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Topics in Magnetism
I. Definitions and Atomic Sources
Anne ReillyDepartment of Physics
College of William and Mary
After reviewing this lecture, you should be familiar with:
1. Definitions of M, H and B2. Magnetic units3. Atomic sources of magnetism4. Paramagnetic and Diamagnetic responses
Material from this lecture is taken from Physics of Magnetism by Chikazumi and Solid State Physics by Ashcroft and Mermin (Chp. 31)
Fundamental Definitions
Magnets have two poles (north and south)Poles exert a force on each other
Definition: magnetic pole m (SI units:Weber, Wb=m2kg/s2A)
NS
NS+m1
-m1+m2
-m2
20
21
4 r
mmF
Magnetic force (N):
0=4 x 10-7 H/m
Fundamental Definitions
Electric current in wire exerts a force on a magnetic pole
Definition: magnetic field H (SI units: A/m)
NS
+m-m
mHF Magnetic force (N):
IH
nIH Field from Solenoid: n = # turns/m, I = current
Fundamental Definitions
What happens to a magnet in a magnetic field?+mH
-mH
sinmlHMagnetic torque:
H
x
HmlFx
Translational force ONLY if there is non-uniform H (gradient):
N
S
+mH-mH
HNSl
Fundamental Definitions +mH
-mH
H
N
S
l
Definition: magnetic moment M = ml (SI units: Wb m)
(dipole moment)
HM
Magnetic torque:
HU
MMagnetic Energy:
Fundamental Definitions
Magnetic materials have a density of magnetic moments
Definition: Magnetization M=NM (SI units: Wb/m2 or Tesla T)
N=moments per unit volume
NSM
Fundamental Definitions
To measure magnetization, use induction (vibrating sample magnetometry)
M
H
B
V
dt
dBNAV Induced Voltage
Definition: magnetic flux density B (SI units: Tesla T)
HMB
0
Fundamental Definitions
Magnetization in materials is proportional to applied field H
HMB
0
HM
Definition: magnetic susceptibility (SI units: H/m)
HHB
0
Definition: magnetic permeability (SI units: H/m)
Fundamental Definitions:Review
M = magnetization (T)H = magnetic field strength (A/m)B = magnetic flux density (also called field) (T)
M
H (externally applied)
B
HMB
0
Fundamental Definitions:Review
In this case, the units of M are A/m
HMB
00
Note that sometimes magnetic flux density is defined as:
Gaussian System of Units:
HMB
4Oersted (Oe) CGS unit of magnetic field (H). The Oersted is defined to be the field strength in a vacuum at a distance 1 cm from a unit magnetic pole.
Gauss (G) CGS unit of magnetic flux density (B). A field of one Gauss exerts a force on a conductor of 0.1 dyne/A cm.
Electromagnetic Unit (emu)CGS unit of magnetic dipole moment (M) equal to 1.256637 x 10-5 Oe.
emu/cm3 or emu/cc CGS unit of magnetization (M) In SI units, one emu/cm3 can be interpreted either as 1.256637 mT as a unit of excess magnetic induction, or as 1000 A/m as a unit of magnetic dipole moment per unit volume.
Common system prior to 1980’s. Defined by magnetic poles.
Unit Conversion:
Gaussian unit
(cgs-emu)
Conversion
(SI/cgs)
SI unit
B Gauss (G) x 10-4 = T or Wb/m2
H Oersted (Oe) x 103/4 A/m
M emu/cm3 x 103 =
x 4
A/m
mT
Note: In free space (M=0), 1 G = 1 Oe
Source of Magnetic Moment: Moving Electric Charge (Current)
Atomic Magnetism arises from electron angular momentum and spin
ML
I
vLe-
S
ML= orbital magnetic moment = IA=eL/me
r
Source of Magnetic Moment: Moving Electric Charge (Current)
Atomic Magnetism arises from electron angular momentum and spin
ML
I
vLe-
S
Atomic magnetic moment: SgL BeB
M
Angular momentum vector Spin vector
Gyromagnetic ratio ge~ 2
ML= orbital magnetic moment = IA=eL/me
r
Bohr magneton
m
eB 2
Source of Magnetic Moment: Moving Electric Charge (Current)
Multi-electron atoms: total magnetic moment determined by total J, L and S
Hund’s rules: electrons fill shells such that1. Largest total S is achieved2. Largest total L is achieved3. J=|L-S| (minimum) in shells less than half full and J=|L+S| (maximum) in shells more than half full.
Example:
m = -2 -1 0 1 2 (lz)
1s
2p
2s
3s
4s
3p
3d
Iron (Fe)
26
n =1
n =4
n =3
n =2Maximum values:L=2+2+1+0-1-2=2S=4/2 = 2J=4
Source of Magnetic Moment: Quantum Derivation(for multi-atom systems)
H
HE
VHM
e
eHMTHM
H
HE
VHM
nn
n
TkEn
TkEn
Bn
Bn
)(1)(
)(),(
)(1)(
/
/
(at T=0)
(at T>0)
where
Magnetization Defined to be:
Source of Magnetic Moment: Quantum Derivation(for multi-atom systems)
2
2
1)(
H
F
V
N
H
M
H
F
VHM
In terms of Helmholtz free energy F:
To calculate magnetic properites, consider Hamiltonian in magnetic field and find energy
Source of Magnetic Moment: Quantum Derivation
Write Hamiltonian for atomic electrons in a Magnetic Field (ignore Vatom)
HrA
SHgrAc
ep
m
i
Beii
i
2
1
)(2
1 2 H
= Vector potential
Consider first term:
22
22
Ac
epAAp
c
eprA
c
ep iiiii
Source of Magnetic Moment: Quantum Derivation
Consider first term: With zHH ˆ
Hyxc
e
Hrc
e
ii
i
222
2
2
2
2
4
Hpr
c
ei
HLc
e
Using these relationships:
ii
i prL
CBACBA
ABBA
)()(
Hamiltonian in a Magnetic Field
SHgyxHmc
eHLp
m Bei
iiBi
i
222
2
2
82
1H
H0
)(8
)(' 2222
2
ii
ieB yxHmc
eHSgL
H
H’
Magnetic field dependent terms considered as perturbation:
nn EEE Magnetic field as a perturbation:
Energy (En is ground state energy)
nn nn
n EE
nnnnE
'
2
'
'H'H'
nn EEE Magnetic field as a perturbation:
Energy (En is ground state energy)
nn nn
n EE
nnnnE
'
2
'
'H'H'
nyxnHmc
e
EE
nSgLHnnSgLnHE
ii
i
nn nn
eB
eBn
)(8
'
2222
2
'
2
'
paramagnetism
diamagnetism
In ground state atoms or ions with closed (filled) shells:
nyxnHmc
e
EE
nSgLHnnSgLnHE
ii
i
nn nn
eB
eBn
)(8
'
2222
2
'
2
'
0000 SLJ 0 0
Larmor Diamagnetism is only response:
006
0012
22
2
20
2
222
2
0
ii
ii
rV
N
mc
e
H
E
V
N
rHmc
eE
3
rzyxlet
Summary of magnetic responses:
BM
BBMparamagnetic(aligns with H)
Diamagnetic(by Lenz’s Law, opposes H)
H
M=magnetization
H <<1, negative
<<1, positive
from http://www.geo.umn.edu
Summary of magnetic responses:
BM
BBMparamagnetic(aligns with H)
diamagnetic(opposes H)
H
H <<1, negative
<<1, positive
BBM
ferromagnetic(even without H!)
>1, positive
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