Ion velocity distribution function

In this section we study how the Doppler spectral line-shape is related to the inhomoge- nious ion velocity distribution function,fi(r,V). This is a six dimensional quantity which

describes the number of particles in a plasma species at position (x, y, z) with the velocity (Vx, Vy, Vz). It may also extend to seven dimensions to include variation in time, t.

22 Spectroscopic and probe diagnostics on MAGPIE

An arbitrary ion velocity distribution function (IVDF) when considered at a single po- sition in the plasma, is a 3D quantity and can be depicted as volume in velocity space as shown in left image of figure 2.9. The color indicates the value of the IVDF and the velocity coordinates are normalised such thatv =V/c. For a cold stationary plasma the distribution function would be a delta function centered at v = (0,0,0). A hot moving plasma will have the distribution shifted in the direction of the particle flow, and the spread will be determined by the thermal energy carried by the particles.

l

Figure 2.9: Left) A Non-Maxwellian IVDF at positionrwithin the plasma. The blue represents

the value of the IVDF in velocity space. The line (red) denotedl is the direction of view defined

by the angles αand ϕ. The length ξ is the coordinate along the viewing chord. Right) the line

shapeg(ξ,r; ˆl) is the plane-integrated IVDF at coordinateξas defined in Equation 2.13.

In this model the direction of observation is shown by the line l (described by the unit vector ˆl) which is defined by the angle of incidenceα and azimuthal angleϕ. The Doppler features of the measured spectral line are determined from the contribution of the intensity- weighted velocities in the direction of the line of observation.

The spectral lineshape for the line lat position r can be called g(ξ,r; ˆl) where the coor- dinate ξ is the normalized frequency coordinate already described in equation 2.7. The values ofg(ξ,r; ˆl) at frequency coordinateξare determined by summing over the values of the IVDF which have a velocity component parallel to ˆlthat is equal to the coordinateξ, see fig 2.9. This 3D Radon transform (integral over planes) is discussed further in chapter 6. Mathematically, this is written as,

g(ξ,r;ˆl) =

Z

fi(r,v)δ(ξ−v·ˆl) dv (2.13)

§2.1 Spectroscopic diagnostics in plasmas 23

plasma.

A state of thermal equilibrium means the distribution function for a species is homogeneous (no r dependence), isotropic (independent of the direction of v) and is time invariant [57]. The Maxwell-Boltzmann distribution function is often used to describe the velocity distribution function of a plasma in thermal equilibrium, where the standard deviation is related to the temperature.

A plasma in local thermal equilibrium (LTE) can also be described by a Maxwell-Boltzmann distribution but relaxes the homogeneity condition so that the temperature and drift may change spatially depending on external parameters such as magnetic fields or sources. For this condition to hold, the plasma density must be high enough that collisional processes dominate radiative processes so that radiative losses (thermal and other) do not signifi- cantly disrupt the local thermal equilibrium [57]. The drifting local thermal equilibrium model is employed as a simple starting point in understanding radiative processes or mod- eling distribution functions in plasmas. The following discussion has been addressed in the works of Howard [42, 58].

Considering the case where the plasma is drifting in local thermal equilibrium, the IVDF is Maxwellian and can be written in the form,

fi(r,v) = πv2_th− 3 2 exp− v−vD vth !2 (2.14)

where the normalised thermal velocity is

v_th2 = kBTS

mc2 (2.15)

and TS(r) is the species temperature. The velocity drift, vD, and the thermal velocity

have been normalised toc.

Substituting the distribution function into the equation 2.13, the spectral line-shape be- comes g(ξ,r;ˆl) =πv_th2 − 1 2 exp− ξ−vD·ˆl vth !2 (2.16) where ξ = ν−ν0 ν0 =vD·ˆl (2.17)

24 Spectroscopic and probe diagnostics on MAGPIE

From the Doppler shift formula (equation 2.1). The Fourier transform of the line-shape gives the local complex coherence (compare with equation 2.11),

G(φ0,r;ˆl) = Z ∞ −∞ g(ξ,r;ˆl) exp (iφˆ0ξ) dξ = πv_th2 −1 2 Z ∞ −∞ exp− ξ−vD·ˆl vth !2 exp (iφˆ0ξ) dξ = expiφˆ0vD·ˆlG0(φ0,r; ˆl) (2.18)

where G0 is the Fourier transform of the Gaussian lineshape

G0(φ0,r; ˆl) = exp− ˆ

φ2₀ξ_th2

4 (2.19)

and where ˆφ0 is the phase delay set by the interferometer. This is distinct to the complex coherence ˜γ(φ0) (as presented in section 2.1.3) which is weighted by the local intensity and subject to line integration effects. The connection between G(φ0,r;ˆl) and ˜γ(φ0) is discussed in section 2.1.7.

It is convenient to introduce the characteristic temperature of the instrument TC (set by

the delay ˆφ0), 1 2mSV 2 C =kBTC where VC = 2c ˆ φ0 (2.20)

The expression for the Gaussian lineshape can be re-written as,

G0(φ0,r;ˆl) = exp−

TS

TC

(2.21)

Plasmas can be inhomogeneous in both space and velocity having multiple features such as fast moving beams or contain both hot and cold species. The velocity distribution functions for such plasmas are non-Maxwellian and therefore the emission line shapes for such plasmas are not expected to be Gaussian.

Figure 2.10 compares the spectral line shapes and associated coherence for the case of (a) a Maxwellian, and (b) a non-Maxwellian IVDF. For a Maxwellian IVDF, the contrast and phase are given by known expressions shown in figure 2.10 (these are derived later in section 4.3.7). As TC is a function of ˆφ0, it is clear that the ion temperature TS and the

flowvD (equivilent to vD·ˆl) can be calculated from measurements of the coherence at an

appropriately chosen delay.

§2.1 Spectroscopic diagnostics in plasmas 25

Non-gaussian envelope

Non-uniform phase

Figure 2.10: Spectral line shape and corresponding real component of the coherence for (a) a Maxwellian and (b) a Non-Maxwellian distribution function.

An example of a spectral line and the corresponding coherence in the temporal domain for a non-Maxwellian plasma is shown in 2.10 (b). In this case, a single measurement of the coherence is not enough to return information about the IVDF. However measurements at multipleφ0 coordinates can help us sample the coherence and therefore identify the shape of the coherence envelope and phase. In theory, tomography on the 3D Radon transform can then deliver the velocity distribution function.

The MAGPIE IVDF has not previously been determined for the plasma conditions exam- ined in this study. Non-Maxwellian IVDFs have been reported in other helicon devices operating at similar conditions [32, 33, 59] and so it is important to consider the extended theory for the non-Maxwellian case.