1. UNIVERSITY OF SOUTHAMPTON
FACULTY OF PHYSICAL AND APPLIED SCIENCES
Physics
An investigation into the main parameters affecting the performance of
Inertial Confinement Fusion
by
Ben Williams
Student ID: 24691925
A final report submitted for continuation towards a MPhys
Keywords:
Inertial Confinement Fusion, Gain, Rayleigh-Taylor Instability
Supervisor: Prof. C.T Sachrajda
April 19, 2015
2. UNIVERSITY OF SOUTHAMPTON
ABSTRACT
FACULTY OF PHYSICAL AND APPLIED SCIENCES
Physics
A final report submitted for continuation towards a Mphys
AN INVESTIGATION INTO THE MAIN PARAMETERS AFFECTING THE PERFORMANCE
OF INERTIAL CONFINEMENT FUSION
by Ben Williams
Inertial Confinement Fusion is the process of using a pulse of radiation to rapidly heat a small
capsule containing fusion material. The outer layer explodes outwards and the resultant force
rapidly compresses the fuel until fusion reactions occur at the core. This project explores
in-depth some of the main parameters that determine the performance of this experimental
method of terrestrial fusion. Two main parameters that were explored are: a) Gain, which in
its simplest terms is the ratio of energy released to the energy delivered, and b) Hydrodynamic
Instabilities, which describe how the plasma flows in the capsule during and after irradia-
tion, a major component that affects capsule performance (seeded from imperfections in the
smoothness of the capsule’s surface). In addition to studying the relationship between these
parameters, the constraints and limits will be calculated and known plots will be replicated
and analysed to show the accuracy of my derivations and research.
4. List of Figures
1.1 4 main stages of ICF (Anatomy of the Universe, 2011) . . . . . . . . . . . . . . 2
1.2 Internal structure of fuel pellet (Brumfiel, 2012) . . . . . . . . . . . . . . . . . 2
1.3 Geometrics of polar and symmetric drive (Hecht, 2013) . . . . . . . . . . . . . 3
1.4 Fuel pellet inside hohlraum Glenzer et al. (2012) . . . . . . . . . . . . . . . . . 4
1.5 Diagram of efficiency of indirect drive LLNL (2013) . . . . . . . . . . . . . . . 5
2.1 Implosion diagram of typical ICF capsule. Notice that shell is particularly thin
in interval 17 < t < 23 ns (Atzeni and Meyer-ter Vehn, 2004, p. 50). . . . . . . 7
2.2 Plot showing the f(x) and its approximations for low and high ablation regimes. 8
3.1 ICF energyy balance (Atzeni and Meyer-ter Vehn, 2004, p. 42) . . . . . . . . . 11
4.1 Rayleigh-Taylor unstable interfaces between fluids of different densities (Atzeni
and Meyer-ter Vehn, 2004, p. 238) . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Progression of RTI with time (Shengtai and Hui, 2006) . . . . . . . . . . . . . . 17
4.3 Plot showing how the growth factor at the outer surface varies with mode
number, plotted for several values of Aif . . . . . . . . . . . . . . . . . . . . . 23
4.4 Plot showing how the growth factor from feedthrough varies with mode num-
ber, plotted for several values of Aif . . . . . . . . . . . . . . . . . . . . . . . . 24
4.5 Plot showing how the total growth factor varies with mode number, plotted for
several values of Aif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1 Graph showing curves of ignition power (for indirect drive) as a function of
ignition energy for different values of Aif , also shown on graph is the Pd for
the maximum allowed temperature. . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Graph showing the maximum growth factor from the most unstable mode ver-
sus Aif . Included on the graph are typical values for Aif and their respective
(GT )max. A trend line calculated using data from the most linear section of
the graph (20 < Aif < 80) is also shown. . . . . . . . . . . . . . . . . . . . . . 30
5.3 Surface roughness versus percentage of fuel-ablator mixing for a number of
different Aif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.4 This graph shows two different shaped laser pulses low-foot and high-foot and
their associated growth factor depending on the mode number (Raman et al.,
2014). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.5 Comparison of the high-foot and low-foot drives (Raman et al., 2014). . . . . . 33
iv
5. Acknowledgements
I would like to thank Professor C.T Sachrajda for his help, supervision and guidance on this
project. I would also like to thank Emma Ditter, my project partner, who’s determination and
hard work made this project possible.
v
6. Nomenclature
Symbol Description Units
Aif In-Flight-Aspect-Ratio -
R Average shell radius m
∆R Outer shell thickness during implosion m
uimp Implosion velocity ms-1
uex Exhaust velocity ms-1
ua Ablation velocity ms-1
Th/r Hotspot/Radiation temperature keV
ρh/c Hotspot/Cold fuel density Tbar
ph/c Hotspot/Cold fuel pressure Pa
Rh/c/f Hotspot/Cold/Fuel radius m
Mh/c/f Hotspot/Cold/Fuel mass mg
Ed/h/f Driver/Hotspot/Fuel energy MJ
Efus Energy released from fusion MJ
ωh/c Internal/Additional internal energy per electron eV
G Target energy gain -
Gf Fuel energy gain -
qDT Fusion energy released per unit mass burnt for single DT reaction 3.3 × 1011 Jg-1
Φ Burn fraction -
HB Burn parameter 7 gcm-2
η Overall coupling efficiency -
α Isentrope parameter -
k Wavenumber m-1
λ Laser light wavelength m
σ Instability growth rate s-1
At Atwood number -
a Acceleration ms-1
L Characteristic density scale length -
l Spherical mode number -
ζ Amplitude of perturbation nm
G Growth factor -
vi
7. Chapter 1
Introduction
1.1 Basic Fusion Principles
Nuclear Fusion is the process of combining light atomic nuclei to form a heavier nucleus, this
reaction releases energy because of Einstein’s mass-energy relationship, Equation 1.1,
Q = (
i=0
mi −
f=0
mf )c2
(1.1)
which states that the mass of the final products is less than the masses of the constituents
(World Nuclear Association, 2013; Atzeni and Meyer-ter Vehn, 2004). Nuclei naturally repel
each other, to achieve fusion this electrostatic repulsion must be overcome. The main issues
which govern terrestrial fusion are the heating of the reactants to fusion temperatures, around
150 million degrees, and the sustained confinement of the plasma, gas of highly ionised parti-
cles (Pitts et al., 2006). There are two main methods of plasma confinement which are currently
under investigation for achieving fusion, magnetic (MC) and inertial (IC) confinement. This pa-
per refers to the latter method, however it is worth noting that there are several experiments
utilizing MC and there has been significant progress in that sector over the last decade (Cowley,
2010).
1.2 Inertial Confinement Fusion
The basic principle of inertial confinement fusion (ICF) is the external irradiation of a static fuel
pellet, typically comprised of Deuterium and Tritium, which ignites fusion in the pellet’s core.
A description of this process is described below, and shown diagrammatically in Figure 1.1,
(Lindl et al., 2004).
• Pellet is radiated by an external source, conventionally X-rays or lasers
1
8. 2 Chapter 1 Introduction
• The Ablator layer (outer layer) vaporises, ionises and explodes outwards
• From Newton’s third law and momentum conservation there is a reactant force which
is directed inwards compressing and heating the core leading to a central hotspot (a
small area that has a higher temperature compared with its surroundings). This takes
place because the imploding material in the hotspot stagnates and the kinetic energy is
converted to internal energy
• If the hotspot temperature and density are sufficient, then the nuclei in the hotspot will
overcome the coulomb barrier and fusion will occur
• A chain reaction, where fusing nuclei transfer energy to other nuclei causing them to
fuse thus causing burn waves to propagate radially outwards igniting the fuel, is able to
happen if the ignition values of the hotspot are achieved
• Overall only 10% − 20% of the fuel is burnt
Figure 1.1: 4 main stages of ICF (Anatomy of the Universe, 2011)
Fuel Pellet
The fuel pellet is comprised of an outer shell of typically plastic ablator, an inside shell of
solid Deuterium-Tritium (DT) and a central cavity containing DT gas, shown in Figure 1.2.
