# Action-angle formulation of generalized, orbit-based, fast-ion diagnostic weight functions

## Abstract

Due to the usually complicated and anisotropic nature of the fast-ion distribution function, diagnostic velocity-space weight functions, which indicate the sensitivity of a diagnostic to different fast-ion velocities, are used to facilitate the analysis of experimental data. Additionally, when velocity-space weight functions are discretized, a linear equation relating the fast-ion density and the expected diagnostic signal is formed. In a technique known as velocity-space tomography, many measurements can be combined to create an ill-conditioned system of linear equations that can be solved using various computational methods. However, when velocity-space weight functions (which by definition ignore spatial dependencies) are used, velocity-space tomography is restricted, both by the accuracy of its forward model and also by the availability of spatially overlapping diagnostic measurements. In this work, we extend velocity-space weight functions to a full 6D generalized coordinate system and then show how to reduce them to a 3D orbit-space without loss of generality using an action-angle formulation. Moreover, we show how diagnostic orbit-weight functions can be used to infer the full fast-ion distribution function, i.e., orbit tomography. In depth derivations of orbit weight functions for the neutron, neutral particle analyzer, and fast-ion D-α diagnostics are also shown

- Authors:

- Univ. of California, Irvine, CA (United States)

- Publication Date:

- Research Org.:
- Princeton Univ., NJ (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1474286

- Alternate Identifier(s):
- OSTI ID: 1374930

- Grant/Contract Number:
- AC02-09CH11466; FC02-04ER54698

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 9; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS

### Citation Formats

```
Stagner, L., and Heidbrink, W. W..
```*Action-angle formulation of generalized, orbit-based, fast-ion diagnostic weight functions*. United States: N. p., 2017.
Web. doi:10.1063/1.4990391.

```
Stagner, L., & Heidbrink, W. W..
```*Action-angle formulation of generalized, orbit-based, fast-ion diagnostic weight functions*. United States. doi:10.1063/1.4990391.

```
Stagner, L., and Heidbrink, W. W.. Mon .
"Action-angle formulation of generalized, orbit-based, fast-ion diagnostic weight functions". United States.
doi:10.1063/1.4990391. https://www.osti.gov/servlets/purl/1474286.
```

```
@article{osti_1474286,
```

title = {Action-angle formulation of generalized, orbit-based, fast-ion diagnostic weight functions},

author = {Stagner, L. and Heidbrink, W. W.},

abstractNote = {Due to the usually complicated and anisotropic nature of the fast-ion distribution function, diagnostic velocity-space weight functions, which indicate the sensitivity of a diagnostic to different fast-ion velocities, are used to facilitate the analysis of experimental data. Additionally, when velocity-space weight functions are discretized, a linear equation relating the fast-ion density and the expected diagnostic signal is formed. In a technique known as velocity-space tomography, many measurements can be combined to create an ill-conditioned system of linear equations that can be solved using various computational methods. However, when velocity-space weight functions (which by definition ignore spatial dependencies) are used, velocity-space tomography is restricted, both by the accuracy of its forward model and also by the availability of spatially overlapping diagnostic measurements. In this work, we extend velocity-space weight functions to a full 6D generalized coordinate system and then show how to reduce them to a 3D orbit-space without loss of generality using an action-angle formulation. Moreover, we show how diagnostic orbit-weight functions can be used to infer the full fast-ion distribution function, i.e., orbit tomography. In depth derivations of orbit weight functions for the neutron, neutral particle analyzer, and fast-ion D-α diagnostics are also shown},

doi = {10.1063/1.4990391},

journal = {Physics of Plasmas},

number = 9,

volume = 24,

place = {United States},

year = {Mon Aug 14 00:00:00 EDT 2017},

month = {Mon Aug 14 00:00:00 EDT 2017}

}

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