Solids (Morrison) Flashcards
what is the layer stacking pattern for hexagonal cp?
ABA
what is the layer stacking pattern for cubic cp?
ABC
How many tetrahedral sites in hcp?
2 x T+ 2 x T-
in ccp what do T+ and T- share?
faces
how many T // O sites per unit cell of ccp?
8 // 4
in hcp what do T+ and T+ share?
corners
in hcp what do T+ and T- share?
edges
in hcp what do O and T sites share?
faces
Rocksalt describe:
ccp (either Na or Cl)
all O sites filled; T sites empty
edge sharing NaCl6 and ClNa6 octahedra
Zinc Blende describe:
ccp S2- with Zn2+ in 1/2 T sites
all O and 1/2 T empty
corner sharing ZnS4 and SZn4 tetrahedra
Wurtzite describe:
hcp S2- with Zn2+ in 1/2 T sites
all O and 1/2 T empty
corner sharing ZnS4 and SZn4 tetrahedra
Nickel Arsenide describe:
hcp As with Ni in all O sites
all T sites empty
Face sharing NiAs6 octahedra, edge sharing AsNi6 trigonal bi-prisms
Fluorite describe:
ccp Ca2+ with F- in all T sites
all O sites empty
edge sharing CaF8 cubes, edge sharing FCa4 tetrahedra
Rutile describe:
hcp O with Ti in 1/2 O sites [distorted]
1/2 and all T sites empty
layers of edge sharing TiO6 octahedra and edge sharing OTi3 trigonal planar
ReO3 describe:
based on primitive cubic
cubic Re lattice with O2- on the midpoints
corner sharing RO6 octahedra
Perovskite describe:
ABO3 - based on primitive cubic
cubic B lattice with O on the midpoints and A central
face sharing AO12 cuboctahedra, corner sharing BO6 octahedra, face sharing OA4B2 distorted octahedra
Spinel describe:
ccp
AB2X4 array with A in 1/8 T sites, B2 in 1/2 O sites
T-T isolated, T-O corner, O-O edge
unconnected tetrahedra and edge sharing octahedral
Inverse spinel describe:
ccp
AB2X4 array with B in 1/8 T sites, A+B in 1/2 O sites
edge sharing BO6 + AO6 octahedra, BO4 tetrahedra unconnected, corner shared with octahedra.
how do you find the radius ratio?
r+/r-
What are some rules to determine radius ratios? (6 things)
based purely on ionic model
hard spheres // only electrostatic forces // maximise CN // avoid C+C+ and A-A- interactions // maintain local electroneutrality
What are some general size relationships for atomic radii? (7 things)
radii increase down a period // isoelectric series - size decreases with increased charge // radius decreases with increased oxidation state // radius increases with increasing CN // for a given OxS radius decreases –> // for a given 3D element, r(HS) > r(LS) // diagonal relationships: r(Na+)~r(Ca2+)
what is the minimum cation size for an octahedral site?
0.414
what is the minimum cation size for an 8-coordination site?
0.732
what is the minimum cation size for a tetrahedral site?
0.225
what is the minimum cation size for a trigonal site?
0.155
what is the minimum cation size for a dodecahedral site?
1.000
What is a mooser-pearson plot?
a graph of n (principle quantum number) vs dx (electronegativity difference between ions) which looks at the type of bonding - non-directional to idealised.
What is Pauling’s electrostatic valence rule?
for a cation Mm+ surrounded by n anions Xx-, the electrostatic bond strength, s = m/n; then for each anion the sum of s = x
this determines the stability of the complex.
What is the tolerance factor?
To determine if something is a perovskite or not.
a perovskite wants to be a perfect cube - when t = 1.
t = (ra + ro)/sqrt2(rb+ro) (for ABO3)
a perovskite has t = 1 +/- 0.05.
what is a simple substitution?
an isovalent substitution where there is no change in charge or interstitial sites or vacancies
in Kroger-Vink notation what is the symbol for filling a vacancy?
v in small
in Kroger-Vink notation what is the symbol for making a vacancy?
V in big
KV what is the symbol for positive effective charge?
.
KV what is the symbol for negative effective charge?
’
KV what is the symbol for neutral effective charge?
x
KV what is the symbol in subscript?
the environment - so what is being replaced in the substitution
KV what is the main symbol?
the type of atom - element or vacancy
KV what is before the arrow vs what is on top of the arrow?
before is what is being substituted in
on top is what is being substituted for
electronic effects: which d orbitals does cfse apply to?
