Charles Kittel Intro Solid State Physics

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SEVENTH EDITION 
Introduction to 
Solid State Physics 
CHARLES KIT TEL 
14 
Diamagnetism and Paramagnetism 
LANGEVIN DIAMAGNETISM EQUATION 417 
QUANTUM THEORY OF DIAMAGNETISM OF 
MONONUCLEAR 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 431 
Nuclear demagnetization 432 
PARAMAGNETIC SUSCEPTffiILITY OF 
CONDUCTION ELECTRONS 433 
SUMMARY 436 
PROBLEMS 436 
1. Diamagnetic susceptibility of atomic hydrogen 436 
2. Hund rules 437 
3. Triplet excited states 437 
4. Heat capacity from internaI degrees of freedom 438 
5. Pauli spin susceptibility 438 
6. Conduction electron ferromagnetism 438 
7. Two-Ievel system 440 
8. Paramagnetism of S =1 system 440 
REFERENCES 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 
CHAPTER 14: DIAMAGNET ISM AND PARAMAGNETISM 
Magnetism is inseparable from quantum mechanics, 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 principal 
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 
hydrogen atpm the orbital moment is zero , and the magnetic moment is that of 
the electron spin along with a small induced diamagnetic 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 pe r unit volume is defined as 
M (SI) X = /-LoM (1) (CeS) x = 13 ' B 
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 system of units . 
Quite frequently a susceptibility is defi ned referred to unit mass Or to a 
mole of the substance. The molar susceptibility is written as XM ; the magnetic 
moment per gram is sometimes written 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 magnetic moments are discussed in Chapter 15; the 
arrays may be ferromagnetic, ferrimagnetic, antiferromagnetic, helical, or 
more complex in form. Nuclear magnetic moments give r ise to nuclear 
3paramagnetism . Magnetic moments of nuclei are of the order of 10- times 
smaller than the magnetic moment of the 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
magnetism we are familiar 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) w = eB/2mc (SI) 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 much lower than the fre­
quency 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 
frequency (2). 
The Larmor precession of Z electrons is equivalent to an electric current 
1 eB)(SI) 1 = (charge)(revolutions per unit time) = (- Ze) (- . -- . (3)271' 2m 
The magnetic moment I.L of a current loop is given by the product 
(current) X (area of the loop). The are a of the loop of radius p is 7TP'2. We have 
ZtfB (S I) JI. = - 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) = (x2) + (y2) + (Z2). For a spherically 
symmetrical distribution of charge we have (x2) = (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) X (5)B 6mc'2 ' 
419 14 Diamagnetism and Paramagnetism 
2 
x = ILQNIl- = ILQNZe (r2 )(SI) 
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 : 
He Ne Ar Kr Xe 
XM in CGS in 10-6 cm3lrnole: -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 conduction elec­
trons is more complicated, as is evident from the de Haas-van Alphen effect 
discussed in Chapter 9. 
QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTEMS 
From (G. 18) the effect of a magnetic field is to add to the hamiltonian the 
terms 
ieh e2 
A2J-C = -(V' . A + A· V') + -- , (6)
2mc 2mc2 
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 , Ay = h B , Az = 0 , (7) 
and (6) becomes 
2iehB(d d) e2BJ-C = -- x- - y- + --(x2 + y2) (8)
2mc dy dx 8mc2 
The first term on the right is proportional to the orbital angular mUlnen­
tum component Lz if r is measured from the nucleus. In mononuclear systems 
this term gives rise only to paramagnetism. The second term gives for a spheri­
cally symmetric system a contribution 
2 2E' = e B 2
-12 (r ) (9) 
n1C2 ' 
The moment is 
netic: 
in with 
ta is in: 
lar oxygen and organic """<'>,","'0 
4. Metals< 
The 1l"15"~;U" moment of an atom or ion in free space is given 
where the total angular momentum IiL and liS 
angular momenta< 
The constant 1'îs the ratio of the moment to the angular momen­
tum; l' is called the 
a g 
defined by 
For an g = 2. as For a free atom the g 
factor is the Landé equation 
g = l + ~---'----~--'---'-
421 14 Diamagnetism and Paramagnetism 
4s:: 1.00 1 
ms IJ.z 10.75 
0. 
8. 0.50'If .02 ( /// 2ILB - IL " s:: 
B025
--', 1 e 
c....
". -2 !J. o1 I ii 1 
o 0.5 1.0 1.5 2.0 
ILBlkBT 
Figure 2 Energy level splitting for one electron 
in a magnetic field B directed along the positive z Figure 3 Fractional populations of a two-level 
axis. For an electron the magnetic moment JL is system in thermal equilibrium at temperature T 
opposite in sign to the spin S, so that JL = in a magnetic field B. The magnetic moment is 
-gJLBS. In th e low energy state the magnetic proportional ta the difference between the two 
moment is paraIJel ta the magnetic field. curves. 
