Chapter Eleven
Diamagnetism and Paramagnetism
The story of magnetism begins with a mineral called magnetite “Fe3O4”.
The first truly scientific study of magnetism was made by William Gilbert.
His book “On the magnet” was published in 1600.
In 1820 Hans Christian Oersted discovered that
an electric current produces a magnetic field.
In 1825 the first electromagnet was made by Sturgeon.
Each stores the same
magnetic energy ~0.4J.
1
ferromagnetism
diamagnetism
M (emu/cm3)
M (emu/cm3)
paramagnetism
H (Oe)
H (Oe)
Magnetic susceptibility per unit volume
M
χ≡
H
M : magnetization, magnetic moment per unit volume
B : macroscopic magnetic field intensity
CGS
In general, paramagnetic material has a positive susceptibility (χ >0).
diamagnetic material has a negative susceptibility (χ<0).
2
P. Curie
1859~1906
1895 Curie proposed the Curie law
1903
1905 Langevin proposed the theory of diamagnetism and paramagnetism
1906 Wiess proposed the theory of ferromagnetism
1920’s The physics of magnetism was developed with theories involving
electron spins and exchange interactions – the beginnings of QM
Three principal sources for the magnetic moment of a free atom :
 Electrons’ spin
 Electrons’ orbital angular momentum about the nucleus
 Change in the orbital moment induced by an applied magnetic field
First two sources give paramagnetic contributions.
The last source gives a diamagnetic contribution.
3
Reviews,
in CGS
 


B = H + 4π M = µ H
  N
M=m
[emu/cm3] V

m : magnetic dipole moment


 
e
e 
L = −µB L
m=−
∑ ri × p i = −
2m i
2m



m = µ Bg o ∑ si = µ Bg o S
i
Bohr magneton
µB =
e
= 9.27 × 10-24 J/T;
2m
Angular
momentum
Spin
1T = 1Vs/m 2
Electronic g-factor go=2.0023
µ
I
v
electron
L and S are quantum operators with no basis in classical physics.
4
Theory of diamagnetism
was first worked out by Paul Langevin in 1905.
Negative magnetism present in an applied field
even though the material is composed of atoms
that have no net magnetic moment.
Motion of the electron is equivalent to current in the loop.
Faraday’s law
dΦ B
Emf = −
dt
[volt]
Paul Langevin
1872~1946
Consider a circular loop with radius r, the effect of applied field H is
to change µ by an amount ∆µ in negative direction.
I
r
µ
N A ρ e2 Z
2
R
χ =−
A 6mc 2
[emu/Oe-cm3]
5
H
The change in moment of a single orbit :
A dH
r dH
π r 2 dH
Emf
=−
=−
=−
E=
2π r dt
2 dt

