paramagnetism: curie law masatsugu sei suzuki department...
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Paramagnetism: Curie law
Masatsugu Sei Suzuki
Department of Physics, SUNY at Binghamton
(Date: 9-13-16)
In a paramagnet, the magnetic moments of electron spins are not aligned. The direction of the
magnetic moment is random. So the total magnetization is zero. However, when the magnetic
field is applied, the situation changes. The magnetization (the magnetic moment per volume) M
appears along the direction of the magnetic field, and obeys the Curie law,
BT
CB
Tk
NBNM
B
BBB
2 .
The thermodynamic and magnetic properties of the paramagnet are discussed using two
approaches (the microcanonical ensemble and the canonical ensemble). The entropy is described
by a scaling function of only a variable TB / . This property of the entropy is used for the cooling
of the system (isentropic demagnetization).
1. Approach from the microcanonical ensemble
We consider the electron spin system with two energy levels in the presence of an external
magnetic field B along the z axis. The spin magnetic moment μ is given by
σS
μ BB ℏ
2,
where S (= )2
σℏ
is the spin angular momentum, mc
eB
2
ℏ (>) is the Bohr magneton, and the
charge of electron is –e (e>0). In the presence of the magnetic field along the z axis, we have a
Zeeman energy given by
z )( BBB BσBμ
Noting that zzz and zzz in quantum mechanics, the energy level splits
into two levels, BB .
(a) The energy BB (higher level),
The spin state z . The spin magnetic moment is antiparallel to the z-axis )( B .
state.
(b) The energy BB (lower level).
The spin state: z . The spin magnetic moment is parallel to the z-axis )( B ; state.
Suppose that
NNN
N: the total number of spins.
N the number of spin magnetic moment parallel to the magnetic field.
N the number of spin magnetic moment antiparallel to the magnetic field.
The total magnetic moment M is
sNNM BBB 2 .
The total energy E is given by
)( NNBE B
Here we introduce the excess spin number s.
sNN 2 .
Then we have
mBB, -mB
-mBB, mB
0
sN
N 2, s
NN 2
.
So the energy E is expressed by
BsE B2
The number of ways for choosing N spins and
N spins out of N spins (identical) is given by
)!2
()!2
(
!!),(
sN
sN
N
NN
NsNW
.
Using the Stirling relation, the entropy can be evaluated as
)]2
ln()2
()2
ln()2
(ln[
]2
)2
ln()2
(2
)2
ln()2
(ln[
])!2
ln()!2
ln(![ln
),(ln
sN
sN
sN
sN
NNk
sN
sN
sN
sN
sN
sN
NNNk
sN
sN
Nk
sNWkS
B
B
B
B
or
)]22
ln()22
(
)22
ln()22
(ln[),(
B
EN
B
EN
B
EN
B
ENNNkENS
BB
BB
B
Using the relation,
]ln2
),(1
BNE
EBN
B
k
E
ENS
T B
B
B
B
we have
BBNE BB tanh
Note that the total magnetization is related to the total energy as
MBE
Then we have
BNM BB tanh
When 1BB , we get
BT
CB
Tk
NBNM
B
BBB
2
showing the Curie law, where
B
B
k
NC
2 .
((Note))
xx
xx
ee
eex
tanh
For 1x , xx tanh .
Fig. Scaling plot of the magnetization. The saturation magnetization is NB; y = 1.
2. Approach from the canonical ensemble
(a) Thermodynamic properties: F, G, E and S
The N- partition function is given by
N
CCN ZZ 1
where N is the number of spins. 1CZ is the one-particle partition function and is given by
)cosh(2)exp()exp(1 BBBZ BBBC .
The Helmholtz free energy F:
)]cosh(2ln[lnln 1 BTNkZTNkZTkSTEF BBCBCNB
Since SdTPdVdF , the pressure P is obtained as
0
TV
FP
The internal energy E:
)tanh(lnln 1 BBNZNZE BBCCN
1 2 3 4 5 6x=
mB B
kB T
0.2
0.4
0.6
0.8
1.0
y=M
NmB
(b). Magnetization M
In the present system, the Gibbs energy is equal to F (M. Kardar, Statistical Physics
of Particles, Cambridge, 2007, p.117). We note that
SdTVdPdG
Here we use the replacement ;
BP , MV
Then we have
SdTMdBdG
So the magnetization is given by
)tanh(ln BNZB
TkB
F
B
GM BBCNB
This expression can be directly derived from a view point of quantum mechanics (see
the Appendix). The magnetization M can be also directly derived from the definition as
)( PPNM BB
where P and P are probabilities given by
BB
B
BB
B
ee
eP
,
BB
B
BB
B
ee
eP
Then we have
)tanh()(
[ BNee
eeNM BBBB
B
B
B
B
BB
BB
In the limit of 0Tk
B
B
B ,
T
B
k
N
Tk
BNM
B
B
B
BB
2 ,
showing the Curie law. Note that
xx
xx
ee
eex
tanh ,
and for 10 x , )(15
2
3tanh 75
3
xOxx
xx
Fig. Scaling plot of the magnetization. The saturation magnetization is NB; y = 1.
(c) Entropy S
The entropy S:
ln( ) ( ln )CN
B B CN
ZU FS k U F k Z
T T
or
[ln(2cosh( ) tanh( )]B B B BS k N B B B
or
1 2 3 4 5 6x=
mB B
kB T
0.2
0.4
0.6
0.8
1.0
y=M
NmB
ln[2cosh( )] tanh( )B B B
B B B B
B B BS
k N k T k T k T
.
