Casimir forces in a Plasma: Possible Connections to Yukawa Potentials
Barry W. Ninham, Mathias Boström, Clas Persson, Iver Brevik, Stefan Y. Buhmann, Bo E. Sernelius
aa r X i v : . [ qu a n t - ph ] S e p Casimir forces in a Plasma: Possible Connections to Yukawa Potentials
Barry W. Ninham, ∗ Mathias Boström,
2, 3, † Clas Persson,
4, 5, 2
Iver Brevik, ‡ Stefan Y. Buhmann, and Bo E. Sernelius § Department of Applied Mathematics, Australian National University, Canberra, Australia Centre for Materials Science and Nanotechnology,University of Oslo, P.O. Box 1048 Blindern, NO-0316 Oslo, Norway Department of Energy and Process Engineering,Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Department of Physics, University of Oslo, P. O. Box 1048 Blindern, NO-0316 Oslo, Norway Department of Materials Science and Engineering,Royal Institute of Technology, SE-100 44 Stockholm, Sweden Physikalisches Institut, Albert-Ludwigs-University Freiburg,Hermann-Herder-Str. 3, 79104 Freiburg, Germany Division of Theory and Modeling, Department of Physics,Chemistry and Biology, Linköping University, SE-581 83 Linköping, Sweden
We present theoretical and numerical results for the screened Casimir effect between perfect metalsurfaces in a plasma. We show how the Casimir effect in an electron-positron plasma can providean important contribution to nuclear interactions. Our results suggest that there is a connectionbetween Casimir forces and nucleon forces mediated by mesons. Correct nuclear energies and mesonmasses appear to emerge naturally from the screened Casimir-Lifshitz effect.
I. INTRODUCTION
The Casimir and Lifshitz theories of intermolecular(dispersion) forces [1–3] have occupied such a vast litera-ture that little should remain to be said. [4–8] However,there exist still many gaps in our knowledge of the theoryof dispersion forces. For instance, we will show in this pa-per that the presence of any non-zero plasma density be-tween two perfectly reflecting plates fundamentally alterstheir long-range Casimir interaction. Such finite plasmadensities are always present near metal surfaces. Theseresults are discussed in detail in Sec. II where we givetheoretical and numerical results for the Casimir interac-tion between two perfect metal surfaces in the presenceof a plasma.The importance of Casimir forces for electron stability[9–13], particle physics, and in nuclear interactions [14],has been predicted over the years. The problem we in-tend to revisit is similar in spirit to the old story called"the Casimir mousetrap" for the stability of charged elec-trons. [10, 13] The negative charges on an electron surfacegive rise to a repulsive force between the different partsof the surface that has to be counteracted by an attrac-tive force in order for the electron to have a finite radius.Casimir proposed that such attractive Poincaré stressescould come from the zero-point energy of electromagneticvacuum fluctuations. [9] A number of attempts have beenmade to compute such Casimir energies. [10–13] However,all concluded that while the magnitude of the interaction ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] was correct, it had the wrong sign. Further it gave a re-pulsive force. [10–13]Finite plasma densities are present between nuclearparticles due to the presence of the plasma of the fluctu-ating electron-positron pairs constantly created and an-nihilated. The magnitude and asymptotic form of thescreened Casimir potential between reflecting surfaces inthe presence of this electron-positron plasma suggest apossible connection between Casimir forces and nucleonforces. [14] In Sec. III we proceed to explore this intrigu-ing similarities of the screened Casimir potential withthe Yukawa potential for nuclear particles as mediatedby mesons. Essentially correct nuclear energies, mesonmasses and meson lifetimes appear to emerge naturallyfrom the Casimir-Lifshitz theory. When taken at facevalue, the screened-Casimir model of the Yukawa po-tential would offer an alternative explanation of nuclearforces as being due to virtual electron-positron excita-tions.A somewhat complementary effect is the Casimir forcedue to electronic wave-function overlap as discussed inRef. [15]. In the latter case, the force results from realplate electrons whose evanescent wave functions expo-nentially decay into the gap between the plates. On thecontrary, in our scenario virtual electron–positron pairsin the space between the plates mediate the force. II. CASIMIR EFFECT BETWEEN PERFECTMETAL SURFACES IN THE PRESENCE OF APLASMA
