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168 Cards in this Set
- Front
- Back
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define crystal lattice |
Construcyed by infinite repetition of identical group of atoms (basis) |
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define primative lattice cell |
a parallelogram with the axes of primative vectors |
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define primative vectors |
for any two points r and r prime look the same |
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how many atoms are in the 3 types of cubic lattice |
simple cubic = 1 atom BCC = 2 atoms FCC = 4 atoms |
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translation vector |
T = U1a1 + u2a2 + u3a3 |
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r prime |
r' = r + T |
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distance between basis |
rj = xja1 + yja2 + zja3 |
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r prime with a basis of 2 atoms |
r' = r + T + rj |
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bands of an insulator |
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bands of a semiconductor |
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bands of a metal |
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directions of r |
characterised by u, v, w that have the ratio of u1 u2 u3 |
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planes of lattices |
defined by the intercepts of plane with crystal lattice. find the reciprocal and reduce the fractions to integers |
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Fermi Dirac distribution |
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dispersion relationship |
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total number of states inside a sphere |
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Fermi k vector |
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Fermi energy |
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density of states |
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momentum |
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Fermi temperature |
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Fermi level at small temperatures |
Fermi level = Fermi energy |
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proof that Y(x+L) = Y(x) |
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principles of free electron model |
ignored crystal lattice valence electrons move almost freely through volume |
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cons of free electron model |
fails to account for: metallic colour |
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define scattering time tau |
time taken for an electrone velocity to relax to zero |
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mobility mu |
a steady state electron velocity acquired in the electric field with a value of 1 V/m |
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electron motion |
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velocity in terms of scattering times |
v = - eEtau/m |
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mobility |
v = - mu E mu = - e tau/m |
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current density |
J = - nev = sigma E |
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conductivity |
sigma = ne^2 tau/ m |
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resistivity |
ro = 1/ne mu |
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free mean path |
L = v tau |
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drude model |
ignores ion cores and interactions. of electrons with periodic crystal lattice |
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electrons in Fermi sphere in absense of E field |
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electrons in Fermi sphere with application of E field |
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phonon scattering |
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defect scattering |
collisions are elastic scattering with large delta k causes resistance |
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matthiesens rule |
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Bragg diffraction |
2asin(theta) = N lamda |
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reciprocal lattice vector |
G = 2pi/a = delta k |
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group velocity |
vg = dw/dk = 1/h bar dE/dk |
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electrons experiencing Bragg diffraction |
have a k vector of k = ±npi/a |
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K vectir in 1st brouillion zone |
±pi/a |
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when are standing waves formed in a 1D chain |
when K = ±pi/a |
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positive and negative wavefunctions. of standing waves |
Y+ = 2cos(pix/a) Y- = 2isin(pix/a) |
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probability of wavefunctions |
Y+*Y- piles up ON ions Y-*Y+ piles up INBETWEEN ions |
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application of E field in a half filled Conduction band |
enables shift of electron distribution with respect to zero non zero current |
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application of E field in a filled Conduction band |
current allows some electrons to jump the band gap |
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overlapping bands |
energy of first state can be higher than that of the second so the states in second band are filled first |
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efficient diffraction |
delta k = 2ksin(theta) |
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Bragg plane |
perpendicular bisector planes that divide boundries of brouillion zones |
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reduced zone schemes |
high order BZ are folded back into 1st BZ by changing k vectors by a multiple of G |
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Construction of Bragg planes |
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define Fermi surface |
surface of a sphere separating full and empty states in K space |
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diagram of Fermi surface |
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deformation of Fermi surface |
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modification of Fermi surface rules |
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how to measure Conduction band distribution of metals |
find Fermi energy by emission of xrays |
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define tight binding model |
constructs electron bands and wavefunctions |
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how does tight binding model work |
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how are energy bands formed |
from different electron orbitals depending on the distance between atoms |
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derivation of effective mass |
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define a hole |
an electron with positive mass and change |
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creation of a hole in a semiconductor |
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do holes create current |
yes because they have positive charge |
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how to make a semiconductor more conductive |
add elements to make more holes or higher electron concentration |
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radius of an atom |
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donors |
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what group will be donors for group IV |
group V |
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acceptors |
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what group will be acceptors for group IV |
group III |
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binging energy |
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number of electrons |
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electron/hole density |
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dependance of Fermi level on temperature |
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Fermi level. at low temperature |
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concentration of electrons at low. temp |
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entrinsic semiconductors |
doped semiconductors where electron concentrationd are given by dopants |
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intrinsic material |
when Na = Nd then both will be ionised and no holes or electrons will be in bands |
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types of semiconductor |
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lorentz force |
F = - ev x B |
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Hall electric field |
eEh=ev x B |
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velocity in direction of Ex |
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what direction is hall field. in |
Y direction |
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Hall coefficient |
Rh = 1/ne = EyB/j |
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current density in terms of width and thickness |
J = I/wt |
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Hall velocity |
Vh=Ey w |
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Hall effect with electrons and holes |
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direction of electrons around B |
will move in anticlockwise direction |
