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6, since Te = 1 x 103 K is high enough for these higher-lying levels. See Fig. 4.2. Example In many practical situations we encounter plasmas which show the population distributions characteristic of the ionizing plasma component. Examples are lowpressure discharges, especially the positive column of a glow discharge. Figure 4.12 shows an example of the experimentally determined populations in an argon positive column plasma. By changing the discharge current, the authors changed ne over three orders of magnitude, while they kept Te almost constant at 5 x 104 K by adjusting the filling pressure. Figure 4.12(a) shows the ne dependence of populations of several excited levels of neutral argon. The reader can locate this plasma on Fig. 1.2 by drawing a short horizontal line. Note that we are dealing with neutral species, z=l. If we compare this figure with Fig. 4.4 we recognize close similarities between them: the density dependences of the populations in Fig. 4.12(a) appear to correspond to those of Fig. 4.4 for p > 3 and 1018 < «e < 1022 m~3. It is noted that, in this experiment, the lowest-lying excited levels, corresponding to p = 2 in Fig. 4.4, are not observed. This is because the population is determined from the emission line intensity observed in the visible region of the wavelength. If our assumption is correct, we may conclude that level 4p'[3/2]1; corresponding to p = 3 in Fig. 4.4, is in the corona phase in the lowest-density region while all the other levels are in the saturation phase, and further that all the populations saturate completely for «e > 1019 m~3. The difference in the boundary «e values by one or two orders of magnitude between the two figures may be explained from

FIG 4.12 Population distribution of excited argon atoms in a positive column plasma. T e ~ 5 x 104 K. (a) Dependences on ne. This figure corresponds to Fig. 4.4 for neutral hydrogen, (b) Population distribution among the levels corresponding to Fig. 4.5. (Quoted from Tachibana and Fukuda, 1973; with permission from The Institute of Pure and Applied Physics.)

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the following two points: (1) we are comparing different atomic species, hydrogen and argon; (2) in the experiment, two of the four lowest-lying levels are metastable, and the resonance lines originating from the other two levels, which correspond to the Lyman a line in the case of hydrogen, are subjected to heavy radiation trapping, which will be dealt with later in Chapter 8. These facts result in a much reduced effective transition probability from the lowest-lying levels, which is not taken into account in the theory and tend to reduce Griem's boundary ne for the lowest excited level. Figure 4.12(b) shows the population distribution over the excited levels for « e = 1 x 1019 m~3, where all the levels are in the saturation phase. Since the levels have various energy values, we take the effective principal quantum number, eq. (1.1 a), as the abscissa. They follow the minus sixth power law except for the lowest-lying levels. Its deviation is in the opposite direction to those in Fig. 4.5. This inconsistency may be explained from the fact that the temperature of the experiment of Te = 5 x 104 K is not sufficiently high for the exponential factor to be neglected entirely. 4.3 Recombining plasma component - high-temperature case In this section we study the recombining plasma component of the excited-level populations. See Fig. 1.9. According to eq. (4.20) this is defined as

We further assume high temperature, i.e. Te/z2 is much higher than 1.5 x 104 K. This boundary temperature is different from that in Section 4.1. This boundary is relevant for the recombining plasma; as we will see later, in this range of temperature, collisional excitation from an excited level is more probable than collisional deexcitation from this level. See Fig. 4.2. We take neutral hydrogen in a plasma with Te= 1.28 x 105 K as an example for the purpose of illustration. The reader may remember the horizontal line drawn on Fig. 1.2 for the plasmas of Section 4.2. Table 4.1(b) gives the result of calculation of the population coefficient r0(p), A salient feature is that, for high densities, virtually all the levels have r0(p)= 1, indicating thermodynamic equilibrium populations, or LTE populations, as we have seen earlier. For lower densities the r0(p)'sdeviate from 1, but the degree of this deviation is rather small. Figure 4.13 shows the population of several levels as functions of ne, where the population per unit statistical weight, nQ(p)/g(p), has been further divided by the ion and electron densities, nzne. The population nQ(p) in this figure is regarded as n0(p)/z3for hydrogen-like ions with nuclear charge z. This is an approximation, but the degree of approximation is better than that for the ionizing plasma, and even exact in the low-density limit. In the following discussions of the recombining plasma component, we sometimes call 0(p)/g(p)nztie]simply the "population". This quantity is almost exactly [n

112 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4.13 The recombining plasma component of excited-level populations against «e [m^1]. The ordinate is n0(p) per unit statistical weight g(p) further divided by the ion and electron densities, nz and ne, respectively. Calculation is for neutral hydrogen with T e =1.28 x 105 K. (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.)

FIG 4.14 Population distribution of the recombining plasma component of the excited-level populations. The ordinate is the same as Fig. 4.13, and the abscissa is the logarithm of the principal quantum number of the level. Calculation is for neutral hydrogen with r e =1.28x!0 5 K. The Saha-Boltzmann population given by eq. (2.la) is shown with the solid line. The dash-dotted lines indicate the boundary between the CRC phase and the saturation (LTE) phase as given by eq. (4.25). (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.) proportional to r0(p) except for the small /^-dependent exponential factor in eq. (2.7) under this high-temperature condition. See also eq. (4.38). Figure 4.14 shows the population distributions for several electron densities. In Fig. 4.14 the population in LTE, or the Saha-Boltzmann population distribution given by eq. (2.la), i.e. r0(p) = 1, is shown with the solid line. As has been noted the actual populations

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in the high-density limit agree with this equilibrium population distribution. It should be noted that the range of the ordinate of these figures is very narrow. All the populations lie within a range of a factor of 1.5 for the whole of the range of ne and/?; this is actually seen in Table 4.1(b): the smallest r0(p) is 0.69. Therefore, we may conclude that, in the high-temperature case, the recombining plasma component is rather close to the Saha-Boltzmann value for low densities (for very high temperatures it is even larger than that - see Fig. 5.10 later; see also Appendix 5C) and tends to it at high densities. Figure 4.15 is similar to Fig. 4.6: the breakdown of the populating (a) and depopulating (b) fluxes for level p = 5. It should be noted that Fig. 4.15(b) is

FIG 4.15 Breakdown of the (a) populating and (b) depopulating fluxes concerning level p = 5 into individual fluxes. This condition corresponds to Figs. 4.13 and 4.14. Other explanation is almost the same as that for Fig. 4.6. Part (b) of this figure is identical to Fig. 4.6(b). (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.)

114 POPULATION DISTRIBUTION AND POPULATION KINETICS identical to Fig. 4.6(b), since the depopulating processes are common to the ionizing plasma component and the recombining plasma component. It is seen that, in the higher-density regions, higher than 1018 m~3, every depopulating flux in (b) is almost exactly balanced in (a) by the corresponding populating flux of its inverse process. In other words, the principle of detailed balance is actually realized. This is consistent with the fact that the populations are almost exactly equal to their Saha-Boltzmann values in these higher-density regions. In Fig. 4.15, we find two puzzling features: 1. For lower-density regions both the populating fluxes and the depopulating fluxes are radiative, i.e. the radiative recombination and the radiative cascade for the populating process in (a) and the radiative decay for the depopulating process in (b). No relationship like the principle of detailed balance is expected among the rate constants of these processes. Still the populations are very close to those given from thermodynamic equilibrium as we have seen. 2. The transition of the populating mechanism in Fig. 4.15(a) from the radiative processes in lower densities to the collisional processes in higher densities takes place at the density approximately equal to that in Fig. 4.15(b) for the transition of the depopulating mechanism, i.e. Griem's boundary given by eq. (4.25). We will investigate whether these features are a mere coincidence or they are a kind of necessary phenomenon stemming from a deeper basis. CRC phase Figure 4.16 is a sketch of the dominant fluxes of electrons in the energy-level diagram for « e = 1012 m~3. This figure together with Fig. 4.15 shows that, in low density, the populating fluxes to all the levels are the direct radiative recombination plus the cascade contributions from the higher-lying levels. The depopulating process is the radiative decay. We name this situation the capture radiative cascade (CRC) phase. The population is given from the above balance relation by

We approximate the populations of the higher-lying levels q to their SahaBoltzmann values, i.e. nQ(q) ~ Z(q)nzne (eq. (2.7a)). We rewrite eq. (4.39) as

The recombination rate coefficient (3(p) is given in Fig. 3.9 and approximately by eq. (3.19a). In the present high-temperature case, we adopt, for large p, the

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FIG 4.16 Sketch of the dominant populating and depopulating fluxes, or flows of electrons, in the energy-level diagram, corresponding to Figs. 4.13-4.15. «e = 1 x 1012m~3. Virtually all the levels are in the CRC phase. At right the total flux of recombination reaching the ground state is given by the downward arrow with the figure in it giving the collisional-radiative recombination rate coefficient. Other explanation is almost the same as that for Fig. 4.7. (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.) approximation for the exponential integral, eq. (3.21),

where Khas been given by eq. (4.12a). See also eq. (2.7). The second term in the numerator of eq. (4.39a) is expressed by use of eq. (3.8) with the approximation gbb = 1. By using a technique similar to that used in deriving eq. (4.12) we obtain the approximation

For the denominator we adopt the approximation

116 POPULATION DISTRIBUTION AND POPULATION KINETICS With the above approximations, eq. (4.39) yields

the near-Saha-Boltzmann population, or

It is thus concluded that the near-Saha-Boltzmann populations in the lowdensity regions in Table 4.1(b) or Fig. 4.14 are the result of the intricate relationships between the radiative recombination rate coefficient and the transition probabilities. It is further noted that the relative contributions in the numerator in eq. (4.39) from the radiative recombination, eq. (4.41), and the cascade, eq. (4.42), is approximately 2:1. This is in accordance with the accurate calculation shown in Fig. 4.15(a). Figure 1.10(b) gives a summary of the recombining plasma component. The left-side region indicates the CRC phase where the simplified population kinetics is depicted. The population distribution at high temperature is given in parentheses. Transition from the CRC phase to the saturation phase In Figs. 4.13 and 4.14, with an increase in «e, populations make a transition from the near-Saha-Boltzmann populations to almost exact Saha-Boltzmann populations. This transition is more clearly seen in Fig. 4.15 as transitions from the radiative processes to the collisional processes both in the populating and depopulating fluxes. The latter phase is called the saturation phase. Figure 4.17 shows a sketch of the dominant fluxes in the energy-level diagram at «e =1017 m~3. For levels lower than p = 5 the feature of the population kinetics is approximately the same as in Fig. 4.16, indicating that these levels are still in the low-density region, or in the CRC phase. Levels higher than p = 6 have entered into the highdensity region, or the saturation phase, so that the situation is completely different. In this example the boundary level which divides the low- and high-lying levels lies between p = 5 and 6. The transition from the CRC to saturation phases in the depopulating mechanism is the same as that for the ionizing plasma as given by eq. (4.25), or with the neglect of minor terms,

Among the populating fluxes the dominant processes at lower densities (or lower-lying levels) in the CRC phase are, as we have seen above, the direct radiative recombination and the cascade. At higher densities (higher-lying levels) in the saturation phase they are mainly the collisional deexcitation from the

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FIG 4.17 Sketch similar to Fig. 4.16 except that «e = 1 x 1017 m 3. The boundary level pG between the CRC phase and the saturation (LTE) phase, given by eq. (4.25), lies between p = 5 and 6, as indicated by the dash-dotted line. The downward recombination flux through a sufficiently high-lying level, level p = 10, is 1% of the net recombination flux as shown with the downward arrow. (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.) higher-lying levels. See Fig. 4.15(a). Thus the transition takes place when the magnitudes of these radiative and collisional fluxes become equal:

It is to be noted that the higher-lying levels q are already in LTE at «e at which this level p makes the transition from the CRC phase to the saturation phase, so that their population is given by n0(q) = Z(q)nzne. A similar procedure has already been employed in approximating eq. (4.39). We further note the principle of detailed balance, Z(q)F(q,p) = Z(p)C(p,q), eq. (3.31a). Then we rewrite the right-hand side of this equation (4.45) as

Equation (4.45) is then rewritten as

This equation is, within the approximations of eqs. (4.41), (4.42), and (4.12), identical with eq. (4.44), which was for the transition in the depopulating

118 POPULATION DISTRIBUTION AND POPULATION KINETICS mechanism. Thus it has been shown that, for both the populating and depopulating mechanisms, the transition takes place at almost the same electron density. The boundary is thus given by eq. (4.25) or (4.29a). Figure 1.10(b) includes this boundary, as labeled "GRIEM". Saturation (LTE) phase For higher densities the levels are in the saturation phase. At the beginning of this section we have already seen that, at high density, for an excited level, the populating fluxes from the higher-lying levels (deexcitation) and the continuum (three-body recombination) are almost exactly balanced by the respective inverse depopulating fluxes (excitation and ionization, respectively, see Fig. 4.15.). If the still higher-lying levels are in LTE and their populations are given by the SahaBoltzmann equation, eq. (2.la), the above balance relationship should result in the LTE population of this level. It is further noted from the discussion around eqs. (4.5)-(4.11) that the most dominant depopulating flux is the excitation to the adjacent higher-lying level, rather than ionization as is sometimes assumed. The most dominant populating flux is accordingly deexcitation from that adjacent level to this level, rather than three-body recombination, again as is sometimes assumed. Roughly speaking, therefore, the dominant flow pattern of the population is (p+ 1)—>/>—>(/>+ 1). This feature is seen in Fig. 4.15, and in Fig. 4.17 for levels/? > 6. Figure 4.18 shows a sketch of the dominant fluxes in the high-density limit. The flux has been further divided by nzn^, and its magnitude is given with

FIG 4.18 Sketch similar to Fig. 4.16 but for the high-density limit, where the magnitude of a flux has been divided by nznl and given by figures. All the excited levels are in the saturation (LTE) phase. (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.)

