1. Lecture: Basics of Magnetism: Magnetic reponse Hartmut Zabel Ruhr-University Bochum Germany
1. Lecture: Basics of Magnetism:
Magnetic reponse
Hartmut ZabelRuhr-University Bochum
Germany
Lecture overview
2H. Zabel, RUB 1. Lecture: Magnetic Response
1. Lecture: Basic magnetostatic properties2. Lecture: Paramagnetism3. Lecture: Local magnetic moments
Content
31. Lecture: Magnetic Response
1. Definitions2. Electron in an external field3. Diamagnetism4. Paramagnetism: classical treatment of
H. Zabel, RUB
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1. Classical magnetic moments
LLmq
21
ωπrmmq
π21
Aπ2
qωIAm
e
2e
e
γ=====
Magnetic dipole moment = current × enclosed area
Loop current generates a magnetic field
Loop current has an angular momentum
π2qω
TqI ==
γ = gyromagnetic ratio, me= electron mass
1. Lecture: Magnetic ResponseH. Zabel, RUB
Torque and precession
5
Zeeman energy of magnetic moment in an external magnetic field: Bm-=E
⋅Energy is minimized for m || B. B is the magnetic induction or the magnetic field density. Applying B, a torque is exerted on m:
BmT
×=If m were just a dipole, such as the electric dipole, it would beturned into the field direction to minimize the energy. However, m isconnected with an angular momentum, thus torque causes thedipole to precess:
BLγdtLdT
×==
Assuming B = Bz, the precessional frequency is:
zL Bγ=ωBz
ωL is called the Lamor frequency. See also EPR, FMR, MRI, etc.
1. Lecture: Magnetic ResponseH. Zabel, RUB
Bohr magneton
6
Bee
Bohr μme
21
mq
21
m -- === L
An electron in the first Bohr orbit with a Bohr radius rBohr has the angular momentum:
Then magnetic moment is:
L
Bµ
Because of negative charge, L and m are opposite.
γ==e
B me
21μ
µB is the Bohr magneton. [µB] = 9.274 x 10-24 Am2.
Magnetic moment: [m] = A m2
== ω2BohrermL
1. Lecture: Magnetic ResponseH. Zabel, RUB
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LOrb SSpin S of the electon contributes to the magnetic moment:
Smqm
e
=spin
Including orbital and spin contributions, the magnetic moment of an electron is:
)S2Lγ()S2L(mq
21m
e
+−=+=
Electron spin
The missing factor ½ is of quantum mechanical origin and will bediscussed later.
1. Lecture: Magnetic ResponseH. Zabel, RUB
Magnetic field and magnetic induction
8
Oersted field H due to dc current: πr2IH =
Any time variation of the magnetic flux Φ = BA through the loop causes an induced voltage: ( )AB
dtdUind
⋅−=
Therefore B is called the magnetic induction or the magnetic fluxdensity B = Φ/A. In vacuum both quantities are connected via the permeability of thevacuum: HμB 0=
-70
V sμ 4 10A m
⋅= π
⋅[ ] [ ]0 2
V s A V s× = = T2 A m m m
IBμπr
⋅ ⋅ = ⋅ = ⋅ 4
2
V s1 1 T 10 Gm
⋅= =
1. Lecture: Magnetic ResponseH. Zabel, RUB
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1. Magnetization is the sum over all magnetic moments in a volume element normalized by the volume element:
2. Thermal average of the magnetization:
3. Magnetic susceptibility:
4. Magnetic Induction: .
∑=i
imV1M
mVNM
=
HM
χ ,HχM magmag ∂∂
==
( ) HμH μμ)χ(1HμMHμB r0mag00
==+=+=
Definitions
H = magnetic field, usually externally applied by a magnet. µ0 = magnetic permeability of the vacuum. µr = relative magnetic permeability µr = (1+χ) (tensor, or a number for collinearity)
1. Lecture: Magnetic ResponseH. Zabel, RUB
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Potential energy (Zeeman – term):
BmEZeeman
⋅−=
BEm Zeeman
∂∂
−=
2
2
0 BE
VN
HM Zeeman
mag ∂∂
µ−=∂
∂=χ
1. Derivative → magnetic moment:
2. Derivative → Susceptibility:
Potential Energy and Derivatives
1. Lecture: Magnetic ResponseH. Zabel, RUB
The susceptibility is the response f
What is more fundamental, H or B?