Consider the Casimir-Lifshitz interaction betweenideal metal surfaces separated by a plasma of dielectricpermittivity ε ( iω ) = 1 + 4 πρe mω = 1 + ω p ω , (1)where the plasma frequency is identified as ω p =4 πρe /m , ρ is the number density of the plasma, m theelectron mass, and e the unit charge. We define some ad-ditional variables ¯ ρ = ρe ~ / (cid:0) πmk T (cid:1) , κ = ω p /c (notethe occurrence of a factor mc in the screening param-eter κ ), and x = 2 kT l/ ( ~ c ) . In these expressions k isBoltzmann’s constant, ~ is Planck’s constant, T the ef-fective temperature of the plasma, c is the velocity oflight, and l the distance between the plates. The exactexpressions for the Casimir-Lifshitz free energy betweenboth real and perfect metal surfaces across a plasma aregiven in Appendix A. We have found (see Appendix B fora derivation) that the asymptotic interaction energy canat high temperatures and/or large separations be writtenas F ( l, T ) = F n =0 + F n> , (2) F n =0 ( l, T ) ≈ − kT κ π e − lκ [ 12 lκ + 14 l κ ] , (3) F n> ≈ ( kT ) l ~ c e − π ¯ ρx e − πx + O ( e − x ) . (4)Here we have separated the zero and finite frequency con-tributions. These expressions may be useful for theoreti-cal comparisons with experimentally measured Casimir-Lifshitz forces [6, 16–22, 25, 26] between metal surfacesinteracting across a high density plasma.We first recall the present understanding of Casimireffect between real metal surfaces in the absence ofany intervening plasma. Fig. 1 shows the experimen-tal result of Lamoreaux [16], compared to the theoret-ical results of Boström and Sernelius [17]. All curvesshow the interaction energy divided by the result of theideal Casimir gedanken experiment at zero temperature, − ~ cπ / (cid:0) d (cid:1) . The lowest curve is for gold at roomtemperature. It was derived using tabulated optical datafor gold as input. Use of the Drude model gives overlap-ping results. To be noted is that theory and experimentclearly disagree for the cluster of experimental pointsaround d = 1 µm . The experimental results agree bet-ter with the zero-temperature results (upper solid curve)and even with the zero- or finite temperatures results forideal metals (the Casimir gedanken experiment, dottedcurves). The agreement is even better with the theoret-ical room temperature result obtained when using theplasma model.This puzzling behavior has given rise to a long-standing controversy in the field. We note that the zerofrequency part of the Casimir interaction between real E n e r gy C o rr ec ti on F ac t o r d ( µ m) d Perfect metal
T=300 K; T=0
Gold
T=300 K; T=0Expt. 300KLamoreaux
Figure 1. Energy correction factor for two gold plates in theabsence of any intervening plasma. The filled squares witherror bars are the Lamoreaux’ experimental [16] values fromthe torsion pendulum experiment. The dashed curves are theperfect metal results. The thick solid curves are the resultsfor real gold plates at zero temperature and at room tempera-ture [17]. The dielectric properties for gold was obtained fromtabulated experimental optical data. metal surfaces depends on how the dielectric functionof the metal surfaces is treated. Different theoreticalgroups have found very different results. [17–21]. A mostvaluable property of the Lamoreaux experiment [16] wasthat it was carried out at large separations. Lamore-aux was also involved in a more recent version of hisold experiment [22] (cf. also the comments of Milton[7]), where plate separations between 0.7 and 7 µ m weretested. Quite convincingly, the theoretical predictionsbased upon the Drude model were found to agree withthe observed results to high accuracy.The thermal Casimir effect is however a many-facettedphenomenon and care has to be taken about the electro-static patch potentials, which cause uncertainties in theinterpretation of the data in the mentioned experiment.There are other experiments, in particular the very accu-rate one of Decca et al. [23], which yield results appar-ently in accordance with the plasma model rather thanthe Drude model. The reason for this conflict betweenexperimental results is not known in the community. Ithas been suggested occasionally that it might have some-thing to do with the so-called Debye shielding, which canchange the effective gap between plates from the geomet-rically measured width. But the experimentalists them-selves turn out to be skeptical towards such a possibility.(An elementary overview of the temperature dependenceof the Casimir force is recently given in Ref. [24].) Thereis clearly an urgent need for more experiments and theo-retical analysis focusing on Casimir-Lifshitz forces in dif-ferent systems that include metal surfaces.As we have shown so far, the presence of any interven-ing plasma is of importance for the long range interactionenergy. We explore next the effect on the energy cor- -5 -4 -3 -2 -1 -1 n = 0, 10 n !
