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Lecture 36
Inertial Confinement Fusion: Adiabatic Compression and Shock Heating

The Sun burns because gravity confines the fuel for times long compared with the nuclear reaction time. In inertial confinement fusion (ICF) one tries to achieve the same basic thermonuclear reaction on the opposite extreme: a sub-millimeter target is compressed so rapidly that its own inertia confines the hot plasma for only a very short time. The confinement mechanism changes completely, but the governing fluid equations do not. The same continuity, momentum, and energy equations that we have already used for stellar gas dynamics also describe an imploding fusion capsule.

That is why ICF belongs naturally in a lecture near shocks. Part of the implosion is designed to be as nearly adiabatic as possible, because low entropy gives high compressibility. But the entropy is set by shocks launched early in the laser pulse, and in some target concepts the final ignition is triggered by a deliberately strong late shock. In ICF, adiabatic compression and shock heating are not competing ideas. They are the two halves of the same design problem.

Overview

ICF is stellar hydrodynamics on a nanosecond clock. The fuel must be compressed almost adiabatically after the early shocks have set its entropy, because low adiabat means high compressibility and large areal density. Breakeven is reached only if the implosion creates a hot spot with enough temperature, enough \(\rho R\), and enough confinement time for alpha-particle self-heating to outrun losses before the target flies apart.

Historical Perspective

The modern ICF story begins with Lawson’s criterion for fusion power balance and with the proposal by Nuckolls and collaborators to compress deuterium–tritium fuel to thermonuclear conditions using intense lasers Lawson (1957); Nuckolls et al. (1972). The central-hot-spot picture was developed in detail by Lindl and collaborators, who made clear that ignition requires both strong compression and careful control of instability, shock timing, and entropy generation Lindl (1998); Lindl et al. (2004). A complementary idea—shock ignition—was later proposed to separate the dense fuel assembly from the final heating by launching a strong late shock into an already compressed target Betti et al. (2007). The recent National Ignition Facility results, culminating in ignition-level performance and then target gain larger than unity, have made the distinction between compression, ignition, and breakeven part of the standard language of the field rather than a purely theoretical discussion Zylstra et al. (2022); Hurricane et al. (2024); Abu-Shawareb et al. (2024).

36.1 The same hydrodynamic equations as in stellar structure

The Euler limit. If magnetic stresses are negligible, the MHD equations reduce to the compressible Euler system together with an energy equation and an equation of state. For an imploding ICF plasma we may write

\[\pp {\rho }{t} + \divergence (\rho \uvec ) = 0, \tag{B.1}\]
\[\rho \left (\pp {\uvec }{t} + \uvec \cdot \grad \uvec \right ) = -\grad p, \tag{B.2}\]
\[\rho \left (\pp {\epsilon }{t} + \uvec \cdot \grad \epsilon \right ) = -p\,\divergence \uvec - \divergence \vect {q} + Q_{\mathrm {dep}} + Q_{\alpha } - Q_{\mathrm {rad}}, \tag{B.3}\]
\[p = p(\rho ,T). \tag{B.4}\]
Here \(\rho \) is the mass density, \(\uvec \) the fluid velocity, \(p\) the pressure, \(\epsilon \) the specific internal energy, \(\vect {q}\) the conductive heat flux, \(Q_{\mathrm {dep}}\) the driver energy deposition, \(Q_{\alpha }\) the deposited \(3.5~\mathrm {MeV}\) alpha heating, and \(Q_{\mathrm {rad}}\) the radiative loss. For order-of-magnitude work it is often enough to use an ideal-gas hot-spot model,
\[p = (\gamma -1)\rho \epsilon , \qquad c_s^2 = \frac {\gamma p}{\rho } \approx \frac {\gamma k_B T}{\bar m}, \tag{B.5}\]
where \(\bar m\) is the mean ion mass.

