![Figure 1. The FIDA intensity function R shows the total FIDA intensity per unit ion density as function of (a) (v‖, v⊥)-coordinates and (b) (E, p)-coordinates. The units are [Nph/(s × sr × m2 ×Ni/m3)]. The Balmer-alpha photons can have any Doppler-shifted wavelength. We computed R using FIDASIM for NBI Q3 at ASDEX Upgrade. Q3 has an injection energy of 60 keV.](/figures/figure-1-the-fida-intensity-function-r-shows-the-total-fida-3kj6xqqv.webp)
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Title
On velocity-space sensitivity of fast-ion D-alpha spectroscopy
Permalink
https://escholarship.org/uc/item/8w89k5hx
Journal
Plasma Physics and Controlled Fusion, 56(10)
ISSN
0741-3335
Authors
Salewski, M
Geiger, B
Moseev, D
et al.
Publication Date
2014-10-01
DOI
10.1088/0741-3335/56/10/105005
Copyright Information
This work is made available under the terms of a Creative Commons Attribution License,
availalbe at https://creativecommons.org/licenses/by/4.0/
Peer reviewed
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Plasma Physics and Controlled Fusion
Plasma Phys. Control. Fusion 56 (2014) 105005 (15pp) doi:10.1088/0741-3335/56/10/105005
On velocity-space sensitivity of fast-ion
D-alpha spectroscopy
M Salewski
1
, B Geiger
2
, D Moseev
2,3
, W W Heidbrink
4
, A S Jacobsen
1
,
S B Korsholm
1
, F Leipold
1
, J Madsen
1
, S K Nielsen
1
, J Rasmussen
1
,
M Stejner
1
, M Weiland
2
and the ASDEX Upgrade Team
2
1
Technical University of Denmark, Department of Physics, DK-2800 Kgs. Lyngby, Denmark
2
Max-Planck-Institut f
¨
ur Plasmaphysik, D-85748 Garching, Germany
3
FOM Institute DIFFER, 3430 BE Nieuwegein, The Netherlands
4
University of California, Department of Physics and Astronomy, Irvine, CA 92697, USA
E-mail: msal@fysik.dtu.dk
Received 19 May 2014, revised 18 July 2014
Accepted for publication 13 August 2014
Published 8 September 2014
Abstract
The velocity-space observation regions and sensitivities in fast-ion D
α
(FIDA) spectroscopy
measurements are often described by so-called weight functions. Here we derive expressions
for FIDA weight functions accounting for the Doppler shift, Stark splitting, and the
charge-exchange reaction and electron transition probabilities. Our approach yields an
efficient way to calculate correctly scaled FIDA weight functions and implies simple analytic
expressions for their boundaries that separate the triangular observable regions in
(v
,v
⊥
)-space from the unobservable regions. These boundaries are determined by the
Doppler shift and Stark splitting and could until now only be found by numeric simulation.
Keywords: fast-ion D-alpha spectroscopy, FIDA, charge-exchange recombination
spectroscopy, fast ions, velocity space
(Some figures may appear in colour only in the online journal)
1. Introduction
Fast-ion D
α
(FIDA) spectroscopy [1–3] is an application
of charge-exchange recombination (CER) spectroscopy [4, 5]
based on deuterium [6–11]. Deuterium ions in the plasma
are neutralized in charge-exchange reactions with deuterium
atoms from a neutral beam injector (NBI). The neutralized
deuterium atoms are often in excited states, and hence they can
emit D
α
-photons which are Doppler-shifted due to the motion
of the excited atoms. As the excited atoms inherit the velocities
of the deuterium ions before the charge-exchange reaction,
spectra of Doppler-shifted D
α
-light are sensitive to the velocity
distribution function of deuterium ions in the plasma. The
measurement volume is given by the intersection of the NBI
path and the line-of-sight of the CER diagnostic. D
α
-photons
due to bulk deuterium ions typically have Doppler shifts of
about 1–2 nm whereas D
α
-photons due to fast deuterium,
which is the FIDA light, can have Doppler shifts of several
nanometers. This paper deals with FIDA light but as the
physics of D
α
-light due to bulk deuterium ions is the same,
our methods also apply to deuterium-based CER spectroscopy.
