- Email: [email protected]
Changes in atomic populations due to edge plasma fluctuations
Journal of Nuclear Materials 438 (2013) S1224–S1227
Contents lists available at SciVerse ScienceDirect
Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Changes in atomic populations due to edge plasma fluctuations R. Hammami a,⇑, H. Capes a, F. Catoire b, L. Godbert-Mouret a, M. Koubiti a, Y. Marandet a, A. Mekkaoui a, J. Rosato a, R. Stamm a a b
PIIM, Aix-Marseille Université and CNRS, centre Saint Jérôme, Marseille 13397, France CELIA, Université de Bordeaux 1 and CNRS, Domaine du Haut Carré, Talence 33405, France
a r t i c l e
i n f o
Article history: Available online 16 January 2013
a b s t r a c t The population balance of atoms or ions in an edge plasma is calculated in the presence of fluctuating density or temperature. We have used a stochastic model taking advantage of the knowledge of the plasma parameter statistical properties, and assuming a stepwise constant stochastic process for the fluctuating variable. The model is applied to simplified atomic systems such as three level hydrogen atoms or the ionization balance of carbon affected by electronic temperature or density fluctuations obeying a gamma PDF, and an exponential waiting time distribution. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Level populations of atom and ions are important quantities for characterizing edge plasmas. Radiation loss, ionization equilibrium, or the intensity of spectral lines used for diagnosing the plasma are directly related to the atomic and ionic populations. The determination of the latter relies on the solution of an atomic population kinetic model which is usually obtained using a stationary collisional-radiative model (CRM), assuming constant plasma parameters such as electronic density and temperature [1]. In this work we assume that these parameters fluctuate as a consequence of edge plasma turbulence. In edge plasmas, density and temperature fluctuations of order unity on time scales of the order of 10 ls have been measured, and characterized with their statistical properties such as the probability density function (PDF) and the correlation function of the fluctuating parameter [2–5]. For such conditions, we adopt a previously developed model which assumes that the fluctuating quantity obeys a stochastic process. The density or temperature is distributed according to a prescribed PDF, and jumps from one constant value to another one according to a given waiting time distribution (WTD) [6,7]. The latter is determined from the correlation function of the fluctuations. An example for such a stochastic evolution of the temperature is shown in Fig. 1, in conditions of a turbulent edge plasma. Assuming such a process for the fluctuations, we may solve the time dependent CRM by an analytical model or a Monte Carlo type simulation. In Section 2, we briefly describe how atomic populations may be ccalculated in the presence of a renewal stochastic process for the fluctuations [8]. Our models are applied to two simple atomic
⇑ Corresponding author. E-mail address: [email protected] (R. Hammami). 0022-3115/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnucmat.2013.01.271
systems in Section 3, a reduced hydrogen atom, and the carbon ionization balance. 2. Modelling the plasma parameters with a renewal process The atomic or ionic populations are described by a vector X(t) = (x1(t), . . . , xN(t)) containing the different population x of each level j. Initial values of the population are given for instance at time t = 0 as X0 = X(t = 0). The kinetics of the atomic populations is governed by a differential equation of the form [6]:
dXðtÞ ¼ MðYðtÞÞXðtÞ; dt
ð1Þ
where M is a time dependent matrix filled with the collisionalradiative rates, and Y(t) is the fluctuating plasma parameter (electronic temperature or density in the following). We now assume that Y follows a renewal process with stepwise constant values separated by instantaneous jumps. The duration on each step obeys a prescribed waiting time distribution (WTD), and the values taken by Y are distributed along a chosen probability density function (PDF) P(Y). Eq. (1) is then a linear stochastic equation, which may be solved by a numerical simulation or an analytic approach [6]. The analytic solution takes a particularly simple form if an exponential WTD / (t) = m exp(mt) is chosen, where m is the typical fluctuation frequency of Y. This choice corresponds to an exponential correlation function, with a correlation time m1. It may be expressed as the Laplace transform of the Green function G associated to Eq. (1):
hGðsÞi ¼ ½1 mGST ðsÞ1 GST ðsÞ;
ð2Þ
where the brackets h i imply that an ensemble average has been taken, and the subscript ST means that a static average over the PDF P(Y) has been taken. This quantity may be written as:
R. Hammami et al. / Journal of Nuclear Materials 438 (2013) S1224–S1227
S1225
the population of the 2s level depend on the typical frequency m and the rate r = DT/hTi of the electron temperature fluctuations, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where DT ¼ hT 2 i hTi2 , and the average electron temperature R1 is given by hTi ¼ 0 TPðTÞdT with P(T) the PDF of the fluctuating temperature. Assuming a gamma function for the PDF P(T):
PðTÞ ¼
Fig. 1. Simulation of a stepwise constant stochastic evolution of the temperature with an exponential WTD of typical fluctuation frequency m = 105 s1, for an electronic density Ne = 1019 m3. The fluctuations obey to a gamma PDF with a fluctuation rate r = 0.9, and an average temperature of 2 eV (see also Sections 2–3).
