Tokamak Plasma Spectroscopy at LLNL
Table of contents
Introduction
Calculation of Ionization and Recombination
Rate Coefficients
Calculation of the Charge State Distribution
Measured Impurity Ion Emission Profiles
Calculations of Radiative Cooling Coefficients
Bolometric Measurement of Impurity Radiative
Cooling
Resonance Enhancement of DWA Impact Excitation
Rate Coefficients
L-shell x-rays in Zr, Nb, Mo, and Pd
M-shell x-rays in highly charged W ions
XUV observations of N-shell W ions
Direct Measurement of Impurity Fluxes
Transport Profiles during auxiliary heating
References and bibliography
Introduction
Non-LTE atomic modeling is crucial for understanding scaled laboratory
experiments for Stockpile Stewardship. Above ground experiments (AGEX)
require detailed atomic physics data. One way that these atomic
data are benchmarked is in their application to tokamak plasma spectroscopy.
The tokamak plasma is a long lived, well characterized and well diagnosed
source containing high-Z ions. Tokamaks field a large compliment
of diagnostics; independent (non-spectroscopic) measurements of the plasma
electron temperature and density can be made. Bolometric and high
resolution spectroscopy measurements directly reflect ionization balance
and excitation processes. The x-ray and XUV observations of
emission from highly charged high-Z ions made in tokamaks compliment experiments
with laser-produced high-Z plasmas, holraum and gasbag experiments on the
Nova laser and the development of spectroscopic diagnostics in ICF fuel
capsules.
This cartoon shows many of the non-LTE physics issues of interest to AGEX
that are tested directly in our tokamak campaigns
Tokamak plasma spectroscopy research in V-Division
is led by Dr. Kevin B. Fournier. The principal tokamak plasma
spectroscopy activities carried out in V-Division
are studies of ionization and recombination rate
coefficients, and impurity radiative cooling.
These activities validate atomic data and collisional-radiative models
for emission from highly charged ions of high-Z elements. We also
have an extensive program of high resolution x-ray and XUV spectroscopy
of highly charged ions. This program is used to validate contributions
to detailed atomic structure calculations from such terms in the Hamiltonian
as vacuum polarization, Lamb shifts and Breit interactions. We also
support ongoing investigations of anomalous
impurity transport at other research facilities. These activities
apply our spectroscopic models to a phenomenological understanding of the
relationship between electrostatic turbulence in the background plasma
and impurity ion transport. Finally, other activities include XUV
and VUV studies of astrophysically abundant elements (Ne, Si, S, Ar, Ca,
Ti, Fe and Ni) and the benchmarking of basic atomic data needed for models
of highly charged high-Z ions. Measurements of radiative power loss
and spectroscopy test non-LTE models and contribute to the design of future
magnetic fusion energy (MFE) facilities.
Dr. Kevin B. Fournier with a XUV spectrometer built at the Johns Hopkins
University and mounted on the Frascati Tokamak Upgrade
Tokamak plasma spectroscopy is carried out in V-Division
in collaboration with the Plasma
Fusion Science Center at the Massachusetts Institute of Technology,
the Plasma Spectroscopy
Group at the Johns Hopkins University, the Frascati
Tokamak Upgrade at the Ente Per Le Nuove Tecnologie L'Energia E L'Ambiente
(ENEA) research center in Frascati, Italy, and the ASDEX
Upgrade Tokamak at the Max Planck Institut für Plasmaphysik in
Garching, Germany. Our recent publications
are listed here.
The inside of the Frascati Tokamak Upgrade vessel. Note, the
inner limiter is made of molybdenum tiles, the rest of the vessel is inconel
stainless steel
One of our collaborators (Dr. Mark May of JHU) in the Alcator C-Mod Tokamak.
All plasma facing components in the Alcator C-Mod Tokamak are made of molybdenum
Calculation of Ionization and Recombination Rate
Coefficients
One of the basic goals of our tokamak spectroscopy program is the benchmarking
of ionization and recombination rate coefficients for the many charge states
of mid- and high-Z impurity ions in the plasma. V-Division
has for years held leadership in the field of calculating the rates of
multistep processes such as dielectronic recombination (DR) and excitation
autoionization (EA). The techniques that we employ will be discussed
below. First, however, some of the basic details about our calculations
are presented.
The presence of impurities in a magnetically confined fusion plasma
is unavoidable; vacuum vessel facing components and limiters contribute
to the total concentration of high-Z (Z > 20) impurities. We have
observed significant differences between measured radial molybdenum ion
distributions in the Alcator C-Mod tokamak (Plasma Fusion Center, Massachusetts
Institute of Technology) [1] and in the Frascati
Tokamak Upgrade (ENEA research center, Frascati, Rome, Italy) [2]
and predictions made with a plasma modeling code that relies on an average
atom model [3] for atomic data such as ionization
and recombination rate coefficients. The reason for the observed
discrepancies is the poor quality of the recombination data used as well
as the neglect of the indirect ionization process, excitation autoionization
in the plasma modeling code. Using the HULLAC package [4]
of fully relativistic atomic structure codes we have computed the level
to level rates of ionization and recombination for nearly all the ions
of argon (Z=18) [5], krypton (Z=36) [6]
and molybdenum (Z=42) [7]. The cartoon below
shows some of the ionization and recombination processes that we investigate;
autoionization and its inverse, radiationless capture (leading to recombination)
are the channels coupling adjacent charge states that receive our principal
efforts.