The FIDA or CER-D
α
light is sometimes obscured by Doppler
shifted D
α
-light from the NBI, unshifted D
α
-light from the
plasma edge, bremsstrahlung or line radiation from impurities.
FIDA spectra can be related to 2D velocity space by
so-called weight functions [2, 3, 12]. Weight functions have
been used in four ways: first, they quantify the velocity-
space sensitivity of FIDA measurements, and hence they also
separate the observable region in velocity space for a particular
wavelength range from the unobservable region [2, 3, 13–29].
Second, they reveal how much FIDA light is emitted resolved in
velocity space for a given fast-ion velocity distribution function
[2, 3, 24–30]. The ions in the regions with the brightest FIDA
light are then argued to dominate the measurement. Third,
weight functions have been used to calculate FIDA spectra
from given fast-ion velocity distribution functions [14, 31–33],
eliminating the Monte-Carlo approach of the standard FIDA
analysis code FIDASIM [34]. Fourth, recent tomographic
inversion algorithms to infer 2D fast-ion velocity distribution
0741-3335/14/105005+15$33.00 1 © 2014 EURATOM Printed in the UK
Plasma Phys. Control. Fusion 56 (2014) 105005 M Salewski et al
functions directly from the measurements rely heavily on
weight functions [12, 31–33, 35].
Here we present a comprehensive discussion of FIDA
weight functions and derive analytic expressions describing
them. FIDA weight functions have often been presented
in arbitrary units, relative units or without any units
[2, 3, 15–28, 30] which is sufficient for their use as indicator of
the velocity-space interrogation region or of the velocity-space
origin of FIDA light. However, correctly scaled FIDA weight
functions, which are necessary to calculate FIDA spectra
or tomographic inversions, have only been implemented in
the FIDASIM code recently [13, 14, 29, 31–33]. Weight
functions are traditionally calculated using the FIDASIM code
by computing the FIDA light from an ion on a fine grid in
2D velocity space and gyroangle. It is then counted how
many photons contribute to a particular wavelength range for
a given observation angle and point in velocity space using
models for the Doppler shift, Stark splitting, charge-exchange
probabilities and electron transition probabilities.
In section 2 we define weight functions and motivate their
interpretation in terms of probabilities. Our viewpoint provides
insights into functional dependencies between wavelength
space and 2D velocity space that are not revealed by the
traditional numerical calculation approach using FIDASIM.
As a consequence we demonstrate how Doppler shift,
Stark splitting, charge-exchange probabilities as well as the
electron transition probabilities contribute to the velocity-
space sensitivity of FIDA measurements. Section 3 focuses
on weight functions implied by the Doppler shift alone as a
relatively simple approximation. In section 4 we additionally
treat Stark splitting and in section 5 the charge-exchange
and the electron transition processes. In section 6 we
present full FIDA weight functions accounting for these four
effects. In section 7 we deduce exact analytic expressions
for the boundaries of FIDA weight functions. We discuss the
applicability of our results to CER spectroscopy and other fast-
ion diagnostics in section 8 and conclude in section 9.
2. Definitions of weight functions
The velocity-space interrogation or observation regions of
FIDA diagnostics are described by weight functions w
vol
which
are determined by charge-exchange probabilities, electron
transition probabilities, Stark splitting and the Doppler shift.
They thereby depend on position space and velocity space.
