Introduction to Solid State Physics

SEVENTH EDITION

Introduction to

Solid State Physics

CHARLES K IT TEL

14

Diamagnetism and Paramagnetism LANGEVIN DIAMAGNETISM EQUATION

417

QUANTUM THEORY OF DIAMAGNETISM OF

MONON UCLEAR SYSTEMS

419

PARAMAGNETISM

420

QUANTUM THEORY OF PARAMAGNETISM 420

Rare earth ions 423

Hund rules 424

Iron group ions 425

Crystal field splitting 426

Quenching of the orbital angular momentum 426

Spectroscopie splitting factor 429

Van Vleck temperature-independent paramagnetism 430

COOLING BY ISENTROPIC DEMAGNETIZATION Nuclear demagnetization

431

432

PARAMAGNETIC SUSCEPTffiILITY OF

COND UCTION ELECTRONS

433

SUMMARY

436

PROBLEMS

436

1. Diamagnetic susceptibility of atomic hydrogen 2. Hund rules 3. Triplet excited states 4. Heat capacity from internaI degrees of freedom 5. Pauli spin susceptibility 6. Conduction electron ferromagnetism 7. Two-Ievel system 8. Paramagnetism of S 1 system

=

REFERENCES

436

437

437

438

438

438

440

440

440

NOTATION : In the problems treated in this chapter the magnetic field B is always closely equal to the applied field Ba, so that we write B for Ba in most instances.

t

+ -------Or---------~T_--------------------------------

Pauli paramagnetism (metals)

Temperature

Diamagnetism

Figure 1 Characteristic magnetic susceptibilities of diamagnetic and paramagnetic substances. 416

CHAPT ER

14: DIAMAGNE T ISM AND PARAMAGNE TISM

Magnetism is inseparable from quantum m echan ics, for a strictly classical system in thermal equilibrium can display no magnetic moment, even in a magnetic field . The magnetic moment of a free atom has three p rincipal sources: the spin with which electrons are endowed; their orbital angular mo­ mentum about the nucleus; and the change in the orbital moment induced by an applied magne tic field. The first two effects give paramagnetic contributions to the magnetization , and the third gives a diamagne tic contribution . In the ground Is state of the h ydrogen atpm the orbital moment is zero , and the magnetic moment is that of the electron spin along with a small induced diamagne tic moment. In the 1S2 state ofhelium the sp in and orbital moments are both zero , and there is only an induced moment. Atoms with filled electron shells have zero spin and zero orbital moment: these moments are associated with unfilled shells. The magnetization M is defined as the magnetic moment per unit volume. The magnetic susceptibility p e r unit volume is defined as M

(Ce S)

x = 13 '

(SI) X = /-LoM B

(1)

where B is the macroscopic magne tic field intensity. In both systems of units X is dimensionless. We shall sometimes for convenience refer to MIB as the sus­ ceptibility without specifying the syste m of units . Quite frequ e ntly a susceptibility is defi ned refe rred to unit mass Or to a mole of the substance . The molar susceptibility is written as XM ; the magnetic moment per gram is sometimes writte n as CT. Subs tances with a negative mag­ netic susceptibility are called diamagnetic. Substances with a positive suscepti­ bility are called paramagnetic, as in Fig. 1. O rdered arrays of magn etic moments are discussed in Chapter 15; the arrays may be fe rromagne tic, ferrimagnetic, antiferromagne tic, helical, or more complex in fo rm. N uclear magnetic moments give rise to nuclea r paramagne tism . Magnetic moments of nuclei are of th e order of 10- 3 times smalle r than the magnetic momen t of th e electron. LANGEVIN DIAMAGNETISM EQUATION

Diamagnetism is associated with the tendency of electrical charges par­ tially to shield the in terior of a body from an applied magnetic field. In electro­ magne tism we are fam iliar with Lenz's law: when the fl ux th rough an electrical circuit is changed, an induced current is set up in such a direction as to oppose the flux change . 417

418

In a superconductor or in an electron orbit within an atom, the induced current persists as long as the field is present. The magnetic fie ld of the induced current is opposite to the applied field, and the magnetic moment associated with the current is a diamagnetic moment. Even in a normal metal there is a diamagnetic contribution from the conduction electrons, and this diamag­ netism is not destroyed by collisions of the electrons. The usual treatment of the diamagnetism of atoms and ions employs the Larmor theorem : in a magnetic field the motion of the electrons around a central nucleus is , to the first order in B, the same as a possible motion in the absence of B except for the superposition of a precession of the electrons with angular frequency

(ces)

(SI)

w = eB/2mc

w

= eB/2m

.