GST ðsÞ ¼
1 : sþmM
ð3Þ
It is possible to obtain the average hX(t)i of the population vector in the limit of long times by using:
limhXðtÞi ¼ limþ shGðsÞiX 0 :
t!1
s!0
ð4Þ
The solution of Eq. (1) may also be obtained by a numerical solution coupled to a simulation of the chosen stochastic process. The fluctuating plasma variable is again assumed to follow a stepwise constant stochastic process with prescribed PDF and WTD, but we now generate histories of the variable on the computer with pseudorandom number algorithms, associated to numerical techniques such as transformation or rejection methods [9,10]. In the following, the calculations have been performed assuming an exponential WTD with an average time on step equal to the correlation time of the fluctuations (the inverse of m). This entails that we have chosen a process which loses all memory of its previous states at every time. Other stochastic processes may be considered by our approach if we expect memory effects to be significant [11]. Starting with an initial population X0 at t = 0, we first apply during a time interval Dt1 a constant value Y1 of the fluctuating parameter. The population X1 at time t = Dt1 may be written as:
X 1 ¼ eMðY 1 ÞDt1 X 0
ð5Þ
For a given time history of the variable Y, we may then iterate this expression until we reach a time for which the value of X is constant and a stationary result has been obtained. We have averaged the values of X(t) over a set of such time histories for a fixed value of m. The number of histories required for obtaining a stable result depends on the number of time jumps required before we reach the stationary result. According to the values of the fluctuation and atomic frequencies, about 10–104 histories may be required for obtaining results being reproducible to within 1%. Our simulation technique, although much less computer efficient than our analytic model, is flexible since it allows to rapidly switch between different PDF or WTD already available in numerical libraries. The two approaches have been used to crosscheck several of our results. 3. Results and discussion 3.1. The gamma PDF The three level system 1s, 2s, 2p of hydrogen has been recently analyzed [6,7] for the case of temperature fluctuations. Changes in
ab CðbÞ
T b1 expðaTÞ;
ð6Þ
the parameters a and b are related to the fluctuation rate r ¼ p1 ffiffi, and b pffiffi b the standard deviation DT ¼ a of the fluctuating variable. In Refs. [6,7], it has been shown that significant changes in the 2s population are observed as a fluctuating plasma temperature is assumed. Such changes depend on the frequency m of the fluctuations, and reach two limits as m is small (static case) or large (diabatic case) compared to the typical atomic frequencies given by the collisional and radiative processes. Note that we assume here that the fluctuations are slow enough so that the local electron velocity distribution remains Maxwellian (this requires m me–e, where me–e is the electron collision frequency). Indeed, we will use Maxwellian rates in the following since this is well justified in edge plasmas. If the analysis of a specific plasma device is performed, the average frequency m is in principle known and determined from the measure of the correlation function of the fluctuating variable. Several measures of the temperature and density fluctuation rates [2–4] indicate that values up to r = 1 are sometimes observed in a tokamak. In the following, we will use values 0.5 < r < 0.9 for this fluctuation rate. 3.2. Three level hydrogen atom As an initial approach for an analysis of the Balmer lines, we study a three level hydrogen atom with only the principal quantum levels n = 2, 3 and 4. The collisional excitation rate employs the simple formula for allowed transitions found in the work of Van Regemorter [12]. That is, for the collisional frequency Cij from a state i to a state j:
C ij ¼ Ne 2:5 1011
Eij 1 pffiffiffi e T fij ; Eij T
ð7Þ
where Eij is the energy separation of the states i and j, fij is the absorption oscillator strength, and T the electron temperature in eV. A principle of detailed balance is used for obtaining de-excitation frequencies. Our calculations below retain only these excitation/deexcitation rates and the Einstein spontaneous emission rates for a closed three level system. Our first result concerns temperature fluctuations obeying a gamma distribution with a fluctuation rate r = 0.9, and an average temperature of 2 eV. The stationary populations for the levels n = 3, 4 are shown in Fig. 2 for a density of Ne = 1019 m3, as a function of the fluctuation frequency m. This couple of density and temperature sets us in