28th ICPIG, July 15-20, 2007, Prague, Czech Republic

On the radiation trapping problem in a finite cylinder: Spatial distribution of resonance and metastable atoms

Yu. B. Golubovskii1, S. Gorchakov2, A. N. Timofeev1, D. Loffhagen2, D. Uhrlandt2

1Saint-Petersburg State University, Ulyanovskaya 1, 198504, Russia

2INP Greifswald, Felix-Hausdorff-Str. 2, D-17489, Greifswald, Germany

A universal solution method for the radiation transport equation in cylindrical geometry with finite dimensions is presented. It is based on transformation of the integral Holstein-Biberman equation [1, 2] to a system of linear algebraic equations using the matrix method [3-5] which was extended for the case of a finite geometry. The four-dimensional matrices, determined numerically, provide sufficiently accurate solutions for various shapes of the excitation source. The results for an excitation source in a form of a delta-like function and for a source concentrated on the periphery of the discharge (skin effect) are presented and discussed. The method can be used for the description of radiation transport phenomena in such objects like classical glow discharge, high frequency, RF and microwave discharges, as well as in plasma-chemical reactors.

1. Introduction.

A theoretical description of discharge plasmas requires the consideration of the non-equilibrium and non-local kinetics of the electrons, of the space-charge confinement and the resulting spatial structure, and of the heavy particle kinetics. In the latter, the transport processes of heavy particles (diffusion) and photons (radiation transport) have to be taken into account. The transport due to the diffusion can be described with sufficient accuracy by transformation of the differential diffusion operator to the system of linear equations on a mesh in coordinate space. The description of the radiation transport requires the solution of integral Holstein-Biberman equation [1, 2]. Here, the so called effective lifetime approximation is often used which corresponds to the local balance of resonance atoms and doesn’t describe their redistribution in space. However, in the case when the spatial distribution of the excited atoms has strong deviations from the fundamental mode of the radiation transport operator the effective lifetime approximation becomes inadequate.

One of the most promising methods for the description of the radiation transport phenomena is the conversion of the integral radiation transport equation to a system of linear equations for fixed geometry and line shape of emission and absorption profiles considering a mesh in coordinate space. This technique was successfully used in the past for the description of plasmas in plane parallel [4], cylindrical [3], and coaxial geometry [5] assuming the inhomogeneous distribution of the excited species in one dimension and spatial homogeneity and infinite size in another dimensions.

The problem of radiation trapping in the case of multi-dimensional system was casually mentioned in [6]. “A possibility to make a computation is to solve Holstein equation for the multi-dimensional geometry, e.g. by the piecewise constant approximation. This leads, however, to an enormous increase in CPU time. Computation of the matrix elements becomes more complicated. For the finite cylinder, we get triple integrals.”

In the present work the spatial distribution of the resonance atoms affected by the radiation transport processes in the cylinder of a finite length and a finite radius is considered. In the cases when one of the sizes of the cylinder (length or radius) is assumed to be infinite, the integral transport operator in the Holstein-Biberman equation can be transformed into a two-dimensional matrix, the elements of which depends only on the geometry and the shape of emission and absorption line [3-5]. Finally, a system of linear algebraic equations should be solved to determine the density of corresponding resonance atoms. Taking into account the finite size in both radial and axial directions (azimuthal homogeneity is assumed to be valid) this matrix becomes four-dimensional. The expression for the matrix elements is given in the present contribution. The application of the method is illustrated by some examples which demonstrate a pronounced difference in the mechanisms of the radiation transfer and particle diffusion for radially and axially inhomogeneous excitation sources. 2. Initial equations and solution method.