The Bohr magneton J-tB is defined as eh/2mc in ces and eh/2m in SI. It is 
closely equal to the spin magnetic moment of a free electron . 
The energy levels of the system in a magnetic field are 
U = - P' B = mjgJ-tBB , (14) 
where mj is the azimuthal quantum number and has the values J, J - l, ... , 
- J. For a single spin with no orbi tal moment we have mj = ± i and g = 2, 
whence U = ± J-tBB. This splitting is shown in Fig. 2. 
If a system has only two levels the equilibrium populations are, with 
T == kBT, 
NI exp(J-tBIT) (15)
N exp(j.LBiT) + exp( - j.LBIT) , 
Nz exp( - J-tB IT) (16)
N exp(J-tBIT) + exp( - j.LBiT) , 
here N j , Nz are the populations of the lower and upper levels, and 
N = N j + N2 is the total number of atoms. The fractional populations are plot­
ted in Fig. 3. 
The projection of the magnetic moment of the upper state along the field 
direction is - J-t and of the lower state is J-t. The resultant magnetization for N 
atoms per unit volume is , with x == J-tB/kBT, 
eX - e-X 
M = (NI - N2)J-t = NJ-t · x + _, = NJ-t tanh x . (17) 
e e 
For x ~ l , tanh x = x, and we have 
M =NJ-t(J-tB/kBT) (18) 
In a magnetic field an atom with angular momentum quantum number J 
has 2J + 1 equally spaced energy levels. The magnetization (Fig. 4) is given by 
M = NgJJ-tB Bj(x) , (x == gJJ-tBB/kBT ) , (19) 
422 
7.00 1TIIInITT':D:o:F:::P:I5F~FiTïi' 
BIT in kG deg- L 
Figure 4 Plot of magnetic moment versus BIT for spherical samples of (1) potassium chromium 
alum, (II) ferric ammonium alum , and (III) gadolinium sulfate octahydrate. Over 99.5% magnetic 
saturation is achieved at 1.3 K and about 50,000 gauss. (ST). After W. E . Henry. 
where the Brillouin function BI is defined by 
2J + 1 ((2J + l)x) 1 ( x )B,(x) = ctnh - - ctnh - (20) 
. 2J 2J 2J 2J 
Equation (17) is a special case of (20) for J = t. 
For x <s:; l, we have 
1 x x3 
ctnh x = - + - - - + (21) 
x 3 45 
and the susceptibility is 
M NJ(J + 1)g2JL~ C 
-= (22)
B 3kBT T 
Here p is the effective number of Bohr magnetons, defined as 
p == gU(J + 1)F /2 . (23) 
14 Dianwgnetism and Paranwgnetism 
40~--------~----'~~-~------~~ 
s 
i 
Temperature, Je 
Figure 5 Plot of l/X vs T for a gadolinium salt, Gd(CzH5 S04h . straight line the 
Curie law, (Aftel' L. C. Jackson and Onnes,) 
Rare Earth Ions 
Even in the 
no other 
atom 
state is charac­
maximum S allowed exclusion 
maximum value of the momentum consistent with 
of S, 
is to IL - SI when the 
shell is more than half fulL 
ruIe L 0, so 
different 
425 14 Diamagnetism and Paramagnetism 
Table l Effective magneton numbers p for trivalent lanthanide group ions 
(Near room tempe rature) 
---­ p(calc) = p(exp), 
Ion Configuration Basic level gU(] + 1)]JJ2 approximate 
__.=:l 
c é+ 4P5s2p6 2F s I2 2. 54 2.4 
Pr3 + 4j25s2p6 3H 4 3. 58 3. 5 
Nd3+ 4P5s2p6 41912 3.62 3.5 
Pm3+ 4f 45s2p6 514 2. 68 
Sm3 + 4fs5s2p 6 6H sf2 0.84 1.5 
Eu3+ 4f65s2p6 7F o 0 3.4 
Gd3+ 4F5s2p6 8S712 7.94 8.0 
Tb3+ 4j'B5s2p6 7F 6 9.72 9.5 
D y 3+ 4f95s2p6 6H 1SI2 10.63 10.6 
Ho3+ 4po5s2p6 sIs 10.60 10.4 
Er3+ 4f1l5s2p6 41 1S12 9.59 9.5 
Tm3+ 4P25s2p6 3H6 7. 57 7.3 
Yb3+ 4P35s2p6 2F7i2 4.54 4.5 
The second Hund rule is best approached by model calculations. Pauling 
and Wilson, l for example, give a calculation of the spectral terms that arise from 
the configuration p2. The third Hund rule is a consequence of the sign of the 
spin-orbit interaction: For a single electron the energy is lowest when the spin 
is antiparallel to the orbital angular momentum. But the Iow energy pairs mL, 
ms are progressively used up as we add electrons to the shell; by the exclusion 
principle when the shell is more th an half full the state of lowest energy neces­
sarily has the spin parallel ta the orbit. 
