On the basis of this model, emission spectra of tungsten plasma were calculated, which are of interest in connection with studying high current mul tiwire liners.
ISSN 20700482, Mathematical Models and Computer Simulations, 2009, Vol. 1, No. 4, pp. 470–481. © Pleiades Publishing, Ltd., 2009. Original Russian Text © I.Yu. Vichev, V.G. Novikov, A.D. Solomyannaya, 2008, published in Matematicheskoe Modelirovanie, 2008, Vol. 20, No. 7, pp. 93–106.
Modeling of the Emission Spectra of Tungsten Plasma I. Yu. Vichev, V. G. Novikov, and A. D. Solomyannaya Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia Received November 11, 2006
Abstract—A model of nonequilibrium radiating multicharge ion plasma, which takes into account the radiation transport and level kinetics of electrons, is constructed. The model allows one to obtain the properties of plasma with arbitrary optical thickness. On the basis of this model, emission spectra of tungsten plasma were calculated, which are of interest in connection with studying highcurrent mul tiwire liners. Computations were carried out by the THERMOS & BELINE program package for plane and cylindrical plasma layers at temperatures T = 30–150 eV and densities ρ = 0.01–0.03 g/cm3. In a number of the most important cases, the computation results were corrected both for the spectral line positions and a more realistic geometry of the liner. DOI: 10.1134/S207004820904005X
INTRODUCTION The magnetic compression of plasma liners in highpower electric generators makes it possible to obtain dense hightemperature multicharge ion plasma and generate highlyintensive radiation [1, 2]. In plasma of relatively light elements, for example, aluminum or argon, Kline radiation in the range of 1– 3 keV is observed [3]. In plasma of heavy elements, in particular, tungsten plasma, the radiation is close to the thermal one with temperature T ~ 70–150 eV [4, 5]. The radiation transfer in such plasma is determined by absorption in strongly overlapped spectral lines. Due to a large number of transitions in multielectron ions as well as to the overlapping of lines and their broadening caused by collision processes and the trapping of radiation in most intensive lines, quanta dif fuse in energy space, forming a practically continuous emission spectrum. Although the radiation is close to equilibrium, its spectral distribution is not Planckian distribution [6]. Usually, the emission spectrum is characterized by the socalled brightness temperature, which corre sponds to the temperature of Planckian reduced to the obtained intensity, and with the spectral tempera ture, i.e., the temperature calculated from the maximum intensity of the emission spectrum. The bright ness and spectral temperatures of radiation may significantly differ from one another [4]. However, even when they are close to each other, the spectral distribution of the radiation cannot be presumed to be in equilibrium. As a rule, the emissivity of heavy elements is determined by groups of overlapped lines, which occupy certain spectral regions. By analogy with emissivity, absorption by plasma is characterized by spectral regions with strong absorption and windows of relative transparency. In spectral regions with high emis sion (in this region, as a rule, absorption is high, too), the spectrum tends to Planckian with a certain effec tive temperature. The extent to which the emission spectrum will be close to Planckian and whether the windows of transparency will be closed depends on the degree of equilibrium of the plasma, its optical thickness, and its inhomogeneity. It should be noted that experimental registration of a spectrum in the entire energy range is hampered by the large length of the range and high intensity of radiation. The experimental study of spectral char acteristics of highcurrent sources is extremely important, since it allows one to obtain additional data on the properties of highcurrent pinch plasma. In particular, the deviation of the spectrum from Planckian can serve as a basis for specifying the homogeneity of plasma and determining the temperature and density of plasma more precisely. Numerical modeling of the dynamics of radiating plasma usually employs rather crude approximations for describing the radiation transfer [7]. In order to find the spectra, a special calculation is made for given temperature and density distributions in which the collisional–radiative equilibrium of plasma in qua sistationary approximation is assumed [8, 9]. In this case, the state of plasma is determined by the free electron temperature and level kinetics in a nonequilibrium radiation field. The radiation field is deter mined by the geometry of the plasma layer and significantly depends on its optical thickness. 470
