On the Correct Implementation of Fermi–Dirac Statistics and Electron

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On the correct implementation of Fermi–Dirac statistics and electron trapping in nonlinear electrostatic plane wave propagation in collisionless plasmas Hans Schamel and Bengt Eliasson

Citation: Physics of Plasmas 23, 052114 (2016); doi: 10.1063/1.4949341 View online: http://dx.doi.org/10.1063/1.4949341 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/23/5?ver=pdfcov Published by the AIP Publishing

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Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.159.82.198 On: Thu, 02 Jun 2016 12:31:32 PHYSICS OF PLASMAS 23, 052114 (2016)

On the correct implementation of Fermi–Dirac statistics and electron trapping in nonlinear electrostatic plane wave propagation in collisionless plasmas Hans Schamel1,a) and Bengt Eliasson2,b) 1Physikalisches Institut, Universitat€ Bayreuth, D-95440 Bayreuth, Germany 2SUPA, Physics Department, University of Strathclyde, John Anderson Building, Glasgow G4 0NG, Scotland, United Kingdom (Received 17 March 2016; accepted 1 April 2016; published online 19 May 2016) Quantum statistics and electron trapping have a decisive influence on the propagation characteristics of coherent stationary electrostatic waves. The description of these strictly nonlinear structures, which are of electron hole type and violate linear Vlasov theory due to the particle trapping at any excitation amplitude, is obtained by a correct reduction of the three-dimensional Fermi-Dirac distribution function to one dimension and by a proper incorporation of trapping. For small but finite amplitudes, the holes become of cnoidal wave type and the electron density is shown to be described by a /ðxÞ1=2 rather than a /ðxÞ expansion, where /ðxÞ is the electrostatic potential. The general coefficients are presented for a degenerate plasma as well as the quantum statistical analogue to these steady state coherent structures, including the shape of /ðxÞ and the nonlinear dispersion relation, which describes their phase velocity. VC 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4949341]

I. INTRODUCTION tunneling effects in the description of electron holes,17,18 but in this paper, we will neglect these effects. We deal with the Plasmas support a large number of nonlinear structures general framework of structures within the Vlasov-Poisson such as shocks, double layers, solitons, envelop solitons, and (VP) description for which Bernstein, Greene, and Kruskal vortices. A particular class of nonlinear structures occurs due (BGK)20 presented the first mathematically correct solution. to particle trapping, where ions or electrons are trapped in Physically, however, another method, the pseudo-potential the self-consistent potential wells in the plasma. Such struc- method19 has proven preferential, as it is mathematically as tures arise naturally in streaming instabilities or other not general and correct but allows from the outset the selection necessarily large amplitude perturbations. Typically, the of additional physically relevant solutions that are not possi- trapping leads to excavated regions in the phase (velocity, ble to obtain within BGK theory.20 Hence, our model is coordinate) space, and hence, they are often referred to as 1–7 semi-classical in that it takes into account the modification ion and electron holes. Electron and ion holes have been of the equilibrium electron distribution function to a Fermi- 8–11 observed in the laboratory and by satellite observations Dirac distribution, while we consider structures large enough 12–14 in space. At much smaller scales and in dense plasmas, that electron tunneling effects can be neglected. the electron density is so high that quantum effects must be 15,16 taken into account. The reason is that the inter-particle II. CONSERVED QUANTITIES AND REDUCTION distance is small enough that the wave-functions of the elec- OF DIMENSIONALITY trons start to overlap, and Pauli’s exclusion principle dictates For pedagogic reasons, we here briefly review some ba- that two electrons, which are fermions, cannot occupy the sic properties of the Vlasov-Poisson system which will be of same quantum state. This changes the equilibrium distribu- importance in Sections III–VII. For brevity, we restrict the tion of the electrons from a Maxwell-Boltzmann distribution discussion to electrons, but the results are easily extended to to a more general Fermi-Dirac (FD) distribution that takes include one or more ion species. The Vlasov equation into account electron degeneracy effects. In the limiting case of zero temperature, the equilibrium distribution for the e @t þ v rþ r/ rv fe ¼ 0 (1) Fermi gas is a flat-topped distribution in momentum space me (or in velocity space for non-relativistic particles). At very small scales, there are also effects due to electron tunneling, describes the evolution of the electron distribution function and in this case, the dynamics of a single electron is fe in 3 spatial and 3 velocity dimensions, plus time, where e described by the Schrodinger€ or Pauli equation, and statisti- is the magnitude of the electron charge, and me is the elec- cal models are based on the Wigner equation. Some works tron mass. The electrostatic potential / is obtained from have taken into account the first-order electron quantum Poisson’s equation ð

