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Black-Body Radiation in Plasmas

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Black-Body Radiation in Plasmas — 155 — UA0000793 BLACK-BODY RADIATION IN PLASMAS Dirk K. CALLEBAUT1 and Ahmed H. KHATER12 1) Phys. Deft., VIA, Univ. of Antwerp. B-S610 Antwerp. Belgium S) Math. Dept., Faculty of Science, Univ. of Cairo. Beni-Suef. Egypt Key words: Black-body radiation, Planck's law, Stefan-Boltzmann law, stellar structure, plasmas. Abstract, In previous papers it was shown that the black-body radiation in a plasma is altered due to the cut-off frequency which depends on the plasma density. Some results are reviewed and commented. It is made clear that the 'mass energy' of the photon in the plasma is indeed the chemical potential. Some physical applications are briefly discussed. 1. Introduction In previous papers [1,2) (ref. 2 will be refered to as paper I), it was shown that Planck's radiation law is changed in a plasma because there occurs a cut-off frequency in a plasma. The change is the bigger the larger the deusity is. In fact the dispersion relation for transver- sal electromagnetic waves (=photons) reads o* = c»fc*+wg (1) where u> is the angular frequency, c the speed of light in vacuum (more generally one might use (^ , i. e. the speed of light in the medium), k the wavenumber and up the plasma 2 l/1 frequency. wp = (e. n/tme) for an electron-ion plasma (neglecting the ionic part) and wp is 21/2 times the previous value for an electron-positron plasma; e, m, and n are the electron charge, mass and density respectively and e the permittivity (SI units). It follows that the energy «7 = fko of a photon may be written as l 3 ty = c(p> + mi<?) > (2) 2 with p = hk the momentum of the photon, rrtjC = hwp the rest energy of the photon (with my its 'mass') in the plasma (and due to the plasma) and ft = h/2ir Planck's constant. The thermodynatnicai potential for the k-th state of a system of bosons is in general {3} 0* =-Tfn(l - «(*"*)/r) (3) with T the temperature in energetic units, /i the chemical potential and «* the energy of a particle in the k-th state (in which there are n> particles). The mean occupation numbers are given by - 156 - 2 Now we take the expression (2) for the energy e* and for /* the mass energy ra7c . Thus the total kinetic energy is given by V r°° (e, - m Eki = ^?J 3T where a = my(?/T = hup/T and where Kn is the Besse! function of the second kind (imagi- nary argument) and of order n. For most of the details, approximations and thermodynam- ical considerations we refer to paper I. For a-tOwc recover the law of Stefan-Boltzmann Ekin(Q) = VaT* (7) where a — ia/c = 7r2/15(fK:)3, with a the constant of Stefan. However in a plasma, a is different froth zero and a is replaced by a function of density and temperature. It may be recalled here that u>p contains the mass of the electron and that this depends on the temperature too at relativistic energies, although we shall not elaborate here that situation. 2. Some comments 2.1. The mass energy as the chemical potential In paper I we called eT — m^<? the kinetic energy but refrained from calling m^<? the chemical potential. This point was questioned by Naudts, Zagorodny and Gorenstcin [4]. In fact Landau and Lifshitz [3] argue that the chemical potential for photons is zero. The latter argue that the number of photons N is a variable quantity and not a given constant as for a usual gas. However in view of the distribution (4) the total average photon number is fixed by the expression • y roo p2dp JT^F/O £(t-,-nv,e»)/r _"J '8' Of course the number of photons fluctuates around this value, but it is clear that N has not to be determined by an independent condition. With the present extension for the photon gas in a plasma it seems plausible to consider m^c2 as the chemical potential, which reduces to the classical case fi = 0 in vacuum.We shall make a verification of the consistency of this below. 2,%. The chemical potential may now be positive In Landau and Lifshitz [3], it is argued that /i for bosons may not be positive because otherwise the geometric series which results in eq. (3) does not converge for all e.^ (including tjt =0). However e-, — m^c2 > 0 and the restriction fi < 0 is unnecessary in the present circumstances. 