Initial Exploration of High-Field Pulsed Stellarator Approach to Ignition Experiments

Abstract

In the framework of fusion energy research based on magnetic confinement, pulsed high-field tokamaks such as Alcator and FTU have made significant scientific contributions, while several others have been designed to reach ignition, but not built yet (IGNITOR, FIRE). Equivalent stellarator concepts, however, have barely been explored. The present study aims at filling this gap by: (1) performing an initial exploration of parameters relevant to ignition and of the difficulties for a high-field stellarator approach, and, (2) proposing a preliminary high-field stellarator concept for physics studies of burning plasmas and, possibly, ignition. To minimize costs, the device is pulsed, adopts resistive coils and has no blankets. Scaling laws are used to estimate the minimum field needed for ignition, fusion power and other plasma parameters. Analytical expressions and finite-element calculations are used to estimate approximate heat loads on the divertors, coil power consumption, and mechanical stresses as functions of the plasma volume, under wide-ranging parameters. Based on these studies, and on assumptions on the enhancement-factor of the energy confinement time and the achievable plasma beta, it is estimated that a stellarator of magnetic field B ~ 10 T and 30 m³ plasma volume could approach or reach ignition, without encountering unsurmountable thermal or mechanical difficulties. The preliminary conceptual device is characterised by massive copper coils of variable cross-section, detachable periods, and a lithium wall and divertor.

Full text

ORIGINAL RESEARCH
Initial Exploration of High-Field Pulsed Stellarator Approach to Ignition
Experiments
V. Queral
1
•F. A. Volpe
2
•D. Spong
3
•S. Cabrera
1
•F. Tabare
´s
1
Published online: 29 October 2018
The Author(s) 2018
Abstract
In the framework of fusion energy research based on magnetic confinement, pulsed high-field tokamaks such as Alcator
and FTU have made significant scientific contributions, while several others have been designed to reach ignition, but not
built yet (IGNITOR, FIRE). Equivalent stellarator concepts, however, have barely been explored. The present study aims at
filling this gap by: (1) performing an initial exploration of parameters relevant to ignition and of the difficulties for a high-
field stellarator approach, and, (2) proposing a preliminary high-field stellarator concept for physics studies of burning
plasmas and, possibly, ignition. To minimize costs, the device is pulsed, adopts resistive coils and has no blankets. Scaling
laws are used to estimate the minimum field needed for ignition, fusion power and other plasma parameters. Analytical
expressions and finite-element calculations are used to estimate approximate heat loads on the divertors, coil power
consumption, and mechanical stresses as functions of the plasma volume, under wide-ranging parameters. Based on these
studies, and on assumptions on the enhancement-factor of the energy confinement time and the achievable plasma beta, it is
estimated that a stellarator of magnetic field B*10 T and 30 m
3
plasma volume could approach or reach ignition, without
encountering unsurmountable thermal or mechanical difficulties. The preliminary conceptual device is characterised by
massive copper coils of variable cross-section, detachable periods, and a lithium wall and divertor.
Keywords Stellarator Ignition parameters Resistive magnets Monolithic support
Introduction
Fusion energy is widely considered a potentially clean and
abundant energy source [1,2]. Current mainline research in
magnetic confinement fusion is based on the tokamak
concept [3], in spite of the important drawback posed by
the possibility of disruptions and the challenge of steady-
state operation. Correspondingly, alternatives based on the
stellarator concept have also been developed [4–6]. Among
them are a design of an ignition experiment (HSR4/18i [7])
and a burning-plasma stellarator concept [8]. Many
tokamaks and stellarators were built and operated to
investigate a variety of fusion plasma problems [9–11].
However, understanding the physics of burning plasmas
remains a research challenge [12,13].
For both concepts, tokamaks and stellarators, a higher
magnetic field leads to a smaller and potentially more cost-
effective experimental device [14,15]. Additionally,
devices equipped with resistive magnets, of moderate cost,
are suited to produce pulses of few seconds (longer or
much longer than the energy confinement time and alpha-
particle slowing down time), which are appropriate to
perform a diversity of burning plasma experiments.
In tokamaks, several high magnetic field devices have
been satisfactorily built and operated to explore and vali-
date this approach, e.g. Alcator and FTU [16,17]. Other
high-field experimental tokamaks have been designed to
reach ignition but not built yet, e.g. IGNITOR and FIRE
tokamaks [18,19]. The IGNITOR design employs massive
cryo-cooled copper magnets and pursues plasma ignition
using a high magnetic field B*13 T in a small plasma
&V. Queral
vicentemanuel.queral@ciemat.es
1
Laboratorio Nacional de Fusio
´n, CIEMAT, 28040 Madrid,
Spain
2
Department of Applied Physics and Applied Mathematics,
Columbia University, New York, NY 10027, USA
3
Oak Ridge National Laboratory (ORNL), Oak Ridge,
TN 37831, USA
123
Journal of Fusion Energy (2018) 37:275–290
https://doi.org/10.1007/s10894-018-0199-5(0123456789().,-volV)(0123456789().,-volV)
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volume V*10 m
3
,atb*1.2% (bis the plasma kinetic
pressure normalized to the magnetic pressure). Similarly,
FIRE is another high-field tokamak design (B*10 T,
V*20 m
3
) aimed at approaching ignition which also uses
cryo-cooled copper magnets.
However, the exploration of high-fields in stellarators
has been scarce. One exception is FFHR2 [20], but that is a
power plant design, not an experimental device. Conse-
quently, a stellarator-based, high-field, high power density
and resistive-magnet approach to the production of plasma
ignition experiments appears fundamental. It would shed
light, rapidly and at modest cost, on essential reactor-rel-
evant physics and technology, and thus, it deserves
exploration.
In this context, the present paper proposes a high-field
stellarator path toward the study of burning plasmas. As an
initial approximation, the work: (1) explores the essential
physics and technological parameters of ignition-capable
experimental stellarators, particularly the operational limits
and difficulties at high fields, and (2) derives an initial
stellarator conceptual design.
