Magnetic Properties of Materials Flashcards
1
Q
What are the two ways in which all materials respond to magnetic fields?
A
Diamagnetism and paramagnetism
2
Q
What direction compared to the field does the diamagnetic response act?
A
From high to low field
3
Q
What direction compared to the field does the paramagnetic response act?
A
From low field to high field.
4
Q
What type of extreme magnetic response do superconductors exhibit?
A
Diamagnetism.
5
Q
What is a typical paramagnet comprised of?
A
Atoms with net magnetic moments.
6
Q
How do paramagnetism respond to applied field and no field?
A
At no field, thermal fluctuations cause random orientation of mag moments.
When field is applied, moments align.
7
Q
What happens to mag moments in a ferromagnetic material?
A
They spontaneously align with no applied field.
8
Q
In vector calculus, what does curl represent
A
The flow of a field.
9
Q
In vector calculus, what does divergence represent?
A
A scalar amount of change in density through a closed surface
Eg
For an incompressible fluid (water) the net flux through a closed surface is 0 meaning div = 0
10
Q
State Lorentz Force Law
A
(F) = q(E) + q((v)x(B))
Used brackets for vector bold.
11
Q
State Maxwell 1 (Gauss Law for E) in words.
A
The divergence of an electric field is relayted to the charge density ρ at that point.
12
Q
Which forms of Maxwell 1 do we use for microscopic and macroscopic charges?
A
Microscopic: div form as true for each point in. Space.
Macroscopic: integral form as uses charge enclosed but is only true in vacuum.
13
Q
State Coulomb’s Law.
A
Check L1 S10
14
Q
State Maxwell 2 (Gauss Law for B) in words.
A
The magnetic field I’d divergence-less.
Therefore magnetic field lines must be closed loops.
15
Q
Which form of Maxwell 2 do we use for micro and macroscopic subjects?
A
Micro: div
Macro: Integral.
Neither is massively useful.
16
Q
State Maxwell 3 (Faraday’s Law of Induction) in words.
A
A changing magnetic field gives rise to an electric field.
17
Q
Which form of Maxwell 3 do we use for micro and macro subjects.
A
Micro: curl form
Macro: Integral form (emf in closed loop = rate of change of flux through loop)
18
Q
How does Maxwell 4 differ from Ampere’s Law?
A
In the curl equation, Maxwell adds a time dependency so that it can be applied to a changin field.
19
Q
State the equation Ampere’s law.
A
Check L1 S13
It is good for symmetric problems.
20
Q
State the Biotech-Savart Law and what situations it is useful for.
A
Check L1 S13
Useful for non-symmetric problems
21
Q
Use both Ampere’s Law and the Biot-Savart Law to derive an expression for the field around a current carrying wire.
A
Check L1 S14-17
22
Q
Sketch the magnetic field around a bar magnet.
A
Check L1 S18
23
Q
Sketch the magnetic field around a current loop and the direction of the magnetic moment.
A
Check L1 S18
24
Q
Give the base equation for magnetic moment.
A
(m) = IA
25
State the equation for energy of a magnetic dipole.
E = -(m).(B)
26
Sketch how energy varies with angle between magnetic moment and B field.
Check L1 S19
27
Define magnetisation
M is the volume density of permanent or induced magnetic dipole moments in a material.
28
Give a formula and unit for magnetisation from dipole moment.
(M) = n(m)
Unit is Am^-1
29
Relate B and M.
B = µ(0)M
30
What is H field?
A measure of magnetic field strength (same units as M Am^-1).
31
Names for H
Magnetic field intensity
Magnetic field strength
Magnetic field
Magnetising field.
32
Names for B
Magnetic flux density
Magnetic induction
Magnetic field
33
Relate B and H in a vacuum.
B(0) = µ(0)H in vacuum
34
Relate B and H in a material.
B = B(0) + B(induced)
B = B(0) + µ(0)M
Local field = applied field + material response
35
Define magnetic susceptibility χ
A dimensionless proportionality constant that indicates the degree of magnetisation of a material in response to an applied magnetic field.
