Magnetism Flashcards
What is the magnetic field B?
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
What is the Lorentz force?
Force that B fields exert on a charged particle: F = mv x B
Force, torque and potential energy of a magnetic dipole?
F = grad(mB)
T = m x B
U = -mB
m = magnetic dipole = magnetic moment
What tells the Ampère-Maxwell’s law?
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
What tells Gauss’s law for magnetism?
The flux of B-field across any closed surface is zero –> magnetic monopoles do not exist –> magnetic field lines always close on themselves
What is the inductance?
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)
What is the magnetization M?
Is the net magnetic moment per unit volume; M = 1/V ∑ mi = ϗ H
M = N < m> where < m> = average magnetic dipole moment
What is the auxiliary field H?
If we consider Ampère’s law, we can write ∇xB= μ0 J = μ0(Jcond + Jbound)
We define ∇xH = Jcond
Why is ∇xH = Jcond ?
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
How do we get ∇⋅H = -∇⋅M ?
From Gauss’s law: ∇⋅B = 0 thus ∇⋅H = ∇⋅(B/μ0 - M) –> ∇⋅H = -∇⋅M
What tells ∇⋅H = -∇⋅M ?
The discontinuity of magnetization M is a source of H-field
∇⋅M ≠ 0 if M is not homogeneous
What are the sources of H-field?
- Current
- Discontinuity of M
H = H_ext + H_dipolar
What is H_ext ?
H-field due to conduction:
∇⋅Hext = Jcond
Integral over a closed surface of H_Ext = I_cond
What is H_dipolar?
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.
What is a magnet?
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
Relationship between B, M and H
- Outside a magnet: B = H
- Inside a magnet: B = H + M where* H* is oriented opposite to M
What is diamagnetism?
Substance that is feebly repelled by a magnet –> is a property of every atom and molecule
Ex: H2O, Cu, H, Air, …
What is paramagnetism?
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, …
What is ferromagnetism?
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
What happens when a B-field is applied to an atom? (Orbital diamagnetism)
Change in angular momentum proportional to B and this change subtract orbital magnetic moment.
What happens when a B-field is applied to an atom? (classical model paramagnetism)
Magnetic moments of the unpaired electrons align themselves with the field, causing the material to become magnetized
What is the magnetic suceptibility ϗ_m?
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
What is the magnetic permeability?
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.
What are the peculiarity of B = μH?
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)
What is a ferromagnetic hysteresis?
Ferromagnetic materials are described by an irreversible non linear response of magnetization M to an imposed magnetic field H
What are three important values in hysteresis curve?
- 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.
Orbital magnetic moment m_l
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
What defines the orbital quantum number l ?
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_Bl_z
µ_B = Bohr magnetron
Spin magnetic moment m_s
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
Spin magnetic moment with z as quantization axis
s_z = ± 1/2 and *m_sz = -eℏ/2m_e x s_z = -2µ_Bs_z
What is the total magnetic moment?
m_tot = m_l + m_s = -µ_B(l + 2g_e s)
Orbital and spin magnetic moment for many electron atom
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
Which are the 3 Hund’s rule?
What are they used for?
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|)
What is the spin-orbit coupling?
An intrinsic interaction between m_l and the magnetic field produced by m_s: H_so = λLS 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
How is described the classical model for paramagnetism?
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)
How is described the QM model for paramagnetism?
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
What is the main difference of non-magnetic and magnetic d-band metals?
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
What is the magnetic coupling? From which observation was discovered?
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
What is the interatomic exchange interaction?
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)
How is Heisenber model for interatomic ex. interaction?
H = -∑J_ij x S_i x S_j where i≠j
* J > 0: parallel orientation (ferromagnetic)
* J < 0: antiparallel orientation (antiferromagnetic)
What is the superexchange interaction?
Usually an antiferromagnetic coupling between two next to nearest neighbouts cations mediated by a non-magnetic anion (usually Oxygen 2-)
What is antiferromagnetism?
exchange interaction J < 0; this type of order exists below a critical temperature T_Néel
What is ferrimagnetism?
Antiferromagnetic exchange interaction J < 0 BUT a net spontaneous magnetization exists M ≠ 0
What is the magnetic anisotropy? What are the three sources?
Preference for magnetization to align along a particular direction in a sample;
3 sources:
* shape anisotropy H_d
* Magnetocrystalline anisotropy
* Induced anisotropy
What determines the magnetocrystalline anisotropy?
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 = λLS* –> easy direction of magnetization
How is shape anisotropy described?
Is related to the difference in energy U_d when the an ellipsoid as example is magnetized along its hard and easy axis
How are magnetic domains formed?
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
How is the domain wall energy defined?
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
What happens to DW in a ferromagnetic hysteresis?
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
Why is domain wall motion a dissipative process?
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)-
What are soft & hard magnets?
- 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
How is the energy density of a magnetic material described?
u_B = ∫ H dB = area of hysteresis