a) 2 e 3 [email protected] r / ¼ ni fed v ; (2) b)[email protected] 0

1070-664X/2016/23(5)/052114/7 23, 052114-1 VC Author(s) 2016.

where ni is assumed to be an immobile and homogeneous with an equilibrium number density n0 andp anffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi electron tem- 2 neutralizing ion background density. Using that the Vlasov perature Te. The plasmap frequencyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xpe ¼ n0e =me0, the and Poisson equations are Galilei invariant, stationary waves thermal velocity vthe ¼ kBT0=me, and the Debye screening in an inertial frame moving with the phase velocity of the length kDe ¼ vthe=xpe will be used for normalization of the wave can be obtained by setting the time-derivative to zero dependent variables, here temporarily being denoted by tilde, in Eq. (3), giving time-independent Vlasov equation into their dimensionless form through t ¼ xpe~t; x ¼ x~=kDe; v ¼ ~v=v ; / ¼ e/~ =k T , and f ðvÞ¼ðv =n Þf~ ð~vÞ, which e the B 0 0 the 0 0 v rþ r/ rv fe ¼ 0: (3) we will use below. me Consider next the underlying 1D Vlasov-Poisson (VP) system (5) and (6), which is perturbed by a coherent plane Solutions of Eq. (3) are constant along particle trajectories in wave structure /ðx; tÞ¼/ðx v tÞ that propagates with ve- ðr; vÞ space, given by E¼m v2=2 e/ ¼ constant, where 0 e locity v in the lab frame. It is then constant in time in a v2 ¼ v2 þ v2 þ v2, and hence a solution can be expressed as 0 x y z frame moving with the wave. In this wave frame, the normal- fe ¼ f0ðEÞ,wheref0ðEÞ is any function of one argument. (f0 can also be a multi-valued, as discussed below.) The con- ized Vlasov equation hence becomes served energy E is essentially due to reversal and translational ½v@x þ /ðxÞ@vfeðx; vÞ¼0; (7) symmetries of the Vlasov equation in time. Depending on the shape of the potential /, there may exist additional symme- where we, for brevity, have dropped the superscript “1D” on tries leading to other conserved quantities. The most important fe and have written vx ¼ v. For our purposes, we write the so- case for our purposes is when / depends only one spatial vari- lution of Eq. (7) as able, which we will take to be in the x-direction. In this case, there is a translational symmetry along the y and z directions, feðx; vÞ¼f0ðnÞhðÞþftðnÞhðÞ; (8) which leads to that the y and z components of the particle mo- where ¼ v2=2 /ðxÞ is the single particle energy and hðxÞ mentum (and velocity) are conserved (vy ¼ constant and is the Heaviside step function. The separatrix, which divides vz ¼ constant along the particle’s trajectory) and can be used to eliminate these quantities in E, so that a “one-dimensional” the phase space into a trapped and untrapped particle region, 2 is given by ¼ 0. The unperturbed distribution function is conserved energy can be found, Ex ¼ mevx=2 e/ ¼ constant. Hence, the solution of the time-independent thereby extended, as a consequence of /ðxÞ, to the inhomo- geneous distribution f ðnÞ, which represents the untrapped Vlasov equation for this case can be written fe ¼ f0ðEx; vy; vzÞ, 0 where f is a function of three arguments. A last step to reduce (or free) particles. The new argument n is a generalized ve- 0 27 the two velocity dimensions perpendicular to the x axis is to locity and is defined by integrate the distribution function as pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ðð n ¼ hðÞsgnðvÞ 2 þ hðÞ 2: (9) 1D fe ðExÞ¼ f0ðEx; vy; vzÞ dvydvz: (4) The solution involves a two-parametric (sgnðvÞ, ) constant of motion of (7), such that (8) solves (7). Note that n ¼ v This one-dimensional distribution function is a solution of when / ¼ 0 and that sgnðvÞ is a constant of motion only for the one-dimensional (1D) Vlasov equation the untrapped particles. Hence, for >0, f0 becomes multi- valued with respect to and depends also on sgnðvÞ. In Eq. e 1D v @ / @ ; v 0; (5) (8), ftðnÞ represents the trapped electron distribution, which x x þ ðÞx vx fe ðÞx x ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi me is restricted to the trapped region, 2/ v 2/, corresponding to 0. Here, we already have adopted a non- obtained by integrating Eq. (3) over v and v , and where / is y z negative /. The electron density is thus obtained as governed by the 1D Poisson equation ð ð pffiffiffiffi ð 1 2/ ffiffiffiffiffi @2/ e p 1D neð/Þ¼ fedv ¼ f0ð 2Þ dv ¼ ni fe dvx : (6) @x2 1 1 0 ðpffiffiffiffi ð 2/ pffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffi If the potential instead has a rotational symmetry (spher- þ pffiffiffiffi ftð 2Þ dv þ pffiffiffiffi f0ð 2Þ dv: (10) ical or cylindrical), an additional conserved quantity to E is 2/ 2/ the angular momentum with respect to the rotation axis, which can be used to construct classes of time-independent solutions of the Vlasov equation.21 Various extensions IV. HALF-POWER EXPANSION OF THE ELECTRON including a constant external magnetic field can be found in DENSITY Refs. 22–24 and relativistic effects in Refs. 25 and 26. In the following, we assume a small but finite amplitude of the perturbation such that 0 /ðxÞw 1, where w is III. GENERAL FORMULATION OF ELECTRON the small amplitude of the potential. The electron density in TRAPPING the presence of a structure is found by a velocity integration Consider a collisionless plasma, in which the unper- of Eq. (8). Utilizing the smallness of /, we get by a Taylor ~ 27,28 turbed homogeneous electron distribution function is f 0ð~vÞ, expansion of Eq. (10) around / ¼ 0, under the

