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Transient ionization time scales for low atomic number elements
Computer Physics Communications 38 (1985) 359—363 North-l-Iolland. Amsterdam
359
TRANSIENT IONIZATION TIME SCALES FOR LOW ATOMIC NUMBER ELEMENTS R. MARCHAND, R. RANKIN and C.E. CAPJACK Department of Electrical Engineering, Unii’ersitr of Alberta. Edmonton. Alberta TOG 2G7. Canada Received 17 December 1984
The transient time for ionization and recombination is estimated as a function of the electron temperature and density, for various low atomic number elements, assuming a collisional radiative model. These estimates are useful for choosing a model for the ionization dynamics in plasmas.
In this paper, a relatively simple model is presented which can be used to estimate transient ionization—recombination timescales of low atomic number laser targets. In simulations of laser plasma interactions, knowledge of these timescales enables an appropriate choice of atomic physics model to be made. Computer modelling is complicated by the multitude of physical processes which occur simultaneously in plasmas. In particular, one major difficulty is to account for the detailed distribution of ionization levels of the various species. In order to precisely account for the ionization dynamics of a given species, for example, one should know at which rate a given atom ionizes or recombines as a function of the local plasma parameters. Ideally, the resulting rate equations should be solved numerically, together with the other equations describing the medium. Such a detailed description, however, is often prohibitively complicated, and simplifications must be made. This is particularly true in large where two- or three-dimensional simulations where the distribution of ionization levels must be calculated in every cell at each time step. A number of techniques have been proposed to accompush this [1—10]. Here we are interested in provid• . . . ing a way to assess the validity of a simple model, one which consists of assuming a local and • instantaneous steady state. Generally, the steady state assumption is valid provided that the time T required to reach that steady state, from a slightly 0010-4655/85/$03.30 © Elsevier Science Publishers (North-Holland Physics Publishing Division)
perturbed state, be smaller than the time
T
11
required for the macroscopic parameters to change significantly. In the opposite limit, the rate equations must be solved, and the solution shows a sensitive dependence on the plasma initial condition or past history. The macroscopic time scale can often be estimated experimentally, or from a crude numerical simulation. It is important. therefore, to have an estimate of the transient time T to compare to the macroscopic timescale T,11. The ionization dynamics are described using the collisional radiative model. This model has been described elsewhere [11—15]. In short, it accounts for collisional ionization, together with radiative, dielectronic and three body recombination. The plasma is assumed to be optically thin and, thus, photoabsorption is neglected. The resulting rate equations are of the form dF/dt=AF,
(1)
F is a vector of components f which represent the fraction of atoms / — I times ionized. For an element of atomic number n — i, t would range from I to n. The electron density n. is computed from F and the ion density n as n.=n
(i-1)f.
The matrix A is tridiagonal of the form
B.V.
(2)
36(1
P. Marc/uind ci id.
0 S
—(S~+R~)
0
5.
Iranoeni ionization (mi’ scale,~
//
0
1
0 ~(S~+R~)
R
4
H5, 0 where the (positive) constants S and R, are the ionization and recombination rates, respectively, for the ionization level /. The rates used here have been computed and tabulated as a function of the electron temperature and density using the cornputcr code CRAFTY [16]. The steady state corresponds to the 1of Asolution associated with eigenvalue eigcnvector F~’ zero. We note that A is singular and indeed has a zero eigenvalue because the sum of all the dcments in each column vanishes. a,,
=
0.
/= 1
.11.
(4)
I
Of course, in order for F111 to represent a physical quantity: i.e.. for its components to represent fraclions of ionization levels, it must he possible to normalize it such that all its components he positive. This is indeed the case, as for any matrix of the form (3). provided that the rates S and R, he positive. More generally. it can he shown th~itif initially all the components of F are positive, they remain so at later times. Also, the sum of the components f can he shown to he independent of tinic.