0, 10 n=0, 10 n !
0, 10 n = 0, 10 n !
0, 10 n = 0, 0n !
0, 0 E n e r gy C o rr ec ti on F ac t o r d ( µ m) Figure 2. Energy correction factor for two perfect metal platesinteracting across a plasma. The curves are the results forperfect metal plates at room temperature for different plasmafrequencies ( ω p in units of rad/s) for the intervening plasma.We show the results for the n = 0 and n ≥ contributions tothe interaction free energy. -1 n = 0 n > 0 n ! A s y m p t o ti c C o rr ec ti on F ac t o r d ( µ m) Figure 3. Asymptotic correction factor for two perfect metalplates interacting across a plasma ( ω p = 10 rad/s ). Theresults show the ratio between numerically calculated energiesand the corresponding asymptotes given in the text. Thereis very good agreement (ratio close to one) for the n = 0 contribution in the entire range considered. For n > and n ≥ the curves go towards one at large surface separations.Note that the asymptotes become more accurate for higherplasma densities. rection factor for different plasma densities between twoideal surfaces, See Fig. 2. Again, all curves show the in-teraction energy divided by the result of the ideal Casimirgedanken experiment at zero temperature in the absenceof a plasma. At large separation the result is stronglyinfluenced by intervening plasma, leading to a consid-erable reduction of the interaction energy. The resultsshow that even weak intervening plasmas can strongly af-fect Casimir force measurements. The possible presenceof spurious plasma densities thus has to be considered carefully.We next investigate the accuracy of the asymptotes (3)and (4) by comparing their predictions with exact numer-ical results.Fig. 3 shows the ratio between numerically calculatedfree energy between two perfect metal plates across aplasma ( ω p = 10 rad /s ) to the corresponding asymp-totes given in the text. We see that in this case theasymptote for the n = 0 term is very accurate. For the n > and n ≥ contributions this ratio only goes to-wards one at large separations. The asymptotes becomefor higher plasma densities. III. A CONTRIBUTION FROM SCREENEDCASIMIR INTERACTION IN NUCLEARINTERACTIONS
We will now point out a potential connection with themeson theory. That is, if we take the Casimir expansionwithout a plasma, the first three terms (see Eq. (7) be-low) are: (1) the usual zero point fluctuation energy (alsoequivalent to current-current correlations); (2) a "chemi-cal potential" term, identifiable as the energy of an elec-tron positron pair sea (see Landau and Lifshitz [27]); (3)the black body radiation in the gap. One can then askhow electromagnetic (EM) theory can give rise to weakinteractions of particle physics. Such contribution fromEM theory comes out if one equates the zero point en-ergy to the black body radiation term. That gives anequivalent density for the electron positron pair sea andthe energy of interaction of about 8 Mev. This agreeswith the experimentally found nuclear interaction energy.The form of the interaction with a plasma in the gap isthe same as that for the Klein-Gordon–Yukawa potentialwith the plasma excitation corresponding to and identi-cal with the π meson mass. (This assumes a plate sizeof 1 Fermi squared in area and that the planar resultstranslated roughly over to that for spheres.)Now we will explore these ideas in more detail. Thescreened Casimir free energy asymptotes in the previoussection can be compared with the Yukawa potential be-tween nuclear particles at distances large compared withthe screening length l π = ~ /m π c ( m π is the mass of themediating meson), F ( l, T ) ∝ e − l/l π . (5)To test if the idea can be correct, we first extract themeson mass by taking the exponents in the F n =0 termgiven in the previous section and the Yukawa potentialto be equal: m π = 4 e ~ c r π ( ρ + + ρ − ) m . (6)Since we know the meson mass (135 MeV) we esti-mate the screening length