UNIVERSITY OF SOUTHAMPTON

FACULTY OF PHYSICAL AND APPLIED SCIENCES

Physics

An investigation into the main parameters a�ecting the performance of

Inertial Con�nement Fusion

by

Ben WilliamsStudent ID: 24691925

A �nal report submi�ed for continuation towards a MPhys

Keywords:Inertial Con�nement Fusion, Gain, Rayleigh-Taylor Instability

Supervisor: Prof. C.T Sachrajda

April 19, 2015

UNIVERSITY OF SOUTHAMPTON

ABSTRACT

FACULTY OF PHYSICAL AND APPLIED SCIENCESPhysics

A �nal report submi�ed for continuation towards a Mphys

AN INVESTIGATION INTO THE MAIN PARAMETERS AFFECTING THE PERFORMANCEOF INERTIAL CONFINEMENT FUSION

by Ben Williams

Inertial Con�nement Fusion is the process of using a pulse of radiation to rapidly heat a smallcapsule containing fusion material. �e outer layer explodes outwards and the resultant forcerapidly compresses the fuel until fusion reactions occur at the core. �is project exploresin-depth some of the main parameters that determine the performance of this experimentalmethod of terrestrial fusion. Two main parameters that were explored are: a) Gain, which inits simplest terms is the ratio of energy released to the energy delivered, and b) HydrodynamicInstabilities, which describe how the plasma �ows in the capsule during and a�er irradia-tion, a major component that a�ects capsule performance (seeded from imperfections in thesmoothness of the capsule’s surface). In addition to studying the relationship between theseparameters, the constraints and limits will be calculated and known plots will be replicatedand analysed to show the accuracy of my derivations and research.

Contents

Acknowledgements v

Nomenclature vi

1 Introduction 11.1 Basic Fusion Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Inertial Con�nement Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Prerequisite Principles 62.1 �e In-Flight-Aspect-Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 �e Rocket Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Lawson Critiera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Plasma Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Gain 113.1 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Isobaric Gain Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Isochoric Gain Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Hydrodynamic Instabilites 164.1 Rayleigh-Taylor Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Classical RTI growth rate derivation . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Ablative RTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Feed-through . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 Inner RTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 �e growth factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Discussion 265.1 RTI’s e�ect on the Driver Energy . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Relation between RTI and Gain . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Exploring the In-Flight-Aspect-Ratio . . . . . . . . . . . . . . . . . . . . . . . 285.4 Reducing the growth factor through pulse shaping . . . . . . . . . . . . . . . . 32

6 Conclusion 34

References 35

iii

List of Figures

1.1 4 main stages of ICF (Anatomy of the Universe, 2011) . . . . . . . . . . . . . . 21.2 Internal structure of fuel pellet (Brum�el, 2012) . . . . . . . . . . . . . . . . . 21.3 Geometrics of polar and symmetric drive (Hecht, 2013) . . . . . . . . . . . . . 31.4 Fuel pellet inside hohlraum Glenzer et al. (2012) . . . . . . . . . . . . . . . . . 41.5 Diagram of e�ciency of indirect drive LLNL (2013) . . . . . . . . . . . . . . . 5

2.1 Implosion diagram of typical ICF capsule. Notice that shell is particularly thinin 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 �uids of di�erent densities (Atzeniand Meyer-ter Vehn, 2004, p. 238) . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Progression of RTI with time (Shengtai and Hui, 2006) . . . . . . . . . . . . . . 174.3 Plot showing how the growth factor at the outer surface varies with mode

number, plo�ed for several values of Aif . . . . . . . . . . . . . . . . . . . . . 234.4 Plot showing how the growth factor from feedthrough varies with mode num-

ber, plo�ed for several values of Aif . . . . . . . . . . . . . . . . . . . . . . . . 244.5 Plot showing how the total growth factor varies with mode number, plo�ed for

several values of Aif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1 Graph showing curves of ignition power (for indirect drive) as a function ofignition energy for di�erent values of Aif , also shown on graph is the Pd forthe 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 ofthe graph (20 < Aif < 80) is also shown. . . . . . . . . . . . . . . . . . . . . . 30

5.3 Surface roughness versus percentage of fuel-ablator mixing for a number ofdi�erent Aif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.4 �is graph shows two di�erent shaped laser pulses low-foot and high-foot andtheir 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

Acknowledgements

I would like to thank Professor C.T Sachrajda for his help, supervision and guidance on thisproject. I would also like to thank Emma Di�er, my project partner, who’s determination andhard work made this project possible.

v

Nomenclature

Symbol Description UnitsAif In-Flight-Aspect-Ratio -R Average shell radius m∆R Outer shell thickness during implosion muimp Implosion velocity ms-1

uex Exhaust velocity ms-1

ua Ablation velocity ms-1

Th/r Hotspot/Radiation temperature keVρh/c Hotspot/Cold fuel density Tbarph/c Hotspot/Cold fuel pressure PaRh/c/f Hotspot/Cold/Fuel radius mMh/c/f Hotspot/Cold/Fuel mass mgEd/h/f Driver/Hotspot/Fuel energy MJEfus Energy released from fusion MJωh/c Internal/Additional internal energy per electron eVG 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 e�ciency -α Isentrope parameter -k Wavenumber m-1

λ Laser light wavelength mσ 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 nmG Growth factor -

vi

Chapter 1

Introduction

1.1 Basic Fusion Principles

Nuclear Fusion is the process of combining light atomic nuclei to form a heavier nucleus, thisreaction 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 �nal products is less than the masses of the constituents(World Nuclear Association, 2013; Atzeni and Meyer-ter Vehn, 2004). Nuclei naturally repeleach other, to achieve fusion this electrostatic repulsion must be overcome. �e main issueswhich govern terrestrial fusion are the heating of the reactants to fusion temperatures, around150 million degrees, and the sustained con�nement of the plasma, gas of highly ionised parti-cles (Pi�s et al., 2006). �ere are two main methods of plasma con�nement which are currentlyunder investigation for achieving fusion, magnetic (MC) and inertial (IC) con�nement. �is pa-per refers to the la�er method, however it is worth noting that there are several experimentsutilizing MC and there has been signi�cant progress in that sector over the last decade (Cowley,2010).