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the numbers. In this figure (and also in Fig. 4.15(a)) it is seen in fact that the direct contribution from the three-body recombination is substantially smaller than the deexcitation. Therefore, the dominant relationship of population balance is given by

and the resulting population ratio is the Boltzmann distribution, eq. (2.3) or eq. (3.32)

We may continue the string of this reasoning along p, (/>+!), (p+2),..., to reach very high-lying levels, denoted by, say, r. As has been discussed already, the atomic characteristics of these negative-energy discrete levels continue smoothly across the ionization limit to low-energy continuum states. Since we assume thermodynamic equilibrium for the "free" electrons, it would be natural to assume that the Maxwell distribution continues smoothly across the ionization limit to the negative-energy discrete levels. In other words, these levels are in thermodynamic equilibrium with the continuum, or they are in LTE, and their populations are given by the Saha-Boltzmann values, nQ(r) = Z(r)nzne, eq. (2.7a). See also the paragraph at the close of Section 2.1. If we now trace back the above reasoning down to level p, then we arrive at the conclusion that this level should be in LTE: Therefore, the saturation phase may be called the LTE phase. In Fig. 1.10(b), the upper region higher than the boundaries "GRIEM" and "BYRON" corresponds to this phase, and the relationship, eq. (4.46), is depicted schematically. The above discussion concerning the continuation of the Maxwell distribution of the "free" electrons to the negative-energy discrete levels may appear less convincing. This is partly due to our assumption of "free" electrons for the continuum state electrons, and partly to the ambiguity in treating very high-lying levels. The first assumption is obviously wrong for low-energy electrons, since the interaction of an electron with the ion cannot be neglected in comparison with its kinetic energy. Remember that the Coulomb force is strong and of long range. The second point is also problematic since we implicitly assume that the principal quantum number of levels can increase indefinitely. We then immediately encounter the difficulty that the total statistical weights of the levels, or the state density of the levels for a finite energy width, or the LTE populations, diverge. These points will be addressed and an adequate resolution will be introduced in Section 9.4. Then, the above arguments gain sound footings. At the close of Section 4.1 an important relationship was introduced:

120 POPULATION DISTRIBUTION AND POPULATION KINETICS which is valid in the limit of high density where all the radiative transitions can be neglected. Another approximate relationship was noted for high temperatures and high densities (Fig. 4.10): It is thus concluded that, in the limit of high density, we have It is easily seen that, for Te/z2^> 1.5 x 104 K, the deviation of r0(p) from 1 is very small except for, say, p = 2. See Fig. 4.3. This is another explanation of the LTE population, eq. (4.47). Equation (4.48) has been derived in the limit of high density. It should be noted, however, that this equation is also valid at much lower densities. As has been shown in Figs. 4.13 and 1.10(b), for example, an excited level is in the saturation phase for densities down to Griem's boundary, and r0(p) continues to take its high-density-limit value until that boundary. In Table 4.1(b), level p = 5, for example, has r0(5) = 1 — 5~6 = 1.00 from the high-density limit down to « e ~10 17 m~ 3 within a 10% deviation. This boundary is nothing but Griem's boundary as clearly seen in Fig. 1.10(b). 4.4 Recombining plasma component - low-temperature case In this section we examine the low-temperature case which is more important in practical situations. We assume the temperature to be low, i.e. Te/z2 -c 1.5 x 104 K. As an example we take neutral hydrogen with Te = 1 x 103 K. The reader can draw another horizontal straight line in Fig. 1.2. Table 4.1 (a) shows r0(p). Unlike the high-temperature case of Table 4.1(b), r0(p) is, in many cases, much smaller than 1. Only very high-lying levels have r0(p)~ 1 in high densities. Figure 4.19 shows the populations of several excited levels against ne, where, as before, the population per unit statistical weight nQ(p)/g(p) has been further divided by the ion and electron densities nzne. This quantity, [«o(/")/ g(p)nzne], is proportional to rQ(p); see eqs. (4.38) and (2.7). For an excited level, we may divide the range of ne into four regions according to the dependence of its population on ne: the low-density limit (for «e < 1012 m~ 3 in the case of level p = 5 taken as an example), the gradual increase in the population, or r0(p), with an increase in ne (10 12 <« e < 1016 m~ 3 for level 5), the steep increase (10 16 <« e <10 19 m~3), and finally saturation (« e >10 19 m~3). The first three regions are named altogether as the capture-radiative-cascade (CRC) phase, and the last region as the saturation phase. This nomenclature follows that of the hightemperature case. On each curve in Fig. 4.19, the closed circle indicates ne at which the dominant depopulating process changes from the radiative decay to the collisional depopulation, as given by eq. (4.25). It is obvious from this figure that this boundary ne gives the transition from the CRC phase to the saturation phase.

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FIG 4.19 The recombining plasma component of excited-level populations for the low-temperature case against ne. The ordinate is n0(p) per unit statistical weight g(p), further divided by the ion and electron densities, nz and ne, respectively. Calculation is for neutral hydrogen with T e = l x l 0 3 K. The boundary ne between the low-density region (CRC phase) and the high-density region (saturation phase) as given by eq. (4.25) is shown with the closed circles. (Quoted from Fujimoto, 1980b; with permission from The Physical Society of Japan.) Figure 4.20 shows the population distribution over the excited levels for several electron densities; Fig. 4.20(a) corresponds to Fig. 4.14, and Fig. 4.20(b) is another plot, the conventional Boltzmann plot, of the populations. The boundary level given by eq. (4.25) is shown with the dash-dotted lines. The excited levels are divided into two groups: the levels lying lower than this boundary and the higherlying levels. From the arguments in the preceding paragraph it is obvious that the former group is in the CRC phase and the latter is in the saturation phase. See Fig. 1.10(b). When we compare Figs. 4.19 and 4.20 with the corresponding figures in the high-temperature case (Figs. 4.13 and 4.14), we recognize that the population characteristics of these two cases have almost nothing in common. Specifically, 1. In the present low-temperature case, the populations in the CRC phase are much less than their Saha-Boltzmann values, eq. (2.7a), and population inversion, i.e. larger nQ(p)/g(p) values for larger p, is established for all the excited levels. 2. In the saturation phase the populations of higher-lying levels tend to the Saha-Boltzmann values, which was also the case for the high-temperature

FIG 4.20 Population distribution of the recombining plasma component for 7~e= 1 x 10 K. The ordinate is the same as Fig. 4.19 and the abscissa is (a) the logarithm of the principal quantum number of the level, and (b) the ionization energy of the level. Part (b) is the conventional Boltzmann plot. The SahaBoltzmann population given by eq. (2.la) is shown in (a) with the solid-anddashed curve, and in (b) with the solid line. In (a), the Saha-Boltzmann distribution at p#, where/IB = 7.25 as given by eq. (4.56), continues smoothly to the line, given by eq. (4.54). The boundary pG between the CRC phase and the saturation phase, or eq. (4.25), is given with the dash-dotted lines and the boundary />B given by eq. (4.55) is shown with the dotted line. (Quoted from Fujimoto, 1980b; with permission from The Physical Society of Japan.)

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case. However, those of the several lowest-lying levels (j> < 7) never reach the Saha-Boltzmann values even in the limit of high electron density. We will examine below these features in detail. CRC phase Figure 4.21 shows a sketch of the dominant fluxes of electrons in the energy-level diagram in the low-density limit. The first point to note is the relative magnitudes of the fluxes of radiative recombination into various levels. In comparison with the high-temperature case, Fig. 4.16, the relative magnitudes are quite different: they have a much weaker dependence on the levels. This difference has already been noted at the close of Section 3.2 and in Fig. 3.9. For the recombination rate coefficient, eq. (3.19a), we adopt the approximation to the exponential integral, eq. (3.22), which is the opposite case to eq. (4.40), i.e.

This is valid for low-lying levels. Equation (3.19a) then reduces to

FIG 4.21 Sketch of the dominant fluxes of electrons in the energy-level diagram. Neutral hydrogen with T e = l x 103 K. The low-density limit, where all the levels are in the CRC phase. The recombination flux through a high-lying level, level p= 10 taken as an example, is shown with the solid arrow. (Quoted from Fujimoto, 1980b; with permission from The Physical Society of Japan.)

124 POPULATION DISTRIBUTION AND POPULATION KINETICS Therefore, /3(p) is proportional top~l and Te~°'5, and these dependences are quite different from the high-temperature case in which /3(p) otp~2'5Te~1'5. Figure 3.5(b) illustrates this point; at this low temperature for low-lying levels the p^1dependence is seen, while for levels />;$> 10, the ^-dependence is stronger. This weak /i-dependence is the reason why the higher-lying levels are more heavily populated in the low-temperature case than in the high-temperature case. Figure 4.22, corresponding to Fig. 4.15, shows the breakdown of the populating and depopulating fluxes for level 5. In the low-density limit, in comparison with the

FIG 4.22 Breakdown of the (a) populating and (b) depopulating fluxes concerning level/? = 5 into individual fluxes, corresponding to Figs. 4.19 and 4.20. The explanation is the same as for Fig. 4.15. (Quoted from Fujimoto, 1980b; with permission from The Physical Society of Japan.)

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high-temperature case, the cascade contribution is more important, about 50% of the total populating flux; this is a consequence of the heavier populations in the high-lying levels. The depopulating flux is the radiative decay. The population balance is given by eq. (4.39). It may be shown by a technique similar to the one leading to eq. (4.12) and eq. (4.42) that the ratio of the contributions from the direct radiative recombination, the first term on the r.h.s. of eq. (4.39), and that from the cascade, the second term, is about equal for any level, being independent of p. (In this derivation, however, the population distribution of higher-lying levels is assumed to be n0(q)/g(q)<xq, instead of eq. (4.51) below; this assumption is more accurate numerically as seen in Fig. 4.20(a).) Thus, for the purpose of deriving the population distribution among the levels we may neglect the cascade contribution in eq. (4.39) to reach the approximation

or

leading to the population inversion. In Fig. 4.20(a) this approximation is compared with the result of the numerical calculation. Figure 1.10(b) contains this distribution in the region of the CRC phase. With an increase in ne, very high-lying levels first enter into the saturation phase (Figs. 4.19 and 4.20). For collisional transitions concerning these levels, the energy differences of the important transitions (excitation and deexcitation to the adjacent levels; see the discussions around eqs. (4.5)-(4.11)) are much smaller than the electron temperature (multiplied by Boltzmann's constant). So, the situation is similar to the high-temperature case. Thus, the arguments in the preceding section concerning the transition of the population from the CRC phase to the saturation phase are valid; that is, Griem's boundary, eq. (4.25) or (4.29a), gives the transition, and the levels lying higher than this boundary are in the saturation phase and thus in LTE. This feature is actually seen in Fig. 4.20(a) and (b). Figure 1.10(b) includes the boundary "GRIEM" and the population kinetics in the upper part of the region of ne higher than this boundary. In the present case of low temperature, the Saha-Boltzmann populations, n(p)/g(p)nzne= Z(p)/g(p),for these levels are substantially higher than their populations in the CRC phase. As a result, with the increase in «e, the downward radiative cascading fluxes from these levels to lower-lying levels in the CRC phase increase substantially. This is the reason why, in Fig. 4.22(a), the relative contribution from the cascade increases with «e. The features described above may be understood in a different way: The lowering of Griem's boundary level with ne may be regarded as if it is the lowering of the ionization limit down to this level. The higher-lying levels are engulfed by the "continuum", and the threshold for "radiative recombination" is lowered to Griem's boundary level, resulting in an increase in the "radiative recombination

126 POPULATION DISTRIBUTION AND POPULATION KINETICS rate". Then, it is natural that, with the increase in «e, all the populations of the lowlying levels increase at almost the same degree; this is actually seen in Figs. 4.19 and 4.20 as parallel movements of the curves and the points, respectively, of the low-lying levels in the CRC phase; in Fig. 4.19 all the curves move almost in parallel for low densities, and in Fig. 4.20 the populations of low-lying levels move upward in parallel. The population inversion is conserved. It should be noted that the above argument has nothing to do with the lowering of the ionization potential, which will be discussed in Chapter 9, or with the merging of levels as discussed in Chapter 7. With a further increase in «e, the boundary between the CRC and saturation phases gradually comes down. See Fig. 1.10(b). Figure 4.19 and 4.20 show that, at « e ~ 1017-1018 m~3, level 6, and higher-lying levels have entered into the saturation phase. Figure 4.22(a) shows that, with this increase in «e, the contribution from the collisional deexcitation to level 5 begins to increase; in particular, that from the adjacent higher-lying level 6 is dominant. On the other hand, the dominant depopulating mechanism is still the radiative decay for « e < 3 x 1018 m~3. The presence of this intermediate density region is characteristic of the lowtemperature recombining plasma. This persistent cascading nature of the depopulating process is the origin of the nomenclature of the capture-radiativecascade phase. The population balance in this highest «e region of the CRC phase is given approximately by

Since the upper level (/>+!) has entered into the saturation phase, nQ(p+l) is almost proportional to nzne. Thus, nQ(p) is approximately proportional to nzn^. This explains the steep slope in Fig. 4.19 of nQ(p) in this highest-density region of the CRC phase. Saturation phase - Byron's boundary As seen in Fig. 4.22(b), with a further increase in «e, the depopulating process changes from radiative decay to collisional depopulation. This transition of the depopulating mechanism is given from eq. (4.25) and the boundary «e value is shown in Fig. 4.19. The boundary between the lower-lying levels in the CRC phase and the higher-lying levels in the saturation phase is shown in Fig. 4.20 with the dash-dotted line. We have already noted this boundary at the beginning of this section. This boundary is also shown schematically in Fig. 1.10(b). Note that the actual boundary «e is higher by about two orders of magnitude in this lowtemperature case. See eq. (4.59) later. Figure 4.19 shows that, with the above increase in «e, level 5, for example, enters into its high-density limit, or into the saturation phase at « e ~ 1019 m~3. Figure 4.23 is a sketch of the dominant fluxes of electrons in the energy-level

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FIG 4.23 Sketch similar to Fig. 4.21 except that «e = 1 x 1020m 3. The boundary pG given by eq. (4.25) is shown with the dash-dotted line and/?B given by eq. (4.55) with the dashed line. (Quoted from Fujimoto, 1980b; with permission from The Physical Society of Japan.) diagram for ne = 1 x 1020 m~3, where levels p > 4 are in the saturation phase. This figure and Fig. 4.22 for level 5 indicate that, in this phase, the dominant populating process is the collisional deexcitation from the adjacent higher-lying level 6. The dominant depopulating process is the collisional deexcitation to the adjacent lower-lying level 4. This feature still holds in the higher «e regions as seen in Fig. 4.22 and Fig. 4.24 for the high-density limit. This feature is contrasted to the high-temperature case in the preceding section, Figs. 4.15 and 4.18, where the dominant depopulating flux was excitation to the adjacent higher-lying level. This new feature is common to many lower-lying levels (Fig. 4.24). For these low-lying levels the exponential factor of the excitation rate coefficient, eq. (3.29), is no longer negligible, and eq. (4.36), F(p,p—1)> C(p,p+l), holds instead of eq. (4.9), F(p,p—T) < C(p,p+l), See Fig. 4.2. In other words, for these levels the energy separation between the adjacent levels (proportional to p~3, see eq. (1.5)) becomes quite significant for the electrons whose average energy is of the order of kTe. The population balance is therefore

Thus, the ladder-like deexcitation flow of electrons is established in the energylevel diagram (Fig. 4.24). It is noted that eq. (4.53) is nothing but eq. (4.33), and

128 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4.24 Sketch similar to Fig. 4.21 except that this is for the high-density limit. The magnitude of the effective recombination flux has been divided by nzr%, and shown with the figure in the arrow at right. (Quoted from Fujimoto, 1980b; with permission from The Physical Society of Japan.) therefore, we have a population distribution similar to eq. (4.31), the minus sixth power distribution,

for low-lying levels for which eq. (4.36) holds. Figure 4.20(a) shows that this approximation is good for high density and for low-lying levels, except for the lowest-lying levels for which some of the above approximations break down. Figure 1.10(b) depicts these features: in the density region higher than Griem's boundary and for low-lying levels, shown are eq. (4.53) for the population kinetics and eq. (4.54) for the population distribution. For higher-lying levels for which eq. (4.9) is valid instead of eq. (4.36), the situation is different. As is seen in Fig. 4.24 and expected from the discussion in the preceding sections, the dominant depopulating process from such a level p is the collisional excitation to the adjacent higher-lying level (/H-l). The balance of the population is approximately given by eq. (4.46), n0(p+l)F(p+l,p)ne= n0(p)C(p,p + l)«e, as was the case in the high-temperature case. Thus, eq. (4.47), n0(p) = Z(p)nzne, holds, and the discussion leading to this equation shows that all of the high-lying levels for which eq. (4.9) holds are in LTE. This is actually seen in Fig. 4.20 to be the case.