36
Give equations for χ
M = χH
Or
µ(0)M = χ B(0)
χ ≈ 10^-7 for non magnetic materials
37
Define permeability, µ(r)
Measure of ability of a material to support the formation of a magnetic field within itself.
38
Derive an expression linking B and H using permeability.
Check L1 S25
39
What happens to the energy of a diamagnetic material when placed in a magnetic field?
It’s energy increases meaning the material wants to move from high to low field.
40
What value of χ do superconductors exhibit?
-1
Meaning magnetic field inside material is zero.
41
State the Langevin theory of diamagnetism.
Check L2 S5
42
Why is χ<0
From Lenz’s Law the direction of motion of electrons in orbitals is such that the magnetic moment opposes the applied field.
43
Do all materials exhibit diamagnetism>
Yes. The external field causes electrons to move in orbits.
Though all materials exhibit diamagnetism, it is weak and if there is any other magnetic mechanism present diamagnetism will not be the dominant effect.
44
Is χ dependent on T?
No direct dependence on T in equation. is weakly dependent on T, but for all practical purposes χ is independent of T.
45
How can we make a material with no susceptibility.
Mix diamagnetic and paramagnetic materials to produce a material with χ = 0
46
Estimate the magnetic susceptibility of diamond (a = 3.57 Angstroms)
Check L2 S8
47
State an expression for the energy of a shell from the principle quantum number.
Check L2 S13
48
State an expression for the magnitude of orbital angular momentum from quantum numbers..
Check L2 S14
49
What values does the magnetic quantum number take?
-l < m(l) < l
50
State an expression for the component of angular momentum along a specified direction in an atom.
L(z) = m(l) h(bar)
51
Derive an expression for the magnetic moment of an electron in an atom.
Check L2 S17
52
What is µ(B)?
The Bohr Magneton, µ(B) is the elemental unit of orbital angular moment in an atom.
53
State an expression for the Bohr Magneton
Check L2 S19
54
Give an expression for the projected magnetic moment on the z-axis.
Check L2 S19
55
State an expression for the energy change when an applied magnetic field interacts with an orbital magnetic moment.
Check L2 S19
56
Why is projected magnetic moment usually shortened to magnetic moment when condisdering orbital magnetic moment
The energy is found with the dot product and if B(0) is in the z direction then only need m(z)
57
What is the Zeeman effect?
In an external magnetic field the 3p orbitals are split in energy while the s-orbital is unchanged. This results in the Zeeman splitting in atomic spectra.
58
Sketch the Zeeman effect.
Check L2 S21
59
Which two quantum numbers are associated with the spin of an atom?
s and m(s)
60
What is the g-factor for orbital and spin angular momentum>?
Orbital g=1
| Spin g=2
61
State an equation for the magnetic moment using spin.
Check L2 S23
62
Why do we want to find the net angular momentum of a multi electron atom (sum of orbital and spin contributions)?
This gives us its total magnetic moment,
63
What doe the Pauli exclusion principle give us in terms of electronic configuration of an atom?
The n and l quantum numbers.
64
What do Hund’s Rules give us in terms of angular momentum configuration?
The m(l) and m(s) numbers for each electron.
65
How do we find L and S for an atom?
L = Σm(l)
S = Σm(s)
66
We can combine L and S to get what?
The total angular momentum of the atom J.
67
State Hund’s rules.
Maximise S
Maximise L
If the shell is less than half full J = |L-S| otherwise J = |L+S|
68
Use Hund’s rules to find the total angular momentum for Ce3+, Ho3+, V3+ and Mn3+.
Check L3 S4
69
State an expression for magnitude of magnetic moment using J and g-factor.
Check L3 S7
70
Calculate their magnetic moment for Ce3+, Ho3+, V3+ and Mn3+.
Check L3 S11
71
When does the calculation of magnetic moment from J work and not?
If J = 0, |m| cannot be found.
f-shell atoms |m| can be calculated closely to the expt value from J.
