Shock ignition John Pasley University of York / CLF
Lecture structure Reading list Primer on shock waves Shock waves in conventional ICF Why shock ignition? Methodology Advantages of shock ignition Outlook for shock ignition
Reading list Shock waves Zel dovich and Razier The Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (Academic Press hardback 2 vols, pub.1966, or Dover paperback version in a single volume, pub. 2002). ICF Lindl Inertial Confinement Fusion: The Quest for Ignition and Energy Gain Using Indirect Drive (As AIP hardcover or Phys. Plasmas volume 2, 3933, 1995)
Shock ignition The original paper: Shock ignition of TN fuel with High Areal Density, R. Betti et al, PRL 98, 15501, 2007 Potential for use at NIF/ LMJ: Shock ignition: A New Approach to High Gain ICF on the NIF, L. J. Perkins et al, PRL 103, 045004, 2009 High-gain shock ignition of direct drive ICF targets for the LMJ, B. Canaud and M. Temporal, New Journal of Physics, 12, 043037, 2010 First experiments: Initial experiments on the shock-ignition ICF concept, W. Theobald et al, Phys of Plasmas, 15, 056306, 2008
Introduction Shock ignition is an advanced central hotspot (CHS) ignition strategy It is sometimes known as Shock fast ignition, for reasons we will consider It differs from conventional CHS ignition (although this also relies upon shock waves for ignition, but in a slightly different way!)
In order to get some useful understanding of Shock Ignition, we need some background in conventional ICF (particularly as regards shock waves!)
A very short introduction to shock wave hydrodynamics
We are familiar with the effects of small pressure disturbances in a gas: sound waves
What happens if the pressure disturbance is large? Higher pressure parts of the wave go faster Profile steepens Shock wave forms
Some pictures of natural shock waves Mach ~30 (at impact) Mach ~30 (near bolt) Mach ~few (much higher near time of event)
Some pictures of man made shock waves ~ Mach 1 ~Mach 1 (goes faster obvs!) Mach ~1 Mach 100+
The shock wave Direction of increasing density, temperature, pressure, internal energy, entropy etc u p T 1,P 1, D T 0,P 0, Spatial dimension A shock wave is a discontinuous jump in the hydrodynamic variables; thickness of shock is dependent on dominant mechanism of energy transport (viscosity, electron heat conduction, radiation) and the relevant mean free paths in the material
Shock waves are discontinuous jumps in the gas dynamic variables Direction of increasing density, temperature, pressure, internal energy, entropy etc u p T 1,P 1, D T 0,P 0, Spatial dimension Shock waves are always: -supersonic with respect to the material ahead of the shock (D>c s0 ) -subsonic with respect to the material behind the shock (D-u p )<c s1 The Mach number of a shock is given by M=D/c s0
Easier to consider in frame in which shock is stationary Direction of increasing density, temperature, pressure, internal energy, entropy etc u 1 =u p -D T 1,P 1, u 0 = -D T 0,P 0, Spatial dimension
Rankine-Hugoniot Relations Mass equation By conservation of mass: u 0 u 1 Thence,
Rankine-Hugoniot Relations Momentum equation The mass entering the shock front in a unit of time dt is given by A.dt.u 0 This mass undergoes a velocity change u 0 -u 1, and so a momentum change A.dt.u 0 (u 0 -u 1 ) Now the force acting is Hence: (P 1 -P 0 )A.dt P 1 -P 0 =u 0 (u 0 -u 1 )=D u p
Rankine-Hugoniot relations Energy equation The kinetic energy entering the shock in unit time dt is mass x u 02 /2 or A.dt.u 03 /2 Conservation of energy requires that the energy leaving the front be the same, divided between KE and internal energy U. The KE leaving the front is just mass x u 12 /2 and the change in U must then be mass x(u 02 -u 02 )/2 Thence: U 1 -U 0 = (u 02 -u 02 )/2 =(2u p D-u p2 )/2 Or alternatively: U 1 -U 0 =(P 1 +P 0 )(V 0 -V 1 )/2 Where V is the specific volume e.g. V 1 = 1/ 1
Hugoniot curves The mass and momentum equations are independent of the material being shocked. In order to solve the energy equation, however, we also need an equation of state (EOS) for the material e.g. P= ( -1) (U/V) Where is the ratio of specific heats C P /C V. For an ideal gas, =5/3. Using the EOS, we can write an expression for the specific volume as a function of P 0,V 0, the shock pressure P 1 and V 1 = [P 0 V 0 ( P 1 V 0 ( [P 0 ( P 1 ( eqn. 1 Plotted in P-V space this describes a Hugoniot curve.