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The aim of this work is to carry out a preliminary theoretical study and analysis of the spectra of tung sten plasma in megaampere Zpinches. The main attention is paid to describing radiation excitations and their transfer with accounting for the local radiation field. In section 1, a short description is given of the model of level kinetics consistent with photon transport processes and based on the averageatom approx imation. In addition, the statement of the problem and method for solving the radiative transfer equation for plane and cylindrical plasma layers are presented, taking into account the dependence of transfer coef ficients over the radiation field. In section 2, the computation results for the emission spectrum of tungsten plasma and its main characteristics—the average ionization degree, radiation power loss, position on the main groups of lines, etc.—are analyzed. The computation results revealed typical features of tungsten plasma, which may be useful for performing experimental measurements. 1. A MODEL OF NONSTATIONARY NONEQUILIBRIUM RADIATING PLASMA 1.1. System of Equations for Ion Concentration In calculating the emission spectra for discharge plasma of heavy elements, it is necessary to solve the radiative transfer equation, in which the leading role belongs to radiation and absorption in spectral lines. In these conditions, level kinetics is coupled with radiation, i.e. radiation transfer influences the level kinetics and, vice versa, the level occupancies, which occurred as a result of the kinetic processes, deter mine the transfer coefficients. The problem requires a selfconsistent solution, which can be obtained by iterations of selfconsistent solutions for energy level occupations and the radiation field. In multicharge ion plasma, the number of ion states exceeds tens of thousands and the number of spec tral lines may be enormous. It is unlikely that a detailed analysis of a spectrum makes sense, since a num ber of factors such as plasma inhomogeneity slur the details of the spectrum and smooth its singularities. In this case, the most effective approaches are those based on the averageatom model, in which ion energy levels and, hence, spectral lines are united in groups, which significantly simplifies computations [10]. It is convenient to use a database computed beforehand that contains all data necessary for the com putations. Restricting oneself to a set of effective ion configurations, one has to calculate beforehand the average energies of the configurations, transition energies, and corresponding oscillator strengths. A spe cial averaging procedure enables one to use the results obtained both in the Hartry–Fock–Slater relativ istic averageatom approximation [10] by the THERMOS code and in more detailed computations taking into account the fine structure of a spectrum, for example, by the FAC code [11]. Along with other codes, the THERMOS and FAC codes took part in international conferences on the comparison of codes with each other and with experimental data [12–15]. In the THERMOS code, an ensemble of relativistic ions [16] is formed on the basis of the average ion by the perturbation theory [17] or on the basis of data computed beforehand by the FAC code. It should be noted that computations of spectra using the FAC code [11] are similar to computations by the model for an ensemble of relativistic ions but, unlike the approach in [16], the FAC code calculates the ion energy levels by the Hartry–Fock multiconfiguration relativistic method with accounting for the fine structure of levels. At present, this code is one of the most accurate codes for computing the atom structures of multicharge atoms. The use of the FAC code enables a more precise determination of spectral line positions, but it does not give a suf ficiently full consideration of configurations, in contrast to the averageatom model, which takes into account all the most probable configurations. Computations by the THERMOS code using the average ion model and data of the FAC code complement each other, enabling one to estimate the general view of the spectrum and correct the positions of the main lines. We will consider plasma obtained from atoms of one element with atomic number Z and consisting of electrons and ions of different multiplicities. We introduce the following notation: xks is the probability of ks
configuration Q with serial number s of an ion of multiplicity k with occupation numbers N nl for a level with the principal quantum number n and orbital quantum number l ( MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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nl
N nl = Z – k).
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We write the equations for the variation of concentrations xks in conditions when the effects of mass transfer may be neglected: ⎛ dx ks = – x ks ⎜ dt ⎝ +
∑x
∑
R ks → k – 1, s' +
s'
∑
I ks → k + 1, s' +
s'
k + 1, s' R k + 1, s' → j, s
s'
+
∑x
∑ s'
k – 1, s' I k – 1, s' → j, s
+
⎞ α ks → ks'⎟ ⎠
∑x
(1)
ks' α ks' → ks .
s'
s'
In the system of equations (1), the rate of recombination of an ion from state ks to state k – 1, s' in the averageatom approximation is reduced to the sum of the rates of the recombination to singleelectron levels nl: R ks → k – 1, s' =
∑ nl
ir
ks
N nl ⎞ ir phr ⎛ 1 – [ α nl ( ks → k – 1, s' ) + α nl ( ks → k – 1, s' ) ], ⎝ 2 ( 2l + 1 )⎠ phr
where α nl (ks → k – 1, s') and α nl (ks → k – 1, s') are the singleelectron rates of the threebody recom bination and photorecombination from state ks to state k – 1, s' with an electron transition from the con k – 1, s' ks = N nl + 1. tinuous spectrum to a singleelectron level with quantum numbers nl; in this case, N nl The ionization rate from state ks to k + 1, s' is written similarly: I ks → k + 1, s' =
∑N
ks ii nl [ α nl ( ks
phi
→ k + 1, s' ) + α nl ( ks → k + 1, s' ) ],
nl
ii
phi
where α nl (ks → k + 1, s') and α nl (ks → k + 1, s') are the rates of collision ionization and photoionization k + 1, s'
from state ks to state k + 1, s' with electron ionization nl; in this case, N nl
ks
= N nl – 1.
