Plasma and Fusion Research: Regular Articles

Volume 14, 3402083 (2019)

Study of Lyman-α Polarization due to Anisotropic Electron
Collisions in LHD∗)
Nilam RAMAIYA1) , Motoshi GOTO1,2) , Tetsutarou OISHI1,2) and Shigeru MORITA1,2)
1)

Department of Fusion Science, SOKENDAI (The Graduate University for Advanced Studies), Toki 509-5292, Japan
2)
National Institute for Fusion Science, Toki 509-5292, Japan(Received 9 January 2019 / Accepted 22 March 2019)
We have investigated the polarization of Lyman-α line in the Large Helical Device (LHD). A theoreticalmodel has been developed for a quantitative understanding of the anisotropy in the electron velocity distributionfunction (EVDF). The present model treats an anisotropic EVDF having diﬀerent electron temperatures in the
directions parallel and perpendicular to the magnetic field. The steady-state time period of an electron cyclotron
heated discharge has been analyzed and the comparison between the theoretical and measured polarization degrees suggests that in the edge plasma the diﬀerence between these two temperatures is approximately 10%.
In addition, the dependence of the observed Lyman-α intensity on the angle of the linearly polarized light has
been studied. The obtained results show that in the experimentally observed intensity the inboard side emission
dominates over the emission from the outboard side and in the edge LHD plasma the electron temperature in theperpendicular direction to the magnetic field is higher than that in the parallel direction.
c 2019 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: Lyman-α line, velocity distribution function, polarization degree, VUV spectrometer, Large Helical
Device
DOI: 10.1585/pfr.14.3402083

1. Introduction
The measurement of anisotropy in the electron velocity distribution function (EVDF) can provide significant information for understanding transport phenomena,equilibria, and current drive in a fusion plasma. Although
the anisotropic EVDF plays an important role in a magnetically confined fusion plasma [1], it has not been actively investigated in plasma experiments. Plasma polarization spectroscopy can be a useful technique to study
the anisotropy in the EVDF [2]. In the present study
we deal with the polarization of Lyman-α line caused byanisotropic collisions with electrons.
The Large Helical Device (LHD) is a heliotron-typemagnetic confinement fusion experimental device. The
measurements of Lyman-α line polarization have been carried out in LHD. On the other hand, the development ofa theoretical model is necessary to correlate the experimentally measured polarization and the anisotropy in the
EVDF. This spectral line at 121.56 nm has been chosen
for this purpose because, owing to the simple energy levelstructure relating to this line, development of an accurate
theoretical model is possible.
The previous study regarding hydrogen emission in
LHD suggests that the dominant emissions of hydrogenare located outside the confined region of the plasma [3,4],
which consists of stochastic magnetic field lines. In such
author’s e-mail: nilam.nimavat@nifs.ac.jp∗) This article is based on the presentation at the 27th International Toki
Conference (ITC27) & the 13th Asia Pacific Plasma Theory Conference(APPTC2018).

regions, the confinement characteristics of the electrons arestrongly dependent on their velocity pitch angle with respect to the magnetic field direction. In LHD the strengthof the magnetic field varies in the toroidal direction andbecause of that electrons move under the influence of amirror eﬀect. Electrons having a small pitch angle, known
as passing electrons, can escape easily from the magnetic
mirror but those with a large pitch angle, known as trappedelectrons, are trapped inside the ripples of the magnetic
field strength. As a result, the confinement time of trappedelectrons should be longer than that of passing electrons,
which would cause the anisotropy in the EVDF.
The theoretical framework for calculating the polarization of Lyman-α due to anisotropic electron collisions
following the methodology proposed by Fujimoto [2] is
described in Sec. 2 with some initial results. For a comparison of the model results with the measured results, somegeometrical aspects of the measurements in LHD need tobe taken into account. The details are described in Sec. 3
and the obtained results are shown in Sec. 4.1. In Sec. 4.2
the comparison between the variation of experimentallyobserved intensity and the simulated intensity with the angle of linearly polarized light is discussed.

