Validation of X-ray Line Ratios for Electron Temperature Determination in Tokamak Plasmas*

A.S. Rosen1, M.L. Reinke2, J.E. Rice2, A.E. Hubbard2, and J.W. Hughes2

1Tufts University, Medford, MA 2Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA*Supported by the Princeton Plasma Physics Laboratory National Undergraduate Fellowship

X-ray imaging crystal spectroscopy (XICS) has been implemented on magnetic confinement

fusion devices as a novel means of measuring local plasma temperature and flow profiles. At

Alcator C-Mod, XICS allows for spatially-resolved, high spectral resolution measurements

between 0.3 nm and 0.4 nm, enabling detailed analysis of He-like and H-like argon x-ray

emission. Electron temperature profiles in the range of 0.5 keV < 𝑇e< 5.0 keV are computed

from ratios of the 𝑛 = 3 dielectronic satellites to the 1𝑠2 − 1𝑠2𝑝 resonance lines in He-like

argon. These data are validated against existing measurements of 𝑇e from electron cyclotron

emission and Thomson scattering. Line ratio data are analyzed via a tomographic inversion

procedure, overcoming the traditional issue of data being averaged over the plasma cross-

section. The implications of utilizing x-ray line ratios as valid local temperature diagnostics

are not limited to Alcator C-Mod; plasma properties in future experiments as well as in

astrophysical phenomena can also be investigated.

AcknowledgementsThis work is supported by US DoE contract DE-FC02-

99ER54512 and in part by an appointment to the National Undergraduate Fellowship

administered by the Office of Fusion Energy Sciences through the Princeton Plasma

Physics Laboratory.

Overview X-ray imaging crystal spectroscopy (XICS) allows for local 𝑻𝐞 profile measurements

Line data are analyzed via a tomographic inversion procedure[1] to overcome averaging

over plasma cross-section

First rigorous analysis for comparing Thomson scattering and electron cyclotron

emission (ECE) data with emissivity ratios for tokamak temperature diagnostics

Different portions of the He-like argon spectrum have unique temperature

dependencies that can be compared

Results are applicable to 𝑇e measurements for astrophysical plasmas and other

laboratory plasmas that do not have Thomson scattering or ECE data but have XICS

Analysis performed using the high resolution x-ray spectrometer with spatial resolution

(HIREXSR[2]) at Alcator C-Mod[3] in conjunction with the HIREXSR analysis code

(THACO[4])

Parameters and Data Set

Ohmic plasma with no external heating

Shot-to-shot plasma current control, toroidal field

control, and density scanning during discharge

allow for large 𝑻𝐞 range

Analyze portion of He-like argon spectrum that is

not vignetted

Six shots: 1100325010, 1100325012, 1100325016,

1100325023, 1100325025, 1100325029

Can be validated against existing 𝑇e measurements

from Thomson scattering and ECE

Compare 𝑻𝐞 profiles with computational theory

via argon K-shell population modeling from ab

initio atomic physics code[12]

The Emissivity Ratio The w-line is produced from electron impact excitation (threshold process) while DR is a

resonance process

Rate coefficient for DR is different as a function of 𝑻𝐞 for a Maxwellian electron

distribution[13] and thus emissivity ratios can provide a 𝑻𝐞 profile

Emissivity can be described by

𝜀 = 𝑛e𝑛𝑞𝑓 𝑇e Line-ratios eliminate density dependency[14]

𝜀1𝜀2

=𝑓1 𝑇e𝑓2 𝑇e

At low 𝑇e, recombination into the w-line (ratio of 𝑛𝑞+1 to 𝑛𝑞) can influence the validity of

𝑇e measurements. This is Ar17+ to Ar16+ for He-like argon

For the emissivity of the 𝑛 = 3 satellites

𝜀sat = 𝑛e𝑛𝑞𝑓sat 𝑇e For the emissivity of the w-line

𝜀𝑤 = 𝑛e𝑛𝑞𝑓exc 𝑇e + 𝑛e𝑛𝑞+1𝑓rec 𝑇e Taking a ratio of the 𝑛 = 3 satellites to the w-line yields

𝜀sat𝜀𝑤

=𝑓sat 𝑇e

𝑓exc 𝑇e +𝑛𝑞+1𝑛𝑞

𝑓rec 𝑇e

∼1

𝑇e

𝑇e was produced from a ratio of 𝒏 = 𝟑 satellites to the wn3 region, which will

quantitatively differ from the above equation but will retain the sensitivity to the ratio of

the charge state density

The wn3 region is used because the starting and ending wavelengths of the w-line are

naturally ambiguous due to the neighboring satellites that blend into the end of the w-line

Producing the Electron Temperature Fitting Procedure: Fits multiple Gaussians to AVESPEC. These will then be integrated to produce a given

brightness

Tomographic Inversion: Converts brightness to emissivity as a function of normalized minor radius, 𝜌, to

directly compare to 𝑇e (Thomson scattering and ECE data were recorded[15] as different functions of 𝜌)

