Post on 25-Dec-2021
1
Chapter Eleven Diamagnetism and ParamagnetismThe 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.
2
H (Oe)
M (e
mu/
cm3 )
H (Oe)
M (e
mu/
cm3 )
diamagnetism
paramagnetism ferromagnetism
Magnetic susceptibility per unit volume
HM
≡χ M : magnetization, magnetic moment per unit volumeB : macroscopic magnetic field intensity
In general, paramagnetic material has a positive susceptibility (χ >0).diamagnetic material has a negative susceptibility (χ<0).
CGS
3
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.
1895 Curie proposed the Curie law
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
1903
1859~1906
P. Curie
4
Reviews, in CGS
H M 4HB
µπ =+=
VNmM
= : magnetic dipole moment
LL
Bi
ii 2mepr
2mem µ−=−=∑ ×−=
S
oBi
ioB gsgm µµ =∑=
[emu/cm3]
Bohr magneton
Electronic g-factor go=2.0023
224-B 1Vs/m1T J/T;109.27
2me
=×==µ
Angular momentum
Spin
I vµ
L and S are quantum operators with no basis in classical physics.
electron
m
5
Theory of diamagnetismwas first worked out by Paul Langevin in 1905.
Motion of the electron is equivalent to current in the loop.
Faraday’s law
Negative magnetism present in an applied fieldeven though the material is composed of atoms that have no net magnetic moment.
dtdΦB−=Emf
[volt]
Consider a circular loop with radius r, the effect of applied field H is to change µ by an amount ∆µ in negative direction.
22
2A
6mce
AN RZρχ −=
[emu/Oe-cm3]
I
H
µr
1872~1946Paul Langevin
6
The change in moment of a single orbit :
dtdH
2r
dtdH
r 2r
dtdHAE
2
−=−=−==ππ
Emf
dtdH
2mcer
mceE
dtdva −=== ∫∫
−= dH
2mcerdv
H2mcerv −=∆
H4mc
rev2cer
2
22
−=∆=∆µ the plane of the orbit is perpendicular to the applied field
R
All possible orientations in the hemisphere
( )( ) 22/
0
22
2
22222
32dsin2 sin
2
AdA sinsinr
RRRR
R
RR
==
==
∫
∫π
θθπθπ
θθθ
7
A single electron with consideration of all possible oriented orbits,
H6mcev
2ce
2
22RR−=∆=∆µ
For an atom that contains Z electrons,
∑∑==
−=−=∆zz
RR1i
2i2
2
1i2
2i
2
6mcHeH
6mceµ Ri is the radius of the ith orbit
Susceptibility, 22
2A
mc6e
AN RZρχ −=
The number of atoms per unit volume : NAρ/A
Avogadro’s number
Atomic mass
Volume density of mass
H6mc
eA
N1i
2i2
2A
−=∆ ∑
=
z
Rρµ
On the volume basis,
Carbon (radius =0.7Å)
( ) ( )( )( ) ( )
36-
2821028
210323
cmemu/Oe101.5
cm107.0cm/s103g1011.966esu1080.4
12.01gg/cm22.21002.6
−×−=
×××
××−= −
−
−
χ
Experimental value : -1.1×10-6 emu/Oe-cm3
Classical Langevin result
8
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.
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.
The other consistency is thatχ is independent of temperaturefor most diamagnetic materials.
Quantum theory of diamagnetismresults in the same expression.
Negative susceptibility
magnitude : 10-5 ~10-6 cm3/mole
9
Diamagnetic materials
Closed-shell electronic structures result in no net momentThe monatomic rare gases He, Ne, Ar, …Polyatomic gases, H2, N2, …Some ionic solids, NaCl, …Some covalent solids, C, Si, Ge, …
Superconductors : Meissner effect
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.