 dt
a=
dv eE
er dH
=
=−
dt mc
2mc dt
 er 
dv
=
∫ ∫  − 2mc dH
er
∆v = −
H
2mc
er
e2r 2
∆µ = ∆v = −
H
2
4mc
2c
the plane of the orbit is perpendicular
to the applied field
All possible orientations in the hemisphere
θR
r2
2
θ dA
sin
= R 2 sin 2 θ = R 2 ∫
A
π /2
R2
2 2
2
(
)(
)
θ
π
θ
θ
=
R
R
=
R
sin
2
sin
d
2 ∫
3
2πR 0
6
A single electron with consideration of all possible oriented orbits,
e2 R 2
eR
H
∆µ =
∆v = −
2
2c
6mc
For an atom that contains Z electrons,
z
e 2 Ri2
e2H z 2
∆µ = ∑ −
R
H=−
2
2 ∑ i
6mc
6mc i =1
i =1
Ri is the radius of the ith orbit
Volume density of mass
The number of atoms per unit volume : NAρ/A
Atomic mass
Avogadro’s number
On the volume basis,
N A ρ e2  z 2 
∆µ = −
 R H
2 ∑ i
A 6mc  i =1 
N A ρ e2 Z
2
R
Susceptibility, χ = −
A 6mc 2
Classical Langevin result
Carbon (radius =0.7Å)
(
)
(
)
2
6.02 ×10 23 2.22g/cm 3
4.80 ×10 −10 esu 6
χ =−
12.01g
6 9.11×10 − 28 g 3 ×1010 cm/s
= −1.5 ×10-6 emu/Oe − cm 3
(
)(
Experimental value :
)
2
(0.7 ×10
-1.1×10-6
−8
cm
)
2
7
emu/Oe-cm3
The agreement between calculated and experimental values for other
diamagnetic materials is generally not that good, but it is at least within
an order of magnitude.
This classical theory of Langevin is a good example of the use of a simple
atomic model to quantitatively explain the bulk properties of a material.
Negative susceptibility
magnitude : 10-5 ~10-6 cm3/mole
The other consistency is that
χ is independent of temperature
for most diamagnetic materials.
Quantum theory of diamagnetism
results in the same expression.
8
Diamagnetic materials
Pierre Weiss
Closed-shell electronic structures result in no net moment
The monatomic rare gases He, Ne, Ar, …
Polyatomic gases, H2, N2, …
Some ionic solids, NaCl, …
Some covalent solids, C, Si, Ge, …
Superconductors : Meissner effect
1865~1940
The first systematic measurements of susceptibility of a large number
of substances over an extended range of temperature were made by
Pierre Curie and reported by him in 1895.
-- mass susceptibility is independent of temperature for diamagnetics,
but that it vary inversely with temperature for paramagnetics.
C
χ=
T
Curie law
C'
χ=
T −θ
in general
Curie-Weiss law (1907)
9
Quantum theory of diamagnetism
2
1  Q 
Hamiltonian
H=
 p − A  + QV
2m 
c 


Terms due to the applied field B = ∇ × A
(
)
 
ie
e2
2
H '=
∇ • A+ A•∇ +
A
2mc
2mc 2

In the uniform magnetic field B = Bk̂
  ∂
∂
∂ 
ĵ + k̂  × Ax î + Ay ĵ + Az k̂
∇ × A =  î +
∂y
∂z 
 ∂x
 ∂Az ∂Ax   ∂Ax ∂Az   ∂Ay ∂Ax 
k̂
î + 
−
−
= 
−
 ĵ + 
∂y 
∂z   ∂z ∂x   ∂x
 ∂y

yB
xB
One solution A = −
î +
ĵ + 0k̂
2
2
ieB  ∂
∂  e 2 B2 2
2
 x − y  +
x
+
y
H '=
4mc  ∂y
∂x  8mc 2
~
Lz
(
)
(
)
10
z-component of angular momentum
 ∂
∂ 
~
L z = −i x − y 
∂x 
 ∂y
paramagnetism
For diamagnetic materials with all electrons in filled core shells,
the sum ∑ L z + 2Sz
over all electrons vanishes
For a spherically symmetric system
x +y
2
2
2 2
= r
3
2
2
e
r
e 2 B2 2
2
2
the additional energy E ' =
x
+
y
=
B
8mc 2
12mc 2
2
2
e
r
∂E '
the associated magnetic moment µ = −
=−
B
2
∂B
6mc
in agreement with the classical Langevin result
11
Vibrating Sample Magnetometer (VSM)
→→ measuring sample’s magnetic moment
oscillate sample up and down
measure induced emf in coils A and B
12
compare with emf in coils C and D from known magnetic moment
13
Classical theory of paramagnetism
In 1905, Langevin also tried to explain paramagnetism qualitatively.
He assumed that paramagnetic materials have molecules or atoms with
the same non-zero net magnetic moment µ.
In the absence of magnetic field, these atomic moments point at random
and cancel one another.
M=0
When a field is applied, each atomic moment tends to align with the field
direction; if no opposing force acts, complete alignment would be
produced and the material demonstrates a very large moment. But
thermal agitation of atoms tends keep the atomic moments pointed at
random.
Partial alignment and small positive χ.
As temperature increases, the randomizing effect of thermal agitation
increases and hence, χ decreases.
This theory leads naturally into the theory of ferromagnetism.
14
A unit volume material contains n atoms with a magnetic moment µ each.
 
