Z. Physik B 37, 193-202 (1980) Zeitschrift for Physik B © by Springer-Verlag 1980 Screening in a One Dimensional Plasma K.D. Schotte and T.T. Truong Institut ftir Theoretische Physik, Freie UniversitM Berlin, Berlin, Germany Received December 3, 1979 We investigate the screening behavior of a neutral one dimensional plasma made up by two types of charges not necessarily of equal values. The thermodynamic potential is given by the lowest eigenvalue of a generalized Mathieu equation. If the latter is interpreted as a Schr6dinger equation, the dielectric constant is related in a simple manner to the band mass of the lowest energy band. We calculate this mass with the W.K.B. method and study the correlation function for a pair of test charges. 1. Introduction Some time ago Baxter [1] has performed calculations on different classical plasmas in one dimension in order to compare and test the validity of the Debye- Hiickel approximation. He showed that the free en- ergy of a two-component plasma and that of a one- component plasma are equal at high temperature and in agreement with the Debye-Htickel approximation. Of course, at low temperatures one encounters de- viations. If however, one studies correlation functions which are a more sensitive tool than thermodynamic potential deviations appear also at high temperatures. Baxter [2] and Kunz [3] found that the one-com- ponent system with compensating background has a density which is periodic in space. In principle it is a crystal also for T~ ~ although the amplitude of the density oscillation tends to zero. Recently, it was pointed out by Zittartz [4] that a two-component system made up of equal and opposite charges is an insulator. The dielectric function e(q) for small wave number (q ---,0) does not become infinitely large. Apel, Everts and Schulz [-5] found, however, that the dielectric constant 5(0) takes on a large value which diverges exponentially when T~ oo. One would expect that the screening effect of an electric field is affected by the discretness of the charges and their tendency to form bound pairs at low temperatures. Then the charges are not free anymore so that an electric field only polarizes these molecules. The two-component plasma reaches the state of a normal plasma only asymptotically as T--, oo when the interaction practically does not count. The one-component system with neutralizing background should exhibit a normal metallic screen- ing effect because a small translation of the discrete charges with respect to the continuous background generates a surface charge. In this work we would like to find out how the discretness of the charges will affect the dielectric function at high temperatures. We shall consider a system of positive charge e and negative charge e/k (where k is an integer) with overall charge neutrality. Thus following a proposal of Choquard [6] we try to interpolate between a two-component system and a one-component system with neutralizing background. We will show that for large k and T we obtain the following dielectric constant ~ ~exp {l~2T2rck3/2 + (-9(kl/2)} (1) where p+ is the density of positive ions of charge +e. As far as calculations are concerned we use the transfer matrix method which reduces the problem of statistical mechanics in one-dimension to an eigen- value problem for a Schr6dinger equation with a complex periodic potential. The derivation and in- terpretation of the important thermodynamical for- 0340-224X/80/0037/0193/$02.00