1.2 Inertial Con�nement Fusion

�e basic principle of inertial con�nement fusion (ICF) is the external irradiation of a static fuelpellet, 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

2 Chapter 1 Introduction

• �e Ablator layer (outer layer) vaporises, ionises and explodes outwards

• From Newton’s third law and momentum conservation there is a reactant force whichis directed inwards compressing and heating the core leading to a central hotspot (asmall area that has a higher temperature compared with its surroundings). �is takesplace because the imploding material in the hotspot stagnates and the kinetic energy isconverted to internal energy

• If the hotspot temperature and density are su�cient, then the nuclei in the hotspot willovercome the coulomb barrier and fusion will occur

• A chain reaction, where fusing nuclei transfer energy to other nuclei causing them tofuse thus causing burn waves to propagate radially outwards igniting the fuel, is able tohappen 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

�e fuel pellet is comprised of an outer shell of typically plastic ablator, an inside shell ofsolid Deuterium-Tritium (DT) and a central cavity containing DT gas, shown in Figure 1.2.�e 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 (Brum�el, 2012)

Chapter 1 Introduction 3

can be accelerated over greater distances, therefore to reach the required implosion velocity forfusion a lower driving pressure is needed, 350kms−1. Also, isentropic compression, requiredfor direct drive, is easier to perform on a hollow shell because the shock waves are similarto plane waves in structure when they reach the centre, compared to a solid pellet where theshock 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 speci�cyield and also the lowest ignition temperature, thus requiring less energy to be provided toinitiate fusion. �ere are other fuels currently being explored but this project only refers to theDT mixture, however it is worth noting that only light elements are used in fusion researchbecause of the higher possibility that quantum tunnelling can occur (Nobelprize.org, 2013).

Main Experimental Methods

�ere are two main experimental methods within ICF these are Direct and Indirect drive, thedi�erences between these drive methods occur predominately with the �rst stage of ICF, theirradiation of the pellet. �e principles which are explored in this project are relevant to bothdrives 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 thesurface of the fuel pellet (the outer shell of the pellet) and cause its ablation (McCrory et al.,2005). �e pellet is irradiated at multiple points on its surface to provide uniform radiation, thisis achieved by spli�ing the laser into a large number of overlapping beams which are focusedby lenses, shown below in Figure 1.3. �e 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 ablationlayer of the pellet. However one disadvantage to direct drive is that as the pellet explodes asigni�cant proportion of the laser light misses the pellet and is instead refracted in the plasmacorona (Atzeni and Meyer-ter Vehn, 2004, p. 49). �e irradiation must take place in a timeinterval in which mass inertia keeps the burning fuel together, this is the time it takes for a

4 Chapter 1 Introduction

sound wave to travel from the surface of the pellet to the centre. �is gives a con�nementtime (τ ) of typically 0.1ns, in this case con�nement time is de�ned as the time the plasma ismaintained above ignition conditions (Atzeni and Meyer-ter Vehn, 2004). To provide maximumenergy release the fuel is compressed quickly and isentropically, this is described by the �rstlaw 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 seriesof superimposed shocks with each subsequent shock having a greater speed than the last, sothat 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 thefuel pellet. �is is the main issue for this type of experimental set up because even a slightvariation from equilibrium will produce instabilities, such as Rayleigh-Taylor and Richtmyer-Meshkov (RT and RM) described in Section 4, which can signi�cantly 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 e�ciency are required. Approximately 20% of thelaser’s energy is consumed to achieve a homogenous spread (Atzeni and Meyer-ter Vehn (2004,p. 65), Raman et al. (2014)).

Indirect Drive

�e key feature of the indirect drive is that instead of directly irradiating the fuel pellet withlaser light, the laser light is converted into X-rays which then interact with the pellet, hencethe name indirect drive. �is is accomplished by placing the pellet inside a cylinder known asa hohlraum, see Figure 1.4. �e 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. �e angles at which the laser light enters the hohlraumare adjusted to give geometric and uniform irradiance in addition to maximum energy trans-fer. As gold has the highest conversion e�ciency of laser light to X-rays, η ≈ 80 − 90% atapproximately 1013Wcm−3, this is the material that typically coats the hohlraum (Atzeni andMeyer-ter Vehn, 2004; Atzeni, 2009).

Chapter 1 Introduction 5

Indirect drives has several advantages over direct drive, however the principal advantage isthat indirect drive produces uniform isotropic radiation. From this RT instabilities are limited,which leads to an overall reduction in instabilities and consequently a signi�cant increase incapsule performance. Nevertheless, the decrease in instabilities comes at a high price, a growthin energy loss from the heating of the hohlraum walls, Figure 1.5 shows a rough diagrammaticrepresentation of the e�ciency of indirect drive (Atzeni and Meyer-ter Vehn, 2004).

Figure 1.5: Diagram of e�ciency of indirect drive LLNL (2013)

Chapter 2

Prerequisite Principles

2.1 �e In-Flight-Aspect-Ratio

During an ICF implosion the laser pulse is shaped in time, this happens for several reasonsone of which is to keep the fuel compressed on a low adiabat (other reasons will be discussedin Section 5.4), this is required to limit the isentrope parameter (Section 3.2) from becomingtoo high and decreasing the gain (Rosen and Lindl, 1984). �is pulse shaping causes the shellto become thinner and denser than it was initially (shown in Figure 2.1), thus it is useful tobe able to compare this thickness with the outer radius of the capsule. A fundamental fusionparameter is the initial aspect ratio of an ICF capsule which is de�ned as the ratio of the initialouter radius to the initial thickness of the outer shell Ar0 = R0/∆R0. From this the ‘in-�ight’aspect-ratio can be de�ned 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 de�ning Aif we can immediately see that because of laser pulsing Aif � Ar0, as duringimplosion the ∆R factor reduces drastically therefore increasing the aspect ratio. �e in-�ight-aspect-ratio is a very important factor in achieving ignition and consequently a signi�cantsection of this paper is devoted to explaining the limits and consequences of Aif .

2.2 �e Rocket Model

�e rocket model is used to describe global features of the implosion process of ICF such as thehydrodynamic e�ciency and implosion velocity. �is paper will follow the partial derivationgiven in Atzeni and Meyer-ter Vehn (2004, p. 230-235) and Lindl (1995, p. 27-30).

