# A code that simulates fast-ion Dα and neutral particle measurements

Abstract: A code that models signals produced by charge-exchange reactions between fast ions and injected neutral beams in tokamak plasmas is described. With the fast- ion distribution function as input, the code predicts the efflux to a neutral particle analyzer (NPA) diagnostic and the photon radiance of Balmer-alpha light to a fast- ion Dα (FIDA) diagnostic. Reactions with both the primary injected neutrals and with the cloud of secondary "halo" neutrals that surround the beam are treated. Accurate calculation of the fraction of neutrals that occupy excited atomic states (the collisional- radiative transition equations) is an important element of the code. Comparison with TRANSP output and other tests verify the solutions. Judicious selection of grid size and other parameters facilitate efficient solutions. The output of the code has been validated by FIDA measurements on DIII-D but further tests are warranted.

## Summary (4 min read)

### 1 Introduction

- Supra-thermal populations of energetic ions play an important role in magnetic fusion research.
- Both NPA and FIDA diagnostics provide valuable information about the fast-ion distribution function but also depend sensitively on other plasma parameters and on atomic cross sections.
- A least-squares minimization scheme that utilizes a weight function can determine which model distribution function agrees best with the data.
- Section 3 describes tests that verify that the code correctly solves the desired equations.

### 2 Model

- The code has four main sections (Fig. 1).
- The first section prepares the data and the second calculates the neutral populations.
- The third and fourth sections both rely on the first two sections but are independent of each other.
- One section computes the NPA flux and the other computes the FIDA radiance.

### 2.1 Input data and coordinate mapping

- The code begins by collecting the input data.
- The code uses the conventions of the NUBEAM module [8] of the TRANSP code [9] to describe the geometry of the viewed neutral beam source (or sources).
- For installations that do not use EFIT, a post-processor that is part of the TRANSP distribution can convert TRANSP output files into the desired format.
- Plasma parameters are one-dimensional functions of flux coordinates.
- Depending on the source of the theoretical fast-ion distribution function, the mapping into the (x,y,z) coordinates can be fairly complicated.

### 2.2 Injected and halo neutral densities

- The second major section of FIDASIM is devoted to calculation of the injected and halo neutral distributions in real space, velocity space, and energy levels.
- The third major simplification in the treatment of the collisional-radiative transitions is to assume that the speeds of the various species follow the ordering ve ≫v f ∼vn ∼vi ≫vI , where the subscripts represent electrons, fast ions, hydrogenic neutrals (both fast and thermal), thermal hydrogenic ions, and impurity ions, respectively.
- From the known source, the code computes the cloud of halo neutrals that surround each injected beam.
- The neutral is followed until it ionizes.
- (For typical parameters, this is true for 99% of the halo neutrals, so this is an excellent approximation.).

### 2.3 NPA flux

- The NPA flux is found in the third stage of the code.
- The detector geometry determines the diagnostic viewing cone but the detected particles have guiding centers that are a gyroradius from the viewing cone.
- The distribution function used in Eq. (2.6) is evaluated at the guiding center position.
- The attenuation of neutrals is computed in the last part of the NPA calculation.

### 2.4 FIDA radiance

- The fourth stage of the code uses a weighted Monte Carlo routine to calculate the FIDA radiance.
- The product n f ∑nn provides a convenient estimate of the probability of a charge exchange reaction (that neglects the computationally intensive dependence of the reaction rate on the relative velocity), so this product is used to determine how many fast neutrals to launch from each cell.
- Next, the trajectory of the fast neutral through the cells is computed by TRACK.
- The code also performs a simple integration over these emissivities ǫi along the specified sightlines.
- Note that the computed spectra neglect instrumental broadening.

### 3.1 Atomic physics

- Atomic physics cross sections are important in two places in the code: in the calculation of neutralization probability and in the solution of Eq. (2.3) in COLRAD.
- The required cross sections and reactivities are available in the literature and in the Atomic Data and Analysis Structure (ADAS) compilation [20, 21].
- An alternative compilation of many of the rates appears in [26] but the effect of these differences is small compared to other uncertainties so the current version of the code uses the older compilation by Janev [24].
- In COLRAD, rates for energy levels up to n=7 are normally employed.
- For the initial neutralization probability, cross sections for the charge-exchange reactions between fast ions and neutrals in states n = 1−4 are given in ADAS [20].

### 3.2 Beam deposition

- The calculation of the injected neutral density was compared with TRANSP for an NSTX case.
- The neutral profiles along the y axis (perpendicular to the neutral beam source) and along the z axis (vertical to the source) axis agree well with the TRANSP simulation results (Fig. 4) and with neutral beam calibration data [27].
- Fig. 5(b) shows the attenuation factors for 60keV neutrals along an NPA sightline for the TRANSP and FIDASIM simulations.
- Fig. 6 compares spectra after integration over the sightline.
- At low density, the agreement is satisfactory when halo neutrals are neglected in FIDASIM.

