- Email: [email protected]
Alternative methods of producing photoionised plasmas in the laboratory
High Energy Density Physics 7 (2011) 377e382
Contents lists available at ScienceDirect
High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp
Alternative methods of producing photoionised plasmas in the laboratory E.G. Hill*, S.J. Rose Plasma Physics Group, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 May 2011 Accepted 14 May 2011 Available online 16 August 2011
We present the conceptual design of a new experiment to reproduce the atomic kinetics of astrophysical photoionised plasmas in the laboratory. In particular the problems of the high densities usually found in laboratory experiments and the production of high colour temperature radiation fields are discussed and solutions presented. Following an analytic discussion, which allows one to find a combination of elements, one for the radiation source and one for the element to be photoionized, the proposed experiment is modelled using existing experimental data for the radiation source and a collisionalradiative model to calculate the photoionisation history. The results indicate that this approach is feasible with current experimental parameters and techniques. Crown Copyright Ó 2011 Published by Elsevier B.V. All rights reserved.
Keywords: Scattering of electromagnetic radiation Laboratory studies of space- and astrophysical-plasma processes Photoionisation and excitation
1. Introduction Many astrophysical plasmas of interest are photoionised by a large incident radiation field, examples include the plasmas surrounding early-type stars, compact objects, and active galaxy nuclei. The radiation emitted by these photoionised plasmas may be used to determine both the conditions of the plasma and the nature of the ionising radiation field, thereby providing a wealth of information about the astrophysical system as a whole. The plasmas are modelled using complex hybrid codes incorporating MHD, radiative transport and atomic physics [8], and therefore calibration of these codes is important to achieve confidence in the modelling. Considering a specific example, in high mass X-ray binaries (HMXBs), a massive star is accreting its stellar wind onto a compact object, a neutron star or black hole, which forms the second component in the binary system. The radiation emitted by the accretion over ionises the stellar wind, for example forcing silicon into hydrogen- or helium-like states at an electron temperature of just 10 eV. The stellar wind in massive stars is often radiatively driven and therefore, since the ionisation stage of the material in the wind has been altered, its motion is correspondingly changed, leading to a complex system of feedback between the radiation field and the stellar wind motion. Another example is the clumpy and optically thick wind around early-type stars where explanations of the observed spectra - both the presence of lines and their Doppler shifting due to the motion of the wind - need to be explained by theory. In this case the
* Corresponding author. E-mail address: [email protected] (E.G. Hill).
calculations are complicated by a distribution of dense clumps and shocks within the wind, see for example Ref. [10]. Exploring analogous plasmas in the laboratory would be highly advantageous to the modelling of these systems [7], and therefore it is important to create and photoionise plasmas in a manner similar to those occurring in the astrophysical case of interest. Low densities and temperatures coupled with a large photoionising radiation field are required; however, these conditions have not yet been recreated in the laboratory. Low density plasmas are found in tokamaks, and illuminate some of the issues involved with their modelling - in particular the effect of the low density ne (1014 cm3 on the atomic kinetics. Metastability of a configuration, the lack of decay channels out of that configuration leading to a build-up of population in it, will become much more likely at lower densities since radiative transitions have strong selection rules, and the collisional rates, which are fne , and would usually provide alternative decay paths are reduced. The low density also leads [9] to a much more complex structure to the cascades that start with a configuration with an electron in a Rydberg state and end with a ground state configuration. These cascades are now restricted by the same radiative selection rules. These plasmas are, however, not photoionised and so experiments have been conducted wherein a radiation field is created and transported to a moderate density plasma, and the effects of photoionisation measured. 2. Recreating the atomic kinetics of a low density plasma When we to attempt to use silicon in astrophysical conditions in a laboratory experiment we find that it would be impossible to obtain silicon at a suitably low density, since the limits of the
1574-1818/$ e see front matter Crown Copyright Ó 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.hedp.2011.05.010
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E.G. Hill, S.J. Rose / High Energy Density Physics 7 (2011) 377e382
