Abstract

Optimum methods for calculating the effects of photon trapping are discussed. An efficient line-transfer algorithm that can calculate trapping when there are overlapping and interacting lines is described. Escape probability formulas are shown to be appropriate for calculating photon trapping for isolated lines and for the highest-energy line in a group of lines in many situations. Major computational savings are achieved by using cylindrical escape probabilities for recombination x-ray laser schemes. For collisional x-ray laser schemes it is shown that the calculation of line transfer in planar geometry is sufficiently fast that one only obtains substantial savings by exploiting the coarser spatial zoning that is possible when using escape probabilities in regions of steep velocity gradients. The use of escape probabilities is shown to be particularly well suited for single-zone parameter studies.

© 1992 Optical Society of America

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References

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  1. See, for example, D. C. Eder, “Hydrogenlike magnesium x-ray laser design,” Phys. Fluids B 2, 3086–3092 (1990); G. J. Pert, S. J. Rose, “Detailed simulation of recombination XUV laser experiments,” Appl. Phys. B 50, 307–311 (1990); R. A. London, M. D. Rosen, M. S. Maxon, D. C. Eder, P. L. Hagelstein, “Theory and design of soft x-ray laser experiments at the Lawrence Livermore National Laboratory,” J. Phys. B 22, 3363–3376 (1989); R. C. Elton, X-Ray Lasers (Academic, San Diego, Calif., 1990).
    [CrossRef]
  2. D. C. Eder, H. A. Scott, “The calculation of line transfer in expanding media,” J. Quant. Spectrosc. Radiat. Transfer 45, 189–204 (1991).
    [CrossRef]
  3. G. B. Rybicki, “Escape probability methods,” in Methods in Radiative Transfer, W. Kalkofen, ed. (Cambridge U. Press, Cambridge, England, 1984), p. 21.
  4. D. G. Hummer, G. B. Rybicki, “A unified treatment of escape probabilities in static and moving media. I. Plane geometry,” Astrophys. J. 254, 767–779 (1982).
    [CrossRef]
  5. A. I. Shestakov, D. C. Eder, “Escape probabilities in a cylindrically expanding medium,” J. Quant. Spectrosc. Radiat. Transfer 42, 483–498 (1989).
    [CrossRef]
  6. Y. T. Lee, R. A. London, G. B. Zimmerman, P. L. Hagelstein, “Application of escape probability to line transfer in laser-produced plasmas,” Phys. Fluids B 2, 2731–2740 (1990).
    [CrossRef]
  7. B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
    [CrossRef] [PubMed]
  8. S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
    [CrossRef] [PubMed]
  9. D. Mihalas, Stellar Atmospheres (Freeman, San Franciso, Calif., 1978).
  10. D. Mostacci, L. M. Montierth, J. Dinguirard, R. L. Morse, “X-ray line emission from laser-produced spherical plasma flows,” Phys. Fluids B 1, 2106–2120 (1989).
    [CrossRef]
  11. J. P. Apruzese, “An analytic Voigt profile escape probability approximation,” J. Quant. Spectrosc. Radiat. Transfer 34, 447–452 (1985).
    [CrossRef]
  12. R. A. London, M. D. Rosen, J. E. Trebes, “Wavelength choice for soft x-ray laser holography of biological samples,” Appl. Opt. 28, 3397–3404 (1989).
    [CrossRef] [PubMed]
  13. A. Zigler, H. Zmora, N. Spector, M. Klapisch, J. L. Schwob, A. Bar-Shalom, “Identification of the spectra of HfXLV, TaXLVI, WXLVII, and ReXLVIII isoelectronic to NiI in laser-produced plasmas,” J. Opt. Soc. Am. 70, 129–132 (1980).
    [CrossRef]

1991

D. C. Eder, H. A. Scott, “The calculation of line transfer in expanding media,” J. Quant. Spectrosc. Radiat. Transfer 45, 189–204 (1991).
[CrossRef]

1990

See, for example, D. C. Eder, “Hydrogenlike magnesium x-ray laser design,” Phys. Fluids B 2, 3086–3092 (1990); G. J. Pert, S. J. Rose, “Detailed simulation of recombination XUV laser experiments,” Appl. Phys. B 50, 307–311 (1990); R. A. London, M. D. Rosen, M. S. Maxon, D. C. Eder, P. L. Hagelstein, “Theory and design of soft x-ray laser experiments at the Lawrence Livermore National Laboratory,” J. Phys. B 22, 3363–3376 (1989); R. C. Elton, X-Ray Lasers (Academic, San Diego, Calif., 1990).
[CrossRef]

