We analyze, in both plane and cylindrical geometries, a collisionless plasma
consisting of an inner region where generation occurs by UV illumination, and
an un-illuminated outer region with no generation. Ions generated in the inner
region flow outwards through the outer region and into a wall. We solve for
this system's steady state, first in the quasi-neutral regime (where the Debye
length ${\lambda}_D$ vanishes and analytic solutions exist) and then in the
general case, which we solve numerically. In the general case a double layer
forms where the illuminated and un-illuminated regions meet, and an
approximately quasi-neutral plasma connects the double layer to the wall
sheath; in plane geometry the ions coast through the quasi-neutral section at
slightly more than the Bohm speed $c_s$. The system, although simple, therefore
has two novel features: a double layer that does not require counter-streaming
ions and electrons, and a quasi-neutral plasma where ions travel in straight
lines with at least the Bohm speed. We close with a pr\'{e}cis of our
asymptotic solutions of this system, and suggest how our theoretical
conclusions might be extended and tested in the laboratory.Comment: 10 pages, 3 figures, accepted by Physics of Plasma

Allen, J. E.

Benilov, M. S.

Franklin, R. N.

Thomas, D. M.

English

arXiv

We analyze, in both plane and cylindrical geometries, a collisionless plasma
consisting of an inner region where generation occurs by UV illumination, and
an un-illuminated outer region with no generation. Ions generated in the inner
region flow outwards through the outer region and into a wall. We solve for
this system's steady state, first in the quasi-neutral regime (where the Debye
length ${\lambda}_D$ vanishes and analytic solutions exist) and then in the
general case, which we solve numerically. In the general case a double layer
forms where the illuminated and un-illuminated regions meet, and an
approximately quasi-neutral plasma connects the double layer to the wall
sheath; in plane geometry the ions coast through the quasi-neutral section at
slightly more than the Bohm speed $c_s$. The system, although simple, therefore
has two novel features: a double layer that does not require counter-streaming
ions and electrons, and a quasi-neutral plasma where ions travel in straight
lines with at least the Bohm speed. We close with a pr\'{e}cis of our
asymptotic solutions of this system, and suggest how our theoretical
conclusions might be extended and tested in the laboratory.info:eu-repo/semantics/publishedVersio