RECOMBINING PLASMA COMPONENT

129

For high enough density, the boundary level p between these high-lying levels and the low-lying levels is given from the boundary between eq. (4.9) and eq. (4.36):

Figure 4.2 shows C(p,p+l)/F(p,p—l) against Te for several levels. For the temperature of the present example, 103 K, this boundary lies between/? = 6 and 7 as shown in Figs. 4.20, 4.23, and 4.24. With the approximate rate coefficient, eq. (3.29) along with appropriate approximations like eq. (3.7), fp,p+\—p/4, and eq. (1.2), E(p,p+l)~2z2R/p3,together with the Taylor expansion of the

exponential factor and the approximation [(/>—1)//>]6~ 1 — (6/p), we arrive at an approximate expression for the principal quantum number of this boundary

The subscript B denotes "Byron et al.", since this expression was first derived by these authors. Byron's boundary, as expressed as the boundary temperature, is given in Fig. 4.2. It is seen that this approximation is good for large p, as is expected from the approximations. Figure 1.10(b) includes the boundary bearing the sign "BYRON", as determined by eq. (4.55) or (4.56). The Saha-Boltzmann population, eq. (4.47), for the high-lying levels as shown in Fig. 4.20(a) has negative slopes in this figure. We define for the abscissa

We differentiate the Saha-Boltzmann populations in this figure with respect to a:

We note that exp(o) is nothing but p, so that at Byron's boundary, pB, as given by eq. (4.56), the slope of the Saha-Boltzmann distribution is —6. This means that the Saha-Boltzmann populations in the higher-lying levels continue smoothly at Byron's boundary to the minus sixth power distribution for the low-lying levels, eq. (4.54). We can thus approximate the whole population distribution: for P >/"B, eq. (4.47) is valid, and for p
130 POPULATION DISTRIBUTION AND POPULATION KINETICS extent. This boundary temperature has yet another basis, which will be given in the next chapter. Griem's boundary We have defined Griem's boundary as the boundary level, or the boundary «e, at which the radiative decay and the collisional depopulation are equally probable. In deriving the numerical formula, eq. (4.29), we relied on the simple approximation for the excitation rate coefficient, eq. (4.7). When Te is low and ne is high, Griem's boundary lies below Byron's boundary; then Griem's boundary should be determined by the deexcitation rate coefficient rather than the excitation rate coefficient as assumed in deriving eq. (4.29). In this case, instead of eq. (4.7) with eq. (4.5), another formula approximates better the deexcitation rate coefficient:

This formula is deduced from a Monte Carlo calculation for classical electron orbitals. By combining this with the approximation for the radiative decay probability, eq. (4.13), we obtain where we have simplified the expression by ignoring the very weak dependence on Te. Here, we have assumed z = 1. We have restricted our discussion to neutral hydrogen because the z-scaling of eq. (4.58) may not necessarily be valid, as understood from the excitation cross-sections in Fig. 3.11. In this low-temperature case the deexcitation rate coefficient is determined largely by the excitation crosssection values near the threshold, which do not follow the z-scaling. Note the Klein-Rosseland relationship, eq. (3.27). Griem's boundary pG as given by eq. (4.59) is about a factor of 2 larger than that given by eq. (4.29a) in the «e region of practical interest. This point should be remembered when Fig. 1.10(b) is applied to low-temperature plasmas. We note here two points: First, as in the case of high temperature, rQ(p) in the high-density limit is given by In this case, however, r^p) in the high-density limit is much larger than eq. (4.35) for all the levels as seen in Table 4.1 (a) and Fig. 4.3, resulting in smaller r0(p) values than the high-temperature case. This high-density limit value of r0(p) continues to lower densities, until ne reaches Griem's boundary. See Figs. 4.19 and 1.10(b) and Table 4. l(a). Secondly, Table 4.1 (a) and Fig. 4.3 suggest that the relative magnitudes of the high-density limit values of rQ(p) and r\(p) have a relationship with/> B . From the discussions in the preceding sections, it may be concluded that for levels lying higher than Byron's boundary, rQ(p) is larger than ri(p) and can be close to 1, and

SUMMARY AND CONCLUDING REMARKS

131

for the lower-lying levels vice versa. This is actually seen to be the case from Figs. 4.3 and 4.2. Examples We now show some examples of the population distributions determined experimentally for recombining plasma. Figure 4.25(a) shows an example of the Boltzmann plots of measured populations for neutral hydrogen in an expanding plasma jet. Te and «e are estimated to be 0.15 eV (1.7 x 103 K) and 2.6 x 1019 m~3, respectively. The reader can locate this plasma in Fig. 1.2. The solid line shows the Saha-Boltzmann population distribution given by eq. (2.7a). This line has been fitted to the experimental populations of high-lying levels. We obtain from eq. (4.59) pG ~ 4.1 and from eq. (4.56) />B ~ 5.5, which are given in the figure with the dashdotted line and the dashed line, respectively. Also shown are the calculated populations by the collisional-radiative model for the recombining plasma component. This distribution illustrates well the characteristics of the low-temperature recombining plasma as discussed in the present section. For levels p>pv the population is given by the Saha-Boltzmann value, for pG
132 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4.25 (a) Boltzmann plot of the measured and calculated populations of neutral hydrogen in an expanding plasma jet. The ionization limit is 13.6 eV. (Quoted from Akatsuka and Suzuki, 1994; copyright 1994, with permission from The American Physical Society.) (b) Boltzmann plot of the measured populations of neutral helium. The ionization limit is 24.5 eV. (Quoted from Akatsuka and Suzuki, 1995, with permission from IOP Publishing.) Boundaries pG (eq. 4.59) and/?B (eq. 4.56) are given by the dash-dotted line and the dotted line, respectively.

SUMMARY AND CONCLUDING REMARKS

133

Byron's boundary, as is seen from eq. (4.56), lies far below p = 2, and does not appear in the present phase diagram for p>2. Therefore, the phase diagram of Fig. 1.10(a) usually lacks Byron's boundary and the "saturation (TE)" phase. See the discussion of the low-temperature case in Section 4.2. Now, we have reached the point where we can interpret the spectra in Fig. 1.5. These are of the resonance-series lines (transitions terminating on the ground state) of singly ionized helium, or hydrogen-like helium with z = 2 (Fig. 1.6). In other words, we try to locate these plasmas on the n&— Te plane of Fig. 1.2, and identify the intensity distributions in terms of the ionizing plasma component, Fig. 1.10(a), or the recombining plasma component, Fig. 1.10(b), or even a combination of them, Fig. 1.9. We have to note two adverse situations here: 1. The spectrometer used in this observation was not calibrated for its relative sensitivity over the wavelength range. Thus, the observed signal intensity for different lines may not directly reflect the number of photons accepted by the spectrometer. However, since the wavelength range is rather narrow, we may assume constant sensitivity over the observed lines. 2. The plasmas may violate one of our assumptions on which we have based our foregoing discussions: that is, in the present case the emission lines may suffer radiation reabsorption within the plasma. The population of the ground-state ions (virtually equal to «e in the present case) is so high that the photons emitted by excited-level ions can be absorbed by them before leaving the plasma. This effect is approximately proportional to the absorption oscillator strength. Remember that this quantity is a measure of the ability of the ions to absorb the light of this transition. See Table 3.1(b) and Fig. 3.4. Therefore, the line 1 <—2, and 1 ^3 to a certain extent, may suffer an apparent decrease in the observed intensity. This problem will be treated in detail in Chapter 8. With the above reservations, especially the second one, in mind, we try to interpret the spectra. Since, these plasmas are known to have «e about 1020 m~3, the reduced density n^/z1 is about 1018 m~3; this is the density to be referred to in interpreting these spectra in terms of the theory developed in this chapter. See Appendix 5B. Figure 1.5(a) reveals that, with the increase in the principal quantum number p of the upper level, the observed line intensity, $(p, 1) (eq. (4.1)), decreases sharply. Note the unusually thick profile of the 1 <— 2 line. This is interpreted as due to the saturation of the detector near the line center. If we correct for this saturation effect the peak would be at around the asterisk marked on this figure. If we ignore the small v dependence in eq. (4.1), the relative population n(p)/g(p) is given from $(p, 1) divided by A(p, l)g(p), which is approximately proportional to p~ , as seen from eqs. (3.1) and (3.6a) and Fig. 3.4. The resulting population distribution which decreases sharply with/? is found to be consistent with the one in Fig. 4.5 at n^/z1 = 1018 m~3. Thus we may conclude that this plasma reveals only the ionizing plasma component of the excited-level populations and that this plasma is located at around « e /z 7 — 1018 m~ 3 in Fig. 1.10(a). Obviously, the recombining plasma component in eq. (4.20a) or Fig. 1.9 is

134 POPULATION DISTRIBUTION AND POPULATION KINETICS virtually absent. Figure 1.5(b), shows that the populations n(p)/g(p) show small values for small p and a broad maximum at around p = 5-7. This distribution may be interpreted as a recombining plasma component, or this plasma is located somewhere in Fig. 1.10(b). This distribution is close to that in Fig. 4.20 with « e /z 7 = 1018 m~3. The ionizing plasma component in eq. (4.20a) or in Fig. 1.9 is absent. The important point here is that, even in the present adverse situation as noted above, we can make a definite conclusion, even if it is qualitative, about the nature of the plasma from its emission line spectrum. This is because the overall pattern of the population distribution is very much different for the ionizing plasma component and the recombining plasma component as clearly demonstrated in Fig. 1.5. In Appendix 8A, the spectra of Fig. 1.5 are interpreted further with the radiation reabsorption effect taken into account. As Fig. 8 A.I (a) shows, the spectrum of Fig. 1.5(a) leads to the population distribution of the ionizing plasma component with ne/z7 = 1018 m~ 3 or a little lower. Figure 8A.l(b) shows that the plasma of Fig. 1.5(b) is the recombining plasma component of ne/z7 = 1018 m~3. Our above speculations will be thus justified. The final identification of these plasmas on Fig. 1.2 will be postponed until Chapter 8. Figure 1.3 is also interpreted as a spectrum of a recombining plasma. The intensity distribution pattern of the series lines shows a typical characteristic of a recombining plasma. The continuation of the series lines to a continuum is also a signature of a recombining plasma. This latter problem will be treated in Chapter 6. In Fig. 1.3, the observed emission lines terminate on excited levels and the radiation reabsorption effects should be minor. However, the sensitivity problem still remains and our conclusion here has to be qualitative again. In this chapter we based our theory on the assumption that excited levels are specified by the principal quantum number as depicted by Fig. 1.6. This assumption is equivalent to assuming the statistical population distribution among the different-/ levels having a common principal quantum number, as depicted in Fig. l.ll(b). However, it is obvious that this assumption is not valid at low densities. In Appendix 4A we examine this problem. It is shown that, in the lowdensity region, lower by some two orders than Griem's boundary, our above assumption breaks down. Therefore, our conclusions deduced in this chapter about the populations in the corona phase of the ionizing plasma component and in the CRC phase of the recombining plasma component should be treated with care in applying them to real plasmas at low density. *Appendix 4A. Validity of the statistical populations among the different angular momentum states As noted in Section 3.6, heavy particle (ion) collisions can be important in inducing a transition between two levels having a small energy difference or even between nearly degenerate levels. Figure 3.23 shows for the transition 3s —> 3p, as

VALIDITY OF THE STATISTICAL POPULATIONS

135

an example, a comparison between the cross-sections by electron impact and by proton collisions. These are the results of calculations in the Born approximation. For higher energies of practical interest, both cross-sections have the energy dependence of \/E, and the magnitude of the cross-section for proton collisions is larger than that for electron impact by about three orders. The reason was given in Section 3.6. We thus have to take into account the former collision processes in considering the population distribution among the different / levels. A collisional-radiative model for neutral hydrogen has been constructed with different / levels resolved. Figure 4A. 1 shows an example of the populations of the ionizing plasma components ni(p)/g(p) forp = 4s, 4p, 4d, and 4f against a change of «e, where the proton density is assumed equal to ne and the electron and ion temperatures are 10 eV. The ground-state atom density is «(1) = 1 m~3. For low densities, the populations (per unit statistical weight) among the different / levels differ by three orders of magnitude. In this density region each of the levels is in its own corona equilibrium and their populations are proportional to ne. With an increase in «e, the population of the 4f level begins to increase with respect to the corona equilibrium value. This increase is found to be due to the substantial increase in the populating flux by cascading transition from the 5g level. With a

FIG 4A. 1 An example of the populations (per unit statistical weight) of the ionizing plasma component with the different / levels resolved for neutral hydrogen with Te= 10 eV and «(!)= 1 m~3. (Quoted from Sawada, 1994.)

136 POPULATION DISTRIBUTION AND POPULATION KINETICS further increase in «e, proton collisions tend to make all the populations n\(p)jg(p) equal; the 4f population comes close to the 4d population, and the 4s population comes close to the 4p population. The most persistent level is 4d. When the radiative decay of 4d becomes dominated by the collisional (by proton) transition 4d —> 4p, the 4d population tends to the 4p population, and all the 41 levels are equally populated, i.e. the stastical populations. For other n levels the situation is found to be similar. The radiative decay rate from the nd level is approximately given by

The rate coefficient for the nd —> np transition by proton collisions is approximated by

Remember that the cross-section was inversely proportional to energy. The critical density is thus given as

Figure 4A.2 shows this critical density with the solid line, where Te= 10 eV is assumed. This figure contains the critical density which is determined from the calculated cross-sections like those shown in Fig. 3.23. The open circles give the density at which the deviation of the populations becomes smaller than 20%. The actual energy-level structure of hydrogen-like ions follows the LS coupling scheme, and an L level (except for S levels, of course) splits into two / (= L ± 1/2) levels owing mainly to the spin-orbit interaction: i.e. fine structure. Sampson considers a similar problem to the above in this case and gives the critical density

This critical density is given in Fig. 4A.2 with the dotted line. As we have seen in Fig. 4A.1, in the lowest-density regions, each of the different / levels are in corona equilibrium. Figure 4A.2 shows with the open squares the upper limit of density below which corona equilibrium is valid for all of the / levels. *Appendix 4B. Temporal development of excited-level populations and validity condition of the quasi-steady-state approximation

In the present chapter we introduced the quasi-steady-state (QSS) approximation and examined various characteristics of excited-level populations on the basis of

QUASI-STEADY-STATE APPROXIMATION

137

FIG 4A.2 The boundary level or boundary ne above which statistical populations can be assumed. Proton density and temperature are assumed equal to the electron density and temperature (10 eV). Q: from the numerical calculation. : eq. (4A.3). : eq. (4A.4). In this figure, the boundary below which corona equilibrium is valid for the individual / levels is shown with D • (Quoted from Sawada, 1994.) this approximation. If our plasma (in the first sense) is stationary, QSS is not an approximation but is exact. If, on the other hand, the plasma, in the first sense as well as in the second sense, is transient, so that its properties change in time, QSS is an approximation and may not be justified under unfavorable conditions. In this appendix we examine the validity condition for this approximation. Ionizing plasma As an example of an ionizing plasma, we consider an ensemble of hydrogen atoms in an electron gas or a plasma with Te = 10 eV and «e = 1018 m~3. The reader may locate this plasma on Fig. 1.2. Neutral hydrogen atoms with «(!)=! m~ 3 are immersed in this plasma, and at t = 0 these atoms begin to be acted upon by this plasma, i.e. excited and ionized. If «(1) is kept constant, after the initial transient this system should reduce to the high-temperature ionizing plasma which was treated in Section 4.2. Figure 4B.1 shows the temporal development of the excitedlevel populations (per unit statistical weight), ni(p)/g(p). The horizontal dashed lines indicate the population values in the final stationary state with «(1) = 1 m~ .

