ISSN 965-545, Computational Mathematis and Mathematial Physis,, Vol. 5, No. 8, pp. 4 446. Pleiades Publishing, Ltd.,. Published in Russian in Zhurnal Vyhislitel noi Matematiki i Matematiheskoi Fiziki,, Vol. 5, No. 8, pp. 56 59. Plasma Waves Refletion from a Boundary with Speular Aommodative Boundary Conditions N. V. Gritsienko, A. V. Latyshev, and A. A. Yushkanov Mosow State Regional University, ul. Radio a, Mosow, 55, Russia e-mail: natafmf@yandex.ru, avlatyshev@mail.ru, yushkanov@inbox.ru Reeived January 4, Abstrat In the present work the linearized problem of plasma wave refletion from a boundary of a half-spae is solved analytially. Speular aommodative onditions of plasma wave refletion from plasma boundary are taken into onsideration. Wave refletane is found as funtion of the given parameters of the problem, and its dependene on the normal eletron momentum aommodation oeffiient is shown by the authors. The ase of resonane when the frequeny of self-onsistent eletri field osillations is lose to the eletron (Langmuir) plasma osillations frequeny, namely, the ase of long-wave limit is analyzed in the present paper. DOI:.4/S9655458 Key words: degenerate plasma, half-spae, normal eletron momentum aommodation oeffiient, speular aommodative boundary ondition, long-wave limit, wave refletane.. BASIC EQUATIONS Researh of degenerate eletron plasma behavior, proesses whih take plae in plasma under eletri field, plasma waves beomes more and more atual at the present time in onnetion with the problems of suh intensively developing areas as miroeletronis and nanotehnologies (see [ 6]). In the works [7, Chapter ; 8 ] the analysis of eletron plasma behavior in external longitudinal alternating eletri field was arried out. In the present work linearized problem of plasma wave refletion from a boundary of a half-spae of ondutive medium is solved analytially. Speular aommodative onditions of eletron refletion from plasma boundary are taken into onsideration. The diffuse boundary onditions were onsidered in [8 ]. Expression for wave refletane is obtained and it is shown that in the ase when normal eletron momentum aommodation oeffiient takes on a value of zero the wave refletane is expressed by the formula obtained earlier in [8, ]. Let us onsider degenerate plasma whih is situated in a half-spae x >. We assume that self-onsistent eletri field E(r, t) inside plasma has one x omponent and varies along the axis x only: E {E x (x, t),, }. In this ase the eletri field is perpendiular to the plasma boundary whih is situated in the plane x. 4πe N Here ω p is the eletron Langmuir plasma osillation frequeny, ω p ------------, N is the eletron numerial density (onentration), m is the mass of the eletron. m Let us take the system of equations whih desribes plasma behavior. As the kineti equation we take τ model Vlasov Boltzmann kineti equation f --- + v--- f + ee---- f t r p f eq ( r, t) f( r, v, t) ----------------------------------. τ (.) The artile was translated by the authors. 4
44 GRITSIENKO et al. Here f f(r, v, t) is the eletron distribution funtion, e is the eletron harge, p mv is the eletron momentum, m is the eletron mass, τ is the harateristi time period between two ollisions, f eq f eq (r, t) is the loal equilibrium distribution funtion of Fermi Dira, f eq Θ( F (t, x) ), where Θ(x) is the funtion of Heaviside, F (t, x) -mv (t, x) is the disturbed kineti energy of Fermi, mv F - is the kineti energy of the eletron. Let us onsider the Maxwell equation for the eletri field where dive( r, t) 4πe ( f( r, v, t) f ( v) ) dω F, dω F Here f is the undisturbed