E = − µ • H = − µH cosθ
H
θ

µ
Let the direction of each moment be presented by a vector,
and let all vectors be drawn thru. the center of the sphere.
dn = KdA e
−
µH cos θ
E
k BT
= K 2π sin θdθ e
µH cos θ
π
n = ∫ dn = K 2π ∫ dθ sin θ e
0
n
k BT
π
k BT
# at angle θ θ+dθ
total #
M = ∫ dnµ cosθ = K 2πµ ∫ dθ cosθ sin θ e
0
k BT
0
π
Let a = µH/kBT, M =
µH cos θ
nµ ∫ dθ cosθ sin θ e a cosθ
0
π
a cos θ
∫ dθ sin θ e
0
−1
=
nµ ∫ dx xe ax
1
−1
∫ dx e
1
ax
 e a + e −a 1 
1


= nµ  a
−
= nµ  coth (a ) − 
−a

a
15
a

e −e

Langevin function L(a)=coth(a)-1/a
L(a) can be expressed in series,
M/Mo
a a 3 2a 5
− ...
L(a) = − +
3 45 945
When a is small (<0.5),
L(a) is a straight line w/. a slope 1/3.
When a is large, L(a) → 1.
a=µH/kBT
Two conclusions :
(1) Saturation will occurs at large a implying that at large H and low T the
alignment tendency of field is going to overcome the randomizing
effect of thermal agitation.
(2) At small a, L(a) depends linearly on a implying that M varies linearly
16
with H.
The Langevin theory also leads to the Curie law.
nµa nµ µH
=
3
3 k BT
M nµ 2
C
χ= =
=
H 3 k BT T
For small a,
M=
nµ 2
C=
3kB
Oxygen is paramagnetic and obeys the Curie law w/. χ=1.08×10-4emu/gOe.
3 k BT χ
3(32g/mole )(1.38 ×10 −16 erg/K )(1.08 ×10 −4 emu/gOe )
µ=
=
n
6.02 ×10 23
= 2.64 ×10-20 erg/Oe − mole
= 2.85µ B
Even in heavy atoms or molecules containing many electrons,
each with orbital and spin moments, most of the moments cancel out
and leave a magnetic moment of only a few Bohr magnetons.
Typically, a =
(
2.64 ×10 erg/Oe )(10000Oe)
=
= 0.0064
(1.38 ×10 erg/K )(300K )
T
µH
kB
smaller enough
−20
−16
L(a)=a/3
<0.5
17
The Langevin theory of paramagnetism which leads to the Curie law, is
based on the assumption that individual carrier of magnetic moment do
not interact with one another, but are acted on only by applied field and
thermal agitation.
However, many paramagnetic materials obey the more general law,
χ=
C
T− θ
Curie-Weiss law (1907)
considering the interaction between elementary moments.
Weiss suggested that this interaction could be expressed in terms of
a fictitious internal field, “molecular field HM”. In addition to H.
The total field acting on material H + HM= H + γM
C
M
M
=
=
χ=
T − Cγ M
H tot H + γM T
(
)
=CH
M
C
C
χ= =
=
H T − Cγ T − θ
where θ (=Cγ) is a measure of the strength of the interaction. 18
1
Theoretical predictions
θ
χ
=
θ
T −θ 1
= T−
C
C
C
Intercept reveals information of θ
θ (FeSO4)<0, θ (MnCl2)>0
θ>0 indicates that the molecular field aids the applied field, makes
elementary moments align with H, and tends to increase χ.
θ<0 indicates that the molecular field opposes the applied field, makes
elementary moments disalign with H, and tends to decrease χ.
19
Precession of atomic magnetic moments about H,

τ = µ×H


like gyroscope, angular momentum in the presence of gravitation.