The processes of dielectronic recombination and excitation autoionization
are the primary focus of our rate coefficients computations
In the high temperature conditions that exist in the bulk plasma of a tokamak,
DR is the dominant recombination mechanism. The rate coefficient
for DR is given by
,
where Vij(Te) represents the rate of radiationless
capture of a free electron by level i of ion (q+1)+ to a level j of ion
q+ in the continuum, the energy of the capture going into the promotion
of a bound electron, and BRj is the branching ratio
for radiative stabilization of level j (discussed below). The rate
of electron capture is given by
where gi represents the statistical weight of level i, AAji
is the autoionization rate from level j of ion q+ to level i of ion (q+1)+,
R is the Rydberg unit of energy, a0 is the Bohr radius, and
DEji is the energy difference between
the initial level of ion (q+1)+ and the doubly excited energy level of
ion q+ (a.k.a. the capture energy). The branching ratio for radiative
stabilization from a doubly excited level j of ion q+ has the form
where Grad and GA
represent sums over all levels reachable from j by radiative and autoionizing
transitions, respectively,
and
where ARjf is the radiative transition rate from
level j to a level f of ion q+, ARjj' is the radiative
transition rate from level j to some other level j' which also lies in
the continuum, and Gstabj
represents a sum of radiative transition rates from level j of ion q+ to
all levels f' below the continuum
.
The second term in the parenthesis of the branching ratio represents the
contribution to recombination from subsequent stabilization following radiative
transitions between energy levels in the continuum (cascades). This
contribution is generally found to be quite small.
For certain iso-electronic sequences, particularly those near to iso-electronic
sequences with filled shell ground configurations (He-, Ne-, Ni-like ions),
EA is the dominant ionization mechanism. The cross section for EA
from a level i of ion q+ to all possible levels of ion (q+1)+ is given
by
where sex(i->j) is the electron impact
excitation cross section from level i to level j (an energy level in the
continuum of ion q+) and BAj is the branching ratio
for autoionization from level j to all possible levels of the next ion.
In the present work, BAj has the form
where Grad and GA
and ARjj' are as above. The second term in
the parenthesis in the branching ratio represents the contribution to the
total EA rate from autoionization following radiative transitions between
energy levels in the continuum. This contribution is found to be
small to negligible for all ions studied here. The rate coefficient
for the EA process from a specific level i is
where Qijex is the impact excitation rate coefficient
from level i to level j, found by averaging the electron impact excitation
cross section over a Maxwellian distribution of free electron velocities
where v is the electron velocity and f(v) is the electron velocity distribution.
The figure below shows the contribution that excitation-autoionization
makes to the total ionization rate from the ground level of certain ions.
The plotted quantity is the ratio of the sum of direct electron impact
ionization (DI) plus EA over the rate of DI alone.
The enhancement of total ionization rates due to the presence of EA channels
in certain M-shell charge states of argon and molybdenum
Calculation of the Charge State Distribution
At a given temperature, the rates of EA and DR, along with the rates for
direct electron impact ionization (including contributions from ionization
from both the valence and inner subshells) and radiative radiative recombination
are used to compute the charge state distribution (CSD) for a given element.
In the low density limit that obtains in tokamak plasmas, the ratio of
the densities of two adjacent charge states is given by
where Sq(Te) and aq+1(Te)
are the total rates for ionization and recombination from the ground levels
of ions q+ and (q+1)+, respectively. The quantity fqCE(Te),
the relative fraction of a charge state in the plasma is given in coronal
equilibrium by
where nZq is the number density of ion q+, and nZ
is the number density of all ions of atom Z. The equation for the
relative densities of two charge states, along with the condition
provides a complete set of equations for the ions of impurity Z.
The CSDs for Ar, Kr and Mo are compared below to calculations that neglect
EA in the total rate of ionization. For Ar, one sees a modest perturbation
to the Na- to S-like charge states. The L- and K-shell ions
are unaffected.
In the case of Kr, the same near Ne-like charge states are strongly
perturbed by the presence of EA in the CSD calculation. The enhancement
of the total ionization rate for the near filled shell ions means they
are "burned through" much more quickly than they would be in a calculation
without EA. The filled shell Ne-like ion, due to the large step in
ionization energies going from the M- to the L-shell does not have strong
EA channels; the greater stability of the Ne-like ion relative to the adjacent
M-shell ions means its abundance is greatly enhanced.
The same phenomenon is observed near Ne-like Mo, and also near the filled
shell Ni-like Mo ion. That is, the large step in ionization energies
going from the N- to the M-shell means that the Cu-, Zn-, Ga- and Ge-like
ions are "burned through" much more quickly than is the filled shell Ni-like
ion.
Measured Impurity Ion Emission Profiles
Now that we have predicted the CSD, we'd like to
test the calculations. This is carried out by observing the equilibrium
distribution of ions in a tokamak plasma. For this, we have an array
of photometrically calibrated, time resolving, spatially scannable spectrometers
at the various tokamak facilities with which we collaborate.
Using these x-ray and X/VUV spectrometers, along with independent diagnostics
of temperature (Thomson scattering, electron cyclotron emission) and electron
density (near IR interferometry, microwave reflectometery) the brightness
of different transitions can be measured along several lines of sight,
each of which samples a different peak temperature. These data can
then be Abel inverted to yield the emissivity profile for the given transition
as a function of radius in the plasma (which is equivalent to temperature).
Shown below is a photo of one of our spectrometers mounted on the Frascati
Tokamak Upgrade. In every case, the spectrometers have been photometrically
calibrated, and thus an absolute signal can be measured. This allows
us to determine the absolute concentration of an impurity in the plasma.
Grazing Incidence, Time Resolving spectrometer, built at the Johns Hopkins
University and mounted on the Frascati Tokamak Upgrade. Show is the
pre-slit before the grating, housing for filters and the multi-channel
plate detector.
An example of our data as measured at the Alcator C-Mod tokamak at MIT
is shown below. In this figure, strong resonance lines from Na- and
Mg-like molybdenum, Mo31+ and Mo30+, along with a
weaker line from Al-like Mo29+ and lines from intrinsic impurities
(C, O and F) are seen. One notices immediately that the molybdenum
features dominate. As the line of sight changes from frame (a) to
frame (c), the absolute signal level drops because the peak plasma temperature
along the line of sight is dropping. Frame (b) shows a simulated
molybdenum spectrum for the central line of sight. The agreement
with the absolute signal is excellent.
M-shell Dn=0 (3-3) transitions in highly ionized
molybdenum as measured with a 2.2m grazing incidence spectrometer along
two lines of sight at the Alcator C-Mod tokamak
Shown below is another example of the kind of data we acquire.
