∞
dE e−βE Ω̃N (E)
ZN (T ) =
0
Z ∞
dE1 e−βE1 Ω̃N1 (E1 )
ZN1 (T ) =
Z0 ∞
ZN2 (T ) =
dE2 e−βE2 Ω̃N2 (E2 )
0
Z
∞
dE ΩN (E) e
0
Change of variables:
Z
−βE
=
Z
∞
dE ΩN (E) e−βE
0
∞
dE2 ΩN2 (E2 )
0
Z
∞
dE1 ΩN1 (E1 ) e−β(E1 +E2 )
0
E = E1 + E2
E1 = E1
Z ∞
Z ∞
ΩN2 (E − E1 ) ΩN1 (E1 ) dE1
dE
=
0
0
113
9.6. IDEAL GAS IN THE CANONICAL ENSEMBLE
⇒
ΩN (E) =
Z
∞
ΩN2 (E − E1 ) ΩN1 (E1 ) dE1
0
The density of microstates at energy E of the total system is given by the convolution of
the density of microstates of the two systems in equilibrium.
Note that this is only valid if H = H1 + H2 and H12 = 0.
9.6
Ideal gas in the canonical ensemble
We consider now the ideal gas in the canonical ensemble, whose Hamiltonian is
H=
N
X
p~i 2
i=1
2m
.
We calculate ZN :
Z
d3N q d3N p −β PNi=1 p~2i
2m
e
h3N N !
3N
Z +∞
dp −β p2
VN
2m
.
e
=
N!
−∞ h
ZN (T, V ) =
In the above expression, we factorize the integral over pi . Now, we make use of the integral
Z +∞
√
2
dx e−x = π.
−∞
With
x2 =
p2
,
kB T 2m
we thus get
Z
+∞
−∞
1p
dp −β p2
2mkB T
e 2m =
h
h
Then,
VN
ZN (T, V ) =
N!
Z
+∞
dx e
−x2
=
√
−∞
√
2πmkB T
h
3N
2πmkB T
.
h
.
With the ”thermal wavelength” λT defined as
λT = √
h
2πmkB T
and pT =
2πh̄
h
=
,
λT
λT
one rewrites (9.11) as
1
ZN (T, V ) =
N!
V
λ3T
N
.
(9.11)
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CHAPTER 9. CANONICAL ENSEMBLE
We are able now to calculate the free energy F :
F (T, V, N ) = −kB T ln
1
N!
V
λ3T
1
+ ln
ln
N!
= −kB T
N !
V
λ3T
N !
V
= −kB T −N ln N + N + N ln 3
λT
V
= −N kB T ln
+1 ,
N λ3T
the entropy S:
′
∂F
V
N λ3T
S = −
= N kB ln
−1
+ 1 − N kB T ln
∂T V,N
N λ3T
V
V
1 λT
V
N 2
= N kB ln
+ 1 − N kB T
· λT 3 · −
3
3
N λT
N λT V
2 T
5
V
+
→ Sackur-Tetrode equation,
= N kB ln
3
N λT
2
and the chemical potential µ:
µ=
∂F
∂N
T,V
V
V
λ3T
= −kB T ln
+ 1 + N kB T
·
N λ3T
N λ3T V
V
.
= −kB T ln
N λ3T
The previous expressions were much simpler obtained than when calculated in the microcanonical ensemble.
⋆ In the thermodynamical limit,
N → ∞,
V → ∞,
N
= const,
V
the average value of an observable is, in general, independent of the ensemble chosen and
therefore one usually chooses the ensemble that is easier to work with.
⋆ BUT, the fluctuations of observables, < O2 > − < O >2 , are ensemble dependent! As
an illustration, let us calculate the energy fluctuations in the microcanonical and in the
canonical ensembles.
115
9.7. ENERGY FLUCTUATIONS
9.7
Energy fluctuations
We want to calculate the probability that the energy of the system takes a value Eα .
In the case of the microcanonical ensemble, one has
R
dΓ δ(E − H)δ(Eα − H)
R
ρ(Eα ) =
= δ(Eα − E),
dΓ δ(E − H)
i.e., the probability of finding the system at the energy E = Eα is 1. → no energy
fluctuations.
For the canonical ensemble, we will have energy fluctuations due to the energy transfer
between the system and the heat reservoir. The probability that the system has energy
Eα = E is given by the distribution function:
ρ(E) = hδ(E − H)i .
Considering the case when H ≥ 0, one has for ρ(E):
R
R
e−βE dΓ δ(E − H)
dΓ e−βH δ(E − H)
R
R
=
ρ(E) = hδ(E − H)i =
dΓ e−βH
dΓ e−βH
1 e−βE Ω(E)
e−βE Ω(E)
=
= R∞
.
h3N N ! ZN (T )
dE e−βE Ω(E)
0
⇒ ρ(E) has a maximum at E = E ∗ (Fig. 9.3).
