Hydrogen Radiation Transport Modeling in
ALCATOR C-Mod
by
Mark Lloyd Adams
Submitted to the Department of Nuclear Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Nuclear Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1999
@ Massachusetts Institute of Technology 1999. All rights reserved.
A uthor ...................
F'
Department of Nuclear Engineering
7 May 1999
Certified by......
Sergei I. Krasheninnikov
Senior Research Scientist, MIT Nuclear Engineering and MIT PSFC
Thesis Supervisor
Certified by.........
I
Jeffrey P. Freidberg
Department Head, MIT Nuclear Engineering
Thesis Supervisor
't
/
D f
t/ V
A ccepted by ..................
..............
Lawrence M. Lidsky
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF RAE
Z*1999
JmLIBRARIES
cne
Hydrogen Radiation Transport Modeling in ALCATOR
C-Mod
by
Mark Lloyd Adams
Submitted to the Department of Nuclear Engineering
on 7 May 1999, in partial fulfillment of the
requirements for the degree of
Master of Science in Nuclear Engineering
Abstract
This thesis represents an effort to better understand hydrogen radiation transport in
the ALCATOR C-Mod tokamak. Through a theoretical discussion and numerical sensitivity study of hydrogen line profiles, in conditions representative of high-density
low-temperature fusion tokamak reactors, information useful for improving experimental estimates of plasma parameters from spectral data has been obtained. A
numerical sensitivity study, using the codes TOTAL and BELINE, is performed to
qualitatively understand how plasma parameters, especially the magnetic field, influence line profile shapes. Through application of a numerical simulation code named
CRETIN to the study of hydrogen radiation transport in ALCATOR C-Mod, further
insight into the effect of opacity on the plasma power balance has been obtained. It
is seen that variations in a line profile effect photon escape rates, which are closely
related to opacity, which has a strong impact on plasma recombination, and hence
influences the plasma power balance. Thus, in plasma regions where recombination
is occurring and there is some trapping of radiation, it is now understood that both
radiative and non-radiative modes of energy transport are equally important in determining the plasma balance.
Thesis Supervisor: Sergei I. Krasheninnikov
Title: Senior Research Scientist, MIT Nuclear Engineering and MIT PSFC
Thesis Supervisor: Jeffrey P. Freidberg
Title: Department Head, MIT Nuclear Engineering
Acknowledgments
I am grateful to the many people who have contributed to my thinking, research and
pursuit of greater proficiency in my undergraduate and graduate studies. Including
my mother, for being loving and supportive, having an open heart, and providing a
non-judgmental approach to my entire educational pursuit; and Terry, for providing
comical relief and witty cynicism when I have been taking myself too seriously. However, I especially want to express my profound gratitude to the following people who
have not only contributed to the foundation of my scientific understanding, but have
provided support and guidance in this research project.
Personnel at Massachusetts Institute of Technology
Alexander Yu. Pigarov, Massachusetts Institute of Technology (MIT) Plasma Science
and Fusion Center (PSFC) research scientist, for his insights regarding the application
of radiation transport theory to ALCATOR C-Mod. Sergei I. Krasheninnikov, thesis
technical supervisor, MIT PSFC Senior Research Scientist, a person who has made
this thesis project possible. Xavier Bonnin, MIT PSFC Postdoc, for the many fruitful
discussions on CRETIN. Dieter J. Sigmar and Jeffrey P. Freidberg, MIT Nuclear
Engineering (NE) faculty members, thanks for making it possible for me to perform
research at MIT.
Personnel at Lawrence Livermore National Laboratory
Howard A. Scott, developer of CRETIN (a code which is at the core of my thesis).
As well, while I was at LLNL Dr. Scott's insight opened my eyes to the purpose of
my research and he gave me the direction which enabled me to understand radiation
transport theory. Alan Wan and Douglas Post, for their significant interest in fusion
energy and support of this research project.
Contents
1
9
Introduction
13
2 Theoretical Line Profiles
3
2.1
Energy Levels of Atomic Hydrogen ...................
13
2.2
Formation of Line Profiles ........................
19
25
Numerical Spectral Line Broadening
3.1
3.2
Line Profile Codes
.........
. . . . . . . . . . . . . .
25
3.1.1
TOTAL .............
. . . . . . . . . . . . . .
26
3.1.2
BELINE ...........
. . . . . . . . . . . . . .
26
. . . . . . . . . . . . . .
27
Line Profile Computations.....
4 1-D ALCATOR C-Mod MARFE Simulations
5
36
4.1
Experimental Data ........
. . . . . . . . . . . . . . . . . . . . .
36
4.2
Numerical Simulations ......
. . . . . . . . . . . . . . . . . . . . .
39
4.3
Recombination and Opacity . . . . . . . . . . . . . . . . . . . . . . .
42
45
Summary
4
List of Figures
3-1
BELINE normalized hydrogen Lyman-y line profile dependence
on angle between the direction of observation and magnetic field
(3). Plasma Parameters: B = 6[T], n, = 100
[cm- 3], Te = 1[eV],
28
....................................
=x...........
3-2
13
X 10
a) BELINE normalized hydrogen Lyman-a line profile dependence
on 3. b) BELINE normalized hydrogen Lyman-0 line profile de-
pendence on 0. Plasma Parameters: B = 6[T], ne = 100 x 1013 [cm-3,
28
Te = 1[eV],/3 = x .....................................
3-3
BELINE normalized hydrogen Balmer-a (He) line profile depen-
dence on /.
Te = 1[eV],
3-4
=x
. . . . . . . . . . . . . . . . ..
X 1013[CM-3
. . . . . . . . . .
29
BELINE normalized hydrogen Balmer-3 (H3) line profile depen-
dence on /.
Te = 1[eV], /=
3-5
Plasma Parameters: B = 6[T], ne = 100
Plasma Parameters: B = 6[T], ne = 100 X 1013[CM-3
x
29
.............................
.........
BELINE and TOTAL normalized HQ line profile dependence on
electron density. Plasma Parameters: B = 0[T], ne = X X 1013 [cm-3]
Te = 2[eV], 0 = 0 .
3-6
..........
...................
30
BELINE normalized H, line profile dependence on electron density as observed parallel to the magnetic field. Plasma Parameters:
B = 6[T], ne = x x 1013 [cm- 3 ], Te = 1[eV],/3 = 0 . . . . . . . . . . . . .
3-7
31
BELINE normalized H 0 line profile dependence on electron density as observed perpendicular to the magnetic field. Plasma Pa3
rameters: B = 6[T], ne = x x 1013 [cm ], Te = 1[eV], / =2..
5
.....
31
3-8
BELINE and TOTAL normalized Ha line profile dependence on
electron temperature.
Plasma Parameters:
B = 0[T], ne = 50 x
1013 [cm- 3 ], T = x[eV], 3 = 0 ......................
3-9
.
32
.
33
BELINE normalized deuterium-a (Da) line profile dependence on
magnetic field and 3. Plasma Parameters: B = x[T], ne = 100 x
1013 [cm- 3 ], Te = 1[eV], 3 = x ......................
3-10 BELINE normalized Ha and Da line profiles as observed parallel
to the magnetic field.
Specific dependence on energy is shown.
