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Dielectric Response of a Relativistic Degenerate
Electron Plasma in a Strong Magnetic Field
A.E. Delsante and WE. Frankel
School of Physics, University of Melbourne
Parkville, Victoria, 3052, Australia
Copies Submitted 2
Manuscript Pages 35
Figures
Tables Relativistic Electron Plasma
Dr. N.E. Frankel,
School of Physics,
University of Melbourne,
Parkville, Victoria, 3052, Australia. 1.
Abstract
The longitudinal dielectric response of a relativistic ultra- degenerate electron plasma in a strong magnetic field is obtained via a relativistic generalization of the Hartree self-consistent field method.
Dispersion relations and damping conditions for plasma oscillations both parallel and perpendicular to the magnetic field are obtained. We also give detailed results for the zero-field case. Applications to white dwarf stars and pulsars are given. 2.
1. Introduction
The dielectric response of a non-relativistic ultra-degenerate electron gas was first given by Linhard [l]. Since then the ground-state properties of the non-relativistic interacting electron gas have been extensively studied. Two main lines were and are pursued: the first is the use of diagram techniques and perturbation series expansions by
Goldstone {2], Hubbard [i] , Gell-Mann and Brueckner [4], Sawada [s] and others. These techniques [6] enable the ground-state energy of the gas to be obtained as a series in the inverse-density parameter r - that is, a high-density expansion - and also enable the dielectric response func tion to be obtained. The second approach, by Nozleres and Pines [f\, for example, derives a generalized dielectric function in the random-phase approximation (RPA), and this is then used to obtain the ground-state energy and collective properties of the gas without resorting to a pertur bation series expansion. Equivalent results for the dielectric function were also obtained by Ehrenreich and Cohen [8J , using the self-consistent field method (as was used by Linhard £l]). These authors also showed that the self-consistent field method and the RPA are rigorously equivalent.
The relativistic, ultra-degenerate electron gas in zero magnetic field has been given a thorough treatment by Jancovici (j)] , who calculated the transverse and longitudinal dielectric constants using a quasi-boson
Hamiltonian approach which took into account pair production as well as the transverse part of the Coulour.ib interaction. These dielectric con stants were used to obtain dispersion relations for transverse and
longitudinal modes, the screened potential about a test charge, and the ground-state energy of the relativistic gas. The problem was also discussed a little earlier by Tsytovich [lo] , who obtained the longitudinal and 3. transverse dielectric functions, taking into account pair production but not the transverse interaction. Recently, a different approach using a quantum generalization of the relativistic Vlasov equation has been given by Hakim and Heyvaerts QlJ, who in some cases obtained results equivalent to those given by Jancovici [9].
For the relativistic electron gas in a magnetic field, a general equation of state has been obtained by Canuto and Chiu Q.2] for non- interacting particles. This equation of state has been used by Canuto and Chou [l3J in a semiclassical-hydrodynamical treatment of the gas. A detailed microscopic dielectric-constant treatment of the non-relativistic ultra-degenerate electron gas in a magnetic field has been given for example by Greene et al Q^Qfcnd by Benford and Rostoker [is], who obtained the appropriate dispersion relations.
Relatively less attention has been given to the dielectric res ponse of a relativistic electron gas in a magnetic field. A generalization of the work of Tsytovich [id] for zero field has been given by Svetozarova and Tsytovich Q&J for non-zero magnetic fields. More recently, Bakshii,
Cover and Kalman Q7] have described a general formalism for obtaining the polarization tensor containing both particle and vacuum contributions.
The vacuum contribution alone has been given in detail Q7j , and just recently [183 these authors have studied the complete polarization tensor, including vacuum contributions, for a relativistically dense electron gas in an intense external magnetic field. Of interest too is the work of Dominguez Tenreiro and Hakim [}0] , who discuss the transport properties of the relativistic ultra-degenerate electron gas in a strong magnetic field.
A good general review of these and related topics has been given 4. by Hakim [20] .
The work in the present paper derives explicit dispersion rela tions for plasma oscillations in a r;lativistic ultra-degenerate electron gas in a strong magnetic field. These results are obtained via a relativistic generalization of the RPA-self-consistent field dielectric constant method. As far as we are aware, this approach and many of the results obtained from it have not appeared in the literature [2l]. For completeness we also give detailed results for the zero-field case and where applicable compare our results with those obtained by Jancovici
00 •
Although this work is of intrinsic interest, the discovery of
astrophysical objects such as magnetic white dwarf stars and pulsars
lends fresh motivation to it. The white dwarf in the classic example
of a dense relativistic ultra-degenerate electron gas. At electron number 30 -3 densities exceeding 10 cm , the electrons are relativistic, and their
Fermi temperature now far exceeds the interior temperature which is
approximately 10 °K. Although the surfaces of some white dwarfs are
thought to support magnetic fields of the order of 10 G [22J , this
value is sufficiently low for them to still be considered in a weak-field
approximation. This is because the field strength B for which the
magnetic energy of the electron is comparable to its rest-mass energy, 2 13
that is, etiB /mc = mc , is approximately 4.4 x 10 G.
In view of this figure, it is interesting to note that a field
strength of approximately this magnitude has recently been measured for
a pulsar [23]. The interior of pulsars are thought to consist of dense ultra-degenerate neutrons, except for the outer layers which may contain 5. tightly bound nuclei and electrons at relativistic densities (24].
Furthermore, because of the enormous field strenghts which exist at the surface, pulsars are surrounded by magnetospheres consisting of dense relativistic electrons (and positrons) in magnetic fields of the order of 10 G (25J. Thus these typical field strenghts and densities make it imperative that a relativistic treatment be used.
The following simple considerations show that qualitatively different results are to be expected for the ultra-relativistic case as compared to the non-relativistic case. Non-relativistically, the kinetic 2 -2 energy of an electron, p /2m, scales as (length) , whereas the Couloumbic 2 -1 potential energy scales as e (lengtn) . Hence at high densities the kinetic energy will dominate the potential energy (since p ' * ~ (length)" ).
However the kinetic energy of an ultra-relativistic electron, pc, also scales as (length)" , so that now the ratio of the potential to kinetic energy is density-independent, but still small, and is in fact the fine- 2 structure constant, e /lie.
The layout of this paper is as follows: in section 2 the form
alism for deriving the dielectric response function via the RPA-self-
consistent potential method is given. In section 3 we present the zero-
field case. The well-known non-relativistic and corresponding ultra-
relativistic results are obtained for the dispersion relations and damping
conditions for plasma oscillations. A detailed treatment of the screened potential about a test charge is also given. Ion acoustic oscillations
for various densities and degeneracy conditions are discussed. In section
4 we discuss the non-zero magnetic field case for various density regimes
and magnetic field strengths. Finally in section 5 we give a general
discussion of our work. 6.
2. Formalism
The Hamiltonian of the system is given in the Couloumb gauge and
Heisenberg representation by
£WP^2/xdV (?•»
where the first term represents the Hamiltcniun of electrons moving in a time-independent external field whose vector potential is given by
A (x), and the second term represents the Couloumb interaction between electrons; a and 8 are the usual Dirac matrices, x = (x, t) and p(x) = e4)T(x)ii*(x) is the charge density. Ke are excluding external elec tric fields.
We note at this point that this formalism neglects the effects of the transverse part of the Couloumb interaction. We will have more to say about this at the appropriate point.
The development now follows closely that given by Harris [26].
The self-consistent potential due to the electrons, A fx), satisfies
Poisson's equation:
or
= r ft*')*y ix.ti) A„(*> X - X' Ife now expand f(x) in terns of the set of positive energy eigenfunctions of the single-particle Dirac Hani1tonian in an external field:
(e H, = C£.(jp-e4 fc)) + f*c*
These eigenfunctions are denoted by *a(x), where {a} is the set of quantum numbers specifying the particle state of energy E :
Of course, $(x) can be expanded in terms of both the positive and negative eigenfunctions if the presence of positrons is to be included. We will I have more to say about this point later.
