Riemann's Contribution to Flight and Laser Fusion

physics involved in the pellet compression and ignition isunderstood and that the process of isentropic compressionby weak shock waves is sufficient to achieve ignition.The FEF's 1979 ProposalTo understand the new approach to laser fusion suggestedby Uwe Parpart and the Fusion Energy Foundationin 1979 requires a more detailed picture of the energyrequirements for the compression process.Figure 1 shows the relationship between pressure andvolume for several different strategies of compression. Onthe left, a, is the trajectory in the state space for a mediumundergoing isentropic compression, where all the energygoes into the compression of the medium and the minimumpossible goes into its heating. In the center, 6, is thetrajectory of a strong shock wave compression, startingfrom the same pressure and volume, with the assumptionthat the medium behaves like a gas (that is, has anequation of state like a gas). In the case of the strongshock wave, it can be shown that the energy used in thecompression is divided equally between compression andheating. Thus, as the figure shows, the final volumeachieved along this trajectory, called a Hugonoit adiabat,is larger than that achieved by the isentropic compression,and so the compression is smaller. In other words, half ofthe compression energy was "wasted" on heating the gas.On the right, c, the figure illustrates the strategy of thenational laboratories for achieving isentropic compression.Strictly speaking, it can be shown that isentropic compressioncan be attained only when there is no discontinuityat the shock front; that is, when there is no shock wave.However, the amount of entropy produced goes as thethird power of the pressure discontinuity: 6(S n - So) = 1/ 12T 0 (a 2 Wd P*) S (P, - P 0 )3where S equals entropy and T is temperature. That is, thechange in entropy before and after the shock wave isproportional to the third power of the pressure change.Therefore, a weak shock wave can very closely approximatethe isentropic trajectory. Note that the slope of theHugonoit adiabat at the initial point in state space is thesame as that of the isentropic trajectory, so that if a weakshock wave is used for a very short compression (a smallchange in pressure), the resulting change in volume willbe very close to that achieved by the isentropic compression.As shown in the figure, a series of weak shock wavescan then provide nearly isentropic compression.Mathematically, the change in pressure provided by theshock wave is related to the change in volume by thefollowing expansion, in terms of volume and entropy, inthe neighborhood of the initial point: 7P, - P„ = (aP/aV) 5 (V, - V 0 ) + y^P/aV^V, - V 0 H+ y^P/aV^v, - V„)3 + (eP/eSMS, -S 0 ) + ....Now, in the case of the isentropic trajectory, by definitionS, - S„ = 0, so that the maximum compression is definedby the conditionP, - P 0 = (eP/aV^V, - V 0 ) + %(8'P/aV') 5 (V, - V 0 )'+ yaWaV'MV, - V 0 P + ....Figure 1TRAJECTORIES FOR COMPRESSION OF A GASThe "state space" plotting the volume against the pressure of a gas is a useful way of comparing differentcompression strategies. These three graphs show the possible compression techniques for a gas starting from agiven volume (X axis) and pressure (y axis) P 0 , V 0 . Figure a shows the path for isentropic compression, whichachieves the maximum volume change for a given pressure difference, P, - P 0 . A shock wave, which mustproduce some entropy during compression, cannot result in as great a compression, as shown in b, where theHugonoit adiabat is traced along with the isentrope (the dashed line). Figure c shows the approximation of theisentrope with three shock waves, each of which propagates along the Hugoniot adiabat, H a , H b/ and H c .'26 FUSION October-November 1981