Plasma Parameters: B = 6[T], ne = 100 x 10
13
[cm
3
], T, = 1[eV],
/3 = 0 34
3-11 BELINE normalized Ha and Da line profiles as observed perpendicular to the magnetic field.
Specific dependence on energy
is shown.
B = 6[T],
Plasma Parameters:
Te = 1[eV],3 ' .=..
.
ne = 100 X 1013[CM-3],
........................
34
3-12 BELINE and TOTAL hydrogen Lyman-/3 line profile scaled to
compare with Lee and Oks Fig. 1. [1] Plasma Parameters: B = O[T),
ne = 3000 x 1013 [cm-3, Te = 3[eV], 3 = 0 ......
4-1
35
................
ALCATOR C-Mod CCD Camera position and viewing area. a)
represents the view that an observer positioned above the tokamak
looking down would have -
the bird's eye view; b) represents a
side view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
4-2
Chord direction and extent for which MARFE spectral data have
been obtained. LCFS stands for Last Closed Flux Surface.
4-3
37
.
.
38
Measured Lyman (lower) and Balmer (upper) series spectral data
from an ALCATOR C-Mod MARFE.
6
. . . . . . . . . . . . . . . .
38
4-4
CRETIN input plasma profiles for MARFE simulations.
electron density in units of 10 2 0 [m-3]; Ng20:
Ne20:
neutral deuterium
density in units of 1020 [m-3; n = 2: first excited state of deuterium
density in units of 10 2 0 [m- 3 ]; Te: electron temperature in units of
[eV]; and Ti: ion temperature in units of [eV]. DLCFS stands for
Distance from the Last Closed Flux Surface and positive numbers
indicate coordinates within the scrape-off layer (SOL); see Fig. 4-2
for an understanding of the geometry. . . . . . . . . . . . . . . . .
4-5
39
CRETIN Lyman (lower) and Balmer (upper) series spectral simu-
lation data for an ALCATOR C-Mod MARFE. The dashed line includes the effect of spectrometer broadening as required for com-
parison with Fig. 4-3. . . . . . . . . . . . . . . . . . . . . . . . . . .
4-6
40
Brightness of Lyman-series and Dc, lines from the 1-D MARFE
model, comparing experimental measurement to two CRETIN
runs, one with standard Te and Ti profiles (noted T = T) and one
with enhanced ion temperature to reproduce the effect of Zeeman
splitting.
4-7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Number of effective recombinations per Dy photon in an optically
thin limit (solid line and diamonds), and for two cases in which
Lya,
are optically thick. The thick, dashed curve and the closed
squares show the results for N 0AL=1 x 1019 m- 2 using escape factor
model and CRETIN, respectively.
The thin, dot-dashed curve
shows the case for N 0 AL=2x1018 m-2. . . . . . . . . . . . . . . . .
4-8
43
Radial opacity and emissivity CRETIN data has been postprocessed to allow for comparison with Lyman-a spectral data collected from a tangential viewing CCD camera.
Absorption and
emission coefficients have been obtained at line center.
7
. . . . .
44
4-9
Radial opacity and emissivity CRETIN data has been postprocessed to allow for comparison with D,
spectral data collected
from a tangential viewing CCD camera. Absorption and emission
coefficients have been obtained at line center.
8
. . . . . . . . . . .
44
Chapter 1
Introduction
Through the study of hydrogen radiation transport one is able to expound salient
characteristics of emerging radiation. If theoretical predictions agree with an experimental spectrum one is then able to infer information about system parameters. For
tokamak plasmas what this amounts to is a non-perturbative means of determining
plasma parameters. This thesis will present theoretical efforts, implemented through
numerical simulation, to obtain a better understanding of hydrogen radiation transport in the ALCATOR C-Mod tokamak.
For many years astrophysicists have studied radiation transport in an endeavor
to understand stars. Astrophysics began with Newton's application of Kepler's third
law to the sun and continues today as scientists seek to interpret phenomena which
even fascinate the general public. An exciting review of seminal radiation transport
papers written at the beginning of this century is given by Menzel [2]. During the
World War II era, primarily due to the usefulness of radiation transport research in
the development of atomic weapons, an enormous amount of progress was made in our
qualitative understanding of the behavior of radiation and quantitative formalism for
obtaining results. Research of this era is reviewed in books written by Chandrasekhar
(1950) [3], Kourganoff (1952) [4] and Sobolev (1963) [5]. In astrophysics today, aside
from exciting theoretical advances, it is interesting to note that numerical simulations
are common and a review of established methods may be found in Mihalas (1978) [6].
Turning our attention to plasma systems and fusion reactors it is safe to say that
9
plasma physicists have always been attune to the presence of radiation. A well established core ion impurity radiation transport model is based on coronal equilibrium
(CE) assumptions and is valid for plasma temperatures greater than 1[eV] and densities below 1015 [cm 3 ]. [7] In CE theory one assumes that the time for an excited atom
to radiate its excess energy is much less than the time for collisional energy transfer
(rad
<- Tco,)
and that the emitted radiation is optically thin. These assumptions
allow one to construct simple temperature dependent ionization-recombination rates
per ion. Thus, spectral data from the core plasma may be collected to determine
temperature profiles from signatures of the dominant ionization level. [8]
In low-temperature high-density regions of a plasma (Te
10
5
-
1[eV] and ne
[cm- 3]), such as those found within a detached divertor or MARFE, the colli-
sional time scale for transfer of excited atomic energy is comparable to the radiative
time scale
(Tco
~
Trad).
Also, the assumption of an optically thin plasma breaks
down and some radiation becomes "trapped". Both effects significantly complicate
the study of radiation transport as will be discussed later in this introduction. Radiation trapping and deviations from local thermodynamic equilibrium (LTE) in plasma
systems is reviewed by Abramov et al. [9]. Krasheninnikov and Pigarov were the first
to consider radiation trapping in fusion tokamaks; they announced that Lya lines
of hydrogen may become trapped in a high-density radiating divertor [10] and then
confirmed their prediction through a one-dimensional transport simulation [11].
At the beginning of the decade, a review article by Stangeby and McCracken
abjured that reducing the divertor target plate incident heat flux would be a major
obstacle in the path toward achieving a fusion reactor. [12] Subsequent experiments in
ALCATOR C-Mod were able to demonstrate majority ion recombination, an inherently collisional phenomenon, with associated divertor detachment and draw attention
toward atomic processes. [13] Mirroring the research performed by Krasheninnikov
and Pigarov, Post et al. [14] reviewed atomic processes relevant to reducing divertor
heat fluxes and Wan et al. [15] presented a multi-dimensional numerical simulation
using CRETIN [16, 17] to further demonstrate trapping of specific hydrogen lines
within a detached divertor.
10
Experimental verification of dominant recombination channels and radiation trapping now exists. ALCATOR C-Mod data indicate that the dominant channel for
recombination is three-body recombination with a small percentage of radiative recombination [18]; data also indicate that the Ly,/ 3 lines are trapped [19]. Molecularactivated recombination (MAR) is also expected to significantly contribute to the
ionization-recombination balance but has not yet been experimentally measured. [20,
21] Further experimental evidence has been reported by ASDEX-U [22], DIII-D [23]
and JET [24].