2. Theoretical Model
The Lyman-α line consists of two components, i.e.,
1 S1/2 − 2 2 P1/2 and 1 2 S1/2 − 2 2 P3/2 . Figure 1 shows
all the possible transitions among the magnetic sublevels
relating to the Lyman-α line. As indicated in the figure,23402083-1

c 2019 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles

Volume 14, 3402083 (2019)
Under the corona equilibrium, the population of theexcited state p is balanced by the electron impact excitationfrom the ground state and the spontaneous radiative decay.
The rate equation for the population can be given asA(p, s) n(p),(4)
C 0,0 (1, p) ne n(1) =sFig. 1 Transitions among the magnetic sublevels relating to the
Lyman-α line. The dashed and dotted lines indicate
σ−light and π−light, respectively.

the Δm J = 0 transition emits π−light, linearly polarized in
the quantization axis direction, and Δm J = ±1 transitions
emit σ−light, circularly polarized in the plane perpendicular to the quantization axis. Here m J means the magnetic
quantum number of the level having the total angular momentum quantum number J. When all the magnetic sublevels are populated uniformly, the emitted light intensity
is isotropic, i.e., the line is not polarized. However, theimbalance of the population among the sublevels results inthe polarization of emitted light.
Here, the quantization axis is taken to be in the magnetic field direction. We assume an axisymmetric systemwith respect to the quantization axis where there is no orientation among the magnetic sublevels, i.e., the populations of m J and −m J sublevels are the same. For this reason the emitted light corresponding to the 1 2 S1/2 − 2 2 P1/2
transition is not polarized, while the 1 2 S1/2 − 2 2 P3/2 transition may emit polarized light. For developing a theoretical model, first the 1 2 S1/2 − 2 2 P3/2 transition is considered and in the final result the influence of the unpolarized
1 2 S1/2 − 2 2 P1/2 transition is taken into account.
The assumptions above mentioned imply that for an
axisymmetric system the density matrix of an excited statep, can be approximately expanded as [2]ρ(p) = ρ00 (p) T 0(0) (p) + ρ20 (p) T 0(2) (p),(1)

where T q(k) (p) is the irreducible tensor operator [5]. The
expansion coeﬃcients ρkq (p) are given by [2]ρkq (p) =(−1) J−N JJM − N | kq ρ M,N (p).
(2)
MN




Here, JJM −N | kq is the Clebsch-Gordan coeﬃcient andρ M,N (p) is the elements of the density matrix in the |J m J 
representation. Thus, two quantities are assigned to eachenergy level: population ρ00 (p) and alignment ρ20 (p). The
conventional population n(p) is expressed byn(p) = 2J p + 1 ρ00 (p).
(3)
The alignment is a measure of the population imbalanceamong the magnetic sublevels in a J level. Henceforth, forsimplicity, a(p) is used in place of ρ20 (p).

where C 0,0 (1, p) is the rate coeﬃcient for the electron impact excitation, A(p, s) is the Einstein A coeﬃcient for thetransition from a level p to a lower level s, and ne is theelectron density. Similarly, the rate equation for a(p) canbe expressed asA(p, s) a(p),(5)
C 0,2 (1, p) ne n(1) =s0,2where C (1, p) is the alignment creation rate coeﬃcient.
The population, n(p), and the alignment, a(p), are then derived as
C 0,0 (1, p) nen(1),n(p) = s A(p, s)(6)

C 0,2 (1, p) nen(1).
a(p) = s A(p, s)(7)