The rate of excitation ks → ks' (Eks < Eks') or quenching (Eks > Eks') is α ks → ks' =
∑∑ nl n'l'
cx
ks
E ks < E ks'
N n'l' ⎞ ks N nl ⎛ 1 – ⎝ 2 ( 2l' + 1 )⎠ abs
× [ α nl, n'l' ( ks → ks' ) + α nl, n'l' ( ks → ks' ) ] +
∑∑ nl n'l'
ks
E ks > E ks'
N n'l' ⎞ dex ks em N nl ⎛ 1 – [ α nl, n'l' ( ks → ks' ) + α nl, n'l' ( ks → ks' ) ]. ⎝ 2 ( 2l' + 1 )⎠ ks'
ks
ks'
Singleelectron dipole transitions nl → n'l' (l' = l + 1) are taken into account; N nl = N nl – 1 and N n'l' = ks
N n'l' + 1. At a given plasma temperature, the rates of collision processes are determined by electron den sity only and the rates of radiation processes substantially depend on the photon density, i.e., on the radi ation field. Expressions for these rates are given, for example, in [17]. dx The solution of the system (1) in the quasistatic approximation ⎛ ks = 0⎞ enables one to calculate the ⎝ dt ⎠ concentrations of ion states xks, necessary for calculating the transfer coefficients, and the average ioniza tion degree of plasma Z0 = ks kx ks .
∑
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1.2. Spectral Coefficients of Absorption and Radiation In the considered approximation, the absorption coefficient expressed in 1/cm has the form – ω/θ ρ ⎧ κ ω = N A ⎨ ( 1 – e ) A ⎩
+
∑x ∑ ks
ks
nl
∑
ks
∑
x ks
ks → ks'
ks
N n'l' ⎞ bb ks N nl ⎛ 1 – σ ks → ks' ( ω ) ⎝ 2 ( 2l' + 1 )⎠ (2)
⎫ bf ff ks N nl [ 1 – n ( ε ) ]σ nl ( ω ) + σ ( ω ) ⎬, ⎭
where ρ is the density of the substance in g/cm3, A is the atomic weight, NA is the Avogadro number, ω is the photon energy, θ is the temperature, n(ε) is the free electron distribution function of energy ε (ω, θ, and ε are expressed in atomic units): ε–µ n ( ε ) = 1 + exp ⎛ ⎞ ⎝ θ ⎠
–1
,
µ is the chemical potential of free electrons, σbb(ω), σbf(ω), and σff(ω) are the reduced cross sections of absorption in the lines, photoionization, and bremsstrahlung absorption, respectively (for example, see [17]). In formula (2), the scattering effects are neglected and the induced radiation is taken into account in the Kirchhoff approximation. The formula for calculating the spectral coefficients of radiation in TW/(cm3 eV steradian) can be writ ten as 3 ⎧ – 4 ρω j ω = 10 N A ⎨ A ⎩
+
∑x ∑ ks
ks
nl
∑
x ks'
∑
ks
ks' → ks
ks'
N nl ⎞ 2 ( 2l + 1 ) bb ks' N n'l' ⎛ 1 – σ ks → ks' ( ω ) ⎝ 2 ( 2l + 1 )⎠ 2 ( 2l' + 1 ) (3)
⎫ bf – ω/θ ff 2 ( 2l + 1 )n ( ε ) 1 – σ nl ( ω ) + e σ ( ω ) ⎬. 2 ( 2l + 1 ) ⎭ ks N nl
The reduced cross section of absorption in the lines is calculated by the formula [18] bb
2
2
σ nl, n'l' = 2π αa 0 f nl, n'l' Φ nl, n'l' ( ω ), where α is the finestructure constant, a0 is the Bohr radius, f nl, n'l' is the oscillator strength of singleelec tron transition nl → n'l', and Φ nl, n'l' (ω) is the spectral line profile. In the calculations, the Voigt profile was used in which, along with the Lorentz and Doppler broadenings, the broadening caused by line overlap ping was taken into account [10]. For the reduced cross section of photoionization and bremsstrahlung absorption, it is convenient to use Kramers approximation [18]. Then, the cross section of oneelectron photoionization from level nl with energy εnl and effective charge Znl has the form 3/2
bf σ nl ( ω )
2 Z nl ε nl 1 . = 64πα a 0 2 3 3 6 2n ω
The bremsstrahlung absorption cross section is determined via the average ionization degree of plasma Z0 and the chemical potential of free electrons µ, which presumably, are in equilibrium: ff 64πα 2 2 µ/θ 1 σ ( ω ) = a 0 Z 0 θe 3 . 3 3 ω
Z0 ρ For nondegenerate plasma, eµ/θ ≈ 0.7 (see [17]). 3/2 Aθ MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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1.3. Solving the Radiative Transfer Equation First, consider the simplest geometry of the plasma layer and the simplest way of taking into account the nonequilibrium of plasma. For a plane layer, the radiative transfer equation has the form dI (4) µ ω = j ω – κ ω I ω , dz where µ = cosϑ and Iω = Iω(z, ϑ) is the radiation intensity (see Fig. 1). If the absorption coefficients and emis sivity do not depend on z and external radiation sources are absent, the solution to Eq. (4) can be written as κ ω z⎞ jω I ω = 1 – exp ⎛ – , ⎝ µ ⎠ κω
µ > 0.