D-shell atoms |m| cannot be found closely to expt value using J.
72
Why doesn’t the measured magnetic moment agree with the theoretical calculated from J for atoms with outer electrons in the 3d orbital?
The orbital angular momentum does not contribute to the observed magnetic moment.
The effect of crystalline environment is that the electric field couples the orbital motion strongly to the lattice - hence it can’t reorientation with an applied field.
This is the orbital angular momentum being quenched.
The only response to an applied magnetic field is by spin angular momentum so we can therefore do magnetic moment from spin.
73
Calciulate the magnetic moment for V3+ and Mn3+ using only spin and compare to expt values.
Check L3 S13
74
What does a paramagnetic material do when placed in a magnetic field?
It moves to decrease its energy. It will move to a region of higher magnetic field.
75
What happens to dipoles in a paramagnetic material when in a magnetic field?
They align with the applied field so the local field is greater than the applied field.
76
Typical value for χ for paramagnetism.
χ = 10^-5
77
How does paramagnetic χ vary in insulators with temperature.
Insulators have localised moments and χ is strongly T dependent. χ decreases strongly with increased T.
78
Derive an expression for Curie’s law with a spin 1/2 paramagnetic.
Check L4 S5
79
Show the condition for the ferromagnetic state.
Check L5 S3
80
Show the effect of temperature on M and relate to M(spon) for a ferromagnetic material.
Check L5 S4
81
Graphically show and derive the Curie temperature of a ferromagnetic material.
Check L5 S6/7
82
Show how only some samples of iron are magnets using domains.
Check L5 S9
83
The formation of magnetic domains is a balance of various energy terms. Which are they?
Magnetic (magenta static)
Anisotropy
Exchange
84
What is the magnetostatic energy’s role in domains?
It is mainly resposible for the formation of domains
85
What is the role of anisotropy and exchange energies in domains?
They control the shape and movement of the domains.
86
Schematically show how domains in iron can have varying magnetostatic energy.
Check L5 S12
87
How does magnetisation tend to align in ferromagnetic?
Along certain crystallographic directions (easy axis).
88
What does exchange energy tend to drive in domains?
Neighbouring spins to align.
89
Sketch a Bloch domain wall
Check L5 S14
90
What tends to dominate, exchange energy or anisotropy energy?
Exchange energy
Anisotropy energy tends to be very small.
91
Derive an expression for the domain wall width using exchange and anisotropy.
Check L5 S15-16
92
What do the exchange and anisotropy energies tend to favour in terms of domain wall width?
Anisitropy: small walls
Exchange: large walls
Typical wall width in bcc iron is ≈500 atoms
93
Sketch how a hysteresis loop forms when a field is applied to a ferromagnetic such as iron.
Check L5 S18-20
94
What is the maximum point on a hysteresis loop known as?
Saturation
95
What is the point at which the hysteresis loop crosses the y-axis known as?
Remanence
96
What is the point at which the hysteresis loop crosses the x-axis known as?
Coercivity
97
What happens to a ferromagnet above the Curie temperature?
The material is paramagnetic.
98
Can both metals and insulators be ferromagnetic?
Yes but not many insulators. EuO is an example.
99
What is the big assumption with Curie-Weiss?
Assume localised atomic moments. This is not the full picture in metals such as Fe.
100
What is a ferrimagnet?
They have very similar properties to ferromagnets (permanent magnet, domain structure etc).
Driven by crystal structure and magnetic moments of ions.
101
Describe an antiferromagnet.
```
Above a critical temperature (Néel Temperature) material is paramagnetic with small positive χ.
Below T(N) magnetic moments anti-align (not a magnet)
```
102
Sketch a plot of χ against T for an anti-ferromagnet.
Check L5 S26
103
Plot 1/χ against T for anti-ferromagnetic and ferromagnetic/ferrimagnetic materials
Check L5 S28
104
What is Pauli Paramagetism?