For this spectrum, Kr gas was introduced through a pizo-electric valve
to the Frascati Tokamak Upgrade plasma [8].
This M-shell Kr spectrum has more lines in it than do the molybdenum spectra
above. This is because there are more emitting charge states (KrXVIII
to KrXXIII) most of which have a more complicated ground configuration
than the (simple) Na- and Mg-like charge states above. The strong
line at 93.35 Å is due to Ar-like Kr18+, which has a simple
ground configuration (3p6) but which has many more excitation
channel than do the Na- or Mg-like ions.
M-shell Kr emission spectrum recorded from a ~ 2keV plasma at the Frascati
Tokamak Upgrade.
To interpret the observed emission from a single ion, a collisional-radiative
(CR) model is constructed that provides information on the net excitation
rate for a given line. In the case of emission from a single charge
state, the CR level populations are found by solving the set of rate equations
where nj is the relative population of level j, Qji
is the collisional rate coefficient from level j to level i for excitation
(e) or de-excitation (d), Aji is the radiative transition probability
from level j to level i and M is the number of levels used in the model.
All electric and magnetic dipole and quadrupole radiative transitions (E1,
M1, E2 and M2) as computed by HULLAC [4] are taken
into account. The CR model includes relativistic calculations of
fine structure energy levels, radiative decay and (semi-relativistic) Auger
decay rates. For the densities that obtain in tokamak plasmas, forbidden
radiative channels (M1, E2 and M2) can give lines that are quite bright
in an ion's spectrum. Direct electron impact excitation rate coefficients
are computed (semi-relativistically) in the distorted
wave approximation. The contribution to a level's population
from ionization and recombination from adjacent charge states can also
be included. In that case the ionization and recombination channels
enter the set of rate equations as additional source and sink terms.
The cartoon below shows the CR processes coupling the He- and Li-like aluminum
atoms. The forward ionization processes (autoionization and direct
impact ionization) are computed using HULLAC; data for the inverse processes
(dielectronic attachment and three-body recombination) are computed by
enforcing the principle of detailed balance. The more general form
of the CR rate equations is shown in the insert in the figure.
Cartoon of Collisional-Radiative processes joining He- and Li-like Al ions
The CR model (neglecting ionization and recombination) is adequate for
interpreting the emission from a single charge state. However, to
interpret the spectrum from a hot plasma where many charge states are present
requires the weighting of the emission from each charge state by the relative
ion abundances. If the plasma were truly iso-thermal, this would
be a direct test of the computed CSD's. However, the emission that
is observed from the tokamak plasma is integrated along the line of sight
of the spectrometer and may sample a large range of temperature and density
conditions.
Since our spectrometers are spatially scannable, the brightness of different
transitions can be measured along several lines of sight, each of which
samples a different peak temperature. These data can then be Abel
inverted to yield the emissivity profile for the given transition as a
function of radius in the plasma (which is equivalent to temperature).
The measured data are shown below, along with two simulations for the line
of sight integrated brightness of each transition. The two simulations
have the same CR excitation model embedded in a plasma transport model
and the different charge state distributions that are contrasted above.
The first figure shows XUV emission profiles from six transitions in four
L-shell argon ions as measured at the Frascati Tokamak Upgrade (left) and
the x-ray emission profile from the Hea
line in Ar16+ (right). The brightness profile computed
using the HULLAC CSD is shown in green, the profile computed without EA
and approximate DR rate coefficients is shown in red. As already
stated above, the effect of EA on the argon CSD is small, and the L- and
K-shell ions are generally unperturbed. This is seen to be the case
in the simulations below; both simulations reproduce the XUV and x-ray
data well.
In the case of Kr, brightness profiles for XUV transitions in five
ions are shown below. These are all M-shell Dn=0
(3-3) transitions. The simulations for the Na- and Mg-like
ions show the greatest difference due to the presence of EA in the HULLAC
CSD calculation. The measured data support the simulation that includes
EA. For the other charge states, Cl-, Ar- and K-like, the simulations
show smaller differences, however, the data still support the HULLAC calculations.
The figure below shows five x-ray emission profiles for three Mo ions,
the Na-, Ne- and F-like charge states. (The XUV spectra of the Na-
and Mg-like ions have been shown above.) In the figure below, in
the panel on the left is the spectrum of three of the x-ray lines [9]
as measured along a central line of sight with a Von Hammos type crystal
spectrograph. The lines are from 2p<-4d transitions in the three
charge states. Below the data is the simulated spectrum with the
HULLAC CR model and the HULLAC CSD (middle) and the simulation with the
HULLAC CR model and the old CSD (bottom). The agreement for the HULLAC
CSD simulation is very good. The radiative brightness profiles for
all five 2p<-4d transitions are shown in the panel on the right.
The simulations agree well with the data for the central (higher temperature)
observations, but less well for the outer parts of the profile. This
may be due to less signal (due to colder plasma temperatures) and effects
due to anomalous transport that are not included in our model.
Calculations of Radiative Cooling Coefficients
Copious impurity radiation from the core of an tokamak plasma will be deleterious
to energy and particle confinement, and can quench the fusion reaction.
On the other hand, exhausting an adequate fraction of an ignited plasma's
power over the surface area of the vessel's entire first wall, rather than
onto a small area of the divertor strike plates is crucial to the successful,
sustained operation of the divertor in a fusion reactor. Impurity
ion radiative processes are the most likely mechanism for achieving this
operating condition; robust, quantitative predictions of impurity radiative
loss rates are therefore essential. Recent work has suggested that
the controlled introduction of a recycling impurity such as krypton (Z
= 36) may be the best solution for enhancing radiative power losses [10]
in the outer part (mantle) of a fusion plasma.