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( ) [ ] N=F ×= BvqF
Lorentz force:
Vector potential: [ ] 2∇ mVsT /AB ==B ×=
Zeeman energy: [ ] WsVAsJ ===E •= BmE-
Oersted field: [ ] mA=H =πr2I
H
Magnetization: [ ] mA=M χH=M
1. Lecture: Magnetic ResponseH. Zabel, RUB
Classification
121. Lecture: Magnetic Response
Application of an external field:a. Paramagnetism: χ>0 und µr >1
b. Diamagnetism: χ< 0 und µr <1
Ideal diagmagnetism, realized in superconductors with M and B antiparallel, for χ = − 1 and µr =0.
Magnetic moments align parallel toexternal field, field lines are moredense in the material than in vacuum.
External field is weakend by inducingscreening currents according to Lenz rule. Field lines are less dense thanin vacuum.
c. Ferromagnetism: Spontaneous Magnetization withoutexternal field due to the interaction of magnetic moments
µr attaines very high values forferromagnets, > 104-105
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Consider a non-relativistic Hamilton operator for electrons in an external magnetic field:
( ) 2
21H Aqpme
+=
A
2. Electron in an external field
The vector potential: is defined by the Coulomb gauge: and using
AB
×=∇
2z
z B~smdiamagnetiB~
orbitalismparamagnet
energykinetic
2222
122H aB
meLB
mp
ze
zzBe
+µ+=
222
32 ayx =+
( )z,B,=B 00
Where we assumed an average over the electron orbitperpendicular to the magnetic field:
*Lz is here a dimensionless quantum number
*
1. Lecture: Magnetic ResponseH. Zabel, RUB
Hamiltonian for electron with spin
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BSmeBSgBmE BssZeeman
⋅=⋅µ=⋅= --
Considering the electron spin in the external field with a Zeeman energy:
224-1027.92
- Amm
eB ×==µ
2=sgLandé factor
Bohr magneton
( )
2z
z B~smdiamagnetiB~
orbital spinismparamagnet
energykinetic
2222
122
2H aB
meSLB
mp
ze
zzzBe
++µ+=
+
Hamilton operator for spin and orbital contributions of a single bondelectron then is:
1. Lecture: Magnetic ResponseH. Zabel, RUB
The gS=2 for the electron is put into the Schrödinger equation by „hand“ but would occur naturally using the Dirac equation. The exact value of 2.0023 isdetermined by QED.
Response functions
15
zBm
∂H∂-=
( ) 02 >+µ zzB SL
ze
Bam
Ze 22
62
2
0 6a
mZe
e
µ−
2
2
0 ∂H∂
zmag BV
Nµ=χ -
1. derivative 2. derivative
Diamagnetic responsefor Z electrons
Paramagnetic response *0
1. Lecture: Magnetic ResponseH. Zabel, RUB
*For single atom we can not define a paramagnetic susceptibility. This is onlypossible for an ensemble of atoms.
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3. Properties of the Langevin diamagnetism
χLangevin is constant, independent of field strength; χLangevin is induced by external field; χLangevin < 0, according to Lenz‘ rule; χLangevin is alway present, but mostly covered by bigger and positive
paramagnetic contribution; χLangevin the only contribution to magnetism for empty or filled
electron orbits; χLangevin yiels ⟨a⟩ and the symmetry of the electron distribution; χLangevin is proportional to the area of an atom perpendicular to the
field direction, important for chemistry; χLangevin is temperature independent.