In metals it is observed that the paramagnetism is independent of temperature. In Curie it is assumed all mag moments contribute, however in a metal only electrons writhing kT of Fermi energy contribute (similar to the heat capacity of gas).
The inner electrons cannot change quantum state so can’t store energy, so only those within kT of Fermi energy able to respond and store heat.
105
State an expression for the χ of a metal.
Check L6 S3
106
Schematically show using the FEG model, DoS and Pauli paramagnetism.
Check L6 S4
107
State an expression for magnetisation in the general case with angular moment J from Curie's Law.
Check L4 S7
108
Derive an expression for Curie's Law from the low field limit of magnetisation with J and the Brillouin function.
Check L4 S8-9
109
Sketch 1/χ against T stating the relation between μ(off) and the gradient.
Check L4 S9
110
State the Curie-Weiss relation.
Check L4 S11.
111
In the Curie-Weiss relation, what happens s when T
The material becomes magnetically ordered (and the material is ferromagnetic).
112
Sketch a plot comparing Curie's Law and the Curie-Weiss relation.
Check L4 S11.
113
Derive the Curie-Weiss Law stating assumptions about an exchange field.
Check L4 S12.
114
What is the exchange field?
A QM concept that allows neighbouring atoms to "communicate".
115
How does exchange work with spins on neighbouring electrons?
The wave function for a set of electrons must be antisymmetrical under the exchange of any two electrons. This leas to the Pauli exclusion principle; two electrons cannot have the same set of quantum numbers. For this reason two opposite-spin electrons can be closer than two same-spin. Being further apart lowers their electrostatic repulsion, and so it is energetically favourable for their spins to be aligned.
116
Expression for exchange energy.
L4 S14
Where J is positive, it is energetically favourable for their spins to align.
117
Show the condition for spontaneous magnetisation in ferromagnetic materials
Check L5 S2
118
Show graphically how temperature impacts magentisation in a material
Check L5 S4
119
Schematically show and derive an expression for the Curie T of a ferromagnetic material based on magnetisation.
Check L5 S6/7
120
Schematically show how only some samples of iron are magnets.
Check L5 S9
121
What energy terms are domains a balance of?
Magnetic (magneto static)
Anisotropy
Exchange
122
What role does magnetostatic energy play in domains?
It is mainly responsible for the formation of domains.
123
What role do the anisotropy and exchange energies play in domains?
They control the shape and movement of the domains.
124
Show schematically how domains can affect the field around a material and their tendency over time.
Check L5 S12
125
How does magnetisation in ferrromagents vary with orientation?
Some preferential alignments (easy axis) for magnetisation.
126
Exchange energy does what to neighbouring spins?
Drives them to align.
127
Sketch a Bloch domain wall
Check L5 S14
128
Derive expressions for domain wall thickness from exchange and anisotropy energies.
Check L5 S15-6
129
What thickness of walls do anisotropy and exchange energies favour?
Anisotropy: small walls
Exchange: large walls
130
How wide is a domain wall in bcc iron?
≈500 atoms.
131
Sketch how domains change shape when a field is applied.
Check L5 S18
132
Sketch the formation of a hysteresis loop in Fe.
Check L5 S19/20
133
The maximum field in a hysteresis loop is known as what?
The saturation field.
134
The point at which the hysteresis loop crosses the y-axis is known as what?
The remanence fiekd
135
What is coercivity in terms of a hysteresis loop?
The field applied to remove the magnetisation.
This is the point at which the hysteresis loop intersects the x-axis.
136
What happens to a ferromagnet above and below the Curie T?
Above the Curie T, the ferromagnet is paramagnetic. Below it is ferromagnetic.
137
What is χ for a ferromagnetic material above the Curie T?
χ > 0, eg 10^4 so is very large.
138
Is it just metals that can be ferromagnetic?
No, insulators can too though not many. Common example of ferromagnetic insulator is EuO.
139
What is the big assumption in Curie-Weiss?
Assume localised atomic moments (not the full picture in transition metals).
140
What is a Ferrimagnet?
