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Calculation of the electron distribution function of a weakly ionized plasma in time-dependent electric and magnetic fields
Volume 27A, number 2
PHYSIC S LETTER S
3 June 1968
including in it the calculation of photoexcitation cross-sections, interference between direct and resonance processes being taken int. account. We have carried out our calculations using &)08
the six-parameter function from [6]. The calcu-. lated resonance profile indices, q~,are presented in table 1. The calculated curve of photoionization near the lowest resonance E 1 60.12 eV is shown in fig. 1 together with the corresponding experimental data [7]. The authors are indebted to Dr. Altick for his kind additional information about the wave functions presented in [1].
002
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-20
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E
0
10
20
-
Fig. 1. Photoionization of He in the region of 60.12 eV resonance. differ somewhat from the results obtained by Altick and Moore [2] but to a considerable less extent than these authors’ results differ from those by Burke and McVicar [4] obtained by the method of close coupling. The success of the diagonalization approach for the description of position and width of resonances (see also [5]) shows that in the problem discussed the coupling between the discrete and continuous spectra is actually weak. This makes it possible to extend significantly the sphere of applicability of the diagonalization procedure by
References
1. R. Propin, Opt, and Spectr. (USSR) 8 (1960) 300; P. L . Altick and E. N. Moore. Phys. Rev. Letters 15 (1965) 100; Lipsky and Phys. Rev.Rev. 142 147 (1966) 59. 2. L. P.L.Altick andA.Russek. E.N.Moore. Phys. (1966) 47. 3. IJ.Fano, Phys. Rev. 124 (1961) 1866. 4. P.G.Burke and D.McVicar, Proc. Phys. Soc. 86 (1965) 988. 5. L.Altick and 6. P. A.L.Stewart andE.N.Moore, T.G.Webb, Proc. Proc. Phys. Phys. Soc. Soc. 92 82 (1963) 532. 7. R. P. Madden and K. Codling, Astrophys. J. 141 (1965) 364.
CALCULATION OF THE ELECTRON DISTRIBUTION FUNCTION OF A WEAKLY IONIZED PLASMA IN TIME-DEPENDENT ELECTRIC AND MAGNETIC FIELDS W. STILLER Arbeitsstelle fUr Statistische Physik der Deutschen Akademie der Wissenschaften zu Berlin, Leipzig, Germany (DDR) Received 6 May 1968
With help of the Boltzmann equation the electron distribution function of weakly ionized plasmas subjected to homogeneous time-dependent electric and magnetic fields is calculated.
Let us assume that the heavy particles of a weakly ionized plasma have a Maxwellian distribution. For the electron distribution function f(r, v,t) the Lorentz ansatz is made. The coupled system of differential equations (obtained from the Boltzmann equation) for the calculation of the parts f°(r, v t) and f’(r, vj, t) is then given by ref. 1. With this system the following field configuration is considered: a homegeneous time-dependent electric field E(t) a(t) = qE(t)/m (q/ m electronic charge-to-mass ratio);
I
102
,
Volume 27A, number 2
PHYSICS
LETTERS
3 June 1968
a homogeneous direction-constant time-dependent magnetic field B(t) : w~(t)= qB~(t)/m. For the induced electric field E~(r, t) one obtains from Faraday’s induction law:
a1(r,t)
=
qE1/m
=
~‘~{~i
-x10}
Making use of the abbreviations w(r,v,t) =f~(r,v,t)÷f(r,v,t)
A~(t)= a~(t)+
ia~(t)
, ,
At~(x,y,t)= a~(y,t) +ia~~(x,t)
=
v
=
v
(-~+
j-~-)
lvi
A~= a~
=
v
the differential equation for the destination of w becomes +
i [Wc(t)
v(v)]W=
-
(.Qwf° + [A~,÷A~~]~_)
=
S~,(r,v,t)
electron-neutral collision frequency). 1 In a similar fashion the differential equation forf~ may be written down. The definition of S~(r,v,1) can be obtained by replacing the subscript w with z in the last equation. Let us now consider the solutions of the inhomogeneous differential equations for u’ and f~: (ii:
a’
=
‘~v(r,v, t)
fl =lflhfl(rvt)=
=
J
d rS~(r,v, T) e~[i fTdt,(wc(t,)
~fdTSz(r,v,T)e~[P(v)r4
- iP(V))]~ exp
-i
f
dt’[w~(t’) - iv(v)]~
exp{-~(v)t}
Putting these formal solutions - which still contain the unknown isotropic partf° - into the differential equation forf° [1], one gets a linear integro-differential equation for f°(r,v, t) [2,3]. With further assumptions (for instance consideration of homogeneous and stationary states) it is possible to obtain analytic expressions forf°. Oncef° is known one can use it for the final calculation of u’ andf~’. It must be pointed out that the hitherto existing theory, which originally is only applicable to direction-constant magnetic fields, makes it possible to treat timely circularly polarized magnetic fields, too [2,3]. The author wishes to express his thanks to Professor Vojta, director of the Arbeitsstelle, for many helpful discussions.
References 1. V.L.Ginzburg and A. V.Gurevi& Fortschr. Phys. 8 (1960) 97. 2. W.Stiller, Beitr~gePlasmaphys. 7(1967) 507. 3. W. Stiller and G. Vojta, Z. Naturforschg, to be published.