(5)
Integrating Eq. (5) with respect to the angles, we obtain the radiation flux density in the direction of the zaxis: π 2
F ωz =
∫I
2π ω cos ϑ sin ϑ dϑ
0
jω
[ 1 – 2E ( κ ∫ dϕ = π κ ω
0
3
ω z ) ],
where Ek(x) is the exponential integral ∞
Ek ( x ) =
∫t
–k –t
e dt.
x
The flux on the boundary of the layer is jω F ω z = L = π ( 1 – 2E 3 ( κ ω z ) ) κω
z=L
jω –τ –τ 2 = π [ 1 – e + τe – τ E 1 ( τ ) ], κω
where τ = κωL is the optical thickness. We assume that plasma is homogeneous in electron temperature and density and that occupation of ion states may depend on the zcoordinate: they are determined by the plasma’s optical thickness. In the case of a large optical thickness, thermodynamic equilibrium takes place and the ion concentration distribu tion satisfies the Saha equations. For low density transparent plasma, the coronal equilibrium takes place (for example, see [8]). In the general case, in order to approximate and take into account the nonequilib rium effects associated with the emission of radiation, we use a simple interpolation for the emission and absorption coefficients [17]. We introduce the effective escapefactor Fω z = L χ = , Fω ( τ → 0 ) where Fω|z = L is the radiation flux density on the boundary of the layer and Fω(τ → 0) is the radiation flux density in the case of transparent plasma: L
∫
F ω ( τ → 0 ) = 4π j ω dz.
L
0
Thus, in accordance with the interpolation method [19] for an arbi trary optical thickness, the absorption coefficients and emissivity take the form ϑ 0
Z
Fig. 1. Plane plasma layer of thickness L.
j ω = χj 0 + ( 1 – χ )j 1 ,
(6)
κ ω = χκ 0 + ( 1 – χ )κ 1 ,
(7)
where j0, κ0 and j1, and κ1 are the absorption coefficients and emissivities of optically thin and optically thick, respectively. It should be noted that plasma can be transparent only in a certain range of photon energy. By iteration, one can obtain a consistent solution to the problem for the radiative transfer equation (4) with coefficients (6) and (7). The method
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can be easily generalized to a nonhomogeneous case, in which the transfer equation must be solved numerically.
∂I sin ϕ ∂I sin ϑ ⎛ cos ϕ ω – ω⎞ + κ ω I ω = j ω , ⎝ ∂r r ∂ϕ ⎠
Ω
ϑ
1.4. Selfconsistent Solution for a Cylindrical Layer For an axially symmetric plasma layer, the radiative transfer equation has the form
x' ϕ r
(8)
where ϑ is the angle between the axis of the cylinder and the radia tion propagation direction, and ϕ is the azimuth angle (see Fig. 2). Equation (8) for given κω and jω was solved by integration along the rays:
Fig. 2. Cylindrical plasma layer of radius R.
τ
I ω ( r, ϑ , ϕ ) =
∫S
ω ( x' )e
τ' – τ
dτ',
(9)
0
where τ = τ(x)
∫
x 0
κ ω ( ξ ) dξ/sinϑ is the optical thickness and τ' = τ(x'). Coordinate x = x(r, ϕ) is taken along
the projection of the ray on the plane perpendicular to the zaxis. It is related to the radius r and the angle ϕ defining the direction of the ray’s projection on this plane by the formula x = where R is the radius of the cylinder. The source function is
2
2
2
R – r sin ϕ + rcosϕ,
jω ( r ) S ω ( r ) = . κω ( r ) In solving Eq. (9), one should take into account that only the outgoing radiation is present on the bound ary r = R, i.e., I ω r = R = 0 for π/2 < ϕ < 3π/2. The main laboriousness in computations was to calculate the ion state concentrations by solving the system of equations (1) and, subsequently, to calculate the absorption and radiation coefficients at each grid point r at a given temperature T and given radiation field Uω(r) = I ω sinϑdϑdϕ obtained in a pre
∫∫
vious iteration. Then, the radiative transfer equation with coefficients jω and κω calculated by formulas (2) and (3) was solved. A selfconsistent solution was obtained by double iteration as a result of the consequent correction of equations for ion concentrations (1) and the radiation field (8). Computations were performed by the MVC100k supercomputer of the Joint Supercomputer Center of the Russian Academy of Sciences. Usu ally, N processors were used, where N is the number of grid nodes with respect to radius r (N ~ 24). In cal culating flux F, the integration of Iω in (9) with respect to angle ϑ was performed analytically and, in the azimuth angle ϕ, 32 directions were taken. 2. COMPUTATION RESULTS The most largescale computations were carried out for a plane tungsten plasma layer at temperatures T ~ 30–150 eV and densities ρ ~ 0.01–0.03 g/cm3. The layer size R ~ 0.05–0.1 cm was chosen in accor Table 1. Average ionization degree Z0 and total radiation flux F for electron density Ne = 1021 1/cm3 and different temperatures T, eV ρ, g/cm3 Z0 F, W/cm2