TC
=χ θχ
−=
TC'
Curie law Curie-Weiss law (1907)
in general
Pierre Weiss
1865~1940
10
Quantum theory of diamagnetism
Hamiltonian VAH QcQp
m21 2
+
−=
Terms due to the applied field
( ) 22
2
2mce
mc2e' AAAiH +∇•+•∇=
A
×∇=B
In the uniform magnetic field kBB =
( )k
yxj
xzi
zy
kjikz
jy
ix
xyzxxz
zyx
∂∂
−∂
∂+
∂∂
−∂∂
+
∂∂
−∂∂
=
++×
∂∂
+∂∂
+∂∂
=×∇
AAAAAA
AAAA
One solution k0j2
xBi2
yB++−=A
( )222
22
yx8mc
Bex
yy
xmc4
Be' ++
∂∂
−∂∂
=iH
zL~
11
z-component of angular momentum
∂∂
−∂∂
−=x
yy
xL~ z i paramagnetism
For diamagnetic materials with all electrons in filled core shells,
the sum over all electrons vanishes∑ + zz 2SL
For a spherically symmetric system 222 r32yx =+
22
2222
2
22
B12mce
yx8mc
Be'r
E =+=the additional energy
the associated magnetic moment B6mce
B'
2
22 rE−=
∂∂
−=µ
in agreement with the classical Langevin result
12
Vibrating Sample Magnetometer (VSM)
→→ measuring sample’s magnetic moment
oscillate sample up and downmeasure induced emf in coils A and Bcompare with emf in coils C and D from known magnetic moment
13
14
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.
15
A unit volume material contains n atoms with a magnetic moment µ each.
θµµ cosHH −=•−=E
TkcosH
TkE
BB dsin2dA dnθµ
θθπ eKeK ==−
∫=∫=π
θµ
θθπ0
TkcosHB sind2dnn eK
∫=∫=π
θµ
θθθπµθµ0
TkcosH
0
B sincosd2cosdnM eKn
Let a = µH/kBT,
−
−+
=∫
∫=
∫
∫=
−
−
−
−
a1n
dx
dx xn
sind
sincosdnM
aa
aa
1
1
ax
1
1
ax
0
cosa
0
cosa
eeee
e
e
e
e
µµ
θθ
θθθµπ
θ
πθ
( )
−=
a1acothnµ
H
θµ
Let the direction of each moment be presented by a vector, and let all vectors be drawn thru. the center of the sphere.
# at angle θ θ+dθ
total #
16
Langevin function L(a)=coth(a)-1/a
a=µH/kBT
M/M
o
L(a) can be expressed in series,
...9452a
45a
3aL(a)
53
−+−=
When a is small (<0.5),L(a) is a straight line w/. a slope 1/3.
When a is large, L(a) → 1.
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 with H.
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The Langevin theory also leads to the Curie law.
For small a,Tk
H3
n3
anMB
µµµ==
Tk3n
HM
B
2µχ ==TC
= B
2
k3nµ
=C
Oxygen is paramagnetic and obeys the Curie law w/. χ=1.08×10-4emu/gOe.
Bµ
χµ
85.2 moleerg/Oe102.64
1002.6)emu/gOe1008.1)(erg/K1038.1)(g/mole32(3
nTk3
20-
23
416B
=−×=
×××
==−−
Even in heavy atoms or molecules containing many electrons, each with orbital and spin moments, most of the moments cancel outand leave a magnetic moment of only a few Bohr magnetons.
Typically, ( )( )( )( ) 0064.0
300Kerg/K1038.110000Oeerg/Oe1064.2
TkHa 16
20
B
=×
×== −
−µ
smaller enough L(a)=a/3
<0.5
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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,
considering the interaction between elementary moments.θ
χ−
=T
CCurie-Weiss law (1907)
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
TMHM
HM
tot
C=
+==
γχ ( ) H M T CC =− γ
θγχ
−=
−==
TTHM C
CC
where θ (=Cγ) is a measure of the strength of the interaction.
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Theoretical predictions
θ
CCCθθ
χ−=
−= T1T1
Intercept reveals information of θθ (FeSO4)<0, θ (MnCl2)>0
θ>0 indicates that the molecular field aids the applied field, makeselementary moments align with H, and tends to increase χ.
θ<0 indicates that the molecular field opposes the applied field, makeselementary moments disalign with H, and tends to decrease χ.
20
Precession of atomic magnetic moments about H,
like gyroscope, angular momentum in the presence of gravitation.
H ×= µτ
dtLdmr
=×= gτ
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.
21
Quantum theory of Paramagnetism
The central postulate of QM is that the energy of a system is discrete.quanta of energyimproving the quantitative agreement
between theory and experiment.