2.3. The total photon energy increases with a In paper I it was clear, analytically and from the graphs (drawn for some values of a), that the kinetic energy density (and a fortiori the total energy density) of the 'plasma-body' - 15? - radiation exceeds the one given by the law of Stefan-Boltzmann; the larger a, the larger this excess, which, when a is larger than unity (cf. White Dwarfs, see below) becomes much, much larger than the value in vacuum. One may wonder why the energy density does not decrease since there is now a cut-off, wp, in the frequency region. However, now the lowest energy levels (corresponding to the u> just above aip) dispose of a much larger volume in the phase space: not the vanishing elementary volume u*ik) around w = 0, but the one around wp. This explains the steep rising of the integrand above wp, while for a — 0 it has zero slope. A study of the asymptotic behaviour for a S> 1 confirms indeed that the photon energy content is much larger than the one for a = 0. 3. Analytic expression and series development for the relevant quantities In order to get a better insight, to verify some relations and for practical use we give here the analytic expressions and the series approximations for small a for several relevant quantities. S.I. The total photon energy Et F -- Introducing x = pc/T yields _ VT1 /•« (x* + a^xHx ' ~ n*(hc)3 h e&W"-" - 1 ( ' Putting x = «sinh0 yields ._ V{TaY i-°°sinh2ecosh'2e<0 l~Tr2(hcYk eot""'1"-1) 1 [ ' ^^ j d$ (12) Now coo jg e-'^'coshWde^Kiz) (13) With sinh20cosh20 = (cosh 40 - l)/8, we obtain Around a = 0 we obtain, from eq. {10), as well as from eq.(I4), the following Taylor expansion: 2 Et = VaT*(l + ^C(3)« + ~~a ) +... (15) E, = VaT*(l + 1.110626a + 0.633257a2) + ... (16) where C is the Zeta function of Riemann with C(3) = 1.2020569... Note that it has not much sense to go to higher order in a as the third derivative of Et yields - 158 - which diverges. 3.2. The kinetic energy Ekin Similarly we obtain for small a, either from eq.(5) or from eq.(6): ^ ^a1-~a3) + ... (17) Ekm = VaT4{l + 0.740418a + 0.126651a'1 - 0.025665a3) + ... (18) Here the Taylor expansion allows the term in a3. S.S. The '771(133 energy' Em te J Em = ^Jfe'« fV «»"sinh <?coshf?<0 (20) - K, ((a)) (21) 1=1 For small a: 4 Em = VoT a(0.370209 + 0.506606a) + ... (23) Clearly Em - 0 for a = 0. 8.4. The total photon number N N = Em/aT (24) In vacuum the number density is about 1.48 10l6/nis at 300 K and 5.5 1028/m3 at 107 K. 5.5. The free energy F We use expression (16) from paper I, where & factor »a is forgotten in the denominator: 3)rs(ftc)3 For small a: • (27) - 159 - F = -^^-(1 + l.U0626a + 0.038497a8) 4 ... (28) o We may rewrite the first two terms as F = F(0) - m,(?N{0) (29) with obvious notations. We intended to interpret this formula as a proof that my(? is the chemical potential and that the 'mass or chemical energy' nN(0) just adds to F(0); however 'just adding' does apparently not correspond to the full change for the free energy: probably ft influences F intrinsically too and there results a minus sign in eq.(29) S.6. The entropy S S = -(drF)v = ^~X:|-(Ml-'«/4)/f3('«)+'Vir1(/«)/4] (30) For small a: J « = + 0.832970a + 0.189977a ) +... (32) .3 3.7. Verification We have, uaing the expressions given above: F + TS + Em = B-gL jg ~(feK3(te) - K3(J«)) (33) Using the recurrence formulae for the Bcssel functions we obtain: E, = F + TS+Em (34) and this confirms the view that /i = m^c2 is indeed the chemical potential. Clearly the series developments satisfy eq. (34) too, which is a verification of their correctness. 5.8. The pressure P P = -(avF)T=g^j£^|(2 + ta)/f1(te)+(2-ta)/^(««)) (35) Tor small a: P = ^(t + ^<(3W + ... (36) /' = -5-(l + 0.555313a) +... (37) 3 4. Some illustrations 4.1. Sun - 160 - M a 6 17 1 In the center of the Sun n = 10 nr and T = 15 10 /C; thus wp = 5.5 lO ^ (more than 2 orders of magnitude higher than visible light!) and a — 0.25. Accordingly the total radiative energy density is about 32% larger than in vacuum. (In ref. 1 and 2 we used only the kinetic energy am! a less good approximation.) The increase in radiative pressure is then about 14% with respect to the vacuum case. If we realize that the radiative pressure is only 0.3% of the gas pressure in the Sun, then the supplement seems very tiny indeed, at least in this case. However several remarks have to be made here, a) The radiative pressure plays an essential role in all stars, much larger than one may expect from its contribution to the total pressure.
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