Power losses via collisionally fed line radiation, dielectronic recombination
line emission and radiative recombination and bremsstrahlung continuum
emission have been evaluated. The power loss coefficient for line
radiation from an ion (with charge q+) at a fixed temperature is given
by
where Jjf(Te) (in units of photon cm3
sec-1) is found from the collisional-radiative model and the
sum is taken over all possible radiative transitions. Because Jjf(Te)
comes from the CR model, emission due to "forbidden" transitions (non-electric
dipole transitions) and emission fed by excitation from levels other than
the ground level (metastable levels and other excited levels) are included
in the sum. The amount of power radiated per unit volume by collisional-radiative
line emission from an impurity atom (with atomic number Z) in a fusion
plasma is given by
where nq is the number density of an impurity ion of charge
q+ and the sum is taken over all charge states of the atom. The sum
in the equation above can be expressed as
where nZ is the total number density of atom Z and fqCE(Te)
is given by calculated charge state distribution.
The radiative cooling coefficient has the units of ergs s-1
cm-3 (impurity density)-1. For the range of
densities encountered in our tokamak experiments, the radiative cooling
coefficient is found to scale linearly with density. Thus, by dividing
the cooling coefficient by the electron density used in the CR model, the
resulting units are ergs s-1 cm-3 (impurity density)-1
(electron density)-1. The power loss coefficient for nearly
all molybdenum ions as computed at six different temperatures is shown
(for an electron density of 5.0x1013 cm-3) in the
left panel of the figure below, the power loss coefficient computed
at six different electron densities is shown (for a temperature equal to
each ion's ionization potential) in the right panel of the figure below
[11]. The fact that all six traces scale
linearly with increasing electron density in the right panel means that
the power loss coefficient is independent of density in this range.
For much lower densities (Ne ~ 108 cm-3)
, the contribution to the power loss by lines excited from metastable levels
falls off quickly. For higher densities (Ne ~ 1016
cm-3), forbidden radiative transitions are quenched by electron
collisions.
Impurity ions in a tokamak plasma can also radiate through channels
other than electron impact excitation of line emission; we also compute
the rate of power loss through line emission fed by dielectronic recombination
and continuum emission via radiative recombination and bremsstrahlung.
The resulting cooling coefficient for all radiative channels, and the sum
over all channels, the total cooling coefficient is shown below for Ar,
Kr and Mo. It is clear from the plots that the CR line emission dominates
the total cooling rate. Also shown in each plot is the ADPAK cooling
coefficient [12] for each element. In the case
of Ar, the ADPAK cooling coefficient has been updated by researchers at
the Los Alamos National Laboratory [13] using
CR models similar to those used here. The Ar cooling coefficient
computed here [14] and that given by the updated
ADPAK agree very well for temperatures between 200 and 2000 eV. For
lower temperatures the calculations show differences of factors of a few.
In the case of Kr [15] and Mo [11],
the two calculations diverge strongly at low temperatures, with the differences
reaching orders of magnitude. Overall, at higher temperatures, there
is general agreement to within a factor of two.
Calculated radiative cooling rate coefficients for Ar, Kr and Mo as a function
of electron temperature
Bolometric Measurements of Impurity Radiative
Cooling
The contribution of each ion to the total radiative cooling rate for an
impurity in a tokamak plasma is benchmarked against spectroscopic observations
as described above. The total cooling coefficient is tested by simulating
bolometric profiles measured at FT-U and C-Mod. The data are collected
with a bolometer array with up to 13 viewing chords through the plasma.
These multiple channels allow us to perform Abel inversion on the measured
signals and get the volumetric radiative emissivity for the plasma.
(A small correction is made for the contribution of neutral particles to
the bolometer's signal.) Below is a cartoon of the bolometric
arrays used for the present work.
In the case where there has been a controlled introduction of the impurity
species, or in the case where the plasma impurity content is dominated
by a single species, we measure directly that impurity's radiative losses.
The inverted bolometric profile as measured in the Frascati Tokamak Upgrade
is shown below. The large bump in the experimental data at the outermost
radii (coldest temperatures) is due to radiative losses from intrinsic
oxygen; aside from oxygen, molybdenum is the overwhelmingly dominant impurity
species in the plasma. Also shown are simulations made using the
cooling coefficient computed here (HULLAC) and using the ADPAK cooling
coefficient. The two experimental traces (labeled m=4 and m=6) are
the inverted signals from the bolometer array. The two numbers, m=4
and m=6, are the orders of the polynomial to which the data is fit before
Abel inversion is carried out. The difference between the two fits
is an indication of the uncertainty in the experimental data. The
measured data confirm the HULLAC cooling coefficient for temperatures between
300 and 2000 eV.
Shown below are preliminary simulations of the bolometer measurements made
during a Kr and Ar injections at FTU. The inverted bolometer data
is given by the squares, and has (at best) a 2 cm spatial resolution.
The simulation made with the cooling curves shown above are given by the
dash-dot traces (labeled HULLAC, new CSD). In both the case of Ar
and Kr, these simulation give the best agreement with the data. In
the case of Kr (the figure immediately below) the ADPAK simulation from
Ref. [12] is shown along with a simulation using the
HULLAC collisional-radiative line emission models and the old CSD without
EA. In both cases, the data for higher temperatures do not favor
these simulations. The bolometer measurements do not have the necessary
spatial resolution to map out features such as the notch near 18 cm in
the case of Kr and 21 cm in the case of Ar.
Simulation of the Kr bolometric profile in FTU
Simulation of the Ar bolometric profile in FTU
Resonance Enhancement of DWA Impact Excitation
Rate Coefficients
At energies below the threshold for direct electron impact excitation,
resonant excitations can make a significant contribution to the total excitation
rate of a given energy level. The resonant excitation process is
nearly the same as the dielectronic recombination process described above,
however, in this case, instead of multiplying the rate of capture of a
free electron by the branching ratio for radiative stabilization, the rate
is now multiplied by the branching ratio for autoionization back to excited
levels of the initial ion [16] . The
distorted wave approximation collisional excitation rate coefficients computed
by HULLAC [4] do not account for this multistep excitation
mechanism; for nearly all highly charged, high-Z ions, neglecting resonant
enhancement of direct electron impact excitation is a good approximation.