Present research efforts, including this thesis, are being directed toward quantifying hydrogen radiation transport effects on the energy balance. Terry et al. [25, 26]
have had great success in experimental studies of recombination rates through use of
a "recombinations per Balmer series photon" concept. Pigarov et al. [27] have been
able to decrease the uncertainty in determining plasma parameters through numerical
simulation of the Balmer continuum. This thesis seeks to make progress toward an
adequate investigation of the full neutral-plasma-radiation relationship. The relation
between ionization-recombination rates, material motion (neutral-plasma) within the
system and photon creation-destruction rates to the energy balance, and hence temperature structure, is well known and solved in codes such as UEDGE. [28] When
radiation becomes trapped and excited states of an ion become important, the energy
balance is significantly effected by photon escape rates and the coupling of excited
states to the radiation field. These relations have yet to be incorporated into codes
like UEDGE [29] and the latter phenomenon has only been investigated apart from
material motion in codes such as CRETIN, a multi-dimensional non-local thermodynamic equilibrium (NLTE) simulation code based on an "isolated atom" treatment of
atomic kinetics. [30, 31] A full self-consistent solution will require an understanding of
how photons are scattered in a plasma system; both how the radiation field interacts
with neutral-plasma particles through photon creation-destruction rates and how escape probabilities, which are purely a field phenomenon, couple excited atomic states
in both frequency and real space and are influenced by boundary conditions. To this
end, much of this thesis will look at line profiles since they are known to significantly
11
contribute to the degree of non-locality in a plasma system; trapped lines of hydrogen
possess wings which are still optically thin.
The outline of this thesis is as follows.
Chapter 2 is a review of line broaden-
ing theory through conceptual arguments; energy levels of the hydrogen atom are
established for low-temperature high-density ALCATOR C-Mod conditions, and an
understanding of line broadening is obtained through a discussion on reference frames
and study of time-dependent perturbations. Chapter 3 presents a compare and contrast deliberation of two existing line broadening codes through a sensitivity study
on dominant lines. In Chapter 4 our interest in obtaining a better understanding of
radiation effects on the energy balance is demonstrated by a presentation of the latest
numerical computations and their comparison with experiment. Chapter 5 concludes
this thesis and motivates future research.
12
Chapter 2
Theoretical Line Profiles
Constructing appropriate theoretical line profiles is an arduous exercise in modeling.
In this chapter an effort is made to qualitatively profile stages of modeling necessary
to achieve accurate line profiles. A dichotomous discussion will begin by considering
an isolated hydrogen atom, looking at first-order perturbations of interest in fusion
plasmas and establishing dominant energy levels.
Focus is then turned to effects,
including those due to a system of particles, that contribute to the broadening of
spectral lines.
2.1
Energy Levels of Atomic Hydrogen
Hydrogen consists of one electron and one proton. A full quantum mechanical energy
level calculation, where an attractive Coulombic force is balanced by a repulsive
centrifugal force, yields E()
=
-a2
for2n2hc
{nlncN, n > 0}.
a =
e2
-137 1
is the
fine structure constant and represents the strength of the Coulombic potential. This
section will look at first-order energy level corrections to the hydrogen atom and
scale their importance in typical ALCATOR C-Mod divertor or MARFE conditions.
We consider the fine structure of a hydrogen atom as well as the effect of strong
external electric or magnetic fields; we will not directly address the Lamb shift or
hyperfine structure but merely remind the reader that they scale like a 5 and mea ,
respectively. Conditions are established which allow one to neglect the fine structure
13
of the hydrogen atom with respect to effects caused by external electromagnetic fields.
A final subsection is included to discuss the hydrogen atom in electric and magnetic
fields.
Fine Structure
In introductory quantum mechanics textbooks it is common to find a section on the
fine structure of a hydrogen atom divided into two parts, one called the relativistic
correction and the other spin-orbit coupling. This branching of the calculation is
misleading since both corrections are due the electron's velocity. To understand the
general method of accounting for the motion of an electron in a hydrogen atom we
briefly walk through the stages of such a calculation.
Let's assume the electron's velocity is significant. Special relativity tells us that
a charged particle in motion produces a magnetic field. To estimate the magnitude
of this magnetic field we could transform to the rest frame of the electron (S') and
calculate the magnetic field at the proton's location. From this result, when in the
rest frame of the proton (S) we now have an idea of the magnitude of the magnetic
field at the location of the electron. The assumption of relativistic velocity leads to a
relativistic correction; the magnetic field in our S reference frame interacts with the
magnetic moment of the electron to yield a what is called the spin-orbit correction. An
almost identical procedure would be followed if one wanted to consider the acceleration
of an electron in a hydrogen atom (Lamb shift). Here one would need to use general
relativity to transform to the accelerating reference frame, as well as quantize the
Coulombic field, to estimate the effects on the proton.
The calculations presented in this subsection have been adapted from Griffiths [32].
For a more complete discussion of the fine structure of the hydrogen atom the reader
is encouraged to peruse Bjorken and Drell [33].
14
Relativistic Correction
The first stage of the fine structure calculation is to let the electron have a relativistic
velocity, hence the classical kinetic energy term becomes:
2
T=
4
2
p2c2+m2c4-mc
-+
2m
2
_-
-
2m
P
8m c2
+
(2.1)
Using the last term on the right hand side (RHS) as our perturbation (JHrei
=
s3c2), the first-order energy level corrections are:
--
E1=
By looking at the ratio of E(
E)0
a2
c
4--
2n 4 11 + )
-
3]4
(2.2)
5 x 10-5 < 1 we see that first-order perturbation
is justified and the relativistic correction is small.
Spin-Orbit Coupling
The moving electron creates a magnetic field, which in the rest frame of the electron
(S') appears as a magnetic field created by the moving proton. In the rest frame of
the proton (S) this magnetic field interacts with the intrinsic magnetic moment of
the electron to give a perturbation of the form:
JHso =
Bp=
-2 3 S
5- L.
m2c r
(2.3)
This leads to a first-order energy correction given by:
E(1 )=
mc
2
j(j + 1) + 1)
+ gl+ 1)
na1~
{
}a 4 'a
(2.4)
which also yields a correction factor of order a 2 .
Stark and Zeeman Effect
How does atomic hydrogen behave in an external electric field (Stark effect) or external magnetic field (Zeeman effect)? When the influence of a single external field
15
is considered we immediately notice that hydrogen possesses a symmetry about the
axis aligned with the field (taken to be the z-axis). This axial symmetry removes the
degeneracy of the total angular momentum (J), leaving only the conservation of Jz
(projection of J along the axis of symmetry). In a later subsection both electric and
magnetic fields are considered and this axis of symmetry is lost along with conservation of J,. In this subsection the Stark effect results are adapted from Landau and
Lifshitz [34] and the Zeeman effect analysis is adapted from Gasiorowicz [35].
Electric Field
Placing our hydrogen atom in an external electric field, which is assumed small enough
that ionization is not a factor, gives a perturbation of the form:
6 HStark =
(2.5)
eF -?= eFz,
where F is the electric field. Landau introduces parabolic quantum numbers (ni and
n2) to simplify the analysis and finds the following first-order energy corrections:
E(1)
2n
3-n ri1 --2
22)
2n
|e|Fh2
2
me2 = -(n,
3
- n 2)IeIFao,
(2.6)
where ao = 5.2918 x 10-11[m] is the Bohr radius. Unlike the fine structure results,
the impact of this term must be determined by considering the system surrounding
the hydrogen atom.