The solid DT layer (DT-ice) compresses the DT gas during an implosion forming the hotspot
(McCrory et al., 2005). A hollow shell is used as opposed to a solid fuel pellet because the fuel
Figure 1.2: Internal structure of fuel pellet (Brumfiel, 2012)
9. Chapter 1 Introduction 3
can be accelerated over greater distances, therefore to reach the required implosion velocity for
fusion a lower driving pressure is needed, 350kms−1. Also, isentropic compression, required
for direct drive, is easier to perform on a hollow shell because the shock waves are similar
to plane waves in structure when they reach the centre, compared to a solid pellet where the
shock waves converge almost adiabatically (Atzeni and Meyer-ter Vehn, 2004, p. 53).
An equimolar DT mixture is the most common fuel used because it has the highest specific
yield and also the lowest ignition temperature, thus requiring less energy to be provided to
initiate fusion. There are other fuels currently being explored but this project only refers to the
DT mixture, however it is worth noting that only light elements are used in fusion research
because of the higher possibility that quantum tunnelling can occur (Nobelprize.org, 2013).
Main Experimental Methods
There are two main experimental methods within ICF these are Direct and Indirect drive, the
differences between these drive methods occur predominately with the first stage of ICF, the
irradiation of the pellet. The principles which are explored in this project are relevant to both
drives and thus a brief description of both will be given.
Direct Drive
Direct drive uses laser light as the external source of radiation, laser beams directly heat the
surface of the fuel pellet (the outer shell of the pellet) and cause its ablation (McCrory et al.,
2005). The pellet is irradiated at multiple points on its surface to provide uniform radiation, this
is achieved by splitting the laser into a large number of overlapping beams which are focused
by lenses, shown below in Figure 1.3. The beams are set up to provide large f-numbers, ratio of
Figure 1.3: Geometrics of polar and symmetric drive (Hecht, 2013)
focal length to diameter of aperture, which deliver maximum energy transfer to the ablation
layer of the pellet. However one disadvantage to direct drive is that as the pellet explodes a
significant proportion of the laser light misses the pellet and is instead refracted in the plasma
corona (Atzeni and Meyer-ter Vehn, 2004, p. 49). The irradiation must take place in a time
interval in which mass inertia keeps the burning fuel together, this is the time it takes for a
10. 4 Chapter 1 Introduction
sound wave to travel from the surface of the pellet to the centre. This gives a confinement
time (τ) of typically 0.1ns, in this case confinement time is defined as the time the plasma is
maintained above ignition conditions (Atzeni and Meyer-ter Vehn, 2004). To provide maximum
energy release the fuel is compressed quickly and isentropically, this is described by the first
law of thermodynamics. In practise a fast and nearly isentropic compression is employed by
‘shaping’ the laser pulse, so that the pulse power is increased in stages, which creates a series
of superimposed shocks with each subsequent shock having a greater speed than the last, so
that they all coalesce at a single destination at the same time (Atzeni and Meyer-ter Vehn, 2004,
p. 52).
Direct drive relies on a homogeneous spread of energy over the whole spherical surface of the
fuel pellet. This is the main issue for this type of experimental set up because even a slight
variation from equilibrium will produce instabilities, such as Rayleigh-Taylor and Richtmyer-
Meshkov (RT and RM) described in Section 4, which can significantly lower capsule perfor-
mance as they degrade the ablator’s ability to compress fusion fuel through mixing the abla-
tor layer material with the fuel. In order to achieve the desired energy spread complex op-
tical chains which drastically reduce the efficiency are required. Approximately 20% of the
laser’s energy is consumed to achieve a homogenous spread (Atzeni and Meyer-ter Vehn (2004,
p. 65), Raman et al. (2014)).
Indirect Drive
The key feature of the indirect drive is that instead of directly irradiating the fuel pellet with
laser light, the laser light is converted into X-rays which then interact with the pellet, hence
the name indirect drive. This is accomplished by placing the pellet inside a cylinder known as
a hohlraum, see Figure 1.4. The hohlraum is designed to absorb the laser light, which enters
Figure 1.4: Fuel pellet inside hohlraum Glenzer et al. (2012)
at the poles, and convert it to X-rays. The angles at which the laser light enters the hohlraum
are adjusted to give geometric and uniform irradiance in addition to maximum energy trans-
fer. As gold has the highest conversion efficiency of laser light to X-rays, η ≈ 80 − 90% at
approximately 1013Wcm−3, this is the material that typically coats the hohlraum (Atzeni and
Meyer-ter Vehn, 2004; Atzeni, 2009).
11. Chapter 1 Introduction 5
Indirect drives has several advantages over direct drive, however the principal advantage is
that indirect drive produces uniform isotropic radiation. From this RT instabilities are limited,
which leads to an overall reduction in instabilities and consequently a significant increase in
capsule performance. Nevertheless, the decrease in instabilities comes at a high price, a growth
in energy loss from the heating of the hohlraum walls, Figure 1.5 shows a rough diagrammatic
representation of the efficiency of indirect drive (Atzeni and Meyer-ter Vehn, 2004).
Figure 1.5: Diagram of efficiency of indirect drive LLNL (2013)
12. Chapter 2
Prerequisite Principles
2.1 The In-Flight-Aspect-Ratio
During an ICF implosion the laser pulse is shaped in time, this happens for several reasons
one of which is to keep the fuel compressed on a low adiabat (other reasons will be discussed
in Section 5.4), this is required to limit the isentrope parameter (Section 3.2) from becoming
too high and decreasing the gain (Rosen and Lindl, 1984). This pulse shaping causes the shell
to become thinner and denser than it was initially (shown in Figure 2.1), thus it is useful to
be able to compare this thickness with the outer radius of the capsule. A fundamental fusion
parameter is the initial aspect ratio of an ICF capsule which is defined as the ratio of the initial
outer radius to the initial thickness of the outer shell Ar0 = R0/∆R0. From this the ‘in-flight’
aspect-ratio can be defined as the maximum value of the ratio between the average shell radius
(R) and outer shell thickness during implosion (∆R):
Aif =
R
∆R
(2.1)
By defining Aif we can immediately see that because of laser pulsing Aif Ar0, as during
implosion the ∆R factor reduces drastically therefore increasing the aspect ratio. The in-flight-
aspect-ratio is a very important factor in achieving ignition and consequently a significant
section of this paper is devoted to explaining the limits and consequences of Aif .
2.2 The Rocket Model
The rocket model is used to describe global features of the implosion process of ICF such as the
hydrodynamic efficiency and implosion velocity. This paper will follow the partial derivation
given in Atzeni and Meyer-ter Vehn (2004, p. 230-235) and Lindl (1995, p. 27-30).
6
13. Chapter 2 Prerequisite Principles 7
Figure 2.1: Implosion diagram of typical ICF capsule. Notice that shell is particularly
thin in interval 17 < t < 23 ns (Atzeni and Meyer-ter Vehn, 2004, p. 50).
First consider a rocket, which is ablation-driven, and has a mass M(t) which decreases over
time dt as the exhaust expels a mass dM. Assume the rocket is ideal, such that the exhaust is
continually heated so that it remains nearly isothermal during its expansion. From momentum
conservation we have d(Mu) − u1dM = 0, where the first term is the change in rocket
momentum and the second term is the momentum of the exhaust, u and u1 are the velocities
viewed in an inertial frame. This can then be rewritten as:
M
du
dt
= uex
dM
dt
(2.2)
uex is the exhaust velocity relative to the rocket given by u1 − u.