138 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4B. 1 Temporal development of the excited-level populations for «(1) = 1 m 3, r e =10eV, and « e =10 18 m~ 3 . The dashed lines show the steady-state values, and the dash-dotted lines show the result of eq. (4B.3). Q: the time at which the transition of eq. (4B.4) takes place. The assumption of constant n(l) is valid until t ~ 1CT4 s, the relaxation time of the ground-state atom density. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.) Figure 4B.2 shows changes of the population distribution with time. The stationary-state values are shown with the closed circles. This distribution approximately corresponds to the one in Fig. 4.5. At early times of t< 10~9 s, all the populations increase linearly with time and »i(/>)/£(/>) is approximately proportional to/>~ 5 . For t> 10~9 s, with the course of time, level 2 reaches its final value first, level 3 second, and then level 4. At t~ 1 x 10~7 s, all the higher-lying levels p>5 reach the final values simultaneously. We define the transient time ttr(p) at which, in Fig. 4B.1, the transient population reaches 63% of its final stationarystate value. In Fig. 4B.3, ttr(p) for each level is shown with crosses. We define the response time ?res as the largest among the ttr(p)'s. We regard our system as being in the initial transient in t < ties, and at t ~ ?res the stationary state is reached. At the early times of t < 1CT9 s, all the levels are populated by the direct excitation from the ground state and depopulation is virtually absent. This is just the situation of eq. (4.3) with An(0)= -n0, or n\(i) = n0(l - e~50 t<^,\/B ~~rt, n0Bt = At; at the start of filling the bucket with water, the flow of water out from the hole at

QUASI-STEADY-STATE APPROXIMATION

139

FIG 4B.2 Population distributions for several values of t. •: the steady-state values, n: the level at which the transition of eq. (4B.4) takes place. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Soceity.) the bottom can be ignored. The population is thus simply accumulating virtually without depopulation: where n(l) = 1 m~3. The fact that nl(p)/g(p) is proportional to p~5 is understood from the p dependence of C(l, p) oc/i;/, <xp~3, eq. (4.23a), and from g(p) = 2p2, Figure 4B.3 shows the depopulation rate by electron impact and that by radiative transitions: on the r.h.s. of eq. (4.4) they are the first three terms and the last term, respectively. For lower-lying levels the latter radiative depopulation rate is dominant and for higher-lying levels the former collisional depopulation rates are dominant. Levels 4 or 5 is the boundary between these levels, i.e. Griem's boundary, pG. (In Figs. 4.4-4.8, pG for « e = 1018 m~ 3 was between p = 3 and 4. This difference is due to the slight difference in Te and slightly different crosssection data used in the numerical calculations.) For a lower-lying level than/?G we can neglect the collisional depopulation and eq. (4.2) is approximated by

140 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4B.3 A: the radiative decay rate; o: the collisional depopulation rate; and n: the total depopulation rate, or the relaxation time tri(p), eq. (4.4), which is referred to the r.h.s. ordinate. For Te = 10 eV. +: the time at which the transition of eq. (4B.4) takes place, x: the time at which the population comes close to the steady-state value. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.)

This is readily solved as

The dash-dotted curves in Fig. 4B.1 indicate eq. (4B.3). Slight differences in the final stationary state come from the neglect of the cascading contributions in eq. (4B.2). See p. 101. For these levels, ttr(p) is given by [^2q 1 x 10~9 s, populations of excited levels become substantial, and the populating process of levels lying higher than/?G changes from direct excitation, eq. (4B.1), to the excitation from the adjacent lower-lying level, which is the dominant populating flux in the final stationary state, as we have seen in Figs. 4.6, 4.8, and 4.9.

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141

In Figs. 4B.1 and 4B.2 the time of this transition or the level which is undergoing this transition is shown. At the time of this transition for level p, the relationship

holds. It may be interesting to note that this equation is exactly the same as eq. (4.27). At this time level (/>—!) is still in the initial phase, eq. (4B.1), so that we have

This is readily solved for t:

where we have used the relation or

for large p. With the aid of eq. may be approximated to

By combining eqs. (4B.3) and (4B.7) we have thus reached a conclusion: tri(p) defined by eq. (4.14) gives ttr(p) for levels p <po, while for levels p >po it gives a transition from the initial phase, eq. (4B.1), to the ladder-like excitation, not ttT(p). Figure 4B.3 indicates that the latter statement is correct within a factor of 2. As we have seen above, all the levels lying higher than the level that is undergoing the transition, eq. (4B.4), are in the ladder-like excitation mechanism and their populations are determined by the population of the lower end of the excitation ladder. This is because the relaxation time tri(p) foip >PG is shorter for larger p as shown in Fig. 4B.3. Therefore these populations appear in Fig. 4B.1 as strictly parallel and in Fig. 4B.2 proportional to p~6. With a further increase in time, the level undergoing this transition comes down. From our discussions in the text, we recognize that the lowest level that undergoes this transition is approximately equal to Griem's boundary level, pG. It is noted that this level has the largest tr\(p). This is the reason why tTi(pG) gives the common ttr(p) for the levels P>Po and why these levels enter into the final stationary state at this time (see Fig. 4B.3). We thus conclude that ties is given by tTi(pG). Until now we have assumed that, in the time scale of tr\(p), the ground-state population «(1) is constant. In Appendix 5B we discuss the temporal development of n(l) and nz. Equation (3.35a) indicates that the ionization rate coefficient in the present example is of the order of 10~14 m 3 s^ 1 . The effective depletion rate coefficient of the ground state atoms is of similar magnitude. The ground-state

142 POPULATION DISTRIBUTION AND POPULATION KINETICS population is thus depleted with a time constant of 1CT4 s. Thus, our assumption of constant n(l) in the time scale of Figs. 4B.1 and 4B.2 is justified. Recombining plasma Protons ofnz=l m~ 3 are immersed in an electron gas or a plasma with Te = 0.1 eV and « e = 1018 m~3. The reader may locate this plasma on Fig. 1.2. At t = 0 they begin to be acted upon by the plasma and to recombine. Figures 4B.4 and 4B.5 show transient characteristics of the populations. The dashed lines in Fig. 4B.4 and the closed circles in Fig. 4B.5 show the final stationary-state values, which approximately correspond to the distribution in Fig. 4.20(a). In Fig. 4B.5, the Saha-Boltzmann, or LTE, populations, eq. (2.7a), are shown with the squares. Figure 4B.6 shows ttr(p) (crosses) determined from Fig. 4B.4 in a similar manner to the ionizing plasma case. This figure also includes the collisional and radiative depopulation rates and tr\(p). We now examine the transient populations. Figure 4B.7 shows the rate coefficients for radiative recombination and three-body recombination. It is noted that

FIG 4B.4 Temporal development of the excited-level populations for nz= 1 m 3, Te = 0.1 eV, and «e = 1018 m~3. The dashed lines show the steady-state values, and the dash-dotted lines show the result of eq. (4B.8). The assumption of constant nz is valid until ?~ 1CT2 s, the relaxation time of the ion density. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.)

QUASI-STEADY-STATE

APPROXIMATION

143

FIG 4B.5 The population distribution for several values of t. •: the steady-state values, n: the LTE populations given by eq. (2.1 a). (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.) the former rate coefficients are almost the same as those in Fig. 3.5(b). For levels p>4 the latter dominates over the former, which in turn is dominant f o r p < 4 . In the early times of t< 1CP10 s the population distribution in Fig. 4B.5 directly reflects Fig. 4B.7, suggesting that the dominant populating process is the direct recombination, radiative, or three-body process. Remember that in Fig. 4B.5 the population has been divided by the statistical weight. The linear increase in the populations with time is seen in Fig. 4B.4. In this figure the dash-dotted lines show

It is interesting to note that population inversion is established for/? > 4 because of the dependence of a(p) <xp6. It is seen in Fig. 4B.4 that, with an increase in t, the population of levels/? > 4 deviates upward from eq. (4B.8), and they reach the final QSS values starting from very high-lying levels. The last point may be regarded as suggesting that ttr(p) is given by tri(p). However, ttr(p) is found to be appreciably longer than tri(p) as seen in Fig. 4B.6. The above two behaviors are explained from the fact that these high-lying levels, when their populations come close to the stationary-state or LTE values, are strongly coupled to each other by collisional transitions. See Figs. 4.22-4.24. As a result, a level, when its population is lower

144 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4B.6 A: the radiative decay rate; O: the collisional depopulation rate; and D: the total depopulation rate; or the relaxation time tri(p), eq. (4.4), which is referred to the r.h.s. ordinate. For Te = Q.l eV. x : the time at which the population comes close to the steady-state value. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.) than the LTE value, receives additional populating flux from the neighboring higher-lying levels, or it is "pulled" upward toward its LTE value. This level, p, reaches its final value only when deexcitation flux from this level to the adjacent lower-lying level (p — 1) is balanced by the excitation flux from (p — 1) to p. In other words, when the population of a level is close to the LTE value, it is "pulled" downward by the lower-lying levels which are still far from LTE. These levels cannot be considered as independent levels. This is the reason of the appreciable deviation of ttr(p) from tr\(p). Among the lower-lying levels, but still higher than pG, there is Byron's boundary level p#. In the present example, />B lies between p = 6 and 7. See also Figs. 4.20-4.24. Between the levels lower than/> B (and still higher than/>0), collisional coupling becomes only downward, eqs. (4.36) and (4.53), and the population of a level is controlled by that of the adjacent higher-lying level. It might thus be assumed that ttT(p) for these levels, as in the ionizing plasma case, is given by ttT(pv). This turns out not to be the case. This is because lower levels have longer tri(p)'s and their populations lag behind n0(pB), In Fig. 4B.4 the slope of the population of level p = 2 first deviates downward from the linear relationship, eq. (4B.8) with a(p)ne replaced by (3(p), at

QUASI-STEADY-STATE

APPROXIMATION

145

FIG 4B.7 Recombination rate coefficients. O: radiative recombination, A: threebody recombination multiplied by « e =10 18 m~3. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.) t~2x 10~9 s, which corresponds to tr\(2); see Fig. 4B.6. If the cascading contributions from levels/? > 2 were absent, the population would have reached a final value at this time. In fact, the cascading contributions are substantial and even dominant in later times. See Fig. 4.23. Depending on the time dependence of populations of the levels that contribute to the cascading population of level 2, its population increases with time. A similar, but less prominent, feature is seen with level 3. This is the reason why these lower-lying levels cannot reach their stationary-state values until Griem's boundary level po = 6, which has the largest tri(p), reaches its stationary-state value; see Fig. 4B.6. We thus again conclude that ?res is given by tA(pG). In Chapter 5 and Appendix 5A it turns out that the effective recombination rate coefficient for the present example is of the order of 10^16m3s^1, so that nz changes appreciably only in 10~ s. Thus, during the transient of the excited-level populations discussed above, nz is virtually constant. Validity condition of QSS We have seen that both for the ionizing plasma and the recombining plasma the overall response time of the excited-level populations, tres, is given by tr\(pG), the

146 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4B.8 The relaxation time of Griem's boundary level pG as calculated numerically are given by the lines. The response time tTes as determined from the temporal development of the populations, O: ionizing plasma of Te = 10 eV and x : recombining plasma of Te = 0.1 eV. +: recombining plasma of T e = 10 eV. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.) longest relaxation time among the excited levels. Figure 4B.8 compares tTes, as determined from Figs. 4B.1 and 4B.4 and from similar calculations in the broad range of «e, and tr\(pG), as determined numerically. First we assume «e and Te are fixed, so that tTes is constant. We have shown that, under the condition of constant n(l) or nz, if the transient population of level pG, which determines the response time tres, reaches its stationary-state value, all the excited levels enter into the final stationary state. We now examine the case of ionizing plasma in which «(1) changes with time. This change may be due to its depletion by ionization or for some other reasons like spatial transport. The temporal development of the population of level pG is approximately expressed by

QUASI-STEADY-STATE APPROXIMATION

147

where n(i) and N(t) stand for n(pG) and n(l), respectively, at t. In the QSS approximation, the time derivative is set equal to zero, eq. (4.18), and the population is given as

We assume that the temporal change of N(f) is expressed in terms of a time constant T as

where NO is the initial value and T is positive for decreasing N(f) and negative for increasing N(f). Under the condition of t, \T\ ;$> tres, eqs. (4B.9)-(4B.11) lead to

This relation indicates that the parameter (ties/T) gives a measure of deviation of the CR population from its actual value. Thus the validity condition of QSS is that

In our example of Figs. 4B.1-4B.3, the change in «(1) is assumed to be only due to ionization. Even if we include the depletion of «(1) at a later time, the requirement (4B.13) is well satisfied, and QSS is valid. Next we examine the temporal change in nz for a recombining plasma. The dominant populating flux of level pG is deexcitation from the adjacent higher-lying level. We now take into account "the time lag" of this population. Then the population of pG is approximately given as

where N(f) is the ion density, in this case, at t, and Z(p) is the Saha-Boltzmann coefficient, eq. (2.7). In the QSS approximation, the population of po is expressed within this approximation as

If N(f) is again expressed by eq. (4B.11), then eqs. (4B.14) AND and (4B.15 lead to

for t, | T\ > tres. The validity condition of QSS is again given by eq. (4B.13).

148 POPULATION DISTRIBUTION AND POPULATION KINETICS When «e changes with time, we may start with eq. (4B.9) or eq. (4B.14). Instead of N(f) in eq. (4B.9), or N(T- tres) in eq. (4B.14) the factor ne now changes. We may proceed with our discussion in a similar way to the above cases. We then reach the same conclusion as for the previous cases. The change in pG or tres with the change in ne, which we have neglected so far, has little effect in the above reasoning. This is because n(pG) is almost linearly dependent on «e. The case of a temporal change in Te is not straightforward because the collision rates have nonlinear dependence on Te. First we examine an ionizing plasma and start with eq. (4B.9). Instead of the change in N(f), C(l,pG) changes this time. When we note that tres is very weakly dependent on Te (see Fig. 4B.8), we can proceed with our discussion in much the same way as for the previous cases. The validity of QSS would then be \ties[dC(l,pG)/dt]/C(l,pG)\ < 1. The excitation rate coefficient is approximately given by eq. (3.29). Except for very high temperature, the temperature dependence is mainly determined by the exponential factor, and the T,T1/2 factor may be neglected. Then the validity condition is written as

For a recombining plasma, except for the case of pG
This equation suggests that the validity condition for QSS would be given by

References The discussions of this chapter are based on: Fujimoto, T. 1979a /. Phy. Soc. Japan 47, 265. Fujimoto, T. 1979b /. Phy. Soc. Japan 47, 273. Fujimoto, T. 1980a /. Phy. Soc. Japan 49, 1561. Fujimoto, T. 1980b /. Phy. Soc. Japan 49, 1569. Fujimoto, T. and McWhirter R.W.P. 1990 Phys. Rev. A 42, 6588. Kawachi, T. and Fujimoto, T. 1995 Phys. Rev. E 51, 1440. Appendix 4A is based on: Sawada, K. 1994 Ph.D. thesis (Kyoto University). Appendix 4B is based on: Sawada, K. and Fujimoto, T. 1994 Phys. Rev. E 49, 5565.

REFERENCES Figure 4.12 is taken from: Tachibana, K. and Fukuda, K. 1973 Japan. J. Appl. Phys. 12, 895. Figure 4.15 is taken from: Akatsuka, H. and Suzuki, M. 1994 Phys. Rev. E 49, 1534. Akatsuka, H. and Suzuki, M. 1995 Plasma Sources Sci. Tech. 4, 125.

