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 anti- matter 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 annihila- tion 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. the macroscopic Maxwell equations, for the y-radiation field, viz. some form of photon Boitzmann equation, and also for
9 where n^ = /fkd w,
- 6 - _ 1
Ck "
3 and Afc J= /(•*•—-}.d w.
By symmetry we have
Ce " Ci ' Ae " Ae
(3).
The second moment of equation (1) leads to the momentum equation, viz. "It
+ V X n m V + + + £ k 5> " k k * ^Ek £Yk ^Ck " ^Ak (4) where the stress tensor is given by
~k
the terms on the right hand side are
^
- 7 - = /m ( L ) d 3w and S> a3w *ck kS 3F ' c . £A]C - '*jiW A
By symmetry we have 5ci = ?Ce and ?Cp = ?Cp <5)
The third moment of eguati'on (1) leads to an equation for the stress tensor, viz.
3P. vk'vIk
(6) where M = M +(transpose of M) and if a,% are unit vectors in the aredirections respectively, thenfP. x B)= ((a«P) x B)*S. The thermal conduction tensor is defined by 2k = /m and the terms on the R.H.S. are 3fk |Ek = ^
- 8 - Once again by symmetry we have Bee = Ice and
Equations (2), (4) and (6) represent the first three moment equations for the primary particles, i.e. the protons, antiprotons, positrons and electrons. These equations are not closed since we have no equations for Q. . The problem of closing these equations is overcome by making various approximations appropriate to various physical situations (Clemmow & Dougherty, 1969), however we will not discuss the problem of closure here, but simply assume that some approxima- tion can be used to obtain a closed set of fluid-type equations.
This point of view contains an implicit assumption concerning the microscopic processes in the ambiplasma. The p - p annihilation lead to the production of high energy (~100 Mev) e - e pairs, which will be represented by a spike on the high energy tail of the electron and positron distribu- tion functions. In order that an approximation to close the moment equations can be found the spikes must be small, there- fore the characteristic time in which these particles lose their energy,either by synchrotron radiation or by collisions with other particles, must be small compared to the characteristic time in which they are produced by annihilation. This condition must hold for the non-relativistic fluid approximation used here to be valid. If the singular particles lose energy by synchrotron radiation then a considerable fraction of the energy released by annihilation will appear as radio waves in the ambiplasma.
- 9 - 2.2) Ambiplasma Equations
Our task now is to combine the individual moment equations tc obtain a convenient set of macroscopic equations in the spirit of chapter 2 of Spitzer (1964). Spitzer defined the total densities of charge and mass and the total electric current and momentum flux, and combined the first two moment equations to obtain the well known set of equations for a normal plasma. We could similarly define
Q - e(ns - ne + np - n->
p - m{n- + ng) + M(np + n-) (8) where m and M are the electron and proton masses respectively, and
£- e(n5v5 - neve + npVp - n-y-) and
i + neye) + M(npXp + n-y^). (9)
The first two moment equations could then be combined to obtain equations for Q, p, £, and v.. However since we have a 4-particle plasma we require four more equations from the first two moment equations. These extra equations could be chosen in a number of ways, and this is a perfectly valid procedure. However a useful property of the set of moment equations for e, e, p and p is their symmetry with respect to mass and charge. By employing the above procedure we would lose sight of this useful symmetry, and involve ourselves in more algebraic manipulation than is necessary. He propose the following method of combining the moment equations which, with
- 10 - suitable notation, keeps the symmetry of the equations fully apparent. The electrons and positrons will be referred to collectively as the leptons (strictly speaking the secondary muons and neutrinos are also leptons, but we will refer only to e and e by the name leptons}, and the protons and antiprotons will fee referred to collectively as the nucleons. We will keep the lepton and nucleon equations separate, and derive equations for the following quantities,
n n = n n * = i " e' Zr e*e " e^e, R * ni + V 2n ° ni*i + ne?e.
R = np - n-, JR = npvp - npv_.
r and R are respectively called the lspton and baryonic numbers in elementary particle theory. We will refer to them here as the lepton and nucleon residues respectively, since, if complete annihilation took place, then r and R would be the residual densities of leptons and nucleons, multiplied by the sign of the appropriate charge. J and JR are respectively the lepton and nucleon residue fluxes, n and N we will refer to as the lepton and nucleon densities respectively, and J_ and J-j are respectively the total lepton and nucleon fluxes.