Repositório Digital da Universidade da Madeira

Phys. Plasmas 20, 123508 (2013); https://doi.org/10.1063/1.4848715 20, 123508© 2013 AIP Publishing LLC.Plasmas generated by ultra-violet lightrather than electron impactCite as: Phys. Plasmas 20, 123508 (2013); https://doi.org/10.1063/1.4848715Submitted: 30 October 2013 • Accepted: 27 November 2013 • Published Online: 11 December 2013R. N. Franklin, J. E. Allen, D. M. Thomas, et al.ARTICLES YOU MAY BE INTERESTED INMultiple solutions in the theory of direct current glow discharges: Effect of plasma chemistryand nonlocality, different plasma-producing gases, and 3D modellingPhysics of Plasmas 20, 101613 (2013); https://doi.org/10.1063/1.4826184 Detailed numerical simulation of cathode spots in vacuum arcs: Interplay of differentmechanisms and ejection of dropletsJournal of Applied Physics 122, 163303 (2017); https://doi.org/10.1063/1.4995368Field to thermo-field to thermionic electron emission: A practical guide to evaluation andelectron emission from arc cathodesJournal of Applied Physics 114, 063307 (2013); https://doi.org/10.1063/1.4818325Plasmas generated by ultra-violet light rather than electron impactR. N. Franklin,1,a) J. E. Allen,2,3 D. M. Thomas,3 and M. S. Benilov41Department of Astronomy and Physics, The Open University, Milton Keynes MK7 6AA, United Kingdom2University College, University of Oxford, Oxford OX1 4BH, United Kingdom and OCIAM,Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom3Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, United Kingdom4Departamento de Fisica, CCCEE, Universidade de Madeira, Largo do Municipio, 9000 Funchal, Portugal(Received 30 October 2013; accepted 27 November 2013; published online 11 December 2013)We analyze, in both plane and cylindrical geometries, a collisionless plasma consisting of an innerregion where generation occurs by UV illumination, and an un-illuminated outer region with nogeneration. Ions generated in the inner region flow outwards through the outer region and into awall. We solve for this system’s steady state, first in the quasi-neutral regime (where the Debyelength kD vanishes and analytic solutions exist) and then in the general case, which we solvenumerically. In the general case, a double layer forms where the illuminated and un-illuminatedregions meet, and an approximately quasi-neutral plasma connects the double layer to the wallsheath; in plane geometry, the ions coast through the quasi-neutral section at slightly more than theBohm speed cs. The system, although simple, therefore has two novel features: a double layer thatdoes not require counter-streaming ions and electrons, and a quasi-neutral plasma where ions travelin straight lines with at least the Bohm speed. We close with a precis of our asymptotic solutions ofthis system, and suggest how our theoretical conclusions might be extended and tested in thelaboratory. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4848715]I. INTRODUCTIONThis paper is concerned to give a description of plasmaswhere the generation is by photo-ionization rather than elec-tron impact.An experimental and theoretical treatment of the situa-tion we have in mind was given by Johnson, Cooke, andAllen1 who in cylindrical geometry passed the output of amercury discharge through a vessel containing mercuryvapour, both vessels joined by a quartz window so passingUV. In their situation, the ionization was associative causedby the excited states 63P0 and 6P31 interacting according tothe scheme Hg(63P0)þHg(63P1) ! Hgþ2 þ e, to produce amolecular ion Hgþ2 in the R state. The relevant term diagramwas given in Forrest and Franklin,2 with earlier measure-ments having been carried out by Tan and von Engel.3Here, we will treat cases where the radiation is sufficientlyenergetic that ionization occurs directly and the schemehþA! Aþþ e applies, that is, direct photo-ionization.For simplicity, we will assume the plasma to be collision-less and we will cover both plane and cylindrical geometries.We begin with the plasma approximation, where resultsnot very different from the positive column are expected.We then consider in detail partial illumination with a maxi-mum in the centre where the most interesting case is that ofa sharp cutoff and it turns out that it is necessary to introducePoisson’s equation