The parameter scan is deliberately broad to provide
rough initial estimates of possible operating points for the
design. Firstly, we estimate the minimum magnetic field
needed for ignition and the fusion power as a function of
the confinement enhancement factor h
E
(as in International
Stellarator Scaling 2004, ISS04 [21]), band V. Subse-
quently, we study the technological parameters: heat load
on the divertor targets, electric power needed to feed the
resistive magnets and stresses on the coil supports, also as a
function of h
E
,band V. Among the potential operating
points, a reasonable one is down-selected at the frontier of
the physics and technological limits. Finally, from the
operating point and the studies performed, the definition of
a possible high-field ignition-capable experimental stel-
larator is presented, called i–ASTER. This is characterised
by massive copper coils of variable cross-section (so as to
reach high fields with feasible power supplies), a lithium
divertor-wall to try to deal with the high power density, and
absence of blankets to lower costs.
The work is organized as follows. In ‘‘Assumptions and
Governing Equations: Ignition Condition’’ section we for-
mulate the governing physics equations. The technological
parameters and constraints are presented in the next sec-
tions: heat load on the divertors (‘‘Power Load on Divertor
Targets’’ section), power needed to operate the resistive
magnets (‘‘Power Dissipated in Resistive Magnets’’ sec-
tion) and stresses in the coil support structure (‘‘Estimation
of Stress in Coil Structures’’ section). Finally, the resulting
specifications of a possible ignition stellarator concept are
presented in ‘‘Definition of i-ASTER’’ .
Assumptions and Governing Equations:
Ignition Condition
A power balance equation and a scaling law for the energy
confinement time are the essential physics equations
involved. Additionally, the fusion power generated under
ignition or the maximum possible plasma density, equal to
the Sudo density limit [22], could have been minimized.
Instead, we decided to minimize the magnetic field since it
clearly correlates with the cost of the coils and their support
structures [23]. Only an initial estimate of possible oper-
ating points is sought here. Detailed plasma calculations
using advanced codes [24,25] are left for future work, as
the design advances.
The governing physics equations assume a scalable
device for scanning the plasma volume and the device size.
Thus, all proportions and all shapes (e.g. of the coils and
their support structures) are preserved, and all dimensions,
such as the distance from plasma edge to the winding
surface, scale with a scaling factor.
Under such premises, two rather extreme values of h
E
(0.75, 1.5) are considered in the remainder, as well as three
values of the volume-averaged beta limit \b[
lim
(2.5%,
5%, and 10%). Values in-between these limits are con-
ceivable and thus, potential operating points. These limits
were selected as follows, according to experimental and
theoretical data.
An enhancement factor h
E
around 1.5 was experimen-
tally achieved in some high-bpulses in W7-AS and slightly
lower in the LHD inward-shifted configuration [21]. Cal-
culations have predicted h
E
*2 for W7-X [24], but this is
yet to be proven experimentally.
Experimentally, W7-AS achieved a maxi-
mum \b[= 3.2% [11] and \b[= 5% was demon-
strated in LHD [26]. Up to b
lim
*7% may be achievable
for the low aspect ratio A*4.5 NCSX [27,28]. b
lim
*10% is calculated for a large aspect ratio A*10 quasi-
helical stellarator [29], and slightly lower b
lim
*8.5% for
A*12 in QIP6 (Quasi-Isodynamic with poloidally closed
contours of constant Bof 6 periods) [30,31]. Second
stability regimes of high beta 7–20% in compact stellara-
tors have been theoretically predicted [32,33] but are yet to
be experimentally proven.
For each combination of h
E
and b
lim,
we estimated the
minimum magnetic field needed for ignition (Fig. 1). This
was done in a way similar to Refs. [34,35]. More specif-
ically, a power balance and an energy confinement scaling
are used, along with expressions for terms to be substituted
in them. Together, they form a set of eight equations
(Eqs. 1–7below, and the definition of b).
276 Journal of Fusion Energy (2018) 37:275–290
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Power Balance
The first equation is the power balance under ignition
conditions (that is, with negligible external heating power):
Paheat PsEPrad ¼0ð1Þ
where the heating power due to aparticles,
Paheat ¼Pað1falossÞ:ð2Þ
is related, via the fraction of alpha particles lost,f
a_loss
,to
the total power generated by alpha particles,
Pa¼fd1=4WaZ\rv[DTn2
edV ð3Þ
where f
d
=n
DT
/n
e
is a dilution factor due to impurities, W
a
the fusion product alpha energy per reaction and n
e
the
electron density. The reaction rate parameter \rv[
DT
is approximated by a sixth order polynomial as in page 30
of Ref. [36]. A fractional loss f
a_loss
= 0.05 and f
d
= 0.84 are
assumed here.
The power loss due to energy transport,P
sE,
is simply
given by the ratio of the plasma stored energy, W
int
=3 k
B
$
Tn
e
dV to the energy confinement time s
E
(electron tem-
perature T
e
=T
i
=T):
PsE¼Wint
sE
ð4Þ
Finally, the power radiated by Bremsstrahlung can be
expressed as:
Prad ¼51037Zeff ZT1=2n2
edV ð5Þ
where an effective charge Z
eff
= 1.3 was assumed, corre-
sponding to about 4% of He ash and 4% of Li. This seems
feasible if Li-coated walls were used, as in TFTR Li shots
[37]. Power radiated by other mechanisms, such as line and
cyclotron emission, can be shown to be negligible.
Energy Confinement
The scaling law for the energy confinement time is
sE¼hEC0RuahBand
ePrij
2=3ð6Þ
Different scaling laws are available in the literature [38],
with different coefficients C
0
and different exponents, but
here we follow the ISS04 international stellarator scaling
[21]. Here Ris the plasma major radius,Pthe effective
heating power (:P
a_heat
) and i
2/3
the rotational transform
at r=2/3a, where ais the plasma minor radius.
An aspect ratio A= 6 is assumed, as a rough average
between A*4.5 in ARIES-CS [39], A*6 in HSR3/15
(Helias Stellarator Reactor of 3 periods) [40] and A*7in
QIP3 (Quasi-Isodynamic stellarator with poloidally closed
contours of 3 periods) [30].
Additional assumptions include ‘intermediate’ temper-
ature and density profiles, similar to HSR4/18i [7]—that is,
neither too flat, nor too peaked. Flatter profiles would yield
higher fusion power but require higher Bfor ignition. More
peaked profiles have been obtained in stellarators [41]but
it is unknown whether they would be feasible in burning
plasmas.