Next we have to consider the ablative implosion of a thin spherical shell in terms of a spherical
rocket. Therefore starting at a time t a shell of mass M(t) and radius R(t) implodes with a
velocity u = dR/dt, the surface ablates with a rate ˙ma which is given by:
dM
dt
= −4πR2
˙ma (2.3)
Using Equation 2.2 and the equation for ablation pressure ( ˙mauex) the above equation can be
recast as:
M
du
dt
= −4πR2
pa (2.4)
Equation 2.3 and Equation 2.4 are not really useful in their current forms, to make use of these
equations we integrate both with respect to time to give:
u = uex ln
M
M0
(2.5)
1 −
M
M0
1 − ln
M
M0
=
3
1 −
R
R0
3
(2.6)
14. 8 Chapter 2 Prerequisite Principles
Here we have introduced the initial mass M0 = 4πR2
0ρ0∆R0, where the density of the shell is
ρ0, the radius is R0 and the thickness is ∆R0. We have also introduced the implosion parameter
(Equation 2.7) into Equation 2.6 which characterises the implosion.
=
˙ma
uexρ0
R0
∆R0
(2.7)
Implosion velocity
We now have the basis for defining the Implosion velocity from the rocket model, using Equa-
tion 2.5 and defining the imploding mass as M1 = M(R = 0) the final implosion velocity is
described by Equation 2.8.
uimp = uexln
M0
M1
(2.8)
At this point it is useful to relate the implosion velocity to the in-flight-aspect-ratio by estimat-
ing that Aif = (ρaR0)/(ρ0∆R0) to give = uaAif /eex. Here we have introduced the ablation
velocity (ua = ˙ma/ρa) which will be explored later on. A function (Equation 2.9) of the implo-
sion velocity and exhaust velocity can now be defined by combining Equation 2.5 and Equa-
tion 2.6. This function can be approximated for two different regimes the high- (0.8 < x < 3)
and low- (x 1) ablation regimes:
f(x) = 1 − (1 + x)exp(−x) ≈
x2 Low-ablation
0.28x High ablation
(2.9)
Figure 2.2: Plot showing the f(x) and its approximations for low and high ablation
regimes.
Thus from this a rough approximation for implosion velocity can be deduced, for direct (low-
ablation regime) and indirect drive (high-ablation regime). The important thing to notice here
is that for direct drive the implosion velocity is proportional to the square root of the in-flight-
aspect-ratio, but for indirect drive it is linearly proportional to the in-flight-aspect-ratio, this
15. Chapter 2 Prerequisite Principles 9
significance of this will be investigated in a later section.
uimp ≈
uauexAif
3
Direct drive
uaAif Indirect drive
(2.10)
2.3 Lawson Critiera
The Lawson Criterion are a set of three variables that determine the conditions for general ther-
monuclear ignition; plasma density (ρ), temperature (T) and confinement time (τ). These con-
ditions were originally derived for use in MCF where all 3 variables can be directly measured.
However, in ICF ρ and τ cannot be measured directly, the traditional procedure to circumvent
this issue is to restrict the Lawson criterion to only describe the hotspot, this is achieved by
replacing the plasma pressure (p) with the ideal gas equation of state (Equation 2.11), where
ρh is the hotspot mass density, Th is the hotspot temperature and mi is the DT average ion
mass (Atzeni and Meyer-ter Vehn (2004, p. 37), Zhou and Betti (2008)).
p =
2ρhTh
mi
(2.11)
This leads to the hot-spot ignition condition (which is valid for hotspot temperatures between
5 and 15kev):1
ρhRhTh > 6
ρh
ρc
0.5
g cm−2
keV (2.12)
This method has two issues a) τ is incorrect as it does not take into account the cold shell and
b) ρhRh, the hotspot areal density cannot be experimentally measured.
Thus a more accurate method, using a dynamic ignition model that relates the hotspot stag-
nation properties to those of the shell, has been presented by Zhou and Betti (2008). However
for the purpose of this paper the above static model for Lawson criteria will be sufficient.
2.4 Plasma Instabilities
Plasma instabilities play a significant role in the performance of ICF. They pose a limit on
the maximum allowed radiation temperature, which in turn places a limit on the maximum
ablation pressure. This culminates in a limiting factor on the maximum power and energy the
lasers can provide to the fusion material (Atzeni and Meyer-ter Vehn, 2004, p. 122).
In ICF, plasma can be treated as a fluid (and thus analysed using magnetohydrodynamics)
and therefore its instabilities can be split into two main categories, hydrodynamic and kinetic.
1
This has been confirmed by numerical simulations and experimental data, the analytical derivation is given
in Atzeni and Meyer-ter Vehn (2004, p. 91).
16. 10 Chapter 2 Prerequisite Principles
Consequently the hydrodynamic instabilities explored in this paper can also be adjusted to de-
scribe some of the instability growth in plasmas (Lindl, 1995). Nearly all laser plasma instability
growth rates scale by
√
Iλ2, which is the reason why laser wavelengths are preferably kept
short in ICF (Pfalzner, 2006). Although this project does not focus on laser plasma interactions,
it is still useful to have an understanding of basic plasma instability growth in order to explore
the upper limits and define the maximum gain.
17. Chapter 3
Gain
General fusion energy gain is described as the ratio of fusion power produced (in a fusion re-
actor) to the power required to confine the plasma at ignition temperatures. Gain is a very
important factor when considering ICF in terms of energy production, Figure 3.1 shows the
energy balance for an ICF reactor. From Atzeni and Meyer-ter Vehn (2004, p. 42) is it calcu-
Figure 3.1: ICF energyy balance (Atzeni and Meyer-ter Vehn, 2004, p. 42)
lated that the gain required for power production is 30 − 100, the gain from uniform heating
at an ignition temperature of 5keV can be estimated at ≈ 20 which is too low for Inertial
Fusion Energy (IFE).1 This is part of the reason why only a small portion of the fuel is ig-
nited (the hotspot), which then through a propagating burn wave, ignites the reservoir of cold
compressed fuel (the solid DT layer) (Atzeni and Meyer-ter Vehn, 2004, p. 44).
1
Calculated by dividing the fusion energy released by a DT reaction (17.6Mev) by the thermal energy of two
ions and two electrons at 5keV (=30keV), and then multiplying this by a burn efficiency of 0.3 and a beam-to-fuel
coupling efficiency of 0.1.
11
18. 12 Chapter 3 Gain
3.1 Definition
Target energy gain is defined as (Atzeni and Meyer-ter Vehn, 2004, p. 102):
G =
Efus
Ed
=
qDT Mf Φ
Ed
(3.1)
Where:
• qDT is the fusion energy released per unit mass burnt for a single DT reaction (3.3×1011
J/g).
• Φ is the burn fraction (or burn efficiency), the ratio between the total number of fusion
reactions and the number of DT pairs initially present in the plasma volume. It can be
approximated as Hf /(HB + Hf ).
– where HB is the burn parameter and takes the value 7g/cm2 for DT fuel.
– Hf = Hh + Hc, where Hc = ρc(Rf − Rh) and Hh = ρhRh.
– ρc/h is the density of the cold and hot fuel respectively.
– Rf/h is the radius of the fuel and the hotspot respectively.
• Ed is the driver energy, related to the fuel energy at ignition (Ef ) by: η = Ef /Ed, where
η is the overall coupling efficiency.
• Mf is the fuel mass, which is the sum of the mass of the hotspot (Mh) and the mass of
the cold fuel (Mc) (Equation 3.2).
Mf = Mh + Mc =
4π
3
ρhR3
h + ρc(R3
f − R3
h) (3.2)
We can also define the fuel energy gain as Gf = Efus/Ef , which is related to the target gain
by:
G(Ed) = ηGf (ηEd) (3.3)
This section explores the two main ICF assemblies, the Isobaric and Isochoric configurations.
In addition to deriving the target and fuel gains we investigate the limiting gain for the target
and fuel cases. The limiting gain is the maximum gain achievable for a specific driver energy.