to obtain closure. We then summarize theuse of matched asymptotic approximations to explore thedouble layer that forms in the partially illuminated case inplane geometry. Finally, we point up ways in which thiswork could be extended and applied.II. THEORETICAL MODELWe show in Figure 1 a schematic diagram of the experi-mental situation we envisage when the plasma is generatedby end illumination. Other geometries are possible.Four equations describe the system at equilibrium. Twoare steady-state fluid equations, namely the continuityequationr  ðnivÞ ¼ G (1)and the ion momentum equationMniðv  rÞvþMvG ¼ nierV (2)while the other two equations are Poisson’s equationr2V ¼ ee0ðne  niÞ (3)and the Boltzmann relationne ¼ n0 expeVkTe ; (4)where ni and ne are the ion and electron densities, respec-tively, v is the ion velocity, G the generation rate, M theion mass, V the electric potential, Te the electron tempera-ture, and n0 the electron density at the system’s centralaxis. The equations are the same in both regions, but G iszero in the un-illuminated region II. We use G rather thanZ in Eq. (1) because the equation with electron impact ioni-zation reads r  ðnivÞ ¼ Zn and is not simply integrable(although it can be solved analytically in the quasi-neutralcase).a)Electronic mail: raoulnf1935@gmail.com1070-664X/2013/20(12)/123508/5/$30.00 VC 2013 AIP Publishing LLC20, 123508-1PHYSICS OF PLASMAS 20, 123508 (2013)Introducing the normalized quantities Uv=cs, Nn=n0,X x=Lion x=ðn0cs=GÞ, and U¼eV=ðkBTeÞ, we rewritethe above equations in normalized form~r  ðNiUÞ ¼ 1; (5)UNiþ ðU  ~rÞU ¼ ~rU; (6)~r2U ¼ Ni  NeK2; (7)Ne ¼ expðUÞ (8)in region I, where K  kD=Lion and ~r is a normalized gradi-ent operator (the spatial derivative when distance is normal-ized by Lion). In region II, Eq. (5) reduces to ~r  ðNiUÞ ¼ 0and Eq. (6) reduces to (U  ~rÞU ¼ ~rU.III. QUASI-NEUTRAL SOLUTION IN PLANEGEOMETRYWe first consider region I in plane geometry under theplasma approximation, which leads to a quasi-neutral solution.Exploiting symmetry, we reduce Eqs. (5)–(8) to scalar equa-tions, and using the plasma approximation ðNi ¼ Ne ¼ NÞ,Poisson’s equation is dispensed with. There are then threeequations to solvedðNUÞdX¼ 1; (9)UNþ U dUdX¼ dUdX; (10)N ¼ expðUÞ: (11)It is readily found that the equations are singular whereU¼ 1, i.e., the Bohm criterion applies. Applying the bound-ary condition that U¼ 0 at X¼ 0, Eq. (9) implies NU ¼ Xand we obtain the analytic solutionX ¼ U1þ U2 ; N ¼11þ U2 ; U ¼ ln 1þ U2ð Þ (12)which implies U ¼ 1, X ¼ 0:5, N ¼ 0:5, U ¼ ln 2 at thesingularity. The corresponding values for electron impactionization, previously given by Franklin,4 are X ¼ 0:571,N ¼ 0:5, U ¼ ln2 ¼ 0:693.In region II, the three equations to solve are more trivialbecause there is no generation: NU is constant, U dU=dX¼ dU=dX, and N ¼ expðUÞ. Eliminating N and U,UdUdX¼ 1UdUdX(13)and this apparently has two possible solutions, dU=dX ¼ 0or U ¼ 1. In either case, the ions are “coasting,” and theonly satisfactory solution is to have the ions coasting at theBohm speed, but then the density is constant and region II isapparently limitless in extent.Thus, this outer region is an unusual type of plasma. It isgenerationless, it has constant density, and, as at a sheath’sedge, its ions travel at the Bohm speed, but unlike a sheath itis not just a few Debye lengths long. Although unusual, itdoes satisfy Langmuir’s definition of a plasma as a “regioncontaining balanced charges of ions and electrons.”5 Wemight call it a quasi-plasma, but perhaps quiescent plasma isa better description. At a certain time historically, there werewhole conferences given over to quiescent plasmas, the lastbeing that in Elsinore in 1971,6 which emphasized the differ-ence then perceived between fusion plasmas and low temper-ature, low pressure plasmas.IV. FINITE DEBYE LENGTH SOLUTION IN PLANEGEOMETRYWe now reintroduce Poisson’s equationd2UdX2¼ Ni  NeK2(14)thus abandoning the plasma approximation. Then, theproblem is tractable only by employing computationalmethods.A natural consequence of introducing Poisson is that thecomputational output gives the space charge differencedirectly, and one finds that at the boundary between the tworegions of plasma, one active and the other quiescent, a dou-ble layer forms.Such structures have been known since the 1970sbetween plasmas with different characteristics of density orelectron temperature due particularly to the work ofAndrews and Allen.7 The size of the double layer is a fewDebye lengths. It is commonly supposed that double layersFIG. 