Consider two examples of the Hund fuIes : The ion c é+ has a single f 
electron; an f electron has l = 3 and s = i. Because the f shell is less than half 
full, the ] value by the preceding rule is IL - SI = L - ! = l The ion Pr3+ has 
two f electrons: one of the mIes tells us that the spins add to give S = 1. Both f 
electrons cannot have ml = 3 without violating the Pauli exclusion principle, so 
that the maximum L consistent with the Pauli principle is not 6, but 5. The] 
value is IL - si = 5 - 1 = 4. 
Iron Group Ions 
Table 2 shows that ~he experimental magneton numbers for salts of the iron 
transition group of the pelt'iodic table are in poor agreement with (18). The 
values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL. Pauling and E. B. Wilson, Introduction to quantum mechanics, McGraw-Hill, 1935, 
pp. 239-246. 
426 
Table 2 E ffective magneton numbers for iron group ions 
Config- Basic p(calc) = p(calc) = 
Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a 
Ti3+, y4+ 3d l 2D 3I2 1.55 1. 73 1.8 
y 3+ 3d2 3F2 1.63 2.83 2.8 
Cr3+, y2+ 3d3 4F 3/2 0. 77 3.87 3.8 
Mn3+, Cr+ 3d4 5DO 0 4.90 4.9 
F e3+, Mn 2+ 3d5 6551 2 5.92 5.92 5.9 
Fe2+ 
Co2+ 
3d6 
3d7 
5D4 
4F 9/2 
6.70 
6.63 
4.90 
3.87 
5.4 
4.8 
Ni2+ 3d8 3F 4 5.59 2.83 3.2 
Cu2 + 3d9 2D5/2 3.55 1.73 1.9 
"Representative values. 
lated as if the orbital moment were not there at ail. We say that the orbital 
moments are quenched. 
Crystal Field Splitting 
The difference in behavior of the rare earth and the iron group salts is that 
the 4f shell responsible for paramagnetism in the rare earth ions lies deep 
inside the ions, within the 5s and 5p sheIls, whereas in the iron group ions the 
3d shell responsible for paramagnetism is the outermost shell. The 3d shell 
experiences the intense inhomogeneous electric field produced by neighboring 
ions. This inhomogeneous electric field is called the crystal field. The interac­
tion of the paramagnetic ions with the crystal field has two major effects: the 
coupling of L and S vectors is largely broken up, so that the states are nO longer 
specified by their J values; further, the 2L + l sublevels belonging to a given L 
which are degenerate in the free ion may nOw be split by the crystal field , as in 
Fig. 6. This split ting diminishes the contribution of the orbital motion to the 
magnetic moment. 
Quenching of the Orbital Angular Momentum 
In an electric field directed toward a fixed nucleus, the plane of a classical 
orbit is fixed in space, so that aIl the orbital angular momentum components Lx> 
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 
427 14 Diamagnetism and Paramagnetism 
@ @ @ 
===== P"Py 
y 
- ---pzy 
@ ® ® 
(a) (b) (c) (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 symmetry the charges on neighboring ions 
will produce an electrostatic potential cp about the nucleus of thJ form 
ecp = AX2 + By2 - (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 V2cp = 0 and compatible 
with the symmetry of the crystal. 
428 
Uy = yf(r) ; Uz = zf(r) 
are normalized. 
= 2Ui , 
= 0 . 
Consider 
dx dy dz ; (28) 
the integral 
the diagonal matrix 
elements: 
+ dx dy dz (29) 
where 
dx dz ; 
The their angular lobes 
o. 
This effect is 
momentum, 
age is zero in 
magnetic moment also 
ParamilgnetÎttm 
(30) 
- À/àl , 
the hetween 
g 
g 
1966; extensive 'See L. Orgel, Introduction to transition 
references are given by D. Sturge, Phys. 