to be 1.458 fm and we alsofind the density of electrons and positrons that wouldbe required to generate this Yukawa potential from theCasimir effect. The equilibrium of electron positron pro-duction can at high temperatures be written as ρ ± =3 ζ (3) k T / (2 π ~ c ) . [27] This means that the requiredeffective temperature of nuclear interaction via a screenedCasimir interaction is . × K.We now address the important question where theenergy to generate this local electron plasma cancome from. [14] Feynman speculated that high energypotentials could excite states corresponding to othereigenvalues, possibly thereby corresponding to differ-ent masses. [28] It turns out that the low-temperatureCasimir interaction, i.e., without an intervening plasma,by itself could be capable of generating the effective tem-perature required to obtain the plasma. The connec-tion between temperature and density of electrons andpositrons given above is exploited in the expression forlow temperature Casimir interaction between perfectlyreflecting surfaces. In the absence of an interveningplasma, it can be written as Eq.(35) in Ref. [29]: F ( l, T ) ≈ − π ~ c l − ζ (3) k T π ~ c + π lk T ~ c . (7)This can further be re-written as F ( l, T ) ≈ − π ~ c l − π ( ρ − + ρ + ) ~ c π lk T ~ c , (8)where the first term is the zero-temperature Casimir en-ergy, the third is the blackbody energy, and the secondhas been rewritten in terms of electron and positron den-sities. If we assume that the entire zero-temperatureCasimir energy is transformed into blackbody energy(which at high temperatures can generate an electron-positron plasma) we can estimate the temperature as T ≈ ~ c/ lk . This will at a distance of 3.6 Fermi givethe required effective temperature (at the distances dis-cussed above the effective temperature is even larger,around . × K). It is intriguing that a cancella-tion of the Casimir zero point energy and the blackbodyenergy term, just like the cancellation of the n = 0 termat low temperatures, gives the right result.The screened Casimir interaction between two per-fectly reflecting surfaces, with estimated cross sectionof 1 fermi squared a distance . Fermi apart, receivesaround 4.25 MeV from the n=0 term and 3.25 MeV fromthe n>0 terms. While the screening length of the n=0term is defined above we find that the screening lengthof the n>0 terms also comes out of the right order ofmagnitude (it is within the crude approximations madeof the order one fermi). The nuclear interaction as ascreened Casimir interaction would thus receive approxi-mately equal contributions from the classical (n=0) andquantum (n>0) terms. The result compares remarkablywell with the binding energy of nuclear interaction thatis around 8 MeV.If the arguments we have given connecting nuclearand electromagnetic interactions have any substance, it is hard to avoid the speculation that the standard de-composition in nuclear physics into coulomb and nuclearforce contributions may not be entirely correct. In theinsightful words of Dyson: "The future theory will bebuilt, first of all upon the results of future experiments,and secondly upon an understanding of the interrelationsbetween electrodynamics and mesonic and nucleonic phe-nomena". [30] The problem is precisely equivalent to thatwhich occurs in physical chemistry where standard theo-ries have all been based on the ansatz that electrostaticforces (treated in a nonlinear theory) and electrodynamicforces (treated in linear approximation by Lifshitz the-ory) are separable. The ansatz violates both the Gibbsadsorption equation and the gauge condition on the elec-tromagnetic field. [31] When the defects are remedied agreat deal of confusion appears to fall into place. If theseresults are not acceptable within the standard model onemust still consider the presence of this additional electro-magnetic fluctutation interaction energy between nuclearparticles.