149

5

IONIZATION AND RECOMBINATION OF PLASMA

In the preceding chapter, we developed our theory of the excited level populations on the basis of the quasi-steady-state approximation to the solutions of the rate equations, eq. (4.2). The rate equation for the ground-state atom density, «(1), and that for the ion density, nz, have been left as they were. Since we have obtained the solutions for the excited level populations we substitute them into these rate equations. Equation (4.2) for p = 1 reduces to

After rearrangments of the terms this equation is rewritten as

with

These quantities express the effective rate of ionization and recombination of the plasma (in the second sense) which is immersed in a plasma (in the first sense). They include the ionization or recombination processes via excited levels. The quantity SCR is called the collisional-radiative (CR) ionization rate coefficient, and O?CR the CR recombination rate coefficient. Appendix 5A shows how SCR and «CR are established in the temporal development of the plasma. Like the population coefficients rQ(p) and ri(p), these coefficients are functions of «e and Te. It is to be noted that, in the expression of eq. (5.3) for SCR, only ri(p) appears and rQ(p) is absent, and vice versa for aCR of eq. (5.4). This is an important feature of our formulation as schematically depicted in Fig. 1.9: the ionizing plasma component

COLLISIONAL-RADIATIVEIONIZATION

151

of the populations gives the effective ionization flux of the plasma SCRn(l)ne (see Figs. 4.7-4.9; the upward arrow on the right-hand side indicates the magnitude of SCR) and the recombining plasma component gives the effective recombination flux aCR«z«e (Figs. 4.16-4.18 and Figs. 4.21, 4.23, and 4.24; the downward arrow on the right is «CR except for Figs. 4.18 and 4.24, in which the rate coefficient divided by ne is given). These features suggest that, from a measurement of n\(p), we may determine ScRn(l)ne, and from nQ(p) we determine aCRnzne. This point will be discussed later on in this chapter. In many practical situations we encounter the cases in which the overall populations summed over all the excited levels are very small in comparison with [«(!) + «J. We saw examples of this situation in the preceding chapter. See Figs. 4.5, 4.14 and 4.20.* The apparent difficulty of divergence of populations of an infinite number of Rydberg levels for a recombining plasma will be resolved in Chapter 9. Thus, we may assume

In the following discussions we are mainly concerned with the high-temperature case, in which Byron's boundary does not play any role. The reasons are given below. The low-temperature case will be treated in Section 5.3 only with regard to the CR recombination rate coefficient; this case is very important in treating actual plasmas. Since the problems of ionization-recombination and ionization balance of a plasma are particularly important in considering ions in various ionization stages rather than neutral atoms, we put some emphasis on hydrogen-like ions with nuclear charge z rather than neutral hydrogen. For ions with a large z, the temperature range of interest tends to be higher. The reader will understand this point later in this chapter. So, we present approximate formulas which fit better the result of numerical calculations for ions of high-temperature plasmas.

5.1 Collisional-radiative ionization Figure 5.1 shows an example of SCR for neutral hydrogen for Te= 1.28 x 105 K, corresponding to the case which we treated in Section 4.2 in the discussion of the ionizing plasma component of the population. The reader may remember the horizontal line drawn in the ne-Te plane of Fig. 1.2 when he or she studied * In Fig. 4.20 in the high-density regions of «e > 1022 m~3, a large population of «(2) > nz could occur, thus violating this statement, so long as we assume the recombining plasma component. However, in many actual plasmas, «(1) can be larger than n2 by many orders of magnitude, and our assumption becomes valid. Rather, if the excited-level population becomes larger than [«(!) +nj, this plasma would constitute an interesting new field to be investigated, which is outside of the scope of this book.

152

IONIZATION AND RECOMBINATION OF PLASMA

Section 4.2. Figure 5.1 shows the breakdown of the net ionization rate coefficient SCR into components: the magnitude of ionization fluxes from various levels is shown for each level from which this final step of ionization takes place. See also Figs. 4.7-4.9. Figure 5.2 shows an example of SCR for a hydrogen-like ion with Te/z2 = 6 x 104 K. In both the cases, starting from the low-density limit, SCR increases with ne, and it finally reaches a constant high-density-limit value. We will investigate why SCR changes this way by examining the details of the ionization process. Region I (ne < nf) We begin our discussion with the low-density region. The upper bound of this region nf will be given later. As we have seen in Section 4.2, in the limit of low density, all the excited levels are in the corona phase as shown in Fig. 1.10(a), and all the excitation fluxes originating from the ground state p=\ eventually return to it through the chain of the radiative cascade decays. See Fig. 4.7. Thus, the flux of electrons into the continuum states comes only from the ground state, i.e. direct ionization. This is also seen in Fig. 5.1.

* In this footnote, we justify Fig. 5.1 as an expression of SCR which was defined by eq. (5.3).Instead of eq. (5.1), we start with

We substitute our solution of the excited-level populations, eq. (4.20) or (4.20a), into eq. (1), and rearrange this equation into two terms, each proportional to «(1) and nz, respectively. Then, we have

where we have adopted the convention that «[(!) = n(l), and r0(l) = 0. The first term on the r.h.s. divided by n(l)ne is nothing but SCR shown in Fig. 5.1. We now show that the second term agrees with our definition of aCR by eq. (5.4). We sum eq. (4.2) over all the excited levels with/; > 2. The excitation and deexcitation fluxes among the excited levels cancel each other and the remaining quantities are

Again, we express «(/>) on the r.h.s. in terms of n(l) and nz. Owing to our assumption of QSS or eq. (4.18), the above eq. (3) vanishes, so that each of the coefficients of n(l) and nz should vanish. For the coefficient of nz we have

It is straightforward to see that the second term on the r.h.s. of eq. (2) agrees with the r.h.s. of eq. (5.4), of course, with the sign reversed. It is also interesting to see the coefficient of n(l) in eq. (3) also leads to the expression of eq. (5.3). The above reasoning is due to Professor M. Otsuka.

COLLISIONAL-RADIATIVE IONIZATION

153

FIG 5.1 Breakdown of the collisional-radiative ionization rate coefficient SCR into the individual ionization processes. The numerals indicate the principal quantum number of the level from which the final step of ionization originates. For neutral hydrogen with Te= 1.28 x 105 K. See also Figs. 4.7-4.9. (Quoted from Fujimoto, 1979b, with permission from The Physical Society of Japan.) We adopt the approximattion

The superscript 0 means the low-density limit. This expression is of a similar functional shape to eq. (3.35a), but fits better to the actual rate coefficient in the temperature range considered here, as seen in Figs. 5.2 and 5.3. Note that, in eq. (3.35a), the factor (z2 R/x(p))7^4 is simply 1 for the ground state. With an increase in «e, Griem's boundary pG comes down (see Fig. 1.10(a)), and the levels lying higher than this boundary enter into the saturation phase. For these levels the multistep ladder-like excitation-ionization mechanism is established, and, roughly speaking, the excitation flux from the ground state into these levels eventually results in ionization. See Fig. 4.8: this figure is for « e = 1018 m~ 3 and 3 <po < 4. As Fig. 5.1 shows, with the increase in «e, the contribution to SCR from ionization from very high-lying levels increases. Both figures show that the contributions from levelsp > 10 are substantial at ne= 1018 m~3. We may express the increasing ionization rate coefficient as

154

IONIZATION AND RECOMBINATION OF PLASMA

FIG 5.2 Collisional-radiative ionization and recombination rate coefficients versus the reduced electron density, «e/z7, for hydrogen-like ions. The reduced electron temperature is Te/z2 = 6 x 104 K. All the quantities are scaled against the nuclear charge z. Approximations according to the formulas in the text, eqs. (5.10), (5.13), (5.18), and (5.24), are compared with the numerical calculation. The scaled ionization ratio [z4«z/n(l)j is also shown. The range of «e/z7 is divided into three regions: I, II, and III. (Quoted from Fujimoto, 1985, with permission from The Physical Society of Japan^) The discussions in the preceding chapter, especially that in p. 125-126, suggest that the lowering of Griem's boundary may be regarded, in effect, as lowering of the ionization limit. Those discussions were concerning a recombining plasma. Here, however, for the ionizing plasma, similar arguments can be made: i.e. the ionization limit comes down to Griem's boundary level. The ionization crosssection, eq. (3.35), may be modified, and if we return to the approximation of eqs. (3.29) and (3.35a), we have the approximation

COLLISIONAL-RADIATIVE IONIZATION

155

FIG 5.3 Comparison of approximate formulas and numerical calculation for the collisional-radiative ionization and recombination rate coefficients in the lowdensity limit and in the high-density limit. Sg^ and a~R are related by eq. (5.23). (Quoted from Fujimoto, 1985, with permission from The Physical Society of Japan.) with

Under the condition ofpo^> 1, the oscillator strength, eq. (3.6a), is approximated to/i^ ~ l.6p~3 (Table 3.1(b)) and/o is given in this approximation as/o — 0.8/>Q2. Since the excitation rate coefficients have a slightly different dependence on Te from that of S(l), we adopt an approximate expression starting from eq.

with pG given by eq. (4.25) or eq. (4.29b). In Fig. 5.2 we compare this approximation with the numerical calculation for Te/z2 = 6 x 104 K, where we have adopted eq. (4.29b) for pG_

156

IONIZATION AND RECOMBINATION OF PLASMA

With a further increase in ne, the boundary pG further comes down and reaches p = 2 at n£°, which is the upper boundary of region I. See Fig. 1.10(a). Note here that, since we assume high temperature, Byron's boundary does not appear in this discussion. This density is approximately given from eq. (4.29a) for pG = 2 as

This boundary is shown in Fig. 5.2. Regions II and III (ne > nf) At about this boundary nf, SCR approaches the high-density-limit value as seen in Figs. 5.1 and 5.2. (See also Fig. 4.4.) As we have seen in Section 4.2, in the highdensity regions of present interest, the multistep excitation-ionization ladder is established among all the excited levels starting from the ground state, as seen in Fig. 4.9. Virtually all the excitation fluxes leaving the ground state contribute to the net ionization flux. In these high-density regions, all the excited levels are in the saturation phase (Fig. 1.10(a)), and the interrelationship between the population coefficients r o(p) + ri(p)= 1> ecl- (4-21), holds. In eq. (5.3), if we neglect the radiative decay terms,* which is justified in the highest-density region as seen in the next section, we have

where we have used eq. (3.31a). The superscript oo means the high-density limit. When we remember that, at high temperature, r0(q) is approximated to (1 —p~6) (eq. (4.48)), or very close to 1, we have

This situation is actually seen in Fig. 4.9. Among the fluxes originating from the ground state the dominant flux is the excitation to the first excited level p = 2, and the following approximation is found to reproduce well the numerical calculation.

* In region II, A(q, 1) in eqs. (5.3) and (5.4) is neglected in comparison with, say, C(q, q + l)«e, that is, the radiative decay is neglected in the depopulating processes. However, A(q, 1) is still larger than the corresponding competing collisional rate F(q, l)«e. See Section 5.2 later.

CR RECOMBINATION - HIGH-TEMPERATURE

157

FIG 5.4 Collisional-radiative recombination rate coefficient aCR for neutral hydrogen. The boundary density «^°+ given by eq. (5.20) with p = 2 is shown with open circles, and that by eq. (5.25) with closed circles. (Quoted from Fujimoto, 1980a, with permission from The Physical Society of Japan.) This approximation is shown in Fig. 5.2 with the horizontal line and also in Fig. 5.3 as converted to a£?R (see later). This simple expression is good to within a factor of 2 in the temperature range of Fig. 5.3. In Fig. 5.2, in the density range beyond n£° the approximation, eq. (5.10), is extrapolated to connect to the highdensity limit value, eq. (5.13). 5.2 Collisional-radiative recombination - high-temperature case Figure 5.4 shows «CR, the collisional-radiative recombination rate coefficient for protons and electrons to form neutral hydrogen atoms in the ground state against «e for several temperatures. The reader may identify the parameter range of this figure in the ne-re plane of Fig. 1.2. Note that we are now treating the case of z=l. Figure 5.2 shows an example of «CR for fully stripped ions to form hydrogen-like ions. Figure 5.5 shows, for the high-temperature case of Te= 1.28 x 105 K treated in Section 4.3, the breakdown of the fluxes of the final steps of recombination reaching the ground state, or each term of eq. (5.4), normalized by the total recombination flux. The hatched areas show the contributions from the radiative transitions, i.e. direct radiative recombination, f3(l)nz «e (denoted by j3), and the spontaneous transitions from excited levels (the upper level is indicated by the numeral).

158

IONIZATION AND RECOMBINATION OF PLASMA

FIG 5.5 Breakdown of the collisional-radiative recombination flux into the individual fluxes of the processes reaching the ground state, normalized by the total collisional-radiative recombination flux. See eq. (5.4). The hatched areas show the radiative transitions and the blank areas the collisional transitions. On the right-end ordinate, the corresponding breakdown of the collisional-radiative ionization rate into the individual rates starting from the ground state (see eq. (5.3)) is shown for the high-density limit. Note that this breakdown is different from that shown in Fig. 5.1. For the relationship, see the footnote in p. 152. See also Fig. 4.9. Neutral hydrogen with Te = 1.28 x 105 K. (Quoted from Fujimoto, 1980a, with permission from The Physical Society of Japan.)

Region I (ne < n£°)

Figures 5.2-5.4 give the low-density limit «CR, which is common to hydrogen-like ions and neutral hydrogen. In this limit (Fig. 4.16) the effective recombination rate coefficient is given as

In the present high-temperature case, /3(p) is approximately proportional to p 2'5 for/? > 2 (Section 3.2; see Fig. 3.9 and also eq. (4.41)), and the summation of f3(p) over these excited levels gives a contribution comparable with (3(1) as actually seen in Fig. 5.5. Note that the cascading contribution as a whole is equal to S(/,>2)/3(/>). The argument of the exponential integral, eq. (3.19a), of /3 (1) in the temperature range of Figs. 5.2 and 5.3 is of the order of 1, and the Te dependence of /3(1) is neither T~1-5 (for extremely high temperature, eq. (3.21) or eq. (4.41)) nor T~°-5 (for low temperature, eq. (3.22) or eq. (4.50)) in these intermediate cases.

CR RECOMBINATION - HIGH-TEMPERATURE

159

See Fig. 3.9. We find that, in the range of Te shown in Fig. 5.3, the following approximation is rather good This approximation is shown in Figs. 5.2 and 5.3. Figure 5.5 shows that, with an increase in ne, the relative contribution from the cascade increases slightly. This is due to the slight increase in the "populations", [Ko(/0/g(p) n z n e], of the excited levels that have entered into the saturation (LTE) phase (Figs. 4.13, 4.14 and 1.10(b)), resulting in a slight increase in aCR in Fig. 5.4. This increase, however, is barely noticeable. This increase is more pronounced for lower temperatures in Fig. 5.4 and also in Fig. 5.2. On the assumption that the populations of the levels in the CRC phase are very low in comparison with the Saha-Boltzmann value in the saturation (LTE) phase (which is incorrect for the case of Te= 1.28 x 105 K and higher but is correct for lower temperatures) the effective rate coefficient may be expressed as

The idea of this approximation is that the levels lying higher than Griem's boundary are strongly collisionally coupled, directly or indirectly, with the continuum, and these levels are the origin of the cascading transitions to the lower-lying levels. This picture may be interpreted in another way: on p. 125-126 we regarded the lowering of Griem's boundary level as an effective lowering of the ionization limit. In the present context, the threshold energy of the radiative recombination is lowered to the energy of Griem's boundary, and the transitions nQ(p)A(p, q) for q<Pa

l, the direct recombination terms Y>pf3(p) for p>pG^>\ are extremely small as compared with, say, (3(1). From eq. (5.17), we have It is found that eq. (5.18) is a good approximation in the temperature range considered here rather than the low-temperature case.* An example for comparison is seen in Fig. 5.2. * At extremely high temperature, the populations in the CRC phase are even higher than the LTE values (see Fig. 5.10 later), and with an increase in «e they decrease to their LTE values. As a result, OCR decreases slightly. In such a case eq. (5.18) does not apply, of course.