Temperature Profile

Error propagation was computed via 100 fits with randomized spectral brightness using photon statistics

Emissivity ratio was interpolated to match 𝜌 values of 𝑇e The data will not be accurate over entire range of 𝜌, notably at the edge of the plasma where

recombination is significant

Outside 𝜌 ≈ 0.7 (𝑇e < 0.8 keV), emissivity ratio no longer monotonically increases and error in emissivity

ratio drastically increases

Therefore, 𝑻𝐞 values were most frequently plotted with 𝟎 ≤ 𝝆 ≤ 𝟎. 𝟕 The 𝟎. 𝟕 ≤ 𝝆 ≤ 𝟎. 𝟗 region carries information about the level of recombination in the plasma

Emissivity Ratio and Normalized Minor Radius

Shot 1100325012, 1.1985 s, 𝑇e ≈ 2 keV

Integrated Spectral Brightness

The spectral brightnesses were integrated over wavelength. This is then passed through the tomographic

inversion procedure[4] to obtain emissivity values

X-ray line-ratio techniques can effectively compute 𝑇e trends locally

For the first time, He-like x-ray line-ratio techniques from XICS have been validated

against conventional 𝑇e measurements in tokamak plasmas

Thomson scattering and ECE data agree with inverted line-ratios for 𝑇e in the range of

0.5 keV ≤ 𝑇e ≤ 3.0 keV An experimental power fit of 𝑇e keV = 0.1645𝑥−0.7642 can be used to describe the

electron temperature of the plasma from an emissivity ratio in the range of 0.0223 ≤𝑥 ≤ 0.2449

Experimental curve can have applications for astrophysical plasma and laboratory

plasma in cases where Thomson scattering or ECE measurements are not available

The trend is accurate for normalized minor radius values below 𝜌 ≈ 0.7 (𝑇e > 0.8 keV)

At higher 𝜌 values, recombination into the w-line and thus the ratio of Ar17+ to Ar16+

becomes an influencing factor

A similar analysis can be performed for He-like Ca, H-like Ca, and H-like Ar data

Conclusion

Difference Between Emissivity and Brightness Ratios

Line-integrated brightness

ratio will not produce an

identical 𝑇e trend

The use of a tomographic

inversion procedure to

compute local 𝑻𝐞 values has

a statistically significant

advantage over the averaging

associated with using a

brightness ratio

Analysis of Recombination

Plot shows a combined 𝑇e trend

containing Thomson scattering

and ECE data where 𝜌 ≤ 0.9 The accuracy of obtaining a 𝑇e

value from an emissivity ratio is

limited at high 𝑇e as the slope is

close to 0 and at low 𝑇e where

recombination is abundant

Emission of wn3 spectrum is a

combination of excitation of the

w-line, the sum of all

dielectronic recombination, and

some recombination population

of the w-line

References[1] M. L. Reinke, et al. Rev. Sci. Instrum. 83, 1135049 (2012)

[2] A. Ince-Cushman, et al. Rev. Sci. Instrum. 79, 10E302 (2008)

[3] E. S. Marmar, et al. Fusion Science and Technology 51, 261 (2007)

[4] M. L. Reinke, et al. Technical Report PSFC/RR-11-9, Plasma Science and Fusion Center (2012)

[5] M. Bitter, et al. Phys. Rev. Lett. 43 129 (1979)

[6] K. D. Zastrow, et al. Phys. Rev. A 41 1427 (1990)

[7] M. Bitter, et al. Phys. Scr. 31 551 (1985)

[8] E. Källne, et al. Phys. Rev. A 44 1796 (1991)

[9] J. E. Rice, et al. Rev. Sci. Instrum. 66 752 (1995)

[10] O. Marchuk, PhD Dissertation “Modeling of He-like spectra measured at the tokamaks TEXTOR and TORE SUPRA” Ruhur-

Universität Bochum (2004)

[11] J. Weinheimer, et al. Rev. Sci. Instrum. 72 2566 (2001)

[12] M. F. Gu, Canadian Journal of Physics 86, 675 (2008)

[13] D. Salzman, Atomic Physics in Hot Plasmas (Oxford University Press, 1998), p. 80

[14] I. H. Hutchinson, Principles of Plasma Diagnostics (Cambridge University Press, 2002), p. 256

[15] N. P. Basse, et al. Fusion Science and Technology 51, 476 (2007)

Dielectronic recombination graphic: Alfred Müller, Advances in Atomic, Molecular, and Optical Physics 55 (2008)

He-like Emission from Mid-Z Ions First detailed comparison of line ratios involving dielectronic recombination (DR) with

independent 𝑇e measurements were done with Ohmically-heated PLT plasmas and He-

like iron

Used ratio of 𝑗 satellite to 𝑤 line with plasma core 𝑇e from 1.65 keV to 2.30 keV

using ECE[5]

𝒏 ≥ 𝟑 satellites that were unresolvable from w line needed to be included to

improve agreement

Systematically predicted ~10% lower 𝑻𝐞 On JET, ratios of 𝑛 ≥ 3 satellites to w line and t satellite to x line were compared to 𝑇e

from ECE for 3.0 keV < 𝑇e < 12.0 keV and ~3000 He-like nickel spectra[6]