Subtracting and adding the continuity equations for electrons and positrons, we obtain. and
- 11 - where Cn = C- + C&, An = A- + Ae. Here we have used equations (3) to obtain zero on the right hand side of (11). By symmetry the corresponding nucleon equations can be obtained from equations (11) and (12) by substituting R and N everywhere for r end n respectively, i.e.
(14)
where CN = Cp + C^, ^ = Ap + A-.
The amount of work saved by writing down equations (13) and (14) using symmetry is rather small, however much more work will be saved by using symmetry to obtain the momentum and energy equations for the nucleons. Equations (11) and (13) show explicitly that both the lepton and nucleon residues are conserved quantities. This is an important property with regard to matter-antimatter separa- tion processes in cosmological considerations. In the usual treatment of a normal plasma C „ and A „ are all zero, and equations (11) and (12), and equations (13) and (14) are equivalent, r and n being equal to minus and plus the electron density respectively, while ft and N are both equal to the proton density. Subtracting and adding the momentum equations for positrons and electrons multiplied by -=— , we obtain
- 12 - 3J £rJn^n^n^r?r 1 5? + V- ^•-::I::^^L::^^ TT~Sr>
•I and
e (rE + J x B) - nV$ + P_ + P + P — P (16) m
Here P , the lepton residue stress, and P . the total lepton stress, are defined by
si -e »e' sn &e =e*
The collision terms are defined by
J?Er = ^Ei " ^Ee' ^En = ^E5 + ^Ee,
P s P ^ •• P P ^ P •• ^h P P """ P ^ "4" P * -yr -ye -ye' -yn ~ye -ye' -»Cn -Ce ^Ce and -Ar = -Ae " -Ae' -An = -Ae + -Ae*
By symmetry, i.e. substituting R, N and H for r, n and m respectively in equations (15) and (16) we obtain the nucleon momentum equations, viz.
- 13 - and
. 1 M
S> -
where, in a similar fashion to the lepton equations,. we have
Sn-Sp- fp' IN= «pp+ Ip' -ER -Ep ^Ep' ^EN -Ep ~Ep' P =P - P —. P =P + P —. £yR -YP ~YP' ~YN ~YP -YP'
and P P . AR - SAP - JAP' SAN = AP
Finally we subtract and add the stress tensor equations for positrons and electrons to obtain,
SP nJ - rJ nJ - rJ -* + 7P • 3t* sn ZZ ^— + 7P^--^^ = n2 r2 -r n2 _ r2 (_n^J - z£__rJ )
nJ. - rJ _ nJ_. .-.' r J. r n n P (P -?( ~ ~ )) + (P^'Vt^S 2^
" Iftr and 8P . . nJ - . . hJ. ..-. rJ._ . . VP • "n vr— n (cont.)
- 14 - ruT - r J nj - r J P V'(— — )) + PnV(-=nS (=£ ) ** n2- r2 «n n2- r2
nJ_ - rJ Dn . nJ . - r J (P -v(E=2 2^ r n2- r2 =n n2- r2
+ 5n i"sr S' = jEn + Jyn + Jen ~ ^An (20) where
Q = Q- ~ Q » Q = Q- + Q ~r ~e ^e P1 £e £e
lyr = lye " -ye' lyn ~ lye + |ye S_ = s - + S and |Ar = JAS " JAe' IAII = #Al
Note that, due to the symmetry of the equations, the left hand side of equation (20) can be obtained from the left hand side of (19) by interchanging r and n. The same is also true for the left hand sides of equations (11) and (12) and equations (15) and (16).
By symmetry we obtain the nucleon stress tensor equations, i.e., 3P NJ - RJ NJ - RJ
_i£ + VP '-^S 2. + VPn* 2 _2S 3t sN N2- R2 aR N2- Ra
NJO - RJM NJ.- - RJ_ P v<-=5 =S—I+,PV.(JJI =5—) =N N2- R2 -R N2- R2
NJ - RJN NJ - RJ »N Mz- R2 sR N2- R2 D V'Q - -|-(PN x B) HS-,+5.- SAR (21)
- 15 - 3P.T NJO - RJW NJM - RJ- and «fi + VP_ ^S =2 + vp __=^L__^R 9t "R N2 - R2 **N N2 - R2
NJ_ - RJM NJM - —^5 =» ) + Pv N^* -
(p ?(^R^ ))D+ ^N^R *R N2 - R2 sN N2 - R2
e D 7'2® " "M~(IR x §} = IEN + lyN + ICN " IAN (22) where
IER = §Ep " IEP' IEN = »
fyR = lyp " ^YP' *YN = l and = + |AR IAP - IAP' IAN " |AP EA?"