eletron distribution funtion of Fermi Dira, f ( ) Θ( F ), is the Plank onstant, F -mv F is the undisturbed kineti energy of Fermi, v F is the eletron veloity on the Fermi surfae whih is onsidered as spherial. Let us searh for the solution of the system (.) and (.) in the following form We obtain (see [9]) linearized system of equations of Vlasov Maxwell d -------------, p d p dp ( π ) x dp y dp z. f f ( ) + F δ( F )H( xμ,, t), μ v x /v. (.) H ----- + μ----- H + H( x, μ, t ) μe( x, t ) t x e( x, t ) ----------------- x Here e(x, t ) is the dimensionless funtion ω p + - H( x, μ', t ) dμ', ------- H( x ν, μ', t ) dμ'. (.) (.4) e( x, t) ev ------E F x ( x, t), ν f x x/l is the dimensionless oordinate, where l v F τ is the average free path of eletrons, t νt is the dimensionless time, ν is the effetive frequeny of eletron sattering, ν /τ. Supposing that k is the dimensional wave number, we introdue the dimensionless wave number k k v F k ----, then we have kx x ν -------, where ε ----. Let us introdue the quantity ω ωτ ω/ν. ε ω p ω p. BOUNDARY CONDITIONS STATEMENT Let the plasma wave move to the plasma boundary situated in the plane x. The eletri field of the wave hanges aording to the following law k e + ( x, t ) E i x exp ------- + ω t. ε The amplitude of this wave E we assume to be given. On the plasma boundary this wave reflets and the eletri field of the refleted wave has the following form (.) k e ( x, t ) E i x exp ------- ω t. ε (.) COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8
PLASMA WAVES REFLECTION FROM A BOUNDARY 45 The amplitude E is unknown and is to be found from the problem solution. The quantities ω and k are not independent, the following dependene ω ω (k ) is determined from the solution of the dispersion equation whih will be introdued below. It is required to determine what part of the wave energy (.) is absorbed under the wave refletion from the plasma boundary, and what part of the energy is refleted, and also to find the phase shift of the wave. It means we have to alulate the refletane whih is determined as square of module of the ratio of refleted and inoming waves amplitudes and to find the argument of the amplitudes ratio φ( k, ωε, ) R( kωε,, ) E /E E --- E arg arge arge. (.) (.4) Let us outline the time variable of the funtions H(x, μ, t ) and e(x, t ), assuming H( x, μ, t ) e iω t h( x, μ), e x, t ( ) e iω t The system of Eqs (.) and (.4) in this ase will be transformed to the following form μ----- h + ( iω )h( x, μ) μe( x ) x de( x ) ------------ dx ω p Further instead of x, t we write x, t. We rewrite the system of Eqs. (.6) and (.7) in the form e( x ). + - h( x, μ' ) dμ', ------- h( x ν, μ' ) dμ'. μ h ---- + z h( xμ, ) μe( x) + - h( xμ', ) dμ', z iω x, (.5) (.6) (.7) (.8) de( x) ---------- dx ----- h( xμ', ) dμ'. ε We onsider the external eletri field outside the plasma limit is absent. This means that for the field inside plasma on the plasma boundary the following ondition is satisfied COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8 e( ). (.9) (.) The non-leakage ondition for the partile (eletri urrent) flow through the plasma boundary means that μh (, μ ) dμ. (.) In the kineti theory for the desription of the surfae properties the aommodation oeffiients are used often. Tangential momentum and energy aommodation oeffiients are the most used. For the problem onsidered the normal eletron momentum aommodation under the sattering on the surfae has the most important signifiane. The normal momentum aommodation oeffiient is defined by the following relation P α i P r p ------------, α p, P i P s (.)
