Claridge - magnetic properties of solids Flashcards
How do magnetic moments arise?
The orbital angular momentum of an electron generates a magnetic moment μl
The spin of an electron generates a magnetic moment μs
What quantum number represents the orbital angular momentum?
What does it describe ab the e-?
lowercase L
which subshell it’s in
Principle quantum number?
What does it describe ab the e-?
n = which shell the e- is in
Magnetic quantum number?
What does it describe ab the e-?
ml (l is subscript L)
gives the allowed orientations of the orbital angular momentum in an applied magnetic field, also how many orbitals in a subshell
What is the magnetic susceptibility?
how responsive a material is to an applied magnetic field
(per unit of molar volume if molar magnetic susceptibility - must be molar to compare them)
Difference between diamagnetic susceptibilities and paramagnetic?
diamagnetic = small magnitude, negative, temp independent, often closed shell semiconductors, no unpaired e-
paramagnetic = positive and temp dependent, includes most metals
Describe diamagnetism
no unpaired e-, so no orbital or spin angular momentum
weakly repelled by an applied field as:
- the orbital motion of the e- in the filled shells is changed
- induced magnetisation opposes the field
the repulsion due to a reduction in density of the magnetic field lines passing through
how does the number of e- affect diamagnetism?
more e- gives a larger repulsion effect
What is the magnetic susceptibility of a perfect diamagnet?
1, completely expels the magnetic field = superconductors
Describe paramagnetism
unpaired e-, net magnetic moment
applied magnetic field induces magnetisation to align the field and gives positive susceptibility
weak attraction to the applied field as the density of the field lines passing through is increased
is Curie’s law for paramagnets or diamagnets?
paramagnets
How can the Curie constant be found from a graph?
the gradient of the plot of inverse molar susceptibility (1/Xm) against temp (x axis)
C = 1/gradient
What is the magnetocaloric effect?
the reversible change in temperature of a paramagnetic material when magnetised or demagnetised with application or removal of an applied magnetic field, respectively
What is magnetic refrigeration?
magnetisation by applied field lines the moments up, heat is removed by a heat sink, demagnetisation leads to cooling, cycle repeats
larger magnetic moment leads to stronger magnetocaloric effect
Describe the Brillouin function
the saturated magnetisation gives the maximum magnetisation obtainable when all the magnetic moments in the sample align with the applied field
What is the saturated magnetic moment?
in large fields when the magnetic moments adopt the lowest energy state by aligning with the field
Hund’s first rule? Why does it act?
arrange e- in their orbitals to maximise S,
to minimise repulsive Coulomb interactions between e- with the same spin
Hund’s second rule? Why does it act?
given the spin arrangement from rule 1, now arrange the e- to maximise L
to reduce the overall energy of an atom by minimising Coulomb repulsion between e-
Hund’s third rule? Why does it act?
For atoms with a less than half full subshell J = |L-S|
For atoms with a more than half full subshell J = L+S as the lowest energy states.
To minimise spin-orbit coupling interaction
in what circumstances is Hund’s 3rd rule only correct?
spherical
Which two ions have discrepancies between the calc and experimental values for effective moments? Why?
Eu^3+ and Sm^3+ because they are low-lying excited states and Hund’s rules only work for ground states,
low-lying excited states can become thermally populated at high temps, changing the magnetic moment
How does μsat compare with μeff?
μsat is always smaller than μeff
For which elements do μcalc and μexp often disagree?
3d transition metals apart from 3d5 and 3d10 where the total orbital angular momentum L = 0
When are the spin-only and spin-only saturated magnetic moment eq used?
for 3d transition metals
What is orbital quenching?
When a strong crystal field lifts orbital degeneracy, causing the orbital angular momentum to be quenched (=0) for many 3d ions, gives spin-only magnetic moments
in which ion is orbital restoration possible?
high-spin Co^2+ 3d7 in an octahedral crystal field
Describe orbital restoration in an example
the d (3x) orbitals in the t2g set are degenerate and related by 90degree rotation symmetry
one electron hole in the t2g^5 config so allows movement of e- which partially restores the orbital angular momentum of Co2+ in a weak octahedral crystal field
giving an μexp of 1μB larger than spin-only
When to use the T term for 3d transition metals?
when the t2g level is neither half full not completely full, so there’s a degree of freedom in filling the t2g level
When to use the E term for 3d transition metals?
when the eg level is neither half full not completely full, so there’s a degree of freedom in filling the eg level
When to use the A term for 3d transition metals?
when the eg and t2g levels are either empty, full, or half-full so there are no other degeneracies
Example of a spin transition material
LaCoO3
at low temp = low-spin 1A
at higher temp = high spin 5T
so has a temp-dependent μeff
How to get total μeff?
calculate for each cation, then do sum of squares
= sqrt[ (μeff A)^2 + (μeff B)^2]
Which term symbols are quenched and which are not?
T not quenched, A and E quenched
Triplet state vs singlet state
Triplet = both e- in overlapping paramagnetic ions are spin up, Stot = 1
singlet = both e- in overlapping paramagnetic ions are spin down, Stot=0
When is the triplet state energetically favoured?
when Jex > 0, Es> Et
When is the singlet state energetically favoured?
when Jex < 0, Es < Et
Which transition metal ion is an exception for paramagnetic metal centre overlap for transition metal oxides?
V4+
Singlet state Goodenough-Kanamari rule?
If two e- of two magnetic metal ions interact with the same anion orbital, then Jex<0
the neighbouring magnetic moments align antiparallel to give the singlet state
Singlet state relation to anti/ferro magnetism?
180degrees interaction gives rise to antiferromagnetism
Triplet state Goodenough-Kanamori rule?