Equations (11) to (22) constitute a set of fluid equations describing an ambiplasma,which we assume can be closed by some approximation appropriate to the physical situation of interest. These equations are supplemented by the Maxwell equations for the collective fields, i.e. V-g = ~ (r + R), V«| = 0,
and e0 J| + e{Jr + JR) = ^^ . (23)
The source terms here represent the collective effect of the ambiplasma. If the high energy electrons produced by annihilation lose a significant fraction of their energy by - 16 - synchrotron radiation, then the collective source terms must be supplemented by terms representing these singular particles. This will have the effect of adding radio wave fields to the collective electromagnetic field. In the non-relativistic limit the cross section for Compton scattering is independent of the velocity of the particle, and we can write P - , for instance, as
3 Py~ - n5 / o5<|k|)Mc#(x,k,t)d k (24) where a-( |k|) is the total scattering cross section for positrons, and where (||r)Y = rate of change of ^ due to Compton scattering with the particles, - (§|r)c - rate of change of ^ due to - 17 - creation of particles and Cg|r)A = rate of change of i> due to annihilation of particles. We have omitted the acceleration term on the left hand side of this equation since the macro- scopic electromagnetic field does not affect the photons, and in the non-relativistic limit the gravitational field also does not affect the photons. If we write then equation (24) obviously leads to P = r¥ , P = ri¥ and similarly (26) When the electromagnetic and neutrino fields have negligibly small energy density compared to the rest, energy density of the primary particles the gravitational potential in the Newtonian approximation is given by V2$ = - 4irG(nm + NM), (27) where G is the gravitational constant. In situations where a significant fraction of the nucleons is being annihilated and the gravitational field is significant, terms representing the electromagnetic field and neutrino energies must appear on the right hand side of this equation. Equations (23), (25) and (27) along with a closed set of ' fluid equations form a complete set of equations describing an ambiplasiria over a wide range of physical parameters. ?o illustrate how the collision terms might be treated we consider some simple approximate forms of the eliStic colli- sion and annihilation terms in the momentum equation. We write - 18 -• Ek = where J'gi.n = momentum transferred from the £ to the k species; assuming that PEk£ = Yj^V^(v^) we have, Since Pftl =-P thus we have reduced the 12 collision constants to 6 independent ones. Now if the thermal motion of the protons and antiprotons is similar, and that of the positrons and electrons is similar, then the form of the collision constant (Spitzer? 1964) gives = = "•p e — Y'p e- Y'p— e Y-'p—e and we hare only 3 independent collision constants, which we write as Y = Y Y = Y and Y n ee' N pp nN ° Ype- The P£r terms of equation (15) can be written as which reduces to - ^.J (28) - 19 - similarly we obtain N ] 29 *En - W»JH " ^n < > and by symmetry £ER - YNtK£N " NJR1 + YnNtRJn " nJR] (30) and m 31 2m WNJn " -2H1 < > We now consider the annihilation terms. If a ,(|u - w| ) is the total cross section for annihilation of a positron and electron of velocities u and w. respectively, and it is a function 3f of |u - w| only, then (jrp—)A is given by where c ( lu - wl ) = lu - w| a_-(|u - wl) and P. is given by f f 3 s 2) e^) i(H>5n(lu - vjl)d ud w If fe(w) = g(w - ve) and f-(u) » g(u - vg) then a change of integration variables to t = w-v,s = u-v ~ •• »e •• *• <»e leads to 3 3 + fin (a - t)g(t)«j(s)5_(»s - t + v- - val)d td *. A The first term here is m(v- - v^)-^= and, assuming g(t)=g(-t), - 20 - the second term is 3 3 2//msg(t)g{s)5n(|s - t + v- - v |)d td a and In the subsonic limit v / v- are much smaller than t and § for most of the range of t and s over which g(t) and g(s) are significant, therefore in this limit the second term is approxi- mately 2//msg(t)g(s)c (|s - t|)d3td*s which is zero since the integrand is odd. Therefore for subsonic motion ^r"rJ«)v (32) Similarly we can obtain (for PAn the subsonic approximation is not necessary); and by symmetry and The total charge and mass densities, and the electric current and momentum flux defined by (8) and (9) are of course simply related to the symmetric quantities, i.e. Q = e(r + R), - 21 - p = mn + MN, pv and we can simply obtain the equations of continuity of charge and mass by combining equations (11) and (13) and equations (12) and (14) respectively i.e. ft + V' (pI> = m(Cn " V •• M(CN " V Similarly we can obtain the Ohm's law for an ambiplasma from a combination (15) and (17), I.e., using equations (26)f (28) to (31), and (32) to (35) r n M + eV. ( ~ SISl££ n2-r2 N2- R2 m M ^ + t-mil + ^ * §J " e(r ,Yn . _„ ,_r T N .r JR.T n T eYnNl~m~ -N iinir ~M~X-n M~ii - 22 - and from a combination of (16) and (18) we obtain the total momentum equation z_ P + P n e (r + R) E +• J x g - pV* -s- n* + N*N "^n —w (nn-Jn + rJ ) + —- (NJ.T + RJr These equations for J and v are no less complex than the equations for J , J , J^ and J , and the symmetry, which is an invaluable aid in handling the equations, has disappeared. Consequently the symmetric quantities defined by (10) are more convenient variables for analyzing the motion of an ambiplasma. While symmetry is a strong motivation for using r, n, R, N etc. it is not the only one, for it turns out that these quanti- ties frequently appear as important parameters characterising the ambiplasma. This can be demonstrated, for instance, by considering the Oebye length in an ambiplasma. 2.3} Debye length in an J'jnbiplasma Applying the Debye-HuckeJ theory to the field of a stationary charge q in an isothermal ambiplasma, the electro- siatic field is given by Poisson's law, i.e. - 23 - ne - ne + np " nf>> n + n )exp( T7 I < I0 po " - n0 - r0 2 Expanding the exponentials on the left hand side and neglecting 2nd order terms in ~ we obtain a simpler equation for § at a distance from g, i.e. V24> = -§7 [r°+ R« + Now r0 + Ro - 0 is the charge neutrality condition for an ambiplasraa, we therefore have 2 2 where X Q = eoKT/(r.o + N0)e . which has the spherically symmetric solution where x is *J»e distance from the charge. - 24 - Hence the total lepton and nucleon numbers appear in the Debye length in an ambiplasma. If we consider the field of a thermally moving nucleon under the assumption that the other nucleons move too slowly to reach equilibrium in it's field (i.e. n = n , n- = n_ ) then we obtain the same solution with q = e and 2 2 \ D = eoKT/noe . (36) Further examples of the lepton and nucleon densities, and also the residues, appearing as important ambiplasma - parameters will be obtained in the following section, in which we will apply the equations derived here to the treatment of small amplitude waves. 3) Waves in an Ambiplasma 3.1) Linearized. Collisionless Equations We wish to consider the simplest, most basic wave phenomena that can occur in an ambiplasma, i.e. plasma waves, acoustic waves, Alfven waves etc. Consequently we assume that the particle densities are sufficiently low for us to neglect all the colli- sion terms in the equations, the characteristic collision times being much longer than the periods of the waves. Of course some of the most distinctive phenomena in ambiplasmas, for cosmological situations in particular, will arise from the annihilation terms, however we will consider here only the very simplest phenomena and put all collision terms to zero. Consis- tently with this assumption we will assume that the Y-ray density is zero, and, considering electromagnetic and acoustic - 25 - waves only, we will assume that the gravitational force is negligible. Under the assumption of no annihilation or creation we can postulate a uniform steady state, and consider the propagation of waves of infinitesimally small amplitude through such an ambiplasma. A set of equations describing such waves is given by the linearized forms of equations (11) to (18) and equation (23). Denoting wave quantities by a bar and steady state quantities by a subscript 0, these equations are || + V3r = 0, (37) + V»5 = 0, (38) —n