160

IONIZATION AND RECOMBINATION OF PLASMA

With a further increase in ne, Griem's boundary comes down to reach the first excited level p = 2 at n£° as given by eq. (5.11). For higher densities all the excited levels have the LTE populations or their high-density-limit values. Region II (nf
An interesting feature is that these equalities, which define the upper boundary Koo+ Qf region ii, appear to hold at almost the same ne, whether it is for recombination, eq. (5.19), or it is for deexcitation, eq. (5.20), and for the latter, almost being independent of p. We now examine this point below. The transition probability A(p, 1) is given by eq. (3.1) with eq. (3.6a), and the deexcitation rate coefficient F(p, 1) is given by eq. (3.31) with the approximation (3.29). We assume level p to be high, so that the energy difference E(l,p) is approximated to z R. Then, eq. (5.20) gives for the boundary ne

where G is given by eq. (3.30). This boundary ne is independent of p because both the radiative transition probability and the collisional rate coefficient have the oscillator strength fif in common. For the temperature of Fig. 5.5, eq. (5.21) gives «e/z7 ~ 8 x 1022 m~3'. In eq. (5.19) we adopt for (3(1) the approximation eq. (3.19a), where the Gaunt factor gbf has been assumed to be 1. For the three-body recombination, the rate coefficient is related to the ionization rate coefficient by eq. (3.40), and the latter is given by eq. (3.35a). Then, eq. (5.19) reduces to

CR RECOMBINATION - HIGH-TEMPERATURE

161

We remember that the oscillator strength/1>c is 0.435 (Table 3.1(b)) and that, for the temperature range of Figs. 5.3 and 5.5 the exponential integral multiplied by the exponential factor is of the order of 1. Then eq. (5.22) gives ne which is very close to that given by eq. (5.21). In the present example of Fig. 5.5, the difference is 16%. This is the reason why the transitions from the radiative processes to the corresponding collisional processes, eqs. (5.19) and (5.20), take place at nearly the same ne for all the final steps of recombination. The above arguments suggest that in Fig. 5.5 the relative magnitude of the radiative flux at lower density should be equal to the corresponding collisional flux at high density. This is only approximately true; e.g. for (3(1) and a(l) the difference is a factor of 3. This inconsistency is due to our crude approximations adopted above. Region II is limited by the boundary «^°+ given by eq. (5.19) or eq. (5.20). We adopt eq. (5.21) for the definition of nf+:

In Fig. 5.4 the boundary nf+ given by eq. (5.20) for p = 2 is shown by the open circles. In Fig. 5.2 the approximation of the constant «CR continues up to this nf+. Note here the difference between nf and nf+. The former boundary comes from the comparison between the radiative decay A(2,1) with the collisional depopulation rate J^ C(2,p)ne at high temperature. We saw that this ne gives complete saturation of the ionizing plasma component of the populations. See Figs. 4.4 and 1.10(a). The latter boundary comes from A(2,1) = F(2, l)«e. Figure 4.2 shows that, for high temperature, C(2,3)/F(2,1)~ 100. This factor gives the difference between n£° and K^°+. This figure suggests that the situation is very different for low temperatures, which turns out to be true as we will see in the next section. It may be worth noting an interesting coincidence: if we put pa = 1 in eq. (4.29a), this equation gives ne which is very close to the present nf+. On the basis of this coincidence, we may extrapolate Griem's boundary in Fig. 1.10(b) to p = 1. Then, pG = 2 gives nf and po=l gives nf+. It is noted that nf and nf+ are almost independent of Te for high temperature. Region III (ne > nf+) In this highest-density region all the terms of radiative transitions in the rate equation, eq. (4.2), or in eq. (5.4), are small in comparison with the terms of the corresponding competing collisional transitions, and they can be entirely neglected. As we have seen in Section 4.1, the interrelationship between the population coefficients r0(/>) + r1(/>) = 1, eq. (4.21), is valid. It is straightforward to see that eqs. (5.3) and (5.4) lead to another interrelationship:

162

IONIZATION AND RECOMBINATION OF PLASMA

Then we obtain from eq. (5.13)

This is the approximation which has been compared in Fig. 5.3. Figure 5.2 also shows the comparison of this approximation with the numerical calculation. Figure 5.6 shows (a~R/»e) calculated for neutral hydrogen. Equation (5.24) is also plotted on this figure. The small discrepancies may be attributed partly to the difference of the scaled excitation cross-sections near the threshold energies for ions which are considered here, and those for neutral atoms as seen in Fig. 3.11. Figure 5.5 shows the high-density limit of the breakdown of the final recombination fluxes. On the right-end ordinate, for the ionizing plasma, the breakdown of the initial steps, excitation and ionization, is shown. See eq. (5.3). This breakdown is different from that in Fig. 5.1. It is seen that the principle of detailed

FIG 5.6 The high-density-limit values of the collisional-radiative recombination rate coefficient divided by «e for neutral hydrogen. The approximation (5.24) is plotted for the high-temperature case of Te Si 1.5 x 104 K. For low-temperature the approximation eq. (5.27) is given. (Quoted from Fujimoto, 1980a, with permission from The Physical Society of Japan.)

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163

balance is actually realized between the individual elementary processes in ionization and in recombination. 5.3 Collisional-radiative recombination - low-temperature case

In this section we examine the recombination process in the low-temperature case (T e /z 2 <1.5xl0 4 K). Figure 5.7 is for neutral hydrogen with Te= 1 x 103 K, which corresponds to Fig. 5.5 for the high-temperature case, and shows the breakdown of the final steps of recombination into the contributions from the individual transitions to the ground state. The reader may remember the horizontal line drawn in Fig. 1.2 when he or she studied Section 4.4. In spite of several substantial differences from Fig. 5.5, with an increase in «e, the transition from radiative processes to collisional processes at about nf+ ~ 1023 m~ 3 appears similar. This boundary, eq. (5.20), for p = 2 is shown on the curves in Fig. 5.4 with the open circles. As noted above, the weak dependence of this boundary on temperature is the result of the small temperature dependence of F(2,1) in eq. (5.20) as seen from eq. (3.31) with eq. (3.29). We notice from Fig. 5.4, however, that, for the present low temperatures, «CR has already reached its high-density-limit value, which is proportional to ne, at much lower densities. This is in contrast to the high-temperature case and may appear puzzling. This point will be considered later. In the limit of low density, all the excited levels are in the CRC phase, and the total recombination rate coefficient is given by eq. (5.14). In this case, as has been

FIG 5.7 Similar to Fig. 5.5 except that this is for Te = 1 x 103 K, low temperature. (Quoted from Fujimoto, 1980b, with permission from The Physical Society of Japan.)

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noted with regard to Fig. 4.21, the relative importance of recombination into excited levels as compared with that into the ground state is much higher in comparison with the high-temperature case, Fig. 4.16. See also j3(p) in Fig. 3.5(b). This feature is reflected in Fig. 5.7 as the small contribution (about 25%) from the direct radiative recombination (denoted with j3) as compared with Fig. 5.5, where it was about a half the total recombination flux. At low temperature, the downward flux of electrons by radiative transitions in the energy level diagram through a very high-lying level is important: Figure 4.21 shows that the radiative cascading flux through level p =10, taken as an example of high-lying levels, is about 25% of the total recombination flux. This means that about half of the total recombination flux is into levels 2

4 plus the direct recombination, up to « e ~10 18 m~3. Remember that, in this region, nQ(p) for levels of p<po has almost common dependences on ne as seen in Figs. 4.19 and 4.20. In the present case, since the temperature is low, the relative increase in aCR is large as compared with the hightemperature case. See eq. (5.17). Figure 5.4 shows that, with the above increase in «e, aCR enters into its highdensity limit at around « e ~ 1018 m~ 3 for T e = 103 K; this density is much lower than nf+ as shown with the open circle. This is obviously a puzzle as mentioned at the beginning of this section: Radiative processes are still dominant in Fig. 5.7 and they depend strongly on «e, and yet aCR does not depend on «e. In order to resolve this puzzle we first look at the sketch of the dominant fluxes of electrons in the energy-level diagram in the high-density limit, Fig. 4.24. Levels above Byron's boundary, p>l, are in LTE, and they are so strongly coupled with each other and, therefore, with the continuum states that the downward fluxes into these levels are almost balanced by the inverse upward fluxes from these levels (Fig. 4.24 and also Figs. 4.15,4.17, and 4.18). In contrast to this, the downward fluxes among the levels lying below Byron's boundary, p < 6, are not balanced; this is because of the relationship F(p,p — 1) > C(p,p + 1), or eq. (4.36), for these levels. Thus, the multistep ladder-like deexcitation flux is established, which eventually reaches the ground state, resulting in recombination. In other words, the electrons which

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originate from the continuum states and reach the levels in LTE (p>pv) can return to the continuum states easily, while those which have crossed Byron's boundary downward and left these high-lying levels can no longer return to these levels and thus to the continuum states. They simply flow down, finally reaching the ground state, completing the process of recombination. Now we look for the critical process that determines the magnitude of «CR, the effective rate of recombination, in this high-density limit. As we have seen the three-body recombination to the ground state never plays a dominant role. See in Fig. 5.7 that the contribution from a is virtually absent. As we have seen already, any collisional process involving the levels lying above Byron's boundary, p >p^, is almost balanced by the respective inverse process, so that none of them can determine the rate. A collisional deexcitation flux between levels lying lower than Byron's boundary, p B lies above PQ. Thus we conclude that, with the increase in «e, when Griem's boundary comes down and reaches Byron's boundary level, aCR enters into its high-density limit. This transition occurs at « e ~ 1018 m~ 3 for Te= 1 x 103 K. This critical density as determined by is shown in Fig. 5.4 with the solid circles, where approximations of eq. (4.29b) for Po (slightly different numerically) and eq. (4.56) for/> B have been adopted. As noted above the effective recombination flux a£?R«z«e should be approximately given by

a 66

IONIZATION AND RECOMBINATION OF PLASMA

We may approximate n0(pB) by Z(pB)nzne, Instead of eq. (4.8) with eq. (4.7) which is good only for high temperature, we adopt another approximation, eq. (4.58), for neutral hydrogen at low temperature:

We then obtain

This is compared in Fig. 5.6 with the result of numerical calculations. Figure 5.8 shows SCR and «CR against Te with «e as a parameter. This is for neutral hydrogen. It may be interesting to note that both coefficients have almost the same value at about T e = 1.5 x 104 K, and that this is approximately true for any «e values. For higher temperatures ionization is faster than recombination, and for lower temperature vice versa. This is the second reason why we chose the temperature of T e /z 2 = 1.5 x 104 K as the boundary between the low-temperature

FIG 5.8 Collisional-radiative ionization and recombination rate coefficients for neutral hydrogen (z= 1) against Te with «e as a parameter. (Quoted from Goto et al., 2002; copyright 2002, with permission from The American Physical Society.)

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case and the high-temperature case. However, we have to be careful when we treat ions. From the scaling law given in Appendix 5B and already adopted in this and previous sections, SCR decreases as z increases while «CR increases. This point will be discussed in more detail in the next section. 5.4 lonization balance Plasma (in the second sense) in ionization balance is important for two reasons: 1. In certain practical situations, plasmas can actually attain this balance, at least approximately. 2. Many plasmas encountered in laboratory and astrophysical observations are far from this balance, but these plasmas are better understood with reference to the idealized situation, i.e. ionization balance. lonization balance is defined by

This balance is established, in principle, when the plasma is stationary, so that there are no temporal changes, and homogeneous, so that spatial transport of plasma particles does not affect the ionization balance. Ionization ratio In the following we call [nz/n(l)] the ionization ratio; it is different from the ionization degree [nz/{n(l) + nz}]. In the preceding sections we have examined the characteristics of SCR and «CR in detail. The ionization ratio is determined by these coefficients. In this section we assume the high-temperature case. This is because, as shown later, in considering ionization balance of ions, the relevant reduced temperature tends to be higher for increasing z. Figure 5.2 shows an example of the ionization ratio, together with these collisional-radiative rate coefficients, for the case of hydrogen-like ions with nuclear charge z in a plasma of Te/z2 = 6 x 104 K, the high-temperature case. Region I (ne < n£°) In the limit of low «e, the collisional-radiative rate coefficients are given by eqs. (5.6) and (5.14). The ionization ratio is given by

where the superscript 0 means the low-density limit. This situation has been customarily called corona equilibrium. It should be noted that the meaning of this term is different from that of the corona phase of the ionizing plasma component of populations at low density (Section 4.2). With an increase in «e, both rate coefficients increase, i.e. eqs. (5.8)-(5.10) for SCR and eqs. (5.16)-(5.18) for aCR. The ratio [nz/n(l)] changes accordingly. If we

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adopt the approximations (5.10) and (5.18), then we obtain

It is straightforward to see that, for low «e or large po (Fig. 1.10), when the exponential factor is expanded, the terms in the square brackets are approximated to \kTe/z2R + 6.8//>QJ. This indicates that, starting from the low-density limit, with an increase in «e, the second term increases. Therefore, the ionization ratio increases. This reasoning explains the increase in the ionization ratio shown in Fig. 5.2. Region II In this region, both rate coefficients are, in the present approximations, constant, as shown in Fig. 5.2. In this case, SCR has increased and reached the high-density-limit value, while «CR is still close to its low-density-limit value. Thus, the ionization ratio takes the maximum value, as seen in Fig. 5.2. Region III Both rate coefficients take the high-density-limit values, and are related by eq. (5.23). The ionization ratio is given by the Saha-Boltzmann equilibrium value, In Fig. 5.2, the approximation of eq. (5.31) is given with the dash-dotted line. We now examine an important consequence of the scaling law of various quantities against nuclear charge z. We have seen in Appendices 3A and 5B, that, in order for the rate equations (4.2) to be independent of z, Te should scale according to z2, eq. (3A.9), and ne to z7, eq. (5B.1).* In this sense, the quantities Te/z2 and ne/z7 are called, respectively, the reduced electron temperature and the reduced electron density. This scaling law means that the horizontal lines drawn for z = 1 in Fig. 1.2, which represents the parameter range of Fig. 5.2, for example, should be shifted for z > 1 parallel according to the oblique scale line shown in this figure. The reader can try this shifting procedure by taking an example of z, say z = 30. A consequence of this scaling is that the collisional-radiative ionization and recombination rate coefficients scale according to eq. (5B.5) and eq. (5B.5a), respectively, i.e. according to z~ 3 and z. The ordinate of Fig. 5.2 follows this scaling. We now suppose that, by keeping Te/z2 and ne/z7 constant, we change z. This procedure is shown in Fig. 1.2, where we follow the oblique scaling line from z = l to z = 30. As a result of the above scaling, the ionization ratio scales according to z~4. This is the reason why the quantity [nzz4/n(l)] has been plotted in Fig. 5.2. For the constant reduced temperature of this figure, Te/z2 = 6 x 104 K, the ionization ratio has a strong z-dependence: for z = 1 (neutral hydrogen) the ionization ratio in the low-density limit is about 104, almost completely ionized. * As seen in Fig. 3.11, the excitation cross-section values near the excitation threshold do not follow the z scalings. Therefore, the scaling in the present context is necessarily approximate, especially for low temperatures.