Magnetic Field
Placing our hydrogen atom in an external magnetic field, which is assumed to be
much greater than the magnetic field due to the electron's motion within the atom,
gives a potential of the form:
6H-=
2mc
B(L + 2S,).
(2.7)
Since n, 1, m, and m, are still "good" quantum numbers the energy correction is of
16
the form:
E(1)
em B(mi + 2m,)
n~lMJM$ = 2mc
=
IBB(m + 2m,),
(2.8)
where PB = 5.788 x 10-5 [eV/T] is the Bohr magneton.
Fusion Reactor Orderings
Having found the dominant first-order energy level corrections to a hydrogen atom
in fusion reactor type plasmas in the subsections above, a rather terse ordering of
corrections seems appropriate. To this end, an estimate of external fields that lead
to comparable energy level corrections is established and then an ordering is deduced
through comparison with approximate plasma parameters.
We begin by collecting first-order energy corrections from above:
ESE(2)4l10-4
- mc a
=
7.25 x i0-
Fao ~ 3.53 x 10-"F[eV]
3
~BB(m + 2ms) 2 1.16 x 10 4 B[T]
E(S
E
(2.9)
(2.10)
(2.11)
From these equations it is easy to see that for a magnetic field of 6.26[T] the Zeeman
effect is comparable to the fine structure. For the Stark effect to be comparable with
the fine structure, merely one electron at a distance of 30.0[nm] from the Hydrogen
atom needs to exist.
For typical fusion reactor plasmas, where ne
-
1014 [cm-3] and Te ~ 1 [eV], the
presence of an electron within 30[nm] is highly probable. At a first glance we can
see that the average interparticle spacing is of the order r ~ (ne)-3
20[nm]. Once
one considers the resultant electric field due to many particles, it is easy to argue
that E( > E(1.
Typical magnetic fields are of order B4 ~ 6[T], hindering us from
simply being able neglect the fine structure correction with respect to the Zeeman
effect. Tradition chooses the latter effect as the next level of hydrogen energy level
17
approximations. Thus, this argument justifies using the energy levels obtained by
Demkov [36], and summarized in the next subsection, in the study of atomic hydrogen
radiation in a fusion reactor.
Electric and Magnetic Field
When solving a time-independent first-order perturbation problem it is best to begin
by looking for a basis of orthogonal vectors which diagonalizes the complete Hamiltonian, thus trivializing the next step of finding energy levels. Although this point
was not explicitly stated in the previous subsections, it was used. In the Stark effect
analysis we used parabolic quantum numbers ni and n 2 (n = ni + n 2 + ml), which
lead to the result (nin 2 mlzlnin 2m) = In(ni - n2) h; in the Zeeman effect analysis the perturbed Hamiltonian was already diagonalized, which is why the "good"
quantum number comment was thrown in. With this solution technique in mind we
follow Demkov [36], switch to using atomic units and present rotation vectors which
diagonalize the Hamiltonian for a hydrogen atom in uniform external electric and
magnetic fields orientated at an arbitrary angle to one another. Fine structure effects
are not considered.
The appropriate perturbation is given by:
JH=-
3
1
-.
-
-
-
(2.12)
nF-A+-B-L,
2c
2
where A is the Runge-Lenz vector:
A=
1
1
{
-2HO2
x
For orthogonal vectors, Demkov defined i,
I2
= I22=
L- L x
=
}(L
(2.13)
) - -}.
r
+ A) and
12 =
i(L - A), where
j(j + 1) and j = 2-1. The perturbation operator becomes:
H =
(2.14)
1+
where the angular velocities are given by ' =
18
Bc -
2n
and
-L=
+
!nF.
First-order energy corrections may now be written as:
E(1)
= win' + w2 n",
(2.15)
where n' represents the projection of 1 on WJ, n" represents the projection of '2 on
c2 and {(n', n")1
2.2
--
...
. "
Formation of Line Profiles
Hydrogen atom energy levels in typical fusion reactor conditions have been established
in prelude to this section. An observer sampling a plasma spectrum will not directly
see the energy levels of hydrogen, but instead will measure transitions between levels.
Also, transitions will not only occur at the precise energy difference between levels
(AE = E, - Ef = hwo) but will have a distribution, or spread, about the central
frequency (wo). Thus, in a spectrum there exist strong peaks (lines) which represent
transitions between bound energy levels of a hydrogen atom, and these lines possess
a definite non-delta function profile about wo.
Transitions between energy levels of a hydrogen atom may be induced by collisions
with the surrounding plasma or through photon-emitter coupling. In this qualitative
discussion of line broadening we neglect collisions completely and only consider transitions associated with the radiation field. Thus, in this section we take a look at the
coupling between a field of photons and an atom from the viewpoint of a laboratory
observer measuring emission from matter.
There is an enormous amount of literature on line broadening. [37, 38, 6] Taking
our simple viewpoint we seek to discuss several factors known to contribute to an
observer measuring a spread about a predicted emission frequency. These factors
will be investigated assuming one mechanism of frequency spreading is not related to
another -
the factors are uncorrelated. For example, we investigate the motion of
an electron within an atom and then the motion of an atom relative to an observer
with an inherent assumption that no relation exists between the two motions. This
correlation assumption is valid provided contributions to the time-varying charge and
19
current distribution are linear. Ordering contributions to time-varying charge and
current distributions is also an effective way of determining dominant modes of line
broadening, a topic which will not be discussed here.
In the subsections below a qualitative discussion of three line broadening mechanisms is given: natural broadening will study an accelerating electron within an atom;
Doppler broadening will study the motion of an atom relative to an observer; and
impact broadening will study an atom in a plasma. Natural broadening is presented
by taking a classical electromagnetic approach to a radiating atom; Doppler broadening is explained as consequence of special relativity and the velocity distribution
of emitters; and impact broadening is discussed in the extreme quantum mechanical
limit leaving it to the reader to extend the argument to more complicated systems.
Natural Broadening
Schwinger [39] presents an enlightening classical discussion of radiation from atomic
hydrogen due to the accelerating electron. The general quantum mechanical result is
discussed in the section on impact broadening and these two subsections are designed
to complement each other; here we use arguments of physics, later we will look at a
general technique and both results take the form of a Lorentzian emission profile.
Let's model the accelerating electron as a damped harmonic oscillator:
i#
= 2r= -wor
- -yr,
-.
where wo is the natural frequency of the bound electron, y
(2.16)
=
2
2
is a damping
parameter and - < 1. This equation has a solution of the from:
fl(t) ~.o cos wot exp [ 2'],
2
(2.17)
with r* representing the electron's displacement from the origin at some initial time
t = 0. If the electron is nonrelativistic our energy spectrum is determined by:
20
E(w)
-2 e2 21= 2e2I(W)
2
2.
(2.18)
v(w) is just the Fourier transform of the derivative of Eq. 2.17:
(
~
2
(
1)
w - w, +i-/2
w + wo +iy/2
(2.19)
Collecting terms, remembering our definition of 7 and dropping the second term on
the RHS of the equation above since it will never compete with the first term, leads
to the following equation:
E(w)
m-r2
27r
7y/2
m
2
"
(w -
LUO)
2
+ (y/2)
2.