40
50
60
70
80
90
100
110
120
130
140
150
0.022 14 0.22
0.018 16.6 0.54
0.017 18.5 1.06
0.015 20.3 1.80
0.014 21.9 2.74
0.013 23.5 3.97
0.012 25.1 5.37
0.011 27.2 6.81
0.011 29 8.66
0.01 30.3 11.2
0.01 31.6 13.8
0.01 32.8 16.2
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Table 2. Same as in Table 1 but for Ne = 3 × 1020 1/cm3 T, eV
40
50
60
70
80
90
ρ, g/cm3 Z0 F, TW/cm2
0.006 15.6 0.15
0.005 17.9 0.37
0.005 20 0.66
0.004 21.9 1.01
0.004 23.6 1.46
0.004 25.4 1.9
Table 3. Same as in Table 1 but for Ne = 3 × 1021 1/cm3 and different temperatures T, eV
100
110
120
130
140
150
ρ, g/cm3 Z0 F, TW/cm2
0.04 23.1 7.5
0.037 24.5 10.4
0.035 26.4 13.6
0.032 28.4 16.8
0.031 29.8 21.2
0.03 30.9 26.8
dance with experimental data [20]. On the basis of the computations, we obtained the parameters of equi librium pinch, for which the thermal equilibrium takes place, i.e., the power loss is compensated by the heating of plasma. With the obtained parameters, the problem for a cylindrical tungsten plasma layer was solved with a selfconsistent consideration of reabsorption effects, as described in subsection 1.4. More accurate computations have shown that the results for plane layers are qualitatively close to those obtained for a more realistic geometry of the plasma layer with an accurate consideration of nonequilibrium effects. Computation results for the average ionization degree and emission spectra of tungsten plasma for a plane layer of thickness L = 0.1 cm are presented in Tables 1–3 and Fig. 3. More detailed results for spectra of tungsten plasma are given in [21]. As is seen in Tables 1–3, the average ionization degree varies from 14 to 33 as the temperature varies from 40 to 150 eV. The dependence of the ionization degree on density is much weaker but noticeable. For typical sizes of a pinch of radius R = 0.1 cm and length l = 1 cm at temperature T = 70 eV and density ρ TW/cm2/eV
TW/cm2/eV T = 60 eV
T = 70 eV
0.004
0.006 0.004
0.002 0.002
0
200
400
600 T = 60 eV
0.004
T = 70 eV 0.006 0.004
0.002 0.002
0
200
400
600
0 Photon energy, eV
400 0
800
Fig. 3. Spectral flux of radiation by a plane tungsten plasma layer for electron density (top) Ne =1021 1/cm3 and (bottom) Ne =3 × 1020 1/cm3. Dashed line represents Planck’s blackbody radiation at a corresponding temperature. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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Table 4. Positions of peculiarities in tungsten spectrum (gaps in comparison with blackbody spectrum). The com putation was carried out for density Ne = 1021 1/cm3 at different temperatures. Values ε1, ε2 and ∆ε1, ∆ε2 characterize the position of centers and widths of two peculiarities (see Fig. 3) L, cm
T, eV
ρ, g/cm3
Z0
ε1, eV
∆ε1, eV
ε2, eV
∆ε2, eV
0.1 0.1 0.1 0.1 0.1
60 70 80 90 100
0.017 0.015 0.014 0.013 0.012
18.5 20.2 21.9 23.4 25
100 108 116 120 130
45 45 65 45 65
257 263 280 300 325
15 30 65 65 85
Table 5. Same as in Table 4 but for Ne = 3 × 1020 1/cm3 L, cm
T, eV
ρ, g/cm3
Z0
ε1, eV
∆ε1, eV
ε2, eV
∆ε2, eV
0.1 0.1 0.1 0.1
60 70 80 90
0.0046 0.0042 0.0039 0.0036
20 22 23.6 25.4
100 108 115 118
65 60 70 75
260 280 307 325
35 65 80 85
= 0.015 g/cm3, the total radiation power P = F2πRl is 1.1TW. Figure 3 shows the emission spectra of a plane tungsten plasma layer of thickness L = 0.1 cm at temperatures T = 60–70 eV. As is seen in Fig. 3, a typical feature of the emission spectrum of tungsten is a sharp decrease in the radiation intensity (gaps in the spectrum) in the ranges of 100–130 eV and 250–325 eV; the position of the gaps depends on temperature and density (see Tables 4 and 5). Let us take a closer look at the peculiarities of the spectrum of tungsten at temperature T = 70 eV and electron density Ne = 1021 1/cm3. Peaks in this given spectrum are characterized by the group of transitions in Cs, Xe, I, and Telike tungsten ions, i.e., in W +19, W +20, W +21, and W +22, respectively. The maxi mum emission peak at បω ~ 200 eV is due to the transitions 4d → 4f, the secondhighest peak in intensity at បω ~ 350 eV is due to transitions 4f → 5g. The main peaks are listed below: 4d 4f → 4d 4f
9
k+1
,
10
k
9
k+1
5s,
4d 4f 5p → 4d 4f
10
k
9
k+1
5p,
10
9
k+1
5s5p,
10
k
4d 4f 5s → 4d 4f
k
4d 4f 5s5p → 4d 4f 9
k
9
k–1
5s5g,
9
k
9
k–1
5p5g.