 
 dL
τ = r × mg =
dt
If the atom was isolated, the only effect of an increase in H would be
an increase in the rate of precession, but no change in θ.
For a system with many atoms, all subjected to thermal agitation,
there is an exchange of energy among atoms. When H is applied,
the exchange of energy disturbs the precessional motion enough so
that the value of θ for each atom decreases slightly, until the
distribution of θ values becomes appropriate to the existing values of
field and temperature.
20
Quantum theory of Paramagnetism
The central postulate of QM is that the energy of a system is discrete.
quanta of energy
improving the quantitative agreement
between theory and experiment.
Classical case
Quantum case
J=1/2
H
H
J=2
θ
θ is continuous variable.
θ is restricted to certain variables.
Space Quantization
21
The orbital and spin magnetic moments for an electron

eh
e h
e 
p orbit
µ orbit =
=
=−
4πmc 2mc 2π
2mc

eh
e 1 h
e 
p spin
µspin =
=
=−
4πmc mc 2 2π
mc
In general

µ = −g
e 
p
2mc
where g is spectroscopic splitting factor, or g-factor
g=1 for orbital motion and g=2 for spin
In a multiple-electron atom, the resultant orbital angular momentum and the
resultant spin momentum are characterized by quantum numbers L and S.
Both are combined to give the total angular momentum that is described by
the quantum number J.
H
µ=g
e
J(J + 1) = g J(J + 1) µ B
2mc
e
m J  = gm J µ B
In the direction of H, µ H = g
2mc
µH θ µ
There are 2J+1 numbers of allowed azimuthal quantum number mJ : 22
J, J-1, … , -(J-1), -J.
For a free atom, the g factor is given by the Landé equation
g = 1+
J(J + 1) + S(S + 1) − L(L + 1)
2J(J + 1)
The energy levels of the system in a magnetic field
ms µs
1/2 -µ
2µB
-1/2 µ
 
E = − µ • H = m J gµ B H
For a single spin with no orbital moment, mJ=±1/2 and g=2.
The equilibrium population for the two level system :
N1
exp( µH/k BT )
=
N exp( µH/k BT ) + exp( − µH/k BT )
lower level
N2
exp( − µH/k BT )
=
N exp( µH/k BT ) + exp( − µH/k BT )
Total number of atoms
is conserved.
upper level
N = N1+N2 = constant
23
The resultant magnetization for N atoms per unit volume
ex − e−x
M = (N1 − N 2 ) µ = Nµ x
= Nµ tanh (x )
−x
e +e
For x<<1, tanh(x)~x and M ≅ Nµ x = N
where x =
µH
k BT
µ 2H
k BT
For a system with known g and J,
Potential energy of each moment in H
U = −m J gµ B H
The probability of an atom w/. energy U
Therefore,
M=N
∑ m J gµ B e
mJ
m gµ
e
∑
J
e
− U/k BT
m J gµ B H/k BT
B H/k B T
=e
m J gµ B H/k BT
Boltzmann statistics
mJ : J, J-1, … , -(J-1), -J
mJ
 2J + 1
 x 
 (2J + 1)x  1
M = NgJµ B 
coth 
 − coth  
 2J 
 2J  2J
 2J
gJµ B H24
= NgJµ B BJ (x)
where x =
Brillouin function
k BT
The saturation magnetization M o = N µ H = N gJµ B
Number of atoms
per unit volume
The maximum moment of each
atom in the direction of H
M
2J + 1
x
 (2J + 1)x  1
−
= B J (x) =
coth 
coth