We therefore write
' ft. where the operators C (t) satisfy the equal-time anti-commutation rela tions
If we now Fourier-expand the density according to 8.
t : fCj.O = •$jd>x.t"' ((*.*) £•*>
then the Haailtonian (2.1) becones
U) - Z Clt'KM^ + *v f f Q-<> Pf-J'*) 7 £<>
Substituting p(x, t) = e<(i (x) V a unere We are of course assuming a smeared-out positive neutralizing background charge. We now introduce an operator B(a, a', t), defined by B(a, a', t) = C (t)C ?(t). Applying the Heisenburg equations of motion for an operator to B(a, a', t) and using (2.4) and (2.6), we obtain -±Zl*£\(ti,i>L<''\<'l*W'l> % d x L _ * [ - We now ensemble average this equation, defining a generalized distribution function where A is the conplete set of plasma states and P. is the probability t finding the system in state A. Furthermore, we make the self-consistent field approximation by writing This approximation neglects exchange and so is a Hartrcc approximation. Finally, the equations are linearized by writing 10. where p} and Fj are small perturbations. F(a) will be taken to be the equilibrium Ferai-Dirac distribution function. Applying (2.S), (2.9) and (2.10) to (2.7), we obtain the linearized Martree equation ^+*^>WO = i f>^0 - HrfZ. Y<*'!e **ll >(**.*> (*'<> where p. has now been written as p. The Fouri'r-transformed Poisson equation for the self-consistent scalar potential A is, from (2.2), Equations (2.6a), (2.11) and (2.12) ny be solved by using Laplace tr.ir.s- fonns wi'.h respect to t to give the sclf-consistent equation for A (q, *): 11. ftp - Ffr) <*|e'i4'5|*> Let the second term on the right-hand side of (2.13) be represented by D(q, q1, w). We now assume that the system is homogenous in space to lowest order, so that D(q, q', to) - 5 , D(q, q, w). Then (2.13) can be solved for A (q, u) : (211) where -iO.X. a l^TTl7- l< • -F "1 L -I is the generalized longitudinal dielectric function. This expression can be compared with the usual non-relativistic expression given for example by Harris £26] : 12. ..in- - *** "** QII.) where the factor of 2 takes into account the sum over spins. Taking the inverse Laplace transform of (2.14) gives A. (*,*> = • KOI ctoo e £ £ V^ CLA' £(j,rt(w-(v: «VO where C lies above any singularity in the integrand. Closing the contour in the lower half-plane and assuming that all collective modes are stable we obtain for A at large times: «hrW* (*•'•• E ay <(t, «V- *)/*X where e(q, w) = e(q, w +in), where n is small and , itive, and is the analytic continuation of e(q, to) beginning from the upper half of the 13. u-plane. Taking a time Fourier transform of A then gives Kit,**) = For non-interacting particles, and hence from (2.6a) we have P..;. &') = or, taking the Fourier transform, 14. Hence, from (2.19) and (2.21), we have A. («,«)= ^-^ O-uj Writing (2.22) as A0(q, u) = where For the case of a relatively slow-moving test particle in the electron gas, we approximate the single-particle potential by its non-retarded, static form. This corresponds to taking a low-(u limit in ((>(q, to): where 2 £ £^,0) 15. 3. Zero Magnetic Field 3.1 The Matrix Elements For a free electron in a box of volume V the solution of the Dirac equation gives the following wave functions [27]: i(f. *-$*)/* %*&'- W**& o-o where p is the momentum of the particle, s = 1, 2 is the spin quantum number, 2 ? ? 4 h EP= (P c~ + m c ) , and for spin up, and sCt> s N for spin down, where N = {(E.+ E )/2E^} * and E = mc . The matrix element occurring in the dielectric constant defined by (2.15) is then 16. and the dielectric constant becomes t(t,«) - 1 + I sy ™ ~ t£+H +Etr where the spin is explicitly accounted for in the matrix elements which appear above. From (3.2a) and 3.2b) we easily obtain rK(r)«5'Cr»-**)r = J-— UAA'T+*t*'ttp+ r V"!] U-V 1 1 where p = p + tiq, A = (E + EQ)/c and (A = (E+E )/c) . As a quick check, we see that in the non-relativistic limit, where E dominates p and p', the right-hand side of (3.5) is just 2, thereby recovering the non-relativistic expression for the dielectvic constant, (2.16). 3.2 Results (i) The Dielectric Constant To calculate (3.4) we take the zero-temperature Fermi-Dirac dis- 17. tribution function for FCp) and write as usual for the thermodynamic limit V I &*f "I We have been able to perform all the integrations except for one remaining p-integral which we unfortunately find intractable. Our result for the dielectric constant is <'•") =1 + *fe/ where J= H^+^) d) K e*i L'{ - < [^ f^j 4 <«^{^j" 18. 1 £\Crr+*dfc + (h+*D *?* tr Ztyt #h n ft»)"-ftoM/r |*-ffr^Tf+C2 (36) li\io •+ -K }A i- UpmjWC*1iJ i 2 2 2 Z where a = fico + (p c + E^ J' , a = -Rtu - (p c + E}Y , and pn is the Fermi O Or 2 1/3 momentum, given by1i(37> p) , where p = N/V. (ii) Plasma frequency for arbitrary density To obtain the dispersion relations we need to solve e(q, u) = 0 for w as a function of q. For arbitrary values of the density we may expand the energy expressions in the integral in (3.6) as follows: [pV-+E* -. y^c* -f tejoi*lJl '4 . - (fvt£.l)*(i + 07) 20. This expression is valid for sufficiently small q - that is, long wave lengths . Expanding (3.6) in this way, integrating and solving e(q, ui) = 0 then gives **- £ffiS&(1 + >,«1 + -> fit) where a = e /1ic. The coefficient A involves a large number of terms, and we will only give it for the non-relativistic and ultra-relativistic limits. We note that the non-relativistic limit of (5.8) is just , the well-known plasma frequency. Equation it E b (3.8) was also obtained by Jancovici [pj (iii) Dispersion relation in the non-relativistic limit In this limit, pc/E « 1, and the leading term in J is then rb ku _ jLfcttf-yn) f*> A — JU- (3?) I ;«o ^ (<»'*%) J which, when integrated, gives the well-known non-relativistic expression for e(q, w) [l, 28]. The n?xt terms give the first relativistic correction to the non-relativistic result. Collecting all relevant terms, performing the 21. p-integrals and solving e(cj, u>) = 0 then gives + •*(*)"—J" -J (}•">) where 1 _ £•//; Co IT **£„ L L ^ as expected from (3.8). (iv) Conditions for damping of plasma oscillations The denominator of (3.4) to be precise should include an addi tional term in, where n is small, according to the standard Landau prescription. Using the Plemelj formula "twn. r(i)-'«%c*) then gives an expression for Im t(q, w), The appropriate 6-functions which will appear in the integral for Ira e are, from (3.4), 22. for the F(p) distribution function, and for the F(p + liq) distribution function. (For the F(p + Hq) distribution function we have made the substitution p* = p + 1lq and then replaced p' by p, so that all integrals range from 0 to p_.) The conditions on q and u which make Im e non-zero are thus obtained by solving inequalities such as [Cr-^cf < t« * O^)"^ [(K*^+<;*]* obtained from the first 6-function. The final results are as follows: let 4£ fc e Then A 1 • [" fcc) - \ko? J (a) If uj > qc, Im e = 0 and there is no damping. (b) Let u < qc. If (i) \\t t-UA\ PFc < t (}IU) then Im e = 0 (ii) if {\Kft-tt»A\ i PF i {(ttc+t*>&) then Ttoitt,*)- ztkclut) __ |^/-v>^«^-^j 23. + (C*-f-»V?)0*-A)] (S,lk) 2 2 2 2 where e = rp„ c + E . and Fc F o ' A = {(ty& ~ **>) • (iii) If ppc > ^(kfC "f too A) t then The regions of damping in the w-q plane are show- in. rig. 1. Region I is where Im E = 0. In Region II, Im £ is given by (3.12b), and in Region III, Im e is given by (3.12c). The well-known non-relativistic results Q, 28] for Im e can be readily obtained from the results (3.12). In particular, for