assumption that feðx; vÞ is continuous across the separatrix relation between l and A is found by a 3D velocity integra- and that ftðnÞ is an analytic function in n that tion using spherical coordinates in velocity space

2 2 3=2 l k0w 3 / A ¼ð2pÞ Li3=2ðe Þ; (17) n ðÞ/ ¼ 1 þ þ a/ þ b/2 þ c þ; (11) e 2 2 where where k0 will later be a measure of the wave’s wavenumber, and the coefficients are X1 zk Li ðÞz ¼ ; jzj < 1(18) ð n kn 1 k¼1 a ¼P @vf0ðÞv dv; (12) v is the polylogarithm function with index n. By a 2D velocity 25=2 integration over ðvy; vzÞ-space, using cylindrical coordinates, b ¼ @2f j þ @2f j ; (13) 3 v 0 v¼0 n t n¼0 we get the reduced, 1D distribution in the lab frame ðð ÀÁ 2p 2 and FD FD vx =2þl f0 ðÞvx ¼ F ðÞE dvydvz ¼ ln 1 þ e : (19) ð A 1 2 c ¼P @v f0ðÞv dv; (14) The Maxwellian case is formally (for more details, see later) v obtained in the limit l !1, which by the small value 3=2 l where P denotes the principal value. In (12)–(14), we made expansion LiðxÞx gives A !ð2pÞ e , which together v2=2þl v2=2þl use of v ¼ n when / ¼ 0, which is valid for untrapped with lnð1 þ e x Þ!e x give the Maxwellian electrons. distribution Deviations from continuity and analyticity have recently 1 2 M ffiffiffiffiffiffi vx =2 been treated in Ref. 29. The coefficient c in Eq. (14) is given f0 ðÞvx ¼ p e : (20) for completeness only and will not be used below. Trapping 2p 3=2 is manifested by the b/ term in Eq. (11). Equation (11) is The free particle distribution function in the wave frame, hence a half-power expansion in /, in which the /1=2 term introducing v ¼ vx v0, then reads vanishes due to continuity at ¼ 0.29 This provides the for- 1 2 mal background under which a wide range of collisionless ðÞvþv0 =2 f0ðÞv ¼ pffiffiffiffiffiffi e : (21) plasmas can be treated formally on the same footing. 2p