However, for certain electric quadrupole transitions in Ni-like Mo14+,
the large manifold of autoionizing resonances in Cu-like Mo13+
enables an enhancement of the direct impact excitation rates that is not
negligible. This process is laid out in a cartoon in the panel on
the right of the figure below.
The use of non-electric dipole ("forbidden") transitions as diagnostics
of plasma density has been a standard technique in spectroscopy for many
years. The slow rate of the forbidden decays relative to E1 decay
rates means that as the density of the plasma increases, the forbidden
decays are quenched due to collisional destruction of their upper levels
much earlier than are E1 decays. The ratio of the forbidden to E1
decays is thus an indication of the local plasma density. In the
case of Ni-like Mo14+, for the low densities that obtain in
tokamaks, the 3d - 4s electric quadrupole (E2) transitions can be brighter
than the nearby 3d - 4p E1 decays. This is the case in the figure
below (left panel). This spectrum was measured by Sugar, Reader and
Rowan in the TEXT tokamak [17]; also seen in the
spectrum are several 3d - 4s E2 lines from Co-like Mo15+.
The reason the the E2 lines are bright, despite their relatively slow decay
rates, is that the levels from which the transitions proceed are highly
populated. In order to calculate the level populations, it is necessary
to include the enhancement due to resonance excitation that is shown in
the panel on the right.
The factor by which the resonance contribution enhances the direct excitation
rate coefficient is shown in the the figure below. This figure is
similar to what is shown above for the case of excitation autoionization
enhancement of direct ionization. That is, the figure shows the ratio
of the resonant plus direct excitation rate coefficient over the direct
excitation rate coefficient alone. The ratios shown here are for
the two upper levels giving rise to the 3d - 4s E2 transitions, and the
two upper levels giving rise to the 3d - 4p E1 transitions.
The levels are indicated by two jj-coupled orbitals, the first being the
3d hole from which the excitation took place, and the second being the
n=4 orbital to which the electron was promoted. The enhancement is
seen to be a strong function of temperature, with the greatest enhancement
coming at the lower temperatures. Investigations in the Alcator C-Mod
tokamak [16] have shown that the MoXV ion achieves
equilibrium in a plasma with an electron temperature near 100 eV, thus
the contribution of the resonances to the excitation of the E2 transitions
must be taken into account.
We have injected Mo (Z=42), Nb (Z=41) and Zr (Z=40) into the Alcator
C-Mod tokamak in order to measure the ratio of the brightness of the Ni-like
3d10 - 3d94s E2 lines to the 3d10 - 3d94p
E1 lines at different densities. We have used an absolutely calibrated
2.2 m grazing incidence spectrometer with approximately 1 cm spatial resolution
and a 4 msec integration time for each frame. The detector was a
microchannel plate coupled to a 1024 element self-scanning Reticon linear
array. A Zr injection is shown below. The background subtracted
data is shown in frame (c) and the line of sight for which the Zr13+
emission is maximal is shown in the insert on the right. The Ni-like
Zr ion exists in a low temperature plasma region near the plasma x-point.
We have computed the ratio of the brightness of the two 3d - 4s
E2 to the two 3d - 4p E1 transitions in MoXV as a function of electron
temperature and density. This result is shown in the figure below.
The sensitivity of the ratio to density is due to the collisional quenching
of the upper levels of the E2 decays as described above. The sensitivity
to temperature is greatly enhanced by the presence of the strong resonance
enhancement channels at low temperatures. The figure also has two
experimental measurements made with photometrically calibrated instruments
in the Alcator C-Mod tokamak and a calibrated measurement by Klapisch et
al. in the TFR tokamak [18]. The density
in the Alcator C-Mod tokamak is much higher than in either the TEXT tokamak
or TFR. Still, even so, we could not reach the density where strong
collisional quenching is observed. The measured electron density
(using microwave reflectometry) is used to place the points from C-Mod
on the curve. The resulting electron temperatures suggested by the
data for the two shots are consistent with the measured bulk plasma temperature
profile and the divertor temperature profile. The spatial location
of the emitting lines is known only to within ~ 1 cm.
L-shell x-rays in Zr, Nb, Mo and Pd
We have conducted a large campaign of impurity injection experiments
at the Alcator C-Mod tokamak with the aim of observing the level structure
of the Ne-like (and adjacent) charge states of Kr (Z=36), Zr (Z=40), Nb
(Z=41), Mo (Z=42) and Pd (Z=46) [19].
The observations are made with an array of five spatially scannable, high
resolution Von Hammos type spectrometers. The radial profiles of
x-ray transitions in argon and molybdenum recorded with this instrument
are shown above. The very high spectral resolution of this
instrument (l/Dl
~ 4000) allows us to test higher order terms in our basic atomic structure
calculations, such as the contributions of vacuum polarization, the Lamb
shift and the Breit interaction to a level's total energy. Shown
in the frame below is a spectrum from a zirconium injection in the range
of the Ne-like Zr30+ 2p6 - 2p5(3/2)7d(5/2)
J=1 and 2p6 - 2p5(1/2)6d(3/2) J=1 transitions.
Also shown, but not labeled are transitions from Na- and F-like zirconium.
Theoretical calculations of transition energies and oscillator strengths
are plotted under the data. The agreement is excellent.
In the frame below, x-ray transitions in four palladium ions are shown.
The strongest line is the Ne-like 2p5(1/2)3d(3/2) J=1 transition
at 3731.7 mÅ. Also shown are 2p - 3d transitions in Na-, Mg-
and Al-like Pd. Again, theoretical calculations for transition
energies and oscillator strengths are plotted under the data.
One result from this campaign is a study of the transfer of strength
from one transition to another as the upper levels of those transitions
approach each other in energy. The Ne-like ions of Zr, Nb, Mo and
Pd are ideal for this task. In the figure below, the 2p(3/2) - 7d(5/2)
and 2p(1/2) - 6d(3/2) neonlike transitions in Mo, Nb and Zr are shown;
as Z increases, the emission features are seen to be moving closer together.