number of photons needed to observe the spectrum would necessitate the use of a prohibitively large volume of plasma, which then poses difficulties for the radiation field uniformity and for the mechanics of confining the plasma. However, if it were the case that silicon, at achievable laboratory densities, was in the same atomic kinetics regime as that at astrophysical densities, the experiments would provide an accurate representation of the astrophysical case. In order to demonstrate that this is not the case, we proceed to examine the atomic kinetics of the helium-like levels 1s2s, 1s2p, and 1s2, and the Lithium-like states that form satellites in the low energy wing of the resonant transition, i.e., 1s2s2, 1s2s2p and 1s2p2. A Grotrian diagram for the system is shown in Fig. 1. This system is of particular interest since the lines produced by the radiative transitions lie at w1 keV for the trace low-Z elements in the astrophysical plasma, for example neon, magnesium, silicon and sulphur. These line groups are observable in many astrophysical spectra and have little absorption from the interstellar medium and also have good separation of the features, making them a valuable and often used diagnostic. Fig. 2 shows an example of a laboratory photoionised silicon spectrum, compared with an astrophysical spectrum. It can be shown by modelling the experiment time-dependently, including energy conservation and opacity effects [5] that the low energy feature in the laboratory spectrum (at 1.84 keV) is not due to the forbidden line but due to the lithium-like satellite lines. Following the discussion in Ref. [5], we can show that the ratio between the low energy and central feature is determined by a parameter P,
P ¼
Cð1s2s3 S1 /1s2p3 P 1 Þ A 1s2s3 S1 /1s21 S0
(1)
to a reasonable approximation, where C is the collisional rate marked in Fig. 1, and A the radiative decay rate from the 1s2s3 S1 is also shown in that figure. For silicon we see that
ne PSi z21013 pffiffiffiffiffi Te
(2)
where ne is in cm3 and Te in eV. If PSi for a given density and temperature is less than 1, we have the low density case, where the forbidden line is visible, for example in the astrophysical case
new1010 cm3 Tew10 eV, PSiw6 1041. We can see, however, that the laboratory case has PSi z 106[1 and so the forbidden line is not visible, and the low energy feature is composed solely of lithium-like emission. In order to solve the problem, that the atomic kinetics at laboratory densities is not comparable with that astrophysically, we propose the use of a higher-Z material in the laboratory experiment. This is because we can write a more general form of P as a function of atomic number Z
ne PZ z0:81pffiffiffiffiffi 11 Te Z
(3)
that shows a very strong Z dependence that arises from the strong Z dependence of the Einstein A coefficient for the M2 forbidden transition. Since we are using higher densities, we might expect the other rates not included in the analytic model, for example three body recombination or collisional ionisation, to become more important, and so we use the collisional-radiative model used in Ref. [5] to verify the general formula for PZ. The model is a Detailed Term Accounting (DTA) model embedded in a Detailed Configuration Accounting (DCA) model, that is, the states of particular interest are considered as a DTA model, with rates in and out of the set of particular states, and are then coupled to a larger DCA model which handles the atomic kinetics of the other states. This allows the generation of spectra, compatible with diagnostics, for atomic systems where extensions of the usual DCA model are required to accurately represent the atomic kinetics, while keeping the runtime reasonable. Fig. 3 shows the dependence of the intensities of the transitions in Silicon at 300 eV and Krypton at 5000 eV on the density, and show the validity to good accuracy of the formula for PZ above. The higher temperatures are needed to ionise the higher-Z elements sufficiently for all densities considered; however, due to the weak Te dependence, the behaviour is little changed when more realistic values, i.e. lower values, are used. In laboratory and astrophysical photoionised plasmas, the high charge state is maintained by the photoionising radiation field rather than the electron temperature. Using this method, that is, using the equivalence of the atomic kinetics of an astrophysically low-density, low-Z plasma with a modest-density, mid-Z laboratory plasma allows us to make the transition from the metastable levels visible in the astrophysical case to laboratory density, and solves the problem of replicating the low density plasmas in the laboratory. However, having increased the value of Z, it has now become even more difficult than in the low-Z case to provide a radiation field of sufficiently high colour temperature to be comparable to that incident on the astrophysical plasma. This problem will now be considered. 3. Producing a high colour temperature radiation field
Fig. 1. A Grotrian diagram for the levels considered, with energies ordered as for silicon. The critical transitions are indicated - the radiative decays which produce the lines of the helium-like triplet, and the radiative decay of the lithium-like satellites (solid arrows), and the collisional excitation of the metastable 3 S1 state. The level energies are normalised to the transition groundstates, so that the coincident energy of the lithium-like transitions and the forbidden line can be seen.
The radiation fields in astrophysical cases are characterised by a higher colour temperature than those produced in photoionisation experiments performed up to this time. The highest yet attained is by Fujioka et al. [1], who produced a radiation field at 480 eV for some 100 ps, somewhat shorter than their atomic kinetic timescale. The radiation fields in astrophysical cases are of a higher though similar colour temperature, i.e., w1000 eV, and persist for a much longer timescale than that of the atomic and plasma kinetics. The colour temperature of a radiation field is only strictly defined for a Planckian, where
BðnÞTcr ¼
2hn3 1 c2 ehn=Tcr 1
(4)
E.G. Hill, S.J. Rose / High Energy Density Physics 7 (2011) 377e382
379
Fig. 2. [Copied from Fujioka et al [1].] The top spectrum is of laboratory density silicon from the experiment of Fujioka et. al., while the second is from astrophysical density silicon in the region surrounding Vela X-1.