For the trapped particles, we use V. THE MAXWELLIAN CASE v2=2 e 0 b 2 ftðÞn ¼ pffiffiffiffiffiffi 1 þ n : (22) We start with the general degenerated state and perform 2p 2 subsequently the classical Maxwellian limit, as follows. For fermions in a degenerate state, the unperturbed single parti- Note that continuity at the separatrix where n ¼ 0 for fi- cle distribution is given by the Fermi–Dirac (FD) distribu- nite / is automatically guaranteed by Eqs. (21) and (22), and tion, which in the three-dimensional (3D) velocity space that (22) uses the lowest order terms of the Taylor expansion 30 b reads in dimensional units of e j/¼0. This latter exponential expression was used earlier for finite amplitude waves, e.g., in Refs. 1–3 and 19, but 3 FD 2m 1 the results are the same as long as w 1. Insertion of F~ ðÞE~ ¼ e ÂÃ; (15) 3 ~ h eðÞEl~ =kBT0 þ 1 expressions (21) and (22) into Eqs. (12) and (13), respectively, then yields ~ 2 2 where l~ is the chemical potential and E ¼ð~vx þ ~vy 2 2 pffiffiffi þ~v Þ=2 ¼ ~v =2 is the single particle energy in the 3D veloc- 1 0 z a ¼ aM Zr v0= 2 ; (23) ity space (in lab frame). In the limit T0 ! 0, one obtains 2 ~ 2 2=3 l~ ¼ eF ¼ðh =8meÞð3n0=pÞ , which is the Fermi energy and and h is Planck’s constant. ÀÁ Returning to dimensionless quantities, we can hence 2 41 b v0 v2=2 4 b ¼ b ¼ pffiffiffi e 0 ¼: bðÞb; v ; (24) write M 3 p 3 0 1 FFDðÞE ¼ ; (16) with ðÞEl Ae þ 1 ð 1 expðÞt2 where the normalization constant A ¼ n h3=½2ðm k T Þ3=2 Z ðÞu ¼ pffiffiffi P dt; (25) 0 e B 0 r p t u is proportional to n0. Moreover, l is now the normalized chemical potential and E ¼ðv2 þ v2 þ v2Þ=2 ¼ v2=2 is the ð x y z ðÞ2 normalized single particle energy in the 3D velocity space 0 2ffiffiffi t exp t ZrðÞu ¼p P dt; (26) (in the lab frame). Assuming a unit equilibrium density, the p t u

which are familiar expressions found in many earlier papers ðÞ2p 3=2 a ðÞ0 ¼ Li ðÞel : (32) of Schamel and coworkers. We note again that the b-depend- FD A 1=2 ence of ft and of bM is a crucial ingredient of getting suited trapped particle equilibria. If, as done by Landau and The function Li1=2ðxÞ is real-valued and negative for x < 0. 31 Lifshitz, b is assumed zero, corresponding to a completely On the other hand, for high phase velocities v0 1, we have ﬂat trapped particle distribution, then no solitary electron the asymptotic expansion holes would exist theoretically,1 to give an example of its ðÞ3=2 limited application (besides its restriction to v0 ¼ 0). The 2p 1 l 3 l aFDðÞv0 ¼ Li3=2ðÞe þ Li5=2ðÞe hollow nature of the trapped particle vortices found in phase A v2 v4 0 0 space, being a general feature of all numerically observed 1 þ O : (33) phase space structures, demands the incorporation of a v6 b < 0. The question is now how in a quantum statistical set- 0