The entropy S is expressed by a scaling function of B/T. This is an essential point to this
system.
We introduce the characteristic temperature T0 and magnetic field B0 as
00 TkB BB
Then we have
)tanh()]cosh(2ln[
)tanh()]cosh(2ln[
t
b
t
b
t
b
Tk
B
Tk
B
Tk
B
Nk
S
B
B
B
B
B
B
B
where
0B
Bb ,
0T
Tt
t
b
T
TTk
B
BB
Tk
B
B
B
B
B
0
0
0
0
We make a plot of Nk
S
B
as function of t, where b is changed as a parameter. In the limit
of t , the entropy reached
2ln)12ln( sNk
S
B
0.693147.
Fig. Plot of Nk
S
B
as a function of a reduced temperature t (= T/T0), where the reduced
magnetic field b (= B/B0) is changed as a parameter. Note that 00 TkB BB . The
highest value of y is ln2 = 0.693147.
3. Proof of 0Nk
S
B
in the limit of 0t
Noting that 1/ tbe and 1/ tbe , we can approximate the expression of entropy
as
tb
tbtb
tb
tbtbtb
tbtb
tbtbtbtb
B
et
b
et
be
t
b
e
e
t
bee
ee
ee
t
bee
t
b
t
b
t
b
Nk
S
/2
/2/2
/2
/2/2/
//
////
2
21
1
1)]1(ln[
)ln(
)tanh()]cosh(2ln[
where we use the approximation
b=1
23
10
2 4 6 8 10x=t
0.1
0.2
0.3
0.4
0.5
0.6
0.7
y=S
kB N
xx )1ln( , xx
x21
1
1
for 10 x
In the limit of 0t , we have
0lim0
S
t
Fig. The entropy of )/( BNkS as a function of t. b = 1. The system settles into ground
state; the multiplicity becomes 1: and the entropy goes to zero.
4. Isentropic demagnetizion
The principle of magnetically cooling a sample is as follows. The paramagnet is first
cooled to a low starting temperature. The magnetic cooling then proceeds via two steps.
Suppose that the spin system is kept at temperature T1 in the presence of magnetic
field B1. The system is insulated (S = 0) and the field removed, the system follows the
constant entropy path AB, ending up at the temperature T2 (isentropic process). If B is
the effective field that corresponds to the local interactions, the final temperature T2
reached in an isentropic demagnetization process is
1
12
B
T
B
T
.
since the entropy is a function of only B/T.
t
0.10 0.50 1
1013
1010
107
104
101
Fig. Point A (tA = 4.29726, yA = 0.3) on the line with 10
B
BA . Point B (tB= 0.859452, ,
yA = 0.3) on the line with 50
B
BB . The path AB is the isentropic process (y = 0.3).
Note that
B
B
A
A
B
t
B
t .
b=1 b=5
AB
2 4 6 8 10x=t
0.1
0.2
0.3
0.4
0.5
0.6
0.7
y=S
kB N
5. Specific heat
The heat capacity is given by
)(sec)( 22
Tk
Bh
Tk
B
Nk
C
B
B
B
B
B
Using the energy gap parameter
Bk BB 2
2/
/22222
)1()(
4
1)
2(sec)(
4
1)
2(sec)
2(
T
T
B e
e
TTh
TTh
TNk
C
Fig. Plot of the heat capacity C/kB as a function of T/. It show a peak at T/ =
0.416778.
The heat capacity as a function of temperature, has a peak at
416778.0T
.
((Schottky anomaly))
The Schottky anomaly is an observed effect in solid state physics where the specific
heat capacity of a solid at low temperature has a peak. It is called anomalous because the
heat capacity usually increases with temperature, or stays constant. It occurs in systems
with a limited number of energy levels so that E(T) increases with sharp steps, one for
each energy level that becomes available. Since Cv =(dE/dT), it will experience a large
peak as the temperature crosses over from one step to the next.
REFERENCES
M. Kardar, Statistical Physics of Particles (Cambridge University Press, 2007)
C. Kittel, Introduction to Solid State Physics, Eighth edition (John Wiley & Sons, 2005).
R. Baierlein, Thermal Physics (Cambridge, 2001).
A. Rex, Finn’s Thermal Physics (CRC Press, 2017).
_______________________________________________________________________
APPENDIX Quantum mechanical treatment for the expression of Magnetization
The spin Hamiltonian for the spin 1/2 system in the presence of a magnetic field
along the z axis, is given by the Zeeman energy as
0.5 1.0 1.5 2.0 2.5 3.0x=T
D
0.1
0.2
0.3
0.4
y=C
NkB
zBB BH
ˆ)ˆ2
(ˆˆ BSBμℏ
where zBB
ˆˆ2
ˆ Sμℏ
is the spin magnetic moment operator. The on-particle
partition function is given by
][][ˆˆ
1zBBH
C eTreTrZ
We can evaluate the average magnetic moment as
zC
B
zBC ZeTrZB
zB
1
ˆ
1 ]ˆ[1
or
1ln1
Cz ZB
The magnetization for the N particle system, we have
1ln CBz ZB
TNkNM
.
_____________________________________________________________________
Bohr magneton
27400915.9B x 10-21 emu
B
B
k
6.71713 x 10-5 (K/Oe)
Nuclear magneton
050783699.5N x 10-24 emu
B
N
k
3.65826 x 10-8 (K/Oe)
scaling
universality