Estimate of Minimum Bfor Ignition
The six equations listed above, together with
dB
dT ¼0 to minimise Bwith respect plasma TðÞð7Þ
and the definition of b,form a set of eight equations in nine
unknowns: V,s
E,
B,n, T, P,P
a,
P
sE,
P
rad
. By eliminating
seven of such unknowns, we are left with a single equation
in two unknowns, for instance Band V, which can be seen
as an expression of B(in fact, the minimum Bfor ignition)
as a function of V. The results are plotted in Fig. 1.As
points of reference, the plasma volume in IGNITOR is
V*10 m
3
and B*13 T, in FIRE V*20 m
3
B*10 T, and in ITER V*840 m
3
.
Density and Temperature Needed for Ignition,
Fusion Power
Figure 2illustrates the line-averaged density needed for
ignition, if the magnetic fields depicted in Fig. 1were used.
Two Sudo limits for radiative collapse [21,22] are also
plotted. The figure shows that, for small enough plasma
volumes (V\400 m
3
for the case h
E
= 1.5 b
lim
= 5%), the
density needed for ignition is lower than the Sudo limit, as
desired. Larger volumes would require densities in excess
Fig. 1 Minimum magnetic field B
0
needed for ignition of a stellarator
plasma (A=6,i= 0.7, f
d
= 0.84) as a function of the plasma volume.
Different curves correspond to different assumptions on h
E,
and beta
limit b
lim
. The i-ASTER operating point is indicated
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of the Sudo limit, which should not necessarily be ruled
out. For example, LHD reached density three times higher
than the Sudo limit in a particular experiment [41].
Figure 3indicates the fusion power generated
(P
f
&5P
a
). The fusion power is nearly constant with
respect to volume. This is due to the reduction of minimum
Bfor ignition at larger plasma volume (Fig. 1) and the
equations involved.
The ignition temperature is independent of h
E,
b
lim
and
V. For the assumed Z
eff
, pressure profile and A, the central
temperature evaluates to T
0.ig
= 14.6 keV.
Power load on Divertor Targets
While physically attractive, some of the data points pro-
jected in Figs. 1,2,3are not necessarily viable. One
technological constraint is posed by the power-load per
unit surface on the divertor targets, P
d
. This is calculated
by dividing the total incident power by the wetted area,
which is smaller than the plasma surface S
p
by a ‘con-
centration factor’ K
d
. In other words,
Pd¼Kd
Pa
Sp
ð8Þ
It is assumed that the incident power equals the alpha
heating power P
a
, in the limit of negligible power radiated
by the divertor mantle and SOL.
K
d
depends on the particular magnetic configuration and
divertor (Table 1).
As known, divertor-related challenges could limit the
attractiveness of fusion as a competitive energy source,
both in stellarators and tokamaks [47,48]. Divertors are
less critical in short-pulse physics experiments, but still
plasma purity and thermal shocks on the walls and divertor
targets are relevant.
Here, in order to calculate P
d
from expression (8), we
make the following assumptions:
1. It is assumed that a reasonable increment of 50% of
wet area relative to W7-X divertor (increase from 2 to
3m
2
in Table 1) is possible by modern optimization,
resulting in K
d
*40.
2. Sweeping of the divertor legs on the targets by slightly
changing the currents in coils. It would change the size
and position of the magnetic islands [11,44], increas-
ing the wet area and smoothing the heat load on the
targets [49–51]. Doubling the wet area of an improved
quasi-isodynamic configuration is assumed in Fig. 4,
K
d
*20.
3. 50% of the power is radiated by the plasma edge, also
considered in Fig. 4.
The resulting heat loads are plotted in Fig. 4.
If such conditions are not met, it can be shown that
ignition could be achieved by reducing bto *2.5% or
less and increasing B. This, however, would largely reduce
the attractiveness of the approach, unless a solution is
adopted—probably based on liquid lithium, which may
withstand high P
d
. As an added benefit, low recycling Li
walls enhanced confinement in TFTR [52], TJ-II [53],
NSTX [54] and other devices [55,56] by various amounts,
ranging between 25% and 100%. Liquid lithium does not
erode or blister. Low Li impurity in the core plasma was
obtained in NSTX and TFTR [55], which allowed low Z
eff
(*1.3), e.g. in TFTR [37]. Drawbacks of lithium
Fig. 2 Line-averaged electron density n
L
and Sudo density limit n
S
as
functions of the plasma volume, for different values of h
E
and b
lim
.
The field is set to the corresponding minimum value needed for
ignition, according to Fig. 1
Fig. 3 Fusion power generated P
f
for the combinations of h
E,
b
lim
and
minimum field for ignition presented in Fig. 1
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utilization, like oxidation, fire risk, tritium retention and
others are cited in Ref. [57].
It goes beyond the scope of the present paper to enter in
excessive details on this aspect. However, it is worth
mentioning several promising Li-based systems:
•Jets of liquid metal droplets flowing on limiters or
divertors. As an example, Ga-In-Sn droplets of 2–4 mm
diameter and 2-5 m/s extracted 5-10 MW/m
2
from the
T–3 M tokamak [58,59].
•Liquid Li limiters or walls based on a Capillary Porous
System (CPS), as tested in FTU [60,61] and TJ-II [62].
In FTU they withstood an average of 2 MW/m
2
and
brief (300 ms) peak values of 5 MW/m
2
(see Fig. 12 in
Ref. [61]). Experiments with a CPS liquid lithium
limiter on T-11 M tokamak [63] achieved 10 MW/m
2
on the limiter (0.3 s pulse), and 30 MW/m
2
including
radiation from Li ions.
•Beams of high speed ([100 m/s) Li droplets [64], as
theoretically proposed for the ITER divertor.
•Molten (tin) shower jets are theorized for the FFHR
reactor [65].
Indeed, promising high power extraction systems could
be properly tested and enhanced in the present high power
density approach.
The average neutron wall load (Fig. 4) is calculated as
the total neutron power divided by the plasma surface.
Power Dissipated in Resistive Magnets
The effective cross section of the coils is maximized in
order to reduce the coil resistance and lower the Ohmic
power dissipated in the coils, as in Refs. [35,66] and in
Fig. 7. As a result of this design, each coil presents variable
cross-section in poloidal direction. The cross sections tend
to be smaller on the inboard of the stellarator and larger on
the outboard (Fig. 7), leading respectively to a local
increase and local reduction of dissipated power, partially
compensating each other.
Ports are not defined in this initial model for electric
calculations, but they will be small as explained in ‘‘Re-
sistive Magnets’’ section and would not hinder the massive
quasi-continuous coils.