Let us now estimate the transient timescale T. In general. the time required to approach steady state depends Ofl the initial plasma conditions. For example. because the rates scale with the electron density, it will take longer to reach a given steady state if the plasma is initially weakly ionized, than it would if it were fully ionized. For simplicity, in what follows, we assume that the plasma is mitiall close to a steady state and linearize the rate equation about that equilibrium. With F= J + fl I) eq. (1) yields to first order in the perturhation: d F’ ‘/dt
=
A,, F’ I
I +
A F”1.
(5)
(3)
0 +
R,
~)
R,,
5,, where A is computed with the assumed steady state electron temperature and density. and A accounts for the changes in the rate coefficients caused by the perturbation. At this point one could imagine several scenarios under which the perturbation could take place. For example. if it is homogeneous in space if the transient is short compared to theand hydrodynamic timetime scale, the ion density should he constant. If, in addition, the physical process involves atoms being ionized (which adds free electrons to the system). the total internal energy per unit volume should be constant. If. on the other hand, it causes electrons to recombine, energy is lost through radiation and it is the electron temperature which should he constant. For simplicity, let us assume that the perturhation only takes place in a small volume and take both the electron density and temperature to he constant. Although of limited validity, this assumption has the advantage that it greatly simplifies eq. (5) and it is sufficient for the order of magnitude estimate required here. Specifically. hecause ti~ and 1~.are constants, the rates S and R, are not affected by the perturbation and A~ vanishes in eq. (5). The solution to the equation can he written as F11 = Ye5’ and the problem reduces to that of finding the non-zero eigenvalues A of the matrix A~.The transient time scale is then the reciprocal of the smallest real part of A in absolute value. It can be shown that all the nonzero eigenvalues have a negative real part, as cxpected if the perturbed distribution is to approach the steady state solution. The results are shown in figs. I to 7 for the elements hydrogen to oxygen, with the exception of nitrogen. Lines of constant timescale r (solid) and constant ion density ii (dashed) are plotted in the T,. n~ plane. Also shown is the boundary (dotted) between the collisional and collisionless regimes. On the low temperature side of that
R. Marchand ci al.
.
102’
/
21 _____________________________________________
,
~i”=1O1~
100
~e
101
361
Transient ionization time scales
102
io~
(cm~)
100
101
102
Te (ev) Fig. 1. Lines of constant ionization—recomhination transient times T (solid) and ion densities n (dashed) in the n~,7, plane for hydrogen.
~
The dotted line shows where the electron ;q equals the transient time r. The values of
1o~
Te (ev) Fig. 2 Same as fig. 1. hut for helium.
//7/~\ /
102’
,.--
units) which fall within the range of electron temperatures and densities considered.
-
-
I ‘
-—
-
1019
boundary, the electron—electron equilibration time
T~
shorter than the transient time T, On the high temperature side, it is longer. The equilibration time is theattime that it would take temperature for a group to of electrons a slightly different equilibrate, or thermalize, with the bulk of surrounding electrons [17]. It is of interest to note that the rates used in the collisional radiative model have been computed assuming a Maxwelhan electron distribution function. In order for the model to be consistent, the macroscopic time scale must therefore be longer than the equilibration time, r,,, > i~. For example, if ;, ;q’ the dcctrons will remain Maxwellian (and the collisional radiative model will be valid) in the collisional regime. They are likely to become non-Maxwelhian in the collisionless regime, where the collisional
/ 1O \ /
~10~
.‘
10 19
/A
4 is
0e (cm3)
.
.
----
------
---
~
1017
.--~ -
--
-_../
~.
~. n=105cm~3 .~‘
1015
100
-
.‘
“
~ 101
102
Te (ev) Fig. 3. Same as fig. 1. but for lithium.
10~
362
R. lIon-hand ci iii.
/
Transient i,,flizaiioii
(lOll’ cia/el
~i~0s101~
1 01 7
1017
,///////•==\\\
1o’~ 100
101
,
102
10~
i0~
1015 100
./~—~-‘‘T~\ 101
102
Te (ev)
Te (ev)
Fig. 4. Same as fig. 1. hut for her hum.