What changes from the Sun to a capsule. In the Sun, pressure gradients are nearly balanced by gravity and one studies a slowly varying stratified object. In ICF, gravity is irrelevant and the inertial term is the whole story. A thin shell is accelerated inward by ablation pressure at its outer surface, a central hot spot forms at stagnation, and the confinement time is set by how long the shell inertia can resist re-expansion. The dominant balance at stagnation is therefore not hydrostatic but dynamic: very roughly,

\[p_{\mathrm {stag}} \sim \rho _d V_{\mathrm {imp}}^2, \tag{B.6}\]
where \(\rho _d\) is the shell density during the deceleration phase and \(V_{\mathrm {imp}}\) is the implosion velocity. This one-line scaling already tells you why adiabat matters: at fixed velocity, a denser shell gives a larger stagnation pressure.

Direct drive, indirect drive, and the common fluid core. In direct drive, the laser interacts more directly with the target surface. In indirect drive, the laser first heats a hohlraum and the target is compressed by x rays. Those engineering details matter enormously in practice, but the central fluid problem is the same in both cases: accelerate a shell inward, keep the entropy low enough that the fuel is compressible, and then create a hot spot that ignites before disassembly Atzeni and Meyer-ter Vehn (2004); Hurricane et al. (2023).

36.2 Adiabatic compression and the meaning of adiabat

Adiabatic evolution of a fluid element. If heat fluxes and explicit source terms are small during some part of the implosion, then (B.3) reduces to

\[\rho \left (\pp {\epsilon }{t} + \uvec \cdot \grad \epsilon \right ) = -p\,\divergence \uvec . \tag{B.7}\]
For an ideal gas this implies that the entropy of a fluid element is constant, or equivalently that
\[K \equiv p\rho ^{-\gamma }, \qquad \dd {K}{t} = 0. \tag{B.8}\]
The quantity \(K\) is an entropy label: for an ideal gas,
\[s = c_v \ln K + \mathrm {const}. \tag{B.9}\]
So when ICF people say that an implosion is on a low adiabat, the clean mathematical statement is that the shell entropy label \(K\) has been kept small.

A note on ICF language. In the ICF literature one often also sees a dimensionless adiabat \(\alpha \), defined as the ratio of the shell pressure to the zero-temperature Fermi pressure at the same density. That is a very useful engineering label for cryogenic DT shells, but the present lecture is really about the more primitive thermodynamic idea. The quantity \(K=p\rho ^{-\gamma }\) comes straight from the Euler equations, and it is the one that most clearly exposes the link between adiabatic compression and shock-generated entropy.

Spherical compression scalings. Now approximate a compressed region as a uniform sphere of fixed mass \(M\) and radius \(R\). If the initial radius is \(R_0\), define the convergence ratio

\[C \equiv \frac {R_0}{R}. \tag{B.10}\]
Mass conservation gives \(\rho \propto R^{-3}\), so
\[\rho = \rho _0 C^3. \tag{B.11}\]
Along an adiabat,
\[p = p_0 C^{3\gamma }, \qquad T = T_0 C^{3(\gamma -1)}, \qquad \rho R = (\rho _0 R_0) C^2. \tag{B.12}\]
For the ideal-gas value \(\gamma =5/3\) this becomes especially simple:
\[T \propto C^2, \qquad p \propto C^5, \qquad \rho R \propto C^2. \tag{B.13}\]
That is one of the most useful back-of-the-envelope results in ICF. In the simplest model, the temperature and the areal density both rise like \(C^2\).

A concrete estimate. A cryogenic DT layer starts at a density of order \(\rho _0\approx 0.25~\mathrm {g\,cm^{-3}}\). If its characteristic initial radius is \(R_0\approx 1~\mathrm {mm}\), then \(\rho _0R_0\approx 2.5\times 10^{-2}~\mathrm {g\,cm^{-2}}\). To reach a hot-spot areal density of order \(0.3~\mathrm {g\,cm^{-2}}\), one therefore needs

\[C \approx \sqrt {\frac {0.3}{0.025}} \approx 3.5. \tag{B.14}\]
For \(\gamma =5/3\), that same compression multiplies the temperature by about \(C^2\approx 12\). So if the material entering the final compression phase is already a few hundred electron volts, adiabatic compression can plausibly bring it into the several-keV ignition range. But if the fuel were still essentially cryogenic, adiabatic compression by itself would be far too weak. The early shock history and the final stagnation heating are therefore not optional details; they are part of the basic thermodynamic budget.