A simple analytical expression is derived in ‘‘Analytic
Approximation to Dissipated Power’’ section for the power
dissipated in the coils. Some factors involved in that
expression are computed in ‘‘Finite Elements Results’’
section with the aid of finite elements.
Analytic Approximation to Dissipated Power
An approximate analytical expression valid for any V,h
E
and b
lim
is sought here. The plasma cross-section is
approximated by a circle (Fig. 5). The vessel and coils are
conformal to the plasma. Let us introduce the ratio f
R
of the
major radius of the magnetic axis to R(R
m
=f
R
R); the
factor nrelating the minor radius of the winding surface,
a
c
, to the plasma minor radius (a
c
=na); the fractional
thickness eof the coils relative to a(e=ea) and the
fractional effective cross-section of the conductor, f
i
(ratio
of copper cross-section S
Cu
to total section S
Cu
plus S
i
,
Fig. 7). Finally, the coil-shape factor f
s
quantifies the
increase of length and reduction of cross-section of the
conductor due to coil twisting. Some of these parameters
are illustrated in Fig. 5. In terms of these geometrical
Table 1 Wetted area and
concentration factor K
d
in
different fusion devices
Device Type of divertor Plasma surface (m
2
) Wet area (m
2
)K
d
Refs.
LHD (torsatron) Helical *90 5 18 [42]
CFNS Super-X div. [43]
W7-X Island divertor 110 2 55 [44]
W7-AS Island divertor 100 [44]
ITER Standard tokamak 678 6 113 [45,46]
Fig. 4 Heat power load on divertor targets, considering improved
divertors and sweeping, with K
d
= 20 and 50% of radiated power at
edge, and average neutron wall load, plotted as functions of the
plasma volume, for combinations of h
E,
b
lim
and minimum field for
ignition presented in Fig. 1
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factors, the Ohmic power dissipated in coils of resistivity q
is given by
Pcoils q4p2fRRB2nf2
s=l2
oefi
 ð9Þ
Accordingly, if we set n=2,e=1, f
s
= 1.3, f
R
= 1.2,
f
i
= 6/7 and adopt the minimum Brequired for ignition
(Fig. 1) we obtain the dissipated power plotted in Fig. 6.
Finite Elements Results
Coil shapes are generated for two quasi-isodynamic mag-
netic configurations (QIP3 and HSR3) by means of the
CASTELL [67] and NESCOIL [68] codes. QIP3 is utilized
as the main modelling magnetic configuration since quasi-
isodynamic configurations have low plasma currents (may
simplify the auxiliary coils and plasma control compared to
quasi-axisymmetric ones), and in particular, QIP3 is a
modern well optimized configuration of intermediate
aspect ratio. The QIP3 coils are shown in Fig. 7.
Power dissipation is calculated by finite elements in the
CASTELL code, using the configuration depicted in Fig. 7
except that we treat the trapezoidal cross-sections of that
figure as rectangular. In addition, values n= 1.75, e= 0.5
are used for QIP3, n=2,e= 1 for HSR3, and f
i
= 6/7 for
both. It results that the analytical expression (9) agrees,
with deviations lower than 20%, with the time-consuming
finite elements calculation for QIP3 and HSR3, taking a
fixed f
s
= 1.3 (in comparison, for a tokamak f
s
= 1). From
the study performed for QIP3 and HSR3 configurations,
1.2 \f
s
\1.4 is expected for typical stellarator magnetic
configurations.
Current Density and Coil Temperature
The current density j
s
in the coils is evaluated at the cross-
sections S located at the major radius R(Fig. 5) and
averaged over all coils. Nevertheless, the current density is
higher in certain locations. We denote by f
c
the concen-
tration factor for the maximum current density relative to j
s
(j
max
=f
c
j
s
). f
c
is calculated by finite elements in CAS-
TELL code as the ratio of the average cross section of all
the finite elements for all the coils to the minimum cross
section found among the coils. As an example, f
c
= 5 for
QIP3 and f
c
= 6 for HSR3 was calculated for the conditions
in ‘‘Finite Elements Results’’ section f
c
\*6 is expected
for non-quasi-isodynamic stellarators since there is not a
mirror-like magnetic field.
The average increase of temperature of the copper at
section S is calculated as
DTave ¼t
Cp
Pcoils
VtotCu
¼tq
Cp
B2f2
s
l2
0e2a2f2
i
ð10Þ
Being, tthe pulse length (5 s
E
), C
p
the volume-specific
heat of the material, P
coils
total power dissipated in coils
from Eq. (9), V
totCu
total volume of copper in all coils, and
the remainder as in ‘‘Power Dissipated in Resistive Mag-
nets’’ section.
The maximum increase of temperature results
DTmax ¼DTavef2
cð11Þ
Fig. 5 Schematic poloidal cross-section of the plasma and a coil, and
definition of the main dimensions used in the calculations
Fig. 6 Approximate electric power consumed in the resistive copper
coils for the parameters and minimum field for ignition presented in
Fig. 1
Fig. 7 Illustration of the concept of massive resistive coils of variable
cross-section (variable-width)
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Limitations and Discussion
The large thickness of the magnets for reasonable power
supplies is a concern. Thickness as wide as the plasma
minor radius (e= 1) is taken for Fig. 6. Despite that,
additive manufacturing can help the fabrication of such
thick layer(s) of conductor and insulation, as being inves-
tigated for stellarator coils in Refs. [69–71].
Also, the fabrication method for the variable cross-sec-
tion coils requires future exploration. Water jet cutting of
copper sheets and winding of the resulting conductors in
additively manufactured grooves is a construction option.
Another alternative is the use of a single, properly grooved
thick metal layer conformal to the vacuum vessel, with
insulating layers in the grooves, similarly to the concept
depicted in Ref. [66].
The massive resistive coils of variable cross-section
involve new calculation methodologies and advanced
magnetic error prediction. The coil width, number of coils
and the number of layers per coil has to be decided
according to: i) finite element analysis of the current paths
in the wide coils, and ii) the non-uniform increase of
copper temperature and thus differential increase of resis-
tivity due to Joule heating. Such advanced calculations will
be investigated in next development phases.
Estimation of Stress in Coil Structures
The yield tensile strength of the coil support materials and
insulation constrain the maximum achievable B.
In this section, first an analytic approximation is
deduced and then a specific finite elements calculation is
performed.
Analytic Approximation of Stress
Let us approximate the stellarator coils as if they were
circular and uniformly distributed, in the toroidal direction,
in a monolithic support of thickness d=wa.