We consider a case in which the emission line is observed from a direction perpendicular to the quantization
axis, using a linear polarizer. The longitudinal alignment,
AL , is defined as [2]AL =I − I⊥,I + 2I⊥(8)

where I and I⊥ are the linearly polarized light intensities inthe direction parallel and perpendicular to the quantization
axis, respectively. For a transition from a level p to a levels, AL and the relative alignment a(p)/n(p), are related as[2]
3
J p +J s(2J p + 1)
AL (p, s) = (−1)
2a(p)Jp Jp 2,(9)×
1
1 J s n(p)where · · · is the 6- j symbol.
Calculations of the coeﬃcients C 0,0 (1, p) and
0,2
C (1, p) are carried out under a certain EVDF. We
assume that the EVDF is axisymmetric with respect tothe quantization axis and the electron temperature has
diﬀerent values in the directions parallel and perpendicular
to the quantization axis or the magnetic field. The explicit
expression for such EVDFs is given as [2]31

m 2 1 22πk
T ⊥2 T  mv2 sin2 θ cos2 θ × exp −+,
2k T ⊥Tf (v, θ) = 2π

(10)where v is the absolute value of the velocity, θ is the pitch
angle of the velocity with respect to the magnetic field, and3402083-2

Plasma and Fusion Research: Regular Articles

Volume 14, 3402083 (2019)from the Q0,00 (1, p) as [2]

J p +J sQ0,2
0 (1, p) = (−1)



Jp
1

×

Jp
1

2
Js

2
(2J p + 1)−1
3

−1

AL (p, 1) Q0,00 (1, p). (15)The data for Q0,00 (1, p) and AL are obtained from Refs.
[6] and [7], respectively. Here, AL is the value for thecase where the hydrogen atoms are excited with a monoenergetic beam of electrons and the emitted Lyman-α radiation is observed from a direction perpendicular with respect to the incident electron beam axis. That means theobservation angle is 90◦ . Figure 2 shows these quantities
for the 1 2 S1/2 − 2 2 P3/2 transition.
The polarization degree for this transition, which may
emit polarized radiation is defined asP1 =Fig. 2 (a) Longitudinal alignment, AL , values under the condition that the hydrogen atoms are excited by a monoenergetic beam of electrons and (b) population creation
cross section, Q0,00 , and alignment creation cross section,22Q0,20 , for the 1 S1/2 − 2 P3/2 transition.

I1 − I⊥1,I1 + I⊥1

where I1 and I⊥1 stand for the intensities in the paralleland perpendicular directions to the quantization axis, respectively.
The unpolarized transition, 1 2 S1/2 − 2 2 P1/2 , is included in the observed line and its influence is evaluated.
We assume that the 2 2 P1/2 and 2 2 P3/2 states are populated
according to their statistical weight, i.e.,n(2 2 P1/2 ) =m and k are the electron mass and the Boltzmann constant,respectively. Here, T  and T ⊥ represent the electron temperature in the directions parallel and perpendicular to themagnetic field, respectively.
The rate coeﬃcients C 0,0 (1, p) and C 0,2 (1, p) are evaluated as [2]

0,0
3
Q0,0
(11)
C (1, p) =0 (1, p) 4π f0 (v) v dv,

C 0,2 (1, p) =


 3
Q0,2
0 (1, p) 4π f2 (v) / 5 v dv, (12)0,2where Q0,00 (1, p) and Q0 (1, p) are the excitation and alignment creation cross sections, respectively, for the corresponding transition, and f0 (v) and f2 (v) are the coeﬃcients of expansion of f (v, θ) by the Legendre polynomials,
PK (cos θ), asfK (v) PK (cos θ).
(13)
f (v, θ) =
K

(16)

1
n(2 2 P3/2 ),2(17)

where n(2 2 P1/2 ) and n(2 2 P3/2 ) are the total populations of
the states 2 2 P1/2 and 2 2 P3/2 , respectively.
The resulting polarization degree, denoted here as
Pres , which takes into account both the transitions is expressed as
Pres =(I1 + I2 ) − (I⊥1 + I⊥2 ).
(I1 + I2 ) + (I⊥1 + I⊥2 )(18)

Here, I2 and I⊥2 are the intensities of the 1 2 S1/2 − 2 2 P1/2
transition in the directions parallel and perpendicular to thequantization axis, respectively.
Because the 1 2 S1/2 − 2 2 P1/2 line is unpolarized, I2 =I⊥2 , and I2 + I⊥2 = (I1 + I⊥1 ) / 2 from the assumption
above, Pres is rewritten as
Pres =

I1 − I⊥1
2
= P1 .
(I1 + I⊥1 ) + (I2 + I⊥2 ) 3(19)

The Pres evaluated for diﬀerent T  and T ⊥ values are shownin Fig. 3. The results indicate that the value of polarizationis negative for T ⊥ > T  and is positive for T ⊥ < T  .