The spatial distribution of resonance atoms in stationary plasma is described by Holstein-Biberman equation [1, 2]

28th ICPIG, July 15-20, 2007, Prague, Czech Republic

( ) ( ) ( ) ( )ArIrdKrNrN

V

rrr

=′′− ∫ 3

)(

ρ (1)

where ( )rN r is the density of the resonance atoms,

)(rI r is the excitation rate, A denotes the

spontaneous transition probability. ( )ρK is the probability for the photons emitted at the point rr′ to pass the distance rr rr ′−=ρ without absorption and to be absorbed at a point rr which has an integral representation

( ) ∫∞ ′−−

′−=

041

rrdekK

rrk

rr

rr

νεπ

ρν

νν , (2)

where νε and kν are the emission and absorption line profiles.

For the solution of the equation (1) it is convenient to divide the total discharge volume V into a number of cells iVΔ inside which the density of the resonance atoms is assumed to be space-independent. Therefore, the integral term in Eq. (1) becomes an approximate representation

( )( )

( ) ( ) ( )( )∫∑∫Δ

′≈′′iVi

iV

rdKrNrdKrN 33 ρρ rr (3)

Lets consider a finite cylinder with its radius divided into M segments with MRr =Δ and its length divided into G segments with GLz =Δ . Thus, the volume VΔ i is a ring lying within the interval 1+<′< jj rrr , 1+<′< ii rzz , πϕ 20 << . The value of the density N corresponds to the centre of this interval ( )2121 , ++ ij zrN . Finally, after all transformations the equation (1) is conversed to a system of linear algebraic equations with the four-dimensional matrix ijnma ,,,

⎟⎠⎞⎜

⎝⎛=⎟

⎠⎞⎜

⎝⎛

++

−

=

−

=++∑∑

21

21

1

0

1

0 21

21 ,,

nm

M

j

G

iijmnji zrIzrNa (4)

( )mnjimnjimnji bAa −⋅= δδ (5) ( ) ( )

∫∫∫Δ+

Δ++

Δ+

Δ

⎟⎠⎞⎜

⎝⎛ ′′′′′=

zi

zinm

rj

rjmnji zrzrKzdrdrdb

1

21

21

12

0

,,,π

ϕ

(6) where mnjiδδ are the Kronecker symbols.

3. Effective transition probability approximation.

The most widespread approximation which describes the radiation transport processes is the effective transition (of effective lifetime) approximation. Following Biberman [1], the effective transition probability, which is a function of coordinates, can be found under the assumption

that the kernel ( )ρK decreases near the point rr much faster than the density ( )rN r

does. Therefore, the value ( )rN r

can be taken out of the integral in Eq. (1). The effective transition probability is determined as

( )( )

( ) ( )rgArdKArAV

effrr

×≡⎟⎟⎠

⎞⎜⎜⎝

⎛′−= ∫ 31 ρ

(7)

where ( )rg r is an escape factor which characterize

the decrease of the probability for a photon to leave the point r due to the radiation trapping. The escape factor increases at the plasma border since it is easier for photons to leave the volume from its boundary.

In the case of plane parallel geometry the calculation of escape factor results in a simple analytical dependence on the coordinate z [4]. When a cylinder of a radius R and an infinite length is considered the quadruple integral in (7) is reduced to a simple integral [3, 5]. Analogous consideration of a cylinder with finite dimensions requires a double integration. For the Lorentzian line shape of emission and absorption profile and large values of the absorption coefficient the effective probability has following representation.

1. Two planes at a distance L ( )Lz <<0 .

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

−+⋅=

zLL

zLLAzA effeff 2

(8)

( )LAA

LLAeff

00

943.03

222/πκπκ

== (9)

2. Infinite cylinder of a radius R ( )Rr <<0 .

( ) ( )( )∫⋅=

π

ϕξϕ

π 0 ,10

rdArA effeff (10)

( ) ϕϕϕξ 22

sin1cos, ⎟⎠⎞

⎜⎝⎛−+=

Rr

Rrr (11)

RAAeff

0

874.0)0(πκ

=

(12)

3. Cylinder of a length L and a radius R (diameter D). In this case the effective probability of a photon leaving from the centre of the cylinder can be represented as a dependence on the ratio L/D.