(2)

If the field is applied slowly, the motion in the rotating reference system will be the same as the original motion in the rest system before the application of the field . If the average electron current around the nucleus is zero initially, the application of the magnetic field will cause a finite current around the nu­ cleus. The current is equivalent to a magnetic moment opposite to the applied field. It is assumed that the Larmor frequency (2) is mu ch lower than the fre­ q uency of the original motion in the central field . This condition is not satisfied in free carrier cyclotron resonance, and the cyclotron frequency is twice the freq uency (2). The Larmor precession of Z electrons is equivalent to an electric current 1 = (charge)(revolutions per unit time)

(SI)

= (-

eB) Ze) ( - 1 . - . 271' 2m

(3)

The magnetic moment I.L of a current loop is give n by the product (current) X (area of the loop). The are a of the loop of radius p is 7TP'2. We have

(S I)

JI.

=-

ZtfB 4m (p'l) ;

(4)

Here (p'2) = (x'2) + (y'2) is the mean square of the perpendicular distance of the electron from the field axis thro tigh the nucleus. The mean square distance of the electrons from the nucleus is (r'2) = (x 2) + (y2) + (Z2). For a spherically symmetrical distribution of charge we have (x 2) = (y2) = (Z2), so that (r 2) = i(p2). From (4) the diamagnetic susceptibility per unit volume is, if N is the number of atoms per unit volume, 2 = NI.L = _ NZe (r'2) (ces) (5) X B 6mc'2 '

14

2

x = ILQNIl- =

(SI)

Diamagnetism and Paramagnetism

ILQNZe (r 2 )

B

6m

This is the classical Langevin result. The problem of calculating the diamagnetic susceptibili ty of an isolated atom is reduced to the calculation of (r 2 ) for the electron distribution within the atom . The distribution can be calculated by quantum mechanics . Experimental values for neutral atoms are most easily obtained for the inert gases. Typical experimental values of the molar susceptibilities are the following :

XM in CGS in 10- 6 cm3lrnole:

He

Ne

Ar

Kr

Xe

-1.9

-7 .2

-19.4

-28.0

-43.0

In dielectric solids the diamagnetic contribution of the ion cores is de­ scribed roughly by the Langevin result. The contribution of con duction elec­ trons is more complicated, as is evident from the de H aas-van Alphen effect discussed in Chapter 9. QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTE MS

From (G . 18) the effect of a magnetic field is to add to the hamiltonian the terms

ieh e2 J-C = -(V' . A + A· V') + - A2 2mc 2mc2

,

(6)

for an atomic electron these tenns may usually be treated as a small perturba­ tion . If the magnetic field is uniform and in the z direction , we may write A x = -~yB ,

hB,

Ay =

Az = 0 ,

(7)

and (6) becomes

iehB(d

J-C = - - x2mc

dy

d)

2

B 2

e - (x 2 + y2) - y- + 2 dx

8mc

(8)

The first term on the right is proportional to the orbital angular mUlnen­ tum component Lz if r is measured from th e nucleus. In mononuclear syste ms this term gives rise only to paramagnetism . The second term gives fo r a spheri­ cally symmetric system a contribution 2

E' =

2

e B2 (r 2 ) -1 n1C 2

(9) '

419

moment is

The netic:

in

with

ta

is

in:

lar oxygen and organic """ Ly, Lz are constant. In quantum theory one angular momentum component, usually taken as Lz, and the square of the total orbital angular momentum L2 are constant in a central field. In a noncentral field the plane of the orbit will move about; the angular momentum components are no longer constant and may average to zero. In a crystal Lz will no longer be a constant of the motion, although to a good approximation L2 may continue to be constant. When Lz averages to zero, the orbital angular momentum is said to be quenched. The

14 Diamagnetism and Paramagnetism

@

@

@

===== P"Py

y

y

@

®

®

(a)

(b)

(c)

-

---pz

(d)

Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis. In the free atom the states mL = ± l, 0 have identical energies-they are degenerate. In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them, as in (b) and (c). The wavefunctions that give rise to these charge densities are of the form zf(r), xf(r) and yf(r) and are called the Pz, Px, Py orbitaIs, respectively. In an axially symmetric field, as shown, the Px and Py orbitaIs are degenerate. The energy levels referred to the free atom (dotted !ine) are shown in (d). If the electric field does not have axial symmetry, ail three states will have different energies.

magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S). In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z; the orbital magnetic moment is quenched if the me­ chanical moment Lz is quenched. When the spin-orbit interaction energy is introduced, the spin may drag sorne orbital moment along with it. If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments, the total magnetic mo­ ment will be larger than for the spin alone, and the g value will be larger than 2. The experimental results are in agreement with the known variation of sign of the spin-orbit interaction: g > 2 when the 3d shell is more than half full, g = 2 when the shell is half full , and g < 2 when the shell is less than half full . We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus, the whole being placed in an inhomogeneous crystalline elec­ tric field. We omit electron spin. In a crystal of orthorhombic sym metry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form ecp = AX2

+ B y2 - (A + B )Z2 ,

(24)

where A and B are constants. This expression is the lowest degree polynomial in x, y, z which is a solution of the Laplace equation V2 cp = 0 and compatible with the symmetry of the crystal.

427

428

Uz = zf(r)

Uy = yf(r) ;

are normalized. =

2Ui

,

=

0 .

Consider

dx dy dz ;

(28)

the integral the diagonal matrix elements:

dx dy dz

+

(29)

where

dx

The

dz ;

their angular lobes

o. This effect is momentum, age is zero in magnetic moment also

ParamilgnetÎttm

(30)

- À/à l

the

,

hetween

g g

'See L. Orgel, Introduction to transition references are given by D. Sturge, Phys.

1966; extensive

430

Van Vleck Temperature-Independent Paramagnetism We conside r an atomic or molecular system which has no magnetic mo­ ment in the ground state, by which we mean that the diagonal matrix element of the magnetic moment operator J.Lz is zero. Suppose that there is a nondiagonal matrix element (slJ.LzIO) of the magnetic moment operator, connecting the ground state with the excited state s of energy  = Es - Eo above the ground state. Then by standard perturbation theory the wavefunction of the ground state in a weak field (J.LzB ~ Â) becomes

°

(32)

and the wavefunction of the excited state becomes (33)

The perturbed ground state now has a moment (34)

and the upper state has a moment (35)

There are two interesting cases to consider: Case (a). Â ~ kBT. The surplus population in the ground state over the excited state is app roximately equal to NÂ/2kB T, so that the resultant magneti­ zation is M = 2BI(slJ.LzIO)1

2

NÂ

2k B T '

Â

(36)

which gives for the susceptibility (37) Here N is the number of molecules per unit volume. This contribution is of the usuaI Curie form , although the mechanism of magnetization here is by polariza­ tion of the states of the system, whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states. We note that the splitting  does not enter in (37). Case (h) .  ;? kB T . Here the population is nearly aIl in the ground state, so that M = 2NBI(slJ.LzIO>1 Â

2

(38)

The susceptibility is (39)

Diamagnetism

type of contribution

Paramagnetism

known as Van Vleck

COOLING DY

The first metbc,d

the partly lined is also lowered if

1) .

in

3The method was suggested by P Debye, Ann. Am, Chem, Soc, 49, 1864 (1927). For many purposes dilution which operates He' play the raIe of atoms in a gas, and 12.

Giauque, by the solution in

SUI)pla'ntt~d

431

432

Total Spin Spin Time-

Lattice Before

1

TimeBe ore

New equili brium

Time at which magnetic field is removed

:\cw equilibrium

Time at which

magnetic field

is l'emoved

Figure 7 During adiabatic dem agnetization the total entropy of the specimen is constant. For effective cooli ng the initi al entropy of the lattice should be small in comparison with the entropy of the spi n sys tem.

The steps carried out in the cooling process are shown in F ig. 8. The field is applied at temperature T l with the specimen in good thermal contact with the surroundings, giving the isothermal path ab. The specimen is then insu­ lated (!la- = 0) and the fi eld removed; the specimen follows the constant en­ tropy path he, ending up at temperature T 2 . The thermal contact at Tl is pro­ vided by helium gas, and the thermal contact is broken by removing the gas with a pump.