10 21
Fig. 6. Same as fig. 1. hut for carbon.
1021
/
m~
1 017
~
1 ~1 100 ~ 101 Te102 (ev) Fig. 5. Same as fig. I. hut for boron.
,.
,..
-
/ --I
-
~o17
1 01 7
10~
10~l
1011 o°
T e (ev)
Fig. 7. Same as fig. 1, hut for oxygen.
10~
4
1o
R. Marchand e~’a!.
/
Transient ionization tIme .1 c-ales
radiative model would he invalid. Of course, the actual macroscopic time scale depends on how rapidly the system is driven externally. As an example of how our figures can be used, let us consider two experiments in which a plasma is formed by irradiating a gas with a laser beam. In one [18], the gas consists of oxygen and the hydrodynamic evolution of the plasma is observed to occur on a time scale of the order of three nanoseconds. The electron density and temperature are of order 0.5 x lots cm and 120 eV, respectively. It follows from fig. 7. that the transient ionization time is longer than the macroscopic time scale, and a computer simulation of this experiment would require a detailed solution of the rate equations. In another experiment [19], performed with helium, the plasma evolves on a time scale of order of hundreds of nanoseconds. with 7~ 3 eV. and n~ IO~cm or more. The transient time scale obtained from fig. 2 in that range of parameters is of order of tens of nanoseconds. The steady state model therefore appears adequate to simulate this experiment. To conclude, we have produced a set of graphics which display the transient tiniescale T for ionization or recombination as a function of the plasma parameters for various low atomic number elements. These times can be compared to the time ‘r,, which characterizes changes in the macroscopic plasma parameters, and can be used to select a particular model to describe ionization dynamics. The figures can also be used to estimate the region of parameter space where the electron distribution function is likely to remain Maxwellian.
363
Acknowledgements The authors gratefully acknowledge the continuing financial support of the Natural Sciences and Engineering Research Council of Canada.
References [I] G.J. Pert. J. Comput. Phys. 27 (1978) 241. [2] J. Magihi. J. Phys. D 10(1977) 2257. [3] J. Magilh. (ornput. Phys. Commun. 16 (1978)1 29. 14] (11 Pert. J (omput. Phys 39 (1980) 251. [5] R.C. Mancini and CF. Fontan. Cornput. Phys. Common. 21(1981) 293. [6] J. Magill. Comput. Ph~s.Commun. 21 (1981) 295. [7] NA. Tahire and LW. Laing. Phys. I.ett. A 79 (1980) 321. [81 A. Shalom. .1. Phys. 1) 15)1982) L21. [91J.C Gauthier. J -P Geindre. N. Grandjouan and .1 Virniont, J. Ph~s.D 16 (1983) 321.
[10] iN. McMullin and CE. Capjack, Comput. ~
Corn-
mun. 32 (1984) II. 111] D. Bates. A. Kingston and R. McWhirter. Proc. Ro~.Soc. London A267 (1962) 297. [12] D. Duston and ii. Duderstadt. J Apph Phvs 49 (1978) 4388. [13] M.K. Matzcn and J.S. Pcarlman. Phys. Fluids 22 (1979) 449.
[141 D.
Duston and J. Davis. Phys. Rev. 23A (1980) 2602. [15] 1). Duston. R.W. Clark, J. Davis and JR Apruzese. Phys. Rev 27A (1983) 1441 [16] D. Quimby.
Mathematical Sciences North
Memorandum CRAFTY
--
West,
Inc
A (‘ollisional-Radiative Atomic
Physics Generator (1979) (unpublished). [17] L. Spitzer Jr., Physics of Fully Ionized Gases (lnterscience. New York, 1956). 118] R Gihes, Ph.D dissertation, Univ of Alberta (1983).
[191 A. Giulietti.