 
2J
2J
2J
Mo

 2J 

Brillouin function, 1927
When J=∞, the classical distribution,
the Brillouin function reduces to the
Langevin function.
M
1
= coth (x ) −
Mo
x
When J=1/2, the magnetic moment
consists of one spin per atom.
M
x<<1 x
= tanh (x )
Mo
2
µ 2H
2 2 µBH
M ≅ Nµ H x = Ng J
=N
k BT
k BT
25
KCr(SO4)12H2O
Henry, Phys. Rev. 88, 559
(1952)
The Brillouin function, like the Langevin, is zero for x equals to zero
and tends to unity as x becomes large. The course of the curve in
between depends on the value of J for the atom involved.
µH
gJµ B H µ H H
classical x =
quantum
=
=
x
≠
k BT
k BT
k BT
µ : net magnetic moment in
the atom
26
When H is large enough and T low enough, a paramagnetic can be saturated.
For example, KCr(SO4)12H2O
at H=5Tesla and T=4.2K
only magnetic ion Cr3+, J=3/2, L=3, and S=3/2
g=2/5
the lower curve shown in the figure
→ data does not follow this predicted curve
The data is well described by the quantum model with J=S=3/2 and g=2 .
L=0 no orbital component
µ = gJμ B = g J(J + 1)μ B
3 3
= 2 ( + 1)μ B = 3.87μ B
2 2
“Spin-only” moment
27
metals
Gd(C2H3SO4)3•9H2O
28
What magnetic moment a metal should have?
Hund’s rule
incorporate coupling of magnetic moment to determine
the magnetic state
(1) Maximize S = ∑ m s subject to the Pauli exclusion principle
ms=±1/2
put spins (in different states) parallel
(2) Maximize L = ∑ m  subject to the Pauli exclusion principle
m= -, -+1, …, -1, 
fill higher m state preferentially
 L − S when shell is less than half full
(3) J = 
L + S when shell is more than half full
J= L-S, L-S+1, …, L+S-1, L+S allowed values
Whether it is a minimum or maximum in the ground state
depends on filling function.
29
Magnetic state
2S+1
LJ
L : 0, 1, 2, 3, 4
S, P, D, F, G
Examples
(1) Full shells : L=0, S=0, and hence, J=0
(2) Half filled shells : L=0 and S=(2+1)(1/2)
Eg. Gd3+ 4f75S25P6
(n=4,=3)
3
2
1
0
-1
-2
-3
State
8
S7/2
L=0 and S=7/2
J=L+S=7/2
30
Eg. Cs3+ 4f15S25P6
(n=4, =3)
3
2
1
0
-1
-2
-3
Eg. Pr3+ 4f25S25P6
(n=4, =3)
2
State
3
2
1
0
-1
-2
-3
F5/2
L=3 and S=1/2
J=L-S=5/2
State
3
H4
L=5 and S=1
Eg.
Dy3+
J=L-S=4
4f95S25P6
(n=4, =3)
3
2
1
0
-1
-2
-3
State
L=5 and S=5/2
J=L+S=15/2
6
H15/2
31
Rare earth ions
p = g J(J + 1)
For Sm3+ and Eu3+, there are discrepancies between
experimental p and theoretically calculated value.
High states of the L-S multiplet.
32
Quenching of the orbital angular momentum : Iron group ions
Effective number of Bohr magneton p = 2 S(S + 1)
p = g J(J + 1)
In the iron group ions the 3d shell, the outermost shell, is responsible
for the paramagnetism and experiences the intense inhomogeneous
electric field produced by neighboring ions.
called “the crystal field” breaking L-S coupling (J is meaningless)
splitting 2L+1 sublevels (diminishing the
33
contribution of orbital motion to magnetic moment)
An atom w/. L=1 is placed in the uniaxial crystalline electric field of
two positive ions along the z-axis. Take S=0 to simplify the problem.
In a free atom, a single electron in a p-state.
 x + iy 
R n (r)Y11 (θφ ) ∝ f(r) 