example, u/q :• V- gives no damping. It is perhaps interesting to note further that the conditions on the modes given in (3.12) can be reduced to the non- 2 2 2 rclativistic ones only when (Htd) << ("ftqc) << 4E . The right-hand inequality is reasonable for a non-relativistic limit, and the left-hand inequality simply states that all phase velocities must be much smaller than c, a condition not required ultra-relativistically. Our general results (3.12) agree with those obtained by Jancovici [p3. There are also contributions to Im c from the electron-positron part of the dielectric constant, which we have not included. However, Jancovici's results (eq. Al' in the appendix of his paper), which do include the posi tron contribution, show that the contribution from positrons to Im e are non-zero only when flu > e„ and/or tiq > ?pp. Clearly in these regions it is 24. energetically possible for the electron-positron pairs to be important, and in particular for the positrons to aid in the danping of plasma oscilla tions. (v) The screened potential Consider a test charge Q immersed in the plasma. We are interested in finding the static electrostatic potential V(r) of this charge, and therefore we may use (2.24): l/ V(r) = C^J^£ «-" (?) t'-'O where w 4*Q I Since we want the value of V(r) at large distances, we need the asyraptotic expansion of the Fourier transform in (3.13). This can be done by the use of a theorem given by Lighthill [29] , which gives an expression for the asymptotic expansion of the Fourier transform of a function in terms of the singularities of that function. In this section we will obtain the screened potential in the non- relativistic limit, but including the first relativistic correction terns. Thus we take (3.6) with to = C and expand for p c « E and small q (since we require V(r) for large r). This gives tCt,<) - 1 + ^U'*> wh ere 25. tt+'fc LU.l)= &3l$-&(*-*M-*Wtt x M(h) &-I1) z E. wncrc z-x_! 1 JW- i- irO-T- )^ ITX j The tern containing the function g(x) is the well-known non-relativistic exprcssicr. [o0\. There is s. logarithmic singularity at liq = * 2pF in L(p_, q), so we may apply Lighthill's theorem to obtain V,„. 2i.iL it I "Mr. ifrfc\\ \ frp^«V) - \Lc- £)] t 26. 2 2 where kp = pp/ti, | = 2me /irfl kp, and C is Euler's constant. The leading 2 term in the coefficient of cos(2k r)/(kpr) is the same as that given by Fetter and Walecka [~3oQ, and the next term in this coefficient is the first relativistic correction term. The coefficient of the sin(2k r)/r terra r given above differs slightly from that given by the above authors, who obtain ^[^C^)-^i(c-i)] As yet we have no explanation of this discrepancy. The parameter ^ here is small as we are in the high-density regime. 3.3 The Zero-Mass Case (i) The dielectrrc constant and dispersion relations In the limit of very high density (that is, the gas is ultra- relativistic), we expect a natural small expansion parameter to be E /ppc It is thus reasonable to assume that the leading term in an ultra-relati vistic expansion will be given by taking E = 0 in the dielectric constant. For E = 0, the integrals in (3.6) can be done easily, and the result is o (i) For fiq > pp, K= U *&Kl+ '(¥)*-«*tf] (ii) For-Tiq < pp, , aA|||[^i(^;-i*i)ii o- Although e(q, w) is continuous at fiq = pp, where will be a discontinuity in some derivative at this point. This discontinuity probably has no physical meaning, since it arises purely from a mathematical procedure, that is, putting E = 0, which gives rise to terms such as |p„ - *fiq|. For a non-zero mass the modulus sign disappears and there will be no discon tinuity. There is also a singularity, of course, in the second derivative of c(g, 0) at 1iq = 2p~, arising from terms such as (2p~ - -ftq) ln|2pF -1iq|. This is a well-known singularity and is also seen in the non-relativistic case, although there it appears in the first derivative. It arises from the fact that in p-space, the two Fermi spheres (arising from the F(p + 1\q) and F(p) distribution functions at zero temperature) no longer overlap when their centres are separated by 2pp, that is, when flq = 2pp. To solve e(q, w) = 0 in the long-wavelength limit, we use (3.15b) and obtain 28. where here CJ1 - -41 (¥-; Equation (3.16) can also be obtained from the results given by Jancovici Q)], who included the effect of pair production and the trans verse interaction in his calculations. He noted that in the limits tiu « e_ and Tiq << p_ the contribution of the electron-positron pairs was unimpor tant. As can be seen from (3.16), the first condition is readily met, and the second condition is met if fiq << p /14. Furthermore, it is interesting to see that when we consider short-wavelength (high-q) modes in the high- density region and just put q = qp in (5.15), t'r.zz in fact "Ru < cp, but if we put q = 2r._ we get fiu > en. Of course (5.16) is valid only in the small-q i' r region, but it should be a precursor if what happens to-ftca when q = 2q . r At any rate our neglect of electron-positron pairs is justified for very long-wavelength ultra-relativistic plasma oscillations. (ii) Conditions for damping Taking E = 0 in (3.12) gives the regions of damping and Im e(q, w) in this limit. The results are as follows: u. I.e = Al . Then (a) If u > qc, Im e = 0. (b) Let a) < qc. Then (i) Forfiq < pp • £[y+-KW-±w" (J-/7*) 29. (ii) Let "tlq > p_. Then 3) if pf < i(ki- %) ,i.o. The regions of damping in the q-w plane are displayed in Fig. 2. In Region 1,1=0. In Region II, I is given by (3.17a) and (3.17b); in Region III, I is given by (3.17c). (iii) The screened potential As far as we are aware, the only calculation of the screened potential for a test charge in an ultra-degenerate relativistic electron gas is that given by Jancovici [Vj. However, in that calculation the Thomas-Fermi limit of the dielectric constant is taken, which gives e(q, 0) = (1 + K q ) , where K is the relativistic Debye screening length. This neglects the relativistic generalization of the Friedel oscillations, which are dominant for large distances. We give this treatment here for the ultra-relitivistic gas. The potential, as given by (3.15), can be rewritten as a two- 30. sided Fourier transform: -«0 G-10 '-JB where H(x) is the Keaviside function, and V(q) is given by **[, * fem] where, from (3.15), we have (3 ia) Ul,°) •- ^-rhf1 ' for "fiq < pp, and KM= H + £ W -'ftJ'J fij -K*-*ft£H •')*!§-'y i *t I ^' for "Rq > pF. Consider now the first integral in (5.18): the integrand is qV(-2Tiq)H(-2iTq) , which has a logarithmic singularity at q = -Pp/Ttfi, (The 31. singularity in the derivative due to the Heaviside function at q = 0 will be discussed later). Near q =-p„/Trft, we can obtain the following expan sion: tV(-*i)H(-»i) - fc •(" ifr) L" f%)0-^'"? i Thus, according to Lighthill's theorem Q>9J, the asymptotic expansion of the Fourier transform of the function given by (3.20) is just the sum of the Fourier transforms of its singular parts. Thus the leading term in V(r) due to the contribution at q =-pc/7rft will be given by the Fourier r transform of -('•?M»$ (apart from some factors which we ignore for the moment), which is JLirhf. , where kp = pF/fi. L Z 3 The second integral in (3.18) is treated in exactly the same way, and the 32. final result for V(r) is £*-*0 where C is Euler's constant. The second tens in (3.21) was obtained by taking the third term in the expansion indicated in (5.20) and taking its Fourier transform. Note that we have dropped the very small number 11a/36- from (3.20). We now need to consider the singularities in the derivatives at q = 0 due to the Heaviside functions in (3.18). *n fact it is easy to show that the contributions from each integral cancel each other by writing qH(-2irq) as We then note that the Fourier transform of axH(ax), where ax = ± 2irq, is, apart from some numerical factors, J_ | i (H.