Similar properties to ferromagnet in that it has a permanent magnet, domain structure etc, but is distinguished by considering crystal structure and magnetic moment of ions.
141
Describe the behaviour of anti-ferromagnets.
Above a critical temperature (Néel T, T(N)) material is paramagneticic with small positive χ.
Below T(N) magnetic moments anti-align making the material not a magnet.
Typcial materials: MnO, FeO, CoO, NiO, Cr
142
Give a modified Curie-Weiss expression for an anti-ferromagnetic material.
Check L5 S27
143
Give an expression for the field in an anti-ferromagnetic material.
Check L5 S27
144
Plot 1/χ against T to compare ferri/ferromagnetic materials to anti-ferromagnetic materials.
Check L5 S28
145
Describe Pauli Paramagentism.
In metals it is observed that the paramagnetism is independent of temperature.
For Curie all magmoments contribute, however in metals only electrons within kT of Fermi energy can contribute (similar to heat capacity for a gas of electrons). Most electrons cannot change quantum state so they cannot store energy meaning only those within kT of the Fermi energy are able to respond and store heat.
146
Show that for metals, paramagnetism is independent of temperature.
Check L6 S3
147
Show how a FEG reacts to an applied B field using DoS.
Check L6 S4
148
Which metals’ magnetic properties are not well defined by the FEG model?
Transition metals.
149
What is itinerant magnetism?
Electrons are able to move around in metal (eg 3d metals) instead of having localised M as in Curie.
150
Sketch the DoS vs E for spin up and down electrons in the d states for an non-magnetic metal and a ferromagnetic metal showing how a metal can be magnetic with no applied field.
Check L6 S8
151
What is the Stoner Criterion?
Large g(E(F)) (DoS at E(F)) means exchange energy dominates thus ferromagnetism
152
Explain why Pt has a larger susceptibility but is not ferromagnetic.
Put has a pretty large DoS at the Fermi energy so it nearly satisfies the Stoner Criterion. A small applied field will give a large magnetic response. (Sometimes called an exchange-enhanced paramagnet)
153
Sketch the result of resistance against T for Hg including the original expected result based on classical physics.
Check L7 S1
154
State an expression for how resistivity varies with T classically including defect and phonon contributions.
Check L7 S4
155
Sketch and state an equation for how critical field varies with T for a superconductor.
Check L7 S5
156
What is the critical field of a superconductor?
The applied magnetic field that ill destroy the superconducting state.
157
Does high critical temperature for an elemental superconductor mean high critical. Field?
No, they can have varying shaped curves.
158
Does the critical temperature of an elemental superconductor depend on isotope?
Yes slightly (dependent on mass)
159
Plot heat capacity against temperature around the critical temperature for a superconductor and normal material
Check L7 S8
160
Schematically show the Meissner effect using lines of flux around a piece of material above and below critical temperature.
Check L7 S7
161
How do superconducting and normal materials behave in optical and microwave frequencies.
Optical: S and N states behave the same.
Microwave: S state transparent below a critical frequency.
162
What are the 4 pieces of evidence for superconducting materials
1. Zero resistivity below a critical T.
b. superconductivity vanishes above a critical magnetic field (and also critical current density)
c. Isotope effect
2. Material expels magnetic field when cooled below T(C)
3. Jump in heat capacity at T(C).
b. Thermal conductivity lower than normal state
4. Optical absorption
163
Is there a structural or magnetic phase change on entering the superconducting phase?
No
164
Is a superconductor just a perfect (loss-less) conductor?
No. Check L7 S14-18 for diagrams but if a perfect loss-less conductor has flux through it above T(C) then is cooled below, flux still passes through the conductor, whereas it is expelled in the superconductor in accordance to the Meissner effect.
165
Show that the superconducting state is thermodynamically favourable for a material below its critical temperature.
Check L8 S4/5
166
Show that the superconducting state is more ordered that the normal state of the material.
Check L8 S6/7
167
Show there is a jump in heat capacity from the normal to superconducting state.