With Z electrons in an atom and an effective radius of <a>
22
0
2
0 6-
6- ∑ a
mZe
VN
me
VN
eeLangevin µµχ ==
i2
ir
1. Lecture: Magnetic ResponseH. Zabel, RUB
Examples for Diamagnetism
171. Lecture: Magnetic Response
Material χLangevin at RT
He -1.9 ⋅ 10-6cm3/mol Xe -43 ⋅ 10-6cm3/mol Bi -16 ⋅ 10-6cm3/g Cu -1.06 ⋅ 10-6cm3/g Ag -2.2 ⋅ 10-6cm3/g Au -1.8 ⋅ 10-6cm3/g
( χ is normalized to the magnetization of 1 cm3 containing one 1 Mol of gas at 1 Oe)
• All noble metals and noble gases are diamagnetic. In case of the nobel metals Ag, Au, Cu mainly the d-electrons contribute to the diamagnetism.
• In 3d transition metals the diamagnetismus is usually exceeded by themuch bigger paramagnetic response.
H. Zabel, RUB
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Anisotropy of diamagnetismus for Li3N
Levitation of diamagnetic materials
1. Lecture: Magnetic ResponseH. Zabel, RUB
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(free = without interactions)Orientation of permanent and isolated magentic moments in an external field Bz = µ0Hz parallel to the z-axis (orientational polarization)
( ) ( )x
xcothxL 1−= Langevin function
Hz
m
θ
4. Paramagnetism of free local moments: classical treatment
TkHm
VN
TkHmLm
VN
mVNM
B
zz
THohe
B
zzz
3
)cos(
2
−
≈
=
= θ
1. Lecture: Magnetic ResponseH. Zabel, RUB
Langevin function
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( )
( )
∫
∫
∫
∫π
π
==
0
0
cos
cos
sin2
cossin2coscos
dθeθπ
dθeθθπ
dΩe
dΩθeθ
TBkθmB
B
B
pot
B
pot TkθmB
TkθE
TkθE
TkB
B
0µ=xθθπ=φθθ=Ω dddd sin2sin
L(x): Langevin-Funktion
µ=
TkBmLNM
B
z 00
( ) ( )xLx
xdsedxd
dse
dssesx
sx
sx
≡−===θ ∫∫
∫−
−
− 1cothlncos1
11
1
1
1
θ= cossusing
1. Lecture: Magnetic ResponseH. Zabel, RUB
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Magnetization of paramagnetic moments in an external field
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
µ0Hz /kBT
Mag
netiz
atio
n
Hz
1. Lecture: Magnetic ResponseH. Zabel, RUB
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Curie-Suszeptibilität χCurie in HTA with the Curie-constant C:
TC
Tkm
VN
HM
B
z
zCurie =
µ=
∂∂
=χ3
20
B
z
km
VNC
3
20µ
=
Paramagnetic Susceptibility
( )zzBz SLm 2+= µMagnetic moment:
χ1
T
Linear dependence fullfilledat high temperatures. At lowT often deviations observeddue to interactions.
But: However, magnetism is not a classical problem, thus Langevin function is only a roughapproximation. As quantum mechanics allows only discrete values for the z-component of the magnetic moments, a different approach has to be chosen ⇒Brillouin function replaces the Langevin function. –The susceptibility of superparamagnetic particles containing a macrospin can betreated classically as the spin orientation of nanoparticles in the field is continuous.
1. Lecture: Magnetic ResponseH. Zabel, RUB
Susceptibility of the Elements
23H. Zabel, RUB 1. Lecture: Magnetic Response
From J.M.D. Coey
Paramagnetic
Diamagnetic
Summary
241. Lecture: Magnetic Response
χ1
T
22
0 6- a
mZe
VN
e
cdiamagnetiLangevin µ=χ
χ
T
3. Paramagnetic response (HTA):
2. Diamagnetic response:
( )
2z
z B~smdiamagnetiB~
orbital spinismparamagnet
energykinetic
2222
122
2H aB
meSLB
mp
ze
zzzBe
++µ+=
+
1. Hamilton operator for an electron in an external field:
Tkm
VN
B
zicparamagnetCurie 3
20µ
=χ
H. Zabel, RUB