^ *. 2L \ ; the 6-function is zero since we are interested in large r, and the a2 /r2 terms will cancel because of the opposite signs of the Heaviside functions. 33. While it is not clear what effect the positron contribution has on the potential (3.21), if any, we do not expect that the general func tional form of the ultra-relativistic (retardation) potential, arising from the structure of the dielectric constant at "fiq = 2p , will be altered. 3.4 Ion Sound So far we have been considering a one-component plasma in the presence of a positive neutralizing background set of ions. We now con sider the allowed modes of oscillation involving both electrons and ions - ion sound. The non-relativistic Tesult for the classical plasma is well- known [j51j , as is also the result for ultra-degenerate, non-relativistic electrons and ions Q>2]. For the latter case a solution for u> from e(q> mass of the electron or ion. In general, however, for the ultra-degener ate case we will require qv << u << qvp , where vp is the Fermi velocity and is given by vp = 3sp/3pp, which equals pF/m in the non-relativistic case. Of course, e(q, u) - 1 is now given by the sum of the appropriate expressions for the electrons and ions. (i) Ultra-degenerate electrons and ions Firstly, we recover the non-relativistic result. For this we use the non-relativistic leading terra for e(q, u) , given by (3.9). For the electrons this is expanded under the con d:-.ion. j << D..c/m , the leading term being 4«C Eoe \P , and for the ions it is expanded under the condition u> » ppq/m. , giving a leading term — &— —\LL-d— i 34. The dielectric constant is then (}1X) •t«. '-' ir|_3 ^(Uj* Solving e(q, w) = 0 for small q then gives the solution = 1 co iirft m C* ') a well-known result [32j. Let the electrons now be ultra-relativistic and the ions still non-relativistic. We then have v Fe c for the electrons, and can use the zero-mass result for e(q, u) given by (3.15). Taking the case tlq < p.. (since we still look for long-wavelength oscillations) the leading term for the electrons is now , and the leading term for the ions is still that given above. The solution for e(q, u) = 0 then is ( ?fC «> = V CM) provi"• - f (•$ »i • (that is, q r.vast r»e very small). This relativistic frequency is smaller than the non-relativistic one, (3.23), j, l by a factor of (E /ppc) . 35. In general u will of course have an imaginary part, which will be a measure of the damping of the ion sound wave. Writing u = uR + i Im u, we use the formula given by Harris [32j to find Im u (assuming it is small): I,M W = (1X5) For the electrons, we use the zero-mass result for Im z given by (3.17a), and for the ions, Im e. is given by the non-reiativistic limit of (3.12): r & (*•») 0 ; ofcUntfise ^ 36. We can now deduce from (3.26) that the ion contribution to (3.25) will be zero using the facts that qc >> wR >> pFq/m. (the condition for ion sound), Pcc/E . « 1 (non-relativistic ions),and —r- ( — J » 1 (small-q condition). The leading term of Im e is then, from (3.17a), just , and so, using (3.25), we find &)1 V *•' (0M e&\ c-j where the imaginary part is small, indicating that the waves are weakly damped. (ii) Degenerate electrons and classical ions (a) Ultra-relativistic electrons and non-relativis tic ions We now consider that the electrons are ultra-relativistic and ultra-degenerate but that the ions are non-relativistic and classical. For the electrons we therefore require p„c >> E and pnc >> k„T , where n rF oe *T B e k is Boltzmann's constant, and for the ions we requir e rp„c << E . and BD ^ F oi 2 p_ /2m. << k„T-. We observe that for a classical electron-ion plasma we rF I B l r require T >> T. to get ion sound, whereas here we have much less restric tive conditions on T and T-, and for simplicity take T = T- = T. e I e l The appropriate conditions en i_ sr-- <^ Co « j{C Q-2S) 37. For the ions we use the classical non-relativistic expression for the dielectric constant, given, for example, by Ichimaru Q31J, and for the electrons we may still use the leading term for e(q, u) given by (3.15), Mfky Then (writing k T = T) 2 2 where u. = 47rne /m. (n = N/V). This gives the solution Jo < \ 'i CO (j-iy - i*f%[l*(> IF ' proviide d that -=r ( «ir- ) "> ^ JL . To obtain the imaginary part of u), we use Im e. for the ions as given for example by Ichimaru [31J: UfX Mi, ^ = (f) -,(T) — e«^- — (s-H and again for the electrons the result in (3.17a) Equations (3.25), (3.29) and (3.30) then give &p = ^ M - t | o-o^y e f -I + (*•*; 38. where Q — " I 1 + ( 1 -f ) J an<* UD *Sno w given by (3.29). Since p„c >> T and ppc « E -, we can show that the imaginary part of u is small. (b) Non-relativistic electrons and non-relativistic ions The electrons are now ultra-degenerate and the ions non-degenerate, 2 and they are both non-relativistic: that is, p^c << E « E -. x>„ /2 The non-relativistic leading tern; for the electron's dielectric constant is now, from (3.9), and so r fry *>L ^--i-^-^- *£-*£*••• -° which gives the solution CO'•¥%£["(.''ifcfl °"> To obtain the imaginary part of w, we use (5.50) for the ions, but for the electrons we now use (5.26) with E . replaced by E . To know which value of Im t from (3.26) to taV.e, wc need to know whether 39. p_c is greater or less than 'jhqc • E «/qc. Using the fact that ; (ppc) » 2E T and (5.35), we can show that the aaxinua value of inqc • E u/qc is 'iRqc • 0.022 p..c, and so, for sufficiently small q we take Jm 6 = 2*E * -£»/ ( Hc) 3 frOB (526)' "cthe n °°tain 6J - U) \ 1 - ' © 0if3 bC -r '*m where her ^ -- > r.a w_ is given by (5.53). .\-ain, since (PpC) >-• 2E T, we can show that the inagimry part of u is snail. 5.5 Sorce Astrophysica 1 Applications (i) Ion sound The density and temperature ranges of the interior of white dwarf stars arc in just the right regions to illustrate the varior.5 case> of ion sound we have just calculated. A rarsj-.- ;" :;-— l.-al n:;-.ber densities for a white dwarf is p = 10 ' - 10"" c:?.~"\ a;;ci ~. r-.rt .cui^.rly hi;"- central temperature is appro.ximatcly T - 10 K "7 "* ~ 7 1/"* (a) Let ,- = 10 ' c~"'\ RcTreT.bc-rin^ that pp - "h(3- p) , 40. we see that p c « E « E - and thus the electrons and ions are both rr oe 01 2 non-relativistic, and the ions are classical, but p /2m kDT = 4.2, so r e D the electrons are fairly degenerate. Thus (3.33) is the most applicable equation, and this gives u = 2.1 x 10"3 qc (b) Let p = 10 cm" . Then the electrons are relativistic (although not ultra-relativistic) and ultra-degenerate and the ions are non-relativistic and semi-classical. Thus (5.29) is a suitable estimate, and this gives u = 1.5 x 10"2 qc. (c) Let p = 10J cm--3. Then the electrons are ultra- relativistic, the ions are non-relativistic, and both are degenerate. Thus (3.27) is a suitable estimate, and this gives to = 4.7 x 10 qc. (ii) Plasraons 30 -3 For a typical white dwarf star density of 10 cm , p c = E , so we use the general plasma frequency given by (3.8). This gives u = 4.9 x 1019 Hz, or IW = 0.03 MeV. We note that it is easy to show from (3.8) that for arbitrary density, c + 2 an fiu << Ep, where ep = QpF ) E 1 , d. so the effect of excitation of positrons from their Fermi sea is negligible. However, collisionless pair production is possible flOj , when "nu > 2E : for a white dwarf thi- corresponds to a rather high density of p > 2.7 x 10 cm""5. 