a regime for which both radiative and collisional processes contribute to the population kinetics. For frequencies m lower than about 106 s1, we are in the static regime, for which the fluctuation frequency m is negligible compared to a typical atomic relaxation frequency mat (m mat). One enters the opposite diabatic limit where m mat for frequencies m higher than about 1010 s1. In between those two asymptotic regimes, it can be seen in Fig. 2 that the population of level n = 3 increases with increasing frequencies by 11%, while in the same range the population of n = 4 decreases by 31%. In our closed system, the population level n = 2 changes in order to maintain a constant total population of atoms. If we use r values lower than about 0.1, we obtain the populations in the fluctuation free case (0.185 for n = 3, and 0.214 for n = 4). These populations are neither equal to the static limit, nor to the diabatic limit
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R. Hammami et al. / Journal of Nuclear Materials 438 (2013) S1224–S1227
Fig. 2. Populations of the level n = 3 and n = 4 of a three level hydrogen atom, as a function of the fluctuation frequency of the temperature, and for a fixed electron density of Ne = 1019 m3. The temperature PDF is a gamma distribution with a fluctuation rate of r = 0.9 and average temperature 2 eV.
Fig. 4. Populations of several carbon ions as a function of the average temperature, and for a fixed electron density of Ne = 1019 m3. Different ion populations (from C+1 to C+3) are plotted with (solid lines) and without (dotted lines) fluctuations.
3.3. Application to the carbon ionization equilibrium
for temperature fluctuations. Other calculations have been made for average temperatures of 1 and 10 eV, or changing the density to Ne = 1020 m3. For each of these cases, an increase of n = 3 and a decrease of the n = 4 populations has been observed with an increasing fluctuation frequency. We now examine electronic density fluctuations which we assume to follow the same gamma distribution as Eq. (6), but for P(Ne) with r = 0.9. In Fig. 3, we have plotted the populations of levels n = 3, 4 for an average density 1019 m3, and a temperature of 2 eV. A change of population is also observed between the static and diabatic limits, with a simultaneous increase of the two levels as the frequency is increased, by 19% and 22% for respectively levels n = 3 and 4. We have also plotted in Fig. 3 the turbulence free populations. It can be seen that since the collisional frequencies are linear in density, the diabatic limit is here identical to the turbulence free populations. From all the cases examined, it appears that a specific study with our model is required for each plasma condition and atomic system under consideration. Here, our simple model predicts that populations of a three level hydrogen atom are changed by temperature or density fluctuations. A comparison to experimental measures would require a more realistic atomic system.
Neglecting particle transport, we have studied the ionization balance of carbon in the presence of temperature fluctuations. The equation of ionization equilibrium without transport can be written for the average population of the ion with charge Z:
dhnZ i ¼ hnZ Ne aZ!Zþ1 i hnZ Ne bZ!Z1 i þ hnZ1 Ne aZ1!Z i dt þ hnZþ1 N e bZþ1!Z i;
ð8Þ
where aZ?Z0 and bZ?Z0 are respectively the ionization and recombination coefficients. These data are taken from the ADAS database [13]. The fluctuation frequency used in our calculation is equal to 105 s1, which corresponds to a correlation time of 10 ls, as typically measured in edge plasma of current fusion devices [3]. The temperature PDF is assumed to be a gamma distribution with a fluctuation rate r = 0.5. We show in Fig. 4 the abundances of several carbon ions (from C+1 to C+3) as a function of the average temperature in the range 1–15 eV, for a density equal to 1019 m3, conditions which can be found in the edge plasma of the Tore Supra tokamak [14]. The relaxation times for ionization/recombination are smaller by at least one order of magnitude than the fluctuation frequency, setting these calculations in a diabatic regime. Changes in the ion populations are clearly seen, with shifts in the position of the maximum abundance modifying the ratio of the different ion species for a given temperature. 4. Conclusion
Fig. 3. Populations of the level n = 3 and n = 4 of a three level hydrogen atom, as a function of the fluctuation frequency of the density, and for a fixed electron temperature of 2 eV. The density PDF is a gamma distribution with a fluctuation rate of r = 0.9 and average density of Ne = 1019 m3.
