a from the quantum Hall effect

1
a from the quantum Hall effect
The everyday Hall effect A current density jx in
the presence of a perpendicular magnetic field Bz
induces an electric field Ey which is
perpendicular to both. From balance of forces
FqvB and FqE the field is equal to vB where v
is the average velocity of charge carriers. The
Hall resistance R is defined to be the ratio of
induced field to applied current
RE/jvB/NvqB/Nq. So the normal Hall resistance
is a measure of N, the density of charge carriers
in the sample. It is sensitive to material type,
impurities, defects, temperature, etc.
The quantum Hall effect If electrons are
confined in a thin layer at low temperature in a
high magnetic field the Hall resistance vs. B
rises in a series of steps which are quantised at
levels given by Rh/ie2 where i is an integer.
2
quantum Hall continued
The quantum Hall effect was discovered by von
Klitzing in 1980 (Nobel prize 1985). It was
totally unexpected and initially unexplained.
A partial explanation A perpendicular magnetic
field splits the states in a 2D electron gas into
Landau levels. The number of current carrying
states in each level is eB/h. The position of the
Fermi level relative to the Landau levels changes
with B. So the number of charge carriers is equal
to the number of filled Landau levels, i, times
eB/h. Plugging this into the classical expression
for R gives Rh/ie2.

• From our point of view the important features
  are
• The resistance can be precisely measured ( 1 in
  108 ).
• It is simply related to fundamental constants.
  Rh/ie2, a e2/2e0hc so R1/2ie0ca. Both e0 and c
  are constants without errors e0 1/m0c2, m04p .
  10-7 (NA-2) and c299,792,458 (ms-1) by
  definition.
• The measurement is done at very low energy so
  higher order corrections are negligible.

B.I.Halperin. Scientific American 1986.
D.R.Leadley http//www.warwick.ac.uk/phsbm/qhe.ht
m
3
GF from the muon lifetime.
Before electro-weak unification, muon decay ( and
all other weak interactions ) were described by
an effective 4-fermion coupling of strength GF.
We now know that weak decays take place via an
intermediate W boson like this However, the
Fermi constant GF is still one of the most
accurately measurable parts of the standard
model. So it is taken as one of the primary
constants. It is related to the other more
fundamental constants by GFsqrt(2).g2/8.mW2 .
Note g is related to other electroweak parameters
through sinqwe/g and cosqwmW/mZ.
The muon lifetime is calculated to be 1/tm GF2
mm5/192p3 h.o.c.
4
The muon lifetime experiment
An accurate measurement of the muon lifetime was
made back in 1984 at Saclay, Paris. Not improved
since. Because GF is accurate enough for present
tests of SM? Method A muon source is built up by
stopping 140 MeV/c p ( and some m ) during a 3
ms burst of the linac. A positron from muon decay
is detected as a triple coincidence in one of the
6 scintillator telescopes. Some time after the
end of each burst a clock is started and the time
distribution of positron signals is recorded.
5
Some questions
How do you stop a 140 MeV/c pion or muon ? Any
charged particle passing through matter loses
energy by ionisation or excitation of some of the
atoms that it passes. If it hits a thick enough
piece of matter it will slow down and stop unless
something else happens first (e.g.. nuclear
interaction, bremsstrahlung or the particle
itself decays). The energy loss rate, -dE/dx, of
a pion of bg1 in aluminium is about 2.5 (MeV g-1
cm2) and is rapidly rising as the momentum falls.
The energy loss in a thickness t of the sulphur
target will be more than DE t (cm) . 2.5 (MeV
g-1cm2) . rsulphur(g cm-3) Sulphur has a density
of 2.1 so we can say that 11 cm will stop the
pions, probably half that is enough. ( K.E.
sqrt(p2m2)-m 58 MeV )
What happens when they stop? 99.99 decay p ?
mnn with a lifetime of 26 ns.
Where do the muons from pion decay go ? In 2 body
decay of M, the energy of decay products in the
rest frame of M is E1 (M2 m22 m12)/2M. mp
139.6 MeV, mm 105.7 MeV so Em 109.8 MeV.
They will stop within a cm or so from where the
pion decayed.
6
Possible sources of error
Backgrounds. Cosmic and Beam-induced The
arrival time of cosmic rays will be random so
they will simply contribute a flat background
which is easy to subtract. The cosmic background
level was measured by taking data at long times
after the linac burst. The beam particles can
create other states with short lifetimes. Their
decays may trigger the telescope and mimic a
positron from muon decay. They will be correlated
with beam bursts in the same way as the muons, so
are a potentially a serious bias to the muon
lifetime. The paper states that there was only
one significant background of this type ( X-rays
from thermal neutron capture ) and that it was
fitted with a single exponential with a time
constant of 160 ms. Random coincidences If two
decays occur within the resolving time of the
electronics then only one of them will be seen.
This is more likely to happen when decays are
happening frequently it is proportional to
instantaneous rate. It is taken into account in
the fit R(t) R0 exp(-lt) A r exp(-2lt) B
l muon lifetime, rrate, Acoincidence
effect, Bbackground.
7
Errors due to muon polarisation

• The muons which are produced from pion decay in
  the target are unpolarised, but some muons reach
  the target after pion decay in flight and these
  ones are partially polarised (Why?). The
  direction of the positron from muon decay is
  correlated with the direction of the muon spin,
  so if the telescope efficiency is not isotropic a
  change of spin direction leads to a change of
  counting rate and therefore a lifetime error.
  This effect is minimised by
• Using sulphur for the target which rapidly
  depolarises the muons (Why?). Some muons will
  stop in other materials so this does not entirely
  eliminate the problem.
• Reducing the magnetic field in the target region
  with Helmholtz coils to 0.1 G ( ie nearly
  cancelling Earths field ). It is the change of
  spin direction by precession that is harmful.
• Measuring the lifetime in each of the 6
  telescopes independently and checking that all
  give the same result.

8
Lifetime result
The result is tm 2197.078 ? 0.073 ns. This is
converted into GF including corrections of up to
(me / mm )8 and (mm / mW )2 and a2. It gives,
together with a similar measurements of tm- , the
result GF/(?c)3 1.16637(1) ? 10-5 GeV-2
9
mZ from LEP lineshape scan
Dr Loebinger has already told how the mass of the
Z boson was measured very accurately at LEP-1 by
scanning the energy of the accelerator across the
resonance and measuring the cross section at each
point. The result is mZ91187.5 ? 2.1 MeV. The
uncertainty is mainly 1.7 MeV due to uncertainty
of the beam energy and common to all LEP
experiments. There is also a common 0.3 MeV due
to theoretical uncertainty of initial state
radiation.