6

Chapter 2 Prerequisite Principles 7

Figure 2.1: Implosion diagram of typical ICF capsule. Notice that shell is particularlythin 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 overtime dt as the exhaust expels a mass dM . Assume the rocket is ideal, such that the exhaust iscontinually heated so that it remains nearly isothermal during its expansion. From momentumconservation we have d(Mu) − u1dM = 0, where the �rst term is the change in rocketmomentum and the second term is the momentum of the exhaust, u and u1 are the velocitiesviewed in an inertial frame. �is can then be rewri�en as:

Mdu

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 sphericalrocket. �erefore starting at a time t a shell of mass M(t) and radius R(t) implodes with avelocity u = dR/dt, the surface ablates with a rate ma which is given by:

dM

dt= −4πR2ma (2.3)

Using Equation 2.2 and the equation for ablation pressure (mauex) the above equation can berecast as:

Mdu

dt= −4πR2pa (2.4)

Equation 2.3 and Equation 2.4 are not really useful in their current forms, to make use of theseequations we integrate both with respect to time to give:

u = uex lnM

M0(2.5)

1− M

M0

(1− ln M

M0

)=ε

3

(1−

( RR0

)3)

(2.6)

8 Chapter 2 Prerequisite Principles

Here we have introduced the initial massM0 = 4πR20ρ0∆R0, where the density of the shell is

ρ0, the radius isR0 and the thickness is ∆R0. We have also introduced the implosion parameter(Equation 2.7) into Equation 2.6 which characterises the implosion.

ε =ma

uexρ0

R0

∆R0(2.7)

Implosion velocity

We now have the basis for de�ning the Implosion velocity from the rocket model, using Equa-tion 2.5 and de�ning the imploding mass as M1 = M(R = 0) the �nal implosion velocity isdescribed by Equation 2.8.

uimp = uexlnM0

M1(2.8)

At this point it is useful to relate the implosion velocity to the in-�ight-aspect-ratio by estimat-ing thatAif = (ρaR0)/(ρ0∆R0) to give ε = uaAif/eex. Here we have introduced the ablationvelocity (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 de�ned by combining Equation 2.5 and Equa-tion 2.6. �is function can be approximated for two di�erent 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 ablationregimes.

�us from this a rough approximation for implosion velocity can be deduced, for direct (low-ablation regime) and indirect drive (high-ablation regime). �e important thing to notice hereis that for direct drive the implosion velocity is proportional to the square root of the in-�ight-aspect-ratio, but for indirect drive it is linearly proportional to the in-�ight-aspect-ratio, this

Chapter 2 Prerequisite Principles 9

signi�cance of this will be investigated in a later section.

uimp ≈

√uauexAif

3Direct drive

uaAif Indirect drive(2.10)

2.3 Lawson Critiera

�e Lawson Criterion are a set of three variables that determine the conditions for general ther-monuclear ignition; plasma density (ρ), temperature (T ) and con�nement time (τ ). �ese 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 circumventthis issue is to restrict the Lawson criterion to only describe the hotspot, this is achieved byreplacing the plasma pressure (p) with the ideal gas equation of state (Equation 2.11), whereρh is the hotspot mass density, Th is the hotspot temperature and mi is the DT average ionmass (Atzeni and Meyer-ter Vehn (2004, p. 37), Zhou and Be�i (2008)).

p =2ρhThmi

(2.11)

�is leads to the hot-spot ignition condition (which is valid for hotspot temperatures between5 and 15kev):1

ρhRhTh > 6

(ρhρc

)0.5

g cm−2 keV (2.12)

�is method has two issues a) τ is incorrect as it does not take into account the cold shell andb) ρhRh, the hotspot areal density cannot be experimentally measured.

�us 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 Be�i (2008). Howeverfor the purpose of this paper the above static model for Lawson criteria will be su�cient.

2.4 Plasma Instabilities

Plasma instabilities play a signi�cant role in the performance of ICF. �ey pose a limit onthe maximum allowed radiation temperature, which in turn places a limit on the maximumablation pressure. �is culminates in a limiting factor on the maximum power and energy thelasers can provide to the fusion material (Atzeni and Meyer-ter Vehn, 2004, p. 122).

In ICF, plasma can be treated as a �uid (and thus analysed using magnetohydrodynamics)and therefore its instabilities can be split into two main categories, hydrodynamic and kinetic.

1�is has been con�rmed by numerical simulations and experimental data, the analytical derivation is givenin Atzeni and Meyer-ter Vehn (2004, p. 91).

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 instabilitygrowth 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 explorethe upper limits and de�ne the maximum gain.

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 con�ne the plasma at ignition temperatures. Gain is a veryimportant factor when considering ICF in terms of energy production, Figure 3.1 shows theenergy 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 heatingat an ignition temperature of 5keV can be estimated at ≈ 20 which is too low for InertialFusion Energy (IFE).1 �is 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 coldcompressed fuel (the solid DT layer) (Atzeni and Meyer-ter Vehn, 2004, p. 44).

1Calculated by dividing the fusion energy released by a DT reaction (17.6Mev) by the thermal energy of twoions and two electrons at 5keV (=30keV), and then multiplying this by a burn e�ciency of 0.3 and a beam-to-fuelcoupling e�ciency of 0.1.

11

12 Chapter 3 Gain

3.1 De�nition

Target energy gain is de�ned as (Atzeni and Meyer-ter Vehn, 2004, p. 102):

G =EfusEd

=qDTMfΦ

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 e�ciency), the ratio between the total number of fusionreactions and the number of DT pairs initially present in the plasma volume. It can beapproximated 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, whereη is the overall coupling e�ciency.

• Mf is the fuel mass, which is the sum of the mass of the hotspot (Mh) and the mass ofthe cold fuel (Mc) (Equation 3.2).

Mf = Mh +Mc =4π

3

[ρhR

3h + ρc(R

3f −R3

h)]

(3.2)

We can also de�ne the fuel energy gain as Gf = Efus/Ef , which is related to the target gainby:

G(Ed) = ηGf (ηEd) (3.3)

�is section explores the two main ICF assemblies, the Isobaric and Isochoric con�gurations.In addition to deriving the target and fuel gains we investigate the limiting gain for the targetand fuel cases. �e limiting gain is the maximum gain achievable for a speci�c driver energy.

