On equilibrium charge distribution above dielectric surface
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y On equilibrium charge distribution above dielectricsurface
Dmytro M. Lytvynenko and Yuriy V. Slyusarenko
Akhiezer Institute for Theoretical Physics, NSC KIPT, 1 Akademicheskaya str., 61108 Kharkiv, Ukraine
The problem of the equilibrium state of the charged many-particle system above dielectric surface is formulated.Weconsider the case of the presence of the external attractive pressing field and the case of its absence. The equilibriumdistributions of charges and the electric field, which is generated by these charges in the system in the case of ideallyplane dielectric surface, are obtained. The solution of electrostatic equations of the system under consideration in caseof small spatial heterogeneities caused by the dielectric surface, is also obtained. These spatial inhomogeneities can becaused both by the inhomogeneities of the surface and by the inhomogeneous charge distribution upon it. In particular,the case of the .wavy. spatially periodic surface is considered taking into account the possible presence of the surfacecharges.
Keywords : charged fermions, surface, solid and liquid dielectrics, equilibrium distribution of charges and electric field.
PACS : 05.30.Fk, 05.70.Np, 41.20.Cv, 71.10.Ca, 73.20.At.
The problems concerned with the research of thecharges above dielectric surface belong to classical elec-trodynamics and electrostatics problems. A specialinterest to such problems appeared due to the phe-nomenon of the Wigner crystallization. These re-searches were initiated in 1934 by Wigner in his theo-retical work [1], per se. In this work, the possibility ofthe existence of periodic structures in the systems withrepulsive forces between particles was demonstratedby the example of the crystallization of the three-dimensional low-density gas of electrons in the fieldgenerated by the spatial-homogeneous positive charge.This field played exactly a role of the compensative fac-tor for the repulsive forces. The Landau-Silin Fermi-liquid theory also enables to predict the existence ofspatially-periodic state of electrons in metals and todescribe its structure (see in this case Ref. [2]). The ex-perimental improvement of Wigners’ prediction of thethree-dimensional crystallic structures still does not ex-ist (see, e.g., Refs. [3, 4]). This is caused by difficul-ties in the achievement of the experimental conditionsfor the mentioned phenomenon, which is also called as”Wigner crystallization”.However, as it is well known, different two-dimensionperiodic electron structures above the surface of afluid helium are experimentally realized (the so-called”Wigner crystals”). The works [5, 6, 7, 8] may be re-ferred as the first publications containing theoreticaland real experimental results of different properties ofthe surface electrons. The large number of works re-lated to the theoretical and experimental research inthis area has appeared by now. The theoretical papers that are devoted to the mi-croscopic description of the charge state above dielec-tric surface are usually based on the conception ofan isolated charge above dielectric surface interact-ing with its electrostatic reflection in dielectric (”lev-itate electron”, see, e.g., Refs. [3, 4, 9]). In thiscase, the quantum-mechanical state ”charge - electro-static reflection” is described as the hydrogen like one-dimensional state with the corresponding energy struc-ture. Very often the localization of such quantum-mechanical object in the ground state is considered(see Refs [3, 4, 5, 6, 9]) occuring at some distance fromthe surface (first ”Bohr radius”). This, particularly inmost cases, allows not to take into account the influ-ence of the surface inhomogeneity on the single chargestate. However, at the description of many-particlecharge system above dielectric surface the mentionedapproach inevitably faces some difficulties. For exam-ple, such difficulty appears when the electron densityabove dielectric surface does not allow to consider thecharged particles as isolated, i.e., it is necessary to takeinto account the interparticle interaction.The references, which are devoted to the two-dimensional Wigner crystallization in the phenomeno-logical approach, predominantly consider the systemthat consists of a large number of charged particles nearthe surface of the fluid dielectric as a two-dimensionalstructure (see, e.g., Refs.[3, 4, 5, 6, 7, 8]).Basing on the premises, it becomes clear that thecomplete description of charges above dielectric sur-face needs to take into account their spatial distribu-tion in vacuum. The possibility of charges adsorptionby the surface must also be taken into consideration(in this case the surface inhomogeneities must play a1rucial role). The possibility of charge spatial distribu-tion above dielectric surface comes from the fact thata charged particle is always attracted by a dielectricsurface. Moreover, in the experiments [3, 4] concernedwith the registration of two-dimensional Wigner crys-tallization an external electric field attracts charges tothe surface and affects on their spatial distribution.The present paper is devoted to the problem of equi-librium charge distribution above dielectric surface asin the external pressing electric field as in its absence.This problem is considered in the case of ideally planevacuum-dielectric boundary and in the case of ”wavy”spatial-periodic surface with account of the possibilityof existence of the”sticked” to the surface charges. Inour opinion, the formulated problem is interesting asfrom purely academic side as from the research sideof the influence of volume charges located closely tothe fluid helium surface on the spatial-inhomogeneousstate of the charges adsorbed on the helium surface.