1. The system to be analyzed, in plane geometry (upper) and cylindri-cal geometry (lower). In both geometries, the system comprises two regionsof plasma: the central, uniformly illuminated region I and the outer, un-illuminated region II.123508-2 Franklin et al. Phys. Plasmas 20, 123508 (2013)are a feature of situations only with counter-streaming ionsand electrons, but that is not so in this case.Thus, having a “plasma” in which the densities are notexactly equal, one can integrate to a point where the electronrandom flux equals the ion directed flux, and this is thestandard condition to determine the position of the wall.We show in Figure 2 results for a specific case (M=me¼ 736744, K ¼ 0:00497) that gives all of the relevant varia-bles and also shows the existence of a double layer. Varyingthe parameters produces results having the same features.V. QUASI-NEUTRAL SOLUTION IN CYLINDRICALGEOMETRYIn cylindrical geometry, we have the continuity equationdðNURÞdR¼ 2R (15)and the ion momentum equationUNþ U dUdR¼ dUdR(16)with the Boltzmann relation remaining the same. The nor-malization is the same but our spatial coordinate is now theradial distance R  Gr=ð2n0csÞ.Singularity occurs again where U¼ 1. In region I, wehaveR ¼ Uð1þ 2U2Þ3=4; N ¼ RU; U ¼ ln N (17)giving values at the boundary between regions I and II ofU ¼ 1, N ¼ 0:439, and U ¼ 0:824. These compare withthe electron impact values of U ¼ 1, N ¼ 0:420, andU ¼ 0:869.We next consider the un-illuminated region II, in thelight of what we have learned from the plane case, using theplasma approximation, and there we now have dðNURÞ=dR ¼ 0, i.e., NUR constant. With R increasing, NU is neces-sarily decreasing, and on physical grounds N is decreasingand U increasing. Going through the algebra of the equa-tions, we find ðU  1=UÞ dU=dR ¼ 1=R. U and R are bothpositive definite functions of R and thus any solution requiresU  1=U and so U  1. It is worth reminding ourselves thatBohm’s original statement8 of what became known as hiscriterion was Mv2  kBTe (in other words, U  1) for asheath to form. Once again we have two different “plasmas”adjoining, but in this case there is continuity of all of the var-iables and their derivatives, and the fundamental wall-sheathrequirement obtains. This is because the equation aboveallows the possibility that simultaneously dU=dR!1 andðU  1=UÞ ! 0, at the illumination edge.VI. FINITE DEBYE LENGTH SOLUTION INCYLINDRICAL GEOMETRYAs in plane geometry, one has to resort to computationto provide a physically reliable solution to the finite Debyelength case, but that we have done. The relevant equationsare again (5)–(8), and as in the plane case one can flattenthem into scalar functions of R by exploiting symmetry. Weshow as Figure 3 the results for M=me ¼ 73 6744 andFIG. 2. Solutions of the model in planegeometry. Dotted curves are the quasi-neutral solution and solid curves thenumerical solution for Hgþ2 ions(kD=xill ¼ 1=100 and M=me ¼ 736744;the wall sheath depends quantitativelyon M=me). The normalized variablesplotted here are (a) electric potential;(b) ion density; (c) ion speed; and (d)net space charge density, demonstratingthe existence of a double layer (insetin (d)).123508-3 Franklin et al. Phys. Plasmas 20, 123508 (2013)K ¼ 0:00758. Again, there is a double layer where regions Iand II meet.VII. JOINING ILLUMINATED AND UN-ILLUMINATEDPLASMAS USING MATCHED ASYMPTOTICAPPROXIMATIONSThe first treatment of transition from an active plasmagenerated by electron impact to a collisionless near-wallspace-charge sheath by means of matched asymptotic expan-sions in both the fluid model and the free-fall model wasgiven by Franklin and Ockendon.9 It was found that therewas a need for an intermediate layer or transition layer inter-mediate in dimension between the Debye length and theplasma dimension, adjacent to the wall.The method of matched asymptotic approximationsinvolves expanding the variables in a series in a small parame-ter, and here we take the illuminated and un-illuminatedregions to be of the same extent to reduce the number