3d (4d and 5d are only low spin)
what is the cfse difference between tetrahedral and octahedral?
dt = 4/9 do
which is bigger (cfse) do or p (pairing energy)
do
in an interstitial mechanism what do you want to maintain?
electroneutrality
what 3 factors do interstitial mechanisms apply to?
thermodynamics, structure and electronic configuration
What are the requirements for a substitutional mechanism of solid solution formation?
same charge similar size (dr~15%)
What is extra the requirement for a continuous substitution mechanism for solid solution formation?
isostructural end members
what are the requirements for an interstitial/vacancy solid solution formation?
ions of similar size but different charge
minimise vacancy formation energy
interstitial sites available (avoid face sharing)
Explain a cation vacancy substitution
introduce new cations with too much positive charge - so you make some vacancies to remove charge to compensate
explain a cation interstitial substitution
introduce cations with insufficient positive charge - so you need to fill some vacant sites with different positive charges or change the oxidation state of the metal to add charge.
give some examples of a cation vacancy substitution
Fe(1-x)O –> balance charge by adding Fe2O3 into FeO
Ca2+ doped NaCl
Al2O3 in MgAl2O4
give some examples of a cation interstitial substitution
stuffed quartz (SiO2 replaced by LiAlO2) tungsten bronze (NaxW(V, 1-x)W(IV, x)O3)
explain an anion vacancy substitution
introduce a less positive cation - need to make vacancies to remove some of the more negative feelings by introducing some positive feeling vacancies where the anion used to be
explain an anion interstitial substitution
either introducing an extra anion and balancing with an extra positive cation which increases number of atoms so need to fill an interstitial site; or balancing by changing the oxidation state of the metal and filling an extra interstitial with the extra atoms.
give some examples of an anion vacancy substitution
ZrO2 replaced by CaO
ZrO2 replaced by Y2O3
CeO2-Ce2O3
give some examples of an anion interstitial substitution
CaF2 - YF3
UO2+x
what is a double substitution?
when both a cation and an anion can switch
give some examples of double substitutions
AgBr-NaCl
Olivine Mg2SiO4
Feldspars CaAl2Si2O4-NaAlSi3O4
Sialons Si3N3-Al2O3
What is Vegard’s law?
describes the effects of replacing A atoms for B atoms and the effects this has on the cell parameters - related to the A-B repulsion (up) and attraction (down)
What causes positive deviations from Vegards law?
due to A-A and B-B forming clusters which increases repulsive energy so curve goes concave up.
what main factors determine electronic properties of transition metal oxides?
extension of transition metal complexes
all about d electrons
in solids d e- are localised in AOs or delocalised in bands
What are the two models used in band theory?
tight binding model and nearly free electron model
what is the tight binding model?
bases electrons in a solid on MO theory.
what is the nearly-free electron model?
analogous to particle in a box –> delocalised electrons are travelling as waves
what type of bonding is present in tight binding model when |ca|^2=|cb|^2
pure covalent
what type of bonding is present in tight binding model when ca=0 and cb=1?
pure ionic
what is quantum number k?
the wave vector = k=2pi/lambda = pi/a
what is e^ikna?
the Bloch function - which is the eigenfunction of the translational operator
is anti bonding or bonding orbitals dominated by anions or cations?
bonding = anion
anti bonding = cation
what is DOS?
density of states
what is the wavelength for the most bonding or most anti bonding orbitals?
infinity
What shape is the energy potential graph for the nearly free electron model and what is on the axes?
E vs k parabola
what is Bragg’s law?
n[lambda]=2asin[theta]
what is the equation for a in braggs law?
a = n[lambda]/2
what is the equation for k in the nfe- model?
k = +/- n[pi]/a
what is then central portion of the band structure graph called for nfe- model?
1st brillouin zone
what are the equations for standing waves in nfe- model?
[psi]+ = exp (ipix/a) + exp (-ipix/a) = 2cos(pix/a) [psi]- = exp (ipix/a) - exp (-ipix/a) = 2isin(pix/a)
what type of orbitals make up the TM valence band?