The coeﬃcient fK (v) is explicitly given as
2K + 1f (v, θ) PK (cos θ) sin θ dθ. (14)fK (v) =2

3. Lyman-α Polarization for LHD
Experiments

The alignment creation cross section Q0,2
0 (1, p) is obtained

We define the “angle of observation” to be the angle
between the line of sight and the quantization axis. For the3402083-3

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Fig. 3 Polarization degree calculated for diﬀerent T  and T ⊥ under the assumption of corona equilibrium. T  and T ⊥ referto the electron temperatures in the directions parallel andperpendicular to the magnetic field, respectively.

Fig. 4 Schematic drawing of magnetic field surfaces for theplasma axis at Rax = 3.75 m along with the observation range covered by the spectrometer (shown with thedashed lines). The reﬀ = ±0.67 m locations are shownwith circles at Z = −0.4 m. Negative and positive valuesof reﬀ correspond to the inboard side and the outboard
side, respectively.

present case, the direction of the magnetic field is taken as
the quantization axis. The results in Sec. 2 correspond tothe cases when the angle of observation is 90◦ . However,in the present experiments in LHD the angle of observation
may be diﬀerent from 90◦ . Therefore, the results of thetheoretical model can not be used directly and the influenceof the angle of observation on the final result must be taken
into consideration.

3.1

Viewing geometry and calculation of theangle of observation

The hydrogen Lyman-α line has been observed fromthe edge region of the LHD plasma with a 3 m normal incidence VUV spectrometer [8]. The optical components
developed and provided by the CLASP (Chromospheric
Lyman-Alpha Spectro-Polarimeter) project [9] team havebeen additionally installed inside the spectrometer to obtain the polarization-resolved spectra of the Lyman-α line.
Figure 4 illustrates a schematic drawing of thehorizontally-elongated cross section of LHD along withthe magnetic surfaces for the magnetic axis position at
Rax = 3.75 m. The field of view is shown with dashed lines.
The variables of Z and R are the vertical and the radial coordinates, respectively. According to the previous study,
the hydrogen emission is located outside the confined region in LHD and it can be approximated that the emission
location is at reﬀ = 0.67 m in the LHD plasma [3, 4]. Here,reﬀ refers to the eﬀective minor radius of the plasma. From
this result, it is clear that the Lyman-α is emitted at boththe inboard and outboard sides of the device, and the intensity observed by the spectrometer is the sum of these
two intensities.
In LHD, the magnetic field is determined accuratelyby the coil currents. The present discussion focuses on
the steady-state time period of an electron cyclotron heatedFig. 5 The definitions of angles of the magnetic field with respect to the line of sight. θ and ζ are the inclination andazimuth angles of the magnetic field vector B, respectively. γ is the angle between the line of sight and B. a is
the unit vector in the same direction as the projection of
B onto the φ − z plane and η is the angle of a with respect
to the z axis.

discharge of shot no. 138764. Using the available data, thedirections and magnitudes of the magnetic field vectors, B,
at the emission locations, i.e., reﬀ = ±0.67 m have been
determined.
Figure 5 shows the angles θ, ζ, and the angle of observation, denoted here as γ. The inclination angle θ represents the angle of the magnetic field B with respect to thez direction. The azimuth angle ζ shows the angle between
the projection of the magnetic field B onto the φ − r plane
and the φ direction. The angle of observation γ can be obtained from the inclination angle θ and the azimuth angle ζusing the relation

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cos γ = sin θ sin ζ.