103
PHYSIC S LETTER S
3 June 1968
including in it the calculation of photoexcitation cross-sections, interference between direct and resonance processes being taken int. account. We have carried out our calculations using &)08
the six-parameter function from [6]. The calcu-. lated resonance profile indices, q~,are presented in table 1. The calculated curve of photoionization near the lowest resonance E 1 60.12 eV is shown in fig. 1 together with the corresponding experimental data [7]. The authors are indebted to Dr. Altick for his kind additional information about the wave functions presented in [1].
002
-40
-30
-20
-10 I
E
0
10
20
-
Fig. 1. Photoionization of He in the region of 60.12 eV resonance. differ somewhat from the results obtained by Altick and Moore [2] but to a considerable less extent than these authors’ results differ from those by Burke and McVicar [4] obtained by the method of close coupling. The success of the diagonalization approach for the description of position and width of resonances (see also [5]) shows that in the problem discussed the coupling between the discrete and continuous spectra is actually weak. This makes it possible to extend significantly the sphere of applicability of the diagonalization procedure by
References
1. R. Propin, Opt, and Spectr. (USSR) 8 (1960) 300; P. L . Altick and E. N. Moore. Phys. Rev. Letters 15 (1965) 100; Lipsky and Phys. Rev.Rev. 142 147 (1966) 59. 2. L. P.L.Altick andA.Russek. E.N.Moore. Phys. (1966) 47. 3. IJ.Fano, Phys. Rev. 124 (1961) 1866. 4. P.G.Burke and D.McVicar, Proc. Phys. Soc. 86 (1965) 988. 5. L.Altick and 6. P. A.L.Stewart andE.N.Moore, T.G.Webb, Proc. Proc. Phys. Phys. Soc. Soc. 92 82 (1963) 532. 7. R. P. Madden and K. Codling, Astrophys. J. 141 (1965) 364.
CALCULATION OF THE ELECTRON DISTRIBUTION FUNCTION OF A WEAKLY IONIZED PLASMA IN TIME-DEPENDENT ELECTRIC AND MAGNETIC FIELDS W. STILLER Arbeitsstelle fUr Statistische Physik der Deutschen Akademie der Wissenschaften zu Berlin, Leipzig, Germany (DDR) Received 6 May 1968
With help of the Boltzmann equation the electron distribution function of weakly ionized plasmas subjected to homogeneous time-dependent electric and magnetic fields is calculated.
Let us assume that the heavy particles of a weakly ionized plasma have a Maxwellian distribution. For the electron distribution function f(r, v,t) the Lorentz ansatz is made. The coupled system of differential equations (obtained from the Boltzmann equation) for the calculation of the parts f°(r, v t) and f’(r, vj, t) is then given by ref. 1. With this system the following field configuration is considered: a homegeneous time-dependent electric field E(t) a(t) = qE(t)/m (q/ m electronic charge-to-mass ratio);
I
102
,
Volume 27A, number 2
PHYSICS
LETTERS
3 June 1968
a homogeneous direction-constant time-dependent magnetic field B(t) : w~(t)= qB~(t)/m. For the induced electric field E~(r, t) one obtains from Faraday’s induction law:
a1(r,t)
=
qE1/m
=
~‘~{~i
-x10}
Making use of the abbreviations w(r,v,t) =f~(r,v,t)÷f(r,v,t)
A~(t)= a~(t)+
ia~(t)
, ,
At~(x,y,t)= a~(y,t) +ia~~(x,t)
=
v
=
v
(-~+
j-~-)
lvi
A~= a~
=
v
the differential equation for the destination of w becomes +
i [Wc(t)
v(v)]W=
-
(.Qwf° + [A~,÷A~~]~_)
=
S~,(r,v,t)
electron-neutral collision frequency). 1 In a similar fashion the differential equation forf~ may be written down. The definition of S~(r,v,1) can be obtained by replacing the subscript w with z in the last equation. Let us now consider the solutions of the inhomogeneous differential equations for u’ and f~: (ii:
a’
=
‘~v(r,v, t)
fl =lflhfl(rvt)=
=
J
d rS~(r,v, T) e~[i fTdt,(wc(t,)
~fdTSz(r,v,T)e~[P(v)r4
- iP(V))]~ exp
-i
f
dt’[w~(t’) - iv(v)]~
exp{-~(v)t}
Putting these formal solutions - which still contain the unknown isotropic partf° - into the differential equation forf° [1], one gets a linear integro-differential equation for f°(r,v, t) [2,3]. With further assumptions (for instance consideration of homogeneous and stationary states) it is possible to obtain analytic expressions forf°. Oncef° is known one can use it for the final calculation of u’ andf~’. It must be pointed out that the hitherto existing theory, which originally is only applicable to direction-constant magnetic fields, makes it possible to treat timely circularly polarized magnetic fields, too [2,3]. The author wishes to express his thanks to Professor Vojta, director of the Arbeitsstelle, for many helpful discussions.
References 1. V.L.Ginzburg and A. V.Gurevi& Fortschr. Phys. 8 (1960) 97. 2. W.Stiller, Beitr~gePlasmaphys. 7(1967) 507. 3. W. Stiller and G. Vojta, Z. Naturforschg, to be published.
103