Magnetism

Question 1

What is the magnetic field B?

Answer 1

Is just an E-field perceived from a different reference frame (special relativity)

See experiment of a neutral wire and the two frame: observer moving and observer still

Question 2

What is the Lorentz force?

Answer 2

Force that B fields exert on a charged particle: F = mv x B

Question 3

Force, torque and potential energy of a magnetic dipole?

Answer 3

F = grad(mB)
T = m x B
U = -mB
m = magnetic dipole = magnetic moment

Question 4

What tells the Ampère-Maxwell’s law?

Answer 4

The sources of B-fields are currents ≠ source of E-fields which are charges.
A time varying E-field generates a B-field and viceversa

Question 5

What tells Gauss’s law for magnetism?

Answer 5

The flux of B-field across any closed surface is zero –> magnetic monopoles do not exist –> magnetic field lines always close on themselves

Question 6

What is the inductance?

Answer 6

Tendency of an electrical conductor to oppose a change in the electric curent flowing through it.
An induced current by a magnetic field will oppose the primary current (AC current)

Question 7

What is the magnetization M?

Answer 7

Is the net magnetic moment per unit volume; M = 1/V ∑ mi = ϗ H
M = N < m> where < m> = average magnetic dipole moment

Question 8

What is the auxiliary field H?

Answer 8

If we consider Ampère’s law, we can write ∇xB= μ0 J = μ0(Jcond + Jbound)

We define ∇xH = Jcond

Question 9

Why is ∇xH = Jcond ?

Answer 9

We notice that ∇xM = Jbound thus replacing in the Ampère law we find that *Jcond = ∇x(B/μ0 - M) *thus Jcond = ∇xH

• In free space: M=0, H = B/μ0
• In a material: ∇xB = μ0(Jcond + Jbound) and J_cond = ∇xH

Question 10

How do we get ∇⋅H = -∇⋅M ?

Answer 10

From Gauss’s law: ∇⋅B = 0 thus ∇⋅H = ∇⋅(B/μ0 - M) –> ∇⋅H = -∇⋅M

Question 11

What tells ∇⋅H = -∇⋅M ?

Answer 11

The discontinuity of magnetization M is a source of H-field
∇⋅M ≠ 0 if M is not homogeneous

Question 12

What are the sources of H-field?

Answer 12

1. Current
2. Discontinuity of M
   H = H_ext + H_dipolar

Question 13

What is H_ext ?

Answer 13

H-field due to conduction:
∇⋅Hext = Jcond
Integral over a closed surface of H_Ext = I_cond

Question 14

What is H_dipolar?

Answer 14

Is called demagnetizing field; it is due to the discontinuity of M:
∇⋅Hd = -∇⋅M
It is created by the magnet itself and exist both inside and outside a magnet.

Question 15

What is a magnet?

Answer 15

A magnet is a collection of microscopic magnetic dipoles, each dipole produces a B-field B_dip
H_d = ∑ B_dip/μ0 - M/3 where M/3 is the H-field produced inside a uniformly magnetized sphere

Question 16

Relationship between B, M and H

Answer 16

• Outside a magnet: B = H
• Inside a magnet: B = H + M where* H* is oriented opposite to M

Question 17

What is diamagnetism?

Answer 17

Substance that is feebly repelled by a magnet –> is a property of every atom and molecule
Ex: H2O, Cu, H, Air, …

Question 18

What is paramagnetism?

Answer 18

Substance that are attracted towards the region of stronger magnetic field –> magnetic moments will aligne to an applied B-field
Ex: Al, Pd, O, alkaline metals, …

Question 19

What is ferromagnetism?

Answer 19

Substance that behaves like iron and magnetite, which are strongly attracted by a magnet –> spontaneous ordered and parallel magnetic moments below T_Curie.
Ex: Fe, Co, Ni and their alloys

Question 20

What happens when a B-field is applied to an atom? (Orbital diamagnetism)

Answer 20

Change in angular momentum proportional to B and this change subtract orbital magnetic moment.

Question 21

What happens when a B-field is applied to an atom? (classical model paramagnetism)

Answer 21

Magnetic moments of the unpaired electrons align themselves with the field, causing the material to become magnetized

Question 22

What is the magnetic suceptibility ϗ_m?