3.2 Isobaric Gain Derivation

�e Isobaric compression is the ICF assembly utilised in direct drive implosions, this paperfollows the derivation given in Atzeni and Meyer-ter Vehn (2004, p. 111) and is known asthe ‘Hot spot ignition model for isobaric compression’. �e compression takes place underconstant 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:

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

�erefore 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

qDTMc

Ef

√Hc

HB(3.5)

Next the cold fuel mass and burn parameter for the cold fuel need to be expressed in terms ofpressure and fuel energy. �is requires the following relations:

ρc = (αAdeg)−0.6p0.6c (3.6)

Here we have introduced the degenerate fuel and isentrope parameter, these are fundamentalfeatures in ICF they refer to extremely high compression involving Fermi-Degenerate fuel. αis the isentrope parameter and is de�ned as the ratio of the pressure of a fuel (as a functionof density and temperature) divided by the degenerate pressure of the fuel as a (function ofdensity), it measures fuel entropy and is constant during isentropic compression (Atzeni andMeyer-ter Vehn, 2004, p. 52). Adeg is a constant which depends on the composition of thefuel and is 2.17 × 1012 (erg/g)/(g/cm3)2/3 for equimolar DT composition. Using the ideal gasequation and the condition for isobarity we get the following relation for fuel and hotspotenergy for a mono-atomic gas:

Eh =3

2pVh = 2πpRh

3 Ef =3

2pVf = 2πpRf

3 (3.7)

Finally by using the relation Ef/Eh = (Rf/Rh)3 and combining the above equations leadsto:

Mc = ρc(Vf − Vh) =( p

αAdeg

)0.6 2Ef3p

(1−

(RhRf

)3)

(3.8)

Hc = ρc(Rf −Rh) =( p

αAdeg

)0.6(Ef2πp

)1/3(1− Rh

Rf

)(3.9)

We now plug these equations into the main equation for fuel gain (Equation 3.5) and introducethe parameter x for convenience, which is Rh/Rf which from Equation 3.7 is also equal to(Eh/Ef )1/3. �is results in the fuel gain, G(Ef , x, p, α,HB,Adeg, qDT ) (Equation 3.10).

Gf =qDT

3H1/2B p4/15

(1− x3)√

1− x(αAdeg)9/10

(Ef2π

)1/6

(3.10)

14 Chapter 3 Gain

�e 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 �nal parameter FDT = phRh which is calculated from theLawson ignition condition (Equation 2.12, HhTh ' 2(g/cm2)keV) and has the value 15Tbarµmfor Th = 8keV and Hh = 0.25g/cm2. �us pressure can be wri�en as:

p =

(2πF3DT

Efx3

)1/2

(3.11)

�e fuel gain is now given by:

Gf = AG

(Efα3

)0.3

f(x) (3.12)

Where AG and f(x) are de�ned by:

AG =qDT

3(2π)3/10H1/2B A

9/10deg F

2/5DT

f(x) = x2/5(1− x3)√

1− x (3.13)

�e limiting fuel gain is found by maximising f(x) this occurs at x∗ ' 0.3485 and givesf(x∗) = 0.507.

G∗f ' 6610

(Efα3

)3/10

f−2/5FDT

fqDT f−1/2HB

(3.14)

Ef is given in megajoules and fFDT, fqDT and fHB

are variations of the �xed parametersaround their reference values (FDT , qDT , HB). Finally the limiting target gain is given bycombining Equation 3.14 and Equation 3.3.

G∗ ' 6610η

(ηEdα3

)3/10

f−2/5FDT

fqDT f−1/2HB

(3.15)

3.3 Isochoric Gain Derivation

We now follow the partial derivation given by Kidder (1976) for the Isochoric assembly, inisochoric compression uniform density is assumed instead of uniform pressure, ρ = ρh = ρc.In this case the fuel gain is given by:

Gf =ΦqDTωc + ωh

(3.16)

• ωc(eV ) = 3αε2/3 (for α ≥ 1) and is the internal energy (per electron) of the fuel whencompressed ε-fold times normal solid density (ρ0 =0.2g/cm3) and α is the isentropeparameter.

• ωh(eV ) = 3x3Th is the addtional internal energy (per electron) due to the hotspot,x = Rh/Rf from the previous section.

Chapter 3 Gain 15

• Φ is the burn fraction from Section 3.1.

ε2/3 can also be given by Hf/H0, with H0 = (3Mfρ20/4π)1/3 which is the burn parameter

‘ρR’ fuel would have at normal solid density (ρ0). �us the isochoric fuel gain can be wri�enas:

Gf =qDTHf

3(HB +Hf )

(αHf

H0+ Th

(Hh

Hf

)3)−1

(3.17)

�is equation transforms into the following equation from Atzeni and Meyer-ter Vehn (2004,p. 123) Equation 3.18.2

G∗f = 0.0828qDT

H1/2B A

7/6degFDT

2/9H4/9h

(Efα3

)7/18

(3.18)

�e 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. �us the limiting gain is simpli�ed to:

G∗f = 2.18× 104

(Efα3

)7/18

(3.19)

2�e proof of this is outside the scope of this paper. However we have veri�ed that both equations are equivalent.

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. �ere are 3 types of hydrody-namic instability that e�ect the stability of ICF:

• Rayleigh-Taylor, the main type of instability and the one that has the largest e�ect onthe performance of ICF.

• Richtmyer-Meshkov, occurs when a shock wave passes through an boundary betweentwo �uids, when the boundary is not �at. It’s relevance to ICF is that RMI can produceseeds which are then ampli�ed by RTI (Pfalzner, 2006).