Hugoniot curves The Hugoniot curve is the locus of all possible shock states (produced by shocks of varying strength) from a given initial state. The Hugoniot curve lies above the isentropic curve given by: P P 1 V 1 P 0 V 0 The change of temperature and entropy in the shock also follow directly from equation 1 using V and Note that weak shocks add very little entropy
Entropy jumps and limiting compression The Hugoniot diverges strongly from the isentropic compression curve only for strong shocks (P 1 >>P 0 ) If compression is achieved by two successive shocks, then a new Hugoniot must be drawn based upon the conditions created by the first shock, in order to establish the conditions that may be reached by the second shock We can see by looking at the equations for temperature and entropy that the temperature and entropy change is a function of the relative pressure change NOT the absolute final pressure Also it is possible to show that for very strong shocks (where the entropy and temperature changes are very great) that the compression approaches a limiting value given by: ( In other words, compression is limited to a factor of 4 for an ideal gas ( =5/3)
The implications are: In order to produce high compressions we must use multiple shocks In order to limit the entropy and temperature change the relative strength of each successive shock (P x+1 /P X ) should be limited
Radiating shocks For very strong shocks, such as are present in ICF targets, the post shock temperature T 1 is high enough that radiation from behind the shock wave can influence the material ahead of the shock (since radiation propagates at the speed of light faster than the shock wave!)
Radiating shocks Shock is said to be critical when T preheat =T 1 Immediately following the viscous jump, the gas is out of equilibrium: superheating T crit ~ 100eV typically x, the preheating depth, is approx. 1 radiation mfp thick for a critical shock
Pressure and density profiles of radiating shocks
In ICF fuel radiative effects are limited Fuel is low-z, and ionized so radiative emission is weak from the thin layers of fuel present in the capsule In indirect drive, the hohlraum wall is high- Z, so radiation can dominate the energy transport here
Weakening of shocks Planar shocks dissipate energy as they pass through a material, so gradually weaken This weakening is greatly exacerbated for expanding shocks (e.g. Spherical exploding shocks) Imploding shocks, on the other hand, strengthen as they propagate (although the total energy in the front diminishes)
Detonation waves A detonation wave is a shock wave that is sustained in strength by a release of energy that results from the passage of the shock E.g. With a chemical explosive, the chemical energy release is triggered by the heating and compression of the shock the energy released then reinforces the shock A similar thing occurs in a thermonuclear detonation wave, the shock is sustained by the release of TN energy behind the shock front The reason why this works is that the shock propagates subsonically with respect to the material behind it. Therefore if we add pressure behind the shock, this pressure can catch up with the shock, and prevent it from decaying in strength
We can apply this learning to ICF To produce a compression >>, it is clear that we need to use multiple shock waves To limit the heating associated with this series of shocks, the relative pressure change across each shock must be limited The question is: what do the words multiple and limited mean in practice?
A very short introduction to ICF capsule design...