Nuclear Demagnetization The population of a magne tic sublevel is a function only of f.LB l kBT , hence of BIT. The spin -system entropy is a function only of the population distribu­ tion ; hence the spin entropy is a function only of BIT. If Bt>. is the effective field that corresponds to the local interactions, the final temperature T 2 reached in an adiabatic demagnetization experiment is 1

T 2 = Tl (Bt>.IB ) ,

1

(41)

whe re B is the initial field and Tl the initial temperature. Because nuclear magne tic moments are weak, nuclear magnetic interac­ tions are much weaker than similar electronic interactions. We expect to reach a tem pe rature 100 times lower with a nuclear paramagnet than with an electron paramagnet. The initial temperature Tl of the nuclear stage in a nuclear spin­ cooling experiment must be lower than in an electron spin-cooling experiment. If we start at B = 50 kG and Tl = 0.01 K, then f.LBlkBT l = 0.5, and the e n­

14

Diamagrwtism and Paramagfletism

0.7,r---------------------------------------------------------~ 0.6 ~ ~ ~ 0.5

§ S ~4

B = 0; BA = 100 gauss

~

g 0. 3 ~

~

S Qi ~

0.1

o6

L

~

10

15

do

~5' j'J

T, mK

·' igure 8 Entropy for a , pin 1 sys tem as a fun etion of te mpera ture, assumin g an intern aI random magne tic field Be:. of 100 gauss. The specimen is magnetized iso th e rmall y along ab , and is th en insulated thermally. The external magnetie field is turned off al ong be. In ord e r to keep the figure on a reasonable seale the initial tem pe rature Tl is lower th an wo uId be us ed in practice, and so is the exte rn al magnetic fi eld .

tropy decrease on magnetization is ove r 10 percent of the maxim um spin en­ tropy . This is sufficient to overwhelm the lattice and from (4 1) we estimate a final te mperature T 2 = 10- 7 K. The first 4 nuclear cooling experiment was car­ ried out on Cu nudei in the metal, starting from a fi rst stage at about 0.02 K as attained by electronic cooling. The lowest tempe rature reached was 1.2 x 10- 6 K. The results in Fig. 9 fit a line of the fonn of(41) : T z = T 1(3.1 /B) with B in gauss, so that B11 = 3.1 gauss. This is the effective interaction field of the mag­ netic moments of the Cu nuclei. The motivation for using nud ei in a metal is that conduction electrons help e nsure rapid thermal contact of lattice and nu­ dei at the tempe rature of the first stage . The present record5 for a spin temper­ ature is 280 pK, in rhodium. PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS We are going to try to show how on the basis of these stati sti cs the fa ct th at many me tals are diamagnetic, or only weakl y paramagnetic, can be brought into agree­ me nt with tb e existence of a magnetic mom e nt of tbe e lectrons . W. Pauli, 1927

Classical fr ee electron theory gives an un satisfactory account of the para­ magnetic susceptibility of the conduction electrons. An electron has associated with it a magne tic mom ent of one Bohr magneton , /-La. One might expect that 4N . Kurti , F . N . H . Robinson, F. E. Simon, and D . A. Spohr, Nature 178 , 450 (1 956); for reviews see N. · Kurti, Cryogenies 1, 2 (1960); Adv. in Cryogenie Engineering 8, 1 (1963). sp. J. Hakonen e t al ., Ph ys . Rev. Lett. 70, 2818 (1993).

433

434 Initial magnetic field in kG

lonr---T5--------~lrO--------~20~---3TO~ 9

8

7

1

6

~

5

e

4

.,1;0 u

Ë .S

3

lL-__L-~~~~~~--------~--~

0.3

0.6

2

Initial BIT in 106 G/K

Figure 9 Nuclear demagnetizations of copper nuclei in the metal, starting from 0.012 K and various fields . (After M. V. Hobden and N. KurtL)

the conduction electrons wo uld make a Curie-type paramagnetic contribution (22) to the magnetization of the metal: M = N/-L~BlkB T. Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature. Pauli showed that the application of the Fermi-Dirac distribution (Chap­ ter 6) w6uld correct the theory as required. We firs t give a qualitative explana­ tion of the situation . The result (18) tells us that the probability an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orien­ tation by roughly /-LBlkB T. For N atoms per unit volume, this gives a net mag­ netization = N/-L2Blk BT, the standard result . Most conduction electrons in a metal, however, have no possibility of turning over when a fiel d is applied, because most orbitais in the Fermi sea with parallel spin are already occupied. O nly the electrons within a range kBT of the top of the Fermi distribution have a chance to turn over in the field; thus only the fraction TIT F of the total number of electrons contribute to the suscep­ tibility. Hence N/-L2B T N/-L2 M =--' -=--B kBT TF kBTF

which is independent of temperature and of the observed order of magnitude. We now calculate the expression for the paramagnetic susceptibility of a free electron gas at T ~ TF. We follow the method of calculation suggested by Fig. 10. An alternate derivation is the subject of Problem 5.