46 GRITSIENKO et al. where P i and P r are the flows of normal to the surfae momentum of inoming to the boundary and refleted from it eletrons, P i μ h(, μ) dμ, P r μ h(, μ) dμ, (.) quantity P s is the normal momentum flow for eletrons refleted from the surfae whih are in thermodynami equilibrium with the wall, P s μ h s ( μ) dμ, h ( s μ ) A s, < μ <. (.4) The funtion h s (μ) is the equilibrium distribution funtion of the orresponding eletrons. This funtion is to satisfy the ondition similar to the non-leakage ondition μh (, μ ) dμ + μh s ( μ) dμ. (.5) We are going to onsider the relation between the normal momentum aommodation oeffiient α p and the diffuseness oeffiient q for the ase of speular and diffuse boundary onditions whih are written in the following form h(, μ) ( q)h(, μ) + a s, < μ <. Here q is the diffuseness oeffiient ( q ), a s is the quantity determined from the non-leakage ondition. From the non-leakage ondition we derive a s q μh(, μ) dμ qa s. After that we find the differene between the flows P i P r q μ h (, μ ) dμ μ a s dμ q μ h (, μ ) dμ q μ A s dμ qp i qp s. Substituting the expressions obtained to the definition of the normal momentum aommodation oeffiient we obtain that α p q. Thus, for the speular and diffuse boundary onditions the normal momentum aommodation oeffiient α p oinides with the diffusion oeffiient q. Together with the speular and diffuse boundary onditions other variants of boundary onditions are used in the kineti theory. In partiular, aommodative boundary onditions are widely used. They are divided into two forms diffuse aommodative and speular aommodative boundary onditions (see []). We onsider speular aommodative boundary onditions. For the funtion h these onditions will be written in the following form h(, μ) h(, μ) + A + A μ, < μ <. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8 (.6) Coeffiients A and A an be derived from the non-leakage ondition and the definition of the normal eletron momentum aommodation oeffiient. The problem statement is ompleted. Now the problem onsists in finding of suh solution of the system of Eqs. (.8) and (.9), whih satisfies the boundary onditions (.) (.6). Further, with use of the amplitudes of refleted and inoming waves found it is required to find the refletane of the inoming wave energy (.) and the argument of the amplitudes ratio (.4).
PLASMA WAVES REFLECTION FROM A BOUNDARY 47. THE RELATION BETWEEN FLOWS AND BOUNDARY CONDITIONS First of all let us find expression whih relates the onstants A, A from the boundary ondition (.). To arry this out we will use the ondition of non-leakage (.) of the partile flow through the plasma boundary, whih we will write as a sum of two flows N μh(, μ) dμ+ μh(, μ) dμ. After evident substitution of the variable in the seond integral we obtain N Taking into aount the relation (.6), we obtain that A A /. With the help of this relation we an rewrite the ondition (.6) in the following form μ[ h(, μ) h(, μ) ] dμ. h(, μ) h(, μ) A μ + -, < μ <. (.) We onsider the momentum flow of the eletrons whih are moving to the boundary. Aording to (.) we have P i P r --- A. (.) 6 It is easy to see further that P s A s /. With the help of the formula (.) we will rewrite the definition of the aommodation oeffiient (.) in the form A α p P r α p --- s + --- ( α p ). 6 Let us onsider the ondition (.5). From this ondition we obtain that Using the ondition (.), we then get With the help of the seond equality from (.) and (.4) we rewrite the relation (.) in the integral form The boundary problem onsists of the equations (.8) and (.9) and boundary onditions (.), (.), and (.5). 4. SEPARATION OF VARIABLES AND CHARACTERISTIC SYSTEM Appliation of the general Fourier method of the separation of variables in several steps results in the following substitution h η ( x, μ) where η is the spetrum parameter or the parameter of separation, whih is omplex in general. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8 A A s μh(, μ) dμ μh(, μ) dμ. A s μh(, μ) dμ. α p μ - h(, μ) dμ --- ( α p )A. 6 z x ----- z exp Φημ (, ), eη ( x) x exp ----- E( η), η η (.) (.4) (.5) (4.)