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 approximately equal to NÂ/2kBT, so that the resultant magneti­
zation is 
M = 2BI(slJ.LzIO)12 NÂ (36)Â 2kBT ' 
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) . Â ;? kBT . Here the population is nearly aIl in the ground state, so 
that 
M = 2NBI(slJ.LzIO>1 2 (38)
 
The susceptibility is 
(39) 
431 Diamagnetism P aramagnetism 
type of contribution 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. Giauque, 
Am, Chem, Soc, 49, 1864 (1927). For many purposes SUI)pla'ntt~d by the 
dilution which operates solution in 
He' play the raIe of atoms in a gas, and 
12. 
432 
Spin 
Total 
Spin 
Lattice Time- Time-
Before 1 New equilibrium Be ore :\cw equilibrium 
Time at which Time at which 
magnetic fie ld magnetic field 
is removed is l'emoved 
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant. For 
effective cooling the initial entropy of the lattice should be small in comparison with the entropy of 
the spin sys tem. 
The steps carried out in the cooling process are shown in Fig. 8. The field 
is applied at temperature Tl 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 T2 . 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 lkBT, 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. IfBt>. is the effective field 
that corresponds to the local interactions, the final temperature T2 reached in 
an adiabatic demagnetization experiment is 
11 T2 = Tl (Bt>.IB) , (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 temperature 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.LBlkBTl = 0.5, and the en­
433 14 Diamagrwtism and Paramagfletism 
B =0; BA = 100 gauss 
0.7,r---------------------------------------------------------~ 
0.6 
~ 
~ ~ 0.5 
§ 
S ~ 4 
~ 
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 funetion of temperature, assuming an internaI random 
magne tic field Be:. of 100 gauss. The specimen is magnetized iso thermally along ab , and is th en 
insulated thermally. The external magnetie field is turned off along be. In order to keep the figure 
on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice, and so is 
the external magnetic fi eld . 
tropy decrease on magnetization is over 10 percent of the maximum spin en­
tropy. This is sufficient to overwhelm the lattice and from (41) we estimate a 
final temperature T2 = 10-7 K. The first4 nuclear cooling experiment was car­
ried out on Cu nudeiin the metal, starting from a fi rst stage at about 0.02 K 
as attained by electronic cooling. The lowest temperature reached was 
1.2 x 10- 6 K. 
The results in Fig. 9 fit a line of the fonn of(41) : Tz = T1(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 ensure 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 stics the fact that many 
metals are diamagnetic, or only weakly paramagnetic, can be brought into agree­
ment with tb e existence of a magnetic moment of tbe e lectrons . 
W. Pauli, 1927 
Classical fr ee electron theory gives an unsatisfactory account of the para­
magnetic susceptibility of the conduction electrons. An electron has associated 
with it a magnetic moment 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 (1956); for 
reviews see N. · Kurti, Cryogenies 1, 2 (1960); Adv. in Cryogenie Engineering 8, 1 (1963). 
sp. J. Hakonen et al ., Phys . Rev. Lett. 70, 2818 (1993). 
434 
Initial magnetic field in kG 
lonr---T5--------~lrO--------~20~---3TO~ 
9 
8 
7 
1 6 
~ 5 
1;0
.,
e 4 
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 would 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 probabili ty 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/-L2BlkBT, the standard result . 
Most conduction electrons in a metal, however, have no possibility of 
turning over when a field is applied, because most orbitais in the Fermi sea 
with parallel spin are already occupied. Only 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. 
--
435 14 Diamagnetism and Paramagnetism 
Total energy, kinetic + 
magne tic, of electrons 
l 1 ~ 
Parallel 
ta field 
, Dffi~~~ .... ~ Density of 
1 orbitais< o~~ 
1 
(a) (b) 
Figure 10 Pauli paramagnetism at absolu te zero; the orbitais in the shaded regions in (a) are 
occupied . The numbers of electrons in the "up" and "down" band will adjust ta make the energies 
equal at the Fermi level. The chemical potential (Fermi level) of the moment up electrons is equal 
to that of the moment down electrons. In (b) we show the excess of moment up electrons in the 
magnetic field. 
The concentration of electrons with magnetic moments parallel to the 
magnetic field is 
l J'F l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF) , 
2 - l'-B 2 0 2 
written for absolute zero. Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 
:2 • 
spin orientation, with allowance fo r the downward shift of energy by - fJ-B . 
The approximation is written for kBT <{ EF • 
The concentration of electrons with magnetic moments antiparallei to the 
magnetic field is 
l JEF l ll'FN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 
21'-B 20 2 
The magnetization is given by M = fJ-(N + - N _), so that 
3N fJ-2 
M = fJ-2 D (EF)B = - k B (42)
2 BTF 
with D(EF) = 3N/2EF = 3N/2kBTF from Chapter 6. The result (42) gives the 
Pauli spin magnetization of the conduction electrons, for kBT <{ EF • 
In deriving the paramagnetic susceptibility, we have supposed that the 
spatial motion of the electrons is not affected by the magnetic field. But the 
wavefunctions are modified by the magnetic fie ld; Landau has shown that for 
436 
B. 