4d 4f 5s → 4d 4f 4d 4f 5p → 4d 4f
Now let us turn to discussing the approximations used in calculating the spectra of tungsten. The results presented in Fig. 3 correspond to the emission spectrum resolved in time for a plane homogeneous plasma layer. Of course, a plasma layer in the experiment is not a plane. A more realistic assumption is that the plasma is axially symmetric. Moreover, only an integral emission spectrum, i.e., the total emission for certain time interval, is usually detected. The simplest approximation of the lifetime τ (s) of a plasma layer of thickness R at temperature T is R τ = , v MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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where v is the plasma expansion velocity (cm/s). We will assume it to be approximately equal to its sound velocity
J/cm2/eV
v = 2.2 × 10
60 40
3θZ 0. 1836A
8
Summing up the spectra for different temperatures T = 40– 80 eV according to formula
20
Pω = 0
200
400 600 800 Photon energy, eV
Fig. 4. Spectral energy distribution of radiation by a plane tungsten plasma layer obtained by time inte gration in temperature interval T = 30–80 eV. Dashed line represents Planck’s blackbody radiation at tem perature T = 66.8 eV, which corre sponds to the total radiation power.
∫F
ω dt
τF ≈ ω dT, T
∫
we obtain an approximate timeintegrated spectrum (see Fig. 4, where the summation for a rough estimate was carried out at a fixed electron density Ne = 1021 g/cm3).
The spectrum obtained by the time integration corresponds to the radiation brightness temperature T = 66.8 eV. As expected, a comparison of the integral spectrum with the instant one at tem perature T = 70 eV (see Fig. 3) shows that, after the time integra tion, the peculiarities of the spectrum become less pronounced. For estimating the equilibrium parameters of the pinch, we use the simplest approximations. First of all, we estimate the equilib rium radius of the pinch (radius of neck) from the condition that the magnetic pressure equals the gas dynamic one (the Bennet condition): 2
B nkT = , 8π or 9 2
N a ( Z 0 + 1 )T = 3 × 10 I ,
(10)
where Na is the linear density in 1/cm and I is the current in A. Setting I = 3 MA, Na = 1018 1/cm, and Z0 ~ 30, we obtain T ~ 1000 eV, which contradicts the experi mental data [5]. Thus, either quasistatic equilibrium (10) is not fulfilled in practice or the plasma and cur rent distributions cannot be assumed to be homogeneous. A more reliable estimate for the temperature of the pinch is based on the thermal balance condi tion for a given sizes of constriction, i.e., on the condition that the Joule heating Qj equals the radi ation loss Qrad: Qj = Qrad. If Fω is the radiation flux from the surface of the cylindrical layer of radius R and length l, the radiation loss is determined by the formula
∫
Q rad = 2πRl F ω dω. The Joule heating is calculated as 2
I l , Q j = ˆ πR 2 σ where the conductivity in the Spitzer approximation [22] has the form 3/2
T ˆ = 85 , σ ΛZ 0 MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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Q, TW/cm Qrad Qj
Qrad Qj
10 I = 3 MA
1
I = 3 MA
I = 1.6 MA
40
80
120
I = 1.6 MA
40 T, eV
80
120
Fig. 5. Joule heating Qj and radiation loss Qrad of a cylindrical tungsten plasma layer of radius R = 0.07 cm versus temper ature T for density (left) Ne = 3 × 1020 1/cm3 and (right) Ne = 1021 1/cm3.