48 GRITSIENKO et al. We substitute the equalities (4.) into the Eqs. (.8) and (.9). We obtain the following harateristi system of equations where z ( η μ)φ( η, μ) ημe( η) z E( η) Let us introdue the following designations γ η + -- Φημ' (, ) dμ', Substituting the integral from the Eq. (4.) into (4.), we ome to the following system of equations The funtion (4.5) is alled eigenfuntion of the ontinuous spetrum. ε 5. EIGENFUNCTIONS OF THE DISCRETE SPECTRUM AND PLASMA WAVES Aording to the definition, the disrete spetrum of the harateristi equation is a set of zeroes of the dispersion equation λ(z)/z. We start to searh zeroes of this equation. Let us take Laurent series of the dispersion funtion --- η -- Φημ' (, ) dμ'. ω ε z ----, η -------, z i--------, + γ η z. ε ω p ( η μ)φ( η, μ) E( η) --------- ( ημ η ), η η E( η) -- Φημ' (, ) dμ'. z Solution of the system (4.4) depends essentially on the ondition if the spetrum parameter η belongs to the interval < η <. In onnetion with this the interval < η < we will all as ontinuous spetrum of the harateristi system. Let the parameter η (, ). Then from the Eq. (4.4) in the lass of general funtions we will find eigenfuntion orresponding to the ontinuous spetrum Φημ (, ) F( ημ, )---------, E( η) (4.5) F( ημ, ) P μη η --------------- (4.6) η μ λη ---------δ ( ) ( η μ), η δ(x) is the Dira delta funtion, the symbol Px means the prinipal value of the integral under integrating of the expression x, the funtion λ(z) is alled as dispersion funtion of the problem, λ( z) + z - η z -------------- zμ d μ z μ. (4.) (4.) (4.4) (4.7) Here λ λ λ( z) λ + --- + --- +, z >. z λ 4 z 4 λ ( ) -- + ---------- z z η λ -- - ------ z 5η γ -----------------------------------, + iε+ γ( γ+ iε) ( + γ + iε) 9+ 5iε( + γ + iε) ------------------------------------, 5( + γ + iε) λ 4 z -- 5 - ------ --------------------------------------. 5 + 7iε( + γ + iε) 7η 5( + γ + iε) (5.) COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8
PLASMA WAVES REFLECTION FROM A BOUNDARY 49 y z Γ R D R γ ε γ ε x R /ε Γ R Fig.. One an easily see that in ollisional plasma (i.e. when ε > ) the oeffiient λ. Consequently, the dispersion equation has infinity as a zero η i, to whih the eigensolutions of the given system orrespond h (x, μ) μ/z, e (x). This solution is naturally alled as mode of Drude. It desribes the volume ondutivity of metal, onsidered by Drude (see, for example, []). Let us onsider the question of the plasma mode existene in details. We find finite omplex zeroes of + the dispersion funtion. We use the priniple of argument. We take the ontour Γ ε Γ R γ ε (see Fig. ), whih is passed in the positive diretion and whih bounds the bionneted domain D R. This ontour onsists of the irumferene {Γ R : z R, R /ε, ε > }, and the ontour γ ε, whih inludes the ut [, +], and stands at the distane of ε from it. Aording to the priniple of argument the number of zeroes N (see [4]) in the domain D ε is determined by the following formula N ------ d lnλ( z). πi Γ ε Considering the limit in this equality when ε and taking into aount that the dispersion funtion is analyti in the neighbourhood of the infinity, we obtain that N ( τ) ------ dln ---------- --- ( τ) dln ---------- πi λ ( τ) πi λ ( τ) λ + λ + - ( τ) arg ---------- π λ ( τ) λ +. Consider the urve γ: z G(τ), τ +, where G(τ) λ + (τ)/λ (τ). It is obvious that G(), lim G( τ). Hene, the number of zeroes N equals to the doubled number of turns of the urve γ round τ the point of origin (doubled index of the problem), i.e. N κ(g), κ(g) Ind [, +] G(τ). Thus, the number of zeroes of the dispersion funtion belonging to the omplex plane out of the segment [, ] of the real axis equals to the double index of the funtion G(τ), alulated on the semi-axis [, +]. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8
