The wall of a plasma vessel can have a strong influence on the overall discharge[61], affecting the rate at which plasma species are lost from the system, the structure of the plasma potential, as well as surface chemistry as discussed in Section 3.2.5. These wall interactions act as the boundary conditions for our model. While we might not expect a magnetically confined plasma to be especially dependent on wall conditions, we will discuss sheath formation and dynamics as well as the atomic loss coefficient in this section.
3.5.1
Plasma Sheath
Due to the high mobility of electrons compared to ions, they are lost to the walls much more rapidly than ions, breaking quasineutrality and creating very strong electric fields which act to accelerate positive species towards the walls and confine negative species to the plasma volume until a stable equilibrium has been established. Typically these sheaths exist on a scale of a few mm from the walls. This leads to extremely rapid spatial variation in densities and electric fields which must be resolved by the solver mesh as discussed in Appendix B. In this model, as we explicitly calculate the transport properties of the electrons and ions, sheath formation occurs as an emergent behaviour.
In general we define the boundary loss condition of each plasma species as
−n·ΓX =
1
4vth,XnX (3.66) however, this does not take electrostatic acceleration and ion inertia into account. Typically for global models this is taken into account by replacing the ion thermal
velocity in the above expression with the Bohm velocity for a given ion[52] vB= r qeTe mi (3.67)
However, this relies on a value of the electron temperature determined in the pre- sheath region. When evaluated at the vessel wall, the electron temperature has undergone electrostatic cooling and is no longer representative of the assumptions used in the derivation of the Bohm velocity. From a computational perspective it is non-trivial to define and evaluate a pre-sheath value of the electron temperature for an arbitrary wall geometry in a 2D-axisymmetric configuration. In order to account for the effect of the sheath on ion losses we therefore define an arbitrary scaling factor γi for postive ions such that
−n·Γi =
γi
4vth,ini (3.68) where γi is typically taken as 4. This is calibrated in order to approximate the
flux we would otherwise expect from the Bohm velocity or else from the explicit inclusion of ion inertia. Under typical operating conditions the plasma volume is not sensitive to this parameter, although numerical stability can be affected for values of γi <4.
3.5.2
Atomic Loss Coefficient
For high ionisation fraction discharges, such as those observed in MAGPIE at high powers, the efficiency with which atomic hydrogen recombines at the walls, described by the atomic loss coefficient, γH can be extremely important in de-
termining the atomic to molecular ratio of the neutral fluid[64], which in turn is important for determining the overall neutral pressure, neutral profiles, negative ion production rates, positive ion composition, and excitation losses. This is due to the fact that wall recombination is the primary mechanism by which molecular hydrogen is replenished in the plasma volume.
The atomic loss coefficient depends on the wall surface properties in a variety of different ways. These include the wall material, surface microstructure (e.g. pitting, metallic fuzz, bubbles, etc.), surface particle loading, temperature, im- purity accumulation, neutral flow velocity, as well as many other properties[64], each of which can vary within plasma shots, between shots, and from day to day, depending on the operating history and cleaning protocols of the plasma device.
Values ofγH can only feasibly be determined experimentally, due to the condi-
tions mentioned above. This is non-trivial but has been performed for a number of relevant surface materials using a variety of different techniques including two pho- ton absorption laser induced fluorescence (TALIF)[158], pulsed induced fluores- cence (PIF)[159], threshold ionisation mass spectrometry (TIMS)[160], photoioni- sation mass spectroscopy (PMS)[161], catalytic probe (CP)[162], actinometric op- tical emission spectroscopy (OES)[163], and vacuum ultra-violet laser absorption spectroscopy (VUVLAS)[164]. Due to the difficulties associated with measuring atomic loss coefficients, existing measurements are limited in both quantity and quality.
During general operation (i.e. without a sample holder present, and with a maximum of one glass plate attached), MAGPIE’s walls consist primarily out of stainless steel in the target chamber and borosilicate glass (Pyrex) in the source chamber. Measurements of γH for stainless steel are highly varied, with values
ranging from 6×10−3 to 2×10−1[64]. Measurements for borosilicate glass are
typically lower than stainless steel, with values ranging from 3 ×10−3 to 4×
10−2[64].
Extensive measurements of γH have been made previously in MAGPIE by
Samuell[64], for powers in the range of 100−800 W. At 800 W, these values are quoted as γH = (1.0±0.3)×10−1 for stainless steel and γH = (4±1)×10−2 for
(and therefore increasing ion flux), with each varying by half an order of magni- tude over the range of powers investigated. As standard operating conditions in this thesis involve total powers of 20 kW, we might expect values of γH to further
increase beyond those quoted values, however we do not currently have any ex- perimental evidence of this, and the power dependence must necessarily plateau as γH approaches unity. As we do not have the appropriate data to extrapolate
out to 20 kW, and we do not self-consistently treat the neutral dynamics within the plasma volume, we assume an approximate uniform value of γH = 1×10−1
for the standard operating conditions of the model. This choice is made with the awareness that the model behaviour must be investigated over a range of values of
γH for any meaningful results to be obtained, owing to the ambiguity associated
with experimental values.