of pa-rameters involved and designate e ¼ ðkD=xillÞ2. The matchingprocess requires much careful manipulation and so we omitthe details here, and one has to proceed order by order. Afirst-order solution describing the illuminated plasma,the double layer, and the un-illuminated plasma represents thequasi-neutral solution found in Sec. III above for the planecase and in Sec. V for the cylindrical case. We will elaboratethe full second-order solution in a subsequent paper, but insummary: we find that the double layer scales as e2=5; thesecond-order terms in expansions of the quantities U, Ni, Ne,and U are of the order of e2=5 in the illuminated region, of theorder of e1=5 in the double layer, and of the orders of,respectively, e1=5 and e2=5 in the un-illuminated region in theplane and cylindrical cases. It is possible to go to higher ordersthan the second but the algebra becomes increasingly tedious,and it is more expedient to use computational methods.VIII. EXTENSIONS AND CONCLUSIONSIt is readily possible to introduce collisions in the abovetheoretical treatments, but this can only be done computa-tionally and in the process the double layer will be smoothedout, typically for ki < kD or cs=i < kD.We have given here only results for a sharp cut-off ofthe illuminated region, but in our earlier exploratory workwe looked at situations where the illumination had a profile,either Gaussian or approximating to a Heaviside function.Those situations gave results consistent with the descriptionwe have given but that then the double layer attenuated anddied in a characteristic parametric manner.Any experimental test of our results would require a highpower UV source, a quite large cylindrical quartz vessel con-taining the target gas, and ideally non-intrusive diagnostics ofthe two plasma regions, e.g., Laser Induced Fluorescence.There would need to be a “dump” for the UV that passedthrough the target gas, but the theory could be readily modi-fied if the attenuation length of the UV was greater than thevessel length, for then the variation with the beam path wouldbe gradual, and not react back on the plasma overall.Producing the situation in which plane geometry pre-vailed would be more complicated, but not impossible, andwould generate the quiescent uniform plasma we haverevealed.FIG. 3. Solutions of the model in cy-lindrical geometry. Dotted curves arethe quasi-neutral solution and solidcurves the numerical solution for Hgþ2ions (kD=rill ¼ 0:00859 and M=me¼ 736744, where rill is the unnormal-ized radius of the illuminated region I).The same variables are plotted here asin Figure 2. An inset detail of (d) againhighlights a double layer.123508-4 Franklin et al. Phys. Plasmas 20, 123508 (2013)We believe that we have found a new and interesting sit-uation in what we have called the partially illuminated casewhen there is an active plasma adjacent to a quiescent onebefore a wall sheath develops. The physical situation is essen-tially identical to that which obtains when an external electronbeam passes through a plasma, only the parameters are sym-bolically different. But, a recent example producing similarsolutions was presented at ICPIG: an ion beam passing througha plasma independently generated, with the same parent gas.10ACKNOWLEDGMENTSWork at Universidade da Madeira was supported by FCTof Portugal through the project PTDC/FIS-PLA/2708/2012.1P. C. Johnson, M. J. Cooke, and J. E. Allen, J. Phys. D 11, 1877(1978).2J. R. Forrest and R. N. Franklin, J. Phys. B 2, 471 (1969).3K. L. Tan and A. H. von Engel, J. Phys. D 1, 258 (1968).4R. N. Franklin, Plasma Phenomena in Gas Discharges (Oxford UniversityPress, 1976), Table 2.1, p. 33.5I. Langmuir, Proc. Natl. Acad. Sci. U.S.A. 14, 627 (1928).6Proceedings of the 3rd International Conference on Quiescent Plasmas,Elsinore, 1971.7J. Andrews and J. E. Allen, Proc. R. Soc. London, Ser. A 320, 459(1971).8D. Bohm, The Characteristics of Electrical Discharges in Magnetic Fields(McGraw-Hill, New York, 1949), Chap. 3.9R. N. Franklin and J. R. Ockendon, J. Plasma Phys. 4, 371 (1970).10A. V. Shumilin, V. P. Shumilin, and N. V. Shumilin, “Low energyion beam propagation in own gas poster 1–110,” in XXXIInternational Conference on Phenomena in Ionized Gases, Granada,Spain, 2013.123508-5 Franklin et al. Phys. Plasmas 20, 123508 (2013)