• Kelvin-Helmholtz, occurs in a strati�ed �uid with the layers in shear motion, small sinu-soidal perturbations grow exponentially in time. KHI plays a minor role in the non-linearevolution 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 �uids separatedby a horizontal boundary, both �uids are subject to gravity (Figure 4.1). When ρ2 > ρ1 smallperturbations of the interface will grow in time. In a short period of time the heavier �uid willsink down in spikes and the lighter �uid will rise in bubbles (Figure 4.2). �is occurs because“any exchange of position between two elements with equal volume of the two �uids 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 stagnationphase. However instead of the �uids being subject to gravity (with the acceleration due togravity providing the driver for the instability growth) we consider two �uids in an acceler-ated frame. In this case the denser �uid is the outer surface of the pellet during the inwardacceleration phase, and the inner surface at the implosion stagnation phase (Pfalzner, 2006).

16

Chapter 4 Hydrodynamic Instabilites 17

Figure 4.1: Rayleigh-Taylor unstable interfaces between �uids of di�erent 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 �uids in which thedensity may change in space, where RTI does not involve a sharp boundary between the �uidsbut 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 wri�en 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)

18 Chapter 4 Hydrodynamic Instabilites

By assuming small perturbations the above equations can be recast as (where tilde denotessmall perturbations):

∂ρ

∂t+ uz

dρ0

dz= 0 (4.5)

ρ0∂ux∂t

= −∂p∂x

(4.6)

ρ0∂uz∂t

= −∂p∂z− aρ (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.

k2p = −σρ0duzdz

(4.9)

�e evolution equation for uz is obtained by substituting the Fourier transformed versionof Equation 4.5 into Equation 4.7 and eliminating p in the resulting equation and the one above.

d

dz

(ρ0duzdz

)− ρ0k

2uz = −k2

σ2adρ0

dzuz (4.10)

Velocity perturbations vanish at large distances from the interface, thus taking solutions whenuz → 0 and z → ±∞ and integrating over z from −∞ to∞ gives the general growth rate( Equation 4.11).

σ2 = k2

∫∞−∞ a

dρ0

dzu2zdz∫∞

−∞ ρ0(z)[(duz

dz

)2+ k2u2

z

]dz

(4.11)

Now starting from the case described in section 4.1, where we have two superimposed homo-geneous �uids characterized by Equation 4.12 (which assumes a sharp interface between the�uids) the classical RTI growth rate can be derived.

ρ0(z) =

ρ2 z > 0

ρ1 z < 0(4.12)

�e condition of continuity of the velocity component normal to the unperturbed boundary islimz→0+(uz) = limz→0−(uz) = uz0. Both the �uids have uniform density with the derivativeof density with respect to time equal to 0, thus the evolution equation for uz ( Equation 4.10),when z 6= 0, is given by Equation 4.13.

uz =

uz0e−kz z > 0

uz0ekz z 6 0

(4.13)

Chapter 4 Hydrodynamic Instabilites 19

�e linear growth rate can be found by inserting Equation 4.10, the densities and the densityderivative (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 de�ned as (ρ2 − ρ1)/(ρ2 + ρ1). Finally we generalisethe classical RTI growth rate to the case of a strati�ed �uids. �us instead of Equation 4.12 wehave:

ρ0(z) =

ρ1 +

ρ2 − ρ1

2exp(2z

L

)z 6 0

ρ2 −ρ2 − ρ1

2exp(− 2z

L

)z > 0

(4.15)

L is a characteristic density scale length (ρ/∇ρ). Inserting this equation into the equation forgeneral growth rate ( Equation 4.11) and assuming Equation 4.13 holds for the strati�ed �uidsthen 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 thegrowth of RTI modes and even fully stabilizes short wavelength modes (Atzeni and Meyer-terVehn, 2004, p. 257). A simpli�ed treatment of ablative RTI can be derived from the observationreported by Kilkenny et al. (1994) which shows that the eigenfunctions of classical RTI expo-nentially grow in time and exponentially decay in space. �e perturbations in ablative RTI alsogrow, but because of ablation the interface changes position, moving into the material with anablation velocity ua. �us, the e�ective perturbation growth is exp(σRT )exp(−kua∆t), fromthis the classical ablative growth rate can be obtained:

σRT =√ak − kua (4.17)

�e classical ablative growth rate can be generalised to �t a range of analytical solutions forvaryingF and ν. WhereF is the Froude number (F = ua

2/aL0) which a�ects the normalizedpressure pro�le andL0 is the ablation-front thickness. For largeF the peak pressure is close tothe ablation front and as F decreases the peak pressure moves away from the ablation front. νis the e�ective power index for thermal conduction and comes from the thermal conductivitypower law (χ = χ0T

ν ) and is dependant on whether the energy is transported by electrons orphotons (Atzeni and Meyer-ter Vehn, 2004, p. 263).

ν

= 2.5 Electron heat di�usion

> 3 Radiative heat di�usion

< 2 Radiative e�ects in direct drive targets

(4.18)

20 Chapter 4 Hydrodynamic Instabilites

�e derivation of the generalised relationship is beyond the scope of this paper but can be foundin full in Be�i et al. (1998) and was �rst proposed by Takabe in 1985. Known as the ‘Takaberelation’ and the ‘generalised Takabe relation’ they form the basic fundamental relations of thelinear 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 ��ing functions depending only on F and ν (Atzeni and Meyer-ter Vehn, 2004,p. 269).

4.4 Feed-through

�e formula for classical RTI deals with �uids that are in�nite or semi-in�nite however for ICFwe use relativity thin shells. Because of this we have to deal with the phenomenon knownas 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 ofthe fuel pellet and surface b is the inner surface of the solid DT section of the fuel pellet. Asthe perturbation of surface a grows, the perturbation of surface b grows at the same rate butwith a reduced amplitude (exp(−k∆z)), meaning that a perturbation from an unstable surfaceis transmi�ed to a stable one. �e implications of this for ICF are such that “perturbationsthat 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). �ese perturbations are the main causeof instabilities occurring at implosion stagnation (when perturbations with λ are much smallerthan 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. Againthis RTI is ablative due to the heat andα-particle �ux from the hotspot which causes ablation ofthe inner surface of the decelerating shell (Atzeni and Meyer-ter Vehn, 2004, p. 278). �e growthrate σin at the inner shell surface (Equation 4.20) is approximated by the same relation usedfor ablative RTI (Equation 4.19) but using the notation for the inner surface of the deceleratingshell. k = l/R (l is the spherical mode number described in the next section),Lin the minimumdensity scale length at the hot spot surface, ua−in is the ablation velocity at the inner shell

Chapter 4 Hydrodynamic Instabilites 21

surface and with βin which is a numerical coe�cient.