θ
H J=1/2 J=2H
Classical case Quantum case
θ is continuous variable. θ is restricted to certain variables.Space Quantization
22
The orbital and spin magnetic moments for an electron
orbitorbit p2mc
e22mc
emc4
e −===
ππµ hh
spinspin pmce
221
mce
mc4e
−===ππ
µ hh
In general
p2mc
e g−=µ 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.
Bgg µµ 1)J(J1)J(Jmc2e
+=+=
In the direction of H, Bgg µµ JJH mmmc2e
==
There are 2J+1 numbers of allowed azimuthal quantum number mJ : J, J-1, … , -(J-1), -J.
θ µµH
H
23
For a free atom, the g factor is given by the Landé equation
1)2J(J1)L(L1)S(S1)J(J1
++−+++
+=g
The energy levels of the system in a magnetic field
2µB
ms µs1/2 -µ
-1/2 µ
HgmHE BJ µµ =•−=
For a single spin with no orbital moment, mJ=±1/2 and g=2.
The equilibrium population for the two level system :
)TH/kexp()TH/kexp()TH/kexp(
NN
)TH/kexp()TH/kexp()TH/kexp(
NN
BB
B2
BB
B1
µµµ
µµµ
−+−
=
−+= lower level
upper level
Total number of atoms is conserved.
N = N1+N2 = constant
24
The resultant magnetization for N atoms per unit volume
( )xtanh NN)N(NM xx
xx
21 µµµ =+−
=−= −
−
eeee where
TkHx
B
µ=
For x<<1, tanh(x)~x andTkHN xNM
B
2µµ =≅
For a system with known g and J,
Potential energy of each moment in H HgmU BJ µ−=
The probability of an atom w/. energy U TH/kgmTU/k BBJB µee =−
Therefore,
Boltzmann statistics
∑
∑=
J
BBJ
J
BBJ
m
TH/kgmm
TH/kgmBJgm
NM µ
µµ
e
e
( )
(x)BJN 2Jxcoth
2J1
2Jx12Jcoth
2J12JJNM
JB
B
µ
µ
g
g
=
−
++
=
Brillouin functionwhere
TkHJx
B
Bµg=
mJ : J, J-1, … , -(J-1), -J
25
The saturation magnetizationBHo JNNM µµ g==
Number of atoms per unit volume
The maximum moment of each atom in the direction of H
( )
−
++
==2Jxcoth
2J1
2Jx12Jcoth
2J12J(x)B
MM
Jo
Brillouin function, 1927
When J=∞, the classical distribution, the Brillouin function reduces to the
Langevin function.( )
x1xcoth
MM
o
−=
When J=1/2, the magnetic moment consists of one spin per atom.
( )xtanhMM
o
= xx<<1
TkHN
TkHJNgxNM
B
2
B
2B22
Hµµµ ==≅
26
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.
TkHx
B
µ=classical
TkH
TkHJx
B
H
B
B µµ==
gquantum≠
µ : net magnetic moment in the atom
KCr(SO4)12H2O
Henry, Phys. Rev. 88, 559 (1952)
27
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=2/5the 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 .
BB μ1)J(JJμ +== ggµ
L=0 no orbital component
BB μ87.3μ1)23(
232 =+=
“Spin-only” moment
28
Gd(C2H3SO4)3•9H2O
metals
29
What magnetic moment a metal should have?
Hund’s rule incorporate coupling of magnetic moment to determine the magnetic state
(1)Maximize S subject to the Pauli exclusion principle
ms=±1/2 put spins (in different states) parallel∑= sm
(2)Maximize L subject to the Pauli exclusion principle
m= -, -+1, …, -1, fill higher m state preferentially
∑= m
(3)
J= L-S, L-S+1, …, L+S-1, L+S allowed valuesWhether it is a minimum or maximum in the ground state depends on filling function.