Let us consider the equilibrium system of charged par-ticles (Fermi-particles) with the charge Q per particlethat is situated in vacuum above dielectric surface withthe permittivity ε . We describe below the surface pro-file by function ξ ( ρ ) ≡ ξ ( x , y ), where ρ ≡ { x , y } is theradius-vector in the plane z = 0 of Cartesian coordi-nates { x , y , z } . The vacuum - dielectric boundary liesin the plane z = 0 and we consider it unbounded below.All physical quantities considered in the area above di-electric, i.e., at z > ξ ( ρ ) we mark by the index ”1” andall quantities concerned with the dielectric ( z < ξ ( ρ ))we mark by the index ”2”. Let us assume that theexternal pressing electric field E acts on particles andis directed along the z-axis. We also assume the exis-tence of some potential barrier that forbids the chargesto penetrate inside the dielectric.As it is mentioned above, the charged particles arealways attracted by the dielectric. Therefore, even inthe absence of the external pressing electric field thereis a reason to believe that there are conditions underwhat the stable equilibrium distribution along z -axisis developed. To avoid the questions on the repulsionof likely charged particles along the plane ρ we shallconsider the system located in a vessel with the wallsat ρ → ∞ . These walls forbid the charges to leave thesystem. Let us describe the equilibrium charge distri-bution above the dielectric surface by the distributionfunction f ( p ; z, ρ ).The electric field potential ϕ i in vacuum above thedielectric surface must satisfy the Poisson’s equation∆ ϕ ( z, ρ ) = − πQn ( z, ρ ) θ ( z − ξ ( ρ )) , (1) where ∆ is the Laplace operator,∆ ≡ ∂ ∂z + ∆ ρ , ∆ ρ ≡ ∂ ∂x + ∂ ∂y , (2) θ ( z − ξ ( ρ )) is the Heaviside function. In eq. (1) thequantity n ( z, ρ ) is the charge density above the dielec-tric surface, which can be expressed in terms of thedistribution function f ( p ; z, ρ ) as n ( z, ρ ) = Z d pf ( p ; z, ρ ) . (3)As charges are considered as Fermi-particles, the dis-tribution function f ( p ; z, ρ ) has the following form: f ( p ; z, ρ ) = g (2 π ~ ) ×× (cid:26) exp β (cid:20) p m + Qϕ ( z, ρ ) − µ (cid:21) + 1 (cid:27) − , (4)where g = (2 S Q + 1), S Q is the spin of the chargedparticle, β = 1 /T , T is the temperature in the en-ergy units, m is the charge mass and µ is the chemicalpotential of charges. We emphasize that taking intoaccount relations (3) and (4) the eq. (1) is often calledthe Thomas-Fermi equation.The electric field potential ϕ is the result of chargeabsence in the dielectric. In the assumption of thedielectric homogeneity and isotropy it must satisfy theLaplace’s equation ε ∆ ϕ ( z, ρ ) = 0 . (5)If the system is placed in the external static homo-geneous electric field, the potentials ϕ and ϕ can bewritten in the form ϕ = ϕ ( i )1 + ϕ ( e )1 , ϕ = ϕ ( i )2 + ϕ ( e )2 , (6)where ϕ ( i )1 , ϕ ( i )2 are the potentials induced by the sys-tem of charges in vacuum and in the dielectric, re-spectively, ϕ ( e )1 and ϕ ( e )2 are the potentials of the ex-ternal field in vacuum and dielectric. According toeqs. (1), (5), these fields satisfy the following equations:∆ ϕ ( i )1 ( z, ρ ) = − πQn ( z, ρ ) , ∆ ϕ ( e )1 = 0 , ∆ ϕ ( i )2 = 0 , ∆ ϕ ( e )2 = 0 . (7)Eqs. (1), (5) for the potentials must be expandedwith the boundary conditions on the vacuum-dielectricborder (as usual, these conditions can be obtained di-rectly from eqs. (1), (5), see e.g.[10]): ϕ ( z, ρ ) | z = ξ ( ρ ) = ϕ ( z, ρ ) | z = ξ ( ρ ) ,n i ( ρ ) { ε ∇ i ϕ ( z, ρ ) − ε ∇ i ϕ ( z, ρ ) } z = ξ ( ρ ) = 4 πσ ( ρ , ξ ( ρ ′ )) , (8)where n ( ρ ) is the unit vector of the surface normal inthe point ρ , σ ( ρ , ξ ( ρ ′ )) is the surface charge density2n the point ρ (here we emphasize the functional de-pendence of this value on the surface profile ξ ( ρ ′ ) ).The surface charge density σ ( ρ , ξ ( ρ ′ )) must satisfy thefollowing relation: Z dS ξ σ ( ρ , ξ ( ρ ′ )) = QN ξ , (9)where N ξ is the complete charge number on the dielec-tric surface and dS ξ is the surface element with theprofile ξ ( ρ ′ ): dS ξ = d ρ p ∂ξ ( ρ ) /∂ρ ) . (10)The surface charge density can appear due to severalreasons. For example, these charges may be speciallyplaced on the dielectric surface and can stay there forarbitrary long time. At that time, the surface chargesand charges above the dielectric surface can differ insign. But in this case we need to consider the possi-bility of the formation of bound states of the oppositecharged particles. Taking into account the presence ofsuch bound states represents a separate rather com-plicated problem. In the present paper such case ofsurface charges is not considered. The case, in whichsome part of charges condenses on the surface from thevolume distribution, stays for some period of time andthen back to the volume, is possible too. In this case,the equilibrium distribution of charges above the sur-face that coexist with the ”sticked” surface charges forsome period of time (the lifetime of the charge stayingon the surface) is possible. Next, we shall take into ac-count only the possibility of the presence of the surfacecharges that have the same sign as the volume ones.It is easy to see that in eq. (8) the directional cosinesof the surface normal vector n ( ρ ) in the point ρ playthe main role: cos ν is cosine of the angle between thenormal and z-axis, cos λ is cosine of the angle betweenthe normal and x-axis and cos µ is cosine of the anglebetween the normal and y-axis. In the case when