σin =

√Rl/R

1 + Linl/R− βin

l

Rua−in (4.20)

�is equation shows that ablative �ow 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 canbe found in Lobatchev and Be�i (2000).

4.6 �e growth factor

From Section 2.3 we know that for ignition to occur a fuel shell with thickness ∆R(t) mustpreserve its integrity during implosion and also create a central hot spot at stagnation with aradiusRh. 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 thefollowing conditions must be satis�ed: ζout(t) � ∆R(t) during implosion, and ζin(t) � Rh

at implosion stagnation. ζout and ζin represent the deformation amplitudes directly relatingfrom the hydrodynamic instabilities.

Using Lindl (1997) as a base and se�ing 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-�ight-aspect-ratio (Aif ), described in Section 2.1, can be retrieved.�is will require examining the e�ect of the hydrodynamic instabilities on target design forthe 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 alteredto 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 aspherical interface we must explore converging �ows, this is achieved using a perturbed poten-tial, through potential theory, and is solved in spherical geometry (r, θ, φ, t) by a superpositionof modes. By solving the perturbed potential we reach the Bell-Plesset equation (Atzeni andMeyer-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

lmkRζl = 0 (4.21)

Here we have introduced a mass variable to simplify the equation (m = ρR2/k), where k isthe wave number given by k = (l+ 1)/R and R is the unperturbed radius. �e mode numberhas a large e�ect on the size of RTI growths, for example it can be shown fast growing RTImodes have mode number of l ≈ 30 whereas modes l ≈ 100 are stable, this is explored in alater section (section 5.3).

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 amplitudeof the outer deformation of a particular mode l, ζoutl , is equal to the initial amplitude, ζoutl0

(due to the pellet surface not being perfectly smooth), times the growth factor which is givenby Equation 4.23.

ζoutl = ζoutl0 Goutl (4.22)

Goutl = exp

(∫ t0

0σl(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

Rua (4.24)

�e integral in Equation 4.23 can be solved by assuming a constant acceleration as the shellimplodes (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 pro�le at the ablation front and f2 is related to thefraction of ablated mass and has typical values:

f2 ≈

0.8 Indirect drive

0.2 Direct drive(4.27)

�us the growth factor is approximately given by Equation 4.28.

Goutl ' exp

[α2

√l

1 + l(f1/2)A−1if

− β2f2A−1if l

](4.28)

R/∆R is a characteristic value ofAif and has been substituted into the above equation. Stablemodes (l > lcut) can be found by substituting Goutl = 1 to give Equation 4.29.

lcut =Aif2f1

(√1 +

4f1α22

f22β

22

Aif − 1

)(4.29)

Chapter 4 Hydrodynamic Instabilites 23

Figure 4.3: Plot showing how the growth factor at the outer surface varies with modenumber, plo�ed 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 adirect result of the pressure exerted by the inner hot gas of the fuel pellet. �ese perturbationsare caused by 2 criterion: a) Defects of the surface of the inner solid DT fuel pellet, and b) feed-through from the ablation front. �e defects have modal amplitudes ζinl00 and the feed-throughhave amplitudes ζin−feedl (Equation 4.30), caused by perturbations at the outer surface whichare transmi�ed to the inner surface.

ζin−feedl = ζoutl Gfeedl (4.30)

Gfeedl ' exp(−lAif

)(4.31)

�e e�ective initial amplitude of mode l (ζinl0 ), assuming random phases, can be approximatedas:

ζinl0 ≈√

(ζinl00)2 + (ζin−feedl )2 (4.32)

�us the perturbation amplitude of the inner surface can be given in a similar manner to the‘Perturbation growth at the ablation front.’

ζinl = ζinl0 Ginl = ζinl0 exp

(∫ tdec+∆tdec

tdec

σinl (t)dt

)(4.33)

Again to calculate the integral a similar assumption must be made, the hot spot radius decreasesat constant acceleration from Rdec to Rh. Also similar parametrizations are made (from abla-tive front perturbations), Atzeni and Meyer-ter Vehn (2004, p. 292):

24 Chapter 4 Hydrodynamic Instabilites

Figure 4.4: Plot showing how the growth factor from feedthrough varies with modenumber, plo�ed for several values of Aif

• Rdec −Rh ≈ f3Rh

• Lin = f4Rh

• ua−in =f5Rh∆tdec

Where f3, f4 and f5 are numerical constants. �us evaluating the integral gives the growthfactor of instabilities at the inner shell surface:

Ginl ≈ exp

[√2f3l

1 + f4l− βinf5l

](4.34)

It is worth noting that the growth of these perturbations is independent ofAif , thus the initialamplitudes must be larger before any degradation to performance is noticed.

In addition, the cut o� for inner surface of the fuel capsule is found through the same methodutilized in the previous section to give:

lcut =1

2f4

(√1 +

8f3f4α22

f25β

2in

− 1

)(4.35)

Total growth factor

Finally, the total growth factor is found by multiplying the individual growth factors calculatedin the previous sections:

GT = Goutl Gfeedl Ginl ≈ exp

[α2

√l

1 + l(f1/2)A−1if

− β2f2l

Aif− l

Aif+

√2f3l

1 + f4l− βinf5l

](4.36)

Chapter 4 Hydrodynamic Instabilites 25

Growth Parameters (Approximate) Typical Valuesf1 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 (plo�ed using typical fusion values, Table 4.1) forseveral values of Aif . �e full anylsis of the growth factors is explored in Section 5.3, howeverit is worth noting that the maximum total growth factor for an in-�ight-aspect-ratio of 50 isapproximately 1000. �is means that due to hydrodynamic instabilities, a �nal perturbationamplitude 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, plo�edfor several values of Aif

Chapter 5

Discussion

5.1 RTI’s e�ect on the Driver Energy

We know from the Lawson criteria, the driver energy Ed must be equal to or greater than theignition energyEign for ignition to occur. If we assume that the fuel energy is roughly approxi-mate to the driver energy times the overall coupling e�ciency (Ed = Ef/η), which is the sameassumption used in the derivation of limiting gain. �en from Equation 5.4 and Equation 5.3the minimium driver energy can be de�ned as:

Ed ≥

kproη−1A−3if p−1.2a Direct drive

kproη−1hohlα

−1.8A−6if T

−8.2r Indirect drive

(5.1)

Where kpro is just a generic constant of proportionality that varies depending on the param-eters used. �is equation shows that for a �xed pa then as the in-�ight-aspect-ratio increasesthe driver energy required reduces. It can be concluded from this that a higherAif is bene�cialto achieving ignition. However as will be seen in the Section 5.3 RTI limits Aif to a maximumvalue, any higher than this value and ignition would not be achieved because of fuel-ablatormixing (mentioned in Section 4.1). �erefore RTI sets a lower bound on the amount of energyrequired to achieve ignition.

