Observation of Critical Fluctuations Associated with Plasma-Wave Instabilities

legd VOLUME «I 1a h. 4 i MARCH 15, 1962 NUMBER OBSERVATION OF CRITICAL FLUCTUATIONS ASSOCIATED WITH PLASMA-WAVE INSTABILITIES S. Ichimaru Departments and D. Pines of Physics and Electrical Engineering, University of Illinois, Urbana, Illinois N. Rostoker John Jay Hopkins Laboratory for Pure and Applied Science, General Atomic Division of General Dynamics Corporation, San Diego, California (Received February 21, 1962) The existence of plasma-wave instabilities as- sociated with the drift of electrons vs ions under the action of an applied electric field in a fully ionized plasma has been considered recently by a. number of authors. ' ' In this Letter we wish to point out the possibility of utilizing the scattering of electromagnetic waves to observe the onset of such instabilities. As the plasma approaches from a region of stability a critical point corre- sponding to the onset of an instability, a phenom- enon analogous to critical opalescence is predict- ed, in which at certain angles enormously en- hanced scattering of the electromagnetic radiation is expected to occur. The scattering of electromagnetic waves by electron density fluctuations in a fully ionized plasma is determined by S(kv), the Fourier trans- form of the electron density-density correlation function. ' The differential cross section, d'o/ diode, for the transfer of momentum Rk (corre- sponding to scattering into a solid angle dQ) and energy S~ from an electromagnetic wave to the electrons in the plasma is given by d dOdu mc' (l sin'8) S(klan), where 8 is the angle between the incident and scattered waves, and we have averaged over di- rections of polarization of both waves. S(kyar) is defined by S(k~) =2mL' t, dr dt I oo (n(r', t')n(r'+r, t'+t)) x exp[-t(k r -~t)], where N is the total number of electrons in a vol- ume L', n(r, t) denotes the electron number den- sity, and the angular brackets refer to a statisti- cal average over the electron states. In many cases of practical importance the fractional change in wavelength of the scattered radiation is small; under these circumstances a measurement of the intensity of the radiation scattered into a given angle yields directly the structure factor, S(k), which is S(k) = (1/N) J d&u S(k&u) S(%) is typically of order unity. We first summarize some results of the theory of fluctuations in a fully ionized plasma. Within the random phase approximation the spectrum of electron density fluctuations S(k, ~) for a two- component plasma (electrons and positive ions with charge e) may be simply obtained by super- posing the fields due to the dressed particles (electrons or ions plus their associated screening