Plasmas generated by ultra-violet light rather than electron impact

We analyze, in both plane and cylindrical geometries, a collisionless plasma consisting of an inner region where generation occurs by UV illumination, and an un-illuminated outer region with no generation. Ions generated in the inner region flow outwards through the outer region and into a wall. We solve for this system's steady state, first in the quasi-neutral regime (where the Debye length ${\lambda}_D$ vanishes and analytic solutions exist) and then in the general case, which we solve numerically. In the general case a double layer forms where the illuminated and un-illuminated regions meet, and an approximately quasi-neutral plasma connects the double layer to the wall sheath; in plane geometry the ions coast through the quasi-neutral section at slightly more than the Bohm speed $c_s$. The system, although simple, therefore has two novel features: a double layer that does not require counter-streaming ions and electrons, and a quasi-neutral plasma where ions travel in straight lines with at least the Bohm speed. We close with a pr\'{e}cis of our asymptotic solutions of this system, and suggest how our theoretical conclusions might be extended and tested in the laboratory

Franklin, RN

Allen, JE

Thomas, DM

Benilov, MS

Spiral - Imperial College Digital Repository

arXiv:1312.0523v1  [physics.plasm-ph]  2 Dec 2013Plasmas generated by ultra-violet light rather than electron impactR. N. Franklin,1, a) J. E. Allen,2, 3 D. M. Thomas,3 and M. S. Benilov41)Department of Astronomy and Physics, The Open University, Mlton Keynes, MK7 6AA,UK2)University College, University of Oxford, Oxford, OX1 4BH,UK; OCIAM,Mathematical Institute, University of Oxford, Oxford, OX26GG,UK3)Blackett Laboratory, Imperial College London, Prince Consrt Road, London,SW7 2BW, UK4)Departamento de Fisica, CCCEE, Universidade de Madeira, Largo do Municipio, 9000,Funchal, Portugal(Dated: 19 July 2018)We analyze, in both plane and cylindrical geometries, a collisi nless plasma consistingof an inner region where generation occurs by UV illumination, and an un-illuminatedouter region with no generation. Ions generated in the innerregion flow outwards throughthe outer region and into a wall. We solve for this system’s steady state, first in thequasi-neutral regime (where the Debye lengthλD vanishes and analytic solutions exist)and then in the general case, which we solve numerically. In the general case a doublelayer forms where the illuminated and un-illuminated regions meet, and an approximatelyquasi-neutral plasma connects the double layer to the wall sheath; in plane geometry theions coast through the quasi-neutral section at slightly more than the Bohm speedcs. Thesystem, although simple, therefore has two novel features:a double layer that does notrequire counter-streaming ions and electrons, and a quasi-neutral plasma where ions travelin straight lines with at least the Bohm speed. We close with aprécis of our asymptoticsolutions of this system, and suggest how our theoretical conclusions might be extendedand tested in the laboratory.a)Electronic mail: raoulnf1935@gmail.com1I. INTRODUCTIONThis paper is concerned to give a description of plasmas where the generation is by photo-ionization rather than electron impact.An experimental and theoretical treatment of the situationwe have in mind was given by John-son, Cooke, and Allen1 who in cylindrical geometry passed the output of a mercury dischargethrough a vessel containing mercury vapour, both vessels joined by a quartz window so passingUV. In their situation the ionization was associative caused by the excited states 63P0 and 6P31interacting according to the scheme Hg(63P0) + Hg(63P1) → Hg+2 + e, to produce a molecular ionHg+2 in the Σ state. The relevant term diagram was given in Forrest and Franklin,2 with earliermeasurements having been carried out by Tan and von Engel.3Here we will treat cases where the radiation is sufficiently energetic that ionization occursdirectly and the scheme hν + A → A+ + e applies, that is, direct photo-ionization.For simplicity we will assume the plasma to be collisionlessand we will cover both plane andcylindrical geometries.We begin with the plasma approximation, where results not very different from the positivecolumn are expected. We then consider in detail partial illumination with a maximum in the centrewhere the most interesting case is that of a sharp cutoff and it turns out that it is necessary tointroduce Poisson’s equation to obtain closure. We then summarize the use of matched asymptoticapproximations to explore the double layer that forms in thepartially illuminated case in planegeometry. Finally we point up ways in which this work could beextended and applied.II. THEORETICAL MODELWe show in Figure 1 a schematic diagram of the experimental situation we envisage when theplasma is generated by end illumination. Other geometries ae possible.Four equations describe the system at equilibrium. Two are steady-state fluid equations, namelythe continuity equation∇ · (niv) = G (1)and the ion momentum equationMni(v ·∇)v+MvG=−nie∇V (2)2I IIwallwallIIIwallwallFIG. 