(2.20)
(
And so the energy spectrum has a Lorentzian from.
Doppler Broadening
Doppler broadening is studied in this section by considering the finite speed at which
a photon may travel, the velocity of emitter relative to an observer, and the distance
of separation between emitter and observer. If we assume the emitter is traveling
in an arbitrary direction to an observer, emitting photons at a constant frequency
v in the rest frame of the emitter, then an observer will measure the time between
emissions to be: T'Im
=
+ (i2
-
, where the position vectors represent the
location of emission in the observers reference frame.
In the observer's reference
frame, if the photons are emitted at an average angle 0 relative to the observer
and the distance between emitter and observer is large compared with the difference
between emission locations, then the measured time separation between signals is
approximately: T'IM
= IT(1
- 3 cos 0). This argument gives us an expression for the
frequency of emission in an observers reference frame: v'Im = v/7 (1 -#3
cos 9). [40, 41]
In the study of spectra from moving atoms one is primarily interested in determining atomic properties from observations. To this end we switch the perspective
in the above argument and study v from knowledge of v'Im:
21
v/ =
-Cos
1
-#/3cos
0) V'|m"' = 1 -
9
V'Im.
(2.21)
Take an emitter traveling directly toward an observer (9 = 0) and perform a Taylor
series expansion in powers of 3: v 2 (1 - 3 - /2/2 + . . .)v'|m. The term of order
/3
is known as the first-order Doppler effect and the term of order 32 is known as the
second-order Doppler effect. [41] It is interesting to note that when the observation is
made perpendicular to the direction of atomic motion (9 = ir/2) only the second-order
Doppler effect survives. [42]
Our knowledge of an emitter in motion relative to an observer may be extended
to the study of fusion plasmas by assuming only the first-order Doppler effect is
important and that the distribution of emitter velocities is Maxwellian. (It would be
more elegant to use an averaging over direction argument.) Further, it is customary to
assume that the Doppler effect is completely uncorrelated with any other phenomena
and form a convolution integral when multiple broadening mechanisms are to be
investigated. The convolution integral that results from considering both Doppler
broadening with Maxwellian velocity distribution for emitters and natural broadening
is known as the Voigt function.
Impact Broadening
Impact broadening describes the effect of surrounding particles and time-dependent
fields on an atom and its ability to emit and absorb photons.
There are many
names applied to approximations of particle interactions and complicated field behavior; rather that attempt to categorize these approximation techniques we will
walk through a simple argument from which more complicated phenomenon may
be understood. What follows is an improved treatment of Weisskopf and Wigner's
general treatment of transition rates as given by Gasiorowicz. [35]
To investigate photon emission from an excited atom we begin by setting up a
system with only one excited atom in an arbitrary ubiquitous scalar potential. From
the subsection on natural broadening it is known that the atom will eventually decay
22
so we further require that the excited atom will decay to the ground state. Using the
language of quantum mechanics we have an initial state given by HI1, 0) = Eli, 0)
and a final state HO, 1) = e(i)j0, 1); there exists a small potential (V) which has
the effect of changing the photon number by one and inducing a transition between
states.
For this problem our Schrbdinger equation has the from:
(2.22)
id ihdt't = (H + V)47,t).
The general solution of this equation takes the from:
-iEt
|b, t) = a(t)l1, 0) exp [
]+
dkb(t)10, 1) exp [
-ie()t
],
(2.23)
with coefficients given by:
-7t I
a(t)=exp[ 2
-i(E + A)t
h
M*(k)
b(t) =*(k
2
+
b(k)- E + f d' IM(k')1
E-E(k')
ihy(
(2.24)
(2.25)
2
where y is the decay rate, A is the energy shift of the excited state due to the
electromagnetic field and M(k) = (0, 1|V 1, 0). Taking the square modules of our
coefficients yields their probabilities as a function of time:
la(t)12 = exp [--yt]
lb(t)12
=
-
E
-
A) 2 + (2Y)2
(2.26)
(2.27)
As expected, the probability of our system remaining in the initial excited state
decreases in time and the emission profile is Lorentzian.
From here it is rather simple (one must merely invoke the dipole approximation)
to arrive at the results of the natural broadening subsection. It is more interesting to
23
discuss the significance of this result and how it applies to plasmas. Atomic transitions
in this simple problem are clearly due to some electromagnetic field, which could be
due to the presence of other particles. When particles are introduced into the space
surrounding our atom, and more energy levels are considered, our electromagnetic
field merely becomes more complicated. This is an extreme oversimplification but
is useful for understanding that all the physics has been placed into defining quantum mechanical states, using appropriate operators and looking at transition matrix
elements. In real problems the collisional time-scales are short compared to system
phenomena and time-dependence is removed. Time-independent models effectively
separate collisionally and radiatively induced transitions. It should be noted that
collisions may be looked at through use of time evolution operators.
24
Chapter 3
Numerical Spectral Line
Broadening
Results from two codes are presented in this chapter. After brief descriptions of the
two codes, a numerical sensitivity study is performed to qualitatively understand how
plasma parameters influence line profile shapes. Much of the terminology used in this
chapter is typical of that found in the literature. For the reader unfamiliar with
such nomenclature, Griem [37, 38] has written the classical text on the subject and
Mihalas [6] would provide a useful first reference.
3.1
Line Profile Codes
The two codes used in this sensitivity study are: TOTAL, which was developed by
R.W. Lee (LLNL) and L. Woltz, C. Hooper, W. Wiese (University of Florida, Physics
Department) and B. Talin, R. Stamm, L. Klein (Universit6 de Provence a Marseille);
BELINE has recently been developed through a joint MIT, Institute for High Temperatures (Moscow) and Keldysh Institute of Applied Mathematics (Moscow) effort.
Both codes are very similar, assuming a LTE plasma system they compute normalized numerical line broadening profiles, and differ primarily with respect to a magnetic
field dependence -
BELINE considers magnetic line broadening.
25
3.1.1
TOTAL
To numerically calculate a line shape TOTAL considers natural broadening, Doppler
broadening and the Stark effect due to plasma-emitter coupling. As the previous
chapter eluded, different collisional approximations are made for the Stark effect due
to electrons and that due to ions. Relative to electron motion, the ions are assumed
to be stationary and only their resultant field is considered by what is known as
the static-ion approximation. For the electrons, it is assumed that binary collisions
dominate and the impact-electron approximation is invoked. [43]
In its current incarnation TOTAL considers an extended line space. This effectively allows previously forbidden transitions, in the Wigner-Eckard time-independent
sense, to contribute to line shapes. For further discussion on effects of an extended
line space see a recent paper by Lee and Oks [1].
3.1.2
BELINE
To numerically calculate a line shape BELINE considers Doppler broadening, Zeeman
effect due to an external magnetic field (B) and the Stark effect due to plasma-emitter
coupling. When introducing an external magnetic field in addition to the static-ion
approximation electric field (F), the energy levels of a hydrogen atom behave as (see
Chapter 2.1.4):
1
22 + Ein'+ E2n" + aHms.
Ev - Ennnm, =
{(n', n")j - In- ,*.., n-},
E1,2 = 12B
T jFI and m,
BELINE computes the following normalized n
<Dnm(w)
= -
41r
dQ
j0
-+
= +j.