Figure 9.3: Distribution function for the canonical ensemble.
e−βE Ω̃(E)
Z
!
1
∂ Ω̃
∂ρ(E)
=
− Ω̃β e−βE = 0
∂E
Z
∂E
!
1 ∂ Ω̃
1
=
kB T
Ω̃ ∂E E=E ∗
ρ(E) =
116
CHAPTER 9. CANONICAL ENSEMBLE
∂ ln Ω̃ ∂E For ∆ → 0
=
E∗
1
kB T
1
∂S =
∂E E ∗
T
This was the equation in the microcanonical ensemble considered in order to obtain the
temperature for a given energy. Since we have in the canonical ensemble that
< E >= U = −
∂
ln Z
∂β
and with
< E >= U = F + T S
U −F
S=
T
we obtain the same thermodynamical potential as in the microcanonical ensemble then
⇒
E ∗ =< E >
Width of the curve: a measure of the width is the root of the variance σ 2
σ 2 =< E 2 > − < E >2
(exercise) with
σ 2 = kB T 2 C V
The relative width between
√
σ
σ2
1p
kB T 2 C V
=
=
<E>
<E>
U
U ∼N
CV ∼ N
1
√
⇒
N
With growing N the fluctuations away from < E > disappear; for N → ∞, the canonical
and microcanonical ensembles coincide.
σ
=O
<E>
9.8
Paramagnetism
We consider a system with N magnetic atoms per unit volume placed in an external
magnetic field H. Each atom has an intrinsic magnetic moment µ = 2µ0 s with spin
s = 1/2. In a quantum-mechanical description, the magnetic moments of the atoms can
point either parallel or antiparallel to the magnetic field.
117
9.8. PARAMAGNETISM
We want to analyze the mean magnetic moment < µH > per atom (in the direction of H)
as a function of the system’s temperature T . Assuming that the atoms interact weakly,
one can consider a single atom as a small system and the rest of the atoms as a reservoir
(as they were introduced in section 9.1) and then treat the whole problem in terms of the
canonical ensemble.
Each atom can be in two states:
- either (+), when the spin of the atom is parallel to H → µH = µ,
- or (–), when the spin of the atom is antiparallel to H → µH = −µ.
The corresponding energies for the states (+) and (–) are:
ε+ = −µH,
ε− = −(−µ)H = µH.
The probability of finding an atom in the state (+) is
P+ = c e−βε+ = c eβµH
and that of finding it in the state (–) is
P− = c e−βε− = c e−βµH ,
where c is a proportionality constant and β =
1
.
kB T
We see that (–) is the state of higher energy and, correspondingly, it is less likely to find
the atom in this state (with the magnetic moment antiparallel to H).
Let us consider the following dimensionless parameter:
y = βµH =
µH
,
kB T
which measures the ratio of a typical magnetic energy to a typical thermal energy, in two
extreme regimes:
(a) for T >> 1 ⇒ y << 1 and the probability that µ is parallel to H is almost the
same as that of its being antiparallel. In this case, µ is almost completely randomly
oriented so that µ̄H ≈ 0;
(b) for small T ⇒ y >> 1 and it is much more probable that µ is parallel to H
rather than antiparallel to it. In this case, µH = µ.
We can quantitatively obtain results (a) and (b) by calculating the mean value of µH in
the canonical ensemble.
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CHAPTER 9. CANONICAL ENSEMBLE
For a system with discrete energy levels such as the magnetic system under consideration
here, instead of integrating over energies one calculates a statistical sum. The mean value
of an observable x is given in this case as
P −βEr
e
xr
< x >= Pr −βEr .
re
For our observable µ, we then write:
< µH >=
⇒
µ eβµH − µ e−βµH
eβµH + e−βµH
< µH >= µ tanh
where we used that
tanh y =
µH
,
kB T
ey − e−y
.
ey + e−y
We define the “magnetization”, or mean magnetic moment per unit volume, as
< M >= N < µH >
and analyze its behavior in the two regimes of interest.
(a) For y << 1,
ey = 1 + y + . . . ,
e−y = 1 − y + . . . .
Then,
tanh y =
(1 + y + . . .) − (1 − y + . . .)
=y
2
so that
< µH >=
µ2 H
.
kB T
For the magnetic susceptibility χ, defined as < M >= χH, we then have
χ=
N µ2
.
kB T
⇒ At high temperatures compared to the magnetic energies, χ ∝ T −1 which is
known as the Curie law.
(b) For y >> 1,
ey >> e−y ,
tanh y = 1
and hence
< µH >= µ,
< M >= N µ.
9.8. PARAMAGNETISM
119
⇒ At low temperatures the magnetization becomes independent of H and is equal
to the maximum magnetization called saturation magnetization.
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CHAPTER 9. CANONICAL ENSEMBLE