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For z = 10 (hydrogen-like neon) it is about 1, and for z = 26 (hydrogen-like iron) it is 2 x 10~2, only 2% ionized. This point can also be understood from Fig. 5.8. Te and ne in this figure are to be understood as the reduced temperature and density, Te/z2 and ne/z7, respectively. This figure is, of course, approximate for ions. With an increase in z starting from 1, SCR shifts downward by z3, while «CR shifts upward by z. Then, for a certain reduced temperature, Te/z2, the ionization ratio changes drastically, as we have seen already. The particular temperature at which the ionization ratio is 1 is of importance; this temperature is called the optimum temperature, Teo. The temperature in Fig. 5.2 is approximately the optimum temperature for z= 10. This temperature has the significance that, for an ionization balance plasma at low density, with a change in Te, intensities of the emission lines from excited levels have their maximum values at around this temperature; this point will be discussed later in this section. Figure 5.9 shows the optimum temperature Teo/z2 against z at «e/z7 = 1016 m~ 3 or in region I (the thick line). The thin line shows an approximation based on approximate expressions for S^R and o^R similar to those given by eqs. (5.7) and (5.15); z2R/kTeo~ 13.5-41nz. Note that the optimum temperature at z = 10 is consistent with Fig. 5.2. The slight decrease of reo/z2 with the increase of z= 1 to 2 is due to the different ionization cross-section values near the threshold for the neutral atom and for ions as seen in Fig. 3.11(c). This point is again understood from Fig. 5.8. With the increase in z, the reduced temperature at which both the coefficients become equal in magnitude, or at which they cross each other, increases.

FIG 5.9 Reduced optimum temperature for low density, at « e /z 7 =10 16 m (thick line). The thin line shows the approximation z2R/kTeo~ 13.5 — 41nz. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.)

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IONIZATION AND RECOMBINATION OF PLASMA

The above features are the important exception of the scaling law which was already mentioned in Chapter 1 and Section 4.1. Excited-level population - high-temperature case The population of excited levels is expressed by eq. (4.20) as a sum of the ionizing plasma component and the recombining plasma component. See Fig. 1.9. We call this n(p) the total population. This equation may be rewritten as

For the purpose of simplicity, we define b(p) as the relative population with respect to its Saha-Boltzmann value. Then eq. (5.32) becomes

Figure 5.10 shows b(p) of hydrogen-like iron (z = 26) in ionization balance for Te/z2 = 5.12 x 105 K for several ne's. Note in Fig. 5.9 that this very high temperature is rather close to the optimum temperature for z = 26. The ground-state ion density in ionization balance, 6iB(l), is given by the solid circles on the left-end

FIG 5.10 Total population, ^IB(^), for a plasma in ionization balance against/? (thick lines), and its recombining plasma component, r0(p) (thin lines), with «e/z7 as a parameter. The power of ten is given. The ground-state population £>(l)iB is given on the left-end ordinate. For hydrogen-like iron (z = 26) with T e /z 2 = 5.12 x 105 K. The horizontal dotted line is regarded as the upper bound of the approximate Saha-Boltzmann population, b(p)= 1+0.1. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.)

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ordinate, and bi^(p) of excited levels are shown with the thick lines. The subscript IB means the ionization balance. The reduced density ne/z7 is given as a parameter. The power of ten is attached to each curve. If the reader tries to identify the parameter range of this figure in Fig. 1.2, not the scaled ones but the real ne and Te, he or she will find them extremely high. Also shown with the thin lines are the recombining plasma component, rQ(p). The horizontal dotted line indicates b(p) = 1 ± 0.1, i.e. the range of approximate Saha-Boltzmann populations. In this very high-temperature case, the recombining plasma component of the population of levels in the CRC phase, or in the case of «e/z7 = 1012m~3 levels/? lying lower than pG ~ 20, is close to the Saha-Boltzmann value, or even slightly larger than that. This is slightly different from Fig. 4.14 for Te/z2= 1.28 x 105 K, where the population was slightly smaller. Note that radiative recombination strictly follows the z scaling. These near Saha-Boltzmann populations of the high-temperature, lowdensity recombining plasma component have been already discussed in Section 4.3. The addition of the ionizing plasma component, ri(p)bi^(l), results in total populations substantially larger than the Saha-Boltzmann values as seen in Fig. 5.10. They are about one order larger, as shown below. We now consider the relative magnitude of the ionizing plasma component and the recombining plasma component in ionization balance plasma. See Fig. 1.9. In the low-density limit the ionization ratio is given by eq. (5.29). The ionizing plasma component is given approximately by eq. (4.23)

The recombining plasma component is given by eq. (4.39), and with the neglect of the cascading contribution (see Fig. 4.15(a)) we have

We now compare the magnitudes of these components for large p, say p of the order of 10. On the assumption of high temperature we finally arrive at

where (9(0.1) means "the order of 0.1". Here we have utilized the approximate relationships, eq. (4.41) or /?(/?) oc/?~2'5, eq. (4.23a) or C(l,/?)oc/?~3, and 5*(1)~ C(l, 2) (see the cross-sections in Fig. 3.1 l(c)). This last relationship comes from the oscillator strengths, /i,2—/i, c (Table 3.1(b)). See also Fig. 4.7. We have already seen an example of this conclusion in Fig. 5.10. Thus, we have come to an important conclusion: In the low-density limit, 1. the recombining plasma component is close to the Saha-Boltzmann equilibrium value, and 2. the ionizing plasma component is larger than that by about one order of magnitude.

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IONIZATION AND RECOMBINATION OF PLASMA

These properties are independent of Te so long as the temperature is high. In particular, the second property is salient. In the case of neutral hydrogen, for example, at Te = 8 x 103 K, which is even outside the range of high temperature, a numerical calculation shows that the ionization ratio nz/n(l) is 3 x 10~5 and "oGOMGO is 0.1-0.7 for^ > 5. At Te= 1 x 106 K, the ionization ratio is 2 x 107, more than 10 orders of magnitude higher, yet nQ(p)/ni(p) is 0.2-1. A similar argument can be done for different ions. Suppose we fix the reduced temperature re/z2 at a certain value and change z. As we have seen in the previous subsection, for Te/z2 = 6 x 104 K the ionization ratio [nz/n(l)] changes by almost six orders of magnitude for the change of z from 1 to 26. Still, eq. (5.35) is valid for all the ions. Figure 5.10 is also an example of this statement. It is sometimes assumed that the excited level population in a plasma is given by eq. (5.33), with the neglect of the contribution from the recombining plasma component. The above arguments show that, so long as the plasma is in ionization balance, this assumption is approximately correct. With an increase in ne, it is seen in Fig. 5.10 that the recombining plasma component (ro(/>)) tends to the Saha-Boltzmann value; this is the problem which we have examined in Section 4.3. The total population, eq. (5.32a) or eq. (4.20), tends to its Saha-Boltzmann value, too, starting from high-lying levels. This feature is related to the problem of the establishment of LTE, which is important in practical plasma spectroscopy. This problem will be discussed in Appendix 5C. *Excited-level population - low-temperature case Figure 5.11 shows the population distribution of neutral hydrogen for Te = 4 x 103 K. The reader may identify the parameter range of this figure in Fig. 1.2. This temperature is, from the practical viewpoint, unrealistically low for a plasma in ionization balance. In this figure, however, an important feature is manifested which is characteristic of the low-temperature case. We note that, at low density, the total population, which is the sum of the ionizing plasma component and the recombining plasma component, of highlying levels, e.g. for/? ~ 10-20 at «e = 1012m~3 at which/? G ~ 20, is very close to the Saha-Boltzmann value. There appears to be no obvious basis for this to happen. We now examine this point. At these low temperatures, the rate coefficients for excitation and ionization from the ground state are determined by the cross-section values near the threshold for each process. We have established the interrelationship between these crosssections values for excitation to very high-lying levels and for ionization, eqs. (3.36)(3.38). As its consequence, we have an approximate relationship between the rate coefficients,

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FIG 5.11 Similar to Fig. 5.10 but for neutral hydrogen with Te = 4x 103K. Other explanations are similar to those for Fig. 5.10. The thick dotted curve shows the populations in slightly recombining plasma for ne= 1016m~3 with b(l) = 9 x 106 instead of bi#(l) = 1-65 x 107. This condition happens to give 6(3)= 1. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.) for large p. The ionizing plasma component of the population of level p in the corona phase is given by eq. (5.33). By using eqs. (5.36) and (5.29) we obtain

We now consider £/3(g). For lower-lying levels q we adopt the approximation (4.50) for (3(q), Then we have

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IONIZATION AND RECOMBINATION OF PLASMA

with

where q\im means the upper limit of q below which the approximation (4.50) is valid. This limit may be 5, 10, or 20, depending on Te. See the example of the /^-dependences of (3(p) in Fig. 3.5(b), where q\im is around 10. See also Fig. 3.9. For levels lying above q\im another approximation, eq. (4.41), is valid so that (3(q)<x q~2'5, and the summation converges rapidly. Therefore, S is of the order of 3. We are concerned with a higher-lying level p, which may be/>~ 10-20, where the radiative decay rate is approximated by eq. (4.12):

Here, \np = ft q ! dq is of the order of 3. By noting that the Saha-Boltzmann coefficient is written as

we conclude that ni(pf of eq. (5.37) is about (2/3)Z(p)nzne, or n(p)bIB(l')~2/3. In the same approximations as the above it may be shown that the recombining plasma component is not far from one-third of the Saha-Boltzmann value, i.e. n0(p)° ~(l/3)Z(p)nztie. This is the reason why the total population at low temperature and low density is close to the Saha-Boltzmann value. Excited-level population - high-density case In the limit of high density, or in region III defined in Section 5.2, the following interrelationship holds:

The ionization ratio is therefore given by the Saha-Boltzmann relationship

We remember

In this limit, the factor n(l)/[Z(l)nzne]

is unity. From another interrelationship,

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we have for all the excited levels. This equation, together with eq. (5.42), indicates that, in this high-density limit, we have the Saha-Boltzmann equilibrium (LTE) populations with respect to the ion density for all the levels including the ground state. This conclusion is independent of Te. This situation is nothing but the complete LTE as defined in Section 2.1. We now examine the above discussion in more detail. Figure 4.3 gives the highdensity-limit values of rQ(p) and r\(p) for severalp's against temperature. We have seen above that the r.h.s. of eq. (5.32) reduces to rQ(p) + ri(p), leading to the LTE populations for all p's. Therefore, Fig. 4.3, for a particular temperature, directly gives the ratio of the recombining plasma component and the ionizing plasma component of the total population for each level. As we have noted toward the end of Section 4.4 (p. 130-131), Byron's boundary/IB determines which of ro(p) or r\(p) is dominant in the total population; for levels lying above /?B the recombining plasma component is dominant, and for levels below, the ionizing plasma component is dominant. Figure 1.9 is a schematic illustration of this situation. Of course, all the total populations are LTE populations. Figure 1.7 is the spectrum of a hydrogen plasma virtually in the high-density limit (ne ~ 2 x 1023 m~3, higher than K^°+), and the emission line intensities as given from the LTE populations are shown with the dotted lines, where line broadening, which will be discussed in Chapter 7, and the effect of radiation reabsorption, which will be discussed in Chapter 8, are taken into account. As Fig. 4.3 indicates, for Tc=\ eV, except for level p = 2, which has a contribution from the ionizing plasma component of about 10%, the population of other levels virtually consists only of the recombining plasma component. Note in passing that the LTE populations for p<6 in Fig. 4.11 of the very low-temperature (Te= 103 K) ionizing plasma component is understood from Fig. 4.3. Figure 5.12 shows an example of the population distribution in lower density «e = 1020 m~3, intermediate between regions I and II, i.e. pG ~ 2. Two salient features are seen: 1. Even for this low density, the total population is rather close to or slightly larger than the LTE population for all the excited levels including p = 2. The groundstate atom density n(l) is larger than the Saha-Boltzmann value by more than three orders of magnitude. 2. The relative contribution from the recombining plasma component and the ionizing component is quite different from that in the high-density limit. For p = 2, for example, in the high-density limit the relative contribution was 86% to 14% (see Fig. 4.3), while at the density of Fig. 5.12 it is 1% to 99%. This means that, with the decrease in «e, the first term in eq. (5.32) decreased by two orders of magnitude, then the second term increased to compensate this decrease. This interesting but puzzling feature belongs the problem of establishment of LTE populations. We will discuss this problem in Appendix 5C.

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FIG 5.12 Boltzmann plot of the total populations of neutral hydrogen in an ionization balance plasma. T e = 1 x 104 K, «(1) = 3.2 x 1022 m~ 3 and n& = nz = 1 x 1020 m~3. The contribution from the ionizing plasma component is dominant for/? = 2 and 3, while the recombining plasma component is dominant forp > 3. We note here an important characteristic of excited level populations in ionization balance plasma. We remember our discussions above and look at Figs. 5.10-5.12. For high temperature, in the low-density limit, where eq. (5.35) is valid the total populations are larger than the Saha-Boltzmann values by an order, or bi^(p) ~ O(10) in terms of eq. (5.32a). For low temperatures, the total populations tend to be close to the Saha-Boltzmann values even in low densities. See Fig. 5.11. An important conclusion is that, in ionization balance plasma excited level populations tend to be larger than the Saha-Boltzmann values in low densities. The only exceptions are that, for extremely low temperature and density, very high-lying levels can have slightly smaller populations than the Saha-Boltzmann values. See Fig. 5.11 with « e =10 12 m~3. With an increase in «e all the total populations come exactly to the Saha-Boltzmann values, or bi^(p) = 1. This problem will be further discussed in Appendix 5C. Ionizing plasma and recombining plasma So far in this chapter, we have assumed that ionization balance is established between the "atoms" n(l) and the "ions" nz. In many practical situations this assumption