C39) (40) 3J 35 " T a - - a x ' -CT-^PR -jj-WoE + JN go) (43) 3J« •• _ _ (44) - 26 - Vl = -^-{r + R) «• Go V'B = 0 (46) If = -7 x I (47) and eo|| + e(Jr + JR) » 2-J-S (48) wha^e we have assumed that go = 0 and J = J = JR = £„ =0. We have not included the stress tensor equations here and the equations must be closed by writing P , P , P and P in terms of the other variables. For simplicity we will assume that these tensors are diagonal, and that the diagonal components are given by a law of adiabatic form, i.e. where a2ko = YkKTk/V In the collisionless limit these approximations certainly do not hold in general (Bernstein et al, 1961). For instance for high frequency longitudinal waves, where the phase velocity is much greater than a, , equation (49) is a good approxima- tion with y. = 3; but when the phase velocity is of the order of a, we cannot neglect thermal conduction and the adiabatic law does not hold. While for high frequency transverse waves the effect of the off-diagonal terms in Pj.is negligible, but for low frequency waves it is not negligible. The complex - 27 - effects of P. and Q. lead usually to damping of the waves, and do not affect the basic nature of the normal wave modes. Consequently we adopt equation (49) for simplicity, assuming that for any given wave mode a suitable value of y. can be found. Assuming further that a- = a. = an eg Gg no and a = a- = a., po Po No i.e. equipartition between the leptons, and between the nucleons, we obtain from (49) _I_V.p = a2 vr Itl ^i XlQ m -$n ng (50) 2 -—-M V«P.a;NT = a NM oVN thus completing our set of equations for the waves. 3.2) Longitudinal Waves with BD = 0 3.2.1) The dispersion relations We assume that all perturbed quantities are proportional to exp(ifa>t - ik.x) . Then equation (46) gives jk.B = 0, and since, for longitudinal waves, - 28 k * B = 0 we have B = 0. Due to the zero magnetic perturbation equations (37), (39), (45), and (48) are not independent and we drop equation (48). Equation (47) is trivially satisfied. (3 f 0) and E If we write k=(k,O,O), 3rfn,R,H" r,nfRfN '°» ' I=< f0,0), the remaining equations, using equations (50), have the form ur - kJr = 0, (51) un - kJn = 0, (52) wR - kJR = 0, (53) ioH - kJN = 0, • (54) iuJ - ika2 r => "^ I, (55) x no m £p f (56) e R ^oR = % -E, (57) 2 iwJN - ika Noi5 - Sp E, (58) and -ikE = -|- (r + R). (59) Using equations (51) and (59) we eliminate J and I from equation (55), obtaining 2 („• - ^sn - u,*pn)r = o» pnR (60) where u)* = a2 k2 sn no 2 a and w pn = noe /meo« - 29 - Similarly using (52) and (59) we obtain from (56} 2 2 where w Dr ~ roe /meo. By symmetry we have <<*2 " "2SN " «V5 = UV (62) 2 2 2 and (co - w sN)N = w pR(r" + 5) (63) 2 c 2 2 2 where w sN = a Hofc ,u) pN = Noe /me0 2 2 and u)po u = Rae /me0. Equations (60) and (62) imply that either «"2 " -'„ " -2pn^ ^2 " u2sN " »V - w2pnwZpN <64> or f = R = 0. (65) When (65) holds, equations (61) and (63) give respectively 2 2 either u> = u> sn (66) or n = 0 2 2 and either u = w N (67) or 5=0. Equations (64), (66) and (67) are therefore the dispersion relations representing the normal modes of oscillation. - 30 - •••.•rim 3.2.2) Plasma type waves Equation (64) can be written as "V i.e. „. where D Assuming that the lepton and nucleon temperatures are of the same order of magnitude, we have 2 HI 2 and assuming ND «. n0, we have 2 m 2 w w PN - M pn* Hence the second term under the square root sign, is of order ra/M relative to the first term, and we can expand the roots, i.e.* 2 sn pn sN pN 2 9 9 9 9 9 - 2 i.e.' we. have two solutions for 2 £.n _ M - 31 - «i is the frequency of lepton plasma waves, corresponding to electron plasma waves in a normal plasma; and (02 is the frequency of nucleon plasma waves, corresponding to ion waves in a normal plasma. The latter correspondence is established by setting n0 =. No/ which is the case for a normal plasma, then k2noe2/mMeo U)2 = 2 2 (k ynKTn/m+noe /meo) If -j2i~r is much greater than the Debye length given by equation (36) then the first term in the denominator is negligible compared to the second term and 2 - ., VnKTn + Y^ w 2 M which is