.. 
.. 
(43) 
the 
by 
The 
UUU1H.l<U.'y high for transition 
Ipl"~r"n,,, heat 
of atomic Z is X 
atomic (Langevin) 
the maximum S 
consistent with this S. The 
and IL - S if the shell is Jess 
is 
437 14 Diamagnetism and Paramagnetism 
8 0 
r iT T TtS 1 1 1 IIT ­
7.0 
6.0 
~ 5.0 
~ 
\\ -r--r-2_ 
w 
E 
" \ ~ --~_ I~ 1 
8 Cr 
__ 	 4.0
" \ // ...... Vg 
~ 
0. ~~/ 	 .......-~w 
'" 
:l " 
V> ~3.0 
'\ 
f-- - r--_ "­
2.0 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-
_ 	 Rhl1'1 
1.0 Na 	 ~ 
'"K-- -+--1f--+_-J-::Hr r--- - - Ta 1R'b"'f---T--t--	 ::l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 
T. in K 
Figure 11 Temperature dependence of the magnetic susceptibility of metals. (Courtesy of C. J. 
Kriessman . ) 
2. 	 Huml mles. Apply the Hund rules to find the ground state (the basic level in the 
notation ofTable 1) of (a) Eu ++, in the configuration 4[1 5S2p6; (b) Yb3+ ; (c) Tb3+ . The 
results fo r (b) and (c) are in Table 1, but you should give the separate steps in applying 
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 rough sketch of entropy versus field , explain how this system might 
be cooled by adiabatic magnetization (not demagnetization). 
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 al50 
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 in zero magnetic field. Find a 
similar + E - with respect to , and solve 
for the value of , in the approximation , ~ 1. Go to show that the 
in agreement with 
6. 	 approximate the eHeet of inter­
aetions among the eonduction electrons if assume that eleetrons with parallel 
with each other \vith energy is positive, while electrons with 
not interact with each other. Show with the of Problem 5 
(1 + () ; 
find a similar expression for the total energy and for in the 
limit {, ~ 1. Show that the magnetization is 
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 
neUc state ({, "'" 0) will have a lower energy th an paramagnetic state. Because of 
the assumption t: ~ l, this is a sufficient condition for but it may 
not be a neccssary condition.It is known 
439 
1 
eNT" = 4.3 x 
0.002 0.004 0006 0008 0.01 
14 Dinmaf!:netism and Paramagnetism 
0.5 r-j--,------r---,---,---,--i 
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 
0006 
;.0 
1 
(3 
E 
0.004E 
.S 
h 
u " 0.002 
Figure 13 The normal-state heat capacity of gallium at T < 0.21 K. The nuclear quadrupole 
(G T 2) and conduction electron (G 0: T) contributions dominate the heat capacity at very low ct: 
temperatures. (After K Phillips.) 
"".°l
.~ 0.3 
" S 
è
,J 
p, '" 
8 0.2 
0.1 
00 
Level21 
,:; 
j Level l 
4 5 6 
x = Tlt. 
TO, in KJ 
u= c= 
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 
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 
1, moment 
as a function 
field and temperature for a system of spins with S 
n. 	(b) Show that in the li mit li-B <{ kT result is ­
A. Abragam and B. Bleaney. Electron resonance tom, Dover, 1986. 
B. 	G. Casimir, Magnetism and very tempe ratu res, DoveT, 1961. A c1assic. 
Darby and K. R. Taylor, Physics of rare earth Halsted, 1972. 
A. J. Freeman, The actinides: electronic structure and related properties, Academie, 1974. 
R. 	 D. Hudson. Princip les and Elsevier, 1972. 
North-Holland, 1970. Knoepfel, Pu/sed 
Lounasmaa, and methods below 1 K, Academie Press, 1974. 
Introduction ta transition metal 2nd ed., Wiley, 1966. 
Van Vleck, The theory Oxford, 1932. deriva­
tions of basic theorems. 
G. K. White, 	 3rd Oxford, 1987. 
R. White, Quantum theory 
A. 	J. Freeman and G. H. Lander, actinides. North-
Holland, 1984-1993. 
Sturge, "Jahn-Teller effect in solids," Solid state 91 (1967). 
O'Brien and C. C. Chancey, "The effect: An introduction and current re­
view," Amer. J, Physics 61, (1993),