D. Giuletti. M Lucchesi and M Vavelli. Opt
Conlnlun. 47)1983)131.
359
TRANSIENT IONIZATION TIME SCALES FOR LOW ATOMIC NUMBER ELEMENTS R. MARCHAND, R. RANKIN and C.E. CAPJACK Department of Electrical Engineering, Unii’ersitr of Alberta. Edmonton. Alberta TOG 2G7. Canada Received 17 December 1984
The transient time for ionization and recombination is estimated as a function of the electron temperature and density, for various low atomic number elements, assuming a collisional radiative model. These estimates are useful for choosing a model for the ionization dynamics in plasmas.
In this paper, a relatively simple model is presented which can be used to estimate transient ionization—recombination timescales of low atomic number laser targets. In simulations of laser plasma interactions, knowledge of these timescales enables an appropriate choice of atomic physics model to be made. Computer modelling is complicated by the multitude of physical processes which occur simultaneously in plasmas. In particular, one major difficulty is to account for the detailed distribution of ionization levels of the various species. In order to precisely account for the ionization dynamics of a given species, for example, one should know at which rate a given atom ionizes or recombines as a function of the local plasma parameters. Ideally, the resulting rate equations should be solved numerically, together with the other equations describing the medium. Such a detailed description, however, is often prohibitively complicated, and simplifications must be made. This is particularly true in large where two- or three-dimensional simulations where the distribution of ionization levels must be calculated in every cell at each time step. A number of techniques have been proposed to accompush this [1—10]. Here we are interested in provid• . . . ing a way to assess the validity of a simple model, one which consists of assuming a local and • instantaneous steady state. Generally, the steady state assumption is valid provided that the time T required to reach that steady state, from a slightly 0010-4655/85/$03.30 © Elsevier Science Publishers (North-Holland Physics Publishing Division)
perturbed state, be smaller than the time
T
11
required for the macroscopic parameters to change significantly. In the opposite limit, the rate equations must be solved, and the solution shows a sensitive dependence on the plasma initial condition or past history. The macroscopic time scale can often be estimated experimentally, or from a crude numerical simulation. It is important. therefore, to have an estimate of the transient time T to compare to the macroscopic timescale T,11. The ionization dynamics are described using the collisional radiative model. This model has been described elsewhere [11—15]. In short, it accounts for collisional ionization, together with radiative, dielectronic and three body recombination. The plasma is assumed to be optically thin and, thus, photoabsorption is neglected. The resulting rate equations are of the form dF/dt=AF,
(1)
F is a vector of components f which represent the fraction of atoms / — I times ionized. For an element of atomic number n — i, t would range from I to n. The electron density n. is computed from F and the ion density n as n.=n
(i-1)f.
The matrix A is tridiagonal of the form
B.V.
(2)
36(1
P. Marc/uind ci id.
0 S
—(S~+R~)
0
5.
Iranoeni ionization (mi’ scale,~
//
0
1
0 ~(S~+R~)
R
4
H5, 0 where the (positive) constants S and R, are the ionization and recombination rates, respectively, for the ionization level /. The rates used here have been computed and tabulated as a function of the electron temperature and density using the cornputcr code CRAFTY [16]. The steady state corresponds to the 1of Asolution associated with eigenvalue eigcnvector F~’ zero. We note that A is singular and indeed has a zero eigenvalue because the sum of all the dcments in each column vanishes. a,,
=
0.
/= 1
.11.
(4)
I
Of course, in order for F111 to represent a physical quantity: i.e.. for its components to represent fraclions of ionization levels, it must he possible to normalize it such that all its components he positive. This is indeed the case, as for any matrix of the form (3). provided that the rates S and R, he positive. More generally. it can he shown th~itif initially all the components of F are positive, they remain so at later times. Also, the sum of the components f can he shown to he independent of tinic.