The effect of temperature and density fluctuations on the population of simple atomic or ionic systems has been investigated. We have shown that significant changes occur in the populations of a three level system modelling hydrogen Balmer emission as we scan the typical frequency fluctuation of temperature or density. For this system, a static limit (m mat) and a diabatic limit (m mat) have been identified, together with an intermediate region where the changes in populations are frequency dependent. The ionization balance of carbon has also been studied in the presence of temperature fluctuations. The abundances of the different ions are modified in the presence of such temperature fluctuations. All these effects are significant only for fluctuation rates r larger than several tens of percent, values which can be found in tokamak edge plasmas. With the aim of achieving a fluctuating edge plasma diagnostic, further studies will concern more realistic atomic system, which will require retaining more atomic processes and a more detailed atomic structure.
R. Hammami et al. / Journal of Nuclear Materials 438 (2013) S1224–S1227
Acknowledgements This work was carried out within the framework of the European Fusion Development Agreement and the French Research Federation for Fusion Studies. It is supported by the European Communities under the contract of Association between Euratom and CEA. The views and opinions expressed herein do not necessarily reflect those of the European Commission. We also acknowledge the support of the French National Research Agency (Contract ANR-11-BS09-023, SEDIBA).
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
References [1] T. Fujimoto, Plasma Spectroscopy, Clarendon Press, Oxford, 2004. [2] J.P. Graves et al., Plasma Phys. Control. Fusion 47 (2005) L1–L9.
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J.A. Boedo, J. Nucl. Mater. 390–391 (2009) 29. H.T. Evensen et al., Nucl. Fusion 39 (1998) 237. G.M. McCraken et al., Nucl. Fusion 38 (1998) 619. F. Catoire et al., Phys. Rev. A83 (2011) 012518. F. Catoire et al., Eur. Phys. J. D 65 (2011) 481. M. Lax, H. Scher, Phys. Rev. Lett. 39 (1977) 781. F.J. Vesely, Computational Physics, an Introduction, Plenum Press, New York, 1994. IMSL, 2012,. J. Rosato et al., Phys. Lett. A 375 (2011) 4187. H. Van Regemorter, Astrophys. J. 136 (1962) 906. H.P. Summers, The ADAS User Manual, version 2.6, 2004,
Contents lists available at SciVerse ScienceDirect
Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Changes in atomic populations due to edge plasma fluctuations R. Hammami a,⇑, H. Capes a, F. Catoire b, L. Godbert-Mouret a, M. Koubiti a, Y. Marandet a, A. Mekkaoui a, J. Rosato a, R. Stamm a a b
PIIM, Aix-Marseille Université and CNRS, centre Saint Jérôme, Marseille 13397, France CELIA, Université de Bordeaux 1 and CNRS, Domaine du Haut Carré, Talence 33405, France
a r t i c l e
i n f o
Article history: Available online 16 January 2013
a b s t r a c t The population balance of atoms or ions in an edge plasma is calculated in the presence of fluctuating density or temperature. We have used a stochastic model taking advantage of the knowledge of the plasma parameter statistical properties, and assuming a stepwise constant stochastic process for the fluctuating variable. The model is applied to simplified atomic systems such as three level hydrogen atoms or the ionization balance of carbon affected by electronic temperature or density fluctuations obeying a gamma PDF, and an exponential waiting time distribution. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Level populations of atom and ions are important quantities for characterizing edge plasmas. Radiation loss, ionization equilibrium, or the intensity of spectral lines used for diagnosing the plasma are directly related to the atomic and ionic populations. The determination of the latter relies on the solution of an atomic population kinetic model which is usually obtained using a stationary collisional-radiative model (CRM), assuming constant plasma parameters such as electronic density and temperature [1]. In this work we assume that these parameters fluctuate as a consequence of edge plasma turbulence. In edge plasmas, density and temperature fluctuations of order unity on time scales of the order of 10 ls have been measured, and characterized with their statistical properties such as the probability density function (PDF) and the correlation function of the fluctuating parameter [2–5]. For such conditions, we adopt a previously developed model which assumes that the fluctuating quantity obeys a stochastic process. The density or temperature is distributed according to a prescribed PDF, and jumps from one constant value to another one according to a given waiting time distribution (WTD) [6,7]. The latter is determined from the correlation function of the fluctuations. An example for such a stochastic evolution of the temperature is shown in Fig. 1, in conditions of a turbulent edge plasma. Assuming such a process for the fluctuations, we may solve the time dependent CRM by an analytical model or a Monte Carlo type simulation. In Section 2, we briefly describe how atomic populations may be ccalculated in the presence of a renewal stochastic process for the fluctuations [8]. Our models are applied to two simple atomic
⇑ Corresponding author. E-mail address: [email protected] (R. Hammami). 0022-3115/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnucmat.2013.01.271
systems in Section 3, a reduced hydrogen atom, and the carbon ionization balance. 2. Modelling the plasma parameters with a renewal process The atomic or ionic populations are described by a vector X(t) = (x1(t), . . . , xN(t)) containing the different population x of each level j. Initial values of the population are given for instance at time t = 0 as X0 = X(t = 0). The kinetics of the atomic populations is governed by a differential equation of the form [6]:
dXðtÞ ¼ MðYðtÞÞXðtÞ; dt
ð1Þ
where M is a time dependent matrix filled with the collisionalradiative rates, and Y(t) is the fluctuating plasma parameter (electronic temperature or density in the following). We now assume that Y follows a renewal process with stepwise constant values separated by instantaneous jumps. The duration on each step obeys a prescribed waiting time distribution (WTD), and the values taken by Y are distributed along a chosen probability density function (PDF) P(Y). Eq. (1) is then a linear stochastic equation, which may be solved by a numerical simulation or an analytic approach [6]. The analytic solution takes a particularly simple form if an exponential WTD / (t) = m exp(mt) is chosen, where m is the typical fluctuation frequency of Y. This choice corresponds to an exponential correlation function, with a correlation time m1. It may be expressed as the Laplace transform of the Green function G associated to Eq. (1):
hGðsÞi ¼ ½1 mGST ðsÞ1 GST ðsÞ;
ð2Þ
where the brackets h i imply that an ensemble average has been taken, and the subscript ST means that a static average over the PDF P(Y) has been taken. This quantity may be written as:
R. Hammami et al. / Journal of Nuclear Materials 438 (2013) S1224–S1227
S1225
the population of the 2s level depend on the typical frequency m and the rate r = DT/hTi of the electron temperature fluctuations, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where DT ¼ hT 2 i hTi2 , and the average electron temperature R1 is given by hTi ¼ 0 TPðTÞdT with P(T) the PDF of the fluctuating temperature. Assuming a gamma function for the PDF P(T):
PðTÞ ¼
Fig. 1. Simulation of a stepwise constant stochastic evolution of the temperature with an exponential WTD of typical fluctuation frequency m = 105 s1, for an electronic density Ne = 1019 m3. The fluctuations obey to a gamma PDF with a fluctuation rate r = 0.9, and an average temperature of 2 eV (see also Sections 2–3).