thesurface profile is given explicitly (in our case z = ξ ( ρ )),these cosines are determined by the following relations:cos ν = 1 p ∂ξ ( ρ ) /∂ρ ) , cos λ = − ∂ξ ( ρ ) /∂x p ∂ξ ( ρ ) /∂ρ ) , cos µ = − ∂ξ ( ρ ) /∂y p ∂ξ ( ρ ) /∂ρ ) . (11)The derived electrostatics equations (1), (5) with theboundary conditions (8), (11) can be solved analyti-cally in a very low case count. Some of these casesare considered below. Before solving eqs. (1), (5), letus consider the simplification of the boundary condi-tions (8) in the case, when the surface profile differsa little from the plane one. In this case we essen-tially have the effective boundary conditions. From eqs. (10), (11) it is obvious that the surface slightly dif-fers from the plane one, when the surface profile slowlyvaries on coordinate, i.e., when the following inequali-ties take place: | ∂ξ ( ρ ) /∂x | ≪ , | ∂ξ ( ρ ) /∂y | ≪ . (12)Let us also consider that the surface profile ξ ( ρ ) canbe given as: ξ ( ρ ) = ξ + ˜ ξ ( ρ ) , | ξ | ≫ (cid:12)(cid:12)(cid:12) ˜ ξ ( ρ ) (cid:12)(cid:12)(cid:12) . (13)It is easy to see that the inequality (12) is provided inthis case by the conditions (cid:12)(cid:12)(cid:12) ∂ ˜ ξ ( ρ ) /∂x (cid:12)(cid:12)(cid:12) ≪ , (cid:12)(cid:12)(cid:12) ∂ ˜ ξ ( ρ ) /∂y (cid:12)(cid:12)(cid:12) ≪ . (14)The directional cosines (11) with accuracy up to thesecond order over ∂ e ξ ( ρ ) /∂ ρ have the following form:cos ν ≈ , cos λ = − ∂ e ξ ( ρ ) /∂x, cos µ = − ∂ e ξ ( ρ ) /∂y. (15)If the relations (13)-(15) take place, we can expectthat the charge and the field distributions in the systemslightly differ from the distributions that take placein the case of the plane dielectric surface. Then, thepotentials ϕ ( z, ρ ) and ϕ ( z, ρ ) (see eqs. (1), (5)) canbe written as ϕ ( z, ρ ) = ϕ ( z ) + e ϕ ( z, ρ ) ,ϕ ( z, ρ ) = ϕ ( z ) + e ϕ ( z, ρ ) , (16)where ϕ ( z ) and ϕ ( z ) are the potentials of some elec-tric field above the dielectric and inside of it (but noton the surface!) in the case of the plane surface. Thesmall distortions of the field above the dielectric andinside of it are described by the potentials e ϕ ( z, ρ ) and e ϕ ( z, ρ ) due to the surface inhomogeneity in the men-tioned above sense. The meaning of the introduced po-tentials ϕ ( z ) and ϕ ( z ), and also e ϕ ( z, ρ ) and e ϕ ( z, ρ )becomes more clear after the obtaining of the Poisson’sequations and effective boundary conditions for them.According to the assumption of small field pertru-bations provided by the wave surface, the following in-equalities take place: | ϕ ( z ) | ≫ | ˜ ϕ ( z, ρ ) | , | ϕ ( z ) | ≫ | ˜ ϕ ( z, ρ ) | . (17)Let us also consider that the distribution of chargesthat can be condensed on surface slightly differs fromthe homogeneous one: σ ( ρ , ξ ) = σ ( ξ ) + e σ ( ρ , ξ ) + ∂σ ( ξ ) ∂ξ e ξ ( ρ ) , | σ ( ξ ) | ≫ | ˜ σ ( ρ ; ξ ) | , | σ ( ξ ) | ≫ (cid:12)(cid:12)(cid:12)(cid:12) ∂σ ( ξ ) ∂ξ ˜ ξ ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12) . (18)In the expressions (18) the quantity e σ ( ρ , ξ ) correspondsto the impact of the weakly inhomogeneous charge dis-tribution on the plane dielectric surface with z = ξ pro-file. The quantity e ξ ( ρ )[ ∂σ ( ξ ) /∂ξ ] in eq. (18) describes3he surface charge inhomogeneity related to the weakirregularity of the surface itself.In the expressions (16), (17) we consider that inthe case of uniformly charged surface, which is ideallyplane and infinitely extended with charge density σ ( ξ )both the field and the charge distributions are homoge-neous along ρ plane. In other words, the spatial chargeand field distribution depends only on z coordinates.From the relations (13)-(18) it is easy to obtain theeffective boundary conditions for field potentials onvacuum-dielectric boundary in the case, when the di-electric surface weakly differs from the ideally plane.To this end, we must develop the perturbation theoryover small values e ξ ( ρ ), e σ ( ρ , ξ ) and ∂ e ξ ( ρ ) /∂ ρ , which,according to the expressions (12)-(18), can be given asfollows: (cid:26) ϕ ( z ) + e ϕ ( z, ρ ) (cid:27) ξ + e ξ ( ρ ) = (cid:26) ϕ ( z ) + e ϕ ( z, ρ ) (cid:27) ξ + e ξ ( ρ ) , (cid:26) ε ∂∂z [ ϕ ( z ) + e ϕ ( z, ρ )] − ∂∂z [ ϕ ( z ) + e ϕ ( z, ρ )] (cid:27) ξ + e ξ ( ρ ) = 4 πσ ( ρ ; ξ + e ξ ( ρ )) . (19)Making the necessary calculations up to the first orderof the perturbation theory from the first relation ofeq. (19) we obtain ϕ ( z ) | z = ξ = ϕ ( z ) | z = ξ , (cid:26) e ϕ ( z, ρ ) − e ϕ ( z, ρ ) , (cid:27) z = ξ == (cid:26) ∂ϕ ( z ) ∂z − ∂ϕ ( z ) ∂z (cid:27) z = ξ e ξ ( ρ ) . (20)The use of the perturbation theory up to the first or-der for the second relation of eq. (19) results in thefollowing equalities: (cid:26) ε ∂ϕ ( z ) ∂z − ∂ϕ ( z ) ∂z (cid:27) z = ξ = 4 πσ ( ξ ) , (cid:26) ε ∂ ϕ ( z ) ∂z − ∂ ϕ ( z ) ∂z − π ∂σ ( ξ ) ∂ξ (cid:27) z = ξ e ξ ( ρ ) −− π e σ ( ρ ; ξ ) = (cid:26) ∂ e ϕ ( z, ρ ) ∂z − ε ∂ e ϕ ( z, ρ ) ∂z (cid:27) z = ξ . (21)Let us remind that in electrostatics the denotations like { ∂ ϕ ( z ) /∂z } z = ξ , { ∂ ϕ ( z ) /∂z } z = ξ have the mean-ing of limits { ∂ ϕ ( z ) /∂z } z = ξ = lim h → { ∂ ϕ ( z ) /∂z } z = ξ + h , { ∂ ϕ ( z ) /∂z } z = ξ = lim h → { ∂ ϕ ( z ) /∂z } z = ξ − h . Then, for the further simplification of the obtained ef-fective boundary conditions (20), (21), according toeqs. (1), (5), (16) we can use the following equations,which are satisfied by the potentials ϕ ( z ), ϕ ( z ): ∂ ∂ ϕ ( z ) = − πQn ( z ) θ ( z − ξ ) , ∂ ∂ ϕ ( z ) = 0 , (22) where n ( z ) = Z d pf ( p , z ) ,f ( p ; z ) = g (2 π ~ ) (cid:26) exp β (cid:20) p m + Qϕ ( z ) − µ (cid:21) + 1 (cid:27) − . (23)Taking into account eq. (21), the conditions (20) can beexpressed in the following form (the first one of themremains the same): (cid:26) ε ∂ϕ ( z ) ∂z − ∂ϕ ( z ) ∂z (cid:27) z = ξ = 4 πσ ( ξ ) , π (cid:26) Qn ( z ) − ∂σ ( ξ ) ∂ξ (cid:27) z = ξ e ξ ( ρ ) − π e σ ( ρ ; ξ ) == (cid:26) ∂ e ϕ ( z, ρ ) ∂z − ε ∂ e ϕ ( z, ρ ) ∂z . (cid:27) z = ξ (24)Thus, we obtain the effective boundary condi-tions (20), (24) for the fields in the system of chargesabove the dielectric surface with the surface profile thatslightly differs from plane ∂ e ϕ ( z, ρ ) ∂z + ∆ ρ e ϕ ( z, ρ ) = 4 πQ ∂n ( z ) ∂µ e ϕ ( z, ρ ) ,∂ e ϕ ( z, ρ ) ∂z + ∆ ρ e ϕ ( z, ρ ) = 0 . (25) It is easy to see that the obtained equations (22), (23)of electrostatics and the effective boundary condi-tions (20), (24) for these equations are much simplerthan the initial electrostatic equations (1), (5) and theboundary conditions (8). Firstly, to solve the equa-tions that determine the charge and the field distribu-tion above the vacuum-dielectric boundary one needsconsider the case of ideally plane surface of this borderthat lies in z = ξ . Let us start withe considering thecase of the surface in the absence of the charge σ = 0.Then, the solution of (22) must satisfy the followingboundary conditions: ϕ ( z ) | z = ξ = ϕ ( z ) | z = ξ , (cid:26) ε ∂ϕ ( z ) ∂z − ∂ϕ ( z ) ∂z (cid:27) z = ξ = 0 . (26)To simplify the further calculations, let us write thefirst formula from eq. (22) in the following form: ∂ϕ ( z ) ∂z = − πQν ∞ Z dεε / { exp β ( ε − ψ ) + 1 } , (27)where we denote ψ ( z ) ≡ µ − Qϕ ( z ) , ν ≡ (2 m ) / / π ~ . (28)4ere we also consider the spin of a charged particleequal to 1 / ψ is so-called electrochemical potential.Multiplying eq. (27) by the derivative ( ∂ϕ ( z ) /∂z )and using the following equality (cid:18) ∂ϕ ∂z (cid:19) e β ( ε − ψ ) + 1 = − βQ ∂∂z ln (cid:20) e − β ( ε − ψ ) + 1 (cid:21) , after simple calculations we obtain the first-order dif-ferential equation: (cid:18) ∂ϕ ∂z (cid:19) = 16 π ν ∞ Z dεε / { e β ( ε − ψ ) + 1 } − + C, where C is an arbitrary integration constant. Thus,the need of the following equation solving arises: ∂ϕ ∂z = ± (cid:26) π ν ∞ Z dεε / { e β ( ε − ψ ) + 1 } − + C (cid:27) / . (29)The sign before the square root in eq. (29) must be cho-sen from the following consideration. The force actingon the charges at z > ξ presses these charges to thedielectric surface. Thus, in the case of positive chargesabove the dielectric we choose the positive sign, andin the case of negative charges we choose the negativeone. Let us consider below the distribution of negativecharges above the dielectric surface, Q = − e , e > ϕ satisfies the relation: ∂ϕ ∂z = − (cid:26) π ν ∞ Z dεε / { e β ( ε − ψ ) + 1 } − + C (cid:27) / . (30)Now we make the following denotations: ϕ ( z = 0) ≡ ϕ , ψ ( z = 0) ≡ µ + eϕ ( z = 0) ,E ≡ − (cid:18) ∂ϕ ( z ) ∂z (cid:19) z =0 (31)Let us remind that we consider the case of the elec-tric forces that attract charges to the dielectric surface.Thus, at z → ∞ there is no charges, f ( p ; z ) → z →∞
0, or { exp β ( ε − ψ ) + 1 } − → z →∞ . (32)The action of the electrostatic image force along z-axismust vanish at z → ∞ : ∂ϕ ( i )1 ( z ) ∂z → z →∞ . As the result, it is essential to say that at z → ∞ thefollowing relation takes place: − ∂ϕ ( z ) ∂z → z →∞ − ∂ϕ ( e )1 ( z ) ∂z ≡ E, (33)where E is the external field intensity that attractscharges to the dielectric surface. At z = 0 from eq. (30) one can get E = 16 π ν ∞ Z dεε / { exp β ( ε − ψ ) + 1 } − + C. On the other hand, from the same equation and takinginto account eqs. (32), (33) at z → ∞ we obtain: C = E . (34)Comparing the last two expressions, we come to therelation between the constants ψ , E (see eq. (31))and the external electromagnetic field E : E − E = 16 π ν ∞ Z dεε / { exp β ( ε − ψ ) + 1 } − . (35)Then, after integration of the first expression fromeq. (23) over z within the limits from ξ to ε and us-ing eqs. (32), (34), we get: E − E = 4 πen s , e > , (36)where n s is the number of the volume charges per unitof the plane dielectric surface: n s = ∞ Z ξ dzn ( z ) ,n ( z ) = ν ∞ Z dεε / { exp β ( ε − ψ ( z )) + 1 } − . (37)Let us emphasize that for the equilibrium charge sys-tem above dielectric the value of the number n s de-pends neither on the coordinates, neither on the fields’distribution. It is determined only by the entire num-ber N of the charges above dielectric. We also pointout that this value characterizes the additional fieldintensity that presses the charges to the dielectric sur-face. Besides that, this field is generated by the chargesthemselves.Thus, eqs. (35), (36) allow to express the unknownquantities ψ and E (integration constants of eq. (27))in terms of the external pressing electric field E and thenumber of charges above the unit item of the dielectricsurface n s (see eq. (37)).The second equation in (38) can be solved triviallyin general case, because the electric field intensity indielectric does not depend on z . Using the boundaryconditions (26), we can express the potential of electricfield in dielectric in the following form: ϕ = − E ε z + ϕ , E = E ε , (38)where E is the electric field intensity in the dielectric, E can be expressed from eqs. (35), (36) and ϕ is thepotential on the surface. As in the electrostatic case,5he field potential is determined accurate within a con-stant. Therefore, we set the potential ϕ equal to