defines Tcr, the colour temperature, which is extracted from a fit to the spectrum. In the laboratory, it is rare to obtain an exact Planckian spectrum, particularly for high Tcr since the frequency-resolved optical thickness of a material generally decreases with increasing frequency and, therefore, even for a nominally Planckian source it is natural to define another colour temperature which reflects the photoionisation processes occurring. Astrophysically the spectrum is also non-Planckian due to absorption, particularly by hydrogen and helium that have high column densities even in small systems, and other effects in the production of the radiation field, for example 0 , the effective colour temperComptonisation (e.g [6].). We define Tcr ature, to be the Tcr of a Planckian such that the photoionisation integrals from two different shells, n and n0 , are in the same ratio for that Planckian as they are for the experimental radiation field, I(n).
ZN 0 ZN
dnsn dnsn0
ZN
BTcr0 ðnÞ
n
BTcr0 ðnÞ
0
¼
n
0 ZN
0
dnsn dnsn0
IðnÞ
n
IðnÞ
(5)
n
here sn is the photoionisation cross-section of the n principal quantum number orbital. The cross-section for all ionisations is approximately of the form
1
sn ðnÞf 3 hn>DE n
(6)
sn ðnÞ ¼ 0hn < DE
(7)
where DE is the energy change involved in the ionisation.
The importance of a high colour temperature radiation field, and the reason for the definition of effective colour temperature, may be seen by considering a ground state configuration with a few electrons, using 1s22s2 as an example for clarity. There are two possible photoionisations which may occur
1s2 2s2 þ hn/1s2s2 þ e
(8)
1s2 2s2 þ hn/ 1s2 2s þ e
(9)
In the latter case we obtain the next ionised ground state configuration, whereas in the former case we obtain a much more exotic, autoionising configuration, which has an appreciable probability of ionising again to produce 1s2 þ e. Therefore, the ratio of the probabilities of ionisation from n ¼ 1 and n ¼ 2, defined by the effective colour temperature, is important for the atomic kinetics. We see that in order to maintain the same atomic kinetics we wish to scale the colour temperature of the incident radiation field with the energy scales in the atom, and therefore to keep the ratio
Z2 Tcr
(10)
approximately constant. In the astrophysical case, we see lines emitted by silicon with an applied radiation field with 0 w1000 eV. Planckian, or close to Planckian, radiation fields with Tcr 0 of 1000 eV are very difficult to achieve in the laboratory, Tcr or Tcr 4 , and so the scaling of since the power radiated by the source is fTcr the colour temperature with the power provided by the laser is very unfavourable. If we propose to model the silicon using krypton, for reasons discussed above, the scaling with Z2 leads to an
380
E.G. Hill, S.J. Rose / High Energy Density Physics 7 (2011) 377e382
Fig. 3. The line intensities of the 1s2s3 S1 /1s2 ‘forbidden’ (green), 1s2p3 P 1 /1s2 ‘intercombination’ (black), and 1s2p3 P 2 /1s2 (red) lines relative to that of the 1s2p1 P 1 /1s2 ‘resonance’ line. The low density regime for krypton extends up to laboratory densities, whereas the difference between the intensities in silicon is very marked, notably the forbidden line is not visible (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
unreasonably high colour temperature for the radiation field. We therefore propose a new method for the production of radiation fields with a high effective colour temperature, which involves using a suitably chosen line radiation field in place of the Planckian. 4. Line radiation fields Line radiation fields have been researched extensively in recent years due to their potential use as absorption sources for probing high density plasmas, for example, for Thomson scattering. Different sources have been proposed and investigated, and it is found that the optimal electron density of the target is z0.1 ncr, where ncr is the critical density of the incident laser. Most successful, in terms of laser energy to K-shell emission conversion efficiency, have been gas and exploded foil targets - the latter having the advantage that they can be used for any non-gaseous element. In order to demonstrate the method of photoionisation we propose we will use an experimental radiation field recorded in Ref. [2] for an exploded foil titanium (Z ¼ 22) target. The spectra, which are shown in Ref. [2], were recorded using two spectrometers that provide absolute, frequency-resolved, measurements of the radiation flux. These spectra covered two adjacent spectral region and were fit together. The He-a line lies just outside of the spectral range to the low energy side, and is included artificially to make the total titanium K-shell flux correct. The continuum between the K and L shell emission of the titanium is included by continuing the free-bound observed background. The reason for photoionising argon with titanium, is that the Kshell emission of the titanium lies just above the K-edge of the argon, and therefore where the argon has a very high absorption crosssection. However, titanium has little emission in the absorption window between the argon K-edge and the L-shell. This leads to the