### 3.3 Halo neutrals

- The halo neutral simulation portion of the code was verified by comparing with a onedimensional diffusion model.
- The leftmost term in Eq. (3.2) represents the halo neutral diffusion.
- It is not easily solved analytically for a spatially varying beam neutral profile but a solution exists for a constant source.
- Fig. 7 shows the halo neu- tral densities calculated from the diffusion model and the Monte-Carlo halo simulation subroutine for identical plasma profiles.

### 3.4 FIDA spectra

- The SPECTRUM subroutine used to compute the Dα spectra and the weighted Monte Carlo scheme were verified as follows.
- As part of the initial investigation of the feasibility of FIDA, a simplified model of the expected spectra was developed that ignores atomic physics and assumes that the magnetic field is purely toroidal; Fig. 2 of [3] shows a result calculated by this code.
- To test the main FIDA simulation loop, the authors replaced the magnetic field with a toroidal field and modified the cross sections to be independent of velocity.
- The resulting spectra were consistent with the output of the simple model.

### 4 Numerics

- Numerical input parameters affect both the accuracy and computational expense of a calculation.
- Uncertainties in plasma parameters (particularly electron density) introduce uncertainties in the predicted radiance or efflux of 20% or more , so extremely fine grids are wasteful and unnecessary.
- The neutral-beam injection energy is 80keV in this plasma.
- The Fortran version of the code is an order of magnitude faster than the IDL version.
- For a typical FIDA simulation with 107 reneutrals this translates to about 28 hours.

### 4.1 Number of Monte Carlo reneutrals launched

- There is a linear dependence between the number of launched neutralized fast ions (or ”reneutrals”) and the computational time in the fourth section of the code.
- Fig. 8(a) compares the spectra at the location of peak Dα emissivity (R = 187.5cm) for several simulations with varying number of launched reneutrals.
- To eliminate the influence of the random number seed choice, the same seed value was used in all five simulations.
- Considering the resulting spectra as the most accurate, it is instructive to compare the ratios of the spectra from simulations with less particles to the spectra from this particular simulation (Fig. 8(b)).
- If the integration starts at half the beam injection energy, ∼3 times more particles are needed.

### 4.2 Simulation volume

- The computational grid must enclose most of the interacting beam ions and neutral particles.
- Further reduction of the grid in the x-direction to 90cm truncates a sizable fraction of the halo neutrals but the effect on the calculated spectra is still small because Further down the x-axis, the plasma density increases and, as more injected neutrals charge exchange with thermal deuterium ions, the beam halo profiles spread in both transverse directions (Fig. 11(c)).
- The authors baseline simulation uses values of the grid half-width of 30 and 40cm in the y- and z-directions, respectively.
- For FIDA chords that view horizontally, expanding the grid along the y-axis is more important than expanding it along the z-axis.

### 4.3 Cell size

- Once the simulation volume in the (x,y,z) space is defined, the size of each cell needs to be determined.
- Physically, the emissivity profiles depend on numerous quantities including the beam injection geometry and the plasma parameters; gradients in any of these quantities impact transverse emissivity gradients.
- Generally speaking, owing to the line integration, the cell size in the approximate direction of the sightlines can be twice as large as in transverse directions.
- To study the effect of coarser or finer grids, the authors use the standard 120×60×80cm computational domain and 107 particles.
- The corresponding computational times are 63% and 22% higher.

### 5 Validation

- The calculation of the injected neutrals was compared with experimental measurements of the beam-emission light in a DIII-D experiment [32].
- After passing through a bandpass filter, two-dimensional images of the light were measured with a CCD camera.
- The code predictions are in good agreement with measurements of the vertical extent of the beam and of the beam penetration as a function of density.
- In MHD-quiescent DIII-D plasmas, code predictions based on the fast-ion distribution function predicted by NUBEAM have the same spectral shape as experiment and the intensity of the FIDA signal agrees to within 25%.
- Comparison of the relative intensity of these four features is a useful check that is independent of any experimental errors in the intensity calibration.

### 6 Outlook

- With plasma profiles and a fast-ion distribution function as input, FIDASIM predicts the flux measured by NPAs and the radiance measured by a Dα spectrometer.
- One possible area of improvement is a post-processor that replaces the approximation of infinitesimal sightlines with an accurate treatment of the collection optics.
- A more challenging upgrade is needed to treat plasmas where the fastion density is comparable to the thermal-ion density.
- A complementary reduced model that is sufficiently fast to make predictions between discharges (for example) is needed.

### Acknowledgments

- This work has benefited from the contributions of a large number of scientists.
- Keith Burrell, Bill Davis, Rainer Fischer, Manuel Garcı́a-Muñoz, Doug McCune, and Mike Van Zeeland gave valuable advice.
- The authors are also indebted to their experimental collaborators on DIII-D and NSTX.
- The originating developer of ADAS is the JET Joint Undertaking.

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