44 GRITSIENKO et al. γ 6 D L L D + ε Fig.. We take on the plane (γ, ε) (see Fig. ) the urve L, determined by equations γ + ( τ), ε L ( τ), τ, where L L ( τ) μ [ s + λ ( + λ )] ------------------------------------------, L λ [ s + ( + λ ) ( τ) ] τ s ----------------------------------. λ [ s + ( + λ ) ] Here λ ( τ) τ - τ + ln ---------, s( τ) + τ In the same way, as was shown in the work [], we an prove that if (γ, ε) D +, then the index of the problem equals to unity, i.e. N is the number of zeroes whih equals to two, and if (γ, ε) D, then the index of the problem equals to zero, i.e. N. Sine the dispersion funtion is even its zeroes differ from eah other by sign. We designate these zeroes as following ±η, by η we take the zero whih satisfies the ondition Reη >. The following solution orresponds to the zero η π - τ. h η ( x, μ) z x ----- E --- ημ η exp ---------------- μ, e ( x ) η η z η exp z x ----- E. η (5.) This solution is naturally alled as mode of Debay (this is plasma mode). In the ase of low frequenies it desribes well-known sreening of Debay (see []). The external field penetrates into plasma on the depth of r D, r D is the radius of Debay. When the external field frequenies are lose to Langmuir frequenies, the mode of Debay desribes plasma osillations (see, for instane, [, ]). From the equalities (.) for the wave e (x, t) with the help of (.6) and the equality (4.) follows the relation between the wave number k and the zero of the dispersion funtion η (ω, ν): i kx z ---- x -----, from ε η where η η (γ, ε) -------- + γ + i ε. k k - The equalities (.5) and (.6) jointly with (5.) mean that the refleted wave orresponds to the zero η η H η μ η ------------------- exp z ( μ η ) i kx ---- ωt, eη ε exp i kx ---- ωt, ε COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8
PLASMA WAVES REFLECTION FROM A BOUNDARY 44 and the wave inoming to the plasma boundary orresponds to the symmetri zero η H η ------------------- + η μ z ( μ+ η ) i kx exp ---- + ωt, e ε η η exp kx i ---- + ωt. ε 6. EXPANSION IN THE TERMS OF EIGENFUNCTIONS In the work [] it was shown that from the non-leakage ondition (.) and the ondition on the eletri field (.) it results that the trivial (equal to zero) solution of the present problem orresponds to the point η i. We will show that the system of Eqs. (.8) and (.9) with the boundary onditions (.), (.5), and (.) has the solution whih an be represented as an expansion by the eigenfuntions of the harateristi system Here E is given, and E is unknown oeffiient. Both of the variables (amplitudes of Debay) orrespond to the disrete spetrum, E(η) is unknown funtion, whih is alled eigenfuntion of ontinuous spetrum. Our purpose is to find the oeffiient of the ontinuous spetrum and the relation whih onnets the oeffiients of the disrete spetrum. Let us substitute the expansion (6.) into the boundary ondition (.). We get the following equation in the interval < μ < where h( xμ, ) E z --- ημ η ---------------- μ exp i kx ---- ε E --- ημ η + + ----------------- + μ exp i kx ---- ε + -- z x -- exp F( ημ, )E( η) dη, η η z Extending the funtion E(η) into the interval (, ) evenly we transform the equation (6.) to the following form Let us substitute the eigenfuntions of the ontinuous spetrum into the Eq. (6.4). We obtain singular integral equation with Cauhy kernel in the interval (, ) 7. SOLUTION OF THE SINGULAR EQUATION We introdue the auxiliary funtion η COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8 z e( x) E exp i kx ---- E ε i kx ---- x + exp + z ε -- exp E( η) dη. η [ F( ημ, ) F( η, μ) ]E( η) dη + ( E + E )ϕ ( μ) -z A + z A μ, ϕ ( μ) η η μ + η ---------------- μ + -----------------. μ η μ+ η η F ( ημ, )E( η) dη + ( E + E )ϕ ( μ) z A μ ( E + E )ϕ ( μ) + ημ ---------------E η ( η μ η ) dη λμ --------- ( ) E( μ) z A μ μ M( z) --------------E ηz η ( η z η ) dη, -z A sgnμ. -z A sgnμ. (6.) (6.) (6.) (6.4) (6.5)