Answer 22

When the magnetization is linear dependent on the applied field, i.e. M = ϗ_m H
ϗ_m is the magnetic suceptibility
For diamagnets is ϗ_m < 0, for paramagnets > 0

Question 23

What is the magnetic permeability?

Answer 23

It tells the ability of a material to support a magnetic field within itself, i.e. the facility of B-field to permeate the material.

For any material which M is proportional to H we have:
B = μ(H+M) = μ0(1+ϗ_m)H
We call μ = μ0(1+ϗm)H the magnetic permeability.

Question 23

What are the peculiarity of B = μH?

Answer 23

This holds only for simple material, i.e. linear, isotropic and homogeneous like paramagnets and diamagnets, i.e.* μ_p and μ_d* are constant

For ferromagnets μ_f is not a constant –> ferromagnetic hysterisis (curve M vs. H)

Question 24

What is a ferromagnetic hysteresis?

Answer 24

Ferromagnetic materials are described by an **irreversible non linear response** of magnetization *M* to an imposed magnetic field *H*

Question 25

What are three important values in hysteresis curve?

Answer 25

* *M_s*: saturation magnetization; all magnetic dipoles are aligned * *M_r*: remanence magnetization; magnetization in absence of an external field * *H_c*: coercive field or coercivity; is the field required to reduce the magnetization to zero.

Question 26

Orbital magnetic moment *m_l*

Answer 26

An electron revolving in an orbit is equivalent to a tiny current loop --> rise of a magnetic dipole = magnetic moment *m_l* = -e/2m_e x **l**; where |**l**| = *ℏ√l(l+1)* is the QM operator for orbital moment, *l* is the orbial quantum number l =0,1,2,3

Question 27

What defines the orbital quantum number *l* ?

Answer 27

It defines the orbital symmetry of the wavefunction (s,p,d,f,...) If we chose z as quantization axis is: *-l ≤ lz ≤ l* and *m_lz* = -eℏ/2m_e x **l_z** = -µ_B**l_z** µ_B = Bohr magnetron

Question 28

Spin magnetic moment *m_s*

Answer 28

An electron posses an intrinsic angular momentum, unrelated to any orbital motion, called spin; *m_s* = -g_e x -e/2m_e x **s**; where |**s**| = *ℏ√s(s+1)* is the QM operator for spin moment, *s* is the spin quantum number *s = 1/2*; *g_e* = Landé factor for an electron ≈ 2

Question 29

Spin magnetic moment with z as quantization axis

Answer 29

*s_z = ± 1/2* and *m_sz = -eℏ/2m_e x **s_z** = -2µ_B**s_z**

Question 30

What is the total magnetic moment?

Answer 30

*m_tot = m_l + m_s = -µ_B(**l** + 2g_e **s**)*

Question 31

Orbital and spin magnetic moment for many electron atom

Answer 31

**S** = ∑ **s_i**, **S_z** = ∑ **s_iz**, **L** = ∑ **l_i**, **L_z** = ∑ **l_iz** We define **J** = **L+S** as the total angular momentum; **|L-S|≤ J ≤ |L+S|** and |J| = *ℏ√J(J+1)*; **J_z** = **-J, -J+1, ..., J**

Question 32

Which are the 3 Hund's rule? What are they used for?

Answer 32

To determine the ground state of a multi-electron atom: * Total spin **S = ∑ s_i** is maximized * Total orbital moment **L = ∑ l_i** is maximized * **L & S** couple parallel (**J = |L+S|**) if the electron shell is more than half filled; if less they couple antiparallel (**J = |L-S|**)

Question 33

What is the spin-orbit coupling?

Answer 33

An intrinsic interaction between *m_l* and the magnetic field produced by *m_s*: *H_so = λ**LS*** where λ = spin orbit coupling parameter. Two reference frame: * nucleus sees an electron orbiting around * electron sees a + charged nucleus orbiting around it, giving rise to an orbital current --> magnetic dipole

Question 34

How is described the classical model for paramagnetism?

Answer 34

Langevin function; the larger the argument in Langevin function (mag.energy/thermal energy) the larger the probability that the average projection of the moment align with the field (Langevin ≈ 1) --> All orientation of the magnetic moment are possible (continuous)

Question 35

How is described the QM model for paramagnetism?