GST ðsÞ ¼
1 : sþmM
ð3Þ
It is possible to obtain the average hX(t)i of the population vector in the limit of long times by using:
limhXðtÞi ¼ limþ shGðsÞiX 0 :
t!1
s!0
ð4Þ
The solution of Eq. (1) may also be obtained by a numerical solution coupled to a simulation of the chosen stochastic process. The fluctuating plasma variable is again assumed to follow a stepwise constant stochastic process with prescribed PDF and WTD, but we now generate histories of the variable on the computer with pseudorandom number algorithms, associated to numerical techniques such as transformation or rejection methods [9,10]. In the following, the calculations have been performed assuming an exponential WTD with an average time on step equal to the correlation time of the fluctuations (the inverse of m). This entails that we have chosen a process which loses all memory of its previous states at every time. Other stochastic processes may be considered by our approach if we expect memory effects to be significant [11]. Starting with an initial population X0 at t = 0, we first apply during a time interval Dt1 a constant value Y1 of the fluctuating parameter. The population X1 at time t = Dt1 may be written as:
X 1 ¼ eMðY 1 ÞDt1 X 0
ð5Þ
For a given time history of the variable Y, we may then iterate this expression until we reach a time for which the value of X is constant and a stationary result has been obtained. We have averaged the values of X(t) over a set of such time histories for a fixed value of m. The number of histories required for obtaining a stable result depends on the number of time jumps required before we reach the stationary result. According to the values of the fluctuation and atomic frequencies, about 10–104 histories may be required for obtaining results being reproducible to within 1%. Our simulation technique, although much less computer efficient than our analytic model, is flexible since it allows to rapidly switch between different PDF or WTD already available in numerical libraries. The two approaches have been used to crosscheck several of our results. 3. Results and discussion 3.1. The gamma PDF The three level system 1s, 2s, 2p of hydrogen has been recently analyzed [6,7] for the case of temperature fluctuations. Changes in
ab CðbÞ
T b1 expðaTÞ;
ð6Þ
the parameters a and b are related to the fluctuation rate r ¼ p1 ffiffi, and b pffiffi b the standard deviation DT ¼ a of the fluctuating variable. In Refs. [6,7], it has been shown that significant changes in the 2s population are observed as a fluctuating plasma temperature is assumed. Such changes depend on the frequency m of the fluctuations, and reach two limits as m is small (static case) or large (diabatic case) compared to the typical atomic frequencies given by the collisional and radiative processes. Note that we assume here that the fluctuations are slow enough so that the local electron velocity distribution remains Maxwellian (this requires m me–e, where me–e is the electron collision frequency). Indeed, we will use Maxwellian rates in the following since this is well justified in edge plasmas. If the analysis of a specific plasma device is performed, the average frequency m is in principle known and determined from the measure of the correlation function of the fluctuating variable. Several measures of the temperature and density fluctuation rates [2–4] indicate that values up to r = 1 are sometimes observed in a tokamak. In the following, we will use values 0.5 < r < 0.9 for this fluctuation rate. 3.2. Three level hydrogen atom As an initial approach for an analysis of the Balmer lines, we study a three level hydrogen atom with only the principal quantum levels n = 2, 3 and 4. The collisional excitation rate employs the simple formula for allowed transitions found in the work of Van Regemorter [12]. That is, for the collisional frequency Cij from a state i to a state j:
C ij ¼ Ne 2:5 1011
Eij 1 pffiffiffi e T fij ; Eij T
ð7Þ
where Eij is the energy separation of the states i and j, fij is the absorption oscillator strength, and T the electron temperature in eV. A principle of detailed balance is used for obtaining de-excitation frequencies. Our calculations below retain only these excitation/deexcitation rates and the Einstein spontaneous emission rates for a closed three level system. Our first result concerns temperature fluctuations obeying a gamma distribution with a fluctuation rate r = 0.9, and an average temperature of 2 eV. The stationary populations for the levels n = 3, 4 are shown in Fig. 2 for a density of Ne = 1019 m3, as a function of the fluctuation frequency m. This couple of density and temperature sets us in a regime for which both radiative and collisional processes contribute to the population kinetics. For frequencies m lower than about 106 s1, we are in the static regime, for which the fluctuation frequency m is negligible compared to a typical atomic relaxation frequency mat (m mat). One enters the opposite diabatic limit where m mat for frequencies m higher than about 1010 s1. In between those two asymptotic regimes, it can be seen in Fig. 2 that the population of level n = 3 increases with increasing frequencies by 11%, while in the same range the population of n = 4 decreases by 31%. In our closed system, the population level n = 2 changes in order to maintain a constant total population of atoms. If we use r values lower than about 0.1, we obtain the populations in the fluctuation free case (0.185 for n = 3, and 0.214 for n = 4). These populations are neither equal to the static limit, nor to the diabatic limit
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R. Hammami et al. / Journal of Nuclear Materials 438 (2013) S1224–S1227
Fig. 2. Populations of the level n = 3 and n = 4 of a three level hydrogen atom, as a function of the fluctuation frequency of the temperature, and for a fixed electron density of Ne = 1019 m3. The temperature PDF is a gamma distribution with a fluctuation rate of r = 0.9 and average temperature 2 eV.
Fig. 4. Populations of several carbon ions as a function of the average temperature, and for a fixed electron density of Ne = 1019 m3. Different ion populations (from C+1 to C+3) are plotted with (solid lines) and without (dotted lines) fluctuations.