(20)

Plasma and Fusion Research: Regular Articles

Volume 14, 3402083 (2019)parallel and perpendicular directions to a: Iσ + ,η and Iσ + ,η⊥ ,respectively.
Iσ + = Iσ + ,η + Iσ + ,η⊥=1(cos2 γ + 1)Iσ+ ,2(22)and the σ− component gives rise to the intensities
Iσ − = Iσ − ,η + Iσ − ,η⊥=1(cos2 γ + 1)Iσ− .
2(23)In the experiments on LHD, a half-waveplate speciallydesigned for Lyman-α is rotated to monitor the linearly polarized light emission at all angles. Here, we define the angle of the linearly polarized light as the polarization angle,
denoted here as α. The intensity observed at the polarization angle α consists of the π component
Io,π = Iπ cos2 (η − α)= sin2 γ cos2 (η − α)Iπ ,Fig. 6 Magnetic field parameters for shot no. 138764 at Z =−0.4 m : (a) the variation of the magnetic field strength,(b) the inclination angle θ of the magnetic field with respect to the z direction, and (c) the azimuth angle ζ whichis the angle between the projection of the magnetic field
vector onto the φ − r plane and the φ direction. The
reﬀ = ±0.67 m locations at the inboard side and the outboard side are indicated with the vertical dashed lines.

The B vectors point upward and downward at the inboard side and the outboard side, respectively. Although
the observation range is 15 cm wide in the Z coordinate,
here we consider only Z = −0.4 m for simplicity. The
parameters of the magnetic field vector B at Z = −0.4 m
are plotted in Figs. 6 (a) - (c). The value of the angle ofobservation is found to be 77.5◦ and 79◦ at the Lyman-αemission locations for the inboard and outboard sides, respectively.

3.2

The Lyman-α line is assumed to be emitted at the position O where the magnetic field B is directed as shown in
Fig. 5. Iπ refers to the linearly polarized light parallel to B.
Iσ+ and Iσ− are the circularly polarized light in the direction
perpendicular to B. For a particular angle of observation γ,
we define a unit vector a which is in the same direction as
the projection of B onto the φ − z plane. η is the angle of awith respect to the z axis.
The π component is observed as the light linearly polarized in the direction of a. We denote its intensity as IπIπ = Iπ sin2 γ.

and the σ+ and σ− components
Io,σ+ = Iσ + ,η cos2 (η − α) + Iσ + ,η⊥ sin2 (η − α)1 2=
cos γ cos2 (η − α) + sin2 (η − α) Iσ+ ,2(25)Io,σ− = Iσ − ,η cos2 (η − α) + Iσ − ,η⊥ sin2 (η − α)1 2=
cos γ cos2 (η − α) + sin2 (η − α) Iσ− .
2(26)The angle η is related to θ and ζ astan η = tan θ cos ζ.

(21)The σ+ component gives rise to the intensities in both the(27)

It should be noted here that the angles θ and η are calculated with respect to the +z axis and the azimuth angle ζ is
calculated with respect to the +φ axis. The total observed
intensity at the polarization angle α is given as
Iobs = Io,π + Io,σ+ + Io,σ− .

Derivation of the intensity observed
from the line of sight

(24)

(28)

We assume axisymmetry with respect to the magnetic field
direction. Therefore, in this case Iσ+ = Iσ− ≡ Iσ . Thus, thefinal expression for Iobs can be given as
Iobs = sin2 γ cos2 (η − α) Iπ+ cos2 γ cos2 (η−α)+sin2 (η−α) Iσ .

(29)

4. Results and Discussion
4.1 Calculation of polarization for the LHD
experiments
The polarization degree, Eq. (16), is defined here withrespect to the unit vector a. Thus, when in Eq. (29), α = η,intensity I is observed and when α = η + π/2, intensity

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Plasma and Fusion Research: Regular Articles

Volume 14, 3402083 (2019)I⊥ is observed. The obtained intensities are given by thefollowing equations
I = Iπ sin2 γ + Iσ cos2 γ,(30)I⊥ = Iσ .