TW/cm2/eV
TW/cm2/eV T = 60 eV
T = 70 eV 0.006
0.003 0.003
0
400
800 0 Photon energy, eV
400
800
Fig. 6. Spectral flux of radiation by a cylindrical column of tungsten plasma of length l = 1 cm and radius R = 0.07 cm for electron density (left) Ne = 3 × 1020 1/cm3 and (right) Ne = 1021 1/cm3. Dashed line represents Planck’s blackbody radi ation at the corresponding temperature.
Λ is the Coulomb logarithm (see handbook [23]) ⎛ Z 0 N 1/2 ⎞ e Λ = 23 – ln ⎜ ⎟ , 3/2 ⎝ T ⎠ Ne is the electron density in 1/cm3, and T is the temperature in eV. Figure 5 shows Qrad and Qj as functions of temperature at currents I = 1.6.and 3 MA. The intersection of the graphs gives the equilibrium parameters of the pinch (see Table 6). The radius of the pinch was taken to be equal to 0.07 cm, which corresponds to the plane’s thickness layer of 0.1 cm. Table 6. Equilibrium parameters of a pinch of radius R = 0.07 cm and length l = 1 cm for different currents I, MA Ne , 1/cm3 T, eV Qj , Qrad , TW/cm
1.6 3 × 1020 52 0.3
3 1021 47 0.2
MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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VICHEV et al. TW/cm2/eV
TW/cm2/eV T = 40 eV
T = 70 eV 0.006
0.001 0.003
0
200
400 T = 60 eV
0
400
800 T = 80 eV
400
800
0.004 0.008 0.002
0
0.004
400
0 800 Photon energy, eV
Fig. 7. Spectral flux of radiation by a plane tungsten plasma layer of thickness T = 0.1 cm for electron density (top) Ne = 3 ×1020 1/cm3 and (bottom) Ne =1021 1/cm3 calculated from FAC code data. Dotted line represents results of com putations by averageatom model. Dashed line represents Planck’s blackbody radiation at the corresponding temperature.
For the obtained equilibrium parameters of the pinch, more detailed computations of the emission spectra of tungsten for a cylindrical plasma layer were carried out. The computations were carried out by the THERMOS and BELINE codes for temperature T ~ 40–150 eV and density ρ ~ 0.01 g/cm3 with the radius of the pinch R = 0.07 cm (see Fig. 6). After taking into account the axial symmetry and radiation capture in a homogeneous pinch, the spectra become close to the blackbody spectrum. For verifying the averageatom model, detailed computations of atomic data were carried out by the FAC code [11] with accounting for the fine structure of the emission spectrum. Using these data, compu tations of the spectra for a plane layer of thickness L = 0.1 cm were carried out for 4 values of temperature. The computation results are shown in Fig. 7, in which, along with the results obtained by the FAC code, the results of computing by the averageatom model are presented. It is seen in Fig. 7 that the aver ageatom model describes the positions of the main groups of lines rather well, although exaggerating their overlap. On the other hand, it is obvious that, despite the huge amount of atom data obtained by the FAC code, this amount is not sufficient for describing the spectrum near បω ~ 400 eV. The spectra obtained in this work correspond to plasma that is homogeneous over temperature and density. Moreover, the computations are based on the quasistatic approximation. However, the selfcon sistent model of radiating plasma used in this work takes into account the main effects associated with the radiation transfer and its influence on the level occupation kinetics. Thus, if radiation plays the leading role in the pinch formation, the results obtained in this work may correspond to a certain stage of pinch formation, namely, when the radiation is maximum. CONCLUSIONS Using the THERMOS and BELINE codes, emission spectra of tungsten plasma at different values of temperature, density, and optical thickness are obtained. For typical parameters of equilibrium pinch at current I ~ 2–3 MA, the results are corrected both for spectral line positions and a more realistic geometry of the liner. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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ACKNOWLEDGMENTS We are grateful to V.I. Zaitsev, E.V. Grabovskii, and G.S. Volkov for their numerous fruitful discussions. This work was supported by the Russian Foundation for Basic Research, project nos. 050100682 and 050216343. REFERENCES 1. Zakharov S.V., Smirnov V.P., Grabovskii E.V., Nedoseev S.L., Oleinik G.M., Zaitsev V.I. Imploding Liner as a Driver for Indirect Driven Target Physics Studies. Proc. of the IAEA Technical Committee Meeting on Drivers for Inertial Confinement Fusion, Paris, 1994 (IAEA, Vienna, 1995), p. 395. 