Although not explored in this paper the upper bound is set by the ablation pressure and limitedby plasma instabilities (Atzeni and Meyer-ter Vehn, 2004, p. 122), shown by the do�ed linein Figure 5.1. From the graph it can be deduced that for ignition to occur the laser parametersmust lie between the ‘maximum allowed temperature’ curve and one of the other curves whichset the minimum driver energy for ignition. As Aif decreases the area between the minimumand maximum curves, the operating window, decreases. �is graph is of great importance inICF as it shows the minimum energy and power required for ignition.

26

Chapter 5 Discussion 27

Figure 5.1: Graph showing curves of ignition power (for indirect drive) as a functionof ignition energy for di�erent values of Aif , also shown on graph is the Pd for themaximum allowed temperature.

5.2 Relation between RTI and Gain

Finding the relationship between the in-�ight-aspect-ratio and Gain (G) �rst requires examin-ing how the implosion velocity (uimp) and fuel energy (Ef ) are e�ected by the in-�ight-aspect-ratio (Aif ). From the rocket model (Section 2.2) the implosion velocity as a function ofAif canbe approximated by:

uimp ∝

α0.6if AifT

0.9r Indirect drive√

α0.6if 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 fordirect drive. �e implosion velocity can then be related to the stagnation pressure.

p ∝ u3impα

−0.9if p0.4

a (5.3)

Within the isobaric model the fuel energy is related to the the stagnation pressure by (thisrelation is derived in the isobaric gain derivation, Section 3.2):

Ef ∝1

p2(5.4)

• �e ablation pressure pa is a�ected by the laser intensity and hohlraum temperature,which are limited by the plasma instabilities.

• �e in-�ight isentrope parameter αif , determines the shock and entropy evolution ofsystem and is subject to preheat and pulse shaping

�us from the de�nitions of limiting fuel gain (Section 3.2) and by combining with Equation 5.4the relationship between Gain and pressure can be found.

Gf ∝ E0.3f ∝ p−0.6 (5.5)

28 Chapter 5 Discussion

From the above relation the gain can be linked with the Aif .

Gf ∝

A−1.8if Indirect drive

A−0.9if Direct drive

(5.6)

�is relationship con�rms our hypothesis of how the Gain is related to the in-�ight-aspect-ratio; which were, asAif increases the gain decreases. From this we can infer that a small valueof Aif is more desirable in ICF. We must now �nd a direct link between Aif and the growthfactor so that we can set a maximum value for the growth factor and therefore determine thecorresponding maximum allowed Aif .

5.3 Exploring the In-Flight-Aspect-Ratio

Equation 4.36 clearly shows that a smallerAif leads to smaller growth factors. Constraints forAif are estimated using the equations mentioned in the previous section. Using �uid codessuch as HYDRA the evolution of a wide spectrum of growth modes for a large sector of afuel capsule can be simulated. “�ese simulations require huge amounts of time and computing

power and are still currently unable to resolve very short wavelengths” (Atzeni and Meyer-terVehn, 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 shelldensity increasing and the entropy decreasing, which is not desirable (Atzeni and Meyer-terVehn, 2004, p. 298).

�e Aif relates directly to the roughness of the shell pellet surface (explored in the next sec-tion). A smallerAif represents rougher targets, which leads to a higher driver energy or a lowerenergy gain, Equation 5.6. Because of this a comprise value for Aif must be found which islarge enough to allow the energy provided by the laser to ignite the fuel, but small enough sothat the capsule remains hydro-dynamically stable and fuel-ablator mixing does not preventthe fuel from igniting.

To �nd this value this we should �rst understand how the growth factor varies with Aif ,Section 4.6 derives the equations for the various growth factors. �e �gures plo�ed in thatsection show how the growth factor varies with mode number, there are several interestingconclusions we can draw from these graphs.

1. �e growth factor at the outer surface is the most signi�cant contributor of growth tothe total growth factor. �is indicates that it is the most important factor to be limitedwhen trying to reduce the overall growth.

Chapter 5 Discussion 29

2. Growth from feed-through actually helps to reduce the overall growth at high modes.�is can be veri�ed by comparing the total growth factor to the growth at the outersurface, for Aif = 50, the maximum value for Goutl is roughly 304 compared with 2000

for GT .

3. �e cut o� mode for GT is much lower than Goutl (∼ 120 compared with ∼ 1200 forAif = 50). �is is due to bothGinl andGoutl Gfeedl having lower cut o� modes thanGoutl .

4. Growth at lower mode numbers can be ignored as the overall growth is very low. �emost unstable modes are around 20 (depending on the value of Aif ).

5. Growth at the inner surface of the pellet is independent ofAif . �erefore the roughnessat outer surface should be considered more important than at the inner surface (whenevaluating the growth factor and instability growth).

6. �e total growth factor GT increases exponentially up to a maximum value then dropssharply away.

We can also calculate the relationship between the in-�ight-aspect-ratio and the total growthfactor. To do this we plo�ed the GT for several values of Aif (similar to Figure 4.5). Nextwe took the maximum growth factor for a speci�ed mode number from each of the plots andplo�ed it against Aif , the resulting graph is shown in Figure 5.2. �e graph shows that for20 < Aif < 80 the relationship is approximately linear, we can ignore values outside thisrange as these would result in either too li�le gain or hydrodynamic instabilities that are solarge that ignition would never occur. We have now determined the �nal relationship betweenthe growth factor and the in-�ight-aspect-ratio (GT ∝ Aif ). Finally we calculated the equationof this linear section to �nd 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 thuscalculate the maximum allowed Aif . Lindl (1995) limits the growth factor to 1000 this leads toan in-�ight-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). Zhouand Be�i (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 e�ect on the outer capsule surface �nish

In an ideal scenario we would like a fuel pellet to be perfectly smooth as the vast majority ofinstability growth is seeded by initial perturbations on the capsule’s outer surface (Section 4.6).