1. The system to be analyzed, in plane geometry (upper) and cylindrical geometry (lower). In bothgeometries the system comprises two regions of plasma: the central, uniformly illuminated region I and theouter, un-illuminated region II.while the other two equations are Poisson’s equation∇2V =eε0(ne−ni) (3)and the Boltzmann relationne= n0 exp(eVkTe)(4)whereni andne are the ion and electron densities respectively,v is the ion velocity,G the genera-tion rate,M the ion mass,V the electric potential,Te the electron temperature, and0 the electrondensity at the system’s central axis. The equations are the sam in both regions, butG is zero in theun-illuminated region II. We useG rather thanZ in (1) because the equation with electron impactionization reads∇ · (niv) = Zn and is not simply integrable (although it can be solved analyticallyin the quasi-neutral case).Introducing the normalized quantitiesU ≡ v/cs, N ≡ n/n0, X ≡ x/Lion ≡ x/(n0cs/G), andΦ =−eV/(kBTe), we rewrite the above equations in normalized form:∇̃ · (NiU) = 1 (5)UNi+(U · ∇̃)U = ∇̃Φ (6)3∇̃2Φ =Ni −NeΛ2(7)Ne= exp(−Φ) (8)in region I, whereΛ ≡ λD/Lion and ∇̃ is a normalized gradient operator (the spatial derivativewhen distance is normalized byLion). In Region II (5) reduces tõ∇ · (NiU) = 0 and (6) reduces to(U · ∇̃)U = ∇̃Φ.III. QUASI-NEUTRAL SOLUTION IN PLANE GEOMETRYWe first consider region I in plane geometry under the plasma approximation, which leads to aquasi-neutral solution. Exploiting symmetry we reduce (5)– 8 to scalar equations, and using theplasma approximation(Ni = Ne = N), Poisson’s equation is dispensed with. There are then threeequations to solve:d(NU)dX= 1 (9)UN+UdUdX=dΦdX(10)N = exp(−Φ) (11)It is readily found that the equations are singular whereU = 1, i.e. the Bohm criterion applies.Applying the boundary condition thatU = 0 at X = 0, (9) impliesNU = X and we obtain theanalytic solutionX =U1+U2, N =11+U2, Φ = ln(1+U2)(12)which impliesU = 1, X = 0.5, N = 0.5, Φ = ln 2 at the singularity. The corresponding valuesfor electron impact ionization, previously given by Frankli ,4 areX = 0.571,N = 0.5, Φ = ln2=0.693.In region II the three equations to solve are more trivial because there is no generation:NU isconstant,U dU/dX = dΦ/dX, andN = exp(−Φ). EliminatingN andΦ,UdUdX=1UdUdX(13)and this apparently has two possible solutions, dU/ X = 0 orU = 1. In either case the ions are‘coasting’, and the only satisfactory solution is to have thions coasting at the Bohm speed, butthen the density is constant and region II is apparently limitless in extent.4Thus this outer region is an unusual type of plasma. It is generationless, it has constant densityand, as at a sheath’s edge, its ions travel at the Bohm speed, but unlike a sheath it is not just afew Debye lengths long. Although unusual, it does satisfy Langmuir’s definition of a plasma as a“region containing balanced charges of ions and electrons”.5 We might call it a quasi-plasma, butperhaps quiescent plasma is a better description. At a certain time historically there were wholeconferences given over to quiescent plasmas, the last beingthat in Elsinore in 1971,6 which em-phasized the difference then perceived between fusion plasmas and low temperature, low pressureplasmas.IV. FINITE DEBYE LENGTH SOLUTION IN PLANE GEOMETRYWe now reintroduce Poisson’s equationd2ΦdX2=Ni −NeΛ2(14)thus abandoning the plasma approximation. Then the problemis tractable only by employingcomputational methods.A natural consequence of introducing Poisson is that the computational output gives the spacecharge difference directly, and one finds that at the boundary between the two regions of plasma,one active and the other quiescent, a double layer forms.Such structures have been known since the 1970s between plasmas with different characteristicsof density or electron temperature due particularly to the work of Andrews and Allen.7 The size ofthe double layer is a few Debye lengths. It is commonly supposed that double layers are a featureof situations only with counter-streaming ions and electrons, but that is not so in this case.Thus, having a ‘plasma’ in which the densities are not exactly equal, one can integrate to a pointwhere the electron random flux equals the ion directed flux, and this is the standard condition todetermine the position of the wall.We show in Figure 2 results for a specific case (M/me = 736744,Λ = 0.00497) that gives allof the relevant variables and also shows the existence of a double layer. Varying the parametersproduces results having the same features.50.0 0.2 0.4 0.6 0.8 1.00123456XΦ(a)0.0 0.2 0.4 0.6 0.8 1.00.20.40.60.81.0XNi(b)0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.02.53.03.5XU(c)0.0 0.2 0.4 0.6 0.8 1.00.000.050.100.15XNi−Ne(d) 0.40 0.50 0.60−0.0100.0000.010FIG. 2. Solutions of the model in plane geometry. Dotted curves are the quasi-neutral solution and solidcurves the numerical solution for Hg+2 ions (λD/xill = 1/100 andM/me= 736744; the wall sheath dependsquantitatively onM/me). The normalized variables plotted here are (a) electric potential; (b) ion density;(c) ion speed; and (d) net space charge density, demonstratig the existence of a double layer (inset in (d)).V. QUASI-NEUTRAL SOLUTION IN CYLINDRICAL GEOMETRYIn cylindrical geometry we have the continuity equationd(NUR)dR= 2R (15)and the ion momentum equationUN+UdUdR=dΦdR(16)6with the Boltzmann relation remaining the same. The normalization is the same but our spatialcoordinate is now the radial distanceR≡ Gr/(2n0cs).Singularity occurs again whereU = 1. In region I we haveR=U(1+2U2)3/4, N =RU, Φ =− lnN (17)giving values at the boundary between regions I and II ofU = 1,N= 0.439, andΦ = 0.824. Thesecompare with the electron impact values ofU = 1, N = 0.420, andΦ = 0.869.We next consider the un-illuminated region II, in the light of what we have learned from theplane case, using the plasma approximation, and there we nowhave d(NUR)/dR= 0, i.e.NURconstant. WithR increasing,NU is necessarily decreasing, and on physical groundsN is decreas-ing andU increasing. Going through the algebra of the equations we find (U −1/U)dU/dR=1/R. U andR are both positive definite functions ofR and thus any solution requiresU ≥ 1/Uand soU ≥ 1. It is worth reminding ourselves that Bohm’s original statement8 of what becameknown as his criterion wasMv2 ≥ kBTe (in other words,U ≥ 1) for a sheath to form. Once againwe have two different ‘plasmas’ adjoining, but in this case th re is continuity of all of the variablesand their derivatives, and the fundamental wall-sheath requi ment obtains. This is because theequation