First we need to establish the meaning of the word cold At the high densities present in the compressed shell, atomic structure gives way and we can consider that our DT comprises an electron gas within which sit the DT ions As long as the temperature does not rise too high, then we can describe the electron gas as being Fermi degenerate (FD) Recall that electrons are fermions, and no two fermions can occupy the same state As the electrons are pushed together, they are forced to increasingly high energy states by the requirement that they be degenerate The electrons would acquire this energy, and an associated pressure, even if the gas was kept at cryogenic temperatures (few Kelvin) throughout the compression
Fermi Pressure The minimum back pressure that the compressed DT fluid will exert to resist implosion is given by the Fermi pressure (of the electron gas): E Fermi =3.65x10-15 n 2/3 ev (N.B. this is the highest energy state in the system of electrons, not the mean energy) P Fermi =2/5 n E Fermi =2.34x10-39 n 5/3 Mbar Note that here n is the electron number density /cc
It is possible to assign a Fermi temperature to the electron gas in the dense fuel blob kt Fermi =E Fermi This Fermi temperature is not a *real* temperature. A highly compressed material could have an actual temperature of only 1 Kelvin, and yet have a Fermi temperature of a few million Kelvin However, the Fermi temperature does serve as a scale for whether the actual temperature of the material matters or not. If the material temperature is below the Fermi temperature, then the Fermi pressure will dominate and we can consider that the material is cold. Here cold = Fermi degenerate Our aim then is to compress the material whilst keeping it cold; not meaning *actually cold* (e.g. like summer often is in England), but cold with respect to the Fermi temperature (e.g. Cold enough that the unavoidable Fermi pressure dominates the back pressure)
Example: Deuterium at 1000g/cc and 1million Kelvin n=3x10 26 E Fermi =1636eV T Fermi ~18.5million Kelvin >> Cold i.e. If deuterium is compressed to 1000g/cc and during the process the temperature rises to only 1 million Kelvin, then we have done very well the implosion has only really had to fight against the Fermi back pressure, which is unavoidable
The picture is more complicated early on in the implosion At <5g/cc, at low temperatures, the DT EOS is dominated by atomic and molecular binding energy, and is poorly approximated by a Fermi gas. Therefore the scaling diverges from the FD scaling of P In addition, our first shock goes into cold, solid fuel. Therefore the relative pressure change for the first shock will be infinite, and significant entropy will be added to the fuel Fortunately, so long as the specific entropy of the fuel is maintained below about 0.4GJ/g/keV, the EOS will not deviate substantially from the FD curve at high densities
Let s look at a family of isentropes for DT These figures made by Steve Haan, who designed the NIF capsule
What kind of pressures do we need to apply, in order to get our fuel to the required conditions for fusion? Let s say that a capsule contains about 1mg of fuel, which at burn fraction =1/3 will give a you 110MJ of yield Fermi statistics tells us that the average energy of an electron in a FD system is E Favg = 3 / 5 E fermi We can calculate then that the absolute minimum energy required to compress 1mg of DT fuel to 1000g/cc is ~33kJ (the average Fermi energy times the number of electrons) I am assuming you know why we need to work with about a milligram of fuel at this kind of density from earlier work (to satisfy the high gain r criteria!)
Let us then equate this to the work done on the fuel during implosion 33kJ = P x V Other lectures have discussed hydrodynamic stability. In order for the capsule to be stable, the fuel layer must not be too thin relative to initial radius of the fuel layer. The NIF baseline target has a fuel layer that is around 1 / 12 th the initial outer radius of the fuel, for this reason. If we take this as being typical, and approximate the volume change to the total initial volume of the capsule (neglecting ablator) we can calculate that for 1mg of fuel: 33kJ ~ P x 0.018 cm 3 This gives P ~ 18Mbar However, we have assumed perfect efficiency in our implosion! In reality we will probably need a rather higher pressure than this. However there is another more serious problem...
Let s look back at our DT isentropes Where you end up with a single shock of ~18Mbar Even assuming we only need 18Mbar, if we apply this all at once, we ll end up way off the FD curve, on a high isentrope (and you can t lose entropy from the fuel during the implosion!) We ve heated the fuel up stagnation will not produce the conditions we desire..
In reality, we must increase the pressure gradually This means, even leaving aside inefficiencies, that our final drive pressure must be far greater than 18Mbar. As the implosion progresses, the opportunity to do work with a given pressure diminishes dramatically since the remaining volume to do work on shrinks with r 3 In real target designs, final drive pressures are required to be on the order of 100Mbar Let s see how we get there...