Why low adiabat matters mathematically. Write the pressure of a uniform compressed mass \(M\) as

\[p = K\left (\frac {M}{V}\right )^{\gamma }, \tag{B.15}\]
where \(V\) is the volume. The compressive work needed to move from \(V_0\) to \(V\) is then
\[W = \int _{V}^{V_0} p\, dV' = \frac {K M^{\gamma }}{\gamma -1} \left (V^{1-\gamma } - V_0^{1-\gamma }\right ). \tag{B.16}\]
At fixed fuel mass and fixed work budget, a smaller \(K\) permits a smaller final volume and therefore a larger final density. This is the cleanest mathematical statement of why a low adiabat is desirable: entropy consumes compression work.

Caution

Low adiabat is good for compression but bad for stability. A shell on a low adiabat is dense and highly compressible, which is exactly what one wants for large \(\rho R\). But the same low pressure support makes the shell more vulnerable to hydrodynamic instability and to errors in shock timing. Much of ICF design is therefore a controlled compromise between compressibility and robustness.

36.3 What breakeven means for an inertially confined plasma

Ignition, scientific breakeven, and engineering gain. The word breakeven is dangerously slippery in ICF. There are at least three related but distinct thresholds. First, ignition is a thermodynamic statement: alpha-particle self-heating must exceed the internal losses so that the burn amplifies itself. Second, scientific breakeven or unity target gain means that the total fusion energy out exceeds the incident laser energy,

\[G_{\mathrm {target}} \equiv \frac {E_{\mathrm {fusion}}}{E_L} > 1. \tag{B.17}\]
Third, an actual power plant would need the wall-plug efficiency of the driver folded in, so engineering gain requires much more than \(G_{\mathrm {target}}=1\) Hurricane et al. (2024); Abu-Shawareb et al. (2024). A capsule can be very interesting scientifically before it is useful economically.

Reaction rate and alpha self-heating. For an equimolar DT plasma,

\[n_D = n_T = \frac {n}{2}, \qquad \mathcal {R}_{DT} = \frac {n^2}{4}\,\langle \sigma v \rangle _{DT}, \tag{B.18}\]
where \(n\) is the total ion density. The fusion power density is then
\[P_f = \mathcal {R}_{DT} E_{DT}, \qquad E_{DT}=17.6~\mathrm {MeV}, \tag{B.19}\]
while the local alpha-heating power density is
\[P_{\alpha } = \mathcal {R}_{DT} E_{\alpha } f_{\alpha }, \qquad E_{\alpha }=3.5~\mathrm {MeV}, \tag{B.20}\]
with \(f_{\alpha }\) the fraction of alpha energy deposited in the hot spot. Since the DT reactivity rises rapidly into the several-keV range and peaks in the rough neighborhood of \(10\)–\(20~\mathrm {keV}\), the temperature target for ignition is naturally in that range rather than at hundreds of electron volts Bosch and Hale (1992); Betti et al. (2010).

A Lawson-like criterion in inertial form. For a hot spot of radius \(R_h\), the hydrodynamic disassembly time is roughly

\[\tau _h \sim \frac {R_h}{c_s}. \tag{B.21}\]
If the burn fraction is still small, then over one confinement time
\[f_b \sim \frac {1}{2} n\langle \sigma v \rangle _{DT} \tau _h. \tag{B.22}\]
Using \(\rho = n\bar m\), one finds
\[n\tau _h \sim \frac {\rho R_h}{\bar m c_s} \qquad \Longrightarrow \qquad \rho R_h \sim (n\tau _h)\sqrt {\gamma \bar m k_B T_h}. \tag{B.23}\]
This is the direct bridge between the usual Lawson product and the ICF language of areal density. In magnetic confinement, one talks about confinement time. In inertial confinement, the confinement time is geometric, so one almost inevitably talks about \(\rho R\) instead.