efficient use of the energy, since the photons produced by the titanium plasma are in the spectral regions where the argon is highly absorbing. This leads to high equivalent colour temperature without the need to generate an entire blackbody. The model is initialised at t ¼ 0 in a steady state, Te ¼ Ti ¼ 30 eV, ne ¼ 1.0 1017cm3, and with no radiation field. The radiation field from a 500 mm titanium foil is applied between t ¼ 103ns and t ¼ 1.000 ns. The resulting average degree of ionisation, temperature and population of the He-like ions in the plasma are shown in Fig. 4. The same run the changes in Te that arise from the radiation deposited in the plasma gave results which were indistinguishable, and shows the independence of the degree of photoionisation on the electron temperature. Therefore, we have a plasma that is photoionisation dominated. 5. Prediction for krypton Having established the feasibility of the scheme for titanium photoionising an argon plasma, we further investigate the photoionisation of krypton. The radiation field will be produced by using Mo or Te which produce K-shell radiation that has a slightly higher energy than the krypton K-edge. For purposes of illustration we chose Mo. Although the radiation from K-shell Mo has never been measured the scaling laws for the K-shell flux from a target are known [3]. The scaling depends on the fact that the low principal quantum number levels in the ions are in coronal equilibrium, and so the power released, for a constant target temperature, scales as
n2e eE=Te
(11)
where E is the transition energy of the K-shell emission [3]. Using this, and the conversion efficiency from previous experiments ([2]
E.G. Hill, S.J. Rose / High Energy Density Physics 7 (2011) 377e382
381
Fig. 4. The average charge state Z*, electron temperature Te and fractional population of the helium-like states are shown, for the time evolution of the argon plasma.
Fig. 26), we find that the K-shell emission of molybdenum, at the same temperature as the titanium experiment, is approximately 0.5% of the incident laser energy, assuming the same laser parameters as used in the titanium experiment. We model the initial krypton plasma conditions as ne ¼ 1.0 1018 cm3 and Te ¼ 200 eV. The radiation field is calculated at a distance of 200 mm from the source, and is found using the scaling described above, giving a conservative estimate of the radiation field produced. We perform a steady-state calculation, and the forbidden line present in the astrophysical plasmas is clearly seen in the predicted spectrum, Fig. 5. At the simulation temperature of 200 eV, we expect krypton to be approximately 20 times ionised, however with the applied radiation field the krypton is helium-like, 34 times ionised, and so it is seen that the plasma is indeed photoionised, that the photoionisation rates far dominate the collisional ionisation rates, and fundamentally change the physics of the plasma. 6. Experimental feasibility Since the plasmas we have proposed to resolve the problems that currently exist when trying to produce meaningful astrophysical collisional-radiative tests in the laboratory are outside the range usually considered in any other problem, these results warrant further investigation. These results indicate that such an
Fig. 5. The steady-state emission spectrum from krypton as described in the text. The total emission (thick, blue), helium-like (thin, black), and lithium-like (thin, red) spectra are shown. The forbidden line is marked with an arrow (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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E.G. Hill, S.J. Rose / High Energy Density Physics 7 (2011) 377e382
experiment, if performed, could reproduce the metastability and radiation-induced atomic kinetics seen in astrophysical photoionised plasmas. Given the tentative nature of the experiment, a major advantage is the possibility of it being performed as a parasitic experiment on existing line radiation experiments, such as those described in Refs. [2,3]. Additionally, the efficient use of the laser energy allows this experiment to be performed on smaller, few kilojoule lasers, of the size of OMEGA, or ORION, rather than on larger and less accessible megajoule laser facilities such as NIF or LMJ. 7. Conclusion We have demonstrated the feasibility of a set of experimental criteria that could provide both the low density kinetics, for example the visibility of the forbidden line, and the inner-shell photoionised nature of an astrophysical plasma. Although the
material being photoionised and radiation field both differ from the astrophysical case, we have shown that the atomic kinetics will be comparable. The experiment outlined here would be the first to allow a direct comparison of the kinetics in a laboratory and astrophysical plasma, and therefore would be a valuable benchmark for calibration of astrophysical density collisional-radiative models. References [1] S. Fujioka, et al., Nat. Phys. 5 (2009) 821. [2] D. Babonneau, et al., PoP 15 (2008) 092702. [3] C.A. Back, et al., PRL 87 (2001) 275003e275011. [5] E.G. Hill, S.J. Rose, PoP 17 (2010) 103301. [6] J.H. Krolik, Active Galactic Nuclei (1999) Princeton. [7] D.A. Liedahl, G.V. Brown, Can. J. Phys. 86 (2008) 184. [8] C.W. Mauche, et al., PTP Supp. 169 (2007) 196. [9] A.K. Pradhan, Ap. J. 288 (1985) 824. [10] W.L. Waldron, J.P. Cassinelli, N.A. Miller, J.J. MacFarlane, J.C. Reiter, Ap. J. 616 (2004) 542.