44 GRITSIENKO et al. the boundary values of whih on the real axis above and below it are related by the Sokhotsky formulas M + With the help of the Sokhotsky formulas for the dispersion and auxiliary funtion we redue the Eq. (6.5) to the boundary ondition of the problem of determination of analyti funtion by its jump on the ontour λ + This equation has general solution (see [4]) ( μ) M ( μ) πi μ ( η )E( μ). ( μ) [ M + ( μ) + ( E + E )ϕ ( μ) z A μ] λ ( μ) [ M ( μ) + ( E + E )ϕ ( μ) z A μ] iπ ------A μη ( μ) sgnμ, η < μ <. (7.) λ( z) [ ϕ( z) ( E + E ) + M( z) z A z] where C is an arbitrary onstant. Let us introdue auxiliary funtion z A μμ ( η ---------- ) sgnμ ------------------------------- μ z dμ + C z, T( z) - Then from the general solution we an easy find M(z): μμ ------------------------------- ( η ) sgnμ d μ z μ. Let us eliminate the pole of the solution (7.) in the infinity. We get that T( z) C M( z) ( E + E )ϕ( z) z A z -z A -------- z + + λ( z) + -------- λ( z). C z A λ. Poles in the points z ±η an be eliminated with the help of one equality sine the funtions onstituting the general solution are uneven ( E z A + E )λ'( η )( η η ) --------------------------------------------------. ( /)T( η ) λ η We substitute the expansion (6.) for the funtion h(x, μ) to the integral boundary ondition (.5). We get the following equation (7.) (7.) E m( η ) + E m( η ) + m( η)e( η) dη In (7.4) the following designations were introdued α ---z A p -----------. 6 α p (7.4) m( ± η ) μ - F ( ± η, μ) dμ, m η ( ) μ - F( ημ, ) dμ. The oeffiient of the ontinuous spetrum we will find from the Sokhotsky formula (7.) after the substitution of the general solution (7.) into it E( η) ------------------------- πi η ( η ) - T+ ( η ----------- ) T ( η) ----------- λ λ + ( η) λ η ----------- -----------. ( η) λ + ( η) λ ( η) (7.5) COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8
PLASMA WAVES REFLECTION FROM A BOUNDARY 44 Let us notie that under the transition through the positive part of the ut (, ) funtions T(z) and λ(z) make jumps, whih differ only by sign. Indeed, let us represent the formula for T(z) in the following form Now from the Sokhotsky formula for the differene of boundary values we obtain that under ondition < η < the following equalities take plae Now one an naturally find that T + With the help of the last relations we find the oeffiient of the ontinuous spetrum from (7.5) We introdue the integral T( z) λ + T + It is evident that in the omplex plane this integral is alulated by the following formula With the help of this funtion we represent the dispersion funtion in the form λ(z) zt (z) + zt ( z), the funtion T(z) we also express in terms of this integral T(z) zt (z) + zt ( z). For the sum of two last expressions we have λ(z) + T(z) + zt ( z). Let us note that the integral T( z) is not singular on the ut < η <. The sum λ(η) + T(η) on the ut < η < is alulated in expliit form without applying to integrals aording to the following formula: We alulate the integrals m(±η ) and m(η) in expliit form. The integrals m(±η ) an be determined easily Let us find the integral m(η). We have - z μ ( η ) -------- + --------- dμ. μ z μ + z ( η) λ iπη( η ( η) λη ( ) η ) ± -------------------------, ( η) T iπη η ( η ( η) T( η) ) ± -------------------------. ( η)λ ( η) T ( η)λ + ( η) ( T( η) + λ( η) ) iπη η ( η ) -------------------------, λ ( η) λ + ( η) iπη η ( η ) -------------------------. E( η) z A Q( η), where Q( η) T ( z) T ( z) -η[ T( η) + λ( η) ] λ η --------------------------------------------------. λ + ( η)λ ( η) - η η ------------- d η z η. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8 z - - z z + + ( η ) ln( /z ). λη ( ) + T( η) - η η η η + [ + ( η ) ln( /η + ) ]. m( ± η ) ( η η ) 6 - η η η - + + --- ln. m( η) μ -μ ( η ημ) ---------- dμ η z η μ η - λη ( )θ+ ( η). η (7.6)