Figure 8, complemented with Fig. 5, illustrate the
notations.
Here f
out
denotes a radial force acting on the outboard
torus, and the field Bis inversely proportional to the major
radius R. Hence, df
out
=B
out
9IdL. After integration, we
obtain the average stress r
s
at section S:
rs¼p
l0
B2A
w
1
2þ2
pfA
arctan
n
A1
fA
!"#
ð12Þ
with
fA¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1n2
A2
s
Values of r
s
are plotted in Fig. 9as a function of the
plasma volume, for w= 0.5, n= 2 and A=6.
Equation (12) can be approximated for n= 2 and A[5
by
rs5
4
B2
wB2
wMPa½;Bin T½ ð13Þ
The maximum stress in the structure is r
max
=f
r
r
s,
where f
r
is a stress concentration factor. Finite element
calculations presented in the next section will show that
f
r
*2 – 3, depending on the type of stellarator.
Finite Element Calculation
A monolithic toroidal support external to the coils for the
QIP3 configuration (Fig. 7) was modelled in CATIA
(Fig. 10) for the sake of the finite element calculations.
This model is somewhat similar to the structures defined
for the ARIES-CS and UST_1 stellarators [72–74].
Fig. 8 Sketch for the analytical approximation
Fig. 9 Approximate average stress in the monolithic support for
w= 0.5 at section S
1
S
2
(Fig. 8), for the combinations of h
E,
b
lim
, and
minimum field for ignition presented in Fig. 1
Journal of Fusion Energy (2018) 37:275–290 281
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Loads due to the weight of the structure are not con-
sidered, and openings through the structure are neglected.
The central ring is modelled as a thin hexagon in order to
avoid impacting the calculation.
To fix the ideas, we set V=30m
3
,B= 9.8 T (see the
h
E
= 1.5, b
lim
= 5% case in Fig. 1), w=0.5 and a current of
1.6 MA in each coil. Under these conditions, 3600 ele-
ments of force on 144 coils were calculated by the CAS-
TELL code, introduced in the Finite Element Analysis
(FEA) module of CATIA and, applied on the support
structure. This model hinders the calculation of the stress in
the coils and intercoil insulation.
The resulting Von Mises stress is shown in Fig. 11.
The maximum stress in the monolithic support
(r
max
*600 MPa) is located at the inboard of the curved
section. Such value is 2.5 times higher than the result
(r
s
= 245 MPa) from Eq. (12), thus f
r
= 2.5.
Limitations and Discussion
This initial stress calculation does not tackle the insulation
stress, which remains for future detailed studies. High
strength insulation might be required.
The type of magnetic configuration changes the location
of the areas of maximum stress, i.e. [72], but the approach
of considering an averaged value r
s
and an stress con-
centration factor f
r
is still helpful.
Local adjustment or optimization of the thickness of the
structure could smooth stress and deformation on the full
structure.
In comparison to tokamaks, the larger aspect ratio of
stellarators decreases the forces in the inboard of the torus
[15] but the stress concentration factor in stellarators is
unfavourable. In spite of this, the maximum stress in the
monolithic support in i-ASTER resulted in similar levels to
the maximum stress in the coil support of a high field
tokamak like IGNITOR, *500 MPa, [18].
Definition of i-ASTER
i–ASTER is a high-field, small size and resistive-magnet
stellarator concept designed to reach ignition and study
burning plasmas. It is not a power plant prototype.
Mission and General Characteristics
i-ASTER aims at, rapidly and at modest cost, achieving
and understanding ignition, and studying alpha-particle
physics in ignited or near-ignited plasmas in a small fusion
device. This physics will be only partially investigated in
ITER. Thanks to its high power-density, i–ASTER could
serve the additional goal of testing and optimizing power
extraction systems (e.g. lithium-based) and studying the
plasma-wall interaction. Indirectly, it would complement
the stellarator research line in the high plasma pressure
range, advance technologies for high field fusion devices
and for the manufacturing of strong stellarator magnets.
Pulses are foreseen to last few seconds (much longer
than the energy confinement, alpha-particle slowing down
time and other timescales of interest) and to be repeated
with a low duty-cycle (*1000 pulses during a *10 year
lifetime). This approach reduces cost and neutronic issues
and still accomplishes the research mission stated above.
The duty-cycle is selected as an initial conservative value
from estimations on neutronics effects (i.e. on copper
resistivity) and, to achieve undemanding and slow cooling
of coils between pulses. The model to perform such
Fig. 10 Section view of the model for the FEA calculation
Fig. 11 Von Mises stress on the surface of the monolithic support
282 Journal of Fusion Energy (2018) 37:275–290
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estimations is an ignition-capable stellarator working at the
frontier of the physics and technological limits (minimum
size device) whose size is independent of the duty cycle.
The optimization of the device size based on the ratio of
number of pulses to facility cost is out of scope of the
present work.
In the spirit of reducing costs, and compatible with short
pulses, i-ASTER adopts resistive magnets, which are faster
to manufacture and simpler to operate than superconduct-
ing coils. Also, resistive magnets allow faster tests, avoid
cryostat, cryoplant and cooldown time, allocate extra space
for the plasma due to thinner shielding, simplify radioac-
tive waste recycling and, thus, moderate costs.
Main Design Features of i-ASTER
The three essential technological characteristics of
i–ASTER (massive resistive magnets, detachable periods
and Li divertors-walls) are described in the three subsec-
tions below. Subsequently, four complementary features
are mentioned.
Resistive Magnets
The external surface of the torus would be covered by a
thick layer or multilayer of copper, forming a series of
wide modular coils of variable cross section (Figs. 7and
10). The magnets would work adiabatically and a minimal
cooling system would remove the heat during the long time
between consecutive pulses. Aluminium is a backup
alternative to copper.
Only one section per period (Fig. 12) will contain small
ports for pumping, diagnostics and plasma heating. The
few ports will be small to maximize the toroidal and
poloidal coverage by the copper coils. This is made
possible by the fact that: (1) ECRH is expected to suffice to
reach ignition (‘‘Heating System’’ section). Several pow-
erful ECRH beams can be concentrated in a small region
(the port area). Thus, port space required for heating is
much reduced. (2) The need for pumping conduits during
the short pulse is almost avoided by the pumping effect of
liquid Li. All the chemical elements reaching the Li-wall,
except for the small amount of He generated during the
short pulse, react fast with liquid Li. Certainly, the whole
vacuum vessel acts as a powerful getter vacuum pump. (3)
Access for maintenance will be provided by detachable
stellarator sectors.