zerobelow. In this case, the value of the electrochemicalpotential on the dielectric surface coincides with thechemical one: ψ ( z = 0) ≡ ψ = µ, ϕ = 0 . (39)Taking into account eq. (34), the spatial distributionof the potential (see eq. (30)) can be written as follows: ∂ϕ ∂z = − π ν ∞ Z dεε / ××{ exp β ( ε − ψ ) + 1 } − + E (cid:27) / ,ψ ( z ) ≡ µ + eϕ ( z ) . (40)It is easy to see that in general case the solution of thisequation can be found only in quadratures (see below).However, the gas of charged Fermi-particles above thedielectric surface is nondegenerate. Therefore, the so-lution of eq. (40) can be obtained analytically. Indeed,in the case of nondegenerate gas its distribution func-tion has the form that weakly differs from Boltzmans’one { exp β ( ε − ψ ) + 1 } − ∼ exp β ( ψ − ε ) . Accordingly, the expression for the density distributionof a gas along the z coordinate (see eq. (37)) becomes: n ( z ) ≈ √ π νβ − / exp( βψ ) . (41)As the Fermi-particle gas is degenerate at low tem-perature and high density ranges (see e.g. [11]) fromeq. (41) one can get the gas nondegeneracy condition:exp( βψ ) ≪ , As the electrochemical potential depends on z, this con-dition is obviously realized in the case when the follow-ing inequality takes place:exp( βψ ) ≪ , (42)where ψ is the electrochemical potential on the dielec-tric surface (see eqs. (28), (39) in this case). The laststatement takes place because of the assumption of theparticle absence at z → ∞ , see above. So, accordingto eq. (41), (42), the formula (40) can be expressed as: ∂ψ∂z = − n π / e β − / ν exp( βψ ) + e E o / . (43)This equation has the analytical solution: √ π νβ − / exp ( βψ ( z )) = β E π χ ( z )(1 − χ ( z )) ,ψ ( z ) ≡ µ + eϕ ( z ) , (44) where the function χ ( z ) is defined by the relation: χ ( z ) ≡ E − EE + E exp {− ( z − ξ ) /z } ,z ≡ ( βeE ) − , β − = T. (45)Let us emphasize that the multiplier before the expo-nent in eq. (45) according to eq. (36) can be expressedin terms of the intensity of the external electric fieldand the number of charges n s in the ”column” abovethe surface unit element: E − EE + E = 2 πen s E + 2 πen s From the eqs. (41), (43) and (44) follows that thecharge density above the dielectric surface has the dis-tribution: n ( z ) = β E π χ ( z )(1 − χ ( z )) , (46)and the electric field intensity above the dielectric E ( z ) is expressed as E ( z ) = E χ ( z )1 − χ ( z ) . (47)It is easy to see that at high values of z , z ≫ z (seeeq. (45)), the charge distribution above the dielectricsurface is close to the Boltzman distribution and theelectric field density exponentially tends to the externalpressing electric field density. This fact confirms theabove assumptions(see eqs. (32), (33)).The inequality (42) that determines the nondegener-acy condition of the charge gas can be written in termsof the obtained solutions: en s ν − β / ( E + 2 πen s ) ≪ . (48)It is obvious that this inequality is not accomplishedin the case of low temperature range or high values ofthe external pressing field.Expressions (44)- (48) allow to make the limit pro-cess at E →
0. In the case of the absence of the externalpressing field, these solutions have the following form: E ( z ) → E → E (cid:26) z − ξ z (cid:27) − ,n ( z ) → E → β E π (cid:26) z − ξ z (cid:27) − , E = E /ε, (49)where z ≡ ( βeE ) − , E = 4 πen s . (50)If to compare the expressions (46), (47) and (49), itis easy to see that in the case of the absence of theexternal pressing field the exponential law of the elec-tric field and charge density above the dielectric surfacechanges to the weaker power depedence. In this case,the inequality (48) can be written as:( en s ) ν − β / ≪ . (51)6ote, that it takes place in the region of the relativelyhigh temperatures and low charge number in the vol-ume above a surface area unit, see eq. (37).The obtained formulae (38), (44)-(51) are the so-lution of the problem of the field and nondegeneratecharged gas distribution in charged particle systemabove the plane dielectric surface as in the externalpressing field as in its absence. Let us emphasize thatthe dielectric permittivity does not appear in these ex-pressions. The reason is that the problem is homo-geneous along the surface coordinate ρ . In the case ofinhomogeneity along ρ , the solution of equations essen-tially depends on the sort of the dielectric, i.e., on itspermittivity ε . These inhomogeneities may be causedby the inhomogeneities of the surface itself or by inho-mogeneity charge distribution on it (or the both rea-sons simultaneously, see eqs. (8), (11), (20), (24)).In the case of the degenerate gas, i.e., when the con-dition (48) or (51) fails, the solution that is obtainedearlier is inapplicable. Let us make the following re-mark relating to this fact. As it is mentioned earlier,the charge density distribution decreases with the dis-tance from the surface. For this reason, in general casedescribed by eq. (40) the gas can be degenerate in thearea near the dielectric surface and nondegenerate farfrom it. The typical distance from the surface that sep-arates these cases can be obtained using the followingconsiderations. As it is well known (see e. g. Ref. [11]),in low temperatures region the temperature expansionsare widely used for the calculus of the thermodynam-ical quantities characterizing the gas. Applying suchexpansion to the integral over the energy in eq. (40),we obtain: ∞ Z dεε / { exp β ( ε − ψ ) + 1 } − ≈≈ ψ / + π β − ψ / − π β − ψ − / + ... (52)From this