Open Book (No phones). Cite reference for formulas used without proof.
1. A long cylindrical floating langmuir probe of radius rp resides in a plasma of
which the ions are completely collisionless, have negligible energy
far from the probe, and are thus attracted to the probe with purely
radial velocity. The electron density is governed by the Boltzmann
factor with temperature Te. The Debye length is negligible with
respect to the probe radius. The ions have charge Ze.
(a)
Derive from basic principles the potential
at which the quasi-neutral solution
has infinite derivative. Take this value as the sheath edge.
(b)
Obtain the full solution of the potential profile expressed
implicitly in the form of a solution for radius r as a function of
potential.
(c)
From this solution, determine how the potential varies
explicitly with r,
asymptotically far from the probe.
(d)
Derive an equation for the value of the probe potential
in units of Te/e, and solve it approximately when the ions are doubly ionized Helium (Z=2,
mi=4 mproton).
2. This question concerns scattering from plasma density fluctuations
due to turbulence that is extremely elongated along the magnetic
field direction. The density fluctuation Fourier components having
k|| ≡ k.∧B ≠ 0 can be considered
to have zero power, while those for which k|| ≡ k.∧B = 0 have substantial power for all perpendicular
wave-vectors such that |k⊥| ≈ k0.
A collective scattering experiment is described in a
coordinate system in which the magnetic field is in the z
direction, and the incident radiation propagating in the x
direction. The incident radiation wavelength is λ( << 2π/ k0)
which corresponds to a wave frequency far larger than any frequency
associated with the density fluctuations. The incident beam can be
taken as having plane wave-fronts with a large transverse extent in
the y and z directions. After passing through the plasma, the
beam and any radiation scattered through small angles passes
through a lens of diameter D (at x=0) which focuses the
radiation onto a detector plane a distance x=L from the lens. The
location in the detector plane can be written as a 2-D vector:
(y,z). The incident beam focus is the origin (x,y)=(0,0).
(a) Where in the detector plane will there be substantial radiation
intensity due to scattering, and where will there be negligible
intensity, and why?
(b) Suppose that an experimenter knows the magnetic field lies in
the y-z plane; does not actually know the direction of the
field (the orientation of the y and z axes); but can determine
in the detector plane where there is scattered intensity and where
not, to a spatial resolution equal to d. (And knows that the
fluctuations have k||=0). To what uncertainty in angle
can the experimenter determine the direction of the magnetic field
from the scattered intensity map?
(c) Suppose the detector resolution could be improved (d
decreased) without limit. What would then be the limiting factor on
the experimenter's ability to diagnose the field direction, and
what would be his best angular accuracy?
3. LDX has a plasma roughly 0.2 m in diameter, and electron density
ne=1017m−3, and for present purposes we can ignore the
small fast electron population. LDX is observed to have a visible radiation
spectrum in which OI and OII lines are comparably intense, but OIII is
not detectably present. We'll take that to mean OIII ions are at least
30 times lower density than OII. The ionization energies (χi) of
oxygen are: OI 13.6eV, OII 35eV (OIII 55eV). In this question
quantitative answers to at most one significant figure are appropriate.
For present purposes, take the collisional ionization rate coefficients to be
〈σv〉 = 5 ×10−14
Ry
χi
⎛ √
Ry
Te
exp(−χi/Te) m3s−1
where Te is the electron
temperature and Ry=13.6 eV.
a.
What is the
probability per unit time in this plasma of collisional ionization of
oxygen atoms OI and OII for Te= 1 and 10 eV?
b.
Assuming that charged oxygen ions have confinement time in the
magnetic field of order 10 ms, estimate
the approximate Te in the LDX plasma. Explain the basis of your
estimate.
c.
Deduce qualitatively the anticipated spatial profile
of OI and OII emission. In particular, would we expect the hotter core of the
plasma to have (i) greater (ii) lesser or (iii) about the same emissivity as
the cooler edge?
d.
Is it a reasonable assumption that the oxygen excited state
populations for a particular ionization stage can be treated using the
coronal approximation, and why?
e.
Take the OII density to be 2% of ne and use your estimated
Te. Estimate the number of photons per second of an OII line from
transitions i→ j collected by an optical system consisting of a
lens of diameter 3cm focusing on to a spectrometer slit of area 0.1mm
by 10 mm,
a distance of 5cm from the lens, viewing across the plasma
diameter. The excitation rate coefficient for its upper level (state
i) from the ground state should be taken approximately equal to the
above ionization rate coefficient, and the spontaneous transition
probabilities satisfy Aij=0.5 ∑kAik.
4. Suppose that Bremsstrahlung processes dominate in emission and
absorption of visible light in the interior of the sun. Take the
electron temperature to be 1 keV, the electron density to be
1030 m−3, and the ions to be all protons. Calculate the
absorption coefficient due to (free-free) Bremsstrahlung at a
wavelength of 400 nm. Hence estimate the "mean free path" of a
photon in the interior of the sun.
If the temperature of a "hypothetical" star like this were uniform equal to 1
keV, but the density had the radial form
ne = 1030 [1+ (r/R0)]−5m−3
where r is the radius and R0=7×105 km, is a nominal
start extent. Roughly how deep (to how small a value of r/R0) could one
"see" into this star at 400nm wavelength?
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version 4.05. On 7 Oct 2014, 11:28.