 2 
R n (r)Y10 (θφ ) ∝ f(r) z
 x − iy 
R n (r)Y1−1 (θφ ) ∝ f(r) 

 2 
Three degenerate p states
34
In a crystal of orthorhombic symmetry the charges on neighboring
ions will produce an electrostatic potential ϕ about the nucleus,
eϕ = Ax 2 + By 2 − (A + B)z 2
Reconstructing unperturbed ground state
U x = xf(r), U y = yf(r), U z = zf(r)
+ perturbed term eϕ
Non-diagonal elements
U x eϕ U y = U y eϕ U z = U z eϕ U x = 0
Diagonal elements
U x eϕ U x = ∫ f(r) x 2 [Ax 2 + By 2 − (A + B)z 2 ]dxdydz
2
= ∫ f(r) [Ax 4 − Ax 2 y 2 ]dxdydz
2
= A(I1 − I 2 )
35
U y eϕ U y = ∫ f(r) y 2 [Ax 2 + By 2 − (A + B)z 2 ]dxdydz
2
= ∫ f(r) [By 4 − By 2 z 2 ]dxdydz
2
= B(I 1 − I 2 )
U z eϕ U z = ∫ f(r) z 2 [Ax 2 + By 2 − (A + B)z 2 ]dxdydz
2
= ∫ f(r) (A + B)[y 2 z 2 − z 4 ]dxdydz
2
= (A + B)(I 2 − I1 )
The orbital moment of each of the states
U x Lz U x = U y Lz U y = Uz Lz Uz = 0
Crystal field splits the original degenerate states into non-magnetic states
separated by energies >> µH.
quenching of the orbital angular momentum
36
Paramagnetic susceptibility of conduction electrons
• Conduction electrons are not bound to atoms so that they are not
included in atomic magnetization.
• Each electron has µ=±µB.
• Curie-type paramagnetic contribution (electron spins align w/. H)
nµ B2 H
where n is # of conduction electrons
M=
k BT
This requires each electron spin align w/. H, independent of all others.
f(ε,T)
Fermi-Dirac distribution
• Pauli pointed out that only electrons near EF (±kBT) can flip over.
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
kBT
T=0
T≠0
εF
H=0
H≠0
εεFF
ε
37
Fraction of electron participating
 T  µ B2 H  nµ B2 H
 =
M =  n 
 TF  k BT  k BTF
k BT
εF
=
T
TF
independent of temperature
Consider the density of states D(ε) in the presence of H
Spin ↑ electrons lowered in energy by µH in H ↑
Spin ↓ electrons raised in energy by µH in H ↑
ε
εF
↑
H
ε
flip spins
µH
↓
ε
↓
↑
2µH
DOS
µH
DOS
38
The concentration of electrons w/. magnetic moment parallel to H
ε
H↑,
T=0K
ε
1 F
1 F
1
N ↑ = ∫ dεD(ε + µH) ≅ ∫ dεD(ε ) + D(ε F ) µH
2 − µH
20
2
The concentration of electrons w/. magnetic moment antiparallel to H
ε
ε
1 F
1 F
1
N ↓ = ∫ dεD(ε − µH) ≅ ∫ dεD(ε ) − D(ε F ) µH
2 µH
20
2
Pauli spin magnetization
3N 2
3N
M = (N ↑ − N ↓ )µ = µD(ε F ) µH =
µ H=
µ 2H
2ε F
2k BTF
valid when kBT<<εF
independent of temperature
39
References :
Kittel, Introduction to Solid State Physics, 8th Ed., Ch.11-12, 2005.
Aschroft and Mermin, Solid State Physics, Harcourt, Ch.31-33, 1976.
Mattis, Theory of Magnetism, Springer, 1985.
Cullity, Introduction to Magnetic Materials, Addison-Wesley, 1972.
Comstock, Introduction to Magnetism and Magnetic Recording,
Wiley-Interscience, 1999.
Spin-dependent Transport in Magnetic Nanostructures, Edited by
Maekawa and Shinjo, Taylor & Francis, 2002.
40
Cooling by isentropic demagnetization
Entropy : A measure of the disorder of a system
The greater the disorder, the higher is the entropy. In the magnetic field
the moments will be partly lined up (partly ordered), so that the entropy
is lowered by the field.
The entropy is lowered if the temperature is lowered, as more of the
moments line up.
When the material is demagnetized at constant entropy, entropy can
flow from the system of lattice vibrations into the spin system.
S = k B ln Z
where Z is the partition function

εM 

Z = ∑  exp
k BT 
m = -I 
m=I
N
41
a
c
b
B/T
Entropy for a spin ½ system depends only for the population distribution.
a→b : the system is magnetized isothermally.
b→c : the magnetic field is turned off adiabatically.
H 
Tf = Ti  ∆  where H∆ is the effective field
 H 
µH
≈ 0.5
If we start at H=5T and Ti=10mK
k BT
The first nuclear cooling was done on Cu, Ti=20mK
Tf ~10-7K
Tf ~1.2×10-6K
42
Cu nuclei in the metal
 3.1
Tf = Ti  
 B
where B∆=3.1Oe.
1.2 µ K
Rd
The present record for a
spin temperature is about
280pK. PRL70, 2818 (1993).
43