2. Sanford T.W.L. et al. Characteristics and Dynamics of a 215eV DynamicHohlraum XRay Source on Z. Pro ceedings of 14th Int. Conf. Beams2002 and 5th Int. Conf. DZP2002 (Albuquerque, 2002). 3. Bailey J.E., Slutz S.A., Chandler G.A. et al. Spectroscopy of Argondopedcapsule Implosions Driven by zpinch Dynamic Hohlraum. 14th Int. Conf. Beams2002 and 5th Int. Conf. DZP2002 (Albuquerque, 2002). 4. Idzorek G.C., Chrien R.E., Peterson D.L., Watt R.G., Chandler G.A., Fehl D.L., Sanford T.W.L. Spectral out put of Zmachine implosions. 28th ICOPS 2001 and 13th IEEE Int. Pulsed Power Conf. (Las Vegas, 2001) 5. V. V. Alekxandrov, E. V. Grabovsky, G. G. Zakakishvili, et al., Currentinduced Implosion of a Multiwire Array as a Radial Plasma Rainstorm, JETP 97 (4), 745–753 (2003). 6. V. G. Novikov and S. V. Zakharov, Modeling of Nonequilibrium Radiating Tungsten Liners, Journal of Quanti tative Spectroscopy and Radiative Transfer 81, 339–354 (2003). 7. R. Benattar, S. V. Zakharov, A. F. Nikiforov, V. G. Novikov, et al., Influence of Magnetohydrodynamic Rayleigh Taylor Instability on Radiation of Imploded Heavy Ion Plasmas. Phys. Plasmas 6 (1), 175–187 (1999). 8. I. I. Sobelman, L. A. Vainshtein, and E. A. Yukov, Excitation of Atoms and Broadening of Spectral Lines (Nauka, Moscow, 1979) [in Russian]. 9. B. F. Rozsnyai, Collisionalradiative Averageatom Model for Hot Plasmas. Phys. Rev B, 55 (6), 7507–7521, (1997). 10. A. F. Nikiforov, V. G. Novikov, and V. B. Uvarov, QuantumStatistical Models of HighTemperature Plasma and Methods for Calculation of Rosseland Paths and Equations of State (Fizikomatematicheskaya Literatura, Mos cow, 2000) [in Russian]. 11. Gu Ming Feng, Atomic Processes in Plasmas: 14th APS Topical Conference on Atomic Processes in Plasmas. AIP Conference Proceedings, 730, 127–136 (2004), http://sprg.ssl.berkeley.edu/~mfgu/fac. 12. Final Report of the Third International Opacity Workshop and Code Comparison Study (Garshing: MaxPlanck Institut fur Quantenoptik. 1995). 13. Final Report of the Fourth International Opacity Workshop and Code Comparison Study (Madrid: Institute of Nuclear Fusion, 1998). 14. A. Rickert, “Review of the Third International Opacity Workshop and Code Comparison Study,” J. Quant. Spectrosc. Radiat. Transf 54 (1/2), 325–332 (1995). 15. G. Winhart, K. Eidmann, C. A. Iglesias, et al., “XUV Opacity Measurements and Comparison with Models,” J. Quant. Spectrosc. Radiat. Transf. 54 (1/2), 437–446 (1995). 16. O. B. Denisov and N. Yu. Orlov, The Theoretical Deduction of the Set of Equations for the Ion Model of Matter under Relativistic Approximation. Matem. Mod. 8 (2), 48–56 (1996). 17. A. F. Nikiforov, V. G. Novikov, and V. B. Uvarov, QuantumStatistical Models of Hot Dense Matter and Meth ods for Computation Opacity and Equation of State (Birkhauser Verlag, 2005). 18. I. I. Sobelman, An Itroduction to the Theory of Atomic Spectra (Nauka, Moscow, 1977) [in Russian]. 19. V. G. Novikov and A. D. Solomyannaya, Spectral Characteristics of Plasma Consistent with Radiation (preprint, Inst. Appl. Mathem., RAS, 1995, No. 85). 20. V. V. Aleksandrov, G. S. Volkov, E. V. Grabovskii, V. I. Zaitsev, G. G. Zukakishvili, S. F. Medovschikov, K. N. Mitrofanov, S. L. Nedoseev, G. M. Oleinik, I. Yu. Porofeev, A. A. Samokhin,P. V. Sasorov, V. P. Smirnov, I. N. Frolov, M. V. Fedulov, Peculiarities of Wire Array Implosion, Dense ZPinches: 6th International Confer ence on Dense ZPinches, 808, 3–8 (2006). 21. V. G. Novikov, A. D. Solomyannaya, and I. Yu. Vichev, Modeling of Emission Spectra of Tungsten Plasma (pre print, Inst. Appl. Math., RAS, 2006, No. 54). 22. S. I. Braginskii, Transfer Phenomena in Plasma, in Problems of Plasma Theory, Ed. by M. A. Leontovich (Gos atomizdat, Moscow, 1963) [in Russian]. 23. J. D. Huba, NRL Plasma Formulary (NRL, Washington, DC, 1994), p. 36. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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