Let us now estimate the transient timescale T. In general. the time required to approach steady state depends Ofl the initial plasma conditions. For example. because the rates scale with the electron density, it will take longer to reach a given steady state if the plasma is initially weakly ionized, than it would if it were fully ionized. For simplicity, in what follows, we assume that the plasma is mitiall close to a steady state and linearize the rate equation about that equilibrium. With F= J + fl I) eq. (1) yields to first order in the perturhation: d F’ ‘/dt
=
A,, F’ I
I +
A F”1.
(5)
(3)
0 +
R,
~)
R,,
5,, where A is computed with the assumed steady state electron temperature and density. and A accounts for the changes in the rate coefficients caused by the perturbation. At this point one could imagine several scenarios under which the perturbation could take place. For example. if it is homogeneous in space if the transient is short compared to theand hydrodynamic timetime scale, the ion density should he constant. If, in addition, the physical process involves atoms being ionized (which adds free electrons to the system). the total internal energy per unit volume should be constant. If. on the other hand, it causes electrons to recombine, energy is lost through radiation and it is the electron temperature which should he constant. For simplicity, let us assume that the perturhation only takes place in a small volume and take both the electron density and temperature to he constant. Although of limited validity, this assumption has the advantage that it greatly simplifies eq. (5) and it is sufficient for the order of magnitude estimate required here. Specifically. hecause ti~ and 1~.are constants, the rates S and R, are not affected by the perturbation and A~ vanishes in eq. (5). The solution to the equation can he written as F11 = Ye5’ and the problem reduces to that of finding the non-zero eigenvalues A of the matrix A~.The transient time scale is then the reciprocal of the smallest real part of A in absolute value. It can be shown that all the nonzero eigenvalues have a negative real part, as cxpected if the perturbed distribution is to approach the steady state solution. The results are shown in figs. I to 7 for the elements hydrogen to oxygen, with the exception of nitrogen. Lines of constant timescale r (solid) and constant ion density ii (dashed) are plotted in the T,. n~ plane. Also shown is the boundary (dotted) between the collisional and collisionless regimes. On the low temperature side of that
R. Marchand ci al.
.
102’
/
21 _____________________________________________
,
~i”=1O1~
100
~e
101
361
Transient ionization time scales
102
io~
(cm~)
100
101
102
Te (ev) Fig. 1. Lines of constant ionization—recomhination transient times T (solid) and ion densities n (dashed) in the n~,7, plane for hydrogen.
~
The dotted line shows where the electron ;q equals the transient time r. The values of
1o~
Te (ev) Fig. 2 Same as fig. 1. hut for helium.
//7/~\ /
102’
,.--
units) which fall within the range of electron temperatures and densities considered.
-
-
I ‘
-—
-
1019
boundary, the electron—electron equilibration time
T~
shorter than the transient time T, On the high temperature side, it is longer. The equilibration time is theattime that it would take temperature for a group to of electrons a slightly different equilibrate, or thermalize, with the bulk of surrounding electrons [17]. It is of interest to note that the rates used in the collisional radiative model have been computed assuming a Maxwelhan electron distribution function. In order for the model to be consistent, the macroscopic time scale must therefore be longer than the equilibration time, r,,, > i~. For example, if ;, ;q’ the dcctrons will remain Maxwellian (and the collisional radiative model will be valid) in the collisional regime. They are likely to become non-Maxwelhian in the collisionless regime, where the collisional
/ 1O \ /
~10~
.‘
10 19
/A
4 is
0e (cm3)
.
.
----
------
---
~
1017
.--~ -
--
-_../
~.
~. n=105cm~3 .~‘
1015
100
-
.‘
“
~ 101
102
Te (ev) Fig. 3. Same as fig. 1. but for lithium.
10~
362
R. lIon-hand ci iii.
/
Transient i,,flizaiioii
(lOll’ cia/el
~i~0s101~
1 01 7
1017
,///////•==\\\
1o’~ 100
101
,
102
10~
i0~
1015 100
./~—~-‘‘T~\ 101
102
Te (ev)
Te (ev)
Fig. 4. Same as fig. 1. hut for her hum.