14 Diamagnetism and Paramagnetism

Total energy, kinetic + magne tic, of electrons

l

~

1

Parallel ta field

Dffi~~~ o~~

,


~

3.0

2.0

--~_ I ~

\

//

~~/



... V

.......-~w

'\ - r--_ "­

f--

I r- - -

_J-_+-_r-zr- v _

1'1 1.0

- -~-

~Nb

-

Rh

l

~

Na

K--

'"

1R'b"'f--- T--t-- -+--1f--+_-J-::H::lr r--- 0

0

200

400

600

J

800

J 1000 1200 T. in K

1400

-

Ta

J 1600

1800

2000

2200

Figure 11 Temperature dependence of the magnetic susceptibility of metals. (Courtesy of C. Kriessman .)

J.

2. Huml mles. Apply the H und rules to find the ground state (the basic level in the notation ofTable 1) of (a) E u + +, in the configuration 4[1 5S 2p6; (b) Yb3+ ; (c) Tb 3 + . The results fo r (b) and (c) are in Table 1, but you should give the separate steps in applyi ng the rules. 3. Triplet excited states. Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state. (a) Find an expression for the magnetic moment (J-L ) in a fie ld B. (b) Show that the susceptibility for T p il is approximately independent of il. (c) With the help of a diagram of energy levels versus field and a rou gh sketch of entropy versus field , explain how this system might be cooled by adiabatic magne tiza tion (not demagnetization).

437

438

4.

Consider two-Ievel system with and Iower states; the splitting may arise from Show that the hoat capacity per system is

c

capacity interaction between nuclear and electronic

electron spin order)

'1'l'",L111!,' are often detected experi­

in the heat capacity in the region T P À.

interaetions (see with fields al 50

spin of a conduction eleetron gas at abso­ another method.

be the eoneentrations eleetrons. Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is

+(), where similar for the

6.

in zero magnetic field. Find a

+ E - with respect to , and solve value of , in the approximation , ~ 1. Go in agreement with

to show that the

approximate the eHeet of inter­ assume that eleetrons with parallel aetions among the eonduction electrons if is positive, while electrons with with each other \vith energy of Problem 5 not interact with each other. Show with the

(1

+ () ;

find a similar expression for the total energy and limit {, ~ 1. Show that the magnetization is

for

in the

so the interaction enhances the susceptibility. (c) Show that with B = 0 the total energy is unstable at' 0 when V > this is satisfied a paramagnetic state. Because of neUc state ({, "'" 0) will have a lower energy th an the assumption t: ~ l, this is a sufficient condition for but it may not be a neccssary condition. It is known

14 Dinmaf!:netism and Paramagnetism

0.5

--,------r---,---,---,--i

r-j

.°l

""

.~

1

Level2

j

Level l

,:;

0.3

"

S

è

,J

'"p,

8

0.2

0.1

00

4

5

6

x = Tlt.

Figure 12 Heat capacity of a two-level system as a function of T/t;,., where t;,. is the level splitting. The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rare­ earth and transition-group metals, compounds, and alloys. 0.008

;.0

0006

eNT" = 4.3 x 1

1

(3

E

E

0.004

.S

h

"

u

0.002

0.002

0.004

0006

0008

0.01

TO, in KJ

Figure 13 The normal-state heat capacity of gallium at T < 0.21 K. The nuclear quadrupole (G ct: T 2) and conduction electron (G 0: T) contributions dominate the heat capacity at very low temperatures. (After K Phillips.)

439

7. Two-level system. The result of Problem 4 is often seen in another form. If the two energy levels are at à and -il, that the energy and heat capacity are

u=

c= of à are proportional to the tem­ to the heat capacity of dilute 1519 It is al50 used in the

8.

Itystem. Find the magnetization as a function field and temperature for a system of spins with S 1, moment n. (b) Show that in the li mit li-B