The practical ignition conditions. The central hot spot must do three things at once. It must be hot enough for significant DT reactivity, dense enough that the alpha particles stop before escaping, and confined long enough that the alpha heating can amplify the initial burn. A useful rule of thumb is therefore a hot-spot temperature of several keV together with a hot-spot areal density of order \(0.2\)–\(0.4~\mathrm {g\,cm^{-2}}\) Betti et al. (2010); Christopherson et al. (2019). That is only the spark. Large gain also requires a surrounding dense shell with still larger total fuel areal density, because the major fusion energy comes from burn propagation into the cold compressed fuel rather than from the tiny hot spot alone Christopherson et al. (2019); Hurricane et al. (2023).

Quantity

Rough target

Why it matters

\(T_h\)

several keV

Needed for appreciable DT reactivity.

\(\rho R_h\)

\(\sim 0.3~\mathrm {g\,cm^{-2}}\)

Needed to trap a substantial fraction of the \(3.5~\mathrm {MeV}\) alpha energy in the hot spot.

\(\tau _h\sim R_h/c_s\)

as large as possible

Sets the time available for self-heating before disassembly.

Shell adiabat \(K\)

as low as stability allows

Low entropy makes the shell dense and raises the achievable stagnation pressure for a fixed work budget.

Total fuel \(\rho R\)

order \(\mathrm {g\,cm^{-2}}\) for gain

Needed for burn propagation and large overall yield, not just ignition of the central spark.


Table 36.1: Useful rule-of-thumb conditions for central-hot-spot ignition and gain.

A simple gain estimate. The DT reaction releases a specific energy of about

\[q_{DT} \approx 339~\mathrm {MJ\,mg^{-1}}. \tag{B.24}\]
So a fuel mass \(M_{DT}\) with burn fraction \(f_b\) gives
\[E_{\mathrm {fusion}} \approx 339~\mathrm {MJ\,mg^{-1}} \left (\frac {M_{DT}}{1~\mathrm {mg}}\right ) f_b. \tag{B.25}\]
For example, a \(0.20~\mathrm {mg}\) DT payload would release about \(68~\mathrm {MJ}\) at full burn. To exceed a laser input of about \(2~\mathrm {MJ}\) therefore requires only a few percent burn, which sounds easy until one remembers that obtaining even a few percent burn demands both ignition and enough surrounding fuel \(\rho R\) for burn propagation. The point is not that breakeven is trivial. The point is that once the ignition thresholds are crossed, the nuclear energy reservoir is enormous.

36.4 Why shocks are both necessary and dangerous

Shocks set the adiabat. Adiabatic compression preserves the entropy label \(K\), but it does not tell us what value of \(K\) the shell started with. That value is set by the shock history. In the purely hydrodynamic limit of the previous shock lecture, a planar shock at rest satisfies

\[\begin{aligned}[\rho u_n] &= 0, \\[4pt] [\rho u_n^2 + p] &= 0, \\[4pt] \left [ u_n \left ( \frac {1}{2}\rho u_n^2 + \frac {\gamma }{\gamma -1}p \right ) \right ] &= 0.\end{aligned} \tag{B.26}\]

These conditions determine the downstream state. The crucial thermodynamic point is that for a genuine shock,

\[\frac {K_2}{K_1} = \frac {p_2/p_1}{(\rho _2/\rho _1)^\gamma } >1. \tag{B.29}\]
Every shock raises the adiabat. In ICF language, every shock that passes through the shell spends some of the implosion budget by converting coherent flow energy into entropy.