IV. CONCLUSIONS
We have explored the effect of an intervening plasmaon the Casimir force between two perfectly conductingplates. The analytically derived asymptotes for largeplate separations show that even spurious plasma densi-ties can considerably reduce the expected Casimir force.The effect of plasmas should therefore carefully be con-sidered in Casimir-force measurements.In addition, the derived asymptotes show an inter-esting structural analogy with the Yukawa potential ofnuclear interactions. We have explored this analogyto discuss whether the electromagnetic Casimir effectcan possibly explain these interactions. The compari-son yields predictions for the required virtual electron-positron plasma density which, however, is only achiev-able at very large ambient temperatures. If the poten-tial connection to nuclear interactions is correct, thenwe speculate that the charged π + and π − mesons wouldcome out to be bound positron-plasmon and electron-plasmon excitations in the electron-positron plasma.Apart from these speculations, our main idea has beento investigate to what extent the screened Casimir ef-fect between perfect metal surfaces, intervened by anelectron-positron plasma, can be applied to estimate nu-cleon forces mediated by mesons. Figures 2 and 3 showthe effect of plasma screening; especially the large sup-pression of the Casimir energy when the plasma densityis large, is clearly shown in Fig. 2. Our main findingsare that nuclear energies and meson masses emerge nu-merically of the right order of magnitude, thus indicatingthat our basic idea is a viable one.Of course, the ideas explored in this paper are some-what speculative. In principle, although the Casimir en-ergy has the right order of magnitude to provide the re-quired temperature, one may object that it is not evidenthow this energy can be converted to thermal radiation.The point we wish to emphasize here is that the presentarguments, although incomplete, may serve as a usefulstarting point for further research in this direction, per-haps within the framework of quantum statistical me-chanics.A final comment: Use of the electrodynamic Casimireffect in the context of nuclear physics is of course notnew. For instance, in hadron spectroscopy viewed fromthe standpoint of the MIT quark bag model it has longbeen known that the zero-point fluctuations of the quarkand gluon fields may generate a finite zero-point energyof the form E zp = − Z /r , for massless quarks. The con-stant Z is not firmly grounded theoretically, but serves aa phenomenological term fitting the experimental data (aclassic review article in this field is that of Hasenfratz andKuti [32]). The phenomenological quark model in whichthe r − dependent part ∆ m ( r ) of the effective quark mass m ( r ) varies according to a Gaussian, ∆ m ( r ) ∝ − e − r /R ,can also be regarded as an example of essentially the samekind [33]. Appendix A: Casimir-Lifshitz Free Energy
One way to find retarded van der Waals or Casimir-Lifshitz interactions between two objects interactingacross a medium is in terms of the electromagnetic nor-mal modes of the system. [34, 35] For planar structuresthe interaction energy per unit area can be written as E = ~ Z d q (2 π ) ∞ Z dω π ln [ f q ( iω )] , (A1)where f q is the mode condition function with f q ( ω q ) = 0 defining electromagnetic normal modes. Eq. (A1) is validfor zero temperature and the interaction energy is theinternal energy. At finite temperature the interactionenergy is a free energy and can be