~10% lower 𝑻𝐞 On TFTR with He-like nickel spectra, 𝑇e derived from line ratios was compared to core

𝑇e from Thomson scattering for 2.5 keV < 𝑇e < 5 keV 𝑇e increased as 𝑛e decreased but Thomson scattering predicted higher 𝑻𝐞 by

~10%[7]

A multi-chord von Hamos spectrometer was used on Alcator C to measure profiles of the

k to w line ratio of He-like argon with r/a < 0.5 (no tomographic inversion)[8]

Agreement between 𝑇e from k/w ratio and ECE for 0.5 keV < 𝑇e < 1.3 keV

Was repeated on Alcator C-Mod[9] with 𝑇e < 1.5 keV Recently, all features of single, chord-averaged spectra were fit with a single 𝑇e and

compared to ECE (0.8–2.2 keV) and Thomson scattering (0.5–1.2 keV) on TEXTOR[10]

and NSTX[11], respectively

Line ratio data agreed within ~10% for both

This is the first time an inversion procedure was used to obtain local line-ratio data

Dielectronic Recombination

Detected by HIREXSR

Satellites to the w-line (1𝑠2 − 1𝑠2𝑝resonance) are produced from dielectronic

recombination from He-like argon into

doubly-excited states of Li-like argon

After DR of Ar16+ that produces an 𝑛 = 3satellite to the w-line, the Ar15+ ion will have

a configuration of 1𝑠2𝑝3ℓ, where ℓ is an

arbitrary spin state

Decay contributes to the satellites near the w-

line and results in a 1𝑠23ℓ configuration

Comparison to Thomson Scattering

Plotted for 𝜌 ≤ 0.7 Points with 𝑥-error bars ≥ 0.3 keV and/or 𝑦-error bars ≥ 0.05 were removed

This was done to maintain a meaningful 𝑇e for a given emissivity ratio

1629/1831 points retained (89.0%)

The red, dashed curves indicate the Ar-K data with unique Ar17+ to Ar16+ ratios: 1.0, 0.1, 0.01, 0.001,

and 0.0001 from bottom to top

Comparison to Electron Cyclotron Emission

Plotted for 𝜌 ≤ 0.7 Points with 𝑦-error bars ≥ 0.05 were removed

The 𝑥-error bars were set to 10% of the temperature in keV at each point

950/956 points retained (99.4%)

HIREXSR and the AVESPEC

HIREXSR is the spectrometer

used to analyze XICS data at

Alcator C-Mod over the entire

plasma cross-section

The spectral data (SPEC) is

binned to form the AVESPEC,

which reduces the number of

spectral points while retaining

accuracy

The AVESPEC is an averaging

of spectral brightness data

through the discretization of

the 2D image in HIREXSR

r/a

AVESPECSPEC

Abstract

Electron Temperature Trend

𝑇e as a function of emissivity ratio (axes

switched) and a power regression of

𝑻𝐞[𝐤𝐞𝐕] = 𝟎. 𝟏𝟔𝟒𝟓𝒙−𝟎.𝟕𝟔𝟒𝟐 with 𝒓𝟐 =𝟎. 𝟗𝟖𝟖𝟕 is shown over the emissivity

ratio range of 0.0223 ≤ 𝑥 ≤ 0.2449 Can predict 𝑻𝐞 given an emissivity ratio

Has applications for accurately measuring

𝑇e of certain astrophysical plasmas

based on line-ratio data

The above plot shows a 100 eV binned 𝑇etrend using the mean value of the data

with 𝜌 ≤ 0.7 The error bars were obtained from taking

the standard deviation of the 𝑥-error and 𝑦-

error over each bin

Theory curves of Ar-K data indicate 1:1

ratio of Ar17+ to Ar16+ is likely too large

Fitting Procedure

Can change the number of Gaussians, the

tying of Gaussians, and the widths/centers

of Gaussians

Reduced 𝜒2 was minimized without over-

constraining the data

Added two Gaussians to original code to

make a total of six and shifted 𝑛 = 3Gaussian center

Reduced 𝜒2 decreased from an average of

7.9 across all 6 shots, 64 channels, and 26

time points to an average of 4.2

Compare the emissivity of the 𝒏 = 𝟑satellites (numerator) to the entire wn3

spectrum (denominator)Shot 1100325012, Ch. 50, 0.5585 s

w-line

𝑛 = 3 satellites𝑛 > 3

He-Like Argon Spectrum

Blue portion of spectrum is the

region of interest (“wn3”) and is

from 3.9440 Å to 3.9607 Å

Isolated wn3 region from rest of

spectrum for analysis after the

AVESPEC was run

Shot 1100325012, Ch. 24, 1.1985 s

“wn3” region

𝑛 = 3 satellites

w-line

𝑥𝑦

𝑞𝑟 𝑘

𝑧

𝜆

𝑛 > 3