Contents lists available at ScienceDirect
High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp
Alternative methods of producing photoionised plasmas in the laboratory E.G. Hill*, S.J. Rose Plasma Physics Group, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 May 2011 Accepted 14 May 2011 Available online 16 August 2011
We present the conceptual design of a new experiment to reproduce the atomic kinetics of astrophysical photoionised plasmas in the laboratory. In particular the problems of the high densities usually found in laboratory experiments and the production of high colour temperature radiation fields are discussed and solutions presented. Following an analytic discussion, which allows one to find a combination of elements, one for the radiation source and one for the element to be photoionized, the proposed experiment is modelled using existing experimental data for the radiation source and a collisionalradiative model to calculate the photoionisation history. The results indicate that this approach is feasible with current experimental parameters and techniques. Crown Copyright Ó 2011 Published by Elsevier B.V. All rights reserved.
Keywords: Scattering of electromagnetic radiation Laboratory studies of space- and astrophysical-plasma processes Photoionisation and excitation
1. Introduction Many astrophysical plasmas of interest are photoionised by a large incident radiation field, examples include the plasmas surrounding early-type stars, compact objects, and active galaxy nuclei. The radiation emitted by these photoionised plasmas may be used to determine both the conditions of the plasma and the nature of the ionising radiation field, thereby providing a wealth of information about the astrophysical system as a whole. The plasmas are modelled using complex hybrid codes incorporating MHD, radiative transport and atomic physics [8], and therefore calibration of these codes is important to achieve confidence in the modelling. Considering a specific example, in high mass X-ray binaries (HMXBs), a massive star is accreting its stellar wind onto a compact object, a neutron star or black hole, which forms the second component in the binary system. The radiation emitted by the accretion over ionises the stellar wind, for example forcing silicon into hydrogen- or helium-like states at an electron temperature of just 10 eV. The stellar wind in massive stars is often radiatively driven and therefore, since the ionisation stage of the material in the wind has been altered, its motion is correspondingly changed, leading to a complex system of feedback between the radiation field and the stellar wind motion. Another example is the clumpy and optically thick wind around early-type stars where explanations of the observed spectra - both the presence of lines and their Doppler shifting due to the motion of the wind - need to be explained by theory. In this case the
* Corresponding author. E-mail address: [email protected] (E.G. Hill).
calculations are complicated by a distribution of dense clumps and shocks within the wind, see for example Ref. [10]. Exploring analogous plasmas in the laboratory would be highly advantageous to the modelling of these systems [7], and therefore it is important to create and photoionise plasmas in a manner similar to those occurring in the astrophysical case of interest. Low densities and temperatures coupled with a large photoionising radiation field are required; however, these conditions have not yet been recreated in the laboratory. Low density plasmas are found in tokamaks, and illuminate some of the issues involved with their modelling - in particular the effect of the low density ne (1014 cm3 on the atomic kinetics. Metastability of a configuration, the lack of decay channels out of that configuration leading to a build-up of population in it, will become much more likely at lower densities since radiative transitions have strong selection rules, and the collisional rates, which are fne , and would usually provide alternative decay paths are reduced. The low density also leads [9] to a much more complex structure to the cascades that start with a configuration with an electron in a Rydberg state and end with a ground state configuration. These cascades are now restricted by the same radiative selection rules. These plasmas are, however, not photoionised and so experiments have been conducted wherein a radiation field is created and transported to a moderate density plasma, and the effects of photoionisation measured. 2. Recreating the atomic kinetics of a low density plasma When we to attempt to use silicon in astrophysical conditions in a laboratory experiment we find that it would be impossible to obtain silicon at a suitably low density, since the limits of the
1574-1818/$ e see front matter Crown Copyright Ó 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.hedp.2011.05.010
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E.G. Hill, S.J. Rose / High Energy Density Physics 7 (2011) 377e382