4. Non-Zero Magnetic Field 4.1 The Matrix Elements For an electron in a uniform magnetic field B in the z-direction, 41. the solution of the Dirac equation gives the following wavefunctions [33]: e ( where n = 0, 1, 2, ..., o = + 1, y = p /mu , w = eB/mc, L = V, the volume of the system, and (E+f.WnW H K 4 'p ' ? K^g] (**> (£*0 H 1 ^f f^E + E,)]- >i where 4.W - MA fe)S]-p(-^) , E = mc* , N b K {itr^" > and H is the Harmonic oscillator function. The single-particle energy is: % E*E --[E>4V + £tc6^-rt/)] The wavefunctions are normalized according to C(5,V>' = SV.' Vf*' VV ^r' To calculate e(q, to) we need the matrix element e ;l" I'* a I ' "-Mi/ '"'fc'.ft'/' Using the above expressions for the wavefunctions, we find 1= i^^ft^V.^ VC^'^«6'A. 43. l + C U^-r,,'-J] * 4 i Cik) 4 V'{ ^>A,n<-f--<^V<.' where / 2 // = ' A/ EH, B'E;T £ = [e*dC^ + 0]Vc , ^= [e^^-^^/)jyc and r oo ^ KAu) = The integral (4.5) has been evaluated by Walters and Harris [34] . However, while the method of evaluation was of course valid, the result suffered from some inconsistencies, so a correct evaluation is given in appendix B. The result for the matrix element is then e l K s <-v„pz/| ~\ '>t*,r*,*'' --i-Wh&^^uO^VWKr.* 44. V ^t>,|M^r|-^ft/|M.-..'|J j (*° and where p ' = p + 1iqv, P-' = P. + MT> X X A ^ *• *• H„. . ^.-'^r^^^^-^ 0-7-; if n > n', and *-* ^-y?;\ ^^--/r -r^--' - K'* fa)e w -U f ej; ^7*; 2 • '^f Lf n' > n, where r = mw,/ft, £ + l% - Q C * > and ,*.'-«. J^ { X ) *s t^!0 generalized Laguerre polynomial. The general expression for e(q, w) given by (2.15) now becomes 45. where «<„ (^*) -- A/y'f v[fe'-v/)KJ •"Vxf 1^,^,1 + -I* + 1 The factor mw /2irfiL in (4.8) comt- from evaluating the rum over p. in (2.IS): writing ]T.] ' "fa gives r.w .L^/2rfi, since the allowed range of p values in a box is J ~ -mCo^L f ~m.(^tL J 46. This along with the 1/V factor in (2.15) then gives the factor mu /2ifnL. While we have not made a detailed comparison, the result (4.8) should agree with the longitudinal part of the general expression obtained using a different formal approach given by Svetozarova and Tsytovich £l6]. 4.2 Dispersion Relations (i) Prelimir.aries In solving £(q,w) = 0 we are interested in the small-q behaviour of w(q). This is convenient since JM J is a function of q /20 = 2 (hq c) /2eficB, a natural small parameter, especially since we are interested in large values of the magnetic field. In fact |M ,j can be expressed as a power series in this parameter. Now in (4.8), the sum over o and o' will give four terms, a typical one being when o = o' = 1. We have dropped the in term in the denominator temporarily. The sum over p in (4.8) is changed to c.n integral in the usual way, and the limits are furnished by the distribution functions. For example, we have = i , 4 £*>,,,,' ± f. 2 2 2 2 Thus p ' ranges from - (e^ - EQ - 2eficBn')'7c to (r.,. - EQ - 2efftcBn')'Vc. Let us now assume that all the electrons are in the lowest Landau level, that 47. 2 1/3 is B > B , where B = *nc(2np ) /e, and p = N/V. (This is derived in appendix A). Apart from simplifying the calculations, this assumption is reasonable from a physical point of view. For example, if we take the 27 -3 surface density of a neutron star to be of the order of 10 cm , this yields B = 3.6 x 10 G; since it is generally agreed that the surface 12 field of a neutron star is of the order of 10 G, we have in this region B > B . If we took B << Bc, then the contribution to e(q, u) due to the relativistic de Haas - Van Alphen oscillations could be studied, and this would be a relativistic generalization of the work of Greene et al. \\A]. 2 "* Now when B > B , we have 2encB > e^ - E , by definition of B . c F c c Hence for the above distribution function, only n' = 0 in the sum over n' is allowed to contribute to the expression for e(q, w). Similarly, any sum which contains F(n, 1, p ) may only take n = 0, and any sum which contains F(n, -1, p ) or F(n', -1, p ') has no allowed values of n or n' when B > B , and therefore does not contribute to the expression. Thus the surviving terms take a = 1 and for these we may perform one of the n or n1 sums by taking only n or n' = 0 as appropriate. Thus the sums over n, n', o and o' in (4.8) reduce to the following terms: h ($•/!) I 48. We now assume q is small. In view of (4.6)- (4.9), to obtain the first q-dependent term in a small-q expansion of w(q), we need only keep those terms |f-l ,| for which |n' - n| < 2, that is, we keep terms up to order |(hqAc) /2eftcB| . Thus, for example, we have Jf'.0.') . »-»"[fei,' "•f,r,')h..H' tw-E0l , +£ .Li I - E. + E >,>>?: *t',i / !li /vV*[(M;*'Vi&')M,,.iJ + -h T = where 49. M„ = l-X + fX: M z= X -X* + --- M o.l '.• + Ms. = M„, = *** for X << 1, where X = (fiq c) /2eficB. Similar results are obtained for the other three terms in (4.11). We are now left with an integral over p7 or -> -> i. 1 p '. For the p7-integral the limits are ifc-" - H ") /c, and the p*inte gral will have the same limits when we replace -•_• bv p_ and p by p - "nq. After some -anipulation, the result for s(q, v; is l[t,u) = 1 + A/V" Erfr'*e'f,ft-1,)|(!'-X + fX1 + J *» -(&{!*?+ (E^«1(^+*t,)M,) l + WXAT(X -X1 +-)[{£rEr'•cV.tfc'fcijJ^rlr,?] fc« - fC+tf'O8,4 (C-Wfc**^*^^)*- + t« + (f.'+fcVJ*- - (£.'+ eY^ftlJ* *- 4W]1 j r.",,V. e + iA/y*(xV..)[{E,Er' + W*!.)f-> ^t'r/ 1 1 1 fc» - (C+flfc ) ' + (e. + cV^^/+ <«=?)•'• ft-tt / 1 2 This is an expansion in powers of X = CnqAc) /2e1icB. (ii) Dispersion relations for oscillations parallel to the field For this case we take q = 0, so that now |M ,| = 6 ,, and the dielectric constant simplifies to v Atr'-tf -A = 1 + 'h t/y-fe^+tb^h}- J -(i'f-d'1t ft«f (e.' + c'Cft^yt-foSt'tf)* )! Qf<3) *"+(t:t Now the range of |p | is 0 to o/c, where from appendix A,o = (2irfic) p/2eB, and we note that for sufficiently high density o will eventually exceed E . Thus in general we expand terms such as [E 2 +. C2 (, p + "fiq )>hh ] as follows |»1 (C+^f (i,-m) This is a valid expansion when q_ is small, since the factor is at most of order 1. Expanding all appropriate terms in this way, inte grating and solving c(q, w) = 0 for to, we obtain the result t'B 1 + f (0(*2')' + • • Crv) 53. where f(o) is a fairly complicated function containing about twenty teras. To siaplify the results, we aay take the following asyaptotic and physi cally useful liaits: (a) For densities and Magnetic fields such that o « E . we have MV CO ~ O, 1 - <£(»£'••)-•] where w is the non-relativistic plasma f; jiified by B-dcpendent P terrcs: . 1 £ *• * u fi-'V t '- This result for the plasaa frequency is not surprising, since the electrons are free to stove in the z-direction, and a « E is essentially a non- relativistic approximation in the density. The presence of negative dis- persion, that is, the q "-dependent tern in (4.16) being negative, is unusual, and is perhaps due to the confinement of the electrons in the lowest Landau level. (b) For densities and magnetic fields such that o >> E , we have «*= where now 3 lot B (*•") iwt HfJb• • • • ] (iii) Dispersion relations for oscillations perpendicular to the field For this case we take q = 0 and obtain z r r !l €(*,") = 1 + •X)K ^k^r'^*(W\ Ute h (i -J-r/c f b.