identical to the ion wave frequency in the long wavelength limit given by Spitzer (1964). From equations (61) and (63) we have 5 = EE— (r + 5) and S = —— (r + R). Therefore if the ambiplasma is symmetric, i.e. if r0 - 0 and =t >Z = and Ro - 0, then it follows that w*pr * DR ^' consequently' n ss ji = 0, i.e. n- == -n and n = -n-. Hence in plasma waves in a symmetric ambiplasma the motion of the positrons and electrons are IT out of phase, and the motion of the protons and antiprotons are also it out of phase. Note that the plasma frequency in a cold arabiplasma is given by /noe2/meo , i.e. the total lepton number takes the place of • ' • • ' • • '. - 32 - the electron density for a normal plasma. 3.2.3) Acoustic type waves The dispersion relations (66) and (67) represent lepton and nucleon acoustic waves respectively. In these waves r = R = 0 (and consequently E = 0), i.e. n- = n and n = n-. Hence in these waves the motions of the particles and corresponding antiparticles are in phase. These modes do not exist in a normal plasma, since for a normal plasma r = -n and R = N therefore the only solutions for a normal plasma are those corresponding to dispersion relation (64) (i.e. plasma type waves), and the trivial solution r=n=R=N=0« 3.3) Transverse waves with Bo * h - 0 3.3.1) The dispersion relation We assume again that all wave quantities ar*> proportional to exp(iu>t - ik.x). Under the assumption that k and Bo are parallel we can consider purely transverse waves, therefore k.J,. = k.5 = k.J_, = k.JM = 0, and equations (37) to (40) imply that r = R = n = N = 0; equations (';0> therefore give Vfo r = VP« m = V-P,sR, = V*P.«N. = 0 Equations (45) and (46) are trivially satisfied' since k.E = k.B = 0 The remaining equations have the form - 33 - 5i ir * So)' (69) -f-(N0§ + JN x B0), (70) wB = k x |, (72) and ioseoE + e (J_ + JD) = - i^§—. (73) Taking the vector product of k and equations (68) to (71), and equation (73), and using equation (72) to eliminate E from the resulting equations, we obtain iwk x Jr = -^-(nou)| + kB0Jn), (74) icok x J = -f-drouB + kBoJ..), (75) x E N iwiS iR -§-< owB + kB0JN), (76) iuk x J^ = -S_(ROUB + kB03R), (77) 2 and i(e0o) - -^i)| + ek x (£r + jR) . o, (78) where we have used k.B0 = kB0 and k.J _ „ „ = 0. We can eliminate 5 from (75) using (74), i.e. -2—iuk x j - nouB t e ^ ,j;—=^ Therefore (79) en where u • By symmetry the nucleon equations give where Substituting from equations (79) and (80) for J and J ^r W» in equation (78) we obtain iB(a>2 - k2c2 00 7 3 "ON"" on where u -, u , u and w are as defined before, and we have 2 used c =l/(p0Eo). The dispersion relation for transverse waves travelling parallel to Bo can easily be derived from this equation, and we will solve the dispersion relation for the asymptotic cases of high and low frequency waves. 3.3.2) High Frequency waves Here we assume that a> » » „ & «cN, and putoLN ** ucN « 0# i.e. we ignore the motion of the nucleons. Equation (81) can then be written as (82) - 35 - which has the form iBf(u>) - g(a>) &£l - 0. If we assume jc » (k, 0, 0) and therefore B =• (0, B , then the matrix form of this equation is therefore the dispersion relation is f2(u>) = g2(w), i.e. f(w) - ±g{u). (83) Then, since (B /B ) = ig (w)/f (u>), we have two wave modes for which (B/B )., - = ±i. y z j.t^ Therefore, if u* V 0, the normal wave modes are circularly polarized with the vectors B, J etc. rotating in opposite senses in the two modes. This is similar to the wave modes in a normal plasma, and the difference in the phase velocities of che two modes leads'to the well known effect of Faraday rotation. The explicit form of (83) is 2 2 2 2 2 2 &>2 u 0) to a) - (m* - k c ) ((u - « cn) ± pr cn ** °° Assuming that u » o> we can neglect second order terms in o> and this equation becomes w2w2 - u2 (u>2 - k2c2) ± w^^cn** *" ©• which can be written as - 36 - 1 (l) j U c -, [1 1 1 pr on I JC f p-j 2 (0 • (0 ' to first order in to / 1 i.e. v =: V ± -2 Av 2 3 u (0 where Av V • 2 pr cn c ft)3 A linearly polarized wave is a combination of the two circularly polarized modes, and due to Av the plane of polariza- tion slowly (relative to ID) rotates. The rate of rotation is given to first order in «L by ^-— » ——— • ^. , where 6 = Gil CIS & v angle of polarization and s = distance traversed by the wave !...