expression it is easy to see that such expan-sion is absolutely useless near the point z obtainedfrom the condition ψ ( z ) = µ + eϕ ( z ) = 0 . (53)The solution of eq. (40) obtained in quadratures isgiven by z − ξ = − z βψ Z βψ dζ (cid:26) π νβ − / E − ∞ Z dyy / ××{ exp ( y − ζ ) + 1 } − + 1 (cid:27) − / , (54)where the distance z is determined by eq. (45). Tak-ing into account eqs. (53), (54), the expression of the border distance z can be written as z = ξ + z βψ Z dζ π νβ − / E − ∞ Z dyy / ××{ exp ( y − ζ ) + 1 } − + 1 (cid:27) − / , (55)where the electrochemical potential ψ as the functionof temperature T = β − and the external electric fieldis obtained from the equation (see eqs. (35), (36), (39))(4 πen s + E ) − E == 16 π ν ∞ Z dεε / { exp β ( ε − ψ ) + 1 } − . (56)As it is mentioned above, the potential ϕ ( z ) is definedaccurate within an arbitrary constant, which can be setequal to zero. Hence, eq. (56) is the expression definingthe chemical potential µ , see eq. (39).As expected, the typical distance z (see eq. (55)) isdefined by the temperature, the external pressing fieldand the number of charges above the dielectric surfacearea unit. Thus, the charge gas is nondegenerate in theregion z ≫ z and degenerate at z ≪ z . Let us pointout that the solutions (45)-(50) are obtained assumingthe charge gas nondegeneracy in the entire area abovethe dielectric surface. Therefore, in general case thementioned expressions describe the charge system onlyin the region z ≫ z . The charge gas above the dielec-tric surface can be degenerate even in the case of theabsence of the external pressing field. It is easy to seeif to analyze the expression (55) at E → ∂ϕ ∂z = − (cid:26) π νψ / + E (cid:27) / ,ψ ( z ) ≡ µ + eϕ ( z ) . (57)However, in this case eq. (57) can not be solved an-alytically, and the numerical integration methods areneeded.Let us show now the influence of the plane dielec-tric surface charges on the obtained results in thepresent section of this paper. It is well known that in-finitely thin homogeneously charged plate with chargedensity σ ( ξ ) induces the homogeneous field intensity E σ = 2 πσ ( ξ ) in vacuum (in this case the expression σ ( ξ ) shows that the surface plane is described by theequation z = ξ ). This field has the opposite directionin the opposite sides of the plane. In the case of thecharged plane dielectric surface the situation is abso-lutely similar. E.g., the negatively charged dielectricsurface induces the field intensity E σ = 2 π | σ ( ξ ) | /ε in7he dielectric and E σ = − π | σ ( ξ ) | above the dielec-tric surface (see eq. (24)). As mentioned earlier in thepresent paper, we consider only the cases of the samesigns of charges as on the dielectric surface, as in thevolume above it (in our case we consider the negativecharges). In this case, the field induced by the surfacecharges repulses the volume charges from the surface.So, the results obtained in the present partition remainuseful if we substitute the external electric field in vac-uum E for E − π | σ ( ξ ) | in the expressions (31)-(56), E → E − π | σ ( ξ ) | . (58)It is easy to see that it is necessary to satisfy the con-dition E − π | σ ( ξ ) | > As already mentioned, the spatial inhomogeneities canbe caused by the surface heterogeneities or by the inho-mogeneous charge distribution on it (or the both rea-sons simultaneously, see eqs. (8), (11), (20), (24))). Letus consider the mentioned surface inhomogeneities thatslightly distort the electric field induced by the chargesystem above the plane dielectric: ϕ ( z, ρ ) = ϕ ( z ) + ˜ ϕ ( z, ρ ) , | ϕ ( z ) | ≫ | ˜ ϕ ( z, ρ ) | ,ϕ ( z, ρ ) = ϕ ( z ) + ˜ ϕ ( z, ρ ) , | ϕ ( z ) | ≫ | ˜ ϕ ( z, ρ ) | , where ϕ ( z ), ϕ ( z ) are the potentials above the dielec-tric and inside of it, respectively, in the case of theideally plane dielectric surface with the equation of theprofile z = ξ (see eqs. (16)- (18)). The obtaining proce-dure for the potentials ϕ ( z ), ϕ ( z ) and charge density n ( z ) is described in details in the previous section ofthe present paper, see eqs. (26)- (59).The next problem is concerned with the potentials˜ ϕ ( z, ρ ) and ˜ ϕ ( z, ρ ) obtaining. For these potentialsone can use the eq. (25) and the boundary condi-tions (20), (24). In terms of the Fourier-transforms˜ ϕ ( z, q ) , ˜ ϕ ( z, q ) over coordinate ρ of the potentials˜ ϕ ( z, ρ ) and ˜ ϕ ( z, ρ )˜ ϕ ( z, ρ ) = Z d q exp ( i q ρ ) ˜ ϕ ( z, q ) , ˜ ϕ ( z, ρ ) = Z d q exp ( i q ρ ) ˜ ϕ ( z, q ) (60)the equations (25) have the following form: ∂ ˜ ϕ ( z, q ) ∂z − q ˜ ϕ ( z, q ) = 4 πe ∂n ( z ) ∂µ ˜ ϕ ( z, q ) ,∂ ˜ ϕ ( z, q ) ∂z − q ˜ ϕ ( z, q ) = 0 . (61) According to eqs. (20), (24), the boundary conditionsconcerned with these equations can be written as: { ˜ ϕ ( z, q ) − ˜ ϕ ( z, q ) } z = ξ == (cid:26) ∂ϕ ( z ) ∂z − ∂ϕ ( z ) ∂z (cid:27) z = ξ ˜ ξ ( q ) , − π (cid:26) en ( z ) + ∂σ ( ξ ) ∂ξ (cid:27) z = ξ ˜ ξ ( q ) − π ˜ σ ( q ; ξ ) == (cid:26) ∂ ˜ ϕ ( z, q ) ∂z − ε ∂ ˜ ϕ ( z, q ) ∂z (cid:27) z = ξ , (62)where ˜ ξ ( q ), ˜ σ ( q ; ξ ) are the Fourier-transforms of thequantities ˜ ξ ( ρ ) and ˜ σ ( ρ ; ξ ), respectively (see eqs. (14),(18)): ˜ ξ ( ρ ) = Z d q exp ( i q ρ ) ˜ ξ ( q ) , ˜ σ ( ρ ; ξ ) = Z d q exp ( i q ρ ) ˜ σ ( q ; ξ ) . (63)Let us consider the electric fields intensity perturba-tions caused by inhomogeneities of the dielectric sur-face rapidly decreasing at z → ±∞ . It is easy to seethat the first equation in eq. (61) in general case can-not be solved analytically. But in two