MagnetismMagnetismPhysics 355

0.00005 T

Soft Gamma Repeater 1806-20, is the most powerful known magnetic object in the universe. Only 10 of these unusual objects have been discovered. With a magnetic-field strength of 100 billion T.

National Magnet Lab (FL) 45 T

From whence does magnetism come?From whence does magnetism come?

1. No answer from classical physics... 1. No answer from classical physics...

2. What is the role of conduction electrons?2. What is the role of conduction electrons?

3. What is the effect of localized electrons? 3. What is the effect of localized electrons?

4. How do we make powerful permanent magnets? 4. How do we make powerful permanent magnets?

Magnetism Electron-Electron

Interactions

QuantumMechanics

QuantumMechanics

Why is magnetism a quantum effect?

• Classically, the change in energy of a particle with respect to time is

...But in a magnetic field, the force on the electron due to the field isalways perpendicular to the electron’s velocity. This would mean thatthe energy of a system should not be able to change in the presence ofa magnetic field – Yet, it does!!

vF dt

dE

Orbital Angular Momentum

All known magnetic properties originate

from the attributes of charged particles.

The right-hand rule determines the direction of the magnetic moment of a current-carrying loop. The direction of the electron's angular momentum vector L can be obtained using the right hand rule for angular momentum.

Lm

eIAm 2

Magnetic

dipole moment

QuantumMechanics

U B

Zeeman EffectZeeman EffectThe Effect of Magnetisation on the Nature of Light Emitted by a Substance

P. ZeemanNature, vol. 5511 February 1897, pg. 347

Zeeman EffectZeeman Effect

When considering atomic spectra, the atomic energy levels, the transitions between these levels, and the associated spectral lines are produced with no magnetic fields influencing the atom.If there are magnetic fields present, the energy levels are split into a larger number of levels and the spectral lines are also split. This splitting is called the Zeeman Effect.

The total Hamiltonian of an atom in a magnetic field is:

                                                            where H0 is the unperturbed Hamiltonian of the atom, and the sums over α are

sums over the electrons in the atom. The term             

is the spin-orbit interaction for each electron (indexed by α) in the atom. If there is only one electron, the sum contains just a single term. The magnetic potential energy

                                              is the energy due to the magnetic moment μ of the α-th electron. It can be written as a sum of the contributions of the orbital angular momentum L and the spin angular momentum S, with each multiplied by the appropriate Landé g-factor, gL or gS. By projecting the vector quantities onto the z-axis, the

Hamiltonian may be written as

                                                                                    where the approximation results from taking the g-factors to be gL = 1 and gS

2. The summation over the electrons was omitted for readability. Here, Jz = Lz +

Sz is the total angular momentum, and the spin-orbit (LS) coupling term has

been combined with H0 and written as Hat.

B

2

g BE

Zeeman EffectZeeman Effect

MJ = +½

MJ = ½1

2

BE g B

The probability that an ion is in a state with Jz = MJ is proportional to

B B BE k T g B k Te e / /

Spin Quantization 2Sm

e

m

QuantumMechanics

Magnetic SusceptibilityMagnetism

B MHB 0

m/AT 104H/m104 770

H – magnetic field intensity (external)

M – magnetization, dipole moment per unit volume (internal)

H

M HHB 10

Magnetic ClassificationsMagnetism

• Paramagnetic

• Diamagnetic

• Ferromagnetic

0

0

0 wheneven 0 HM

ExampleExample

• Suppose a long cylindrical specimen with a uniform magnetization M parallel to its axis is placed in a uniform applied magnetic field BA, applied along the axis of the cylinder. Suppose the specimen has a magnetic susceptibility .

• Find the magnitude of the magnetization M, the magnetic induction B and the magnetic field intensity H.

M

Example - AnswersExample - Answers

• If the material is paramagnetic, then the magnetization is in the same direction as the applied field and it produces a field 0M in the same direction as the applied field.

A0A 1 BMμBB )(

0

A

0

0A

0 11

BMBB

M)()(

0

A

0

BM

BH

Example - AnswersExample - Answers

• If the material is diamagnetic, then the susceptibility is negative. The results are the same as before, except that B < BA.