3.2 Isobaric Gain Derivation
The Isobaric compression is the ICF assembly utilised in direct drive implosions, this paper
follows the derivation given in Atzeni and Meyer-ter Vehn (2004, p. 111) and is known as
the ‘Hot spot ignition model for isobaric compression’. The compression takes place under
constant pressure such that the pressure of the cold fuel is at the same pressure as the hotspot
(p = ph = pc). Near maximum gain the following inequalities are true:
19. Chapter 3 Gain 13
• Mh Mf , only a tiny amount of fuel contributes to the hotspot.
• Hh Hf ≈ HB, this arises from the assumption that Mf Mc and Hf Hc
Therefore the burn fraction can be approximated by:
Φ =
1
2
Hc
HB
(3.4)
From this, Equation 3.1, in terms of fuel energy can be estimated as:
Gf
1
2
qDT Mc
Ef
Hc
HB
(3.5)
Next the cold fuel mass and burn parameter for the cold fuel need to be expressed in terms of
pressure and fuel energy. This requires the following relations:
ρc = (αAdeg)−0.6
p0.6
c (3.6)
Here we have introduced the degenerate fuel and isentrope parameter, these are fundamental
features in ICF they refer to extremely high compression involving Fermi-Degenerate fuel. α
is the isentrope parameter and is defined as the ratio of the pressure of a fuel (as a function
of density and temperature) divided by the degenerate pressure of the fuel as a (function of
density), it measures fuel entropy and is constant during isentropic compression (Atzeni and
Meyer-ter Vehn, 2004, p. 52). Adeg is a constant which depends on the composition of the
fuel and is 2.17 × 1012 (erg/g)/(g/cm3)2/3 for equimolar DT composition. Using the ideal gas
equation and the condition for isobarity we get the following relation for fuel and hotspot
energy for a mono-atomic gas:
Eh =
3
2
pVh = 2πpRh
3
Ef =
3
2
pVf = 2πpRf
3
(3.7)
Finally by using the relation Ef /Eh = (Rf /Rh)3 and combining the above equations leads
to:
Mc = ρc(Vf − Vh) =
p
αAdeg
0.6 2Ef
3p
1 −
Rh
Rf
3
(3.8)
Hc = ρc(Rf − Rh) =
p
αAdeg
0.6 Ef
2πp
1/3
1 −
Rh
Rf
(3.9)
We now plug these equations into the main equation for fuel gain (Equation 3.5) and introduce
the parameter x for convenience, which is Rh/Rf which from Equation 3.7 is also equal to
(Eh/Ef )1/3. This results in the fuel gain, G(Ef , x, p, α, HB, Adeg, qDT ) (Equation 3.10).
Gf =
qDT
3H
1/2
B p4/15
(1 − x3)
√
1 − x
(αAdeg)9/10
Ef
2π
1/6
(3.10)
20. 14 Chapter 3 Gain
The next step is to rewrite the pressure in terms of the fuel energy, this is achieved by us-
ing Equation 3.7 and introducing a final parameter FDT = phRh which is calculated from the
Lawson ignition condition (Equation 2.12, HhTh 2(g/cm2)keV) and has the value 15Tbarµm
for Th = 8keV and Hh = 0.25g/cm2. Thus pressure can be written as:
p = 2π
F3
DT
Ef x3
1/2
(3.11)
The fuel gain is now given by:
Gf = AG
Ef
α3
0.3
f(x) (3.12)
Where AG and f(x) are defined by:
AG =
qDT
3(2π)3/10H
1/2
B A
9/10
deg F
2/5
DT
f(x) = x2/5
(1 − x3
)
√
1 − x (3.13)
The limiting fuel gain is found by maximising f(x) this occurs at x∗ 0.3485 and gives
f(x∗) = 0.507.
G∗
f 6610
Ef
α3
3/10
f
−2/5
FDT
fqDT f
−1/2
HB
(3.14)
Ef is given in megajoules and fFDT
, fqDT and fHB
are variations of the fixed parameters
around their reference values ( ˆFDT , ˆqDT , ˆHB). Finally the limiting target gain is given by
combining Equation 3.14 and Equation 3.3.
G∗
6610η
ηEd
α3
3/10
f
−2/5
FDT
fqDT f
−1/2
HB
(3.15)
3.3 Isochoric Gain Derivation
We now follow the partial derivation given by Kidder (1976) for the Isochoric assembly, in
isochoric compression uniform density is assumed instead of uniform pressure, ρ = ρh = ρc.
In this case the fuel gain is given by:
Gf =
ΦqDT
ωc + ωh
(3.16)
• ωc(eV ) = 3αε2/3 (for α ≥ 1) and is the internal energy (per electron) of the fuel when
compressed ε-fold times normal solid density (ρ0 =0.2g/cm3) and α is the isentrope
parameter.
• ωh(eV ) = 3x3Th is the addtional internal energy (per electron) due to the hotspot,
x = Rh/Rf from the previous section.
21. Chapter 3 Gain 15
• Φ is the burn fraction from Section 3.1.
ε2/3 can also be given by Hf /H0, with H0 = (3Mf ρ2
0/4π)1/3 which is the burn parameter
‘ρR’ fuel would have at normal solid density (ρ0). Thus the isochoric fuel gain can be written
as:
Gf =
qDT Hf
3(HB + Hf )
α
Hf
H0
+ Th
Hh
Hf
3
−1
(3.17)
This equation transforms into the following equation from Atzeni and Meyer-ter Vehn (2004,
p. 123) Equation 3.18.2
G∗
f = 0.0828
qDT
H
1/2
B A
7/6
degFDT
2/9
H
4/9
h
Ef
α3
7/18
(3.18)
The Lawson ignition condition, Equation 2.12, in the isochoric case is given by ρhRhTh =
6(g/cm2)keV, this leads to the following conditions FDT = 46Tbarµm, Th = 12keV and Hh =
0.5g/cm2. Thus the limiting gain is simplified to:
G∗
f = 2.18 × 104 Ef
α3
7/18
(3.19)
2
The proof of this is outside the scope of this paper. However we have verified that both equations are equivalent.
22. Chapter 4
Hydrodynamic Instabilites
As mentioned in Section 1.2 hydrodynamic instabilities severely impact the overall perfor-
mance of ICF and can prevent fusion from occurring altogether. There are 3 types of hydrody-
namic instability that effect the stability of ICF:
• Rayleigh-Taylor, the main type of instability and the one that has the largest effect on
the performance of ICF.
• Richtmyer-Meshkov, occurs when a shock wave passes through an boundary between
two fluids, when the boundary is not flat. It’s relevance to ICF is that RMI can produce
seeds which are then amplified by RTI (Pfalzner, 2006).
• Kelvin-Helmholtz, occurs in a stratified fluid with the layers in shear motion, small sinu-
soidal perturbations grow exponentially in time. KHI plays a minor role in the non-linear
evolution of RTI bubbles (Atzeni and Meyer-ter Vehn, 2004, p. 243).
4.1 Rayleigh-Taylor Instability
A simple way to imagine Rayleigh-Taylor Instability (RTI) is by picturing two fluids separated
by a horizontal boundary, both fluids are subject to gravity (Figure 4.1). When ρ2 > ρ1 small
perturbations of the interface will grow in time. In a short period of time the heavier fluid will
sink down in spikes and the lighter fluid will rise in bubbles (Figure 4.2). This occurs because
“any exchange of position between two elements with equal volume of the two fluids leads to a
decrease of the potential energy of the system” (Atzeni and Meyer-ter Vehn, 2004, p. 238).
In ICF RTI occurs at two stages, the inwards acceleration phase and the implosion stagnation
phase. However instead of the fluids being subject to gravity (with the acceleration due to
gravity providing the driver for the instability growth) we consider two fluids in an acceler-
ated frame. In this case the denser fluid is the outer surface of the pellet during the inward
acceleration phase, and the inner surface at the implosion stagnation phase (Pfalzner, 2006).