particular casesthe analytical solution exists. In the first case, we solveeq. (61) at z ∼ ξ setting ∂n ( z ) /∂µ equal to its valueon the plane surface, z = ξ : ∂n ( z ) ∂µ ≈ ∂n ( ξ ) ∂µ . (64)Such consideration is possible in the case when the typ-ical size of spatial inhomogeneities of the unperturbedcharge density n ( z ) considerably larger than the typi-cal size of the spatial inhomogeneities of the potential˜ ϕ ( z, q ) along z -axis: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂n ( z ) ∂µ (cid:19) − ∂∂z ∂n ( z ) ∂µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = ξ ≪≪ (cid:12)(cid:12)(cid:12)(cid:12) { ˜ ϕ ( z, q ) } − ∂ ˜ ϕ ( z, q ) ∂z (cid:12)(cid:12)(cid:12)(cid:12) z = ξ . (65)Let us return to the discussion of the condition (64)below.Then, taking into account the assumption of rapidlyfading field densities at z → ±∞ , the solution ofeq. (60) can be given in the following form:˜ ϕ ( z, q ) = A ( q ) exp ( − zb ( q )) , ˜ ϕ ( z, q ) = A ( q ) exp ( zq ) , (66)where (see eq. (64)) b ( q ) ≡ s q + 4 πe ∂n ( ξ ) ∂µ , (67)and A ( q ), A ( q ) are obtained from the boundary con-ditions (62). To this end, we put the expressions (66)8nto the boundary conditions (62) and obtain the fol-lowing relations for the potentials ˜ ϕ ( z, q ), ˜ ϕ ( z, q ):˜ ϕ ( z, q ) = exp ( − ( z − ξ ) b ( q )) εq + b ( q ) { [ εq ( E ( ξ ) − E ( ξ )) ++4 π ( en ( ξ ) + ( ∂σ ( ξ ) /∂ξ ))] ˜ ξ ( q ; ξ ) + 4 π ˜ σ ( q ; ξ ) o , ˜ ϕ ( z, q ) = − exp (( z − ξ ) q ) εq + b ( q ) { [ b ( q ) ( E ( ξ ) − E ( ξ )) −− π ( en ( ξ ) + ( ∂σ ( ξ ) /∂ξ ))] ˜ ξ ( q ; ξ ) − π ˜ σ ( q ; ξ ) o , (68)where (see eqs. (22), (26)) E ( ξ ) ≡ − (cid:18) ∂ϕ ( z ) ∂z (cid:19) z = ξ , E ( ξ ) ≡ − (cid:18) ∂ϕ ( z ) ∂z (cid:19) z = ξ . Let us emphasize that according to the boundary con-ditions (20), (21) (see also eqs. (58), (59)) the valuesof the quantities E ( ξ ), E ( ξ ) can be expressed as E ( ξ ) = E − π | σ ( ξ ) | > ,E ( ξ ) = ( E + 2 π | σ ( ξ ) | ) /ε, (69)where the field intensity E is defined by the rela-tion (36): E = E + 4 πen s . Let us remind that the values of the potentials˜ ϕ ( z, q ), ˜ ϕ ( z, q ) at z = ξ do not coincide due to thefact that the potential continuity on the surface in thecase of its inhomogeneous wavy structure is providedby the inequalities (see eqs. (16)-(20)): ϕ ( z ) | z = ξ = ϕ ( z ) | z = ξ , δϕ ( ξ, q ) = δϕ ( ξ, q ) , where δϕ ( ξ, q ) ≡ ˜ ξ ( q ) ( ∂ϕ ( z ) /∂z ) z = ξ + ˜ ϕ ( ξ, q ). Ac-cording to eq. (68), one can get: δϕ ( ξ, q ) = − εq + b ( q ) (cid:26) [ εqE ( ξ ) + b ( q ) E ( ξ ) −− π ( en ( ξ ) + ( ∂σ ( ξ ) /∂ξ ))] ˜ ξ ( q ; ξ ) − π ˜ σ ( q ; ξ ) (cid:27) , (70)where E ( ξ ), E ( ξ ) are still defined by the relations(69). It is easy to see from the obtained formulae (68),(70) that the gas of the volume charges can sufficientlyaffect on the potential of the electric field near the di-electric surface.Now let us show that the solution of eq. (61) in theforms (66), (68) is correct. As it is mentioned above,the condition of the existence of such solution is definedby the relation (65). According to eq. (67) it can beexpressed as follows: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂n ( ξ ) ∂µ (cid:19) − ∂∂z ∂n ( ξ ) ∂µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ s q + 4 πe ∂n ( ξ ) ∂µ . (71)The explicit expression for the derivative ∂n ( ξ ) /∂µ can be obtained from eqs. (31), (35), (36), (41), (46), (52), (56) as in the case of degenerate charge gas abovethe dielectric surface, as in the case of nondegenerateone. In the second case the condition (71) has a rathersimple form: q ≫ β e n ( E + 2 πen s ) + 4 π e n s o . (72)In the case, when the gas of charged Fermi-particles isdegenerate at z ≪ z (see eq. (55)) and low tempera-ture expansions (52) take place, we can obtain the fol-lowing expressions for the volume charge density n ( z )at z ∼ ξ and the electrochemical potential ψ at z = ξ (see eqs. (35), (52), (56)): n ( z ) ≈ νψ / , ψ ≈ (cid:26) E − E πν (cid:27) / , (73)where ν and ψ are still defined by the relations (28)with Q = − e , and E , E are expressed by (36). Bythe use of the expression (73), one can write the con-dition (71): E − πen s E − π e n s ≪≪ q e − (cid:26) en s ( E + 2 πen s ) ν (cid:27) / . (74)Accounting the solutions (68) are the Fourier-transforms of the potentials ˜ ϕ ( z, ρ ), ˜ ϕ ( z, ρ ) (seeeq. (60)), the relations (71)-(74) in general case are cor-rect for any value of q , including also the value q = 0.It is easy to see that the relation (72) does not satisfysuch a requirement. The relation (74) can take placeat all values of q in the case of the external pressingfield E that satisfies the following inequality:0 ≤ E ≤ E ′ , E ′ ≈ πen s . (75)At E > E ′ the expressions (68) do not take place. Inthis case, as in the case of the condition (72) realiza-tion, the equations (61) must be solved by the use ofthe numerical methods.The case of the particular interest is the spatially pe-riodic inhomogeneities caused by the dielectric surface.As it is already mentioned in the present paper, suchinhomogeneities are concerned with two-dimensionalWigner crystallization. In the most simple case of spa-tial periodic inhomogeneities the Fourier-transforms ofthe quantities ˜ ξ ( ρ ), ˜ σ ( ρ ; ξ ) (see eq. (63)) can be ex-pressed in the form:˜ σ ( q ; ξ ) = 12 X α =1 ˜ σ ( q ασ ; ξ ) { δ ( q + q ασ ) + δ ( q − q ασ ) } , ˜ ξ ( q ) = 12 X α =1 ˜ ξ ( q αξ ) { δ ( q + q αξ ) + δ ( q − q αξ ) } , (76)where q ασ ( α = 1 ,