Detachable (Half)Periods
The periods or half-periods of the stellarator shall be easily
separated from adjacent periods for easy assembly and
maintenance. A (half)period would be removed from the
torus and immediately, a refurbished or new one would be
installed in order to minimize the maintenance downtime,
e.g. coil replacement, which will be critical in the future
power plants. Detachable periods were previously studied
for superconducting coils [75] and appear equally advan-
tageous and easier to realize for resistive magnets. The
accuracy of the re-assembly is a concern, but appropriate
remote maintenance techniques are highly accurate
[76,77]. For example, a circular central ring (Fig. 12)
would facilitate accurate reassembly. Larger twisted mod-
ular coils located at the vacuum vessel interfaces would
facilitate (dis)assembly and port allocation (Fig. 12). Large
modular coils were also planned in certain versions of
NCSX stellarator [78].
Lithium Divertor-Wall
An island divertor [11,79] and a first-wall almost entirely
covered with low-temperature (low recycling) liquid
lithium is planned for i-ASTER. The latter could be real-
ized by electrostatic/centrifugal spraying or by evaporation
[80] of lithium on a thin Capillary Porous System (CPS)
mesh (*0.2 mm thickness), similarly to the approach in
Ref. [62]. The mesh is locally heated during coating from
inside the vacuum vessel for proper Li deposition in the
capillary mesh. The CPS is located on a thick copper
substrate (the first wall) coated with a thin protective film
of a Li compatible material (W or Mo). The lithium in the
CPS is solid before the plasma discharge, at room tem-
perature or slightly higher, and it is liquefied after the pulse
start. For simplicity, heaters [62] are not planned in the
copper substrate. The copper substrate at the divertor target
areas would reach surface temperature 1200–1300 C (for
30 MW/m
2
thermal load and 2 s pulse), which would melt
Cu and volatilize Li. Dry (tungsten or CFC) divertor targets
Fig. 12 Concept of detachable (half)periods. The depicted large coils
and vacuum interfaces are only a reference to understand the concept
Journal of Fusion Energy (2018) 37:275–290 283
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enduring *30 MW/m
2
heat load [81,82] or, advanced
Li-based systems (jets of droplets, beams of droplets or
shower jets, ‘‘Power Load on Divertor Targets’’ section) to
dissipate a fraction of the heat load before reaching the Li-
CPS, would allow withstanding the intense heat load.
Pulse Length
Ignition conditions are to be maintained for few energy
confinement times s
E
(5 s
E
assumed here, comparable to
10 s
E
in FIRE [19]). The discharge is approximately 40
times longer than the alpha-particle slowing down time
[83], thus enabling the study of alpha particles and their
confinement.
Distance from Plasma to Coils
The copper coils are as thick (e= 1) and as far from the
Last Closed Flux Surface (LCFS) (n= 2) as reasonably
possible for a smooth plasma shape of the HSR3 type.
The distance from the LCFS to the internal surface of
the modular coils is (‘‘Power Dissipated in Resistive
Magnets’’ section)
D0¼naea=2a¼ne=21ðÞa
For n=2,A= 6 and V=30m
3
, this gives D’ = 0.3 m.
No space is allocated for the breeding blankets in
i–ASTER because breeding Tritium goes beyond the scope
of the device. Besides, D’ is too small to accommodate a
breeding blanket.
Heating System
The heating systems would only be used to ignite the
plasma. The frequency needed for ECRH heating at
B= 9.8 T, even at first harmonic, is unusually high
(275 GHz), which will increase the cost of the gyrotrons.
The cut-off density for O-mode ECRH is 9.2 910
20
m
-3
,
slightly lower than required (Fig. 2). This implies that the
plasma will be slightly overdense and will require the
excitation of Electron Bernstein Waves by means of
Ordinary-eXtraordinary-Bernstein mode conversion—a
technique well-established in the W7-AS stellarator and
elsewhere [84].
Essential Diagnostics Strategy
Detailed integration of plasma physics (e.g. magnetic
configuration, experimental plan) and technology (e.g. coil
design, access for diagnostics) shall be produced. In the
current initial design, two main ports (Fig. 12) are con-
sidered available for diagnostics (‘‘Resistive Magnets’’
section), which will be complemented with some small
ports. The diagnostics shall be designed and accommo-
dated in each port in a fully integrated manner, for
miniaturization. In a first stage, the diagnostics would be
committed to plasma operation and machine protection
(characterization of density and temperature profiles, neu-
tron diagnostics, monitoring Li divertor-wall conditions,
and the few plasma control diagnostics needed in a stel-
larator). In a 2
nd
stage, they would be mostly dedicated to
study energetic particle dynamics (e.g. alpha-particle
induced instabilities, alpha-particle losses and confine-
ment). The FIRE tokamak diagnostics [85] are a reference
for i-ASTER.
Size and Materials for i-ASTER.v1 According
to Limits
Values of h
E
= 1.5 and b
lim
= 5% are selected according to
available experimental and theoretical data, ‘‘Assumptions
and Governing Equations: Ignition Condition’’ sec-
tion. Those values were experimentally proven in W7-AS
and LHD respectively. The achievement of both values
simultaneously is predicted for the W7-X stellarator,
‘‘ Assumptions and Governing Equations: Ignition Condi-
tion’’ section.
Concerning divertors, and considering the hypothesis
and calculations in ‘‘Power Load on Divertor Targets’’
section, 30 MW/m
2
thermal power load on targets is
obtained for V=30m
3
, Fig. 4. This power load is the
practical limit for solid divertor targets [81,82,86], and a
prospect for advanced Li-based systems as divertor targets,
‘‘ Lithium Divertor-Wall’’ section.
A Zamak alloy (a commercial alloy of zinc, aluminium,
copper and magnesium) is selected for the coil support
structures. Zamak is non-ferromagnetic, easy to cast at low
temperature (400–420 C) in high-precision shapes, and
has high yield strength S
yield
= 360 MPa for the ‘Zamak 2’
alloy.
