Magnetic work

Masatsugu Sei Suzuki

Department of Physics, SUNY at Binghamton

(Date: September 25, 2017)

When we discuss the magnetic work in the thermodynamics, we realize that the internal

energy is expressed in terms of different forms,

MdBTdSdUR , BdMTdSdUK

where the suffixes R and K denote the names of Reif and Kittel, for convenience. There are two

gedanken experiments on the magnetic work, (1) Kittel and (ii) Reif. The expression of the

internal energy is different, depending on the experiments. First we show such gedanken

experiments.

In the quantum mechanics, the mutual energy should be equal to MdB as is expected from

the Zeeman energy. This means that Reif’s expression is close to the prediction from the

quantum mechanics. Note that the origin of the magnetic energy lies in the quantum mechanics,

but not in the classical physics since no work can be done for the Lorentz force.

Here we discuss the validity of the two expressions based on the quantum statistical

mechanics. To this end, we use the simple model of spin 1/2 paramagnet. We show that the

Reis’s expression can be derived from this method.

It is important to stress that the results of the thermodynamic properties based on the Reif

notation are the same as those based on the Kittel’s notation. The thermodynamics of

superconductivity can be discussed mainly in terms of the Kittel’s notation.

1. Gedanken experiment on the magnetic work by Kittel (classical method)

Fig. Magnetic material (cylinder) inside the solenoid with area A1 and distance l.

The magnetic material is put inside the coil. The magnetic field is produced by current

through the coil. The magnetic material can be magnetized.

For a coil having 1n turns per unit length, we apply the Ampere’s theorem.

JHc

4 ,

lHlInc

dc

dd 111

44)(

aJlHaH ,

or

lHlInc

111

4

or

111

4In

cH

Fig. Ampere’s law for the magnetic field of the solenoid

The magnetic flux density B inside the magnetic material is

01 4 MHB

where 0M is the magnetic moment per unit volume. The magnetic flux 0 is given by

BAHAA 21210 )(

When 1I is increased, 1H and 0M cahnges. Then B also changes. Thus the voltage V is

generated inside the coil,

])[(11

21

2110

1dt

dBA

dt

dHAAln

cdt

dln

cV

The work done by the power supply during the time dt,

dBH

VdHH

VV

ldBAH

ldHAAH

dt

dBA

dt

dHAAdt

lH

dt

dBA

dt

dHAAln

cdtI

VdtIdWK

44)(

]4

)(4

])[(4

]))[(1

(

)(

121

121

21

1211

21

211

21

2111

1

where

111

4In

cH

, lAV 11 , lAV 22

Noting that

01 4 MHB

we have

012

2

11

11

211

1

121

121

8

)(44

44)(

dMHVHV

d

dHdBH

VdHH

V

dBH

VdHH

VVdWK

This is the work done on the system consisting of the sample plus field. We use M which is the

total magnetic moment of the sample, the magnetic field H, and volume V as

MMV 02 , HH 1 , VV 1

So we have

HdMVH

ddWK

8

2

(1)

The internal energy is given by

8

2VHdHdMTdSdWTdSdU KK (2)

When we use B instead of H in Eq.(2), we have the final form,

8

2VBdBdMTdSdUK (2’)

2. Gedanken experiment on the magnetic work by Reif

We consider the force exerted by a magnetic field H on a magnetic moment represented by a

small rectangular current loop. There is a net x component of force shown in Fig. below, where I

is the current and M is the magnetic moment of the loop. The force on a large sample can be

regarded as due to the superposition of forces on many such infinitesimal moments.

Fig. ])(

)([1

2 dxx

xHxHIdy

cF

, )(1

1 xIdyHc

F

The magnetic moment is defined by

Idxdyc

IAc

M11

Net x-component for the force

x

HM

dxdyx

H

c

I

dyHc

Idx

x

HHdy

c

IFx

)(

Then the work done by the system is

MdHdxx

HMdxFW xout

The work done on the system (Reif) is

MdHdWdW outR (3)

dx

dy

I

F2F1H x

This is the work done on the system in some given magnetic field. It is the work done on the

system consisting of the sample alone. The internal energy is given by

MdHTdSdUR (4)

When we use B instead of H in Eq.(4), we have the final form

MdBTdSdUR (4’)

or

BdMMBdTdSdUR )(

3. Relation between KU and RU

We now consider the relation between two expressions KU and RU . We introduce *U as

MBUU R * .