(3.1)
Using these energy levels
m line profile:
dFP(F) 1 G,,py,(w)
/
(3.2)
by performing a numerical average over F with the corresponding distribution P(F)
and over the microfield directions (angle between B and F). Each allowed transition
is broadened by considering a Maxwellian velocity-distribution of emitters (Doppler)
26
and binary electron collision; thus p,(w) represents the uncorrelated convolution of
the two broadening profiles.
Gv =e
" Ev- Ep,
is the relative intensity of the v
-p
_#(V1
2
Ie(vb-'I)12
)1
(3.3)
components for non-polarized light and e' is the
unit polarization vector.
3.2
Line Profile Computations
Numerical line profiles from TOTAL and BELINE are presented for principle quantum number (PQN) transitions 2 -- + 1 (Lyman-a), 3 --
1 (Lyman-,3), 4 -+ 1
(Lyman-y), 3 -+ 2 (Balmer-a) and 4 -+ 2 (Balmer-3); as previously mentioned,
TOTAL calculations include an extended line space. The sensitivity of these lines
with respect to electron density, electron temperature and external magnetic field
strength will be investigated. Various hydrogen isotope lines are presented, as well
as profile variations in angle between the direction of observation and magnetic field
(13). Comparisons are also made with existing literature.
In radiation trapping studies one is primarily concerned with the area closest
to line center, traditionally called the core and defined by a FWHM (or HWHM
depending on authors). In Fig. 3-1 hydrogen Lyman--y line profiles are presented
with variation in 3. This figure indicates that for Lyy, variation in 3 doubles the
core region as the direction of observation changes from perpendicular to parallel to
the magnetic field to parallel. Figs. 3-2 to 3-4 present four more lines of hydrogen,
under identical plasma conditions, which do not exhibit such behavior. Balmer lines
are clearly sensitive to the magnetic field, demonstrating significant magnetic line
broadening, and appear very different from the Lyman lines. In the paragraph that
follows, we look at Ha and vary density, temperature and magnetic fields. As the first
few figures show, comments made for Ha do not necessarily apply to other hydrogen
isotope lines.
Variation in the normalized H, line profile with electron density is considered in
27
600
____=0
...............
3 irI4
----
400
ir/2
'A
0
Z
200
0
-0.0025
0
0.0025
0.005
Energy [eV]
Figure 3-1: BELINE normalized hydrogen Lyman-y line profile dependence on
Plasma
angle between the direction of observation and magnetic field (3).
Parameters: B = 6[T], ne = 100 x 1013[cm-3], Te = 1[eV], / = x
900
600
-
p.o
......... _... .... .
--
750
s..
i 4
A
500
0
Z
300
250
0
a)
A
-0.001
0
0.001
A
b
0.002
b)
Energy [eVj
-
-0.0015
i
0
0.0015
0.003
Energy [9V]
Figure 3-2: a) BELINE normalized hydrogen Lyman-a line profile dependence
on 3. b) BELINE normalized hydrogen Lyman-0 line profile dependence on 3.
Plasma Parameters: B = 6[T], ne = 100 x 10 13 [cm- 3], T, = 1[eV], / = x
28
1600-
P=O
61200
800-
eV
0
'
''
00
-0.0005
0
Energy [eV]
0.0005
0.001
Figure 3-3: BELINE normalized hydrogen Balmer-a (HQ) line profile dependence
= X
6[T], ne= 100 x 10 13 [cm- 3], T = 1[eV],
on f. Plasma Parameters: B
750-
-
P=O
................
....... ...... =0.73
-------
=2
500-
2500
'
'
-0.0015
0
Energy [eV]
0.0015
0.003
Figure 3-4: BELINE normalized hydrogen Balmer-O (H3) line profile dependence
on 0. Plasma Parameters: B = 6[T], ne = 100 x 10 13 [cm-3], T = 1[eV], 3 = x
29
4200
....... ...................
------
TOT n,=10
TOT n ,=50
BEL n,=1 0
BEL n0=50
BEL n,=1 000
.2800
\
/
1400
0
-
0
\
II
~
-
-
.
-0.0005
\
0
Energy [eV]
0.0005
0.001
Figure 3-5: BELINE and TOTAL normalized H, line profile dependence on
electron density. Plasma Parameters: B = O[T], n, = x x 10 13 [cm- 3 ], T, = 2[eV],
0=0
Figs. 3-5 to 3-7. Fig. 3-5 shows very little change in the core region as density is
increased; BELINE and TOTAL compare well the FWHM. When a magnetic field
is turned on, the Ha line is split and density seems to effect the central component
more that the o components. There is very little dependence on 3.
Fig. 3-8 looks at the normalized H, line profile dependence on electron temperature using TOTAL and BELINE. Both codes yield similar profiles. As temperature increases the core region widens. This is expected since temperature effects are
included through the plasma Maxwellian velocity-distribution function (Gaussian).
An increase in temperature is proportional to an increase in thermal velocity and
broadening of the Gaussian FWHM. Fig. 3-9 looks at the normalized Ha line profile
dependence on magnetic field. This figure demonstrates the importance of includ-
30
2100
..................
1400
n. =10
n. = 100
= 1000
-n
700
0
z
-0.0005
0
0.0005
0.001
Energy [eV]
Figure 3-6: BELINE normalized Ha line profile dependence on electron density
as observed parallel to the magnetic field. Plasma Parameters: B = 6[T], ne =
x x 10 13 [cm- 3 ], T, = 1[eV], 3 = 0
2550
........-------------
-"=10
n, = 100
n, =1000
5.~1700
VI
0
21
'0 850
/- *-
0
Z
----
0
-. \
-0.0005
0
.
0.0005
0.001
Energy [eV]
Figure 3-7: BELINE normalized Ha line profile dependence on electron density
as observed perpendicular to the magnetic field. Plasma Parameters: B = 6[T],
n, = x x 1013 [cm- 3], Te = 1[eV], )3 = '
31
4200
..................
-------
TOT
TOT
BEL
BEL
T,=1
T,=5
T.=1
T =5
.. 2800
I\
aD
/1\
1400
0
0
-
\/
-0.00025
0
Energy [eV]
0.00025
0.0005
Figure 3-8: BELINE and TOTAL normalized H, line profile dependence on
electron temperature. Plasma Parameters: B = o[T], ne = 50 x 10 13 [cm- 3 ], Te =
x[eV], 3 = 0
ing magnetic line broadening effects on line profiles as the core area is increased by
approximately 6.5 with change in magnetic field from B = O[T] to B = 6[T].
Fig. 3-10 and 3-11 show BELINE normalized Ha and D, line profiles as observed
parallel and perpendicular to the magnetic field, respectively. The difference in iso-
tope mass clearly changes the central frequency of emission and the Doppler width
of the individual Zeeman components. In Fig. 3-11 it is easy to see how interpreting
spectral data from ALCATOR C-Mod is very sensitive to theoretical modeling; notice
how the central hydrogen component nearly overlays the deuterium a- component.
A final figure is included for comparison with recent literature.
According to
Lee and Oks [1], considering higher PQN's leads to an increase in the probability of
forbidden transitions. For the Lyman-#3 line shown this translates into an filling of
32
4500 r
............................