Answer 35

Brillouin function; If we consider a level described by *n, l, **S, L, J*** then it has a (**2J+1**) degeneracy, which is removed by an external magnetic field that *split the states* according to **J_z** --> discrete set of J_z values

Question 36

What is the main difference of non-magnetic and magnetic d-band metals?

Answer 36

The d-band of a magnetic metal tend to split into spin up and spin down states to maximize the spin moment and gain exchange energy --> m_s = (m_down - m_up)µ_B s-states are delocalized, i.e. no energy gain in maximizing the spin

Question 37

What is the magnetic coupling? From which observation was discovered?

Answer 37

To explain how certain materials have a permanent magnetization and a Curie temperature = 1000K. --> There should exist an internal B-field (called Weiss field) which orders the moment against the thermal motion --> k_BT = µBxBw ---> Bw = 1300 T

Question 38

What is the interatomic exchange interaction?

Answer 38

Interplay between Pauli principle and Coulomb interaction --> two electron of opposite (same) spin can (cannot) share the same orbital and come close (stay further apart)

Question 39

How is Heisenber model for interatomic ex. interaction?

Answer 39

*H* = -∑J_ij x S_i x S_j where i≠j * J > 0: parallel orientation (ferromagnetic) * J < 0: antiparallel orientation (antiferromagnetic)

Question 40

What is the superexchange interaction?

Answer 40

Usually an antiferromagnetic coupling between two next to nearest neighbouts cations mediated by a non-magnetic anion (usually Oxygen 2-)

Question 41

What is antiferromagnetism?

Answer 41

exchange interaction J < 0; this type of order exists below a critical temperature T_Néel

Question 42

What is ferrimagnetism?

Answer 42

Antiferromagnetic exchange interaction J < 0 BUT a net spontaneous magnetization exists *M ≠ 0*

Question 43

What is the magnetic anisotropy? What are the three sources?

Answer 43

Preference for magnetization to align along a particular direction in a sample; 3 sources: * shape anisotropy H_d * Magnetocrystalline anisotropy * Induced anisotropy

Question 44

What determines the magnetocrystalline anisotropy?

Answer 44

The crystal field determines the symmetry of the wavefunction with lower energy --> fixes **L** relative to the crystal lattice --> **L** has slightly different values along different crystal directions --> direction with the largest component of **L** --> lowest spin-orbit energy H_so = λ**LS*** --> easy direction of magnetization

Question 44

How is shape anisotropy described?

Answer 44

Is related to the difference in energy U_d when the an ellipsoid as example is magnetized along its hard and easy axis

Question 45

How are magnetic domains formed?

Answer 45

A magnetic system tends to minimize its total free energy by reducing the magnetostatic (demagnetizing energy) *Ud = 1/2µ0VNM^2* (since *V *reduced) --> Domain wall cost energy to the system (exchange energy), this prevents the formation of ∞ domain wall

Question 46

How is the domain wall energy defined?

Answer 46

Is a balance between **exchange energy**, which will favor the formation of longer DW and **magnetic anisotropy** which doesn't like long domains with spins on hard axis

Question 47

What happens to DW in a ferromagnetic hysteresis?

Answer 47

Domains rotate because the magnetization aligns coherently to the easy crystallographic direction closer to the external field, irrespective of easy/hard axis. --> The non linear growth of magnetization occurs through domains expansion

Question 48

Why is domain wall motion a dissipative process?

Answer 48

The presence of hysteresis and associated energy loss shows the dissipative nature of mangetization process --> dissipations due to expansions (shrinkage) of magnetic domains in a sample that is initially unmagnetized (magnetized)-

Question 49

What are soft & hard magnets?

Answer 49

* Soft magnets: require small applied H field to reach saturation; have small H_c = less energy loss during an hysteresis loop * Hard magnets: retain their magnetization after removing the H-field (e.g. permanent magnets) --> high remanance and coercivity --> "energy product" designated as the area of the largest B-H rectange that can be constructed in the 2nd quadrant of hysteresis curve

Question 50

How is the energy density of a magnetic material described?

Answer 50

u_B = ∫ H dB = area of hysteresis