number of photons needed to observe the spectrum would necessitate the use of a prohibitively large volume of plasma, which then poses difficulties for the radiation field uniformity and for the mechanics of confining the plasma. However, if it were the case that silicon, at achievable laboratory densities, was in the same atomic kinetics regime as that at astrophysical densities, the experiments would provide an accurate representation of the astrophysical case. In order to demonstrate that this is not the case, we proceed to examine the atomic kinetics of the helium-like levels 1s2s, 1s2p, and 1s2, and the Lithium-like states that form satellites in the low energy wing of the resonant transition, i.e., 1s2s2, 1s2s2p and 1s2p2. A Grotrian diagram for the system is shown in Fig. 1. This system is of particular interest since the lines produced by the radiative transitions lie at w1 keV for the trace low-Z elements in the astrophysical plasma, for example neon, magnesium, silicon and sulphur. These line groups are observable in many astrophysical spectra and have little absorption from the interstellar medium and also have good separation of the features, making them a valuable and often used diagnostic. Fig. 2 shows an example of a laboratory photoionised silicon spectrum, compared with an astrophysical spectrum. It can be shown by modelling the experiment time-dependently, including energy conservation and opacity effects [5] that the low energy feature in the laboratory spectrum (at 1.84 keV) is not due to the forbidden line but due to the lithium-like satellite lines. Following the discussion in Ref. [5], we can show that the ratio between the low energy and central feature is determined by a parameter P,
P ¼
Cð1s2s3 S1 /1s2p3 P 1 Þ A 1s2s3 S1 /1s21 S0
(1)
to a reasonable approximation, where C is the collisional rate marked in Fig. 1, and A the radiative decay rate from the 1s2s3 S1 is also shown in that figure. For silicon we see that
ne PSi z21013 pffiffiffiffiffi Te
(2)
where ne is in cm3 and Te in eV. If PSi for a given density and temperature is less than 1, we have the low density case, where the forbidden line is visible, for example in the astrophysical case
new1010 cm3 Tew10 eV, PSiw6 1041. We can see, however, that the laboratory case has PSi z 106[1 and so the forbidden line is not visible, and the low energy feature is composed solely of lithium-like emission. In order to solve the problem, that the atomic kinetics at laboratory densities is not comparable with that astrophysically, we propose the use of a higher-Z material in the laboratory experiment. This is because we can write a more general form of P as a function of atomic number Z
ne PZ z0:81pffiffiffiffiffi 11 Te Z
(3)
that shows a very strong Z dependence that arises from the strong Z dependence of the Einstein A coefficient for the M2 forbidden transition. Since we are using higher densities, we might expect the other rates not included in the analytic model, for example three body recombination or collisional ionisation, to become more important, and so we use the collisional-radiative model used in Ref. [5] to verify the general formula for PZ. The model is a Detailed Term Accounting (DTA) model embedded in a Detailed Configuration Accounting (DCA) model, that is, the states of particular interest are considered as a DTA model, with rates in and out of the set of particular states, and are then coupled to a larger DCA model which handles the atomic kinetics of the other states. This allows the generation of spectra, compatible with diagnostics, for atomic systems where extensions of the usual DCA model are required to accurately represent the atomic kinetics, while keeping the runtime reasonable. Fig. 3 shows the dependence of the intensities of the transitions in Silicon at 300 eV and Krypton at 5000 eV on the density, and show the validity to good accuracy of the formula for PZ above. The higher temperatures are needed to ionise the higher-Z elements sufficiently for all densities considered; however, due to the weak Te dependence, the behaviour is little changed when more realistic values, i.e. lower values, are used. In laboratory and astrophysical photoionised plasmas, the high charge state is maintained by the photoionising radiation field rather than the electron temperature. Using this method, that is, using the equivalence of the atomic kinetics of an astrophysically low-density, low-Z plasma with a modest-density, mid-Z laboratory plasma allows us to make the transition from the metastable levels visible in the astrophysical case to laboratory density, and solves the problem of replicating the low density plasmas in the laboratory. However, having increased the value of Z, it has now become even more difficult than in the low-Z case to provide a radiation field of sufficiently high colour temperature to be comparable to that incident on the astrophysical plasma. This problem will now be considered. 3. Producing a high colour temperature radiation field
Fig. 1. A Grotrian diagram for the levels considered, with energies ordered as for silicon. The critical transitions are indicated - the radiative decays which produce the lines of the helium-like triplet, and the radiative decay of the lithium-like satellites (solid arrows), and the collisional excitation of the metastable 3 S1 state. The level energies are normalised to the transition groundstates, so that the coincident energy of the lithium-like transitions and the forbidden line can be seen.
The radiation fields in astrophysical cases are characterised by a higher colour temperature than those produced in photoionisation experiments performed up to this time. The highest yet attained is by Fujioka et al. [1], who produced a radiation field at 480 eV for some 100 ps, somewhat shorter than their atomic kinetic timescale. The radiation fields in astrophysical cases are of a higher though similar colour temperature, i.e., w1000 eV, and persist for a much longer timescale than that of the atomic and plasma kinetics. The colour temperature of a radiation field is only strictly defined for a Planckian, where
BðnÞTcr ¼
2hn3 1 c2 ehn=Tcr 1
(4)
E.G. Hill, S.J. Rose / High Energy Density Physics 7 (2011) 377e382
379
Fig. 2. [Copied from Fujioka et al [1].] The top spectrum is of laboratory density silicon from the experiment of Fujioka et. al., while the second is from astrophysical density silicon in the region surrounding Vela X-1.