A strength safety factor of 1.5 accounts for uncertainties
on the materials, stress concentration due to the ports and
other uncertainties. From ‘‘Analytic Approximation of
Stress’’ section and Eq. (12) with w= 0.5, it is calculated
r
s
=240 MPa = S
yield-Zamak2
/1.5. However, r
max
(‘‘Finite
Element Calculation’’ section) exceeds S
yield–Zamak2
. For
Zamak 2 (E &85 GPa) the maximum displacement cal-
culated by finite element analysis is 11 mm for w= 0.5.
This displacement would be too large since coil positioning
and shapes should have a tolerance of 0.1% or better
[87,88], corresponding to about 4 mm for i-ASTER.
Therefore, it will be necessary to locally increase the
thickness of the structure to w[0.5 and to install a central
support ring so as to balance the stresses and reduce the
maximum displacement. These matters will be studied in
future development stages.
284 Journal of Fusion Energy (2018) 37:275–290
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From the estimations and calculations performed in the
paper, V=30m
3
is selected for i–ASTER.v1 as a lower
limit for the plasma volume, under the conditions and
materials taken into account. Indeed, the main constraining
parameters (power load on divertor targets, thickness of the
coils, electric power to fed the coils, stress in the coil
support, and maximum DTof insulation, see Table 2) are
demanding, but technically possible.
i-ASTER Specifications
Table 2summarises the specifications of i–ASTER.v1.
Discussion of the Specifications
Line-averaged plasma density up to n
line
=4910
20
m
-3
was achieved in the High Density H-mode in W7-AS [11]
Table 2 i–ASTER.v1 specifications
Element i–ASTER.v1 Ref.
V30 m
3
‘‘ Size and Materials for i-ASTER.v1 According to
Limits’’ section
B9.8 T ‘‘Assumptions and Governing Equations: Ignition
Condition’’ section and Fig. 1
R3.8 m
a0.63 m
A6‘‘Assumptions and Governing Equations: Ignition
Condition’’ section
Plasma surface 95 m
2
n
line
1.1 910
21
m
-3
‘‘ Density and Temperature Needed for Ignition, Fusion
Power’’ section and Fig. 2
T
0
14.6 keV ‘‘Density and Temperature Needed for Ignition, Fusion
Power’’
Fusion energy gain Q Q ?1
(ignition)
‘‘ Power Balance’’ section
Fusion power 1.4 GW ‘‘Density and Temperature Needed for Ignition, Fusion
Power’’ and Fig. 3
h
E
(ISS04) 1.5 ‘‘Size and Materials for i-ASTER.v1 According to
Limits‘‘ section
\b[5% ‘‘Size and Materials for i-ASTER.v1 According to
Limits’’ section
s
E
0.4 s
Pulse length 2 s 5 s
E
Load on divertor targets (50% improvement, factor 2
sweeping, 50% radiation)
30 MW/m
2
‘‘ Power Load on Divertor Targets’’ section and Fig. 4
Average neutron wall load 12 MW/m
2
‘‘ Power Load on Divertor Targets’’ section and Fig. 4
Relative magnet thickness e1‘‘Power Dissipated in Resistive Magnets’’ section
Weight of the copper magnet *1000 Ton
Current per coil (one turn/coil, 144 coils) 1.6 MA
Power consumed in the resistive copper coils *750 MW ‘‘Power Dissipated in Resistive Magnets’’ section and
Fig. 6
Total magnetic energy stored *4.6 GJ
Material of the monolithic support (initial selection) Zamak 2 ‘‘Size and Materials for i-ASTER.v1 According to
Limits’’ section
Relative thickness of monolithic coil support W0.5 ‘‘Size and Materials for i-ASTER.v1 According to
Limits’’ section
Ave. stress on coil support at S 240 MPa ‘‘Analytic Approximation of Stress’’ section and Fig. 9
Max. local stress on coil support (QIP3 configuration, uniform
W)
600 MPa ‘‘Finite Element Calculation’’
DT
max
copper coils *insulation, only Ohmic
(QIP3 *f
c
=5)
100 K ‘‘Current Density and Coil Temperature’’ section
D’ (distance LCFS-coil) 0.3 m ‘‘Distance from Plasma to Coils’’ section
Journal of Fusion Energy (2018) 37:275–290 285
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and a central plasma density of 10
21
m
-3
was reached in
LHD [89]. The feasibility of n
line
*10
21
m
-3
should be
experimentally proved, but, certainly, a high-field stel-
larator would favour high densities, according to the Sudo
limit [21].
i-ASTER considers reactor-relevant b(5%) and adopts a
high magnetic field B. As a result, the power density
(*b
2
B
4
) and the heat load on the divertor is high.
This will be an opportunity to test and enhance high power
extraction systems and plasma purity, for example, by
lithium-based systems.
The evaluation of intermediate Q regimes and implica-
tions on the results (e.g. different divertor load) is beyond
the scope of the present paper. These intermediate Q
regimes might occur if ignition or near-ignition could not
be achieved in i-ASTER.
The electric power required for the magnets is sub-
stantial, but appears tractable. For example, TFTR fly-
wheels provided up to 0.7 GW [90].
The use of steel would reduce the thickness of the
monolithic structure. Nonetheless, steel requires more
expensive casting and machining than Zamak. Alterna-
tively, laminated composite (S
yield
[1000 MPa) shaped on
additive manufacturing structures is envisaged, inspired by
Refs. [69,70].
Discussion on Neutronics
Neutron damage lower than 0.1 dpa is roughly estimated
for the most exposed copper of the coils after 10 years
lifetime (total of 1000 pulses, no shielding). This would
produce some Cu embrittlement, but minor resistivity
reduction and feasible insulation materials [91]. The esti-
mation is based on the ratio r
dpa-NLW
of dpa per full-power-
year (fpy) to the average neutron wall load (NWL), which
is calculated from data in Refs. [92,93] for ferritic–
martensitic steels, resulting r
dpa-NLW
*10 (dpa/fpy) /
(MW/m
2
). For the i-ASTER wall surface and total neutron
power, with duty cycle 6 x 10
-6
, ten years operation, peak
NWL twice the average NWL [92], and dpa’s in copper
60% higher than in ferritic-martensitic steel [94], it results
0.03 dpa.
Concerning the neutron heating (‘n-heat’) of coils, a first
approximation is obtained as: i) the DEMO n-heat at the
first wall for ferritic-martensitic steel is taken, 8 W/cm
3
[93], ii) n-heat for copper and iron are similar [95], iii)
scaling n-heat to the plasma surface and neutron power in
i-ASTER, with neutron shielding of 80%, resulting in
n-heat *14 W/cm
3
. For copper coil, an average
DT
aveNWL
*8C is calculated at the end of the 2 s pulse
(DT
peakNWL
*16 C).