444 GRITSIENKO et al. Here θ + (η) is the harateristi funtion of the interval < η <, i.e. θ + ( η), < η <, < η <. Calulating the integral in the preeding equality we obtain that the integral m(η) an be determined by the formula where where m( η) - η ( η η 6 ) η - η ηη + [ + ( η )f + ( η) ], The Eq. (7.4) with the help (7.6) we rewrite in the form f + ( η) ln ---------, + η < η < η η ln ---------, < η <. η α E m( η ) + E m( η ) z A p ----------- + Q m, 6α p (7.7) Q m m( η)q( η) dη. Replaing the quantity z A in the Eq. (7.7) aording to (7.), we obtain the following equation E m( η ) + E m( η ) from whih we find the amplitude E required The following designations were introdued in (7.8) α ----------- p + Q E + E m -----------------------------, 6α p -T( η ) λ η α E p m( η )A( η ) + B( η )C( α p ) -------------------------------------------------------------E α p m( η )A( η ) + B( η )C( α p ). A( η ) α -T( η ) λ η, C( α p ) ----------- p + α p Q m, 6 (7.8) B( η ) ( η η )λ'( η ). Thus, all the oeffiients of the expansions (6.) and (6.) are determined unambiguously, and this ompletes the proof of these expansions. From the equality (7.8) it is seen that under α p we have E E, i.e. under ondition of pure speular refletion of eletrons from the boundary the wave refletane equals to unity: R, and the phase shift of the inoming and refleted waves is equal to 8, i.e. φ π, from where arge arge + π. Let us represent the formula (7.8) in the form whih is more onvenient for numerial analysis. Let us designate the ratio of amplitudes as K, K E /E, then α K p [ m( η ) m( η )] + α ----------------------------------------------, p m( η ) + C( α p )D( η ) COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8 (7.9)
PLASMA WAVES REFLECTION FROM A BOUNDARY 445 R. R..96.96 4.9 4.. k Fig...9.5.ε Fig. 4. 5 R. argk.4.96.8.9 4.5. αp Fig. 5..5.5. k Fig. 6. where D( η ) B( η ) ----------- A( η ) ( η η )λ'( η ) -------------------------------------. ( /)T( η ) λ η 8. LONG-WAVE LIMIT For study of the inoming wave refletane R K and the phase shift φ argk we will use the formula (7.9). We onsider the dispersion Eq. (5.) with small values of the wave number k λ i εz ----- k λ λ k --------. ε z (8.) We assume the frequeny ω is omplex: ω ω + iω. Then the quantity γ is omplex also: γ γ + iγ. Here γ ω /ω p, γ ω /ω p. From the Eq. (8.) we find that when k is small we have γ.k, γ.5ε. We express the parameters of the problem in terms of k and ε as λ.6k ( iε), +.k z - -----------------, η i ε ε - +.k + i.5ε ( +.k ), η --------------------------------. k With the help of these parameters let us arry out the study of the refletane and the phase shift in long-wave limit (when k is small by magnitude). COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8
446 GRITSIENKO et al. On the Fig. one an see the dependene of the refletane R on the wave number k for the ase when ε. The urves,, orrespond to the following values of the aommodation oeffiient α p.,.5,.. The urve 4 orresponds to the diffuse boundary onditions (see []). From the graph it is seen that when k takes on small values the urve (orresponding to α p ) oinides pratially with the urve 4 (to whih orresponds the ase q ), whih was obtained by the linearization of the refletane value by k. Taking into aount that under α p the refletane is equal to refletane for speular boundary onditions (i.e. when q ) we an onlude that speular aommodative boundary onditions approximate speular and diffuse boundary onditions under α p q very well. On the Fig. 4 the dependene of the refletane R on the quantity ε for the ase α