defines Tcr, the colour temperature, which is extracted from a fit to the spectrum. In the laboratory, it is rare to obtain an exact Planckian spectrum, particularly for high Tcr since the frequency-resolved optical thickness of a material generally decreases with increasing frequency and, therefore, even for a nominally Planckian source it is natural to define another colour temperature which reflects the photoionisation processes occurring. Astrophysically the spectrum is also non-Planckian due to absorption, particularly by hydrogen and helium that have high column densities even in small systems, and other effects in the production of the radiation field, for example 0 , the effective colour temperComptonisation (e.g [6].). We define Tcr ature, to be the Tcr of a Planckian such that the photoionisation integrals from two different shells, n and n0 , are in the same ratio for that Planckian as they are for the experimental radiation field, I(n).
ZN 0 ZN
dnsn dnsn0
ZN
BTcr0 ðnÞ
n
BTcr0 ðnÞ
0
¼
n
0 ZN
0
dnsn dnsn0
IðnÞ
n
IðnÞ
(5)
n
here sn is the photoionisation cross-section of the n principal quantum number orbital. The cross-section for all ionisations is approximately of the form
1
sn ðnÞf 3 hn>DE n
(6)
sn ðnÞ ¼ 0hn < DE
(7)
where DE is the energy change involved in the ionisation.
The importance of a high colour temperature radiation field, and the reason for the definition of effective colour temperature, may be seen by considering a ground state configuration with a few electrons, using 1s22s2 as an example for clarity. There are two possible photoionisations which may occur
1s2 2s2 þ hn/1s2s2 þ e
(8)
1s2 2s2 þ hn/ 1s2 2s þ e
(9)
In the latter case we obtain the next ionised ground state configuration, whereas in the former case we obtain a much more exotic, autoionising configuration, which has an appreciable probability of ionising again to produce 1s2 þ e. Therefore, the ratio of the probabilities of ionisation from n ¼ 1 and n ¼ 2, defined by the effective colour temperature, is important for the atomic kinetics. We see that in order to maintain the same atomic kinetics we wish to scale the colour temperature of the incident radiation field with the energy scales in the atom, and therefore to keep the ratio
Z2 Tcr
(10)
approximately constant. In the astrophysical case, we see lines emitted by silicon with an applied radiation field with 0 w1000 eV. Planckian, or close to Planckian, radiation fields with Tcr 0 of 1000 eV are very difficult to achieve in the laboratory, Tcr or Tcr 4 , and so the scaling of since the power radiated by the source is fTcr the colour temperature with the power provided by the laser is very unfavourable. If we propose to model the silicon using krypton, for reasons discussed above, the scaling with Z2 leads to an
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Fig. 3. The line intensities of the 1s2s3 S1 /1s2 ‘forbidden’ (green), 1s2p3 P 1 /1s2 ‘intercombination’ (black), and 1s2p3 P 2 /1s2 (red) lines relative to that of the 1s2p1 P 1 /1s2 ‘resonance’ line. The low density regime for krypton extends up to laboratory densities, whereas the difference between the intensities in silicon is very marked, notably the forbidden line is not visible (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
unreasonably high colour temperature for the radiation field. We therefore propose a new method for the production of radiation fields with a high effective colour temperature, which involves using a suitably chosen line radiation field in place of the Planckian. 4. Line radiation fields Line radiation fields have been researched extensively in recent years due to their potential use as absorption sources for probing high density plasmas, for example, for Thomson scattering. Different sources have been proposed and investigated, and it is found that the optimal electron density of the target is z0.1 ncr, where ncr is the critical density of the incident laser. Most successful, in terms of laser energy to K-shell emission conversion efficiency, have been gas and exploded foil targets - the latter having the advantage that they can be used for any non-gaseous element. In order to demonstrate the method of photoionisation we propose we will use an experimental radiation field recorded in Ref. [2] for an exploded foil titanium (Z ¼ 22) target. The spectra, which are shown in Ref. [2], were recorded using two spectrometers that provide absolute, frequency-resolved, measurements of the radiation flux. These spectra covered two adjacent spectral region and were fit together. The He-a line lies just outside of the spectral range to the low energy side, and is included artificially to make the total titanium K-shell flux correct. The continuum between the K and L shell emission of the titanium is included by continuing the free-bound observed background. The reason for photoionising argon with titanium, is that the Kshell emission of the titanium lies just above the K-edge of the argon, and therefore where the argon has a very high absorption crosssection. However, titanium has little emission in the absorption window between the argon K-edge and the L-shell. This leads to the