Strong-shock heating. For a strong hydrodynamic shock in an ideal gas,

\[\frac {\rho _2}{\rho _1} \to \frac {\gamma +1}{\gamma -1}, \qquad p_2 \approx \frac {2}{\gamma +1}\rho _1 u_1^2, \tag{B.30}\]
where \(u_1\) is the upstream speed in the shock frame. The downstream temperature is then
\[k_B T_2 \approx \frac {2(\gamma -1)}{(\gamma +1)^2}\,\bar m u_1^2. \tag{B.31}\]
For \(\gamma =5/3\) this reduces to the familiar estimate
\[k_B T_2 \approx \frac {3}{16}\bar m u_1^2. \tag{B.32}\]
If one inserts a DT mean ion mass and an upstream speed of \(300~\mathrm {km\,s^{-1}}\), the result is only about \(0.44~\mathrm {keV}\). That is enough to matter greatly for the entropy budget, but not enough by itself to create a conventional ignition hot spot. Strong shocks are therefore excellent at setting the adiabat and at producing local heating, yet the full several-keV ignition state still relies on geometric convergence, stagnation work, and in some designs additional late-time shock amplification.

Why multishock pulse shaping matters. A practical ICF drive launches not one shock but several. The early weak shocks are timed so that they coalesce at carefully chosen locations inside the shell. If they merge too early, the shell is preheated and the adiabat rises, making later compression less efficient. If they merge too late, large velocity gradients survive into the deceleration phase and the shell does not assemble cleanly. This is why shock timing became a classic experimental problem in its own right: it is a direct diagnostic of whether the pulse has put the shell on the intended adiabat Boehly et al. (2011).

Shock heating versus adiabatic compression. The two processes play sharply different thermodynamic roles.

Process

Entropy change

Characteristic scaling

ICF consequence

Adiabatic compression

\(K\) constant

\(T,\,\rho R \propto C^2\) for \(\gamma =5/3\)

Efficient way to turn convergence into both temperature and areal density once the entropy has already been set.

Shock heating

\(K_2>K_1\)

\(T_2\propto u_1^2\) for a strong shock

Rapid local heating and a powerful way to launch or ignite, but paid for by an irreversible increase of entropy.


Table 36.2: The essential thermodynamic contrast between adiabatic compression and shock heating in ICF.

In a good implosion, one wants just enough shock heating to launch the right trajectory and set the correct shell adiabat, followed by as much nearly adiabatic compression as possible. Too little shock control and the implosion is mistimed. Too much shock heating and the shell becomes too stiff to compress.

Shock ignition. The central-hot-spot strategy tries to obtain the final temperature mainly from adiabatic compression and stagnation work. Shock ignition changes the division of labor. The shell is assembled at comparatively lower velocity and lower hot-spot temperature, and then a strong late-time shock is launched into the already compressed fuel to trigger ignition Betti et al. (2007); Baton et al. (2012); Nora et al. (2015). Conceptually, this is very attractive because the dense assembly and the final ignition are partially separated. Thermodynamically, however, the price is obvious from (B.29): the ignitor shock is a deliberately irreversible event. The gain comes only if the shock arrives late enough that most of the fuel has already been assembled at high density.

36.5 Experimental perspective

Shock timing as a diagnostic discipline. One of the cleanest experimental realizations of the abstract shock-timing problem came from OMEGA measurements of multiple spherically converging shocks in liquid deuterium. Those experiments directly measured shock velocity and coalescence timing at pressures of relevance to ignition designs, turning an apparently abstract pulse-shaping problem into a quantitative hydrodynamic diagnostic Boehly et al. (2011).

Adiabat engineering on NIF. The National Ignition Facility made the entropy trade-off especially visible. High-foot, high-adiabat implosions deliberately raised the early shock pressure to reduce instability and improve robustness, at the cost of lower ultimate compression Park et al. (2014). Later adiabat-shaped pulses used a more carefully tailored multishock sequence to recover higher \(\rho R\) while keeping the implosion acceptably stable Casey et al. (2015). This is exactly the design dialectic predicted by (B.16) and (B.29): stability likes a higher adiabat, but compression does not.