3.3. Application to the carbon ionization equilibrium
for temperature fluctuations. Other calculations have been made for average temperatures of 1 and 10 eV, or changing the density to Ne = 1020 m3. For each of these cases, an increase of n = 3 and a decrease of the n = 4 populations has been observed with an increasing fluctuation frequency. We now examine electronic density fluctuations which we assume to follow the same gamma distribution as Eq. (6), but for P(Ne) with r = 0.9. In Fig. 3, we have plotted the populations of levels n = 3, 4 for an average density 1019 m3, and a temperature of 2 eV. A change of population is also observed between the static and diabatic limits, with a simultaneous increase of the two levels as the frequency is increased, by 19% and 22% for respectively levels n = 3 and 4. We have also plotted in Fig. 3 the turbulence free populations. It can be seen that since the collisional frequencies are linear in density, the diabatic limit is here identical to the turbulence free populations. From all the cases examined, it appears that a specific study with our model is required for each plasma condition and atomic system under consideration. Here, our simple model predicts that populations of a three level hydrogen atom are changed by temperature or density fluctuations. A comparison to experimental measures would require a more realistic atomic system.
Neglecting particle transport, we have studied the ionization balance of carbon in the presence of temperature fluctuations. The equation of ionization equilibrium without transport can be written for the average population of the ion with charge Z:
dhnZ i ¼ hnZ Ne aZ!Zþ1 i hnZ Ne bZ!Z1 i þ hnZ1 Ne aZ1!Z i dt þ hnZþ1 N e bZþ1!Z i;
ð8Þ
where aZ?Z0 and bZ?Z0 are respectively the ionization and recombination coefficients. These data are taken from the ADAS database [13]. The fluctuation frequency used in our calculation is equal to 105 s1, which corresponds to a correlation time of 10 ls, as typically measured in edge plasma of current fusion devices [3]. The temperature PDF is assumed to be a gamma distribution with a fluctuation rate r = 0.5. We show in Fig. 4 the abundances of several carbon ions (from C+1 to C+3) as a function of the average temperature in the range 1–15 eV, for a density equal to 1019 m3, conditions which can be found in the edge plasma of the Tore Supra tokamak [14]. The relaxation times for ionization/recombination are smaller by at least one order of magnitude than the fluctuation frequency, setting these calculations in a diabatic regime. Changes in the ion populations are clearly seen, with shifts in the position of the maximum abundance modifying the ratio of the different ion species for a given temperature. 4. Conclusion
Fig. 3. Populations of the level n = 3 and n = 4 of a three level hydrogen atom, as a function of the fluctuation frequency of the density, and for a fixed electron temperature of 2 eV. The density PDF is a gamma distribution with a fluctuation rate of r = 0.9 and average density of Ne = 1019 m3.
The effect of temperature and density fluctuations on the population of simple atomic or ionic systems has been investigated. We have shown that significant changes occur in the populations of a three level system modelling hydrogen Balmer emission as we scan the typical frequency fluctuation of temperature or density. For this system, a static limit (m mat) and a diabatic limit (m mat) have been identified, together with an intermediate region where the changes in populations are frequency dependent. The ionization balance of carbon has also been studied in the presence of temperature fluctuations. The abundances of the different ions are modified in the presence of such temperature fluctuations. All these effects are significant only for fluctuation rates r larger than several tens of percent, values which can be found in tokamak edge plasmas. With the aim of achieving a fluctuating edge plasma diagnostic, further studies will concern more realistic atomic system, which will require retaining more atomic processes and a more detailed atomic structure.
R. Hammami et al. / Journal of Nuclear Materials 438 (2013) S1224–S1227
Acknowledgements This work was carried out within the framework of the European Fusion Development Agreement and the French Research Federation for Fusion Studies. It is supported by the European Communities under the contract of Association between Euratom and CEA. The views and opinions expressed herein do not necessarily reflect those of the European Commission. We also acknowledge the support of the French National Research Agency (Contract ANR-11-BS09-023, SEDIBA).
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
References [1] T. Fujimoto, Plasma Spectroscopy, Clarendon Press, Oxford, 2004. [2] J.P. Graves et al., Plasma Phys. Control. Fusion 47 (2005) L1–L9.
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