(In every case, there is a small sodiumlike feature that is close to the
2p(1/2) - 6d(3/2) neonlike line.) As the 2p(1/2) - 6d(3/2) upper
level moves closer to the upper level of the 2p(3/2) - 7d(5/2) transition,
the strength of the 2p(1/2) - 6d(3/2) is seen to diminish relative to that
of the 2p(3/2) - 7d(5/2) transition [19]; the
reason for the transfer of strength is due to the configuration interaction
between the upper levels.
In the figure below, the calculated oscillator strengths are plotted
for the four 2p(1/2,3/2) - 6d(3/2,5/2) and 2p(1/2,3/2) - 7d(3/2,5/2) transitions
for all elements from Y (Z=39) to Pd (Z=46). In the lower frame of
the figure is plotted the calculated energy difference between the 2p(1/2)
- 6d(3/2) and 2p(3/2) - 7d(5/2) upper levels; between Mo and Tc (Z=43)
the order of the levels switches and the 2p5(1/2)6d(3/2) level becomes
higher in energy than the 2p5(3/2)7d(5/2) level. As the levels approach,
the higher energy 2p(3/2) - 7d(5/2) transition oscillator strength is seen
to be strengthened at the expense of the 2p(1/2) - 6d(3/2) transition.
After the levels cross, the transfer of strength is in the other direction.
The data shown above confirm this result.
M-shell x-rays in highly charged W ions
Shown below are two spectra recorded by Dr. Rudolf Neu during tungsten
(Z=74) injections into ASDEX Upgrade tokamak plasmas [20].
The tungsten injections into the ASDEX plasma are made by laser ablation
of tungsten from a glass slide at a rate of 20 Hz, resulting in a nearly
constant tungsten inventory during the discharge. The central temperature
in the two cases differs by nearly 30%. The strong emission near
7Å is due to transitions of the form 3d - 4p, the higher energy features
near 5.5Å are due to 3d - 4f transitions [21].
The spectra are acquired with a rotating crystal Bragg spectrometer; the
constancy of the plasma conditions and the tungsten inventory with time
are important considerations. The relative strength of the 3d - 4p
transitions to the 3d - 4f transitions is at odds with collisional-radiative
predictions. Understanding the origin of this discrepancy is an ongoing
part of our research program. In the upper frame, the strongest emission
features are from charge states with ground configurations iso-electronic
to ions in the middle of the 4pk subshell (W41+,
As-like 4p3, W40+, Se-like 4p4).
In the lower frame, the emission has shifted to lines from higher charge
states, including the last of the N-shell ions, Zn- and Cu-like W44+
and W45+ (4s2 and 4s1). In the lower
frame, the emission from the M-shell Ni-like W46+ is seen to
be strongly enhanced due to the increased temperature.
XUV observations of N-shell W ions
The Dn=0 (4-4) counterparts to the Dn=1
(3-4) x-ray lines discussed above have been measured with an absolutely
calibrated McPherson type spectrometer by Dr. Knut Asmussen at the ASDEX
Upgrade tokamak [22]. These Dn=0
spectra have been developed into a sensitive diagnostic of the tungsten
concentration in the plasma, the details of that development are given
elsewhere [22]. The tungsten injections
into the ASDEX plasma are made by laser ablation of tungsten from a glass
slide at a rate of 20 Hz. The spectrum in the 120 to 140Å range
is shown in the figure below. The digitized data from the multichannel
plate is shown in the top frame, the time of the first tungsten injection
is clear in the data. A background subtracted line-out of the data
is shown in the middle frame. The CR calculations done with HULLAC
are shown in the bottom frame [21]. Nearly
all of the transitions in this range are 4s - 4p transitions; there is
a much larger discrepancy between the HULLAC wavelengths and the observed
wavelengths for these transitions than for the x-ray transitions modeled
above.
The 4p-4d transitions in the N-shell W ions are shown in the figure
below. The transitions near 60Å are nearly coincident with
4d-4f transition arrays from lower charge states. These transition
arrays make a quasi-continuum whose shape evolves strongly with increasing
temperature. By monitoring the time history of the tungsten injections,
we can see the quasi-continuum burn through and these unambiguous emission
lines [22] emerge. The calculated wavelengths
in this spectral region agree better with the observations than do those
in the figure above. Precise observations of radiative transitions
in highly charged high-Z ions help us to test directly the methods employed
by our atomic structure codes.
Direct Measurement of Impurity Fluxes
Often the ions in a tokamak plasma are seen to be distributed in
a way that is different than what the coronal equilibrium would predict.
Tokamak plasma spectroscopy in V-Division
supports investigations into anomalous transport, the cause of this departure
from equilibrium. One way research in V-Division
has made an impact in this field is by providing highly detailed models
for the emissivity of x-ray transitions in molybdenum ions near the neonlike
charge state [2]. With these models, researchers
at the Frascati Tokamak Upgrade (FTU) have been able to use the measured
x-ray emission profiles for three adjacent molybdenum ions to construct
the impurity flux profile across the plasma [23].
With this profile, the strength of the transport effects on individual
ions, at nearly all radii in the plasma, can be determined; this is a major
step forward in the understanding of the phenomena that drive anomalous
impurity transport.
The x-ray profiles measured at FTU have been recorded with a
rotating crystal Bragg spectrometer coupled to a multi-wire proportional
counter detector [24] that were designed and built by
researchers at FTU. The spectrometer has mountings for six crystals,
which allows for sensitivity to different energy regions during the discharge.
The detector provides a fast count rate, good energy resolution, and, by
exploiting differences in sensitivity to different orders of diffraction,
discrimination of the different orders that appear in the spectrum.
A photograph of the spectrometer and the housing for the six crystals is
shown in the figure below. A cartoon showing the principles by which
the spectrometer and detector work is shown below the photograph.