Shock-focused ignition studies. Shock ignition has also developed a substantial experimental life of its own. Planar experiments have isolated the basic ignitor-shock physics, while spherical experiments on OMEGA have demonstrated the production of gigabar-class converging shocks in implosion-like geometry Baton et al. (2012); Nora et al. (2015). These studies are valuable even when they fall short of ignition, because they test the part of the design where a strongly nonlinear shock must propagate through already compressed plasma without ruining the assembly it is supposed to ignite.

From compression to ignition to gain. The recent NIF milestones are best understood as triumphs of the same hydrodynamic logic. The August 2021 shot achieved ignition-level behavior and clear signatures of self-heating Zylstra et al. (2022). The December 5, 2022 shot, reported in 2024, demonstrated target gain larger than unity and gave a clean laboratory example of scientific breakeven Hurricane et al. (2024); Abu-Shawareb et al. (2024). What made those implosions work was not a single magic ingredient. It was the successful coordination of drive symmetry, instability control, shell entropy, shock history, and stagnation confinement. In other words, the experimental frontier ended up validating the same simple lecture message: compression must be as adiabatic as possible right up until one needs shocks to do something only shocks can do.

Takeaways
  • ICF uses the same compressible-fluid equations as stellar hydrodynamics, but with inertia rather than gravity providing confinement.
  • Along an adiabatic trajectory, \(K=p\rho ^{-\gamma }\) is constant, so low entropy is the central route to high compressibility and large \(\rho R\).
  • Breakeven requires more than a hot plasma: one needs a hot spot with enough temperature, enough alpha trapping, and enough confinement time for self-heating.
  • Shocks are indispensable because they launch the implosion and can ignite the hot spot, but every shock also raises the adiabat and therefore reduces the maximum compression available from a fixed work budget.
  • Much of ICF target design is the art of using shocks only where irreversibility is worth the price.

Bibliography

    J. D. Lawson. Some criteria for a power producing thermonuclear reactor. Proceedings of the Physical Society. Section B, 70(1):6–10, 1957. doi:10.1088/0370-1301/70/1/303.

    J. Nuckolls, L. Wood, A. Thiessen, and G. Zimmerman. Laser compression of matter to super-high densities: Thermonuclear (CTR) applications. Nature, 239(5368):139–142, 1972. doi:10.1038/239139a0.

    John D. Lindl. Inertial Confinement Fusion: The Quest for Ignition and Energy Gain Using Indirect Drive. Springer-Verlag, New York, 1998.

    John D. Lindl, Peter Amendt, Richard L. Berger, S. G. Glendinning, Siegfried H. Glenzer, Steven W. Haan, Robert L. Kauffman, Otto L. Landen, and Larry J. Suter. The physics basis for ignition using indirect-drive targets on the national ignition facility. Physics of Plasmas, 11 (2):339–491, 2004. doi:10.1063/1.1578638.

    R. Betti, C. D. Zhou, K. S. Anderson, L. J. Perkins, W. Theobald, A. A. Solodov, et al. Shock ignition of thermonuclear fuel with high areal density. Physical Review Letters, 98(15): 155001, 2007. doi:10.1103/PhysRevLett.98.155001.

    A. B. Zylstra et al. Experimental achievement and signatures of ignition at the national ignition facility. Physical Review E, 106(2):025202, 2022. doi:10.1103/PhysRevE.106.025202.

    O. A. Hurricane, D. A. Callahan, D. T. Casey, A. R. Christopherson, A. L. Kritcher, O. L. Landen, S. A. MacLaren, R. Nora, P. K. Patel, J. Ralph, D. Schlossberg, P. T. Springer, C. V. Young, and A. B. Zylstra. Energy principles of scientific breakeven in an inertial fusion experiment. Physical Review Letters, 132(6):065103, 2024. doi:10.1103/PhysRevLett.132.065103.

    H. Abu-Shawareb, R. Acree, P. Adams, J. Adams, B. Addis, R. Aden, P. Adrian, B. B. Afeyan, M. Aggleton, L. Aghaian, et al. Achievement of target gain larger than unity in an inertial fusion experiment. Physical Review Letters, 132(6):065102, 2024. doi:10.1103/PhysRevLett.132.065102.