Thus we have

BdMTdSdU *

The internal energy KdU can be rewritten in the exact way, as

)8

(

8

2*

2

VBUd

VBdBdMTdSdUK

which leads to

8

2* VB

UUK , MBUUR *

For now we omit the term 8

2VB. Then we have

BdMTdSdUdUK * ,

MdBTdSMBUddUR )( *

4. Thermodynamic relations by Reif

MdBTdSPdVTdSdUR

leading to the corresponding relation

MP , BV

Fig. Born diagram using the definition by Reif; MP , BV

where

P, B (intensive variables)

M, V (extensive variables)

S

B

T

M

U

FG

H

Helmholtz free energy:

STUF RR , MBFG RR

MdBSdT

TdSSdTMdBTdS

STddUdF RR

)(

B

R

T

FS

, T

R

B

FM

Gibbs free energy:

MBFG RR

BdMSdT

BdMMdBMdBSdT

MBddFdG RR

)(

RG is a function of T and M.

((Thermodynamic analysis))

BT T

M

B

S

(Maxwell’s relation)

and

B

T

S

T

S

B

S

BT

BS

TB

TS

BT

SB

BT

ST

SB

ST

B

T

),(

),(

),(

),(

),(

),(

),(

),(

),(

),(

or

BBTB

B

T

S T

M

C

T

B

S

C

T

T

ST

B

ST

B

T

where BC is the heat capacity at fixed B.

((Example))

For the paramagnet, M is given by

BTM )(

with

T

CT )(

where C is the Curie constant. Using the expression of M, we have

2

)(

T

CB

T

TB

B

S

T

<0 (Isothermal process)

This means that in the isothermal process, the entropy decreases with increasing magnetic field.

We also mote that

0

SB

T (Isentropic cooling)

which means that the temperature decreases with decreasing B during the adiabatic process.

5. Thermodynamic properties by Kittel (Stanley)

BdMTdSPdVTdSdUK

leading to the corresponding relation

BP , MV

Fig. Born diagram using the definition by Kittel; BP , MV

S

M

T

B

U

FG

H

where

P, B (intensive)

M, V (extensive)

Helmholtz free energy:

STUF KK ,

BdMSdT

TdSSdTBdMTdS

STddUdF KK

)(

M

K

T

FS

, T

K

M

FH

KF is a function of T and M.

Gibbs free energy:

MBFG KK

MdBSdT

BdMMdBBdMSdT

MBddFdG KK

)(

KG is a function of T and B.

B

K

T

GS

, T

K

B

GM

((Maxwell relation))

BT T

M

B

S

, MT T

B

M

S

MS S

B

M

T

, HS S

M

B

T

(a) Isothermal process

Maxwell’s relation:

BT T

M

B

S

(2)

For the paramagnet, M is given by

BTM )(

with

T

CT )(

where C is the Curie constant. Using the expression of M, we have

2

)(

T

CB

T

TB

B

S

T

<0

This means that in the isothermal process, the entropy decreases with increasing magnetic field.

(b) Isentropic process

TB

T

B

T

B

S

B

S

C

T

B

S

T

ST

T

B

S

T

S

BT

ST

BT

BS

SB

ST

B

T

1

),(

),(

),(

),(

1

,(

),(

where CB is the heat capacity at constant B and is positive and 0

TB

S. Thus we have

SB

T

which means that the temperature decreases with decreasing B during the adiabatic process.

So we note that the Maxwell’s relation (1) for the Reif type is the same as that (2) for the Kittel

type.

6 Magnetic work derived from quantum statistics (MacDonald)

The magnetic work is equal to zero in classical physics since the scalar product of the

Lorentz force and the displacement vector is always equal to zero. On the other hand, in quantum

mechanics, the magnetic work is equal to a Zeeman energy given by a form BM . In this

sense, the magnetic work should be treated in terms of the quantum mechanics. To this end we

consider a simple case such as a paramagnet where the electron with the spin (S = 1/2) is under

the presence of an external magnetic field B.

There are two states (up state and down state) of the magnetic moment B ). Both states are

assumed to be nondegenerate. We apply an external magnetic field B in the up direction. So that

a positive magnetic moment has an negative energy BB , while a nagative magnetic moment

has a positive magnetic energy BB . The energy difference is BB2 .