Rotating Crystal Bragg Spectrometer (RCS) at FTU
Rotating Crystal Bragg Spectrometer (RCS) at FTU
A spectrum covering the full energy range of the RCS at FTU is shown
below. The n=2 to n=1 Hea transitions
in Ti (Z=22), Cr (Z=24), Mn (Z=25), Fe (Z=26) and Ni (Z=28) are visible
at the highest energies. The n=3 to n=2 transitions from the F-like
Mo33+ to the Mg-like Mo30+ ions are seen at the lower
energy side of the spectrum. There is a sharp edge in the detector
efficiency near 3.8 Å due to an absorption edge in argon, the detector
fill gas [24].
Full range spectrum recorded with the RCS at FTU
The two panels below show a blow up of the molybdenum emission between
4.2 and 5.4 Å (top), and a synthetic spectrum for the line of sight
of the spectrometer on this discharge, #8001 (bottom). The dominant
Ne-like Mo32+ emission features are labeled in the bottom panel
with the conventional notation. The bottom panel shows a very high
spectral resolution simulation as well as the same simulation degraded
to the resolution of the spectrometer and detector. The blending
of the weak lines from Mo31+ and Mo30+ with the strong
neonlike transitions accounts for nearly all the features seen in the data
in the top panel. With an assumed smoothly increasing background
level (shown by the dashed line in the top panel), the relative integrated
areas under the experimental and synthetic peaks agree very well [2].
Given our knowledge of the emission spectra of near neonlike molybdenum
ions, the brightness of specific lines in Mo32+, Mo31+
and Mo30+ have been measured by Dr. Danilo Pacella at FTU in
a series of reproducible discharges [23].
The measured brightnesses for the three lines have been Able inverted to
yield the emissivity profile for the lines as a function of radius in the
plasma. The spectrometer has been absolutely calibrated, so the measurement
can be converted into an absolute emissivity. The raw data (before
inversion) for the observations are shown in the three panels below.
Given that the absolute emissivity of the transition in each ion is measured
as a function of radius, and that the electron temperature and density
profiles are independently measured, the densities of the specific ions
can then be found. The collisional-radiaitive model developed by
Dr. Kevin Fournier in V-Division
confirms that the three transitions chosen for this experiment are fed
overwhelmingly by direct excitations from the ground level of the respective
charge state. The excitation rates are computed using HULLAC for
several temperatures in the FTU, and the ion densities are derived.
The results are shown in the figure below. Also shown are the predicted
ion density profiles for the case of coronal equilibrium. The experimental
profiles extend much further out in the plasma and are less sharply peaked
than in the coronal equilibrium case.
Given that we know the absolute density for three adjacent charge states,
and we know the rates of ionization and recombination
between the charge states, we can use the equation of continuity
to derive the absolute flux of Mo31+ across the plasma.
In steady state, the explicit time derivative is zero and the divergence
of the flux can be derived from the knowledge of the source terms,
where S and a represent the total ionization
and recombination rates for each ion and are functions of the local electron
temperature, Te(r), and Ne is the electron density.
The flux is derived by radial integration of the equation above,
.
In the present case, the flux G31+ of
Mo31+ can be derived using the density profiles N32+,
N31+ and N30+ that have been obtained from the soft x-ray
spectra. Thus [23], 
.
The result is shown in the figure below.
The resulting Mo31+ flux represents the transport effect of
the plasma on this impurity ion. We can then define a radial drift
velocity associated with the flux G31+,
.
This quantity is plotted for Mo31+ in the figure below, along
with the electron temperature and density profiles, as measured with electron
cyclotron emission and DCN interferometery, respectively. The radial
drift velocity has the advantage that it does not depend on the absolute
amount of Mo31+ present and thus is independent of the absolute
calibration of the RCS. The result of the present analysis is, as
expected, that the transport effect is small in the central region and
very large in the intermediate region of the plasma. In fact, the
transport effect is more than one order of magnitude greater than the neoclassical
prediction in the intermediate region of the plasma (well outside the error
bars on the radial drift velocity due to the inversion of the ion density
profiles).
Near the maximum of Vr in the figure below, the thermal drift
velocity for Mo31+ is approximately 5 x 106 cm s-1,
so the drift velocity is about 0.4% of the thermal one. Starting
from the measured Vr, we can estimate the order of magnitude
of the impurity confinement time along the radius,
.
The result is that tt is of the order
of tens of ms in the central region (having taken for Vr the
maximum value given by the error bars) and it falls to about 1 ms at r
= 15cm, where the transport is very large. As can be seen, the radial
outgoing velocity Vr is negligible in the central region and
indicates a large anomalous transport effect in the region around the plasma's
half radius, a region where many of the plasma parameters exhibit strong
radial gradients (see the figure above).
Transport profiles during auxiliary heating
Progress in magnetic confinement fusion research has produced plasmas with
continuously higher temperatures. Moreover, interest in advanced
tokamak scenarios is continuously growing, requiring more refined diagnostic
tools to investigate particle transport processes. All this has stimulated
the study of the atomic physics of highly ionized medium Z elements, like
molybdenum (Z=42). On one hand, the tokamak plasma provides a "laboratory"
where atomic calculations, relevant in many others fields, can be experimentally
validated and, on the other hand, these models allow the understanding
of many plasma processes.
Atomic calculations performed at LLNL have been used to understand the
molybdenum x-ray emission from the FTU plasma in two different scenarios
with electron cyclotron resonance heating (ECRH), providing important information
about the plasma impurity content [25]. In
the first case, with on axis heating and a central electron temperature
of 8 keV [26], the analysis is performed
during the current ramp up when the magnetic shear is still negative or
zero [27]. In a toroidal configuration with
nested magnetic surfaces, the magnetic shear s is a measure of the radial
variation of the safety factor q, i.e. of the field lines winding index.
The magnetic shear is defined by s = (1/r) dq/dr so that shear reversal
(i.e. negative shear in the plasma core) implies the presence of an off-axis
minimum in the q profile. In the second case, with off axis heating
and a peak temperature of 7 keV, spectra are analyzed when the current
reaches the plateau and sawtooth activity occurs. These results will
be discussed and compared in the following sections. The radial profiles
of the electron temperature (Te) are shown for both discharges, as measured
by ECE, in the figure below.