If two e- of two magnetic metal ions interact with two orthogonal intermediate non-magnetic anion orbitals, then Jex>0
the neighbouring magnetic moments align parallel to give the triplet state
Triplet state relation to anti/ferro magnetism?
90degree interaction gives rise to ferromagnetism
Which orbitals interact in the singlet state diagram?
2pz with two dz^2 ones
Which orbitals interact in the triplet state diagram?
2py with two dx^2-y^2 ones
ferromagnetic v antiferromagnetic
ferromagnetic: the magnetic moments align parallel
antiferromagnetic: the magnetic moments align antiparallel
paramagnet v ferromagnet v antiferromagnet
paramagnet: weakly interacting, randomly orientated neighbouring magnetic moments
ferromagnet: strongly interacting, parallelly ordered neighbouring magnetic moments
antirferromagnet: strongly interacting, antiparallelly ordered neighbouring magnetic moments
What is required for a material to enter its ferromagnetic or antiferromagnetic state?
cooling below its critical magnetic ordering transition temperature, the Curie temperature Tc
What is the state between paramagnet and ferromagnet called?
cooperative paramagnet (in the liquid state)
Define the Weiss constant
the sum of all of the exchange interactions in any magnetic system
What is the sign of the Weiss constant for antiferromagnets?
negative
What is the sign of the Weiss constant for ferromagnets?
positive
What is ferrimagnetism?
When the neighbouring magnetic moments align antiferromagnetically
but the moment sizes are different so the magnetisation is only slightly cancelled and occurs below Tc
strong interaction, antiparallel ordering
What is Hm?
the internal molecular field used in mean field theory
What does a positive value of theta mean for mean field theory?
the molecular field acts in the same direction as the applied field, alignment of moments is parallel = ferromagnetism
When will magnetic susceptibility diverge according to mean field theory? What does this cause?
at T= theta
causes phase transition
Why do ferromagnets not behave as magnets until magnetised?
bc the sample will break into ferromagnetic domains to minimise the total energy, moments are aligned ferromagnetically in each domain, but the domains are randomly aligned,
magnetisation aligns the domains
What are hard/soft magnets?
soft: small remanence, small coercivity
hard: large remanence, large coercivity
What is coercivity?
how difficult a magnet is to magnetise/demagnitise, small coercivity = easily magnetised and demagnetised
What is remanence?
the remaining magnetic field after demagnetisation
Describe doped perovskite manganites
two antiferromagnetic insulators
doping Sr2+ into LaMnO3 oxidised Mn3+ to 4+
eg electron can move between Mn ions so the material becomes metallic below Tc = metal-insulator transition
also has magnetoresistance
What is magnetoresistance?
change in resistance of a material when a field is applied
giant magnetoresistance is used to detect magnetic fields
what is the critical ordering temperature for antiferromagnets?
Neel temp TN (subscript N)
What does a negative value for theta mean for mean field theory?
negative molecular field, acts in the opposite direction as the applied field
causes neighbouring moments to align anti-parallel = anti ferromagnetism
Why is pi-type 180 degree superexchange weaker than sigma type?
less orbital overlap
What are the two different superexchange interactions for LaMnO3? What type of structure does this give?
sigma-type 90degrees and pi-type 180degrees
antiferromagnetically coupled ferromagnetic layers
Why are neutrons better at determining the positions of lighter atoms within a crystal structure?
neutral charge prevents interaction with electrons so deeper penetration is possible
neutrons have a spin so the magnetic moment interacts with unpaired e- and gives magnetic structure info
what is powder neutron diffraction used for?
antiferromagnet analysis
What causes magnetic diffuse scattering in the powder neutron diffraction pattern?
the magnetic correlations are only short-range
General formula for ferrites?
MFe2O4 where M is a 2+ cation
Describe normal spinel ferrite structure
cubic spinel, FCC, M2+ occupy 1/8 tetrahedral and Fe3+ 1/2 octahedral sites,
dominant interaction = AFM but bc Zn2+ and Cd2+ are diamagnetic = paramagnetic overall
Describe inverse spinel ferrite structure
M2+ occupy 1/4 tetrahedral and Fe3+ 1/4 octahedral sites and 1/8 tetrahedral
dominant interaction = AFM
examples = NiFe2O4, CoFe2O4, Fe3O4
Describe magnetite
Fe3O4, strong antiferromagnetic superexchange interactions between Fe3+ ions = net moment of 0μB
ferromagnetic double exchange interactions between Fe3+ and Fe2+ ions = net moment of 4μB
Use of ferromagnetism? Why?
memory storage, magnetisation can be switched with an applied field to give magnetisation up/down
What is a multiferroic material?
where two or more primary ferroic orderings occuring spontaneously
Describe the 3 primary ferroic orders in solids
Ferromagnetic: spontaneous magnetisation develops below Tc, switchable by applied magnetic field
Ferroelectric: spontaneous electric polarisation develops below Tc, switchable w/ applied electric field
Ferroelastic: spontaneous strain develops in crystal structure below Tc, switchable by applied mechanical stress
Describe an example of a ferroelectric
BaTiO3
the cooperative displacement of the Ti4+ 3d0 ions in the TiO6 network gives electric polarisation and tetragonally distorts the cubic perovskite
Where do ferroelastics occur?
in materials that have 2 or more orientations of crystal structures which are energetically equivalent when without stress
Describe example of ferroelastic
NiTi
high temp cubic phase
low temp monoclinic phase
ferroelastic formed during cooling phase transition in the absence of stress
Piezoelectric material? use and eg?
combination of ferroelectric and ferroelastic, generates electric charge under pressure, used in sensors and motors
perovskite lead zirconate titanate (PZT)