Y. T. Lee, R. A. London, G. B. Zimmerman, P. L. Hagelstein, “Application of escape probability to line transfer in laser-produced plasmas,” Phys. Fluids B 2, 2731–2740 (1990).
[CrossRef]

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

1989

S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

D. Mostacci, L. M. Montierth, J. Dinguirard, R. L. Morse, “X-ray line emission from laser-produced spherical plasma flows,” Phys. Fluids B 1, 2106–2120 (1989).
[CrossRef]

A. I. Shestakov, D. C. Eder, “Escape probabilities in a cylindrically expanding medium,” J. Quant. Spectrosc. Radiat. Transfer 42, 483–498 (1989).
[CrossRef]

R. A. London, M. D. Rosen, J. E. Trebes, “Wavelength choice for soft x-ray laser holography of biological samples,” Appl. Opt. 28, 3397–3404 (1989).
[CrossRef] [PubMed]

1985

J. P. Apruzese, “An analytic Voigt profile escape probability approximation,” J. Quant. Spectrosc. Radiat. Transfer 34, 447–452 (1985).
[CrossRef]

1982

D. G. Hummer, G. B. Rybicki, “A unified treatment of escape probabilities in static and moving media. I. Plane geometry,” Astrophys. J. 254, 767–779 (1982).
[CrossRef]

1980

Apruzese, J. P.

J. P. Apruzese, “An analytic Voigt profile escape probability approximation,” J. Quant. Spectrosc. Radiat. Transfer 34, 447–452 (1985).
[CrossRef]

Bar-Shalom, A.

Busch, G.

S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

Charatis, G.

S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

Da Silva, L. B.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

Dalhed, S.

S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

Dinguirard, J.

D. Mostacci, L. M. Montierth, J. Dinguirard, R. L. Morse, “X-ray line emission from laser-produced spherical plasma flows,” Phys. Fluids B 1, 2106–2120 (1989).
[CrossRef]

Eder, D. C.

D. C. Eder, H. A. Scott, “The calculation of line transfer in expanding media,” J. Quant. Spectrosc. Radiat. Transfer 45, 189–204 (1991).
[CrossRef]

See, for example, D. C. Eder, “Hydrogenlike magnesium x-ray laser design,” Phys. Fluids B 2, 3086–3092 (1990); G. J. Pert, S. J. Rose, “Detailed simulation of recombination XUV laser experiments,” Appl. Phys. B 50, 307–311 (1990); R. A. London, M. D. Rosen, M. S. Maxon, D. C. Eder, P. L. Hagelstein, “Theory and design of soft x-ray laser experiments at the Lawrence Livermore National Laboratory,” J. Phys. B 22, 3363–3376 (1989); R. C. Elton, X-Ray Lasers (Academic, San Diego, Calif., 1990).
[CrossRef]

A. I. Shestakov, D. C. Eder, “Escape probabilities in a cylindrically expanding medium,” J. Quant. Spectrosc. Radiat. Transfer 42, 483–498 (1989).
[CrossRef]

Fields, D. J.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

Hagelstein, P. L.

Y. T. Lee, R. A. London, G. B. Zimmerman, P. L. Hagelstein, “Application of escape probability to line transfer in laser-produced plasmas,” Phys. Fluids B 2, 2731–2740 (1990).
[CrossRef]

S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

Hummer, D. G.

D. G. Hummer, G. B. Rybicki, “A unified treatment of escape probabilities in static and moving media. I. Plane geometry,” Astrophys. J. 254, 767–779 (1982).
[CrossRef]

Keane, C. J.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

Klapisch, M.

Lee, Y. T.

Y. T. Lee, R. A. London, G. B. Zimmerman, P. L. Hagelstein, “Application of escape probability to line transfer in laser-produced plasmas,” Phys. Fluids B 2, 2731–2740 (1990).
[CrossRef]

London, R. A.

Y. T. Lee, R. A. London, G. B. Zimmerman, P. L. Hagelstein, “Application of escape probability to line transfer in laser-produced plasmas,” Phys. Fluids B 2, 2731–2740 (1990).
[CrossRef]

S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

R. A. London, M. D. Rosen, J. E. Trebes, “Wavelength choice for soft x-ray laser holography of biological samples,” Appl. Opt. 28, 3397–3404 (1989).
[CrossRef] [PubMed]

MacGowan, B. J.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

Matthews, D. L.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

Maxon, S.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

Mihalas, D.