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

== D

LfLA

DLf

RAA LR

Lzreff

002/0

943.0874.0πκπκ

(13)

28th ICPIG, July 15-20, 2007, Prague, Czech Republic

Figure 1 shows the functions ( )DLf R /

and ( )DLf L / . The first term in the right-hand side of Eq. (13) characterize the contribution of the photons which leave the plasma through the cylindrical surface, the second term corresponds to the photons which escape through the flanges. The limit transitions from a finite cylinder to an infinite one are realized for cases with L/D<0.01 (infinite length), and with L/D>10 (infinite radius). The spatial dependence of the escape factor is shown in Fig. 2. The value of g is found to increase when approaching the surface of the cylinder.

4. Spatial distribution of excited atoms in a finite cylinder.

The developed solution method of radiation transport equation will be now illustrated on the examples of a given excitation sources for a cylinder with L=D. Two cases are considered: (A) the excitation source is concentrated in the centre of the volume and has a shape of a delta-like function, and (B) the maximum of the excitation rate is shifted to the periphery of the plasma (skin-effect). In the case A the excitation source has a form

( )⎩⎨⎧ Δ<<Δ+<<

= otherwise ,0

0 ,22 ,1,

rrzLzLzrI

(14)

Fig. 2. Escape factor for a finite cylinder.

Fig. 3a. Spatial distribution of the density of the resonance atoms for a delta-like excitation source.

Fig. 3b. Spatial distribution of the density of the metastable atoms for a delta-like excitation source.

Fig. 4a. Spatial distribution of excitation source for skin effect.

Fig. 4b. Spatial distribution of the density of the resonance atoms for skin effect.

Fig. 1. Dependence of the functions fR and fL on the parameter L/D.

28th ICPIG, July 15-20, 2007, Prague, Czech Republic

The results for this excitation source are shown in Fig. 3a. In order to illustrate the differences in the mechanisms of particle transport (diffusion) and radiation transport the spatial distribution of the metastable atoms for the same excitation source is presented in Fig. 3b. A distinct difference of the spatial distributions of the resonance and metastable atoms is obvious. The widening of the spatial distribution of the excited atoms in comparison with the shape of the excitation source is more pronounced in the case of diffusion.

The results for the case B are presented in Fig. 4

and Fig. 5. The form of the excitation source is shown as a 3D graph in Fig. 4a and as a contour plot in Fig 5a. Corresponding representations of the spatial distribution of the density of the resonance atoms are given in Fig. 4b and 5b. Due to the transport effect a non-zero density of the resonance atoms appears in the regions where the excitation is absent. However, the shape of the spatial distribution outside the centre is very similar to that of the excitation source.

Thus, using the matrix method the solution of the radiation transport equation for different shapes of the excitation sources becomes possible. Developed method can be applied for modeling of real gas discharge plasmas. Using these matrices and the method presented in [1] the set of two-dimensional eigenvalues and eigenfunctions of the radiative transport operator could be determined.

5. References [1] Holstein T., Phys. Rev., 72 (1947) 1212. [2] Biberman L. M., Zh. Exp. Teor. Phys., 17 (1947) 416. [3] Yu. B. Golubovskii, I. A. Porokhova, H. Lange, D. Uhrlandt, Plasma Sources Sci. Technol. 14 (2005) 36. [4] Yu. Golubovskii, S. Gorchakov, D. Loffhagen, D. Uhrlandt, Eur. Phys. J. Appl. Phys. 37 (2007) 101. [5] I. A. Porokhova, Yu. B. Golubovskii, C. Csambal, C. Wilke, J. F. Behnke, Phys. Rev. E 65 (2002) 046401. [6] A. F. Molisch, B. P. Oehry, Radiation Trapping in Atomic Vapours, Oxford (1998).

Fig. 5a. Contour plot of excitation source for skin effect.

Fig. 5b. Contour plot of the density of the resonance atoms for skin effect.