-(E.l*p)* + @*liPudcs)* L f ! -f r = J_ XHlN'l[(E B ' c^)\(c' ; )\ X T T + F h *" - (e<,1+^)'/'' + (E.l+fi.V+«efe«j,i cr=(T '«! +• l 15 4 :» +(s. +lf0 - te'+ft-V* *'-*<*)' n-o, n *L The regions of interest here are large and small values of o = (2irhc) p/2eB, and large and small values of B (where by small we mean B > B ), keeping in mind that the parameters a and B are not independent. A. Low "Density" For this case we assume that the density p is sufficiently small so that o « E when B i B . However, we will not assume anything about 2 the relative sizes of E and 2e1icB. Hence the natural large quantity will 2 2 be S = E + 2eTicB, and the energy terms are expanded as follows: The energy denominators in (4.20), after expansion, will contain terms such as and 0 + '* UEQ IS J 56. I -I fc«-(S-E.) + **(£-£) The first type of term gives a straightforward integral; however, in the second type above we must consider the two cases tiu > S-E and HIM < S-E , since this o o will affect the functional form of the integrals. The results are as follows: (a) Let -hid > S-F. . Then v ' o J «(*>") = 1 + i4 (l-x)J * xX + where S+E, + ^g r + _j 1 J- + J = CSaF I *" + S-E. J cS(*«*-o 0 + (f'0 + c5 J ^(T)[A(S-I^ 5 where 57. A*= ^-fcc-fS-f.*SE. ) , and J' has the same form S-E, as J except that 2eticB is replaced by 4eficB wherever it occurs, Let Hw < S-E . Then (b) o •3(*-0 5 + E„ £T -f 1 + H- 2 **-£-£, J = cS(WS-0 6S E„ /\ + r SE< ft-22) S + E £ + 2S + t XS s A- We now solve £(q, to) = 0 by making the following two assumptions: (i) "fito > S-E and (ii) a « A. The solution will then be checked for self-consistency. The result for the leading term in u(q) is ^ Hence assumption (i) is certainly satisfied. Assumption (ii) yields the condition , so now the condi tions on o are \ 58. f«M(H?)J Note that the assumption Ika < S-E yields an inconsistent solution. 2 To obtain the term proportional to q we write (W) (fc*-(5-tf+£^*'X + and solve for y. (It is sometimes necessary to assume that X is sufficiently small so that yX is negligible compared to terms independent of X, to maintain the spirit of a small q expansion). The general result for y is 1 _e_ idcBt (V*0 /- - 1 ^ ' irkt 5 J 2 U where S = (E + 4eTicB)2. Taking the high and low-field limits respec tively, wo have ,- 2 (a) When 2eficB >> E , we have \ j O *£„ tu 2 ae cg •+ ... X + & Q (few/* ( * ) 1- (lekcey rkc (iekcfi)v> (b) When 2eficB << E (this condition can still be consistent with B > B provided that the density is sufficiently low), we have 59. ^ (o^^l ± + 27 -t L*' J where u, = eB/mc is the familiar non-relativistic Larmor frequency, and 2 1-2 u = (4ire p/m) ,the non-relativistic plasma frequency. Ke note that for very low density, ID, » u and the coefficient of X will be negative; there will also be an intermediate density region (which still satisfies a << E ) for which u > w, and where now the coefficient of X will be positive. B. High "density" Ke now consider the case n >> E . This limit excludes the possi- 2 bility of taking the limit 2encB « E as we did before. This can be 2 1 3 2 seen as follows: recalling that Br = 7rhc(2irp ) ' /e and a = (2irnc) p/2eB, 2 2 we have o = (2eficBc) (Bc/B) . Hence a >> E0 implies (2eticBc) » E0(B/BC), 2 ! but 2eficB « E implies (2o!i^Bc) - << "-0C3c/3j * ^-- tho^e two conditio are contradictory since B > B . Thus we take a high-B limit by assuming (o1, + E ) 2 « (2e?icB) ". The energy expressions are expanded as follows: 60. (ldS,^^f^(^cBf(l, J£l 2cJ>tE> and the results for the dielectric constant are: 2Tr/rc L> J where I has the following structure (I' is the same as I except that 2e1\cB is replaced by 4eficB wherever it occurs; . Let (in?- ~ itkcB £ = E^/Cletice)^ K - (a) If |K| > a, then Z ~- JM - 9 . 2 2 2 where V = (£p-E0)/;eF + EQ) and ep = o + EQ . (b) If |K| < a, then «.-*• K€p/£( T = - ar^ (-£ ) - .a/r*( * fr*5) ft - fc /- ,' r=~ / We note here that if K < 0, then |K| > Ep/(2e?icS) 2 or \<\ < a ensures that c(q, u) has the simple form given by (4,23) and (4,29) or (4.30). 61. If a < |K| < Ep/(2e1\cB),h2 the integral for e(q, u>) needs careful treatment because of poles in the integrand. However, the solutions for u we are interested in, that is w = (2ecB/ti) , the relativistic Larraor frequency, satisfy the above conditions on K. The results we have given in (4.29) and (4.30) are in fact valid for any value of o/E , provided that o « (2encB)2, and so we can recover our result (4.26) by assuming K < 0 and writing | Interestingly, the limit a >> F. divides itself further into two regions which have different dispersion relations. Zirnc/e- _. . for 1 << a/E << e UJ o IT2- 2B0 ~Z7T*"C (W =. Meg 1 - — -&» L (i^cBy- (itkc$)*-_ e* n-3 i,^kc Q-») tuMy X + r--\ r ic- _ 2irnc/e . (n) For o/E >> e , we have 7 xr -mkc/t - CW = **'« [l + j£^ t irrtc/c% + fa-**) £UktB)\ 62. Again we note that different density regions yield positive or negative dispersion. Ke also note that the number which often crops up 2irnc/e^ 374 in this region is e =10 , a "very large" and equally, "very unphysical" number indeed! 4.3 Conditions for damping of Plasma Oscillations (a) Parallel to the field: q = 0 Referring to the expression for e(q, u) given by (4.13) and reinserting the in term in the energy denominators yields, via the Pleirelj formula, the two 6-functions for In F. (q, u) . These are where I [to + (e.S ftV)* - ( E0S eth+Hty)*] ' p ranges from -a/c to o/c. The first 6-function takes care of the case q < 0, and the second q > 0. The results are as expected: if w > |q,lc> then Im £(q, u) = 0 and there is no damping. If ID < |q |c, damping exists if /. c i \ '/. 4- f, ^ \K These results are the one-dimensional analogue of the results obtained for the zero-field case. (bj Perpendicular to the Fiolc: q, = 0 Here we need not restrict ourselves to the expansion of c(q, u) in powers of q given by (4.12), but instead we can use the expressions 63. given in (4.11) to obtain the appropriate 5-functions for Im e(q, oi). From (4.11), when q = 0, these take the form &[*w - O^+f/)*- + (f»Cl + f/- - ze^ogn.)'4] , and S[fcw + (gclt E/)V- - (frV* £„* + 2Bkc6*) *] , where n = 0, 1, 2 ... or n = 1, 2, as appropriate for each of the four terms in (4.11). Clearly the argument of the first 6-function has no zero for any non-negative value of n and positive hu, so we need only consider solutions for p of 1 z This has a solution for p provided that Ze£c5* - (kv)z > o (r'Q Then V/x %1X 6C = + E: and p c is real if z ZttcK - (fc«f > zfcw£< (frW) The above condition incorporates (4.54). I.inaliy, me condition r»c < tr = (£,*-£.*) gives the condition Mc^ - ^af < rt» h ft-") 64. Therefore, from (4.35) and (4.36), we can say that damping exists if there exists some value of n = 1, 2, ... which satisfies (c) Regions of Damping For oscillations parallel to the field, q = 0, it is clear from the results given in (4.33) that the modes given by (4.16) and (4.18) are undamped for small |q_|, since they satisfy u > |q_|c. For oscillations perpendicular to the field, q, = 0, we use (4.37) to test whether the modes given by (4.26), (4.27), (4.51) and (4.32) are damped. To do this it is convenient to write B = f3B , where 8 > 1. Con sider the mode given by (4.26). For this mode to exist we must have 2eficB » E 2 and — <<. | ^ -S~ ( " ' *-- ^ { . Since Q [* •i &(&)] 2 J' 2 S = (E + 2chcB)2, we must therefore have o « aE /n,where a = e /He. Using o = (2ifhc) «/2eB and B = Trftc(2Trp ) /e, the conditions for the mode to exist reduce to These conditions can be satisfied for sufficiently large values of 8. For very long wavelength modes, we use from (4.26) and insert this into (4.57). We find that if 65 fcrffef**/l < &)"$Y) (tr») is satisfied, then there is no value of n for which (4.37) can be satisfied, and so the mode is undamped. Clearly the conditions (4.38) and (4.39) are compatible. 