D. Mihalas, Stellar Atmospheres (Freeman, San Franciso, Calif., 1978).

Montierth, L. M.

D. Mostacci, L. M. Montierth, J. Dinguirard, R. L. Morse, “X-ray line emission from laser-produced spherical plasma flows,” Phys. Fluids B 1, 2106–2120 (1989).
[CrossRef]

Morse, R. L.

D. Mostacci, L. M. Montierth, J. Dinguirard, R. L. Morse, “X-ray line emission from laser-produced spherical plasma flows,” Phys. Fluids B 1, 2106–2120 (1989).
[CrossRef]

Mostacci, D.

D. Mostacci, L. M. Montierth, J. Dinguirard, R. L. Morse, “X-ray line emission from laser-produced spherical plasma flows,” Phys. Fluids B 1, 2106–2120 (1989).
[CrossRef]

Osterheld, A. L.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

Rosen, M. D.

R. A. London, M. D. Rosen, J. E. Trebes, “Wavelength choice for soft x-ray laser holography of biological samples,” Appl. Opt. 28, 3397–3404 (1989).
[CrossRef] [PubMed]

S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

Rybicki, G. B.

D. G. Hummer, G. B. Rybicki, “A unified treatment of escape probabilities in static and moving media. I. Plane geometry,” Astrophys. J. 254, 767–779 (1982).
[CrossRef]

G. B. Rybicki, “Escape probability methods,” in Methods in Radiative Transfer, W. Kalkofen, ed. (Cambridge U. Press, Cambridge, England, 1984), p. 21.

Schwob, J. L.

Scofield, J. H.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

Scott, H. A.

D. C. Eder, H. A. Scott, “The calculation of line transfer in expanding media,” J. Quant. Spectrosc. Radiat. Transfer 45, 189–204 (1991).
[CrossRef]

Shestakov, A. I.

A. I. Shestakov, D. C. Eder, “Escape probabilities in a cylindrically expanding medium,” J. Quant. Spectrosc. Radiat. Transfer 42, 483–498 (1989).
[CrossRef]

Shimkaveg, G.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

Spector, N.

Stone, G. F.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

Trebes, J. E.

Zigler, A.

Zimmerman, G. B.

Y. T. Lee, R. A. London, G. B. Zimmerman, P. L. Hagelstein, “Application of escape probability to line transfer in laser-produced plasmas,” Phys. Fluids B 2, 2731–2740 (1990).
[CrossRef]

Zmora, H.

Appl. Opt.

Astrophys. J.

D. G. Hummer, G. B. Rybicki, “A unified treatment of escape probabilities in static and moving media. I. Plane geometry,” Astrophys. J. 254, 767–779 (1982).
[CrossRef]

J. Opt. Soc. Am.

J. Quant. Spectrosc. Radiat. Transfer

J. P. Apruzese, “An analytic Voigt profile escape probability approximation,” J. Quant. Spectrosc. Radiat. Transfer 34, 447–452 (1985).
[CrossRef]

A. I. Shestakov, D. C. Eder, “Escape probabilities in a cylindrically expanding medium,” J. Quant. Spectrosc. Radiat. Transfer 42, 483–498 (1989).
[CrossRef]

D. C. Eder, H. A. Scott, “The calculation of line transfer in expanding media,” J. Quant. Spectrosc. Radiat. Transfer 45, 189–204 (1991).
[CrossRef]

Phys. Fluids B

See, for example, D. C. Eder, “Hydrogenlike magnesium x-ray laser design,” Phys. Fluids B 2, 3086–3092 (1990); G. J. Pert, S. J. Rose, “Detailed simulation of recombination XUV laser experiments,” Appl. Phys. B 50, 307–311 (1990); R. A. London, M. D. Rosen, M. S. Maxon, D. C. Eder, P. L. Hagelstein, “Theory and design of soft x-ray laser experiments at the Lawrence Livermore National Laboratory,” J. Phys. B 22, 3363–3376 (1989); R. C. Elton, X-Ray Lasers (Academic, San Diego, Calif., 1990).
[CrossRef]

D. Mostacci, L. M. Montierth, J. Dinguirard, R. L. Morse, “X-ray line emission from laser-produced spherical plasma flows,” Phys. Fluids B 1, 2106–2120 (1989).
[CrossRef]

Y. T. Lee, R. A. London, G. B. Zimmerman, P. L. Hagelstein, “Application of escape probability to line transfer in laser-produced plasmas,” Phys. Fluids B 2, 2731–2740 (1990).
[CrossRef]

Phys. Rev. Lett.