16
23. Chapter 4 Hydrodynamic Instabilites 17
Figure 4.1: Rayleigh-Taylor unstable interfaces between fluids of different densi-
ties (Atzeni and Meyer-ter Vehn, 2004, p. 238)
Figure 4.2: Progression of RTI with time (Shengtai and Hui, 2006)
4.2 Classical RTI growth rate derivation
To derive the classical RTI growth rate we must consider incompressible fluids in which the
density may change in space, where RTI does not involve a sharp boundary between the fluids
but where the density changes gradually in the direction of the acceleration.
First we start by considering 2D perturbations in the x and z directions, with the accelera-
tion in the negative z-direction (−aez), the conservation equations (the continuity equations,
Equation 4.1, conservation of energy, charge conservation) can be written as:
∂ρ
∂t
+ · (ρu) = 0 (4.1)
ρ
∂ux
∂t
+ ρ ux
∂ux
∂x
+ uz
∂ux
∂z
= −
∂p
∂x
(4.2)
ρ
∂uz
∂t
+ ρ ux
∂uz
∂x
+ uz
∂uz
∂z
= −
∂p
∂z
− ρa (4.3)
∂ux
∂x
+
∂uz
∂z
= 0 (4.4)
24. 18 Chapter 4 Hydrodynamic Instabilites
By assuming small perturbations the above equations can be recast as (where tilde denotes
small perturbations):
∂˜ρ
∂t
+ ˜uz
dρ0
dz
= 0 (4.5)
ρ0
∂˜ux
∂t
= −
∂˜p
∂x
(4.6)
ρ0
∂˜uz
∂t
= −
∂˜p
∂z
− a˜ρ (4.7)
∂˜ux
∂x
+
∂˜uz
∂z
= 0 (4.8)
Using Fourier transforms in x and t on the above equations, makes the tilde quantities propor-
tional to eikxeσt, when combining into a single equation gives Equation 4.9.
k2
˜p = −σρ0
d˜uz
dz
(4.9)
The evolution equation for ˜uz is obtained by substituting the Fourier transformed version
of Equation 4.5 into Equation 4.7 and eliminating ˜p in the resulting equation and the one above.
d
dz
ρ0
d˜uz
dz
− ρ0k2
˜uz = −
k2
σ2
a
dρ0
dz
˜uz (4.10)
Velocity perturbations vanish at large distances from the interface, thus taking solutions when
˜uz → 0 and z → ±∞ and integrating over z from −∞ to ∞ gives the general growth rate
( Equation 4.11).
σ2
= k2
∞
−∞ a
dρ0
dz
˜u2
zdz
∞
−∞ ρ0(z)
d˜uz
dz
2
+ k2 ˜u2
z dz
(4.11)
Now starting from the case described in section 4.1, where we have two superimposed homo-
geneous fluids characterized by Equation 4.12 (which assumes a sharp interface between the
fluids) the classical RTI growth rate can be derived.
ρ0(z) =
ρ2 z > 0
ρ1 z < 0
(4.12)
The condition of continuity of the velocity component normal to the unperturbed boundary is
limz→0+ (˜uz) = limz→0− (˜uz) = ˜uz0. Both the fluids have uniform density with the derivative
of density with respect to time equal to 0, thus the evolution equation for ˜uz ( Equation 4.10),
when z = 0, is given by Equation 4.13.
˜uz =
˜uz0e−kz z 0
˜uz0ekz z 0
(4.13)
25. Chapter 4 Hydrodynamic Instabilites 19
The linear growth rate can be found by inserting Equation 4.10, the densities and the density
derivative (dρ/dz = δ(z)(ρ2 − ρ1)) into the general expression for growth rate Equation 4.11:
σRT = Atak (4.14)
Where At is the Atwood number and defined as (ρ2 − ρ1)/(ρ2 + ρ1). Finally we generalise
the classical RTI growth rate to the case of a stratified fluids. Thus instead of Equation 4.12 we
have:
ρ0(z) =
ρ1 +
ρ2 − ρ1
2
exp
2z
L
z 0
ρ2 −
ρ2 − ρ1
2
exp −
2z
L
z 0
(4.15)
L is a characteristic density scale length (ρ/ ρ). Inserting this equation into the equation for
general growth rate ( Equation 4.11) and assuming Equation 4.13 holds for the stratified fluids
then the growth rate is given by:
σRT =
Atak
1 + kL
(4.16)
4.3 Ablative RTI
RTI at a laser- or radiation-driven front (Ablative RTI) is very important in ICF as it reduces the
growth of RTI modes and even fully stabilizes short wavelength modes (Atzeni and Meyer-ter
Vehn, 2004, p. 257). A simplified treatment of ablative RTI can be derived from the observation
reported by Kilkenny et al. (1994) which shows that the eigenfunctions of classical RTI expo-
nentially grow in time and exponentially decay in space. The perturbations in ablative RTI also
grow, but because of ablation the interface changes position, moving into the material with an
ablation velocity ua. Thus, the effective perturbation growth is exp(σRT )exp(−kua∆t), from
this the classical ablative growth rate can be obtained:
σRT =
√
ak − kua (4.17)
The classical ablative growth rate can be generalised to fit a range of analytical solutions for
varying F and ν. Where F is the Froude number (F = ua
2/aL0) which affects the normalized
pressure profile and L0 is the ablation-front thickness. For large F the peak pressure is close to
the ablation front and as F decreases the peak pressure moves away from the ablation front. ν
is the effective power index for thermal conduction and comes from the thermal conductivity
power law (χ = χ0Tν) and is dependant on whether the energy is transported by electrons or
photons (Atzeni and Meyer-ter Vehn, 2004, p. 263).
ν
= 2.5 Electron heat diffusion
> 3 Radiative heat diffusion
< 2 Radiative effects in direct drive targets
(4.18)
26. 20 Chapter 4 Hydrodynamic Instabilites
The derivation of the generalised relationship is beyond the scope of this paper but can be found
in full in Betti et al. (1998) and was first proposed by Takabe in 1985. Known as the ‘Takabe
relation’ and the ‘generalised Takabe relation’ they form the basic fundamental relations of the
linear theory of RTI in ICF.
σ =
α1(F, ν)
√
ak − β1(F, ν)kua F > F∗(ν)
α2(F, ν)
ak
1 + kLmin
− β2(F, ν)kua F ≤ F∗(ν)
(4.19)
Lmin is the minimum value of the density-gradient scale length (Lmin = L/2At). And α1, α2,
β1 and β2 are fitting functions depending only on F and ν (Atzeni and Meyer-ter Vehn, 2004,
p. 269).
4.4 Feed-through
The formula for classical RTI deals with fluids that are infinite or semi-infinite however for ICF
we use relativity thin shells. Because of this we have to deal with the phenomenon known
as feed-through, the mathematics of which are described in Atzeni and Meyer-ter Vehn (2004,
p. 254) but are not necessary for the level of detail explored in this paper.
If we consider two surfaces characterised by a and b, where surface a is the outer surface of
the fuel pellet and surface b is the inner surface of the solid DT section of the fuel pellet. As
the perturbation of surface a grows, the perturbation of surface b grows at the same rate but
with a reduced amplitude (exp(−k∆z)), meaning that a perturbation from an unstable surface
is transmitted to a stable one. The implications of this for ICF are such that “perturbations
that grow at the ablation front are fed to the inner surface of the solid DT fuel during inward
acceleration”(Atzeni and Meyer-ter Vehn, 2004, p. 254). These perturbations are the main cause
of instabilities occurring at implosion stagnation (when perturbations with λ are much smaller
than the thickness of the shell feed-through is negligible) (Atzeni and Meyer-ter Vehn, 2004,
p. 255).