10 21
Fig. 6. Same as fig. 1. hut for carbon.
1021
/
m~
1 017
~
1 ~1 100 ~ 101 Te102 (ev) Fig. 5. Same as fig. I. hut for boron.
,.
,..
-
/ --I
-
~o17
1 01 7
10~
10~l
1011 o°
T e (ev)
Fig. 7. Same as fig. 1, hut for oxygen.
10~
4
1o
R. Marchand e~’a!.
/
Transient ionization tIme .1 c-ales
radiative model would he invalid. Of course, the actual macroscopic time scale depends on how rapidly the system is driven externally. As an example of how our figures can be used, let us consider two experiments in which a plasma is formed by irradiating a gas with a laser beam. In one [18], the gas consists of oxygen and the hydrodynamic evolution of the plasma is observed to occur on a time scale of the order of three nanoseconds. The electron density and temperature are of order 0.5 x lots cm and 120 eV, respectively. It follows from fig. 7. that the transient ionization time is longer than the macroscopic time scale, and a computer simulation of this experiment would require a detailed solution of the rate equations. In another experiment [19], performed with helium, the plasma evolves on a time scale of order of hundreds of nanoseconds. with 7~ 3 eV. and n~ IO~cm or more. The transient time scale obtained from fig. 2 in that range of parameters is of order of tens of nanoseconds. The steady state model therefore appears adequate to simulate this experiment. To conclude, we have produced a set of graphics which display the transient tiniescale T for ionization or recombination as a function of the plasma parameters for various low atomic number elements. These times can be compared to the time ‘r,, which characterizes changes in the macroscopic plasma parameters, and can be used to select a particular model to describe ionization dynamics. The figures can also be used to estimate the region of parameter space where the electron distribution function is likely to remain Maxwellian.
363
Acknowledgements The authors gratefully acknowledge the continuing financial support of the Natural Sciences and Engineering Research Council of Canada.
References [I] G.J. Pert. J. Comput. Phys. 27 (1978) 241. [2] J. Magihi. J. Phys. D 10(1977) 2257. [3] J. Magilh. (ornput. Phys. Commun. 16 (1978)1 29. 14] (11 Pert. J (omput. Phys 39 (1980) 251. [5] R.C. Mancini and CF. Fontan. Cornput. Phys. Common. 21(1981) 293. [6] J. Magill. Comput. Ph~s.Commun. 21 (1981) 295. [7] NA. Tahire and LW. Laing. Phys. I.ett. A 79 (1980) 321. [81 A. Shalom. .1. Phys. 1) 15)1982) L21. [91J.C Gauthier. J -P Geindre. N. Grandjouan and .1 Virniont, J. Ph~s.D 16 (1983) 321.
[10] iN. McMullin and CE. Capjack, Comput. ~
Corn-
mun. 32 (1984) II. 111] D. Bates. A. Kingston and R. McWhirter. Proc. Ro~.Soc. London A267 (1962) 297. [12] D. Duston and ii. Duderstadt. J Apph Phvs 49 (1978) 4388. [13] M.K. Matzcn and J.S. Pcarlman. Phys. Fluids 22 (1979) 449.
[141 D.
Duston and J. Davis. Phys. Rev. 23A (1980) 2602. [15] 1). Duston. R.W. Clark, J. Davis and JR Apruzese. Phys. Rev 27A (1983) 1441 [16] D. Quimby.
Mathematical Sciences North
Memorandum CRAFTY
--
West,
Inc
A (‘ollisional-Radiative Atomic
Physics Generator (1979) (unpublished). [17] L. Spitzer Jr., Physics of Fully Ionized Gases (lnterscience. New York, 1956). 118] R Gihes, Ph.D dissertation, Univ of Alberta (1983).
[191 A. Giulietti.
D. Giuletti. M Lucchesi and M Vavelli. Opt
Conlnlun. 47)1983)131.