(31)

In the present case, the angle of observation is 77.5◦
and 79◦ for the inboard and outboard sides, respectively.
The electron temperature in this region is roughly 10 eV.
The polarization degree has been calculated by assuming
T  = 10 eV and with the angle of observation 77.5◦ . The
result of the calculation is shown in Fig. 7 together withPres obtained for the angle of observation 90◦ . The perpendicular electron temperature, T ⊥ , is varied from 3 eV to
30 eV. The results indicate that the polarization values forthese two angles of observation are nearly the same. The
details of the experimental setup on LHD and of the calculation of polarization degree are given in Ref. [10]. The
observed P value is in the order of 0.01, which suggeststhat the diﬀerence between T  and T ⊥ is about 10%.

4.2

Simulation of experimentally observed
intensity variation

The derived expression for the intensity observed fromthe line of sight, Eq. (29), and the obtained values of the angle of observation, γ, can be used to understand the variation of the experimentally monitored intensity with the polarization angle. From such investigation, the information
regarding the relation between the magnetic field direction
and the intensity profile, and regarding the dominant emission region can be inferred.
The simulated intensity profiles of Lyman-α havebeen obtained using Eq. (29) and the parameters of themagnetic field at the inboard side and the outboard side.
In the present case, we focus on the shot no. 138764 of
LHD and the magnetic field values at reﬀ = ±0.67 m for
Z = −0.4 m have been used. These values have been shownin Fig. 6. Positive and negative values of reﬀ are for the

Fig. 7 Polarization degree for the angle of observation 90◦ (bluedashed line) and 77.5◦ (red dash-dotted line).

outboard side and the inboard side, respectively. Here thevalue of polarization degree has been taken to be the sameas the experimental value.
Figure 8 shows the variation of the simulated intensities with the polarization angle α for the inboard andoutboard sides with the experimentally obtained intensity
variation. It is clearly seen that the experimental result
agrees with the simulated result for the inboard side. Although the experimental intensity generally consists ofemissions from both the inboard and outboard sides, and
it is not possible to evaluate local P values at these twosides, this observation suggests that the main contribution
to the polarized emission comes from the emission at the
inboard side plasma. Even when emission at the outboard
side exists, it may be unpolarized and works to lower theobserved polarization degree. Due to this reason, although
the observed P value is in the order of 0.01, there is a possibility that the local P value at the inboard side may behigher than 0.01 and thus the diﬀerence between T  andT ⊥ at the inboard side is larger than 10%.
It is also found that the experimentally obtained intensity is explained only when P is negative. This fact indicates that the electron temperature in the direction perpendicular to the magnetic field is higher than that in the
direction parallel to the magnetic field, i.e., T ⊥ > T  .
The error in the measured intensity itself is smaller
than the symbol size in Fig. 8, and the actual uncertaintyof the present measurement is mainly attributed to the unsteadiness of the discharge which is recognized as the discrepancy of the measured intensities from the fitted curvein Fig. 8. For the purpose of estimating the uncertainty ofthe present measurement, the root mean squared error of
the measured intensity with respect to the fitted curve isevaluated and given as the error bars in Fig. 8.

Fig. 8 Lyman-α intensity as a function of polarization angle.
The experimental result is plotted with circles and thefitted result is shown with the dashed-dotted line. The
solid and dashed lines show the simulated results for the
inboard side and the outboard side, respectively.

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Acknowledgments
We are grateful to the LHD Experiment Group and thetechnical staﬀ of LHD for their support of this work. We
would also like to thank the CLASP project team for providing the optical components for the LHD experiments.
This work is partly supported by JSPS KAKENHI Grant(Number 26287148) and by the National Institute for Fusion Science grant administrative budget (ULHH028).
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