written as F = ∞ X n =0 F n = 1 β Z d q (2 π ) ∞ X n =0 ′ ln [ f q ( iξ n )] ; (A2)where β = 1 /kT , and the prime on the summation signindicates that the term for n = 0 should be divided bytwo. The integral over frequency in Eq. (A1) has beenreplaced by a summation over discrete Matsubara fre-quencies ξ n = 2 πn ~ β ; n = 0 , , , . . . (A3)For planar structures the quantum number that char-acterizes the normal modes is q , the two-dimensional(2D) wave vector in the plane of the interfaces. Two modetypes can occur: transverse magnetic (TM) and trans-verse electric (TE). These dictate the form through thewave amplitude reflection coefficients, r . For instance, for two planar objects in a medium, corresponding tothe geometry 1|2|1, the mode condition function is givenby f q = 1 − e − γ d r , (A4)where d is the thickness of intermediate medium, and thereflection coefficients for a wave impinging on an interfacebetween medium and from the -side given as r TM12 = ε γ − ε γ ε γ + ε γ , (A5)and r TE12 = ( γ − γ )( γ + γ ) , (A6)for TM and TE modes, respectively. Here, γ j stands for γ i ( ω ) = q q − ε i ( ω ) ( ω/c ) . (A7)where ε i ( ω ) is the dielectric function of medium i , and c the speed of light in vacuum.For two perfectly conducting plates ( r TE12 = − , r TM12 = 1 ), the Casimir energy (A2) across a dissipation-free plasma takes the simple form F ( l, T ) = kTπ ∞ X n =0 ′ Z ∞ dqq ln h − e − l √ q +( ξ n /c ) + κ i , (A8)recall that κ = ω p /c . By a simple variable substitution,the first term in the Matsubara sum can be cast int thealternative form F n =0 ( l, T ) = kT π Z ∞ κ dtt ln(1 − e − lt ) . (A9) Appendix B: Asymptotic Casimir Free Energy in aPlasma
Exact treatments of Casimir forces between perfectmetal surfaces across a plasma using the above expres-sions typically give asymptotic expansions that are notuniformly valid. The treatment we present here givesa different result. Our starting point is the above for-mula (A8) for the ineraction of two perfectly conduct-ing plates accross an intervening plasma as re-written byNinham and Daicic. [29, 35, 36] F ( l, T ) = − − kT πl πi Z c ds Γ( s ) ζ ( s + 1)( s − πx ) s − ∞ X n =0 ′ ( n + ¯ ρ ) − s/ (B1)The zero frequency ( n = 0 ) term gives the followingcontribution, F n =0 ( l, T ) = − − kT πl πi Z c ds Γ( s ) ζ ( s + 1)( s − κl ) s − , (B2) F n =0 ( l, T ) = kT π Z ∞ κ dtt ln(1 − e − lt ) , (B3)which at large separations becomes: [29] F n =0 ≈ − kT κ π e − lκ [ 12 lκ + 14 l κ ] . (B4)The interaction free energy can be written as: [29] F ( l, T ) = − − kT πl πi R c ds Γ( s ) ζ ( s +1)( s − πx ) s − × ζ G ( − s/ , ¯ ρ ) , c > , (B5) ζ G ( z, a ) = 2 ζ EH ( z, a ) + a − z = 2 ∞ X n =1 n + a ) z + a − z (B6)where as discussed in detail by Ninham and Daicic thegeneralized Epstein-Hurwitz ζ function ζ G is meromor-phic and has simple poles in the complex plane at z=-k+1/2 (k=0,1,2,..) [29]. In the limit of low temperaturesor distances x << they found (see Ref. [29] for thecomplete expression) F ( l, T ) = − π ~ c l (cid:20) − ρe ~ ( πmk T ) ( 2 kT l ~ c ) − ...... (cid:21) (B7)where at low temperatures the n = 0 term cancels out acontribution from the higher frequency terms.It is possible with some algebra to express the Lifshitzfree energy between two ideal metal plates with interven-ing plasma in the following form (useful for deriving theasymptotes considered in this contribution): F ( l, T ) = − − kT πl η ( l, T ) = − kT πl (cid:2) πx I (¯ ρ, x ) (cid:3) . (B8)The integral I consists of two parts, I = I + I , (B9)where I = Z ∞ dye − π ¯ ρy y − / ¯ ω ( x /y ) , (B10)and I = Z ∞ dye − π ¯ ρy y − / ¯ ω ( x /y )2¯ ω ( y ) , (B11)respectively.The function ¯ ω ( y ) appearing in both integrands is de-fined as, [36] ¯ ω ( y ) ≡ ∞ P n =1 e − n πy ≡ (cid:8) − y − / [1 + 2¯ ω (1 /y )] (cid:9) . (B12) The sum converges faster the larger the y -value. To makeuse of this fact we divide the integration range for I intotwo parts and use the two different expressions for thesum in the two resulting integrals. Thus, I = H + H , (B13)where H = Z ∞ x dye − π ¯ ρy y − / [ 12 ( − p y/x )+ p y/x ¯ ω ( y/x )] , (B14)and H = Z x dye − π ¯ ρy y − / ¯ ω ( x /y ) , (B15)respectively.The integrand in I has a product of two ¯ ω functionswith different arguments. Here we divide the integrationrange into three regions and choose the form of the sumthat gives the fastest convergence. Under the assumptionthat x > we have I = R dye − π ¯ ρy y − / ¯ ω ( x /y )[ − y − / (1 + 2¯ ω (1 /y ))]+2 R x dye − π ¯ ρy y − / ¯ ω ( x /y )¯ ω ( y )+ R ∞ x dye − π ¯ ρy y − / ¯ ω ( y )[ − p y/x (1 + 2¯ ω ( y/x )]= J + J + J + J + J + J + J , (B16)where J = 2 Z x dye − π ¯ ρy y − / ¯ ω ( x /y )¯ ω ( y ) , (B17) J = 2 x Z ∞ x dye − π ¯ ρy y − ¯ ω ( y )¯ ω ( y/x ) , (B18) J = 2 Z dyy − ¯ ω (1 /y )¯ ω ( x /y ) e − π ¯ ρy , (B19) J = − Z ∞ x dye − π ¯ ρy y − / ¯ ω ( y ) , (B20) J = 1 x Z ∞ x dye − π ¯ ρy y − ¯ ω ( y ) , (B21) J = − Z dye − π ¯ ρy y − / ¯ ω ( x /y ) , (B22)ans J = Z dye − π ¯ ρy y − ¯ ω ( x /y ) , (B23)respectively.We now add the two I terms and recombine the inte-gral terms to find I = I + I = J + J + J + K + K + K + K + K , (B24)where K = 1 x Z ∞ x dye − π ¯ ρy y − [ 12 + ¯ ω ( y )] , (B25) K = − x Z ∞ x dye − π ¯ ρy xy − / [ 12 + ¯ ω ( y )] , (B26) K = J , (B27) K = 1 x Z ∞ x dye − π ¯ ρy y − ¯ ω ( y/x ) , (B28)and K = R x dye − π ¯ ρy y − / ¯ ω ( x /y )= x R x dye − π ¯ ρx /y √ y ¯ ω ( y ) , (B29)respectively.In K we let y → x ξ , K = 1 x Z ∞ dξe − π ¯ ρx ξ ξ − ¯ ω ( ξ ) , (B30)and let ξ → /y , use the definition of ¯ ω ( y ) , and separateinto three terms, K = M + M + M (B31)where M = − x Z dye − π ¯ ρx /y = − x Z ∞ x dye − π ¯ ρy y − , (B32) M = 12 x Z dy √ ye − π ¯ ρx /y = 12 Z ∞ x dye − π ¯ ρy y − / , (B33)and M = 1 x Z dy √ ye − π ¯ ρx /y ¯ ω ( y ) , (B34)respectively. M and M exactly cancel with the terms with / in K and K . Now we combine the expressions for K and K and insert into the expression for I , I = I + I = J + J + J + K + N + N , (B35)where N = 1 x Z ∞ x dye − π ¯ ρy ( y − − xy − / )¯ ω ( y ) , (B36) and N = 1 x Z x dye − π ¯ ρx /y √ y ¯ ω ( y ) , (B37)respectively.Of these terms J , J , K and the term R ∞ x are all O ( e − x ) and we may drop them. So we have apart froma term O ( e − x ) the following expressions: η ( l, T ) = η + η = πx ( I + I ) , (B38)where η ≈ π Z x dy √ ye − π ¯ ρx /y ¯ ω ( y ) , (B39)and since η ≈ π Z ∞ dy √ ye − π ¯ ρx /y ¯ ω ( y ) − O ( e − x ) , (B40)we may write η ≈ π Z ∞ dy √ ye − π ¯ ρx /y ¯ ω ( y ) . (B41)Using the definition of ¯ ω ( y ) and the following repre-sentation: e − y = 12 πi Z C + i ∞ C − i ∞ dpy − p Γ( p ) , Re( p ) = C > , (B42)we obtain η ≈ π Z ∞ dy √ y Z C + i ∞ C − i ∞ dp ∞ X n =1 Γ( p )( n πy ) p e − π ¯ ρx /y . (B43)With a variable substitution κ l = π ¯ ρx =4 πρe l / ( mc ) we find η ≈ √ π Z ∞ dy √ y Z C + i ∞ C − i ∞ dp ∞ X n =1 Γ( p )( n y ) p e − κ l /y (B44)and using the Riemann ζ function, η ≈ √ π Z C + i ∞ C − i ∞ dp Γ( p ) ζ (2 p ) ( κl ) ( κl ) p Z ∞ dyy p − / e − y . (B45)Integration over y results in η ≈ ( κl ) √ π Z C + i ∞ C − i ∞ dp Γ( p ) ζ (2 p )( κl ) p Γ( p −