B. J. MacGowan, S. Maxon, L. B. Da Silva, D. J. Fields, C. J. Keane, D. L. Matthews, A. L. Osterheld, J. H. Scofield, G. Shimkaveg, G. F. Stone, “Demonstration of x-ray amplifiers near the carbon K edge,” Phys. Rev. Lett. 65, 420–423 (1990).
[CrossRef] [PubMed]

S. Maxon, S. Dalhed, P. L. Hagelstein, R. A. London, B. J. MacGowan, M. D. Rosen, G. Charatis, G. Busch, “Calculation for Ni-like soft x-ray lasers: optimization for W (43.1 Å),” Phys. Rev. Lett. 63, 236–239 (1989).
[CrossRef] [PubMed]

Other

D. Mihalas, Stellar Atmospheres (Freeman, San Franciso, Calif., 1978).

G. B. Rybicki, “Escape probability methods,” in Methods in Radiative Transfer, W. Kalkofen, ed. (Cambridge U. Press, Cambridge, England, 1984), p. 21.

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Figures (4)

Fig. 1
Fig. 1

Energy-level diagram for Ni-like Ta showing the important lasing lines between the 4d and 4p levels.

Fig. 2
Fig. 2

Time dependence of the gain at the center of the expansion for the J = 0 to J = 1 lasing line at 44.83 Å for no trapping, trapping calculated with the line-transfer package, and trapping calculated by using escape probabilities.

Fig. 3
Fig. 3

Time dependence of the gain at the center of the expansion for the J = 2 to J = 1 lasing line at 74.42 Å for no trapping, trapping calculated with the line-transfer package, and trapping calculated by using escape probabilities.

Fig. 4
Fig. 4

Spatial dependence of the gains at t = 4.9 × 10−10 s for the J = 0 to J = 1 lasing line at 44.83 Å for no trapping, trapping calculated with the line-transfer package, and trapping calculated by using escape probabilities. The spatial dependence of the velocity at this time is also shown.

Equations (17)

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g = π e 2 m e c f osc Ψ ( ν ) N U ( 1 - h U N L h L N U ) ,
Δ ν e ν c [ ( 1 + Z ) k T e / M i ] 1 / 2 ,
Δ ν e / Δ ν d ( 1 + Z ) 1 / 2 ( T e / 2 T i ) 1 / 2 .
Δ ν e > Δ ν 1 > Δ ν d .
I ν ( τ ν ) = I ν ( 0 ) exp ( - τ ν ) + 0 τ ν S ν ( t ) exp ( - t ) d t ,
S ν = η ν c + Σ l η l ψ ν l χ ν c + Σ l χ l ϕ ν l ,
J ¯ l = d Ω 4 π d ν ϕ ν l I ν .
Δ I ν = [ λ ν J ¯ S ν ( J ¯ ( 0 ) ) + λ ν ( J ¯ ( 0 ) ) S ν J ¯ ] Δ J ¯ ,
Δ J ¯ = d Ω 4 π d ν ϕ ν Δ I ν + [ J ¯ ( 1 ) - J ¯ ( 0 ) ] ,             J ¯ ( 1 ) = d Ω 4 π d ν ϕ ν I ν ( 0 ) .
( 1 - Λ ˜ ) Δ J ¯ = J ¯ ( 1 ) - J ¯ ( 0 ) ,
Λ ˜ = d Ω 4 π d ν ϕ ν ( λ ν J ¯ S ν + λ ν S ν J ¯ ) .
P S = 1 4 π d Ω 1 - exp ( - τ s ) τ s ,
P S ( R ) = 1 3 π 0 π d λ f ( x ) , f ( x ) = 1 x - 2 exp ( - x ) [ x K 0 ( x ) + ( 1 2 - x ) K 1 ( x ) ] ,
x τ s ( 1 + α cos λ ) ,             τ s = ξ · v ,             α ( v - v / R ) · v .
P S ( R ) = 1 3 τ s - 2 3 exp ( - τ s ) [ τ s K 0 ( τ s ) + ( 1 2 - τ s ) K 1 ( τ s ) ] .
P S ( τ s ) = 1 3 τ s { 1 + ( 2 τ s - 1 ) exp ( - τ s ) + 2 π τ s 3 / 2 [ erf ( τ s ) - 1 ] } ,
P static ( τ ) = - + ϕ ( x ) E 2 [ ϕ ( x ) τ ] d x ,

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