Weight functions are defined to obey [2, 3, 12]
I(λ
1
,λ
2
,φ)=
vol
∞
0
∞
−∞
w
vol
(λ
1
,λ
2
,φ,v
,v
⊥
, x)
×f(v
,v
⊥
, x)dv
dv
⊥
dx. (1)
I(λ
1
,λ
2
,φ) is the intensity of FIDA light in the wavelength
range λ
1
<λ<λ
2
with a viewing angle φ between the line-of-
sight of the FIDA diagnostic and the magnetic field. (v
,v
⊥
)
denote velocities parallel and perpendicular to the magnetic
field, respectively, and
x denotes the spatial coordinates. Here
we use (v
,v
⊥
)-coordinates rather than the more widespread
(E, p)-coordinates (energy, pitch) since our mathematical
expressions are simpler in (v
,v
⊥
)-coordinates. The energy
and the pitch are defined as
E =
1
2
m
D
v
2
+ v
2
⊥
(2)
p =−
v
v
(3)
where m
D
is the mass of a deuteron and v =
v
2
+ v
2
⊥
is the velocity magnitude. Note that the pitch is positive
for co-current particles as usual. Key expressions are
given in (E, p)-coordinates in the appendix. We assume
w
vol
(λ
1
,λ
2
,φ,v
,v
⊥
, x) and the fast-ion distribution function
f(v
,v
⊥
, x) to be spatially uniform within the small
measurement volume V . This may be violated near the foot
of the pedestal where the density gradient length scale could
be comparable with the mean free path of the emitters, but it
should be fulfilled in the core plasma. With
w(λ
1
,λ
2
,φ,v
,v
⊥
) = Vw
vol
(λ
1
,λ
2
,φ,v
,v
⊥
, x) (4)
equation (1) becomes
I(λ
1
,λ
2
,φ)
=
∞
0
∞
−∞
w(λ
1
,λ
2
,φ,v
,v
⊥
)f (v
,v
⊥
)dv
dv
⊥
. (5)
Weight functions w relate the FIDA intensity I(λ
1
,λ
2
,φ)with
units [N
ph
/(s×sr×m
2
)] to the 2D fast-ion velocity distribution
function with units [N
i
/(m
3
× (m/s)
2
)]. The units of FIDA
weight functions w are hence [N
ph
/(s×sr ×m
2
×N
i
/m
3
)], i.e.
FIDA weight functions w quantify the FIDA intensity per unit
ion density in the wavelength range λ
1
<λ<λ
2
for a viewing
angle φ as a function of the ion velocity (v
,v
⊥
). The units of
FIDA weight functions w
vol
are [N
ph
/(s ×sr ×m
2
×N
i
)], i.e.
the FIDA intensity per ion in λ
1
<λ<λ
2
for a viewing angle
φ as function of (v
,v
⊥
). We will split FIDA weight functions
w into a FIDA intensity function R(v
,v
⊥
) and a probability
prob(λ
1
<λ<λ
2
|φ,v
,v
⊥
) according to
w(λ
1
,λ
2
,φ,v
,v
⊥
)
= R(v
,v
⊥
)prob(λ
1
<λ<λ
2
|φ,v
,v
⊥
). (6)
R(v
,v
⊥
) determines the total FIDA intensity for any
wavelength of the photons per unit ion density. It depends only
on the charge-exchange and electron transition processes, but
not on the Doppler shift or Stark splitting that only change
the wavelength of the photons. prob(λ
1
<λ<λ
2
|φ,v
,v
⊥
)
determines the probability that a randomly selected detected
photon has a wavelength in a particular range λ
1
<λ<λ
2
for a given projection angle φ and (v
,v
⊥
)-coordinates. The
conditioning symbol ‘|’ means ‘given’. The subject of this
paper is the derivation of this probability. prob(λ
1
<λ<
λ
2
|φ,v
,v
⊥
) depends on the Doppler shift and Stark Splitting
as well as on the charge-exchange and electron transition
processes which in turn all depend on the gyroangle γ of
the ion at the time of the charge-exchange reaction. We treat
γ ∈ [0, 2π] as a random variable since we do not know the
phases of all ions in the plasma, i.e. the initial conditions of
any set of equations determining the ion motion are unknown
as always in problems with a very large number of degrees
of freedom. Since λ is determined by γ , it is also treated
2
Plasma Phys. Control. Fusion 56 (2014) 105005 M Salewski et al
−4 −2 0 2 4
0
1
2
3
4
v
||
[10
6
m/s]
v
⊥
[10
6
m/s]
100
200
300
400
500
(a)
050100
−1
−0.5
0
0.5
1
Energy [keV]
Pitch [−]
100
200
300
400
500
(b)
Figure 1. The FIDA intensity function R shows the total FIDA intensity per unit ion density as function of (a) (v
,v
⊥
)-coordinates and (b)
(E, p)-coordinates. The units are [N
ph
/(s ×sr × m
2
× N
i
/m
3
)]. The Balmer-alpha photons can have any Doppler-shifted wavelength. We
computed R using FIDASIM for NBI Q3 at ASDEX Upgrade. Q3 has an injection energy of 60 keV.
as random variable. Probabilities are always dimensionless
numbers in the interval [0,1], and hence the FIDA intensity
function R(v
,v
⊥
) has the same units as weight functions.