    Stefano Atzeni and J"urgen Meyer-ter Vehn. The Physics of Inertial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Matter. Oxford University Press, Oxford, 2004. doi:10.1093/acprof:oso/9780198562641.001.0001.

    O. A. Hurricane, P. K. Patel, R. Betti, et al. Physics principles of inertial confinement fusion and U.S. program overview. Reviews of Modern Physics, 95(2):025005, 2023. doi:10.1103/RevModPhys.95.025005.

    H.-S. Bosch and G. M. Hale. Improved formulas for fusion cross-sections and thermal reactivities. Nuclear Fusion, 32(4):611–631, 1992. doi:10.1088/0029-5515/32/4/I07.

    R. Betti, P.-Y. Chang, B. K. Spears, K. S. Anderson, J. Edwards, M. Fatenejad, J. D. Lindl, R. L. McCrory, R. Nora, and D. Shvarts. Thermonuclear ignition in inertial confinement fusion and comparison with magnetic confinement. Physics of Plasmas, 17(5):058102, 2010. doi:10.1063/1.3380857.

    A. R. Christopherson, R. Betti, and J. D. Lindl. Thermonuclear ignition and the onset of propagating burn in inertial fusion implosions. Physical Review E, 99(2):021201, 2019. doi:10.1103/PhysRevE.99.021201.

    T. R. Boehly, V. N. Goncharov, W. Seka, S. X. Hu, J. A. Marozas, M. A. Barrios, P. M. Celliers, D. G. Hicks, G. W. Collins, and D. D. Meyerhofer. Velocity and timing of multiple spherically converging shock waves in liquid deuterium. Physical Review Letters, 106(19):195005, 2011. doi:10.1103/PhysRevLett.106.195005.

    S. D. Baton, M. Koenig, E. Brambrink, et al. Experiment in planar geometry for shock ignition studies. Physical Review Letters, 108(19):195002, 2012. doi:10.1103/PhysRevLett.108.195002.

    R. Nora, W. Theobald, R. Betti, et al. Gigabar spherical shock generation on the OMEGA laser. Physical Review Letters, 114(4):045001, 2015. doi:10.1103/PhysRevLett.114.045001.

    H.-S. Park, O. A. Hurricane, D. A. Callahan, et al. High-adiabat high-foot inertial confinement fusion implosion experiments on the national ignition facility. Physical Review Letters, 112(5): 055001, 2014. doi:10.1103/PhysRevLett.112.055001.

    D. T. Casey, J. L. Milovich, V. A. Smalyuk, et al. Improved performance of high areal density indirect drive implosions at the national ignition facility using a four-shock adiabat shaped drive. Physical Review Letters, 115(10):105001, 2015. doi:10.1103/PhysRevLett.115.105001.

Problems

Problem 36.1.
Starting from (B.1) and (B.7), derive (B.8). Then use mass conservation for a uniform sphere to derive (B.12).
Problem 36.2.
Use (B.23) with \(\gamma =5/3\), a DT mean ion mass \(\bar m \approx 2.5 m_p\), a temperature \(T_h=10~\mathrm {keV}\), and a Lawson product \(n\tau _h = 10^{15}~\mathrm {s\,cm^{-3}}\) to estimate the hot-spot areal density required for ignition. Compare your result with the rule of thumb quoted in the lecture.
Problem 36.3.
Starting from the hydrodynamic jump conditions (B.26)–(B.28), derive the strong-shock compression ratio and the temperature estimate (B.31). Then evaluate the post-shock temperature for \(u_1=500~\mathrm {km\,s^{-1}}\) and a DT plasma.
Problem 36.4.
Use (B.16) to show explicitly that for fixed \(M\), \(V_0\), and \(W\), a lower adiabat \(K\) gives a smaller final volume. Explain in words why this is the thermodynamic reason that low-adiabat implosions are attractive.
Problem 36.5.
Compare central-hot-spot ignition with shock ignition. In each case, identify where most of the temperature rise comes from, where the entropy is generated, and why the required shock timing is different.