ting the corresponding structural equilibria turn out to be. Figure 1 shows line-plots of aFDðv0Þ given by (29) for different values of v0 and l. While aFDðv0Þ is a decreasing function of l for v ¼ 0, the dependence of a ðv Þ on l is more VI. THE FERMI–DIRAC CASE 0 FD 0 complicated for larger values of v0 with either local maxima For fermions in a degenerate state, we got (16) as the or minima (cf. Fig. 1(b)). FD FD FD dimensionless Fermi-Dirac distribution F ðEÞ in 3D veloc- To obtain bFD, the two functions f0 ðvÞ and ft ðvÞ have FD ity space, from which the reduced 1D distribution f0 ðvxÞ in to be differentiated twice and, according to (13), the result the lab frame (19) was obtained. has to be evaluated at v ¼ 0 and n ¼ 0, respectively. After The reduced free particle distribution function for 0 some algebra, we get in the wave frame, introducing v ¼ vx v0, then reads "#ÀÁ 7=2 2 ÂÃ 2 p r v0 1 r 2p 2 b ¼ bFDðÞv0 2 þ b lnðÞ 1 þ r ; (34) f FDðÞv ¼ ln 1 þ eðÞvþv0 =2þl : (27) 3A ðÞ1 þ r 0 A lv2=2 For the trapped particle distribution with <0, we choose where we defined r ¼ e 0 . For a wave with zero velocity the same function as in the Maxwellian case but with a dif- with respect to the lab frame, v0 ¼ 0, we hence get with l ferent factor that accounts for continuity at ¼ 0 (and at r0 ¼ e n ¼ 0) 27=2p r b ðÞ0 ¼ 0 þ b lnðÞ 1 þ r ; (35) ÀÁ FD 3A 1 r 0 FD 2p v2=2þl b 2 þ 0 f ðÞn ¼ ln 1 þ e 0 1 þ n : (28) t A 2 a relative simple expression, which stands for the electron Equations (27) and (28) are the two desired functions needed trapping nonlinearity in case of a non-propagating wave pat- to derive a and b for the density expression (11) in the FD tern in a degenerate plasma. The dependence of bFDðv0Þ on case. By insertion of (27) into (12), we first get v0, l, and b is shown in Fig. 2. pffiffiffi 1 l a ¼ aFDðÞv0 Z 0 v0= 2 ; (29) 2 r VII. CNOIDAL ELECTRON HOLES AND THEIR SPEED To close the system, i.e., to find self-consistent solutions where we defined the Fermi-Dirac plasma dispersion for both systems, we have to solve the second part, the function ÀÁ Poisson equation, which in the immobile ion limit becomes ð 2 ð 23=2p ln 1 þ elt ZlðÞu P dt; (30) 00 0 r A t u / ðxÞ¼ f ðx; vÞdv 1 ¼Vð/Þ: (36)

which is by definition real valued and whose derivative is In the second step, we have introduced the pseudo-potential given by Vð/Þ. Its knowledge allows by a quadrature to find the final ð 5=2 shape of the potential structure /ðxÞ via the pseudo-energy l 2 p t Zr 0ðÞu ¼ P 2 dt: (31) A ðÞ1 et l ðÞt u 1 0 2 þ / ðÞx þVðÞ/ ¼ 0; (37) 2 The generalized complex plasma dispersion function ZlðuÞ with a complex argument u would then be a one in which the where we without loss of generality have assumed Vð0Þ¼0. principal value is replaced by the Landau contour. Insertion of the electron density (11) into Eq. (36) and a l l 0 subsequent /-integration yield We note that Zr ðuÞ!ZrðuÞ and Zr 0ðuÞ!ZrðuÞ in the classical limit l 1. In the case of a standing wave pat- 2 tern, v ¼ 0; a can be obtained by a direct integration and k 1 2 2 5=2 0 FD VðÞ/ ¼ 0 w/ þ a/ þ b/ þ: (38) can be expressed in terms of a polylogarithm function as 2 2 5

FIG. 1. The linear coefﬁcient aFD for different values of v0 and l. Note that aFD ! aM in the classical limit l 1.

