Research Report

INSTITUTE OF PLASMA PHYSICS NAOOYA UNIVERSITY Equations for a Plasma Consisting of Matter and Antimatter A. H. Nelson and K. Ikuta IPPJ-131 August 1972 RESEARCH REPORT NAGOYA, JAPAN Equations for a Plasma Consisting of Matter and Antimatter A. H. Nelson end K. Ikuta IPPJ-131 August 1972 Further communication about this report is to be sent to the Research Information Center, Institute of Plasma Physics, Nagoya Uniyersity, Nagoya, JAPAN. Contents Page Abstract 1)Introduction 1 2)Basic Equations 3 2.1) Individual Particle Equations 5 2.2) Ambiplasma Equations 10 2.3) Oebye length in an Ambiplasma 23 3)Waves in an Ambiplasma 25 3.1) Linearized, Cnllisionless Equations 25 3.2) Longitudinal Waves with Bo = 0 28 3.2.1) The dispersion relations 28 3.2.2) Plasma type waves 31 3.2.3) Acoustic type waves 33 3.3) Transverse waves with Bo * k = 0 33 3.3.1) The dispersion relation 33 T.3.2) High Frequency waves 35 3.3.3) Low Frequency waves 39 References 40 ' Figure 1 41 Acknowledgement 42 Abstract A set of fluid type equations is derived to describe the macroscopic behaviour of a plasma consisting of a mixture of matter and antimatter •> The equations ?xe written in a form which displays the full symmetry of the medium with respect to particle charge and mass, a symmetry absent in normal plasmas. This symmetry of the equations facilitates their manipulation and solution, and by way of illustration the equations are used to analyze the propagation of electromagnetic and acoustic waves through a matter-antimatter plasma. Some differences from the propagation of such waves in a normal plasma are noted. 1) Introduction In the theory of elementary particles the properties of matter and antimatter are symmetric, and this symmetry has to a large extent been verified by laboratory experiments (DeBenedetti and Corben, 1954; Segre", 1958). As a consequence interest has recently arisen in the possibility of matter and antimatter being equally, or almost equally, abundant in the observable universe (Alfven, 1965; Laurent & Malm, 1967; Harrison, 1968; Omnes, 1970). However, due to the lad: of observational evidence for the existence of antimatter in astronomical objects, the cosmological matter-antimatter symmetry principle does not enjoy a great deal of popularity. Alfve'n (1965) has pointed out though that we cannot distinguish between regions of pure matter and regions of pure antimatter by observation, but rather it would be the distinctive properties of regions of mixed matter and antimatter that would offer the possibility of unambiguously estabilishing the existence of antimatter in the universe. Such mixed regions must exist, for instance, where pure matter and pure antimatter come into contact, and Omnss (1971) has recently discussed the theoretical properties of such a contact boundary layer using a simple hydrodynamical model. On the observational side discussions of the data from possible regions of mixed matter and antimatter have been given by Burbridge & Hoyle (195S), Frye (1969), and Ekspong et al (1966). In addition Vlasov (1965) recently discussed the -ii T_ — possibility of observing the characteristic line radiation of positronlum and protonium from mixed matter-antimatter regions. In order to interpret observations properly, and also to discuss the detailed processes, such as matter-antimatter separation, that will occur in a matter-antimatter symmetric universe, we need to have a sound knowledge of the basic properties of a mixed matter-antimatter gas. Alfve"n hast given the name ambiplasma to the ionized state of such a gas,, and it is our purpose in this paper to propose a set of fluid-type equations describing a fully ionized ambiplasma. These equations we hope will be suitable for analysing the properties of an ambiplasma in a wide range of physical situations. The physics of ordinary plasmas, containing matter only? is notoriously complicated, and we may expect ambiplaisma physics to be doubly so. However the existence of heavy particles of negative charge, viz. antiprotons, and light particles of positive charge, viz. positrons, gives the ambiplasma a degree of symmetry, with respect to mass and charge, which is absent in ordinary plasmas, i.e. containing protons and electrons only. As a consequence, although the number of particle equations: is doubled, they acquire a symmetry which facilitates their manipulation and solution. The equations used are the usual fluid-type equations derived from the moments of the Boltiiraami equation for charged particles, however they are combined in a novel way which displays the full symmetry of the ambipllasma system. These equations are discussed in section (2). The usefulness of the equations can only be judged by applying them to specific situations, and as an example of - 2 - their use we discuss the linear propagation of longitudinal and transverse waves through a uniform ambiplasma in section (3). Not surprisingly the number of normal wave modes is greater than in an ordinary