First shock must keep the specific entropy below 0.4GJ/keV/g 0.9Mbar 1 st shock strength below ~1Mb will achieve this
Next shock takes us to about 4Mb 4Mbar 0.9Mbar Here we can see how Hugoniot Curves are used in practice!
After the first shock, we typically find that we can increase each subsequent shock by no more than a factor of 4 in pressure relative to the previous shock, if we are going to stay close to the isentrope set by the first shock
Typically four shocks are employed, the final shock having a strength of ~100Mbar Trajectory of main fuel in P- space Note the isentropes are labelled slightly differently in this diagram (taken from Lindl)
Ablator DT Ice DT gas This shows four shocks going through a direct drive capsule Shocks are timed to converge on the inside of the ice layer (This is just an illustrative rad hydro simulation that I ve run- the real thing can tend to be a bit more messy!)
The fuller picture: The effect of convergence Spherical convergence strengthens shocks as the propagate (according to simple shock theory they would reach infinite strength at the centre of the capsule!) Spherically implosive material flow also results in compression
At stagnation the picture is somewhat complicated Imploding shocks rebound off the capsule centre, then partially reflect off the inside of the imploding fuel shell. Reflection coefficient is proportional to the relative density jump, so most of the shock energy is reflected back into the hot spot Transmitted shock energy acts to decelerate shell and compress it further Shock bouncing in hotspot raises its final temperature, whilst the shell is rapidly brought to stagnation The combination of multiple shocks, spherical convergence and stagnation result in very high densities being obtained in the fuel shell (~1000g/cc)
A condition that must be satisfied prior to stagnation is that implosion velocity must be high enough that the fuel KE is greater than the total Fermi energy of the fuel in the desired final state We can extend our earlier working for the capsule with 1mg of fuel, to show that our 33kJ corresponds to an implosion velocity of 2.6x10 7 cm/s (using 1 / 2 mv 2 = 33kJ with m=1mg) In real conventional CHS schemes we need ~4x10 7 cm/s... Partly because there is another criteria we need to satisfy...
The hot-spot needs to be... HOT!! As may be apparent from the scale of conventional ignition laser systems, there is more to ignition than achieving high compression We also need to raise the hotspot temperature to ~ 10keV It is this requirement that results in the required energy being so high
Conventional CHS ignition = iso-baric central ignition Conventional CHS ignition results from a hotspot that has a similar pressure to that of the surrounding fuel (at the moment of ignition) In conventional CHS ignition the hotspot is brought to ignition conditions by being rapidly compressed by the dense fuel layer that is imploding onto it (piston-like) in addition to being heated by the shock waves that have accelerated the main fuel This is quite inefficient: we need a big laser to do it
Summary of conventional ignition... Conventional CHS ignition requires a high implosion velocity (~4x10 7 cm/s) for the dense fuel, in order that the compressive heating of the hotspot to ignition conditions results from stagnation This acceleration is (intentionally) performed somewhat inefficiently, and the velocity used is higher than would be needed simply for compression. The by product of this inefficiency and high implosion velocity is a 10keV hotspot (as well as other less desirable things, like a dense surrounding fuel mass that is rather warmer than it could be...) Accelerating the shell and heating the hotspot in this way requires sustained high driver intensities near the end of the laser pulse=> high driver energy required What would we need if all we wanted to do was compress the fuel to high density?
Our very naive calculation showed that we needed ~ 33kJ to compress a useful mass of fuel to 1000g/cc Allowing for driver-capsule coupling inefficiencies and unavoidable hydro inefficiencies (the small physical size of the capsule means we must shock the fuel to some degree)... We need an absolute minimum of around 100kJ. This doesn t seem like very much...?