4.5 Inner RTI
Inner RTI is the the RTI that occurs during the deceleration and stagnation of ICF target. Again
this RTI is ablative due to the heat and α-particle flux from the hotspot which causes ablation of
the inner surface of the decelerating shell (Atzeni and Meyer-ter Vehn, 2004, p. 278). The growth
rate σin at the inner shell surface (Equation 4.20) is approximated by the same relation used
for ablative RTI (Equation 4.19) but using the notation for the inner surface of the decelerating
shell. k = l/R (l is the spherical mode number described in the next section), Lin the minimum
density scale length at the hot spot surface, ua−in is the ablation velocity at the inner shell
27. Chapter 4 Hydrodynamic Instabilites 21
surface and with βin which is a numerical coefficient.
σin
=
¨Rl/R
1 + Linl/R
− βin
l
R
ua−in (4.20)
This equation shows that ablative flow in fact stabilizes the deceleration phase and this culmi-
nates in a growth reduction as l and ua−in increase, the full discussion of this phenomena can
be found in Lobatchev and Betti (2000).
4.6 The growth factor
From Section 2.3 we know that for ignition to occur a fuel shell with thickness ∆R(t) must
preserve its integrity during implosion and also create a central hot spot at stagnation with a
radius Rh. We also know that hydrodynamic instabilities “cause deformations of the shell’s outer
and inner surfaces” (Atzeni and Meyer-ter Vehn, 2004, p. 291). Because of this it is clear that the
following conditions must be satisfied: ζout(t) ∆R(t) during implosion, and ζin(t) Rh
at implosion stagnation. ζout and ζin represent the deformation amplitudes directly relating
from the hydrodynamic instabilities.
Using Lindl (1997) as a base and setting typical values for target and beam parameters (Ta-
ble 4.1), the above conditions can be evaluated more precisely and a relationship between in-
stability growth and the in-flight-aspect-ratio (Aif ), described in Section 2.1, can be retrieved.
This will require examining the effect of the hydrodynamic instabilities on target design for
the 3 areas that we have previously described (Ablative RTI, feed-through and Inner RTI).
In previous sections we have considered RTI at plane boundaries, this must now be altered
to examine what happens when RTI occurs at spherical interfaces. Takabe’s Formula (Equa-
tion 4.19) refers only to RTI in an equilibrium state or steady state. To extend this model for a
spherical interface we must explore converging flows, this is achieved using a perturbed poten-
tial, through potential theory, and is solved in spherical geometry (r, θ, φ, t) by a superposition
of modes. By solving the perturbed potential we reach the Bell-Plesset equation (Atzeni and
Meyer-ter Vehn, 2004, p. 275), which shows that the amplitude ζl of the lth perturbation mode
(the spherical mode number) evolves according to:
∂t(m∂tζl) −
l − 1
l
mk ¨Rζl = 0 (4.21)
Here we have introduced a mass variable to simplify the equation (m = ρR2/k), where k is
the wave number given by k = (l + 1)/R and R is the unperturbed radius. The mode number
has a large effect on the size of RTI growths, for example it can be shown fast growing RTI
modes have mode number of l ≈ 30 whereas modes l ≈ 100 are stable, this is explored in a
later section (section 5.3).
28. 22 Chapter 4 Hydrodynamic Instabilites
Perturbation growth at the ablation front
From linear theory and Atzeni and Meyer-ter Vehn (2004, p. 292) it is shown that the amplitude
of the outer deformation of a particular mode l, ζout
l , is equal to the initial amplitude, ζout
l0
(due to the pellet surface not being perfectly smooth), times the growth factor which is given
by Equation 4.23.
ζout
l = ζout
l0 Gout
l (4.22)
Gout
l = exp
t0
0
σl(t)dt (4.23)
σl (Equation 4.24) is the linear growth rate of mode l from Equation 4.19, where k = l/R
accounting for spherical geometry.
σl = α2
al/R
1 + lLmin/R
− β2
l
R
ua (4.24)
The integral in Equation 4.23 can be solved by assuming a constant acceleration as the shell
implodes (R0/2 < R < R0), this occurs in a time interval t0 = R0/a. Also R and ∆R must
be assumed to be constant. Lmin and ua are then parametrized according to Atzeni and Meyer-
ter Vehn (2004, p. 292) by the following relations (where f1 and f2 are numerical constants).
Lmin = f1∆R (4.25)
ua = f2∆R(t0)/t0 (4.26)
f1 depends on the shape of the density profile at the ablation front and f2 is related to the
fraction of ablated mass and has typical values:
f2 ≈
0.8 Indirect drive
0.2 Direct drive
(4.27)
Thus the growth factor is approximately given by Equation 4.28.
Gout
l exp α2
l
1 + l(f1/2)A−1
if
− β2f2A−1
if l (4.28)
R/∆R is a characteristic value of Aif and has been substituted into the above equation. Stable
modes (l > lcut) can be found by substituting Gout
l = 1 to give Equation 4.29.
lcut =
Aif
2f1
1 +
4f1α2
2
f2
2 β2
2
Aif − 1 (4.29)
29. Chapter 4 Hydrodynamic Instabilites 23
Figure 4.3: Plot showing how the growth factor at the outer surface varies with mode
number, plotted for several values of Aif
Perturbation growth at the inner shell surface
Inner surface perturbations occur when the imploding material starts to decelerate this is a
direct result of the pressure exerted by the inner hot gas of the fuel pellet. These perturbations
are caused by 2 criterion: a) Defects of the surface of the inner solid DT fuel pellet, and b) feed-
through from the ablation front. The defects have modal amplitudes ζin
l00 and the feed-through
have amplitudes ζin−feed
l (Equation 4.30), caused by perturbations at the outer surface which
are transmitted to the inner surface.
ζin−feed
l = ζout
l Gfeed
l (4.30)
Gfeed
l exp
−l
Aif
(4.31)
The effective initial amplitude of mode l (ζin
l0 ), assuming random phases, can be approximated
as:
ζin
l0 ≈ (ζin
l00)2 + (ζin−feed
l )2 (4.32)
Thus the perturbation amplitude of the inner surface can be given in a similar manner to the
‘Perturbation growth at the ablation front.’
ζin
l = ζin
l0 Gin
l = ζin
l0 exp
tdec+∆tdec
tdec
σin
l (t)dt (4.33)
Again to calculate the integral a similar assumption must be made, the hot spot radius decreases
at constant acceleration from Rdec to Rh. Also similar parametrizations are made (from abla-
tive front perturbations), Atzeni and Meyer-ter Vehn (2004, p. 292):
30. 24 Chapter 4 Hydrodynamic Instabilites
Figure 4.4: Plot showing how the growth factor from feedthrough varies with mode
number, plotted for several values of Aif
• Rdec − Rh ≈ f3Rh
• Lin = f4Rh
• ua−in =
f5Rh
∆tdec
Where f3, f4 and f5 are numerical constants. Thus evaluating the integral gives the growth
factor of instabilities at the inner shell surface:
Gin
l ≈ exp
2f3l
1 + f4l
− βinf5l (4.34)
It is worth noting that the growth of these perturbations is independent of Aif , thus the initial
amplitudes must be larger before any degradation to performance is noticed.
In addition, the cut off for inner surface of the fuel capsule is found through the same method
utilized in the previous section to give:
lcut =
1
2f4
1 +
8f3f4α2
2
f2
5 β2
in
− 1 (4.35)
Total growth factor
Finally, the total growth factor is found by multiplying the individual growth factors calculated
in the previous sections:
GT = Gout
l Gfeed
l Gin
l ≈ exp α2
l
1 + l(f1/2)A−1
if
−
β2f2l
Aif
−
l
Aif
+
2f3l
1 + f4l
− βinf5l
(4.36)
31. Chapter 4 Hydrodynamic Instabilites 25
Growth Parameters (Approximate) Typical Values
f1 0.1
f2β2 0.8
α2 1
f3 1
f4 0.03
βinf5 0.09
Table 4.1: Approximate values for the growth factor numerical constants
Figure 4.5 shows the total growth factor (plotted using typical fusion values, Table 4.1) for
several values of Aif . The full anylsis of the growth factors is explored in Section 5.3, however
it is worth noting that the maximum total growth factor for an in-flight-aspect-ratio of 50 is
approximately 1000. This means that due to hydrodynamic instabilities, a final perturbation
amplitude is a maximum of 1000 times greater than the initial perturbation.
