Transcript ppt file

Superconductivity
• Resistance goes to 0 below a critical temperature Tc
element Tc resistivity (T=300)
Ag
--.16 mOhms/m
Cu
-.17 mOhms/m
Ga 1.1 K
1.7 mO/m
Al
1.2
.28
Res.
Sn
3.7
1.2
Pb
7.2
2.2
Nb
9.2
1.3
T
• many compounds (Nb-Ti, Cu-O-Y mixtures) have
Tc up to 90 K. Some are ceramics at room temp
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Superconductors observations
• Most superconductors are poor conductors at
normal temperature. Many good conductors are
never superconductors
•  superconductivity due to interactions with the
lattice
• practical applications (making a magnet), often
interleave S.C. with normal conductor like Cu
• if S.C. (suddenly) becomes non-superconducting
(quenches), normal conductor able to carry current
without melting or blowing up
• quenches occur at/near maximum B or E field and
at maximum current for a given material. Magnets
can be “trained” to obtain higher values
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Superconductors observations
• For different isotopes, the critical temperature
depends on mass. ISOTOPE EFFECT
M 0.5Tc  cons tan t ( Sn115,117,119 )
Evibrations 
K
M
• again shows superconductivity due to interactions
with the lattice. If M  infinity, no vibrations, and
Tc 0
• spike in specific heat at Tc
• indicates phase transition; energy gap between
conducting and superconducting phases. And what
the energy difference is
• plasma  gas  liquid  solid  superconductor
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What causes
superconductivity?
• Bardeen-Cooper-Schrieffer (BCS) model
• paired electrons (cooper pairs) coupled via
interactions with the lattice
• gives net attractive potential between two electrons
• if electrons interact with each other can move from
the top of the Fermi sea (where there aren’t
interactions between electrons) to a slightly lower
energy level
• Cooper pairs are very far apart (~5,000 atoms) but
can move coherently through lattice if electric field
 resistivity = 0 (unless kT noise overwhelms 
breaks lattice coupling)
atoms
electron
electron
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Conditions for
superconductivity
• Temperature low enough so the number of random
thermal phonons is small
• interactions between electrons and phonons large
( large resistivity at room T)
• number of electrons at E = Fermi energy or just
below be large. Phonon energy is small (vibrations)
and so only electrons near EF participate in making
Cooper pairs (all “action” happens at Fermi energy)
• 2 electrons in Cooper pair have antiparallel spin 
space wave function is symmetric and so electrons
are a little closer together. Still 10,000 Angstroms
apart and only some wavefunctions overlap (low E
 large wavelength)
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Conditions for
superconductivity 2
•
2 electrons in pair have equal but opposite
momentum. Maximizes the number of pairs as
weak bonds constantly breaking and reforming. All
pairs will then be in phase (other momentum are
allowed but will be out of phase and also less
 
probability to form)
e
Ppair  p1  p2  0
 
ip  r
different times
different pairs
• if electric field applied, as wave functions of pairs
are in phase - maximizes probability -- allows
collective motion unimpeded by lattice (which is
much smaller than pair size)
|  total |2 |  1   2  .... n |2
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Energy levels in S.C.
• electrons in Cooper pair have energy as part of the
Fermi sea (E1 and E2=EFD) plus from their
binding energy into a Cooper pair (V12)
E12  E1  E2  V12
• E1 and E2 are just above EF (where the action is). If
the condition E12  2EF is met then have
transition to the lower energy superconducting state
2 EF
Egap
E12
normal
s.c.
TC
Temperature
• can only happen for T less than critical
temperature. Lower T gives larger energy gap. At
T=0 (from BCS theory)
E gap  3kTC
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Magnetic Properties of
Materials
• H = magnetic field strength from macroscopic
currents
• M = field due to charge movement and spin in
atoms - microscopic
 

B  0 (H  M )


M  cH  c  magnetic susceptibi lity
can be : c (T ), c ( H ), scalar , vector
• can have residual magnetism: M not equal 0 when
H=0
• diamagnetic  c < 0. Currents are induced which
counter applied field. Usually .00001.
Superconducting c = -1 (“perfect” diamagnetic)
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Magnetics - Practical
• in many applications one is given the magnetic properties of a
material (essentially its c) and go from there to calculate B
field for given geometry
beamline
sweeping
magnet
spectrometer airgap analysis
magnet
D0 Iron
Toroid
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Paramagnetism
• Atoms can have permanent magnetic moment
which tend to line up with external fields
• if J=0 (Helium, filled shells, molecular solids with
covalent S=0 bonds…)  c = 0
c  104 most, c  105 Fe
• assume unfilled levels and J>0
n = # unpaired magnetic moments/volume
n+ = number parallel to B
n- = number antiparallel to B
n = n+ + n• moments want to be parallel as
 
E    B
  B ( antiparall el )
  B ( parallel )
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Paramagnetism II
• Use Boltzman distribution to get number parallel
and antiparallel
B / kT
n  Ce
n
n  Ce B / kT  n
M   (n  n )
• where M = net magnetic dipole moment per unit
volume
M
e B / kT  e  B / kT
  average 
  B / kT
n
e
 e  B / kT
if B  kt 
(1  B / kT )  (1  B / kT )  2 B
 

(1  B / kT )  (1  B / kT ) kT
• can use this to calculate susceptibility(Curie Law)
B  0 H  0M  0 H
( c small )
M n n 2 B  0 n 2
c



H
H
kTH
kT
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Paramagnetism III
• if electrons are in a Fermi Gas (like in a metal) then
need to use Fermi-Dirac statistics
n  C 
n  C 
1
e
( B  E F ) / kT
1
n
1
e
(  B  E F ) / kT
1
n
• reduces number of electrons which can flip,
reduces induced magnetism, c smaller
antiparallel
B  0 kT  EF
parallel
2 B
EF
turn on B field.
shifts by B
antiparallel states drop to
lower energy parallel
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Ferromagnetism
• Certain materials have very large c (1000) and a
non-zero B when H=0 (permanent magnet). c will
go to 0 at critical temperature of about 1000 K (
non ferromagnetic)
4s2: Fe26 3d6
Co27 3d7
Ni28 3d8
6s2: Gd64 4f8
Dy66 4f10
• All have unfilled “inner” (lower n) shells. BUT lots
of elements have unfilled shells. Why are a few
ferromagnetic?
• Single atoms. Fe,Co,Ni
D subshell L=2.
Use Hund’s rules  maximize S (symmetric spin)
 spatial is antisymmetric and electrons further
apart. So S=2 for the 4 unpaired electrons in Fe
• Solids. Overlap between electrons  bands
but less overlap in “inner” shell
overlapping changes spin coupling (same atom or
to adjacent atom) and which S has lower energy.
Adjacent atoms may prefer having spins parallel.
depends on geometry  internuclear separation R
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Ferromagnetism II
• R small. lots of overlap 
broad band, many possible
energy states and magnetic
effects diluted
EF
• R large. not much overlap,
energy difference vs 
small
P A
EF
P A
• R medium. broadening of
energy band similar to
magnetic shift  almost all
in  state
EF
vs 
E(unmagnetized)-
Fe Co
Ni
E(magnetized)
R
Mn
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