VQLUME 25, NUMBER 25 PHYSICAL REVIEW LETTERS 21 DECEMSER 1970 N. F. Ramsey and E. M. Po.rcell, Phys. Rev. 85, 143 (19M). ¹ F. Ramsey, Phys. Rev. 91, 303 (1953). 3E. A. G. Armour, J. Chem. Phys. ~49 5445 (1968). E. Ishiguro, Phys. Rev. 111, 203 (1968). T. P. Das and R. Bersohn, Phys. Rev. 115, 897 (1959) . M. J. Stephen, Proc. Roy. Soc. , Ser. A 243, 274 {1957}. D. E. O'Reilly, J. Chem. Phys. 36, 274 (1962). J. Schaefer and R. Yaris, J. Chem. Phys. 46, 948 {1967). S. Ray, M. Karplus, and T. P. Das, unpublished. J. Goldstone, Proc. Boy. Soc. , Ser. A 239, 267 {1967). C. M. Dutta, ¹ C. Dutta, and T. P. Das, Phys. Rev. A 1, 661 (1970). D. R. Bates, K. Ledsham, and A. L. Stewart, Phil. Trans. Roy. . Soc. London, Ser. A 246, 216 {1963). D. R. Bates and B. H. G. Beid, in A.defences in Atom- ic and Molecular Physics, edited by D. R. Bates (Aca- demic, New York, 1968), Vol. 1V, p. 13. ' D. R. Bates, U. Opik, and G. Poots, Proc. Phys. Soc. , London, Sect. A 66, 113 {1963). 5K. Riidenberg, J. Chem. Phys. 19, 1459 (1951). 8M. Kotani, A. Amemiya, E. Ishiguro, and T. Kimura„ Tables of Molecular Integrals {Maruzen Co. Ltd. , Tokyo, 1966). H. P. Kelly, Phys. Rev. 131, 684 (1963). H. P. Kelly, Phys. Rev. 173, 142 (1968}, aud 180, 66 (1969) . N. C. Dutta, C. Matsubara, B. T. Pu, and T. P. Das, Phys. Rev. 177, 33 (1968), and Phys. Rev. Lett. 21, 1139 (1968). T. F. Wimmett. Phys. Rev. 91, A476 (1963). Sign determined hy I. Ozier, P. N. Yi, A. Khosha, and ¹ F. Ramsey, Bull. Amer. Phys. Soc. 12, 132 (1967). S. D. Mahanti and T. P. Das, Phys. Bev. 170, 426 (1968) . 22R. I, . Matcha and W. B. Brown, J. Chem. Phys. 48, 74 (1968). J'. Goodisman, J'. Chem. Phys. 48, 2981 (1968). Formation and Strncture of Electrostatic Collisionless Shocks* D. %. Forslund and C. R. Shonk Los Alamos Scientific Laboratory, University of California, Los Alamos, Neav Mexico 87544 (Received 4 November 1970) A one-dimensional, two-species numerical-simulation code has been used to study the formation and structure of coQisionless electrostatic shocks formed by two colliding plasmas and by a plasma striking a perfectly reflecting piston. Collisionless shocks are formed in hydrogen up to a maximum piston velocity of M — = vD/e s - 3.5. Shocks with ve- locities up to M 4 have been produced which have a predominantly laminar structure accompanied by strong collisionless dissipation. Theoretical models for electrostatic shocks in the absence of a magnetic field have been dis- cussed by a number of authors. '' Lom-Mach- number [M=vo/cs&2, where cs=(T, /m;)'~'] shocks have been experimentally produced by Alikhanov, Belon, and Sagdeev' and Taylor, Bak- er, and Ikezi, ' and numerically simulated by Mason' and Sakanaka, Chu, and Marshall. ' In this Letter me discuss formation of stable, high- Mach-number electrostatic shocks by means of the particle-in-cell simulation technique treat- ing both the ions and electrons exactly with the mass ratio of hydrogen (1836). Since the elec- trons are treated exactly instead of isothermally, the critical Mach number of I-1,6 does not ap- ply. The shocks described here are mell approxi- mated by a rapid change from one equilibrium state to another with little or no fluctuations in the shock front (although perhaps having large fluctuations behind the front), and thus corre- spond most closely to the "laminar"' ~ rather than "turbulent" ' model. Initial conditions. — The shocks discussed here are created in tmo mays. The fiI st method con- sists of a plasma of ions and electrons moving to the right with Mach number M, as shown in Fig. 1(a), with the plasma initially (T =0) just in contact with the right-hand, perfectly reflecting mall. The plasma is sustained at the left by con- tinually injecting a Maxwellian plasma. The sec- ond method ls similar but with the above plasma filling only the left half of the box and an equal- density plasma with Mach numbeI of opposite sign occupying the right half. This configuration clearly should generate tmo shocks which dlffel' only statistically. For M &1 a shock will readily form on the basis of the ion-ion instability'0 in one dimension. Shocks formed with M &1 do so by means of the nonlinear process described be- low. Theory of formation and structure. — In the tmo lllltlal conditions used the method of shock for-