R(v
,v
⊥
) is a common factor of all weight functions for
agivenφ at any wavelength. On the contrary, the probability
function depends on the wavelength range and the projection
angle φ and hence contains the spectral information. We
compute R(v
,v
⊥
) using FIDASIM by modeling the charge-
exchange and the electron transition processes. Examples
of the FIDA intensity function for NBI Q3 at ASDEX
Upgrade, which is used for FIDA measurements, are shown in
figure 1(a)in(v
,v
⊥
)-coordinates and in figure 1(b)in(E, p)-
coordinates. The sensitivity of FIDA is low for very large ion
energies where few photons are generated per ion. Ions with
positive pitch generate more photons per ion than ions with
negative pitch for Q3.
Usually one measures spectral or specific intensities I
λ
, i.e.
the intensity per wavelength with units [N
ph
/(s × sr × m
2
×
nm)]. The intensity and the spectral intensity are related by
I(λ
1
,λ
2
,φ)=
λ
2
λ
1
I
λ
(λ, φ) dλ. (7)
The spectral intensity I
λ
(λ, φ) can likewise be related to
f(v
,v
⊥
) by a probability density function pdf(λ|φ,v
,v
⊥
)
that then leads to a differential weight function dw as
I
λ
(λ, φ) =
∞
0
∞
−∞
dw(λ, φ, v
,v
⊥
)f (v
,v
⊥
) dv
dv
⊥
(8)
with
dw(λ, φ, v
,v
⊥
) = R(v
,v
⊥
)pdf(λ|φ,v
,v
⊥
). (9)
However, the weight functions we discuss here are related
to a wavelength range rather than a particular wavelength
since FIDA intensity measurements can only be made
for a wavelength range and not for a single wavelength.
Mathematically this is reflected in the always finite amplitudes
of w whereas dw is singular at its boundary.
3. Doppler shift
An approximate shape of FIDA weight functions can be found
by considering only the Doppler shift λ − λ
0
where λ
0
=
656.1 nm is the wavelength of the unshifted D
α
-line and λ
is the Doppler-shifted wavelength. In this section we derive
this approximate shape by neglecting Stark splitting and by
assuming that the D
α
-photon emission is equally likely for all
gyroangles γ of the ion at the time of the charge-exchange
reaction. The probability density function in γ of randomly
selected detected D
α
photons is
pdf
D
α
(γ | v
,v
⊥
) = 1/2π. (10)
Stark splitting and an arbitrary pdf
D
α
describing charge-
exchange and electron transition probabilities will be
introduced into the model in the next two sections. The
Doppler shift depends on the projected velocity u of the ion
along the line-of-sight according to
λ − λ
0
= uλ
0
/c (11)
where c is the speed of light. Equation (11) assumes u c.
Consider a gyrating ion with velocity (v
,v
⊥
) in a magnetic
field. The ion is neutralized in a charge-exchange reaction
which ultimately leads to emission of a D
α
-photon. We define
a coordinate system such that for γ = 0 the velocity vector of
the ion is in the plane defined by the unit vector along the line-
of-sight ˆ
u and B such that v ·ˆu > 0. Then the ion velocity is
v = v
ˆ
B + v
⊥
cos γ ˆv
⊥1
− v
⊥
sin γ ˆv
⊥2
(12)
and the unit vector along the line-of-sight is
ˆ
u = cos φ
ˆ
B + sin φ ˆv
⊥1
. (13)
The velocity component u of the ion along the line-of-sight at
a projection angle φ to the magnetic field is then given by [12]
u =
v ·ˆu = v
cos φ + v
⊥
sin φ cos γ. (14)
3
Plasma Phys. Control. Fusion 56 (2014) 105005 M Salewski et al
Figure 2. Projection of the ion velocity (v
,v
⊥
) and the unit vector ˆv ×
ˆ
B onto the line-of-sight. The latter is required for the treatment of
Stark splitting discussed in section 4.
