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