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

. Gro

wth

Fac

tor

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 modeversus 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 containsa wide spectrum (with random phases) of initial perturbations with a root-mean-squared am-plitude (Lindl, 1995):

ζrms =

√∑l

ζ2l (2l + 1)−1 (5.8)

By combining the equations for perturbation amplitudes in Section 4.6 with the total growthfactor we can show how the resulting �nal perturbation amplitude is a�ected by the surfaceroughness.

ζinl =√

(ζinl00Ginl )2 + (ζoutl0 GT )2 (5.9)

ζoutl0 is the initial perturbation amplitude at the outer surface of the fuel pellet and thus equal toζrms and ζinl00 is the initial perturbation amplitude of the inner surface of the fuel pellet. If wechoose 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)

Chapter 5 Discussion 31

�is is obviously an overestimation as it assumes that all modes have the maximum growthfactor, which is not true, but for the purpose of this analysis this assumption does not overlye�ect 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 theinitial perturbations) changes with surface roughness, the importance of this is explained inthe section on basic ICF principles. �e 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 thereforea lower gain. To achieve this we �rst assume that perturbations with initial amplitude ζrmscover the entire surface of the DT fuel. Next we consider a shell on the inner surface of thefuel with radius equal to ζrms. We now assume that ∼ 1/2 the volume of this shell is fuel andthe other half is the plastic ablator layer introduced because of RTI fuel-ablator mixing. 1 �enthe percentage of fuel-ablator mixing (with 0% meaning there is no mixing) as a function ofsurface roughness and the in-�ight-aspect-ratio is:

100

2

(ζrms(34.776Aif + 346.63)

Rf

)3

(5.11)

�is formula allows for the calculation for the percentage of fuel-ablator mixing for di�erentvalues of surface roughness, in-�ight-aspect-ratio and fuel radius (Rh +Rc). Figure 5.3 showsfuel mixing curves for the minimum fuel radius for ignition with maximum gain (with a driverenergy of 1MJ and a fuel mass of 1mg) Rf = 143µm.

Figure 5.3: Surface roughness versus percentage of fuel-ablator mixing for a numberof di�erent Aif .

Lindl (1995) states that the surface �nish must have an ζrms of between 200− 300 angstroms= 20− 30nm, this was later re�ned by Lindl et al. (2004) to 10− 20nm, which is the currentlimiting value for ICF pellet design used at NIF. It is apparent from the graph why these limitswere chosen as there is almost no fuel-ablator mixing in that range. However, in reality this is

1�is assumption originates from the prime assumption that all the perturbations on the surface are all iden-tical in height, width and spacing. �is is obviously a gross overestimation and is an area that requires furtherinvestigation.

32 Chapter 5 Discussion

not true because of the assumption made in the footnotes and the initial perturbation amplitudeof 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 shapedin time to provide the fast and nearly isentropic compression required. However the NIF is cur-rently experimenting with shaped pulses that have signi�cantly lower instability growth, Fig-ure 5.4. �e low-foot pulse uses the type of pulse shaping that is currently utilised in ICF

Figure 5.4: �is graph shows two di�erent 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 compressionof any shot to date” (Raman et al., 2014). �e pulse currently underdevelopment at NIF isdubbed the high-foot pulse, and has “produced the highest neutron yields to date, nearly 10times higher than what was achieved during NIC” (Raman et al., 2014). �e high-foot di�ersfrom the low-foot pulse in several aspects (Figure 5.5):

1. �e 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 largersustained peak towards the end. However, the low-foot pulse has an additional smallpeak just before the main sustained peak.

3. �e high-foot pulse has an overall lower peak radiation temperature.

�e reasoning behind the these di�erences in pulse shape is beyond the scope of this paper,however they are explained in detail in the paper by Raman et al. (2014). �e resulting bene�tsfrom using this drive are obvious, by reducing the maximum growth factor through pulseshaping the relationship between the in-�ight-aspect-ratio and the growth factor is changed

Chapter 5 Discussion 33

(Equation 5.7). �erefore a largerAif can be used while achieving the same gain, in addition tothis the ignition energy is reduced (Equation 5.1). �e consequence of this is that less energyis required, thus increasing the overall e�ciency and performance of ICF.

Figure 5.5: Comparison of the high-foot and low-foot drives (Raman et al., 2014).

Chapter 6

Conclusion

�e e�ect 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 thedirect and indirect cases and also examined the connection between RTI growth and the laserenergy 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 ultimatelyleads to a lower gain. �erefore a comprise between ignition energy and gain is required andas we have seen this results in an in-�ight-aspect-ratio of approximately 40.

In addition to exploring the hydrodynamic instability growth in terms of Aif , the relationshipbetweenAif and the maximum growth factor has been discovered. �is has also allowed us toinvestigate the association of the surface roughness of the fuel pellet to the maximum growthfactor and thus the Aif and the overall performance of ICF. From this we have con�rmed thereasoning behind current limits for the surface roughness. �e model we created is only the�rst step in examining the full e�ect of fuel-ablator mixing in relation to surface roughness.A further extension of this project could improve on this model by re�ning some of the as-sumptions made and also by including the e�ect of the initial inner surface perturbations onthe �nial perturbation amplitude.

Finally methods of reducing instability growth and its e�ect on ICF performance have beenbrie�y looked into. �ese new methods have mainly arisen from the fact that we have almostreached our technical and manufacturing limit regarding the smoothness of the fuel pellets.�us methods such as pulse shaping o�er an achievable method of reducing instability growthwhile keeping the input energy su�ciently 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 usone step closer to achieving sustained fusion. �is in turn will eventually lead to completion ofthe principal aim of terrestrial fusion, providing almost limitless ‘clean’ energy and eliminatingthe use of fossil fuels.

34

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