Two molybdenum spectra from shot #12658, one during ECRH and one after,
are compared in the figure below. Since the wavelengths are scanned
in approximately 5 ms, a time shorter than the time evolution of the density
and temperature, the plasma can be assumed stationary on this time scale.
It is seen that the three features that we label a,
b and g, which are
dominated by transitions from Mo35+, Mo36+ and Mo37+
respectively, decrease strongly when the temperature drops, while the remaining
spectrum (dominated by Mo32+ transitions) is less affected.
The detailed modeling of these transitions is discussed below.
High resolution synthetic spectra for Te = 3.8, 5.5 and 8.0 keV are shown
in the three panels below; it should be noted that even if Gaussian shapes
with a fwhm of 1.0 mÅ are assumed, blending still persists.
As can be seen, the spectra are sensitive to the temperature, both because
of changing ionization state distribution and, to a lesser extent, due
to the temperature dependence of the intensity of individual lines.
Each ion (from Mo30+ to Mo39+) carries information
from the radial layer where it exists; studying these x-ray spectra we
obtain spatial information over a large fraction of the minor radius (0<r/a<0.7
and Te(0)=1-8 keV). In particular, the features labeled
a, b and g
are typical of the higher temperature regime (Te > 3keV) and
drop in intensity when the ECRH is switched off (see the spectra above);
g is dominated by B-like Mo37+ which
exists in appreciable amounts only in plasmas with Te > 5 keV, while a
and b are dominated by lines from N-like
Mo35+ and C-like Mo36+, respectively.
The collisional-radiative model described
above is used to reconstruct the measured molybdenum spectra knowing the
electron temperature and density profiles. This is possible since the typical
times, t, required for these ions to reach ionization
equilibrium in this temperature and density range are a few milliseconds.
Since the central electron temperature reaches nearly its peak value
20 ms (Dt) before the spectrum above is recorded
(t ~ 0.1 s), the ions have enough time (Dt >>
t) to reach equilibrium. Our goal is to simulate the full spectrum,
and in particular the relative ratio of these features (a,
b and g) with respect
to the resonant lines at 4.8-4.9 Å from Mo32+, Mo31+
and Mo30+. (These features are better understood than
those at 4.5 - 4.6Å from Mo34+ and Mo33+.)
This ratio depends strongly on the total impurity profile, Nmo
(r). The synthetic and experimental spectra are shown in the frames
below for a peaked impurity profile (top) and for a flat one (bottom).
In the figure the continuous line represents the experimental spectrum,
while the shadowed areas are the contributions of Mn and Cr Ka
in second order.
The (normalized) peaked impurity profile, Nmo (r), is shown
in the figure below together with the electron density profile. The
peaked profile produces a much better fit to the experimental spectrum.
This assessment of the impurity radial density profile is robust because
it relies on agreement from all the emitting ions (Mo39+ to
Mo30+), over the entire radial range 0 < r/a < 0.7 and
not just on the "high temperature" features. The calculated feature
b in the spectra above is narrower and higher
with respect to the measured one, but the integral is the same; this is
due to spurious blending of several of the weak lines making up this feature.
There are two results from this analysis that should be emphasized.
We find an impurity transport level much lower with respect to that one
usually measured at FTU in the steady state. For the spectra measured
early in the discharge, the derived molybdenum charge state distribution
is close to that given by coronal equilibrium. Also, we find a central
impurity accumulation early in the discharge. These two results taken
together are compatible with a neoclassical transport regime. The
absence in the core of impurity anomalous transport is an important result
for this plasma regime of strong auxiliary heating. This result should
be taken together with the good energy confinement property observed in
previous work [27] studying the energy balance and
the thermal conduction for the same shot, #12658.
In contradistinction to the case above, a second discharge was studied
in a different transport regime. In this shot (#15505) ECRH heating
off axis was used with double power (800 kW) on the same plasma target.
The Mo emission was analyzed at a later time (t=0.135 s) after the current
flat top had been reached and the onset of the sawtooth instability (t=0.120
s). The off axis heating and the presence of the sawtooth crashes
produce a plateau in the central electron temperature (the temperature
profile is shown above) and a much broader region emitting at high temperature.
We have applied the same method to simulate the experimental spectrum (thick
pink line in the figure below). The features a,
b and g are well
simulated with the emitting ions (Mo35+ to Mo39+)
in coronal equilibrium. Further, the relative ratio of these high
temperature features with the line of Mo32+ at 4.8 Å is
well reproduced with a flat total molybdenum profile, Nmo (r).
However, the features between 4.6 and 4.9 Å are not well simulated
with the assumption of coronal equilibrium, in particular the lines emitted
by Mo31+ and Mo30+ are underestimated. This implies
that in the radial region where these two ions exist (r/a>0.3) anomalous
transport is now present, reducing the degree of ionization. Unlike
the previous case, in this phase of the discharge the shear has become
positive and the anomalous transport is growing, particularly in the half
radius region. This observation is in agreement with the results
found above by deriving the impurity flux
for Mo31+. In other work not reported upon here, the impurity
transport is modeled as being driven by electrostatic turbulence [28]
during the current flat top phase.
We remark that molybdenum is an excellent diagnostic tool because it has
several ions that emit strongly and have appropriate ionization times (
t ~ ms) in the temperature range up to 20 keV.
In particular the resonant lines of the B-, Be- and Li -like ions are bright
and can be used as diagnostics of very high temperature plasmas in the
same way the N-, C- and B-like ions have been used in the present work.
A synthetic spectrum for Mo40+ to Mo37+, both at
high and low spectral resolution, is shown in the figure below. The
present experiments at FTU were performed with one and two gyrotrons; the
installation of 4 tubes with a total effective full power of 1.6 MW is
now in progress. Consequently, in future experiments temperatures
greatly in excess of 8 keV are expected and other plasma scenarios will
be accessible; these ions will provide an excellent diagnostic tool to
study these plasmas.
________________________________________________________________________
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UCRL: MI-135159