B=0
B= 3;= 0
B = 3; = /2
B=6; p=0
B= 6; p
n/2
U3000
A
Z
S1500
E
(0
0
,_! I
, - J-.,.Mo
-0.0005
II
0
III
I
I
I
II
0.0005
-w
0.001
Energy [eV]
Figure 3-9: BELINE normalized deuterium-a (Do) line profile dependence on
13
magnetic field and 3. Plasma Parameters: B = x[T], n, = 100 X 10 [cm-3]
Te = 1[eV], 3 = x
33
2100
Ha;f0=0
Dc; p = 0
................--
f
f
L1400
V
f1
l 700
z
OL,
1.8137
.
1.888
1.89
1.889
Energy [eV]
I
1.891
Figure 3-10: BELINE normalized H, and D, line profiles as observed parallel to
the magnetic field. Specific dependence on energy is shown. Plasma Parameters: B = 6[T], n, = 100 x 101 3 [cm- 3 ], Te = 1[eV], )3 = 0
2100-
Ha; J=Wr2
Da; =r2
......--.......
1400
A
7000
z
. . ......
,-
0
1.887
1.888
-....
1.889
Energy [eV]
1.89
.
.
1.891
Figure 3-11: BELINE normalized HQ and Da line profiles as observed perpendicular to the magnetic field. Specific dependence on energy is shown. Plasma
Parameters: B = 6[T], ne = 100 x 1013 [cm- 3 ], Te = 1[eV], 3 = I
34
135
................
90
BELINE
TOTAL
-V
@1
45
0
-0.01
0
0.01
0.02
Energy [eV]
Figure 3-12: BELINE and TOTAL hydrogen Lyman-3 line profile scaled to
compare with Lee and Oks Fig. 1. [11 Plasma Parameters: B = 0[T], ne =
3000 x 1013[cm- 3], T = 3[eV], 3 = 0
the central dip. Since BELINE does not consider such effects this phenomenon is
clearly visible. Study of extended line spaces is important to future modeling efforts
since one needs to decide how magnetic line broadening should be treated in radiation
transport simulations. Following the implications of this research, support of a unified
line treatment instead of identifying three separate lines for radiation transport would
be justified and could lead to a significant reduction in computational cost.
35
Chapter 4
1-D ALCATOR C-Mod MARFE
Simulations
This chapter will present significant results in simulating high-density low-temperature
plasma regions of ALCATOR C-Mod. Results have been obtained through use of
a multi-dimensional non-local thermodynamic equilibrium (NLTE) simulation code
named CRETIN. [16, 15, 17] MARFE simulations have been able to reproduce experimental data, estimate recombination rates and demonstrate deuterium Ly" and
D, trapping.
4.1
Experimental Data
In ALCATOR C-Mod large stable midplane deuterium MARFEs may be developed by
puffing D 2 through a midplane capillary. Diagnostic equipment, including Chromex
and McPherson spectrometers, allow co-linear spectroscopic data collection in the
visible (Balmer series) and VUV (Lyman series) frequency ranges. Fig. 4-1 shows the
system geometry and includes shaded regions depicting position and viewing area of
a CCD camera; Fig. 4-2 depicts the location of MARFE formation, near the inner
wall, and chords directions and extent for which spectral data have been obtained.
Spectral data are displayed in Fig. 4-3.
36
a)
inner wall
inner div. nose (51.1 cm)
x-point (56.5 cm)
Nose of outer divertor (61.2 an)
outer edge of divertor tiles (75.8
as)
Lower edge of outer limiter (79.6 cm)
b)
"---4.00
em
cm
61.15 cm
118.86 cm
51.00
10.6c
51
m-
Figure 4-1: ALCATOR C-Mod CCD Camera position and viewing area.
a)
represents the view that an observer positioned above the tokamak looking
down would have - the bird's eye view; b) represents a side view.
37
'
0.4
...
'"".."""'" ' r""""'
'""
..
."'I"
+
0.675
-0.011
0.2
MARFE
Location
0.0
L
LOES
-0.2
0.870
03
d
-0.4
0.40
0.50
0.60
0.70
0.80
0.90
R (m)
Figure 4-2: Chord direction and extent for which MARFE spectral data have
been obtained. LCFS stands for Last Closed Flux Surface.
100.0
(a)
-
10.0
S1.0
g
0.1
440
420
400
Wavelength (nm)
380
(b)
10
23
A
-4
-
a 1022
1021
85
90
95
Wavelength (nm)
100
Figure 4-3: Measured Lyman (lower) and Balmer (upper) series spectral data
from an ALCATOR C-Mod MARFE.
38
MARFE profiles
3
102
101
10
0
10 -
-- Ne2O -...
Ne2Q
Ne2O -
--
N
I
to-
10-3
-6
.
. . , . .
-4
-2
0
DLCFS (cm)
2
Figure 4-4: CRETIN input plasma profiles for MARFE simulations. Ne20: electron density in units of 10 2 0 [m- 3 ]; Ng20: neutral deuterium density in units of
1020[M- 3]; n = 2: first excited state of deuterium density in units of 10 2 0 [m- 3]; Te:
electron temperature in units of [eV]; and Ti: ion temperature in units of [eV].
DLCFS stands for Distance from the Last Closed Flux Surface and positive
numbers indicate coordinates within the scrape-off layer (SOL); see Fig. 4-2 for
an understanding of the geometry.
4.2
Numerical Simulations
Numerical simulations have been performed using CRETIN, a multi-dimensional nonlocal thermodynamic equilibrium (NLTE) simulation code based on an "isolated
atom" treatment of atomic kinetics.
One-dimensional results have been obtained
through specification of the plasma parameter profiles found in Fig. 4-4.
Using a
ten-level deuterium atomic data set, CRETIN self-consistently solves a set of atomic
rate equations along with the radiation field through a complete linearization procedure. [44, 6] Using TOTAL-like line profiles the spectrum of Fig. 4-5 is obtained.
A few comments in behalf of the results are in order. When comparison is made
between the measured and simulated Balmer series it is seen that although the location and slope of the lines appear to be very similar, their simulated widths are
too narrow and magnitudes are at least an order too high -
39
even after allowing for
Balmer series spectrum
103
101
t o~1
10
10-1
36 0
10
10L
460
0
r
to 8
S107
440
Lyman series spectrum
1
4)
420
400
Wavelength (nm)
380
1
. . . . . . . . .1
I~
85
I
90
95
100
Wavelength (nm)
Figure 4-5: CRETIN Lyman (lower) and Balmer (upper) series spectral simu-
lation data for an ALCATOR C-Mod MARFE. The dashed line includes the
effect of spectrometer broadening as required for comparison with Fig. 4-3.