+−
=full half than more is shell when SLfull half than less is shell when SL
J
30
Magnetic stateJ
12S L+ L : 0, 1, 2, 3, 4S, 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)
3210-1-2-3
L=0 and S=7/2
J=L+S=7/2
Eg. Gd3+ 4f75S25P6
(n=4,=3)
7/28SState
31
3210-1-2-3
L=3 and S=1/2
J=L-S=5/2
Eg. Cs3+ 4f15S25P6
(n=4, =3)
5/22 FState
3210-1-2-3
L=5 and S=1
J=L-S=4
Eg. Pr3+ 4f25S25P6
(n=4, =3)
43 HState
3210-1-2-3
L=5 and S=5/2
J=L+S=15/2
Eg. Dy3+ 4f95S25P6
(n=4, =3)
15/26 HState
32
Rare earth ions
1)J(Jgp += For Sm3+ and Eu3+, there are discrepancies between experimental p and theoretically calculated value.
High states of the L-S multiplet.
33
Quenching of the orbital angular momentum : Iron group ions
1)S(S2p +=Effective number of Bohr magneton 1)J(Jgp +=
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 contribution of orbital motion to magnetic moment)
34
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.
−
∝
∝
+
∝
− 2yxf(r) )((r)YR
z f(r) )((r)YR2
yxf(r) )((r)YR
11
10
11
i
i
n
n
n
θφ
θφ
θφ
In a free atom, a single electron in a p-state.
Three degenerate p states
35
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential ϕ about the nucleus,
222 B)z(AByAxe +−+=ϕ
Reconstructing unperturbed ground state zf(r) Uyf(r), Uxf(r),U zyx ===
+ perturbed term eϕ
0UUUUUU xzzyyx === ϕϕϕ eee
)IA(I
]dxdydzyAx[Axf(r)
]dxdydzB)z(ABy[Axxf(r)UU
21
2242
22222xx
−=
−=
+−+=
∫∫ϕe
Non-diagonal elements
Diagonal elements
36
)IB(I
]dxdydzzBy[Byf(r)
]dxdydzB)z(ABy[Axyf(r)UU
21
2242
22222yy
−=
−=
+−+=
∫∫ϕe
)IB)(I(A
]dxdydzzzB)[y(Af(r)
]dxdydzB)z(ABy[Axzf(r)UU
12
4222
22222zz
−+=
−+=
+−+=
∫∫ϕe
The orbital moment of each of the states
0ULUULUULU zzzyzyxzx ===
Crystal field splits the original degenerate states into non-magnetic states separated by energies >> µH.
quenching of the orbital angular momentum
37
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)
• Pauli pointed out that only electrons near EF (±kBT) can flip over.
TkHnM
B
2Bµ= where n is # of conduction electrons
This requires each electron spin align w/. H, independent of all others.
Fermi-Dirac distribution
0.00.10.20.30.40.50.60.70.80.91.01.1
ε
f(ε,T
) T=0T≠0
εF
kBT
εFεF
H=0 H≠0
38
Fraction of electron participating FF
B
TTTk
=ε
FB
2B
B
2B
F TkHn
TkH
TTnM µµ
=
= independent of temperature
2µH
DOSDOS
Consider the density of states D(ε) in the presence of HSpin ↑ electrons lowered in energy by µH in H ↑Spin ↓ electrons raised in energy by µH in H ↑
HεF
ε
↑↓
µH
↑
↓
µH
flip spins
ε ε
39
( ) HT2k
3NH23NH)(DM 2
FB
2
FF µµ
εµεµµ ===−= ↓↑ NN
Pauli spin magnetization
independent of temperature
The concentration of electrons w/. magnetic moment parallel to H
The concentration of electrons w/. magnetic moment antiparallel to H
H↑, T=0K
H)(D21)(Dd
21)H(Dd
21
F0H
FF
µεεµεεε
µ
+≅+= ∫∫−
↑ εεN
H)(D21)(Dd
21)H(Dd
21
F0H
FF
µεεµεεε
µ
−≅−= ∫∫↓ εεN
valid when kBT<<εF
40
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.
41
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.
ZlnkS B=where Z is the partition function
∑=
=
=
Im
-Im
N
B
M
TkexpZ ε
42
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.
B/T
= ∆
HHTT if where H∆ is the effective field
If we start at H=5T and Ti=10mK 5.0Tk
H
B
≈µ Tf ~10-7K
The first nuclear cooling was done on Cu, Ti=20mK Tf ~1.2×10-6K
a
bc
43
=
B3.1TT if
Cu nuclei in the metal
where B∆=3.1Oe.
The present record for a spin temperature is about 280pK. PRL70, 2818 (1993).
Rd
1.2 µK