2) are the vectors of the recipro-cal two-dimensional lattice concerned with the spatial9eriodic charge distribution on the dielectric surface, q αξ ( α = 1 ,
2) are the vectors of the reciprocal two-dimensional lattice concerned with the spatial periodicwavy surface type, and ˜ σ ( q ασ ; ξ ) , ˜ ξ ( q αξ ) are the am-plitudes of the corresponding surface heterogeneities.Of course, it is necessary to consider that the condi-tions (14) take place, which in this case can be writtenas: q αξ ˜ ξ ( q αξ ) ≪ , q ασ ˜ ξ ( q ασ ) ≪ . (77)Then, putting the expressions (76) into eq. (68) andmaking inverse Fourier transformation according toeq. (60), it is easy to obtain the following expressionsfor the potentials ˜ ϕ ( z, ρ ), ˜ ϕ ( z, ρ ):˜ ϕ ( z, ρ ) = X α =1 exp ( − ( z − ξ ) b ( q αξ )) εq αξ + b ( q αξ ) (cid:26) εq αξ ( E ( ξ ) −− E ( ξ )) + 4 π ( en ( ξ ) + ∂σ ( ξ ) /∂ξ ) (cid:27) ˜ ξ ( q αξ ) cos ( q αξ ρ ) ++4 π X α =1 exp ( − ( z − ξ ) b ( q ασ )) εq ασ + b ( q ασ ) ˜ σ ( q ασ ; ξ ) cos ( q ασ ρ ) , ˜ ϕ ( z, ρ ) = − X α =1 exp (( z − ξ ) q αξ ) εq αξ + b ( q αξ ) (cid:26) b ( q αξ ) ( E ( ξ ) −− E ( ξ )) − π ( en ( ξ ) + ∂σ ( ξ ) /∂ξ ) (cid:27) ˜ ξ ( q αξ ) cos ( q αξ ρ ) ++4 π X α =1 exp (( z − ξ ) q ασ ) εq ασ + b ( q ασ ) ˜ σ ( q ασ ; ξ ) cos ( q ασ ρ ) . (78)The obtained expressions represent the solution of thepotential distribution problem (so, the charge densitydistribution, too) in the area near the dielectric sur-face with the weak (see eq. (77)) spatially periodic in-homogeneities. Let us emphasize that for the expres-sions (78) validity it is no longer necessary to satisfythe conditions (71) for all values of q . Its sufficiency isprovided by the accomplishment of the conditions (71)(or eqs. (72), (74)) for two-dimensional reciprocal lat-tice distances q ασ , q αξ .Let us remind that we consider the simplest type ofthe spatial periodic inhomogeneities related to the di-electric surface. In the case of more complicated struc-ture of spatial-periodic homogeneities it is necessary touse the coefficients of two-dimensional Fourier seriesexpansion for ˜ ξ ( ρ ) and ˜ σ ( ρ ; ξ ), which describe theseinhomogeneities. Thus, the problem of equilibrium state of the chargedparticles above the dielectric surface is solved. Equi-librium distributions for the charge and electric fieldinduced by these charges in the system are obtainedas in the case of ideally plane dielectric surface, asin the case of weak spatial inhomogeneities concerned with the dielectric surface. The weak spatial inhomo-geneities caused as by the inhomogeneities of the sur-face itself, as by inhomogeneous charge distributions onit are taken into account. The case of ”wavy” surface,in particular, the spatially periodic one, is concernedtaking into account the possibility of the surface chargepresence on it. The influence of the external pressingelectric field acting on the system is also taken intoconsideration. It is shown that the presence of the gasof volume charges essentially influences on the value ofthe electric field potential in the area near the dielec-tric surface. Mostly, this fact plays an important rolein the description of the deformation of the liquid di-electric surface caused by near-surface charges pressureon it. Authors of the present paper are working on thisproblem now.However, in our opinion, the solved problem is use-ful not only in two-dimensional Wigner crystallizationaspect. The problem is worth concerning purely withthe academic purposes as it can be related to the num-ber of classical problems of electrodynamics and sta-tistical physics. Due to this fact, in this paper wedo not use the results of the real experiments on two-dimensional Wigner crystallization research. The for-mulations and obtained results in this paper can beused for the research of the influence of the volumecharges near the liquid helium surface on the spatialinhomogeneous states of charges, which are adsorbedon the helium surface at the system parameters closeto the experimental ones.
Acknowledgements
Authors acknowledge financial support from the Con-solidated Foundation of Fundamental Research ofUkraine under grant No. 25.2/102 and thank S.V.Peletminsky for valuable discussions.
References [1]
E. Wigner.
On the interaction of electrons in met-als. - Phys. Rev., , no.11, p. 1002, (1934).[2] A.S. Peletminsky, S.V. Peletminsky, Yu.V.Slusarenko.
On phase transitions in a Fermi-liquid.II Transition assotiated with translational symme-try breaking. - Low Temp. Phys., , no.5, p. 303,(1999).[3] Yu.P. Monarkha and V.B. Shikin.
Low-dimensional electronic systems on a liquidhelium surface (Review). - Sov. J. Low Temp.Phys. , no. 6, p. 279, (1982).[4] Y. Monarkha and K. Kono.
Two-dimensionalCoulomb liquids and solids. - Berlin: Springer -Verlag, p. 346, (2003).105]
M.W. Cole, M.H. Cohen.
Image-potential-induced surface bands in insulators. - Phys. Rev.Lett., , no. 21, p. 1238 , (1969).[6] V.B. Shikin.
On helium ions motion near vapour-liquid boundary. - JETP, , no. 5, p. 1748,(1970)[in Russian].[7] T.R. Brown, C.C. Grimes.
Observation of cy-clotron resonance in surface-bound electrons inliquid helium. - Phys. Rev. Lett., , no. 18, p.1233, (1972).[8] C.C. Grimes, T.R. Brown.
Direct spectroscopyobservation of electrons in image - potential stateoutside liquid helium. - Phys. Rev. Lett., , no.6,p. 280, (1974).[9] V.S. Edelman.
Levitated Electrons. - Sov. Phys.Usp. no.4, p. 227,(1980).[10] I.E. Tamm.
Fundamentals of the theory of elec-tricity, Central Books Ltd, 684 pages, (1980).[11]