The speed of the potential structure is obtained from the non- In case of a Maxwellian unperturbed plasma, where we linear dispersion relation (NDR), VðwÞ¼0, which becomes have to replace (a; b)by(aM; bM), the NDR can be written 6,27–29,32,33 pffiffiffiffi as 2 4b k0 þ a þ w þ ::: ¼ 0: (39) pffiffiffi pffiffiffiffi pffiffiffiffi 5 2 1 4 16 k Z0 v = 2 ¼ b w bðÞb; v w; (41) 0 2 r 0 5 M 15 0 Eliminating a in Eq. (38) by Eq. (39) we get where we used (24) in the last step to show coincidence with k2 2b pffiffiffiffi pffiffiffiffi VðÞ/ ¼ 0 /wðÞ / /2 w / þ :::: (40) the previous result. In the more general degenerate electron 2 5 state, we replace (a; b)by(aFD; bFD) in Eq. (39) to obtain the NDR It represents a typical cnoidal wave solution, a one being rep- pffiffiffi pffiffiffiffi resented by Jacobian elliptic functions such as cnðxÞ or 2 1 l 4 k Z 0 v0= 2 ¼ bFDðÞv0 w; (42) snðxÞ. The first term in (40) represents a purely harmonic 0 2 r 5 wave, and the second one is a solitary wave, provided that b < 0. The general two-parametric solution, which involves with bFDðv0Þ given by (34). both terms, has been analyzed in Refs. 32 and 33. It is a peri- Figure 3 shows plots of the NDR (42), where we used odic function with an in general high content of Fourier har- x0 ¼ k0v0,forb ¼1 and different values of l. Both the þ monics. As said before, its phase velocity v0 is determined zero amplitude limit w ’ 0 in Fig. 3(a) and the small am- by the NDR (39). plitude case w ¼ 0:1 in Fig. 3(b) show similar features with

FIG. 2. The coefﬁcient of fractional nonlinearity bFD for different values of v0, l, and b. Note that bFD ! bM in the classical limit l 1.

þ FIG. 3. The nonlinear dispersion relation (NDR) for (a) w ¼ 0 and (b) w ¼ 0:1, and different values of l with b ¼1, using x0 ¼ v0k0. The classical, non- degenerate case corresponds to l ¼1.

