Open Book. Cite reference for formulas used without proof.
1. Consider diagnosis of a plasma by fast charge-exchange neutrals, as
shown schematically here.
The plasma is a uniform cylinder of radius a. It has only hydrogen
ions, electron density ne, temperatures Te and Ti, and a
density of neutral hydrogen atoms nh that can be taken as having
zero energy. The detection system has a collimator consisting
of two apertures, each 5mm by 5mm a distance 0.5m apart. The neutrals
are then energy-analysed into a spectrum that gives the rate of
arrival of neutrals accepted by the collimator per unit time per unit
energy, F(E), where E is the energy. For the cases where
evaluation is required, consider a plasma: a=0.2m,
ne=1019m−3, Te=Ti=2keV, nh=1015m−3.
(a)
Write down an expression for the probability that a fast neutral
created in the plasma will reach the edge without experiencing an
ionizing collision, pointing out any approximations involved.
(b)
Evaluate your expression for a 10 keV energy neutral created at
the plasma center.
(c)
Show how to obtain a general expression for F(E).
(d)
Suppose Ti is deduced ignoring attenuation, as the inverse of
the slope of ln|F/σcE| in the vicinity of 10keV, estimate
quantitatively the error that arises from the actual attenuation of
neutrals.
(e)
Calculate the rate of detection of neutrals in an energy range of
0.5 keV at 10keV.
(f)
If the spectrum is collected for a time of 1 ms, estimate the
fractional statistical fluctuation in the observed detection rate.
(g)
How are these quantities (d), (e) and (f) different if the plasma
density is a factor of ten higher: 1020m−3?
2. Reflectometry is performed on a linear density profile that may be taken
as given by
ne = n0
⎛ ⎝
1 −
r−r0
l
⎞ ⎠
for r < r0+l and ne=0 for r > r0+l.
The reflectometer launches an ordinary wave at r=a, where
a > r0+l, and it propagates in one dimension along the
decreasing radius r until it reaches the cutoff density, at a radius
rc say, and is then
reflected, returning to be received again at r=a. The parameter
l describes the scale-length of the density variation. The
wave frequency is ω and the cut-off density is nc. We are
interested in the way changes in density profile change the
reflectometer observations.
(a)
Obtain an expression for the effective path-length (L)
observed by the reflectometer, defined as the phase-shift of the
reflected wave (ϕ) divided by ω/c.
(b)
If the density profile parameter l changes, with
no and r0 constant, the path-length, L, will either increase or
decrease. Deduce the range of values of nc/no over which the change
of L has the same sign as the change in the distance
(a−rc) between the launch point and the critical radius.
3. ITER has approximate parameters R=6.2 a=2m, ne ≈ 1020m−3, Te=Ti ≈ 10keV. Write a summary of the
specific opportunities or challenges that these parameters (or
other aspects of ITER operation) present
for each of four proposed diagnostics:
(a)
Density interferometry;
(b)
Incoherent thomson scattering;
(c)
Neutral beam based diagnostics (such as charge exchange
spectroscopy);
(d)
Electron cyclotron emission.
Explain as quantitatively as possible in each case, the
reasons why diagnostic challenges are different (or the same) for the
ITER case, compared with smaller, present-day tokamaks.
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