(low number) p (from oxygen)
what type of orbitals make up TM conduction band
d
what type of orbitals make up the TM anti bonding band?
metal s-, p- bands
what do you do if the conduction band isn’t accurate?
fill it with normal d orbital MOs.
what is d-d overlap?
many orbitals of the same energy overlap = to make a band of orbitals covering a range of energies over the entire crystal. (called the d band)
what is d-p ovelap?
eg orbitals on TM can sigma overlap with p orbitals on an anion ---> splits into a bonding and anti bonding band and t2g (dxy, dxz, dyz) orbitals can pi overlap with with O2- 2p orbitals
how does a band form?
continuous overlap of d-p-d-p-d orbitals
what are the conditions for d-d orbital overlap?
size of d orbitals must be large enough to overlap with neighbouring cations (favours LHS)
what are the conditions for d-p orbital overlap?
size not important but energy of d and p is the controlling factor - if d orbitals are low in energy - similar to near the top of the O2- 2- band - gives strong overlap (favours RHS)
what are the two criteria for the theory of the hubbard model?
electron transport would require transfer of an electron and spin pairing –> energetically unfavourable
e-e repulsion favours e- localisation in atomic orbitals —> opposed band formation
What does the Hubbard Parameter, U, require for electron transfer?
the opening of an energy gap
what is the equation for the quantum hubbard? and what does it explain
U = Ie- - x (ionisation energy minus electronegativity)
to move an electron around it costs energy which is measured by U
what is the equation for charge fluctuation?
U = dn + dn —-> dn-1 + dn+1
what is the Hubbard hamiltonian? and which is the potential term and the kinetic term?
H = t(sum)c+c + U(sum)n(up)n(down)
kinetic potential
in the Hubbard Hamiltonian, what does it mean when t>U
delocalisation occurs –> solid conducts
in the Hubbard Hamiltonian, what does it mean when t<u></u>
localisation occurs –> solid insulates
what is the equation for bandwidth?
w = 2zb (where z = number of nearest neighbours)
in the equation for bandwidth, what is b?
the exchange integral - degree of interaction/overlap
bij = (psi, H’psi) = eij(psii, psij)
eij = one e- energy; psii, psij is the overlap integral; H’ is the perturbation of atomic potential bt neighbouring atoms at ri and rj
At what point in the hubbard model does an insulator become a conductor?
when U=W –>
what are the two conditions for electron delocalisation according to hubbard model?
d orbitals are diffuse and overlap AND U is small
d-p overlap is continuous AND U is small
what is the alternative classification? (ZSA)
for charge transfer insulators: delta dn —> dn+1 + L+
U or delta control - two options
Mott-Hubbard insulators - U
charge transfer semiconductors - delta
what is a mott-hubbard insulator? and some examples
when Eg~U - there is a hole and e- in d band (p d d)
v2O3, Ti2O3, Cr2O3 + halides
what is a charge transfer semiconductor? and some examples
Eg~delta - hole in p band, electron in d band (d p d)
CuX2, NiX2
what are some examples of a mixture of the U or delta control?
NiO, NiF2, CuF2
what are the five types of magnetism?
dia, para, ferro, antiferro, ferri
what is the magnetic susceptibility of the 5 types?
small and neg; small and pos; large and pos; small and pos; large and pos
what are the magnetic behaviours of the 5 magnetic types?
no moment; random; parallel; antiparallel; mixed (anti)parallel
what type of magnetism has the curie temperature?
ferromagnetism
what type of magnetism has the Niell temperature?
ferrimagnetism
What is the Heisenberg exchange hamiltonian?
Hex = -(sum)JijSiSj
in the HEH, what happens when J>0
parallel spin –> ferromagnetic ordering
in the HEH, what happens when J<0
antiparallel spin –> antiferromagnetic ordering
what is the equation fo rJ in the HEH
J = -2t^2/U
What are the 3 types of exchange?
direct (cation-cation), indirect (double exchange e- transfer), superexchange (cation-anion-cation)
how do unpaired electron spins couple?
through covalent interactions with intervening ligand
what type of magnetism present in a 1/2 filled metal - 1/2 filled metal orbital interaction?
AFM
what type of magnetism present in a 1/2 filled metal - empty metal orbital interaction?
FM
what happens to the strength of the superexchange interaction as covalency increases?
it increases
how does delocalisation occur in neighbouring atoms
spins must be aligned in a certain way (usually ferromagnetically)
what are the two main mechanisms for determining the magnetism of magnetic oxides?
superexchange and indirect double exchange