Check L8 S9/10
168
What kind of behaviour happens to heat capacity when a material is in the superconducting reginme and what does it suggest?
The exponential behaviour of heat capacity when superconducting suggests a new excitation mechanism.
169
What happens to microwave absorption as we approach T(C)?
The gap at E(F) reduces as T(C) is approached. This is in the meV regime.
170
Describe the essential ideas behind BCS theory.
Electrons can lower their energy by pairing up (Cooper pairs) - each pair of electrons has 1 spin up and 1 spin down.
A cooper pair is a “composite boson”. Bosons can occupy the same quantum state (unlike fermions).
It takes energy to break apart a Cooper pair. They travel through a material without losing energy.
171
Give an approximate expression for the distance between electrons in a cooper pair.
Check L9 S4
172
Explain how electrons can be paired to form Cooper pairs in a superconductor.
The solid ionic lattice mediates an attractive interaction.
As the 1st electron passes through, it polarises the material (by drawing positive ions together). This causes a net positive charge which attracts a second electron.
The response time of the lattice is much slower than the electrons hence the electrons can be coupled.
173
State an expression for the binding energy of a Cooper pair.
Check L9 S5
174
Sketch the DoS for a normal material and a superconductor, showing the binding energy of Cooper pairs.
Check L9 S6
175
State an expression for the binding energy of Cooper pairs in the weak coupling limit.
Check L9 S7
176
Why is the weak coupling limit used for finding the binding energy of Cooper pairs?
Often in this limit as electron-phonon interaction is weak.
177
State an expression for the predicted superconducting critical temperature from BCS theory.
Check L9 S8
178
State an expression for the predicted superconducting critical temperature from BCS theory in the weak coupling limit.
Check L9 S8
179
From the BCS prediction of critical T, what do we need for a high T(C).
Large Debye frequency (α 1/sqrt(m))
Large V
Large g(ε(f))
180
State an expression and sketch a plot for how binding energy of Cooper pairs depends on T from BCS theory.
Check L9 S11
181
In what way is superconductivity a cooperative phenomena?
The binding energ of Cooper pairs is largest when all the pairs are in the same state (at 0K).
182
What type of particle is a Cooper pair and how does that affect its wave function?
It is a composite Boson and has no net spin or angular momentum. Each pair has the same quantum numbers/quantum state and the wave function is the same for all particles (check L9 S13)
183
Derive an expression for Cooper pair concentration
Check L9 S14
184
Derive the London equation.
Check L9 S21-23
185
Derive the London penetration depth of a superconductor in a magnetic field.
Check L9 S21-26
186
Resistivity in metals is caused by what two types of scattering and mechanisms?
Elastic (change in direction only) - defects
| Inelastic (change in energy and momentum) - phonons.
187
In the superconducting state is the number of Cooper pairs constant?
It is in a dynamic equilibrium with electrons. Constant splitting and forming of Cooper pairs
188
Are elastic or inelastic scattering events allowed for Cooper pairs?
No as they affect the coherency of the wave function.
Elastic - can’t change k for the individual pair, only the whole coherent state therefore forbidden (all Cooper pairs have the same mtm).
Inelastic - need to supply 2Δ energy therefore forbidden.
189
What is critical current for Cooper pairs?
The current at which the momentum is so high that the excess energy of the Cooper pair is > binding energy causing breakdown of Cooper pairs and resistivity.
190
Derive an expression for the critical current of a superconducting material.
Check L10 S9/10
191
What is the difference between type I and II superconductors.
Type 1 show perfect Meissner effect then immediately breakdown to a normal material.
Type 2 show a penetration depth before breakdown. (Can go to larger fields before complete breakdown so more useful).
192
What is coherence length of a superconductor?
Measure of uniformity of superconducting wave function.
193
Give an expression for coherence length of a superconductor.
Check L10 S12
194
Describe the difference between type I and II superconductors in terms of London penetration and coherence length.
Type I: coherence length > London penetration depth (positive boundary energy)
Type II: coherence length < London penetration depth (negative boundary energy)