« ds 2czzoo>> 2 -)• 2m2eoc If the ambiplasma is symmetric, i.e. r0 - 0, it follows that T- = 0, therefore there is no Faraday rotation in a symmetric ambiplasma. This is consistent with equation (82), _ a. which reduces to Bf(o))= 0 when - 37 - simple to explain. If we assume that the electromagnetic wave field is varying as in a linearly polarized wave then the electrons and positrons describe ellipses in such a field, rotating with the frequency of the wave. However they rotate in opposite senses so that if we consider the motion of an e-e pair at four stages in a wave period we have the sequence shown in Fig. 1 (see page 41), where + and - represent the positron and electron respectively, and J - is the electric current due to this pair only. We see that J - does not rotate, it oscillates. Hence if the residue is zero, and therefore for every electron there is a positron, the total ambiplasma current will oscillate, and does so in a manner compatible with the assumed linear polarization of the electromagnetic wave field, i.e. in the polarization plane of E. Linearly polarized waves therefore propagate with no Faraday rotation. However if there is an excess of either electrons or positrons, then the excess will introduce a component of the total current perpendicular to the electric field and the assumed linear polarization must be modified, i.e. Faraday rotation takes place. In this case the normal modes of wave motion are the usual circularly polarized waves, as we have found, in which the electrons and positrons describe circles, this time rotating in the same sense, though IT out of phase with each other. If significant regions of ambiplasma exist in the observable universe then the reduced Faraday rotation in these regions should have an important effect on the rotation of linearly polarized radio waves from distant astronomical objects. For instance the variation of the root mean square rotation (for - 38 - groups of radio sources) with red shift might provide a test for the Alfven-Klein cosmology, in which vhe lepton and necleon residues are assumed to be uniformly zero at early times. 3.3.3) Low Frequency Waves We assume that to << to „ s u and also w << kc, then equation (81) is approximately t,io2 ,.,2 - k2c2] en CN pr . 0. (84) "en "cN er0 R eR0 NOW e°B° and therefore the charge neutrality condition, i.e. re + Ro gives en CN and equation-(84) becomes to2 u)2 „ Therefore we have two linearly polarized normal modes with a common dispersion relation, viz. 2 Bo eo UoPo i.e. normal Alfv£n waves propagate in an ambiplasma. - 39 ••-• References Alfven,H.: 1965, Rev. Mod. Phys. 37_, 652. Bernstein,I.B., and Trehan,S.K.: I960, Nuclear Fusion jL, 3. Burbridge,G.R., and Hoyle,F.: 1956, I& Nuovo Cimento, IV, 558. Chapman,S., and Cowling,T.G.: 1939, "The Mathematical Theory of Non-Uniform Gases", Cambridge U.P. Clemmow,P.C, and Dougherty, J.P.: 1969, "Electrodynamics of Particles and Plasmas", Addison-Wesley, London. DeBenedetti,S., and Corben,H.C: 1954, Annual Review of Nuclear Science, 4_, 191. Delcroix,J.L.: 1960, "Introduction to the Theory of Ionized Gases", Interscience, New York. Ekspong, A.G., Yamdagni,N.K., and Bonnevier,B.: 1966/ Phys. Rev. Letts. 1£, 664. Frye,G.M.: 1969, Can. J. Phys. 46_, 5448. Harrison,E.R.: 1968, Phys. Rev., 167, 1170. Laurent,B.E., and Malm,B.E.: 1969, Arkiv Fysik 3£, 325.. Omnes,R.: 1971, Astron & Astrophys., 10_f 228. Omnes,R.: 1971, Astron & Astrophys. 11, 450. Segre,E.: 1958r Annual Review of Nuclear Science 8_, 127. Spitzer,L.: 1962, "Physics of Fully Ionized Gases", Intersciencs, New York. Vlasov, N.A.; 1965, Soviet Aatr-AJ 8_, 715. - 40 - I© »l o II o 0 ON 5 iS 4-r Acknowledgement During the course of this work one of us (A.H.N.) was supported by a Royal Society Oversea* Research fellowship, and would like to thank Prof. K. Husimi and Prof. Y. Terashima for their kind hospitality at the Kagoya Institute of Plasma Physica. - 42 -