2 For the mode given by (4.27), we must have 2eficB << E and , which reduces to All these conditions can be summarized bv the condition (trftt^ « •$\ (9-4°) L f Then, if the mode is to be undamped, we must satisfy (2TT)l3tc^ > it* + jJs^Wf + ;fefV;V/ + &-w ^ where the next terms are small compared to the first three terms. Again, it is clear that (4.40) and (4.41) are compatible. For the modes given by (4.51) a;;d (4.52), it is easy to show th.it for small q they satisfy (4.57) for n = 1, and hence are always dar.ped. The strength of the damping, where it exists, can of course be determined in a straightforward but tedious fashion by calculating Im e(q, w) and then using (5.25) to find Im u. We do not give this here, 66. but have merely indicated the regions where damping is to be expected. 5. Discussion and Conclusion Using a generalized longitudinal dielectric response function obtained via the self-consistent field approximation, we have studied the ultra-degenerate relativistic electron gas in zero and non-zero magnetic fields. For the zero-field case we have obtained some well-known non- relativistic results along with relativistic correction terms. Correspond ing results in the ultra-high density (ultra-relativistic) limit have also been obtained. Where applicable, our results have been compared with those of Jancovici [9] . For the non-zero magnetic field CJ.se, we have obtained the disper sion relations for plasma oscillations parallel and perpendicular to the magnetic field. The plasma frequencies obtained were found to be either density-dominated or field-dominated according to the appropriate region. An unusual feature of these results was the appearance of negative disper sion for parallel and perpendicular propagation for almost all regions of density and magnetic field. This negative dispersion (indicating a local minimum in w(q) ) is we believe a manifestation of a combination of rela tivistic effects along with the fact that all electrons are restricted to the lowest Landau level, that is, B > B . Another interesting/amusing ? fi / 2 feature was the appearance of thv "ult ra-i,irje" r.'...:ber e~ TT c e as ;•. natural division of the very high density, high-field region. The disper sion relations differed according to whether c/EQ was larger or smaller than this number. Although it is easy to see how this number arises mathemati cally, the physical reasons behind it are certainly not clear. In any case, 67. the results in this exotic regime should only be taken as indicative, as here multi-quantua-electrodynamical (QED) effects are important and no doubt dominant. It is perhaps worthy of note that this same "mammoth" number appears in attempts to sua the perturbation series in quantum electro dynamics [3SJ . The formalism we have used allows for the natural inclusion of the presence of electron-positron pairs by directly including the positron states in (2.5a). However, as we h:ive discussed in sections 5.2 and 5.3, the dispersion relations and icaginary pari of c(q, -} for the zero-field case are unaffected by electron-positron pairs in th-_- icw-q and flu < cf region. Ke expect the dispersion relations for non-nero isagnctic field to be similarly unaffected by the presence of pairs in this sisular region. 12 To be specific, if wc tike a aagnctic field value of 10 G, characteristic of the surface field of a pulsar, the maxima number density for which all electrons are in the lowest Landau level is approximately 10 ' cm" , which is again characteristic of the electron number density at the surface [2\\. Thus for these values, wc have B > B , and it is easy to show from oir results in sections 4.2 (ii) and (iii) that the appropriate plasma frequencies, both for parallel and perpendicular propagation, are less than E /ft, which is itself always less than Cp/Tl, as can be seen fron (A6). Therefore we now use the above values, B = 10 ~6, p = 10 en""*, to illustrate the difference h^fven - p and B, we find a << F.Q, and inserting a surface temperature of 10 K, 68. d = 10. Thus for oscillations parallel to the field, (4.17) gives 18 a) = 1.7 x 10 Hz, or ftw =1.2 keV. For oscillations perpendicular to the field, (4.23) gives u> = 2.1 x 1019 Hz, or fta> = 13.1 keV. However, for very large values of B it is in principle possible for hu to be greater than Gp. We first consider our results for perpendicular 2 propagation. When o « E and 2eftcB >> £ , we see from (4.26) that s 2 2 ftw = (2e?RcB)' , and from (A6) that ep = E (1 + a /2EQ + ...), and sofiu > ep. When a << E and 2eftcB « E , we see from (4.27) that ftu> = EC"".2 + C^u )2]*> and it can be shown that ftu is always less than e_. When c >> E . (4.31) r o v ~> 2 2 and (4.32) were obtained subject to the conditioii 2eftcB >> a" + E = e_ , 2 3 and so here also ftw > £p. Thus whenever ft^ "> c^, we must have B » m c /eft. Turning to parallel propagation, when a << E it can be shown from (4.16) 2 3 that for ftuj to exceed ep (> E here), B must be greater than B = n c /eft = 13 4.4 x 10 G: this value is obtained by equating the electron's magnetic ^ 2 energy (eftcB)* to its rest-mass energy mc , as mentioned in the introduction. Again, when a >> E ftu is given by (4.19), and for ftw to be greater than Ep (= a now) , it can be shown that B must be greater than B . Thus we see that for all frequencies given in section 4, when ftu exceeds e then B must be greater than B . Thus for these regions we should, because ftu > ep, first of all include the effect of electron-positron pairs directly in the calculations, which we are currently doing. Furthermore, the associated condition B > B heralds the onset of numerous non-linear quantum-elect rodyna;:iical effects. A d-_>t:t: lii z::.:^il~l^r. :•: many of these types of processes is presented in the comprehensive review by Erber j36j. It gees without saying that the non-linear QED processes that occur for the vacuum in the presence of a very intense (B » B ) field arc extraordinarily difficult to handle. A particularly relevant set of 69. examples is: (i) for a relativistic electron gas in a strong magnetic field, the radiative shift in the energy of an electron at rest in an arbitrary magnetic field [37J (ii) vacuum polarization in an intense magnetic field [l7, 38j and (iii) photon splitting in an intense magnetic field [39] . Now, when concerned with how these effects change the behaviour of a highly degenerate electron gas in such an intense field, we face a truly Herculean task. In principle the method in this paper can be generalized to include such processes by adding to the Har.iItalian additional inter action terms of the form where jv is the appropriate current and A the appropriate vector potential for the appropriate process. We do not mean to be glib here; obviously these harmless-looking additions are only the tip of an iceberg of a multitude of difficulties. This procedure would in principle ther allow for a treatment of the external field and non-linear medium polarization in a unified manner. Essentially, in this region of magnetic field we are dealing with the highly non-linear response of a dense electron-posi tron-photon plasma. In this context we would like to mention the work of Steinert [45*J . A different direction which we also find appealing is the perturbative Green's function