32 ) , Re( p ) = C > . (B46)We now exploit relations for the Γ function: Γ( p − /
2) = Γ( p + 1 / p − / p − / , (B47)and Γ( p )Γ( p + 1 /
2) = √ π − p Γ(2 p ) , (B48)to obtain η ≈ κl ) Z c + i ∞ c − i ∞ dp Γ( p ) ζ ( p + 1)(2 κl ) p ( p − κl ) Z c + i ∞ c − i ∞ dp Γ( p ) ∞ X n =1 n p +1 (2 κl ) p ( p − κl ) Z c + i ∞ c − i ∞ dp Γ( p ) Z ∞ dxx ( x + 1) p/ (2 κl ) p × ∞ X n =1 n p +1 = 2( κl ) Z ∞ dxx ∞ X n =1 n − e − κln √ x +1 = − κl ) Z ∞ dxx ln(1 − e − κl √ x +1 )= − l Z ∞ κ dtt ln(1 − e − lt ) . (B49)The free energy from η is then seen to give a contri-bution equal to the zero frequency part of the Lifshitz-Casimir energy between ideal metal surfaces with an in-tervening plasma, [29] F ( l, T ) = kT π Z ∞ κ dtt ln(1 − e − lt ) . (B50)The remaining η term is η ≈ πx Z x dyy − / e − π ¯ ρy ¯ ω ( y )¯ ω ( x /y ) . (B51)Now, since ¯ ω ( y ) = ∞ X n =1 e − n πy = ∞ X n =0 e − π ( n +1) y , (B52)we have e − πy < ¯ ω ( y ) < e − πy − e − πy , (B53)and R x dyy − / e − π ¯ ρy e − πy e − πx /y < I < R x dyy − / e − π ¯ ρy e − πy e − πx /y (1 − e − πy )(1 − e − πx /y ) < R x dyy − / e − π ¯ ρy e − πy e − πx /y (1 − e − π ) . (B54)Apart from a very small uncertainty (1 − e − π ) wehave η ≈ πx Z x dyy − / e − π ¯ ρy e − πy e − πx /y , (B55)and with the substitution y → yx we have ≈ πx / Z x /x dyy − / e − π ¯ ρxy e − π ( y +1 /y ) x , (B56)which for x → ∞ (large separations or high tempera-tures) produces a simple final expression. To find thiswe notice that the integral has a steep maximum. Take f ( y ) = y + 1 /y , then f ′ ( y ) = 1 − /y is equal to zeroat y = 1 and f ( y ) = 2 and f ′′ ( y ) = 2 . Thus, we maywrite η ≈ πx / Z ∞−∞ dye − π ¯ ρx e − πx e − πx ( y − y ) , (B57)and η ≈ πx / e − π ¯ ρx e − πx Z ∞ dte − πxt = 2 πxe − π ¯ ρx e − πx . (B58)The free energy from η gives a contribution at high x (large separations or high temperatures) F = − ( kT ) l ~ c e − ¯ ρx e − πx = − ( kT ) l ~ c e − ρ ~ c e mc l e − πkT l/ ( ~ c ) . (B59)The whole Casimir free energy in the high x =2 kT l/ ( ~ c ) limit is F ( l, T ) = kT π R ∞ κ dtt ln(1 − e − lt ) − ( kT ) l ~ c e − ρ ~ c e mc l e − πkT l/ ( ~ c ) + O ( e − x ) . (B60)This is the correct limit for either high temperature atfixed separation or for large distances at fixed temper-ature. The given expression can also be valid at smallseparations or low temperatures. This is a crusial pointbut one should remember that the derivation of plasmadensity from the equating of black body radiation tozero point energy and subsequent use of that densityrequires "high" temperatures. [27] The situation for twonuclear particles is one with very high effective tempera-ture and separations being "large", at least compared tothe screening length of the high density plasma. Appendix C: ζ functions in physics We would like to point out that that zeta functionshave been applied to many physical problems in thepast. [36–40] Elizalde considered for example the sum S ( t ) , defined by S ( t ) = ∞ X n =1 e − n t , with t a parameter. This is transformed into the equation S ( t ) = −
12 + 12 r πt + ∞ X k =1 ( − t ) k k ! ζ ( − k ) + ∆ ( t ) , where ∆ ( t ) is a remainder. The zeta-function termdoes not contribute, and the reminder reduces to thesum/integral ∆ ( t ) = 2 ∞ X n =1 Z ∞ dxe − x t cos(2 πnx ) = r πt ∞ X n =1 e − π n t . It means that S ( t ) = −
12 + 12 r πt + r πt ∞ X n =1 e − π n t . This formula was a key component in our derivations. [41]
ACKNOWLEDGMENTS
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