the typical thumb shape of the dispersion curves in the (x0, half-power expansions of the potential, which we find ques- k0) space. In general, the wave frequency increases with tionable, are found by not following the recipe of first defin- increasing values of l. The high-frequency Langmuir branch ing trapped and free populations of electrons before takes the frequency x0 ! 1ask0 ! 0 as can also be shown calculating the electron density and by an incorrect reduction by using the NDR (39) together with the asymptotic expan- in the distribution function from 3D to 1D. sion (33) for a ¼ aFD together with Eq. (17) for A, and that Finally, a mention on the validity of the present approach b ¼ bFD ! 0 in the limit v0 !1. is necessary. We note that hot and non-degenerate plasmas 1=3 We note that the character of the dispersion relation are characterized by kn kdB, where kn ¼ð4pn0=3Þ is changes and becomes “non-dispersive” with a lower cut-off the mean particle distance and kdB ¼ h=ðmevtheÞ is the ther- for k0 in the high frequency branch when the right hand side mal de Broglie wavelength with the reduced Planck’s con- of the NDR is fixed rather than b and w (see Fig. 5 of Ref. stant h ¼ h=ð2pÞ, while cold and degenerate plasmas are Շ 33). In case of a Maxwellian plasma, denoting the rightffiffiffiffiffiffiffi hand characterized by kn kdB. In the latter limit, the equilibrium ^ p side of (41) by BM, this cut-off is given by k0 :¼ BM and distribution changes from a Maxwell-Boltzmann distribution 2 1 ^þ to a Fermi-Dirac distribution. Moreover, the collisionless na- the phase velocity v0 diverges as v 2 as k0 ! k .A 0 k BM 0 0 pffiffi ture of the plasma requires that the mean kinetic energy of 15 pffiffiffi 2 fixed BM then corresponds to a b ¼ p BM expðv =2Þ, the particles is much larger than the mean potential energy 16 w 0 due to Coulomb interactions between the particles. For a which assumes extremely large negative values. In such a Maxwell-Boltzmann distributed plasma, this leads to the situation, the trapped electron range is more or less empty. 2 requirement 3kBT=2 > e kn=ð4p0Þ,orkn < kDe, while in the cold limit of Fermi-Dirac distributed electrons, we have VIII. CONCLUSION 2 3EFe=5 > e kn=ð4p0Þ, which is fulfilled for sufficiently In conclusion, we have presented a general framework տ 3 2 2 high densities n0 a0 where a0 ¼ 4p0h =ðmee Þ5:3 11 30 3 of weak coherent structures in partially Fermi-Dirac degen- 10 m is the Bohr radius, giving n0 տ 6:3 10 m . erate plasmas. These coherent structures are necessarily This is larger than typical densities of conduction electrons in determined by the electron trapping nonlinearity and hence metals, 5 1028 m3, but this restriction may be relaxed are beyond any realm of linear Vlasov theory. This implies due to Pauli blocking reducing the collisionality of the elec- that even in the infinitesimal amplitude limit, linear Landau trons.35,36 The use of the Vlasov approach against the more and/or van Kampen theory fail as possible descriptive general Wigner framework of quasi-distributions6 necessi- 29,33 approaches to these structures. The spatial profiles of the tates the semi-classical limit DxDv h=me, where Dx is the potential /ðxÞ in the small amplitude limit are of cnoidal spatial length-scale of the structure and Dv is the velocity hole character, similar is in the classical case, and are given scale. As an example, if Dx 1=k0 is the typical length-scale by the already known expressions (see Ref. 33 and referen- of a periodic wave-train and the velocity scalepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in an ultra- ces therein). The NDR on the other hand is for given b; w cold plasma is the Fermi speed Dv vFe ¼ 2EF=me, then again of thumb shape type but appears to be rather sensitive the Wigner approach has to be used when the wavelength to the electron degeneracy characterized by the normalized k ¼ 2p=k0 is small enough to be comparable to the inter- chemical potential l. An important aspect of the nonlinear particle distance kn. If the velocity scale is instead given by theory in the small but finite amplitude limit is that the elec- the thermal speed vthe in a warm, Maxwellian plasma, then tron density is described by half-power expansions of the the Wigner approach would only be necessary at extremely 34 electrostatic potential. Different results not giving rise to short wavelengths when k is comparable to kdB kn. The

Maxwellian limit l !1is only a formal one, since rela- “Cluster observations of electron holes in association with magnetotail reconnection and comparison to simulations,” J. Geophys. Res. 110, tivistic effects have to be taken into account when kBT 2 A01211, doi:10.1029/2004JA010519 (2005). becomes comparable to the electron rest mass energy mec , 14J. P. McFadden, C. W. Carlson, R. E. Ergun, F. S. Mozer, L. Muschietti, I. 9 which occurs at temperatures higher than 10 K, and in Roth, and E. Moebius, “FAST observations of ion solitary waves,” which case relativistic effects have to be taken into account J. Geophys. Res. 108, 8018, doi:10.1029/2002JA009485 (2003). 15 in the treatment of electron holes.25 A similar constraint P. K. Shukla and B. Eliasson, “Nonlinear aspects of quantum plasma physics,” Phys. -Usp. 53, 51–76 (2010). applies to very high densities when the electron Fermi energy 16P. K. Shukla and B. 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