40
6
4
Line Brightnesses:
M
E Lyman data
0
U
E
2
1B
0
E
CRETIN
with Te=T
Balmer data
CRETIN
with Te=Ti
E 1008
8
6
.~
4
0
0)
--
*C
-
I
I
2
-*
10
8
-I
6
4
2
3
4
5
6
7
8
Upper level principal quantum number
Figure 4-6: Brightness of Lyman-series and Da lines from the 1-D MARFE
model, comparing experimental measurement to two CRETIN runs, one with
standard Te and Ti profiles (noted Te = T) and one with enhanced ion temperature to reproduce the effect of Zeeman splitting.
instrumental broadening. Also, the simulated continuum is too high and does not exhibit the smooth discrete-continuum transition seen in measured data; an atomic data
set including approximately twenty levels would be necessary to see this phenomenon
with CRETIN. [27] Similar comments apply for the Lyman series simulation. The
results presented were obtained by optimizing the ion temperature profile so that
the experimental brightnesses of both Lyman and Balmer series were matched (see
Fig. 4-6). T was treated as a free parameter to mimic the Zeeman effect splitting
through enhanced Doppler broadening. In a physical MARFE, one expects T = Te,
but when CRETIN was given such equilibrated conditions, no match could be found
between the experimental and simulated brightnesses. The best results are shown as
squares for this T = T case. When allowing for a different T radial dependence,
thereby widening the Lyman-series lines to an amount comparable to their Zeeman
splitting, the results labeled as "CRETIN" were obtained.
41
4.3
Recombination and Opacity
CRETIN MARFE simulations have been able to demonstrate deuterium Ly, and D,
trapping and quantify the importance of opacity on recombination rates. This latter
result clearly indicates the need to couple CRETIN with a code such as UEDGE.
Fig. 4-7 applies the number of effective recombinations per D^ photon concept,
promoted by Terry et al., to quantify the effect of opacity on plasma recombination. [25, 30] Since opacity is seen to significantly effect the recombination rate, this
implies that photon escape rates are important in determining the plasma energy
balance. Another way of seeing this relationship is by stating that the plasma energy
balance is directly related to ionization-recombination rates, material motion within
the system and photon creation-destruction rates. The radiation field is dependent on
scattering phenomenon which include photon creation-destruction rates and photon
escape rates -
the latter of which is does not directly effect the energy balance but
is what opacity effect may be attributed to.
While paying close attention to several factors that contribute to the energy balance, CRETIN does ignore material motion. As the energy balance strongly effects
the temperature structure in a system this would explain the strange temperature
profile necessary to achieve good agreement with experiment in the previous section.
Codes such as UEDGE, which do not address photon escape rates but consider other
factors of the energy balance also do not tell the whole story. It has been suggested
that merging of the two codes could be performed through simple iterations between
the two codes, with CRETIN calculating escape probabilities and sending them to
UEDGE. Such an idea may be feasible if the arguments laid out here are really as
simple as they seem.
In Fig. 4-8 deuterium Ly 0 is shown to be very trapped and in Fig. 4-9 Da appears
marginally trapped. Fig. 4-8 is a plot of the intensity as might be seen by a tangential
CCD camera, see the shaded region of Fig. 4-1a. Opacity and emissivity values of the
Lya line have been taken from CRETIN at line center as a function of major radius
and postprocessed to quantify the anticipated spectral signal magnitude. What is
42
S250
For n_1021 m-3
Spatial diffusion oni
200 --
Using radiation transfe
code CRETIN_
150
Optically
- 100
6
100--
:
~
50
-U
10
S
0
0.0
~
109m
c
-.
. . . .
0.5
1.0
1.5
2.0
2.5
Te (eV)
Figure 4-7: Number of effective recombinations per Dy photon in an optically
thin limit (solid line and diamonds), and for two cases in which Lya3 are optically thick. The thick, dashed curve and the closed squares show the results for
N 0 AL=1x 109 m- 2 using escape factor model and CRETIN, respectively. The
thin, dot-dashed curve shows the case for N 0 AL=2x10 1 8 m-2.
important is that this plot displays no change in intensity as the spatial observation
length doubles when the viewing chord no longer intercepts the inner wall. Fig. 4-9
presents a similar plot for D, along with actual tangential CCD camera data. In this
case there is a slight change in intensity about R = 44[cm]. Since this intensity does
not double, it would appear that the line is partially trapped.
43
240
a160
g 80
Inner
-
Wall
. . . .. .. . .. .. .. .. . . .. . . .
u
42
44
46
Major Radius [cm]
48
50
Figure 4-8: Radial opacity and emissivity CRETIN data has been postprocessed
to allow for comparison with Lyman-a spectral data collected from a tangential
viewing CCD camera. Absorption and emission coefficients have been obtained
at line center.
240
............
-
6160
80
CRET..
CC
Cam ra
CRETIN
'
..........
Inner
Wall
-
42
44
46
Major Radius [cm]
48
50
Figure 4-9: Radial opacity and emissivity CRETIN data has been postprocessed
to allow for comparison with Da spectral data collected from a tangential viewing CCD camera. Absorption and emission coefficients have been obtained at
line center.
44
Chapter 5
Summary
Through a theoretical discussion (Chapter 2) and numerical sensitivity study (Chapter 3) of hydrogen line profiles, in conditions representative of high-density lowtemperature fusion tokamak reactors, information useful for improving experimental
estimates of plasma parameters from spectral data has been obtained. Through application of a numerical simulation code named CRETIN to the study of hydrogen
radiation transport in ALCATOR C-Mod (Chapter 4), further insight into the effect
of opacity on the plasma power balance has been obtained.
Characteristics of a spectrum, such as the ratio of a specific line intensity to the
continuum intensity (Te) or the FWHM of a specific line (ne), are relied upon to yield
information about plasma parameters. Recent research has indicated that measurement of the deuterium Balmer-3 transition might be useful in determining the electron
density. [45] Through numerical broadening calculations including magnetic field effects, they found the HWHM (ai/2 ) of Do behaves in a fashion directly proportional to
the linear Stark effect at high densities (ne > 2 x 10"[m 3 ]): a 1 / 2 =
Z2
4n
In a similar manner this thesis has probed the sensitivity of line profiles to variations in plasma parameters, collecting information that will aid future theoretical
and experimental.
Having verified the reliability of BELINE, by comparison with TOTAL and available experimental data (which has not been discussed in this thesis), methods are also
being looked at to incorporate the results in CRETIN. BELINE is currently too com45
putationally expensive to be directly included in radiation transport codes and thus
line dependent fits are under development. Generation of tables has been considered,
however since the line profiles are sensitive to so many factors (density, temperature,
magnetic field, polarization, angle of observation) this is almost as computationally
expensive as BELINE itself. By generating fits, we will enable one to more easily
quantify how variations in a line profile directly effect the plasma energy balance,
which is of experimental interest, and improve CRETIN.
Although effects of finite photon mean-free-paths in stellar atmosphere have been
known for some time, and the relation of such effect to system dynamics have been
equally investigated, implementing such knowledge in the study of fusion reactors is
relatively new. This thesis has made progress toward the development a self-consistent
neutral-plasma-radiation solution in high-density low-temperature tokamak regions.
More specifically, it has been shown that variations in a line profile effect photon
escape rates which are closely related to opacity which has a strong impact on plasma
recombination and hence influences the plasma power balance. Hence, opacity effects
the plasma energy balance and photon escape rates must be included in codes such
as UEDGE.
In plasma regions where recombination is occurring and there is some trapping of
radiation, it is now understood that both radiative and non-radiative modes of energy
transport are equally important in determining the plasma balance.
Simulations
focused on non-radiative energy transport modes, with simple radiation formulas
which neglect NLTE effects, will not accurately predict plasma parameters or describe
the power balance in detached divertor or MARFE tokamak regions. The next step is
to couple existing radiative (CRETIN) and non-radiative (UEDGE) energy transport
mode codes with the hope that iterations between the to will converge to the correct
self-consistent solution.
46
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