treatment utilized by Dubois and Bezzerides and Dubois [Vl] , and similarly by Tsytovvrh ['nj . In particular, we note that the Coulou-ib interaction should be generalized to include the transverse (radiative) field effects, even when B = 0, which would be important for the very high density regime such as found in the interior of very dense white dwarfs. In this case, in (5.1) 70. j would be the single-particle current and A the transverse part of the radiation field. The calculation would then proceed along standard QED lines as outlined for example by Bethe and Salpeter [4lJ. As we have noted before, the transverse field effect has been included in the calculations by Jancovici \p]. However, it contributes only to the transverse part of the dielectric tensor, and since we have only dealt with the longitudinal dielectric response function, we have not included it here. The simple relativistic generalization of the Hartree method for an ultra-degenerate electron gas in a strong magnetic field yields results which, as we have noted, are applicable to the interior of white dwarf stars and to the surface of pulsars. An additional useful application, in this context, of the dielectric constant would be to calculate the energy loss due to collective oscillations in dense relativistic plasmas in order to assess energy dissipation processes involving plasmons. As noted in section 4, the formalism can also be used to study the relativistic de Haas Van Alphen oscillatory contributions to the longitudinal dielectric constant for relatively weak B (B << B ) and relativistic densities. Similarly, the dielectric tensor could be readily obtained by generalizing the calculations given herein, with the inclusion of the transverse field effects but in a relatively weak-field regime where the effects of pair production and other QED modifications are not important, to obtain the ground-state energy of a relativistically dense electron gas in a relatively weak mag netic field. This would generalize the work of Gla^ser i_42 for the non- relativistic gas in a weak magnetic field and that of Jancovici [9j for the relativistic gas in zero magnetic field. In conclusion, it is pleasing to observe that the results of the theory presented here for a relativistic quantum plasma are applicable to 71. the density and magnetic field values found in such fascinating astrophysi- cal objects such as white dwarfs and pulsars. The results also give an indication of changes in the modes that begin to occur in a linear theory when B > B . Of course when B » B one cannot overlook the numerous non- o o linear processes which occur in a fully magnetized, positron-electron- photon system. These enormously complicated effects begin to take us far away from the relatively simple relativistic results presented here. Acknowledgement. We thank G.R. Fox for his valuable collaboration in the early stages of this work. One of us (N.E.F.) wishes to thank D. ter Haar for his kind hospitality and generous financial a^sistar.oe of an S.R.C. grant which helped make this work possible. One of us (A.E.D.) acknowledges the financial support of a C.S.I.R.O. studentship. Appendix A The statistical mechanics of a relativistic electron gas in a magnetic field has been treated by Visvanathan [45], more recently in a series of papers by Canuto and Chiu [if] , and by Buot [jl4j , who also included the electron's anomalous magnetic moment in his calculation of the susceptibility. We give here an expedient derivation of the Fermi energy for the ultra-degenerate case. For a gas in a box of side length L, the number of particles is given by 72. mcj L c ]T H*>h>r) «>h,c where OJ = eB/mc and F is the Fermi-Dirac distribution function. Taking the sum over p to an integral in the usual way, and taking a - +1 to represent particles with spin up and a = -1 particles with spin down, we have (M<) A/_ IHW, d (hi I.) V 6'*>\ k Y. ?(•«'?*,-') n. At zero temperature, = 1 , If E.. < t, ft r Consider N+/V: the integral over p , from (4.3), is just 2 2 t* 2(ep - E - 2eficBn)Vc. Clearly the sum over n cannot now be unrestric- 2 ~> ted since we require that cp - E " - 2e?icBn be non-negative. Hence n ranges from 0 to n where n is given by -i // -i1/, '/t (A2) z [E + XetcBn, < €f < 73. and so 7 \'/ 2m ue (A3) V '•) c**y in=o A similar result for N/V then gives the equation for the Fermi energy: A/, fL V V V f 2w«, i tf-£.f-uZ(t,-s:-!***)'• CW K-l This result is of course exactly equivalent to that obtained by Canuto and Chiu [li] , provided that we note that the Fermi energy used by them, say e is related to ours by Ep = ep + EQ. Since u is proportional to B, it is clear from (A4) that as B increases, z~ must tend to a constant. In other words, for sufficiently large B, the condition (A2) can only be satisfied for n = 0. Thus, for B > B , we have **"* • * rM'4 (A*) e = (ivt) H*-*y or -12 X XlirUff l (A0 ^ = E„ f llh 74. Note that for n = 0, N = N and N = 0, so that B = B is the value of o + ' c the field for which all the spins point up. An expression for B may be obtained by putting n = 1 in (A2) and making the left-hand inequality an equality. Then (A 7) (**& As E decreases from B more Landau levels are excited, and it can be shown from (A4) that eF(B) behaves as displayed in Fig. 3. Appendix B The integral we want to evaluate is Mv,U,0 - (BO j -oo where M(J \ ^ wv. C ex(- H (**; kb) - VA f IK and the various other symbols are defined in 4.1. The evaluation of (Bl) which wc give below follows almost identically that given in the paper by Walters and Harris [34], with corrections where appropriate. 2 Let 3 = imjcAh, a = yQ = •fikx/muc, b * 1iqx/m^c = yQ' - yQ, and x = y-a. Then 75. , eO M = A/.A/, h <<* t.^)^[K*-o] J -eO **r(-if*1 - i*x('-1? -%('**)) (S3) We use the generating function ex ff-s'tzs*) - Y_r *.ft>' n^o as follows: multiply M by and sum over n and m. Then *> /i'v'1 i"'1 M, s = II V n.«v» n^o m = o (n'.^M'^ CO c^ »o f ft, WkL #<-*>] -irr rr v/ir ^— /•— n-o ^-o h! "i( ~<0 76. exp^-ijV - if'L*-Oz-^[xf)} .<*> exp[-sS2fsx-kx-*itrlz-i)-iflxl-J f fr-*)1-*,MJ rf* ft J by use of the generating function. Completing the square in the exponen tial above and integrating, we have S = txtU-Xfk-ift-^ + fal-ij'f+OnXf1-* o° -t[Iv l using •k The t and s- P dcpcnd.jiit parts can be written a.- fTll^wi: xts - j&ft + ft+OO'* - HE) = 77. -«(«-^*^ •'{'*--ft Hence we can rewrite S by expanding the exponentials involving t and s: r : 4/V 0 where «-(?-jr-c?'v) Wc now equate coefficients of snt in (B4) with those of (B5). If n > m, we rewrite the triple power series in (B5) as follows: let r + p = n, r + i. = m, and let the new summation variables be n, m and I (where I < m, n > m-£). Then (nO^K; h~o M~Q 78. oO /*l oO ^ = e + and so nfi nt/i. M. (A.')W"-' '"'^ £* >»*»,) where c: is the generalized Lagtierre polynomial, defined by fc!(r»»-A)!(n-« + fc)! A*. c?£ Writini g £ e * - ^ + ^ , we have finally * *j 79. 01 > i*V If m > n, we still write r + p = n, r + i = m, but now make the new summation variables n, m and p (where n > p, i?. > n-p). 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Steinert, Nuovo Cimento 36_ (1965) 935; Nuovo Cimento B 40_ (1965) 345. 85. Figure Captions Fig. 1. Regions of damping in the w-q plane for the non-zero mass case. Regions II and III have non-zero damping. Fig. 2. Regions of damping in the w-q plane for the zero-mass case. Regions II and III have non-zero damping. Fig. 3. Behaviour of e„ as a ^unction cf the magnetic field. n "fcu I t u 2? hq, '0