efficient use of the energy, since the photons produced by the titanium plasma are in the spectral regions where the argon is highly absorbing. This leads to high equivalent colour temperature without the need to generate an entire blackbody. The model is initialised at t ¼ 0 in a steady state, Te ¼ Ti ¼ 30 eV, ne ¼ 1.0 1017cm3, and with no radiation field. The radiation field from a 500 mm titanium foil is applied between t ¼ 103ns and t ¼ 1.000 ns. The resulting average degree of ionisation, temperature and population of the He-like ions in the plasma are shown in Fig. 4. The same run the changes in Te that arise from the radiation deposited in the plasma gave results which were indistinguishable, and shows the independence of the degree of photoionisation on the electron temperature. Therefore, we have a plasma that is photoionisation dominated. 5. Prediction for krypton Having established the feasibility of the scheme for titanium photoionising an argon plasma, we further investigate the photoionisation of krypton. The radiation field will be produced by using Mo or Te which produce K-shell radiation that has a slightly higher energy than the krypton K-edge. For purposes of illustration we chose Mo. Although the radiation from K-shell Mo has never been measured the scaling laws for the K-shell flux from a target are known [3]. The scaling depends on the fact that the low principal quantum number levels in the ions are in coronal equilibrium, and so the power released, for a constant target temperature, scales as
n2e eE=Te
(11)
where E is the transition energy of the K-shell emission [3]. Using this, and the conversion efficiency from previous experiments ([2]
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Fig. 4. The average charge state Z*, electron temperature Te and fractional population of the helium-like states are shown, for the time evolution of the argon plasma.
Fig. 26), we find that the K-shell emission of molybdenum, at the same temperature as the titanium experiment, is approximately 0.5% of the incident laser energy, assuming the same laser parameters as used in the titanium experiment. We model the initial krypton plasma conditions as ne ¼ 1.0 1018 cm3 and Te ¼ 200 eV. The radiation field is calculated at a distance of 200 mm from the source, and is found using the scaling described above, giving a conservative estimate of the radiation field produced. We perform a steady-state calculation, and the forbidden line present in the astrophysical plasmas is clearly seen in the predicted spectrum, Fig. 5. At the simulation temperature of 200 eV, we expect krypton to be approximately 20 times ionised, however with the applied radiation field the krypton is helium-like, 34 times ionised, and so it is seen that the plasma is indeed photoionised, that the photoionisation rates far dominate the collisional ionisation rates, and fundamentally change the physics of the plasma. 6. Experimental feasibility Since the plasmas we have proposed to resolve the problems that currently exist when trying to produce meaningful astrophysical collisional-radiative tests in the laboratory are outside the range usually considered in any other problem, these results warrant further investigation. These results indicate that such an
Fig. 5. The steady-state emission spectrum from krypton as described in the text. The total emission (thick, blue), helium-like (thin, black), and lithium-like (thin, red) spectra are shown. The forbidden line is marked with an arrow (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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experiment, if performed, could reproduce the metastability and radiation-induced atomic kinetics seen in astrophysical photoionised plasmas. Given the tentative nature of the experiment, a major advantage is the possibility of it being performed as a parasitic experiment on existing line radiation experiments, such as those described in Refs. [2,3]. Additionally, the efficient use of the laser energy allows this experiment to be performed on smaller, few kilojoule lasers, of the size of OMEGA, or ORION, rather than on larger and less accessible megajoule laser facilities such as NIF or LMJ. 7. Conclusion We have demonstrated the feasibility of a set of experimental criteria that could provide both the low density kinetics, for example the visibility of the forbidden line, and the inner-shell photoionised nature of an astrophysical plasma. Although the
material being photoionised and radiation field both differ from the astrophysical case, we have shown that the atomic kinetics will be comparable. The experiment outlined here would be the first to allow a direct comparison of the kinetics in a laboratory and astrophysical plasma, and therefore would be a valuable benchmark for calibration of astrophysical density collisional-radiative models. References [1] S. Fujioka, et al., Nat. Phys. 5 (2009) 821. [2] D. Babonneau, et al., PoP 15 (2008) 092702. [3] C.A. Back, et al., PRL 87 (2001) 275003e275011. [5] E.G. Hill, S.J. Rose, PoP 17 (2010) 103301. [6] J.H. Krolik, Active Galactic Nuclei (1999) Princeton. [7] D.A. Liedahl, G.V. Brown, Can. J. Phys. 86 (2008) 184. [8] C.W. Mauche, et al., PTP Supp. 169 (2007) 196. [9] A.K. Pradhan, Ap. J. 288 (1985) 824. [10] W.L. Waldron, J.P. Cassinelli, N.A. Miller, J.J. MacFarlane, J.C. Reiter, Ap. J. 616 (2004) 542.