above allows the possibility that simultaneouslydU/dR→ ∞ and(U −1/U)→ 0, at theillumination edge.VI. FINITE DEBYE LENGTH SOLUTION IN CYLINDRICAL GEOMETRYAs in plane geometry one has to resort to computation to provide a physically reliable solutionto the finite Debye length case, but that we have done. The relevant quations are again (5)–(8),and as in the plane case one can flatten them into scalar functions ofR by exploiting symmetry.We show as Figure 3 the results forM/me = 736744 andΛ = 0.00758. Agan there is a doublelayer where regions I and II meet.VII. JOINING ILLUMINATED AND UN-ILLUMINATED PLASMAS USINGMATCHED ASYMPTOTIC APPROXIMATIONSThe first treatment of transition from an active plasma generated by electron impact to a col-lisionless near-wall space-charge sheath by means of matched asymptotic expansions in both the70.0 0.2 0.4 0.6 0.80246RΦ(a)0.0 0.2 0.4 0.6 0.80.20.40.60.81.0RNi(b)0.0 0.2 0.4 0.6 0.80123RU(c)0.0 0.2 0.4 0.6 0.80.000.020.040.06RNi−Ne(d)0.30 0.40 0.50−0.0100.0000.010FIG. 3. Solutions of the model in cylindrical geometry. Dotted curves are the quasi-neutral solution andsolid curves the numerical solution for Hg+2 ions (λD/r ill = 0.00859 andM/me = 736744, wherer ill is theunnormalized radius of the illuminated region I). The same variables are plotted here as in figure 2. An insetdetail of (d) again highlights a double layer.fluid model and the free-fall model was given by Franklin and Ockendon.9 It was found that therewas a need for an intermediate layer or transition layer intermediate in dimension between theDebye length and the plasma dimension, adjacent to the wall.The method of matched asymptotic approximations involves expanding the variables in a seriesin a small parameter, and here we take the illuminated and un-illuminated regions to be of the sameextent to reduce the number of parameters involved and designateε = (λD/xill )2. The matching8process requires much careful manipulation and so we omit the details here, and one has to proceedorder by order. A first-order solution describing the illuminated plasma, the double layer, and theun-illuminated plasma represents the quasi-neutral solution found in Section III above for the planecase and in Section V for the cylindrical case. We will elaborte the full second-order solution ina subsequent paper, but in summary: we find that the double layer scales asε2/5; the second-orderterms in expansions of the quantititesU , Ni , Ne, andΦ are of the order ofε2/5 in the illuminatedregion, of the order ofε1/5 in the double layer, and of the orders of, respectively,ε1/5 andε2/5 inthe un-illuminated region in the plane and cylindrical cases. It is possible to go to higher ordersthan the second but the algebra becomes increasingly tedious, and it is more expedient to usecomputational methods.VIII. EXTENSIONS AND CONCLUSIONSIt is readily possible to introduce collisions in the above th oretical treatments, but this can onlybe done computationally and in the process the double layer will be smoothed out, typically forλi < λD or cs/νi < λD.We have given here only results for a sharp cut-off of the illuminated region, but in our earlierexploratory work we looked at situations where the illumination had a profile, either Gaussian orapproximating to a Heaviside function. Those situations gave results consistent with the descrip-tion we have given but that then the double layer attenuated and died in a characteristic parametricmanner.Any experimental test of our results would require a high power UV source, a quite largecylindrical quartz vessel containing the target gas, and ideally non-intrusive diagnostics of the twoplasma regions, e.g. Laser Induced Fluorescence. There would need to be a ‘dump’ for the UVthat passed through the target gas, but the theory could be readily modified if the attenuation lengthof the UV was greater than the vessel length, for then the variation with the beam path would begradual, and not react back on the plasma overall.Producing the situation in which plane geometry prevailed would be more complicated, but notimpossible, and would generate the quiescent uniform plasma we have revealed.We believe that we have found a new and interesting situationin what we have called thepartially illuminated case when there is an active plasma adj cent to a quiescent one before awall sheath develops. The physical situation is essentially identical to that which obtains when an9external electron beam passes through a plasma, only the parameters are symbolically different.But a recent example producing similar solutions was present d at ICPIG: an ion beam passingthrough a plasma independently generated, with the same parent gas.10IX. ACKNOWLEDGEMENTWork at Universidade da Madeira was supported by FCT of Portugal through the projectPTDC/FIS-PLA/2708/2012.REFERENCES1P. C. Johnson, M. J. Cooke, and J. E. Allen,J. Phys. D11, 1877 (1978).2J. R. Forrest and R. N. Franklin,J. Phys. B2, 471 (1969).3K. L. Tan and A. H. von Engel,J. Phys. D1, 258 (1968).4R. N. Franklin,Plasma Phenomena in Gas Discharges, table 2.1, p. 33 (Oxford University Press,1976).5I. Langmuir,Proc. Natl Acad. Sci., 14, 627 (1928).6Proceedings of the 3rd International Conference on Quiescent Plasmas, Elsinore (1971).7J. Andrews and J. E. Allen,Proc. Roy. Soc. A320, 459 (1971).8D. Bohm,The characteristics of electrical discharges in magnetic fields, chapter 3 (McGraw-Hill, New York, 1949).9R. N. Franklin and J. R. Ockendon,J. Plasma Phys., 4, 371 (1970).10A. V. Shumilin, V. P. Shumilin, and N. V. Shumilin. Low energyion beam propagation in owngas, poster 1-110, XXXI International Conference on Phenomena in Ionized Gases, Granada,Spain (2013).10

Physics of Plasmas

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Plasmas generated by ultra-violet light rather than electron impact

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