Figure 4.5: Plot showing how the total growth factor varies with mode number, plotted
for several values of Aif
32. Chapter 5
Discussion
5.1 RTI’s effect on the Driver Energy
We know from the Lawson criteria, the driver energy Ed must be equal to or greater than the
ignition energy Eign for ignition to occur. If we assume that the fuel energy is roughly approxi-
mate to the driver energy times the overall coupling efficiency (Ed = Ef /η), which is the same
assumption used in the derivation of limiting gain. Then from Equation 5.4 and Equation 5.3
the minimium driver energy can be defined as:
Ed ≥
kproη−1A−3
if p−1.2
a Direct drive
kproη−1
hohlα−1.8A−6
if T−8.2
r Indirect drive
(5.1)
Where kpro is just a generic constant of proportionality that varies depending on the param-
eters used. This equation shows that for a fixed pa then as the in-flight-aspect-ratio increases
the driver energy required reduces. It can be concluded from this that a higher Aif is beneficial
to achieving ignition. However as will be seen in the Section 5.3 RTI limits Aif to a maximum
value, any higher than this value and ignition would not be achieved because of fuel-ablator
mixing (mentioned in Section 4.1). Therefore RTI sets a lower bound on the amount of energy
required to achieve ignition.
Although not explored in this paper the upper bound is set by the ablation pressure and limited
by plasma instabilities (Atzeni and Meyer-ter Vehn, 2004, p. 122), shown by the dotted line
in Figure 5.1. From the graph it can be deduced that for ignition to occur the laser parameters
must lie between the ‘maximum allowed temperature’ curve and one of the other curves which
set the minimum driver energy for ignition. As Aif decreases the area between the minimum
and maximum curves, the operating window, decreases. This graph is of great importance in
ICF as it shows the minimum energy and power required for ignition.
26
33. Chapter 5 Discussion 27
Figure 5.1: Graph showing curves of ignition power (for indirect drive) as a function
of ignition energy for different values of Aif , also shown on graph is the Pd for the
maximum allowed temperature.
5.2 Relation between RTI and Gain
Finding the relationship between the in-flight-aspect-ratio and Gain (G) first requires examin-
ing how the implosion velocity (uimp) and fuel energy (Ef ) are effected by the in-flight-aspect-
ratio (Aif ). From the rocket model (Section 2.2) the implosion velocity as a function of Aif can
be approximated by:
uimp ∝
α0.6
if Aif T0.9
r Indirect drive
α0.6
if Aif (IL/λL)4/15 Direct drive
(5.2)
Tr is the radiation temperature, IL is the intensity and λL the wavelength of the laser light for
direct drive. The implosion velocity can then be related to the stagnation pressure.
p ∝ u3
impα−0.9
if p0.4
a (5.3)
Within the isobaric model the fuel energy is related to the the stagnation pressure by (this
relation is derived in the isobaric gain derivation, Section 3.2):
Ef ∝
1
p2
(5.4)
• The ablation pressure pa is affected by the laser intensity and hohlraum temperature,
which are limited by the plasma instabilities.
• The in-flight isentrope parameter αif , determines the shock and entropy evolution of
system and is subject to preheat and pulse shaping
Thus from the definitions of limiting fuel gain (Section 3.2) and by combining with Equation 5.4
the relationship between Gain and pressure can be found.
Gf ∝ E0.3
f ∝ p−0.6
(5.5)
34. 28 Chapter 5 Discussion
From the above relation the gain can be linked with the Aif .
Gf ∝
A−1.8
if Indirect drive
A−0.9
if Direct drive
(5.6)
This relationship confirms our hypothesis of how the Gain is related to the in-flight-aspect-
ratio; which were, as Aif increases the gain decreases. From this we can infer that a small value
of Aif is more desirable in ICF. We must now find a direct link between Aif and the growth
factor so that we can set a maximum value for the growth factor and therefore determine the
corresponding maximum allowed Aif .
5.3 Exploring the In-Flight-Aspect-Ratio
Equation 4.36 clearly shows that a smaller Aif leads to smaller growth factors. Constraints for
Aif are estimated using the equations mentioned in the previous section. Using fluid codes
such as HYDRA the evolution of a wide spectrum of growth modes for a large sector of a
fuel capsule can be simulated. “These simulations require huge amounts of time and computing
power and are still currently unable to resolve very short wavelengths” (Atzeni and Meyer-ter
Vehn, 2004, p. 296).
Limits of the In-Flight-Aspect-Ratio
RTI growth can be reduced by decreasing ua (Equation 4.24), however this leads to the shell
density increasing and the entropy decreasing, which is not desirable (Atzeni and Meyer-ter
Vehn, 2004, p. 298).
The Aif relates directly to the roughness of the shell pellet surface (explored in the next sec-
tion). A smaller Aif represents rougher targets, which leads to a higher driver energy or a lower
energy gain, Equation 5.6. Because of this a comprise value for Aif must be found which is
large enough to allow the energy provided by the laser to ignite the fuel, but small enough so
that the capsule remains hydro-dynamically stable and fuel-ablator mixing does not prevent
the fuel from igniting.
To find this value this we should first understand how the growth factor varies with Aif ,
Section 4.6 derives the equations for the various growth factors. The figures plotted in that
section show how the growth factor varies with mode number, there are several interesting
conclusions we can draw from these graphs.
1. The growth factor at the outer surface is the most significant contributor of growth to
the total growth factor. This indicates that it is the most important factor to be limited
when trying to reduce the overall growth.
35. Chapter 5 Discussion 29
2. Growth from feed-through actually helps to reduce the overall growth at high modes.
This can be verified by comparing the total growth factor to the growth at the outer
surface, for Aif = 50, the maximum value for Gout
l is roughly 304 compared with 2000
for GT .
3. The cut off mode for GT is much lower than Gout
l (∼ 120 compared with ∼ 1200 for
Aif = 50). This is due to both Gin
l and Gout
l Gfeed
l having lower cut off modes than Gout
l .
4. Growth at lower mode numbers can be ignored as the overall growth is very low. The
most unstable modes are around 20 (depending on the value of Aif ).
5. Growth at the inner surface of the pellet is independent of Aif . Therefore the roughness
at outer surface should be considered more important than at the inner surface (when
evaluating the growth factor and instability growth).
6. The total growth factor GT increases exponentially up to a maximum value then drops
sharply away.
We can also calculate the relationship between the in-flight-aspect-ratio and the total growth
factor. To do this we plotted the GT for several values of Aif (similar to Figure 4.5). Next
we took the maximum growth factor for a specified mode number from each of the plots and
plotted it against Aif , the resulting graph is shown in Figure 5.2. The graph shows that for
20 < Aif < 80 the relationship is approximately linear, we can ignore values outside this
range as these would result in either too little gain or hydrodynamic instabilities that are so
large that ignition would never occur. We have now determined the final relationship between
the growth factor and the in-flight-aspect-ratio (GT ∝ Aif ). Finally we calculated the equation
of this linear section to find the precise relation, Equation 5.7.
(GT )max = 34.776 Aif + 346.63 (5.7)
Now that we have this base relationship we can choose limits for the growth factor and thus
calculate the maximum allowed Aif . Lindl (1995) limits the growth factor to 1000 this leads to
an in-flight-aspect-ratio of between 30−40, this is “about the maximum tolerable growth factor
for most high convergence ratio inertial fusion target designs”(Logan et al., 2007, p. 11). Zhou
and Betti (2008) state that “typical values of Aif in conventional ICF target designs range from
35 − 45,” this corresponds to a growth factor limited from 850 to 1230.
