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Unformatted text preview: 7 Magnetic properties
B = µ0(H + M) Magnetic induction (7.1) (magnetic flux density)
H: magnetic field density,
µ0 = 4π × 10-7 Vs/Am: permeability of free space
Vacuum: M = 0
In matter: M= 1
V ∑m sum of magnetic i i moments / volume (7.2) Fig. 7.1 A loop current gives rise to a
torque in a magnetic field given by
IA × B. The quantity IA is defined as
magnetic moment. Lorentz force on conduction loop ⇒ torque
with magnetic moment: m=IA
7-1 (7.3) Potential energy of a dipole in a homogeneous magnetic field E (α ) = α α ∫ Tdα ’ = π∫ mb ⋅ sin α ’⋅dα ’ = −mB ⋅ cos α
/2 /2 ↑ choise of zero point: E = 0 for m ⊥ B
E = -m ⋅ B
Force in inhomogeneous B-field:
F = -grad E = grad(m ⋅B) (7.6) Magnetic susceptibility χmag: M = χmag H (7.7) ⇒ B = µ0(H + M) = µ0(1+χmag)H (7.8) 1+χ = µ: permeability
χmag = µ -1 M = χmag H = (µ - 1)H (index ″mag″ often omitted) χ, µ: generally symmetric tensors,
here: only isotropic media and cubic crystals. Diamagnetism:
(7.10) χ<0⇔µ<1 Magnetization opposite to applied field χ << 1 ⇔ µ ≈ 1 e.g., Cu: χ = -7×10-6 Generally χ = M/H = const.
χ(T) = const. F = grad(m ⋅B), m ∝ -B ⇒
• Diamagnetic materials are driven out of regions of high field
• Homogeneous field: orientation such that M minimum
E = -m B > 0
All materials exibit diamagnetism.
Can be masked by other magnetic properties. Paramagnetism
χ>0⇔µ>1 Magnetization in directiion to applied field
7-3 χ << 1 ⇔ µ ≅ 1
χ strongly temperature dependent.
Orientation of magnetic moments against thermal disorder.
Al, T = 293K: χ = + 21×10-6 Pt, T = 293K: χ = + 264×10-6 O2, T = 90K (liquid): E.g.: χ = + 3620×10-6 F = grad(m ⋅B), m ∝ B ⇒
• Paramagnetic materials are driven in regions of high field
• Homogeneous field: orientation such that M maximum.
Generally χ =M/H = const. Ferromagnetism
χ = χ(H) ⇔ µ = µ(H)
7-4 dependent on T and history ⇒ hysteresis loop
χ(H) ≈ µ(H) >> 1
E.g.: Fe, Co, Ni
initial susceptibility ≈ 104
Also antiferromagnetism and ferrimagnetism (see Fig. 7.2) Fig. 7.2 Schematic illustration of the arragement of magnetic moments in ferromagnetism,
antiferromagnetism and ferrimagnetism. Connection between orbital momentum
and magnetic moment 7-5 Magnetic moment µ = I A
Electron on spherical orbit: I= dQ − e
= − ω
µ = − r 2ω
2 A = πr (7.11) Fig. 7.3 Electron on a spherical orbit. l = r × p = m0 r × v l = m0 r 2 ω Angular momentum:
v =ω ×r (7.12) ⇒ µ =− e
2m 0 (7.13) Also holds in atomic range (Einstein-De Haas experiment)
l= l ( l + 1) h , lz = m lh, l = 0 ,..., n − 1 m l = − l ,... l magnetic quantum number 7-6 µB = e
2 m0 µ l = − gl µ B Bohr magneton (7.14) l
h gl: Lande’s g-factor (7.15) orbital momentum: gl = 1 µ l ,z = − g l µ B lz
= − ml µ B
h ⇒ splitting of energy levels in B-field Diamagnetism
7-7 (7.16) Fig. 7.4 an electron in an atom can be
regarded as an atomic gyroscope. Electron in atom with momentum l: atomic gyroscope.
Magnetic moment: ⇒ T= µ = − g l ( µ B / h )l dl
= µ × B = gl (µ B / h) B × l
dt (7.17) T ⊥ l ⇒ only direction of l changed.
⇒ precession of the atomic gyroscope with T= dl
=ω p ×l
dt (7.18) follows from dl = lsinα ⋅ ωpdt (see Fig. 7.4) Analogy to motion on spherical orbit: 7-8 F= (7.17), (7.18) dp
dt ⊥ p ⇒ F= dp
dt =ω × p gl µ B
⇒ ωL =
h (7.19) ωL: Larmor frequency (precession frequency ωp)
γ: gyromagnetic ratio
ωL independent of α !
e.g., same ωL for µl and µ-l.
⇒ precession of all Z electrons with ωL about B axis. (7.11) ⇒ magnetic moment due to precession of i-th electron: 7-9 µ Larmor e2
= − r⊥i ω L = − ( xi + yi2 )ω L
2 (7.20) Approximation: charge distribution nearly spherical ⇒
xi2 + y i2 = 22
3 Ze 2 natoms
⇒ M = − eZn atoms ω L < r >= −
< r2 > B
6m 0 ⇒ where χ diamag .
natoms = µ 0 Ze 2 n atoms
< r2 >
6m0 (7.21) (7.22) N atoms
V Interpretation of diamagnetism in terms of Lenz’s law:
When magnetic flux through an electrical circuit is changed,
an induced current is set up in such a direction as to oppose
the flux change (χ < 0).
⇒ Electrons try to screen the interior of a body against an
external magnetic field. Spin of the electron 7-10 s = s ( s + 1)h,
s z = m s h, s= 1
2 ms = ± (7.23) 1
2 µ s = −gs µ B spin quantum number s
h (7.24) gs = 2.0023 (7.25) gs from splitting of the energy levels in B-field and from
relativistic quantum field theory (Dirac 1928). µs,z = -gsµBms (7.26) = ± 1.00116 µB
2 possible spin
splitting of energy
levels in magnetic
field. Fig. 7.5 Illustration of the two possible spin orientations
in a magnetic field (in z-direction). 7-11 ...
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This note was uploaded on 01/19/2010 for the course MATERIALS M504 taught by Professor Adelung during the Spring '02 term at Uni Kiel.
- Spring '02