The probability for finding the system with the positive magnetic moment is

)exp()exp(

)exp(

BB

BP

BB

B

The probability for finding the system with the negative magnetic moment is

)exp()exp(

)exp(

BB

BP

BB

B

The internal energy U is given by

)tanh(

)exp()exp(

)]exp()[exp(

)(

BB

BB

BBB

PBBPU

BB

BB

BBB

BB

The magnetization M:

)tanh(

)exp()exp(

)]exp()[exp(

)(

B

BB

BB

PPM

BB

BB

BBB

BB

The entropy S:

)tanh()]cosh(2ln[

)lnln(

BBkBk

PPPPkS

BBBBB

B

Then we have

)]}cosh(2ln[(

)]}cosh(2ln[()tanh({)tanh(

BTk

BBBTkBB

STUF

BB

BBBBBB

In another approach, using the partition function Z for the canonical ensemble defined by

)cosh(2

)exp()exp(

B

BBZ

B

BB

we can also calculate the Helmholtz energy F, the entropy S, and the magnetization M as follows.

The Helmholtz free energy:

)]cosh(2ln[ln BTkZTkF BBB

The entropy S:

)}tanh()]cosh(2{ln[ BBBkT

FS BBBB

B

The magnetization M:

)tanh( B

B

FM

BB

T

The expressions of S and M thus obtained are the same as those derived above. Since F is a

function of B and T, dF can be expressed by

dTT

FdB

B

FdF

BT

The derivatives TB

F

and BT

F

can be evaluated as

MBB

FBB

T

)tanh(

S

BBkBkT

FBBBBB

B

)tanh()]cosh(2ln(

Thus we have

SdTMdBdF

The internal energy U is related to the Helmholtz free energy F as

STUF

We now consider the Gibbs free energy defined by

MBFG

leading to the relation

BdMSdT

BdMMdBMdBSdT

MBddF

MBFddG

)(

)(

So the Gibbs free energy G depends on T and M,

dMM

GdT

T

GdG

TM

We now derive the expression of magnetic field B in terms of M and T. To this end, we start with

1)2exp(

1)2exp()tanh(

B

BB

M

B

BB

B

The magnetic field B can be uniquely obtained as

)]1ln()1[ln(2

1

1

1

ln2

1

ln2

1

BBB

B

B

B

B

B

B

MM

M

M

M

MB

Since

M

MB

B

BB

)exp(

F can be also evaluated as

)]1)(1ln[(2

12ln

1

]

)1)(1(

2ln[

1

]))((

2ln[

1

]ln[1

)]exp()ln[exp(1

BB

BB

BB

B

B

B

B

B

BB

MM

MM

MM

M

M

M

M

BBF

The Gibbs free energy is obtained as

)]1ln()1()1ln()1[(2

12ln

1

1

1

ln2

1)]1)(1ln[(

2

12ln

1

BBBB

B

B

BBB

MMMM

M

M

MMM

MBFG

or

)]1ln()1()1ln()1[(2

12ln

1

BBBB

MMMMG

Thus we have

B

MM

M

G

BBBT

)]1ln()1[ln(2

1

and

S

MMMMkk

Gk

T

G

BBBB

BB

M

B

M

)]1ln()1()1ln()1[(2

12ln

2

Then dG can be expressed by

BdMSdT

dMM

GdT

T

GdG

TM

Here we note that

B

B

B

BB

B

B

B

B

B

B

B

B

B

BB

B

B

B

B

BBB

BBB

B

M

M

M

MM

M

M

M

M

M

M

M

M

MM

M

M

M

M

MBBB

BBBk

S

1

1

ln]

11

2ln[

1

1

ln]

1

1

1

1

ln[

ln]ln[

})]exp()ln[exp(

)tanh()]cosh(2ln[

or

)]1ln()1()1ln()1[(2

12ln

BBBB

BB

MMMMkkS

Using the relations

BdMSdTdG

SdTMdBdF

MBSTUMBFG

we get

MBSTGU

and

MdBTdS

BdMMdBTdSSdTBdMSdT

MBdSTddGdU

)()(

In summary, it is reasonable to conclude that from a view point of quantum statistical physics,

MdBTdSdU and BdMSdTdG

The expression for dU is the same as that derived by Reif and the expression for dG is the same

as that derived by Kittel. We agree with the comment by Robertson that the magnetic work done

on the system by the mutual energy must be MdB

7. Problem and solution (1)

K. Huang

Introduction to Statistical mechanics, second edition (CRC Press, 2010)