The projections of the ion velocity v and the unit vector
ˆ
v ×
ˆ
B (relevant for Stark splitting) onto the line-of-sight in
this coordinate system are illustrated in figure 2. Equation (14)
shows that u is a random variable which depends on the random
variable γ ∈ [0, 2π]. We now calculate the probability
prob(u
1
<u<u
2
|φ,v
,v
⊥
) that the ion has a projected
velocity between u
1
and u
2
at the time of the charge-exchange
reaction and therefore a Doppler-shifted D
α
-line wavelength
between λ
1
and λ
2
according to equation (11). For given
(v
,v
⊥
) with v
⊥
= 0 and projection angle φ = 0, the projected
velocity depends on the gyroangle γ . Conversely, we can
calculate the gyroangles that lead to a given projected velocity
u by solving equation (14) for γ :
γ = arccos
u − v
cos φ
v
⊥
sin φ
. (15)
The arccos function is defined for 0 <γ <π, and a second
solution in π<γ
< 2π is given by
γ
= 2π − γ. (16)
Using equations (15) and (16) we can calculate gyroangles γ
1
and γ
2
and γ
1
and γ
2
corresponding to the limits u
1
and u
2
and
transform the calculation of the probability to γ -space:
prob(u
1
<u<u
2
|φ,v
,v
⊥
)
= prob(γ
2
<γ <γ
1
|v
,v
⊥
) + prob(γ
1
<γ <γ
2
|v
,v
⊥
)
=
γ
1
γ
2
pdf
D
α
(γ | v
,v
⊥
)dγ +
γ
2
γ
1
pdf
D
α
(γ | v
,v
⊥
) dγ.
(17)
As we here assume a uniform probability density, we can
integrate equation (17) analytically:
prob(u
1
<u<u
2
|φ,v
,v
⊥
) =
γ
1
− γ
2
2π
+
γ
2
− γ
1
2π
=
γ
1
− γ
2
π
. (18)
The probability prob(u
1
<u<u
2
|φ,v
,v
⊥
) is thus the
fraction of the gyro-orbit that leads to a projected velocity
between u
1
and u
2
. Substitution of γ using equation (15)
gives
prob(u
1
<u<u
2
|φ,v
,v
⊥
)
=
1
π
arccos
u
1
− v
cos φ
v
⊥
sin φ
− arccos
u
2
− v
cos φ
v
⊥
sin φ
.
(19)
Equation (19) is singular for v
⊥
= 0orφ = 0. If φ = 0, the
projected velocity is just the parallel velocity as equation (14)
reduces to u = v
. Then the probability function becomes
prob(u
1
<u<u
2
|φ = 0,v
,v
⊥
) =
1 for u
1
<v
<u
2
0 otherwise
(20)
which is identical to equation (19) in the limit φ → 0. For
v
⊥
= 0, i.e. on the v
-axis corresponding to ions not actually
gyrating, equation (14) reduces to u = v
cos φ, and the
probability function becomes
prob(u
1
<u<u
2
|φ,v
,v
⊥
= 0)
=
1 for u
1
/ cos φ<v
<u
2
/ cos φ
0 otherwise.
(21)
Lastly, we note that the argument of the arccos function
is often outside the range [−1;1]. In this case the
arccos is complex, and we take the real part to obtain
physically meaningful quantities. Equation (19) is a weight
function describing just the projection onto the line-of-sight.
We have previously derived the corresponding probability
density function pdf(u|φ,v
,v
⊥
) to describe the velocity-
space sensitivity of collective Thomson scattering (CTS)
measurements [12]. The pdf can be found from the probability
function by letting u
1
,u
2
→ u:
pdf(u|φ,v
,v
⊥
) = lim
u
1
,u
2
→u
prob(u
1
<u<u
2
|φ,v
,v
⊥
)
u
2
− u
1
=
1
πv
⊥
sin φ
1 −
u − v
cos φ
v
⊥
sin φ
2
. (22)
Equations (19)to(22) have been used to interpret CTS
measurements at TEXTOR [36] and should have great utility
4