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cyclotron resonance |
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absorption coefficient |
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complex velocity |
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circularly polarised E field |
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FWHM of cyclotron resonance |
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define electrical conductivity |
depends on carrier concentration and their distribution of thermal velocities |
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where is Fermi level |
somewhere in band gap |
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kinetic energy of electrons in Conduction band |
1/2 mv^2 = 3/2 kT |
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charge defects |
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free mean path wrt T at low T |
proportional to T^2 |
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scattering time with respect to T at low T |
tau is proportional to T^3/2 |
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free mean path wrt T at high T |
proportional to T^-1 |
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scattering time wrt T at high T |
proportional to T^-3/2 |
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optical absorption |
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what is an exciton |
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moss-burstein effect |
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derive dielectric function |
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what happens when w>wp |
wave will propagate |
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what happens when w<wp |
wave will be reflected |
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electrons under electric field - electrostatic screening |
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dielectric function of semiconductors |
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dispersion relation wrt dielectric function |
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define plasmon |
a quantum or longitudinal plasma oscillation |
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plasma frequency |
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wp |
n^2/epsilon naught me* |
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where is electronic paramagnetic found |
atoms, molecules and lattice defects with odd number of electrons free atoms and ions with partially. filled inner shell metals |
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total angular momentum |
J = angular momentum + spin |
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hunds rules |
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magnetic dipole of orbital motion |
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magnetic moment for spin |
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equation for bohr magneton |
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resultant magnetic momentum |
precesses around J component of mu R that lies parallel to J gives actual value of dipole moment perpendicular component averages to zero |
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magnetic moment of J |
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landé splitting factor g |
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what are the values of the splitting factor g |
when S=0 g=1 when L=0 g=2 |
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magnetic energy of parallel and antiparallel dipoles |
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probability of finding an electron in parallel and antiparallel states |
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what happens to probability when thermal energy is larger than the magnetic energy |
probability of parallel is roughly equal to probability of perpendicular magnetic dipole is roughly zero |
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what happens to probability when thermal energy is smaller than magnetic energy |
probability of parallel is larger than probability of perpendicular
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brillouin function |
this gives the average dipole component in the field direction as a function it B and T and takes a value between 1 and 0 |
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derive an expression for the magnetic susceptibility in the limit of small. applied fields and high temperatures |
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equations for magnetic susceptibility |
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bulk properties of a ferromagnetic |
magnetisation is large and positive its contribution to total B field is significant magnetism is a complex function of the applied magnetic H field magnetism depends on history of sample |
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what causes the increasing magnetism in a paramagnet |
increasing alignment of magnetic dipoles |
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how to produce large values of magnetisation in a ferromagnet |
very small H field direct alignment of magnetic dipoles |
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edges of domains |
called walls width of a few 100 atoms wide |
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two main mechanisms by which a ferromagnet is magnetised |
growth of domains with favourably. orientated magnetisation vectors rotation of magnetisation vectors |
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hysteresis effects |
irreversible domain wall motions give rise to these effects |
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domain model for the magnetisation of a ferromagnet |
displacement of domain walls eat away the unfavourable domains whose direction is not aligned with the applied magnetic field |
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what makes a domain wall motion irreversible |
motion of walls is strongly influenced by defects or impurities in the sample |
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the total energy of a ferromagnet crystal has a contribution from 3 mechanisms |
magnetostatic energy
anisotropy energy
exchange energy sum of these must be minimised |
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temperature dependance of a ferromagnet |
if you heat a permanent magnet past the curie temperature and then cool it, it will destroy domain structure |
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sources of an magnetic field |
electric current that flows in a conductor from intrinsic magnetic properties of particles having a spin |
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define H field |
can only be produced by a free current |
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define B field |
related to total current, bound and free |
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what is a magnetic dipole ina in wire |
current times area |
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define magnetisation |
where a non magnetic material is converted to a magnetic material magnetic dipole moment per unit volume |
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relationship of B and H |
B = mu naught H |
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relationship between B and H when there's magnetisation |
B = mu naught (H+M) B = mu naught mu R H |
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2 measures of susceptibility |
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force on a conductor in a magnetic field |
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circulating negative charge |
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magnetic dipole moment of a circulating electron |
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energy when placing a magnetic dipole in a B field |
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torque when placing a magnetic dipole in a B field |
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magnetisation and H field |
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theory of diamagnetism |
all materials are diamagnetic have a small negative values of susceptibility magnetism will oppose an applied magnetic field |
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precession |
the motion of a magnetic dipole in a constant magnetic field |
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precessional motion |
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langeins theory of diamagnetism |
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what does the precessional motion mean. |
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what can cause spontaneous magnetisation |
need alignment of neighbouring magnetic dipoles |
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pauli exclusion principle |
for 2 identical fermions, total wavefunction is antisymmetic under exchange of particles |
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4 ways of writing a wavefunction |
one singlet = spins are antiparallel = symmetric three triplets = spins are parallel = antisymmetic |
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overlap in distributions |
smaller for two electrons in an antisymmetic spatial wavefunction therefore a smaller coulombic repulsion in a triplet wavefunction |
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energy comparison for states |
triplet state is lower than the singlet state |
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energy difference between singlet and triplet |
exchange energy best condition for ferromagnetiv state is when J is positive |