p is represented. The urves,,, 4, 5 orrespond to the following values of the wave number k.,.,.5,.,.. The more aurate analysis shows that with the growth of the aommodation oeffiient the value of the refletane dereases. On the Fig. 5 the dependene of the refletane R on the value of the aommodation oeffiient α p for the ase ε is presented. The urves,,, 4 orrespond to the following values of the wave number k.5,.,.5,.5. On the Fig. 6 the dependene of the angle φ argk (of the phase shift) on the quantity φ for the ase k. is represented. The urves,, orrespond to the following values of the aommodation oeffiient α p.,.5,. The analysis shows that the dependene between the values of the angle φ and the wave number and the aommodation oeffiient as well is small. 9. CONCLUSION In the present work new boundary onditions for the problems of plasma wave refletion from the plane boundary of a half-spae of degenerate plasma were proposed. These boundary onditions are naturally alled as speular aommodative onditions. Suh boundary onditions are most adequate for the problems of normal propagation of plasma waves (perpendiular to the boundary), sine aommodation oeffiient under suh boundary onditions is normal eletron momentum aommodation oeffiient. In the present paper the analytial solution of the problem of plasma wave refletion from a boundary with normal eletron momentum aommodation is obtained. The analysis of the main parameters of the problem in long wave limit is arried out. This analysis shows that the boundary onditions proposed are intermediate between pure speular and pure diffuse boundary onditions. Indeed, from the Fig. it is seen that all the graphs showing the dependene between the refletane and the wave number are loated between graphs orresponding to speular and diffuse boundary onditions. REFERENCES. The Enylopaedia of Low Temperature Plasmas 5. Vols. 7, Ed. by V. E. Fortov (Nauka, Mosow, 997 9) [in Russian].. V. V. Vedenyapin, Boltzmann and Vlasov Kineti Equations (Fizmatlit, Mosow, ) [in Russian].. A. A. Abrikosov, Fundamentals of the Theory of Metals (Nauka, Mosow, 977; North Holland, Amsterdam, 988). 4. M. Dressel and G. Grüner, Eletrodynamis of Solids. Optial Properties of Eletrons in Matter (Cambridge. Univ. Press, ). 5. T. J. N. Boyd and J. J. Sanderson, The Physis of Plasmas (Cambridge Univ. Press, ). 6. R. L. Liboff, Kineti Theory Classial, Quantum, and Relativisti Desription (Springer Verlag, New York, In., ). 7. A. V. Latyshev and A. A. Yushkanov, Analytial Solution of the Problem on Behavior the Degenerate Eletroni Plasmas, Chapter : in Enylopaedia of Low Temperature Plasma) (Nauka, Mosow, 8), Vol. 7, pp. 59 77 [in Russian]. 8. A. V. Latyshev and A. A. Yushkanov, Refletion and Transmission of Plasma Waves at the Interfae of Crystallites in Computational Mathematis and Mathematial Physis (7), Vol. 47, No. 7, pp. 79 96. 9. A. V. Latyshev and A. A. Yushkanov, Refletion of Plasma Waves from a Plane Boundary, Theor. Math. Phys., 5 (), 45 45 (7).. A. V. Latyshev and A. A. Yushkanov, Refletion of a Plasma Wave from the Flat Boundary of a Degenerate Plasma, Tehnial Physis 5 () 6 (7).. A. V. Latyshev and A. A. Yushkanov, Boundary value problems for degenerate eletroni plasma. Monograph (Mosow State Regional University, 6) [in Russian].. A. V. Latyshev and A. A. Yushkanov, Moment boundary onditions in problems of rarefied gas slip, in Proeedings of the Russian Aademy of Sienes. Mehanis of fluid and gas No., 9 8 (4).. P. M. Platzman and P. A. Wolf, Waves and interations in solid state plasmas (Aademi Press, New York and London, 97). 4. F. D. Gakhov, Boundary Value Problems (Nauka, Mosow, 977) [in Russian]; (English transl., Dover, New York, 99). COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 5 No. 8