plasma, since the number of degrees of freedom has increased, and some waves propagate in ambiplasmas which do not exist in ordinary plasmas. An interesting result obtained is that the Faraday rotation of high frequency, linearly polarized,' transverse waves does not occur in a symmetric ambiplasma, i.e. one where the number of antiparticles equals the number of particles. This may provide a test of the Alfven-Klein cosmological model by considering the Faraday rotation of >*»dio waves from distant objects, which would be emitted at a time when, according to the model, matter and antimatter were mixed. Lastly it should be noted that it is in principle possible, though probably beyond the bounds of present technology, to produce an ambiplasma in the laboratory, using either positronium or antiproton sources, or both, and magnetic confine- ment. Hence the subject of this paper cannot be totally devoid of physical significance. 2) Basic Equations The principal components of a fully ionized ambiplasma are positrons (e), electrons(e), protons ^p), antiprotons (p)f and electromagnetic field. The microscopic processes involving these components are very similar to that of an ordinary plasma, but with the important addition that in an ambiplasma - 3 - annihilation and creation of particles and antiparticles occur. The annihilation of a protori-antiproton pair is a rather complex event {Segre, 1958), but essentially it produces neutrinos, pions, muons { and their antiparticles) and y-rays, while the annihilation of an electron-positron pair produces only y-rays. The pions and muons produced by p - p annihilation decay to produce further neutrinos and y ~ rays plus electron-positron pairs. In addition to the creation of e - e pairs from p - p annihilations, various other collision processes of high energy particles can lead to e - e pair production, and if very high energy particles are present we can also have p - p pair production. In an ambiplasma therefore we have many additional microscopic processes of a complex nature which do not arise in the classical treatment of a normal plasma. However for sufficiently small densities these processes can be simply represented by additional terms in the classical equations of motion, while the neutrinos, piotis and muons can be regarded as secondary particles under the following two assumptions. Firstly we assume that the scale lengths of interest are much smaller than the collision length of a neutrino in the ambiplasma, a condition that will hold for a wide range of situations due to the small interaction between neutrinos and other particles. Under this assumption the neutrinos represent simply a sink of momentum and energy provided <:he gravitational field' is negligible. If the gravitational field is not negligible then the possible influence of the neutrinos in determining that field must be considered. - 4 - Secondly we assume that the characteristic time for p - p annihilation is much larger than the decay time of the pions and inuons. Under, this assumption the number density of the pions and muons will be negligibly small compared to the number density of the primary particles and we can neglect their presence. They represent only a short-lived transition state between p - p pairs and neutrinos, y "* rays and e - i pairs. Our purpose in this section is therefore to derive fluid- type equations describing the motion of the primary components of an ambiplasma under the above two assumptions. We will follow the well worn path of taking moments of the Boltzmann equation for individual primary particles (Chapman & Cowling, 1952; Delcroix, 1961), with extra collision terms representing the annihilation and creation processes. The moment equations of the individual particles will then be combined to produce a set of equations which conveniently describe,the collective motion of an ambiplasma. 2.3) Individual Particle Equations If fj.(2» w, t) is the distribution function of the fc» species then the Boltzmann equation is 3fk afk — 5 •» 8fk where (•><. )„ =» rate of change of f. due to elastic (i.e. k = electromagnetic) collisions with other particles, (w^—)v rate of change of f. due to Compton collisions with y - rays, k 3fk („•. -)c = rate of change of fk due to particle creation, and - ( " )- = rate of change of f. due to particle annihilation; a, is the force on a k - particle of velocity w due to the macroscopic, i.e. collective, fields, therefore mk where qk and m, are the charge and mass of the k-particle respectively, E and B are the macroscopic electric and magnetic fields, and $ is the gravitational potential. It should be noted that in equation (1) the electromagnetic field has been split into three components, viz, the microscopic 8fk ' particle fields implicit in (•%?—)E, the collective fields E and B, and the Y-radiati°n field, arising from annihilation, 9f. which is implicit in (,,. • • ) . Supplementary equations will be required for E and B, viz.

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