In-Flight-Aspect-Ratio’s effect on the outer capsule surface finish
In an ideal scenario we would like a fuel pellet to be perfectly smooth as the vast majority of
instability growth is seeded by initial perturbations on the capsule’s outer surface (Section 4.6).
36. 30 Chapter 5 Discussion
Lindl1995
Atzeni2004
Zhou2008
y = 34.776x - 346.63
0
500
1000
1500
2000
2500
3000
0 10 20 30 40 50 60 70 80 90 100
Max.GrowthFactor
In-Flight-Aspect-Ratio
Max. growth factor for a specific in-flight-aspect-ratio
Figure 5.2: Graph showing the maximum growth factor from the most unstable mode
versus Aif . Included on the graph are typical values for Aif and their respective
(GT )max. A trend line calculated using data from the most linear section of the graph
(20 < Aif < 80) is also shown.
However, in the ‘real world’ due to manufacturing and technological constraints it is not pos-
sible to have a perfectly smooth fuel pellet. In practise the surface of the fuel pellet contains
a wide spectrum (with random phases) of initial perturbations with a root-mean-squared am-
plitude (Lindl, 1995):
ζrms =
l
ζ2
l (2l + 1)−1 (5.8)
By combining the equations for perturbation amplitudes in Section 4.6 with the total growth
factor we can show how the resulting final perturbation amplitude is affected by the surface
roughness.
ζin
l = (ζin
l00Gin
l )2 + (ζout
l0 GT )2 (5.9)
ζout
l0 is the initial perturbation amplitude at the outer surface of the fuel pellet and thus equal to
ζrms and ζin
l00 is the initial perturbation amplitude of the inner surface of the fuel pellet. If we
choose to ignore the 1st term, resulting from inner surface roughness, and substituting Equa-
tion 5.7 the from the previous section we get:
ζin
= ζrms(34.776 Aif + 346.63) (5.10)
37. Chapter 5 Discussion 31
This is obviously an overestimation as it assumes that all modes have the maximum growth
factor, which is not true, but for the purpose of this analysis this assumption does not overly
effect the end results, as we are looking for a general relationship.
We now want to use this equation to examine how fuel mixing (fuel-ablator mixing due to the
initial perturbations) changes with surface roughness, the importance of this is explained in
the section on basic ICF principles. The motivation for this is obvious as the greater the fuel-
ablator mixing is, then the less ‘clean’ fuel there is, leading to a smaller hotspot and therefore
a lower gain. To achieve this we first assume that perturbations with initial amplitude ζrms
cover the entire surface of the DT fuel. Next we consider a shell on the inner surface of the
fuel with radius equal to ζrms. We now assume that ∼ 1/2 the volume of this shell is fuel and
the other half is the plastic ablator layer introduced because of RTI fuel-ablator mixing. 1 Then
the percentage of fuel-ablator mixing (with 0% meaning there is no mixing) as a function of
surface roughness and the in-flight-aspect-ratio is:
100
2
ζrms(34.776Aif + 346.63)
Rf
3
(5.11)
This formula allows for the calculation for the percentage of fuel-ablator mixing for different
values of surface roughness, in-flight-aspect-ratio and fuel radius (Rh + Rc). Figure 5.3 shows
fuel mixing curves for the minimum fuel radius for ignition with maximum gain (with a driver
energy of 1MJ and a fuel mass of 1mg) Rf = 143µm.
Figure 5.3: Surface roughness versus percentage of fuel-ablator mixing for a number
of different Aif .
Lindl (1995) states that the surface finish must have an ζrms of between 200 − 300 angstroms
= 20 − 30nm, this was later refined by Lindl et al. (2004) to 10 − 20nm, which is the current
limiting value for ICF pellet design used at NIF. It is apparent from the graph why these limits
were chosen as there is almost no fuel-ablator mixing in that range. However, in reality this is
1
This assumption originates from the prime assumption that all the perturbations on the surface are all iden-
tical in height, width and spacing. This is obviously a gross overestimation and is an area that requires further
investigation.
38. 32 Chapter 5 Discussion
not true because of the assumption made in the footnotes and the initial perturbation amplitude
of the inner surface of the fuel pellet must be taken into account.
5.4 Reducing the growth factor through pulse shaping
From basic fusion principles we know that the laser pulse delivered to the fuel pellet is shaped
in time to provide the fast and nearly isentropic compression required. However the NIF is cur-
rently experimenting with shaped pulses that have significantly lower instability growth, Fig-
ure 5.4. The low-foot pulse uses the type of pulse shaping that is currently utilised in ICF
Figure 5.4: This graph shows two different shaped laser pulses low-foot and high-
foot and their associated growth factor depending on the mode number (Raman et al.,
2014).
implosion experiments at NIF, it is the pulse that has “achieved the highest fuel compression
of any shot to date” (Raman et al., 2014). The pulse currently underdevelopment at NIF is
dubbed the high-foot pulse, and has “produced the highest neutron yields to date, nearly 10
times higher than what was achieved during NIC” (Raman et al., 2014). The high-foot differs
from the low-foot pulse in several aspects (Figure 5.5):
1. The high-foot pulse is quicker than the low-foot (about ∼ 6ns)
2. Both have the same general shape, a small peak at the beginning followed by a larger
sustained peak towards the end. However, the low-foot pulse has an additional small
peak just before the main sustained peak.
3. The high-foot pulse has an overall lower peak radiation temperature.
The reasoning behind the these differences in pulse shape is beyond the scope of this paper,
however they are explained in detail in the paper by Raman et al. (2014). The resulting benefits
from using this drive are obvious, by reducing the maximum growth factor through pulse
shaping the relationship between the in-flight-aspect-ratio and the growth factor is changed
39. Chapter 5 Discussion 33
(Equation 5.7). Therefore a larger Aif can be used while achieving the same gain, in addition to
this the ignition energy is reduced (Equation 5.1). The consequence of this is that less energy
is required, thus increasing the overall efficiency and performance of ICF.
Figure 5.5: Comparison of the high-foot and low-foot drives (Raman et al., 2014).
40. Chapter 6
Conclusion
The effect of hydrodynamic instabilities on target design and the gain of ICF have been ex-
plored. We have found a general relationship linking RTI growth with the limiting gain in the
direct and indirect cases and also examined the connection between RTI growth and the laser
energy and power. It is clear that while greater hydrodynamic instabilities (in terms of RTI)
lead to a lower ignition energy it also increases the instability of the capsule which ultimately
leads to a lower gain. Therefore a comprise between ignition energy and gain is required and
as we have seen this results in an in-flight-aspect-ratio of approximately 40.
In addition to exploring the hydrodynamic instability growth in terms of Aif , the relationship
between Aif and the maximum growth factor has been discovered. This has also allowed us to
investigate the association of the surface roughness of the fuel pellet to the maximum growth
factor and thus the Aif and the overall performance of ICF. From this we have confirmed the
reasoning behind current limits for the surface roughness. The model we created is only the
first step in examining the full effect of fuel-ablator mixing in relation to surface roughness.
A further extension of this project could improve on this model by refining some of the as-
sumptions made and also by including the effect of the initial inner surface perturbations on
the finial perturbation amplitude.
Finally methods of reducing instability growth and its effect on ICF performance have been
briefly looked into. These new methods have mainly arisen from the fact that we have almost
reached our technical and manufacturing limit regarding the smoothness of the fuel pellets.
Thus methods such as pulse shaping offer an achievable method of reducing instability growth
while keeping the input energy sufficiently low to allow acceptable gains for sustainable ICF.
In conclusion, the reduction of hydrodynamic instability growth and thus the increase in per-
formance of ICF, through exploring the areas this project has reviewed, will ultimately bring us
one step closer to achieving sustained fusion. This in turn will eventually lead to completion of
the principal aim of terrestrial fusion, providing almost limitless ‘clean’ energy and eliminating
the use of fossil fuels.
34
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