Regarding the n-heat in the first-wall, following the
previous procedure, without shielding, it results DT
ave
*40 C(DT
peak
*80 C).
No major neutronics difficulties are envisioned, thanks
in part to the favourable high ratio of plasma surface to
plasma volume in the relatively large aspect-ratio and small
size i–ASTER.
Limitations and Discussion
Limitations
Different quasi-isodynamic magnetic configurations (QIP3,
HSR3) were utilized for the models. A definitive magnetic
configuration for i–ASTER is not yet decided and it will
have some impact on the resulting parameters. For exam-
ple, the magnetic configuration impacts the areas of stress
concentration (‘‘Limitations and Discussion’’ section) and
the current density factor (‘‘Current Density and Coil
Temperature’’ section).
Calculations by complex systems codes [25] have not
been carried out yet, and will be the subject of future work.
However, the rough estimates presented may be sufficient
for this initial stage of development.
It is unknown if the assumptions performed for the
estimation of the power load on divertor targets (large
wetted area, sweeping, 50% edge radiation) can be simul-
taneously achieved. Lowering bto *2.5% or less and
increasing Bcould still achieve ignition at lower divertor
loads.
The initial stress calculation does not tackle the insula-
tion stress. Also, the (small) ports have not been modelled.
The strength safety factor considered in the study may
cover the uncertainties. However, further calculations will
be required as the geometrical design advances.
Refined neutronics calculations are required to estimate
the neutron damage to coil insulation, activation and
damage on copper, and neutron heating of first wall and
coils.
Discussion
A quasi-isodynamic configuration was assumed for
i–ASTER in order to advance the design. Currently, there
is no universally accepted criterion to decide a best type of
quasi-symmetry, and it advises against an early decision on
the definitive i–ASTER magnetic configuration.
Optimization of stellarator magnetic configurations
continues worldwide [25,96–98] and new stellarator con-
cepts continue to emerge [96,99]. Hence, future versions
of i-ASTER might have larger A, which usually gives
higher beta limit b
lim
(‘‘Assumptions and Governing
286 Journal of Fusion Energy (2018) 37:275–290
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Equations: Ignition Condition’’ section, [30]), or higher
number of field periods. There is not any property (number
of periods, type of quasi-symmetry) of the selected QIP3
and HSR3 configurations that makes them unique for the
mission and engineering approach of i–ASTER. Only, the
intended small size of the device favours moderate aspect
ratio.
Power extraction systems (e.g. solid divertor targets,
flowing liquid metals) are critical for the attractiveness of
fusion as a competitive energy source [47]. The liquid–
metal option has been favoured for i-ASTER due to its high
theoretical potential, e.g. high speed metal droplet beams
[64] or molten tin shower jets [65], despite the compara-
tively limited level of development.
The massive resistive coils of variable cross-section
involve new calculation methodologies that have only been
initiated and represent a novel field of study.
Resistive magnets may not be the best option for stel-
larator power plants. Nevertheless, the requirement of
simplification suggests this option for a first ignition
experimental device.
If it is reasonable to study high-field ignition-capable
tokamaks like IGNITOR and FIRE, it appears reasonable
to explore the potential of high-field stellarators of com-
parable size and magnetic field.
Summary and Conclusions
Wide ranges of physics and engineering parameters have
been explored, in search for the conditions enabling igni-
tion in a small-size, high-field stellarator experiment. The
magnets are resistive to contain construction costs.
Specifically, massive copper coils of variable cross-section
are envisaged to reach high fields with feasible power
supplies. A monolithic toroidal coil support structure,
external to the coils, is also proposed. Analytic expressions
and finite-element calculations were produced for the
power consumed in the magnets and the stress in the
monolithic support. Plots were generated for all the rele-
vant parameters, under a variety of assumptions on the
energy confinement enhancement factor h
E
, stability beta
limit b
lim
and plasma volume. From this parametric study,
a preliminary conceptual design of a high-field ignition-
capable experimental stellarator (i–ASTER) has emerged,
based on a quasi-isodynamic magnetic configuration. i–
ASTER presents three distinctive features: massive resis-
tive coils of variable cross-section, detachable periods and
lithium-coated walls and divertors. i–ASTER.v1 has a
plasma volume of 30 m
3
and an average magnetic field
B*10 T on axis, comparable with the IGNITOR and
FIRE tokamak designs.
No unsurmountable difficulties have been found for this
high-field pulsed stellarator approach to ignition experi-
ments. The main concern is the possibly intractable power
load on divertor targets and subsequent impurity influx.
This could be tackled by lowering the operating band
using lithium-based power extraction systems. The con-
siderable radial thickness of the magnets is also a concern,
but additive manufacturing could lessen this issue.
This work is undertaken in order to fill a gap in the
knowledge of high-field ignition-capable fusion devices of
the stellarator type, which were significantly studied for
tokamaks in the IGNITOR and FIRE tokamak concepts,
and proposes a high-field resistive-magnet stellarator path
towards the study of burning plasmas.
The definition and detailed calculation of the magnetic
configuration and the 3D coil structure will be the subject
of future work. Additive manufacturing of the coil support
structure will also be further investigated. Detailed neu-
tronics and more detailed mechanical and electric calcu-
lations will be performed in the next development stages.
Acknowledgements The authors are grateful to M.I. Mikhailov, J.
Nu
¨hrenberg et al. [30] for supplying the QIP3 magnetic configuration,
to A. Werner, J. Baldzuhn and J. Geiger for providing the coil defi-
nition of HSR3, and to E. Blanco and K.J. McCarthy for proof
reading. The first author acknowledges J.A. Romero and J.A. Ferreira
for longstanding discussions about fusion and stellarators. The work is
partially funded by the Spanish ‘Ministry of Economy and Compet-
itiveness’ under the grant number ENE2015-64981-R (MINECO /
FEDER, EU). This work is partly supported by the US Department of
Energy under Contract DE-AC05-00OR22725 with UT-Battelle, LLC
and the US DOE.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creative
commons.org/licenses/by/4.0/), which permits unrestricted use, dis-
tribution, and reproduction in any medium, provided you give
appropriate credit to the original author(s) and the source, provide a
link to the Creative Commons license, and indicate if changes were
made.
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