((Problem 3-8))

Thermodynamic variables for magnetic system are B, M, T, where B is the magnetic field, M is

the magnetization, and T the absolute temperature. The magnetic work is

BdMdW ,

and the first law states BdMTdSdU . The equation of state is given by Curie’s law

T

BCM ,

where C (>0) is the Curie constant. This is valid only at small B and high T. The heat capacity at

constant B is denoted by BC , and that at constant M is MC . Many thermodynamics relations can

be obtained from those of a PVT system by the correspondence PB , MV

(a) Show

M

MT

UC

, BB

BT

MB

T

UC

(b) From the analog of the energy equation and Curie’s law, show that

0

TM

U

This is the analog of the statement that the internal energy of the ideal gas is independent of the

volume.

(c) Show

C

M

T

CBCC MB

2

2

2

____________________________________________________________________________

Here we use the correspondence relations such that

BP (intensive quantity)

MV (extensive quantity)

PdVTdSdU HdMTdSdU

____________________________________________________________________________

((Solution))

(a)

HdMTdSdU

The heat capacity:

M

MT

STC

,

HHH

HT

MH

T

U

T

STC

(b)

MTT T

HTH

M

STH

M

U

where we use the Maxwell’s relation:

MT T

H

M

S

M is given by the Curie law as

S

M

T

B

U

FG

H

BT

CM

Noting that

T

B

C

M

T

B

M

we have

0

MT T

BTB

M

U

(c)

T

M

M

B

M

BT

MS

T

BT

MT

BT

MS

TMT

MST

T

STC

),(

),(

),(

),(

),(

),(

),(

),(

or

T

BT

B

BTTB

T

TB

TB

T

M

B

M

T

M

B

ST

T

ST

T

M

B

S

B

M

T

S

B

M

T

B

M

T

M

B

S

T

S

B

M

TC

][

Using the Maxwell’s relation

BT T

M

B

S

Then we get

T

BBM

B

M

T

MT

CC

2

For the paramagnetic system,

BT

CM

2

2

T

CBCC BM

or

C

M

T

CBCC MB

2

2

2

8. Problem and solution (2)

K. Huang

Introduction to Statistical mechanics, second edition (CRC Press, 2010)

((Problem 3-9))

Define the free energy ),( TMF and Gibbs energy ),( TBG of a magnetic system by analogy

with the PVT system.

(a) Show

BdMSdTdF , MdBSdTdG

(b) Show

BT T

M

B

S

hence

2T

CB

B

S

T

.

(c) With the help of the chain rule, show

TC

CB

B

T

BS

((Solution))

Helmholtz free energy F

Gibbs free energy G

HdMTdSPdVTdSdU

STUF , HMFPVFG

where BP , MV (Kittel)

(a)

BdMSdT

TdSSdTBdMTdS

TdSSdTdU

STUddF

)(

S

M

T

B

U

FG

H

MdBSdT

MdBBdMBdMSdT

BMFddG

)(

(b)

BT

GS

, TB

GM

B

TB

BTT

T

M

B

G

T

BT

G

TB

G

T

G

BB

S

2

2

Since

BT

CM ,

2T

CB

T

M

B

2T

CB

B

S

T

(b)

Noting that

1),(

),(

),(

),(

),(

),(

TS

TB

BT

BS

SB

ST

we get the relation

1

TBS S

B

T

S

B

T

Since

CB

T

B

SS

B

T

T

21

B

BB

CT

Q

T

ST

we have

TC

CB

T

CT

CB

T

S

B

S

B

S

S

T

B

T

BB

B

T

TBS

2

___________________________________________________________________________

REFERENCES

C. Kittel and H. Kroemer, Thermal Physics, second edition (W.H. Freeman and Company, 1980).

F. Reif, Fundamentals of statistical and thermal physics (McGraw-Hill, 1965).

H.E. Stanley, Introduction to Phase Transition and Critical Phenomena (Oxford, 1971).

H.S. Robertson, Statistical Thermodynamics (PTR Prentice Hall, 1993).

D.K.C. MacDonald, Introductory Statistical Mechanics for Physicists (Dover, 2006).

H.B. Callen, Thermodynamics (John Wiley, 1960).