This is where Shock Ignition and also other forms of Fast Ignition enter the arena... Both aim to firstly COMPRESS the fuel efficiently without heating Then HEAT by an additional (and more efficient) process
Shock ignition = non iso-baric central ignition In shock ignition the central hotspot is driven to final ignition conditions by the convergence of a strong shock wave. NOT with piston-like compression by the shell. This hotspot is not in pressure equilibrium with the surrounding dense fuel at the moment of ignition If we equate non iso-baric ignition to FI then we can see shock ignition as a form of FI
(Reminder: Fast Ignition) iso-baric non iso-baric
How is shock ignition achieved Efficient (very cold) implosion of fuel Short duration (~200ps) high intensity (~10 15-10 16 W/cm 2 ) spike in driver intensity when the fuel is near stagnation This drives a very strong shock into the dense imploded fuel blob, which converges on capsule centre and heats the hotspot to ignition conditions directly Total energy required ~350kJ
Taken from Perkins PRL (see reading list)
Advantages of Shock ignition over other forms of FI Hotspot/main fuel geometry places much less extreme demands on the driver for ignition (next slide) These relaxed requirements result in less extreme physics being involved. We hope we understand this physics better than the physics involved in other forms of FI. Spherical symmetry is maintained (no cone) We don t need a fancy (read: expensive) CPA laser system
The fact that the hotspot ignites at a higher pressure than the surrounding fuel increases hydrodynamic cooling of the hotspot, but not to the same degree as in other forms of fast ignition The pressure difference at the surface of the hotspot is much less than in other forms of FI since the surrounding fuel is much denser than the hotspot The work done by the hotspot on the surrounding fuel due to the pressure gradient causes enhanced hydrodynamic cooling relative to conventional CHS, but this is not such a critical issue as in other forms of fast ignition where the pressure difference is ~100x Related criteria are relaxed compared to other forms of FI (time scales for heating of h.s. and ignition)
However the requirements are not so relaxed that shock ignition can work with indirect drive - hohlraum acts a bit like a capacitor in an electronic circuit - it takes a while to heat up the hohlraum wall to a new temperature (large heat capacity; limited realistic laser power > slow temperature rise) - consequently the soft x-ray drive intensity cannot be cranked up with sufficient rapidity - (and there may be other problems when it comes to the physics of heating of the capsule!)
Shock ignition looks promising but... Laser interaction at intensities suggested (up to a few 10 16 W/cm 2 ) not well so explored as for conventional CHS ICF concern that promising simulation results do not take into account details of this interaction properly (particularly as regards parametric instabilities) Symmetry is critical for shock ignition: shock needs to converge spherically to generate extreme temperatures required: codes tend to produce more symmetry than reality again fear that promising simulation results may not be reflected by reality Optics damage may be a problem at the shock generating peak of the pulse Effective zooming is required since the final shock is launched when the fuel capsule is already at quite small radius
Nevertheless, as you can see from the material in the reading list, shock ignition has been considered as a possibility for adding a high-gain capability to both NIF and LMJ With an existing Mega Joule scale laser, the higher efficiency of the shock-ignition approach may enable much higher gains to be achieved than are expected with the planned indirect drive experiments
Getting into the details.. In Shock Ignition, the ignitor shock is timed to collide with the expansion shock as it exits the hotspot. This results in optimal heating, but means the shock must be carefully timed (and the capsule quite uniform in order to get the desired outcome) Shock ignition allow large drivers (such as NIF and LMJ) to assemble larger masses of fuel to high density using efficient compression > much higher gain upon ignition Although parametric instabilities may be significant at peak drive intensities, the highly compressed capsule is far less sensitive to suprathermal electron preheat than a conventional capsule (electrons can t penetrate to great depth in the compressed fuel blob). Actually it has been suggested that suprathermal electrons can help drive the ignition shock
Pressure (a.u.) Also note: From Betti et al, proc. 5 th IFSA conference (IFSA2007) that the final shock does, of course result in compression of the fuel a non-igniting hotspot is inevitably formed during the collapse of the shell (prior to the ignitor shock)
Conclusions Shock ignition is an exciting new avenue in ICF research, potentially enabling ignition with smaller drivers or high gain with current MJ laser systems Shock ignition offers many of the benefits of other Fast Ignition schemes without many of the associated risks Hydro-instabilities are a potential pit-fall since, unlike other approaches to FI, shock ignition does require quite a uniform implosion