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is not correct; rather the ionization ratio is far from the ionization balance. If the plasma is time dependent, or if the plasma is spatially inhomogeneous so that transport of the plasma particles affects the ionization-recombination, then ionization balance is not established. It should be noted that, in the above discussions of the excited level populations in a plasma in ionization balance, both the contributions from the ionizing plasma component and the recombining plasma component could be substantial, or even comparable in magnitude; i.e. at low density and high temperature the former contribution is larger than the latter by about an order, and at high density and rather low temperature both the components are necessary for the levels to have the LTE populations, eq. (5.43). Now, we come to an important conclusion: If the plasma is far from ionization balance, either of the components tends to predominate over the other, and we may neglect the other contribution entirely in the total population. We introduce here the concepts of an ionizing plasma and a recombining plasma: an ionizing plasma is defined by [nz/n(l)] [«z/«(l)]iB- In this case, nz is highly "overpopulated", so that the recombining plasma component will be much larger than the ionizing plasma component, which could be neglected entirely. Consider typical examples of actual plasmas. The first is a hydrogen (z= 1) positive-column plasma in a low-pressure discharge with Te = 5 x 104K, nz = « e = 1017 m~ 3 and «(!)= 1022 m~3. See Fig. 1.2. This is the condition which corresponds approximately to a helium-neon gas laser discharge. This plasma has nz/n(l)= 10~5, which is to be compared with [nz/n(l)]i# = 103. This overpopulation of «(1) by eight orders of magnitude is the result of the transport or diffusion of ions to the discharge cell wall, owing to the spatial inhomogeneity of the plasma. In order to take this effect into account the terms describing the spatial transport of the ions and atoms should be added to the r.h.s. of eq. (5.2). The recombination term aCRnzne is, in this case, smaller than the ion diffusion loss term by eight orders of magnitude. The ionization term balances with this diffusion loss term, in this case. Thus, this is a typical example of an ionizing plasma. Remember that the example cited in Section 4.2 (Fig. 4.12) was a positive-column plasma. In fact, Te of that plasma was 5 x 104 K and «e was 1017-1020 m~3. Similar situations are realized for hydrogen and impurity ions in a magnetically confined plasma, and in a plasma heated by a shock wave during the process of ionization relaxation. Figure 5.13 shows an example of an ionizing plasma, where [nz/n(l)] is smaller than [«2/K(l)]iB by two orders of magnitude. For the lower-lying levels than/? = 7, the ionizing plasma component is predominant, while for the higher-lying levels this is not the case. The reason why the recombining plasma component becomes dominant in this ionizing plasma is that, with an increase in/?, the ionizing plasma component decreases very rapidly, r^p) <xp~6, while the recombining plasma component, r0(p), stays almost constant (~ 1). It should be noted that the total

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FIG 5.13 Similar to Fig. 5.12 except that the plasma is slightly ionizing, nz/n(\) ~ KT2[«Z/«(1)]IB. re = 4 x 104 K, n(l)= 1 x 1019 m~3and ne = nz=lx 1020 m~3. Even in this ionizing plasma, the populations of the high-lying levels are dominated by the recombining plasma components. population for low-lying levels is larger than the Saha-Boltzmann values. This situation of b(p) > 1 for low-lying levels is the common feature for ionizing plasmas. An example of a recombining plasma is an afterglow plasma of neutral hydrogen, Te = 2 x 103 K, and nzn&= 1018 m~3. See Fig. 1.2. This condition gives [«Z/«(!)]IB ~ 1CT27, or K(!)IB ^ 1045 m~3, which is too large to be realistic. Since n(l) is "underpopulated" by more than 20 orders of magnitude, the ionizing plasma component is almost completely predominated over by the recombining plasma component. The examples cited in Section 4.4 (Fig. 4.25) were the flowing afterglow plasmas. There are many kinds of plasmas which fall in this class of recombining plasma. For instance, the plasma produced by illumination of intense laser light on a solid target sometimes shows this characteristic. A plasma surrounding a star which emits strong ultraviolet radiation, and being photoionized by it is another example. Highly ionized impurity ions diffusing from the central plasma to the outer region of a magnetically confined plasma are also an example of the recombining plasma. Divertor plasmas sometimes fall into this class.

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lonization flux, recombination flux, and emission-line intensity In Chapter 4 we examined various features of the populations of excited levels, and in this chapter we have studied the features of ionization and recombination of a plasma in the second sense. Both the ionizing plasma component, eq. (4.22), and the ionization flux, eq. (5.2), are proportional to the ground-state atom density, n(l). Both the recombining plasma component, eq. (4.38), and the recombination flux, eq. (5.2), are proportional to the ion density, n2. See also Fig. 1.9. These observations suggest that from the measurement of an excited level population, or of an emission line intensity, we can infer the magnitude of the flux of ionization or that of recombination, or even both. Figure 5.14 shows an example of the relationships between these quantities. Suppose we have neutral hydrogen atoms of «(1) = 1 m~ in a stationary plasma of « e = 1018 m~ 3 with varying temperatures. Shown are the ionization flux and the corresponding emitted photon numbers of the Balmera a(p = 2-3) transition per unit time and volume. It is seen that these two quantities have a similar dependence on temperature. Figure 5.15 shows the proportionality factor [5>cR«(l)«e/»i(3)^4(3,2)], or the number of ionization events per Balmer a photon emission. Figure 5.14 also shows similar quantities for a recombining plasma: for

FIG 5.14 The ionization flux and the Balmer a line intensity, or the number of photons emitted by excited atoms, originating from the ionizing plasma component; «(1) = 1 m~ 3 and «e = 1018 m~3. The recombination flux and the line intensity from the recombining plasma component; nz=\ m~ 3 and « e = 1018 m~3. For the situation where n(l) = nz=l m~ 3 is assumed, the sum of the line intensities is given with the solid curve. (Quoted from Goto et al., 2002; copyright 2002, with permission from The American Physical Society.)

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IONIZATION AND RECOMBINATION OF PLASMA

FIG 5.15 (a) Proportionality factor for the number of ionization events per Balmer a photon emitted, (b) Similar quantity for the recombining plasma. (Quoted from Goto et al., 2002; copyright 2002, with permission from The American Physical Society.) protons of KZ = 1 m 3 in a plasma of ne = 1018 m 3 the recombination flux and the corresponding emitted photon numbers are shown. Again both quantities have a similar temperature dependence. Figure 5.15 shows the proportionality factor [aCRnzne/n0(3i)A(3,2)], or the number of recombination events per Balmer a photon emission. It is interesting to note that, for recombination, the efficiency of producing photons per recombination event has weak dependences on Te and ne in the parameter range of this figure. We now return to a plasma in ionization balance in the present context. Figure 5.16 shows an example. The total number density of atoms and ions is assumed constant, i.e. The atom density

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FIG 5.16 A plasma in ionization balance, where n(l) + nz= 1 m 3 is assumed. The densities, n(l) and nz, are shown. Also shown are the magnitude of the flux of ionization-recombination and the number of the emitted Balmer a photons with its breakdown into the rates coming from the ionizing plasma component and the recombining plasma component. (Quoted from Goto et al., 2002; copyright 2002, with permission from The American Physical Society.) and the ion density against a change in temperature, as calculated from SCR and aCR (Fig. 5.8), are given by the thin solid lines. For low temperatures the atoms are dominant and for high temperatures the ions are dominant. In this figure, also shown are the flux of ionization, ScR_n(l)ne, or that of recombination, aCRnzne. Since the ionization balance is defined by eq. (5.28) or ScRn(l)ne = acRnzne, both fluxes are equal. It is to be noted that the magnitude of the ionizationrecombination fluxes have the maximum at about the optimum temperature at which «(1) = nz holds. The reason may not be obvious, but it can be understood from physical considerations; For low temperatures, the atom density is high, but the CR ionization rate coefficient is small. See Fig. 5.8. For high temperatures, the CR ionization rate coefficient is large, but the atom density is now low. A similar argument can be made concerning the recombination flux. At the beginning of this section we showed that, for a plasma in ionization balance, the relative magnitude of the ionizing plasma component is larger than the recombining plasma component by about an order of magnitude, eq. (5.35). The Balmer a line intensity as given by eq. (4.1), or the photon number, coming from the ionizing plasma component [Ki(3)J(3,2)] is given in this figure by the upper dashed line and the line intensity from the recombining plasma component [n0(3)A(3,2)] is given by the lower dashed line. The above statement is seen to hold, especially at high temperatures.

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IONIZATION AND RECOMBINATION OF PLASMA

The total intensity, i.e. the intensity we observe from this plasma, is given by the dash-dotted line. It is noted that the total line intensity moves approximately in parallel with the magnitude of the flux of ionization or recombination. Thus, the line intensity takes its maximum near the optimum temperature Teo. If we look at this figure more closely, we recognize that the actual temperature of the maximum emission intensity is slightly higher than the optimum temperature. The ionization-recombination flux takes the maximum at a still higher temperature. This problem is examined in Appendix 5D. We now consider a plasma out of ionization balance, or an ionizing plasma or a recombining plasma. In Fig. 5.14 suppose we fix «(1) = nz=\ m~3, and the temperature changes. The total intensity, eq. (4.20) or Fig. 1.9, is the sum of the intensities from the ionizing plasma component (the dashed curve) and from the recombining plasma component (the dashed curve), to yield the total intensity given by the thin solid line. The important point to note here is that the total intensity takes the minimum at about the optimum temperature. The situation at this particular temperature is rather close to the situation of the ionization balance plasma, Fig. 5.16; as mentioned above, near the optimum temperature, the intensity took the maximum. Note the consistency of the intensities of these minimum and maximum in these figures, respectively. In Fig. 5.14, for temperatures higher than the optimum temperature, the line intensity becomes high, indicating that the ionization flux is large because this plasma is an ionizing plasma. For temperatures lower than the optimum temperature, the line intensity is again high indicating that, this time, the recombination flux is large for this recombining plasma. We have now come to another important conclusion. When a plasma emits line radiation this is an indication that this plasma is undergoing ionization or recombination, or even both of them simultaneously in the case of ionization balance. A plasma out of ionization balance tends to emit intense radiation, indicating that this plasma is undergoing strong ionization or strong recombination. The line intensity is thus a measure of the magnitude of this ionization flux or the recombination flux. In other words, a plasma in ionization balance emits the weakest radiation among the plasmas in various ionization-recombination states. 5.5 Experimental illustration of transition from ionizing plasma to recombining plasma As has been noted in the preceding section, a plasma cannot attain ionization balance if it is spatially inhomogeneous or time dependent. The example of experimental observations of the ionizing plasma shown in Section 4.2 and that of the recombining plasma in Section 4.4 were both stationary. So the origin of the deviation from the ionization balance was that these plasmas are inhomogeneous and the spatial transport of the plasma particles was important in determining the ionization-recombination state of the plasma. In this section we take an example of a time-dependent plasma, a pulsed gas discharge. When a pulsed current (Fig. 5.17(a)) is drawn through a gas, helium in

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FIG 5.17 A discharge plasma is produced from a helium gas of 2torr filled in a discharge tube of 5 mm inner diameter, (a) Discharge current, (b) Emission line intensity of the neutral helium line Hel A587.6 nm (23P — 33D) measured from the side of the tube. See the dotted line in Fig. 1.4. The first peak is called peak A and the second peak B. (Quoted from Hirabayashi et al., 1988; copyright 1988, with permission from The American Physical Society.) this case, a plasma is produced, and the emission line intensity shows two peaks during the course of time as is seen in Fig. 5.17(b). This is a neutral helium line Hel A587.6 nm(23P - 33D). See the energy-level diagram of Fig. 1.4. We now call the former and latter peaks A and B, respectively. Peak A corresponds to the time during the current rise. Peak B appears just after the finish of the discharge current and the intensity decays rather slowly. From the discussions at the close of the previous section, we may suppose that peak A indicates that the "gas" is undergoing ionization during this period and its intensity is proportional to the ionization flux, and that peak B and the subsequent emission indicate the recombination flux. We observe the emission line spectra from these plasmas, as shown in Fig. 5.18 (a) and (b) for peak A and peak B, respectively, and deduce the excited level populations according to eq. (4.1). We plot the population distribution in a graph similar to Figs. 4.5 and 4.20(a). Figure 5.19(a) is the result. Peak A clearly shows the minus sixth power law, and the distribution at peak B looks similar to Fig. 4.20(a) at high densities. Figure 5.19(b) is another plot, the Boltzmann plot like Fig. 4.20(b), of the peak B populations. The minus sixth power population distribution is characteristic of the highdensity ionizing plasma for levels lying higher than Griem's boundary (Figs. 4.5 and 1.10(a)), or it also occurs to levels lying lower than Byron's boundary in the

184

IONIZATION AND RECOMBINATION OF PLASMA

FIG 5.18 (a) Observed spectrum at peak A. (b) at peak B. The series lines of 23P — «3D and, in (b), the recombination continuum are seen. See also Chapter 6. (Quoted from Hirabayashi et al,, 1988; copyright 1988, with permission from The American Physical Society.) high-density, low-temperature recombining plasma (Figs. 4.20(a) and 1.10(b)). From the fact that (1) the minus sixth power distribution in Fig. 5.19(a) seems to extend to very high-lying levels, and (2) peak A appears when the plasma is in the build-up process from a gas during the current rise, we may conclude that the plasma at peak A comes from an ionizing plasma and the emission intensity indicates the magnitude of the ionization flux. From the facts that peak B appears in the period of plasma recombination and that the population distribution matches the low-temperature recombining plasma, we may conclude that the plasma at peak B is a recombining plasma and the emission intensity indicates the magnitude of the recombination flux.

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185

FIG 5.19 Excited-level population distribution for «3D levels for peak A and peak B (O). See Fig. 1.4. The calculated result is shown with +. (a) Plot similar to Figs. 4.5 and 4.20(a). (b) The Boltzmann plot of the population distribution for peak B. Note that the experimental populations extends to the continuum states. See Chapter 6. (Quoted from Hirabayashi et al., 1988; copyright 1988, with permission from The American Physical Society.)

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We now construct a collisional-radiative model for helium. The ground-state atom density is given from the filling pressure of the gas, i.e. n(l :S) = 6 x 1022 m~3. An excited-level population is given by eq. (4.20), and for peak A we take only the second term, the ionizing plasma component, and for B only the first term, the recombining plasma component. By fitting the calculated populations to experiment (see Fig. 5.19), we obtain Te and ne for these plasmas. The peak A plasma has Te of 4-5 x 104 K and ne ~ 1020 m~ 3 and the peak B plasma has Te = 5.1 x 103 K and ne= 1.25 x 1021 m~3. It is noted that the ion density is virtually equal to the electron density under the present condition. The reader may locate these plasmas on Fig. 1.2. We are concerned here with neutral helium, z = 1. We can thus explain the temporal development of the excited-level population in Fig. 5.17(b) as follows. During the current rise, Te is high so that energetic electrons ionize the gas; this means that the situation similar to Figs. 4.9 and 4.12 is actually realized. Therefore, the ionizing plasma component dominates over the recombining plasma component, and the population is high, indicating a large flux of ionization. Figure 5.20(a) shows the calculated total population distribution in this plasma including both components. The above conclusion is clearly seen. After the current takes the maximum the population begins to decrease. Until this time a plasma with a sufficient ionization ratio has been formed. The total number of ionization events during this initial stage, or the resulting ion density of this plasma at its peak, may be inferred from the area under the peak A. During the current decay, there is no need for the plasma electrons to ionize the gas further so that Te decreases to about 2 x 104 K, close to the optimum temperature. An ionization state close to the ionization balance is established, and the intensity shows a minimum, being consistent with Fig. 5.17(b). When the current ceases, there is no mechanism to sustain high Te, and it drops very rapidly. With this decrease in Te the recombining plasma component increases, and the ionizing plasma component becomes small by many orders of magnitude, as is seen in Fig. 5.20(b). Thus, the intensity is high, again suggesting a high recombination flux. From the area under peak B we may infer the total number of recombination events during this afterglow. If we adopt the proportionality factor for the hydrogen Balmer a line, Fig. 5.15, for our case of He (23P — 33D), the number of ionization events during peak A is almost equal to the number of recombination events in peak B. We could even deduce the time history of ne. Unfortunately, however, the intensity is measured only relatively (Fig. 5.17(b)), so that we cannot determine the absolute value of ne. It is noted that the situation realized in this experiment is similar to the situation of Fig. 5.14, where both the ground-state atoms and ions are present simultaneously and temperature changes over a wide range. In the present example, we start with a high temperature during the current rise leading to the strong emission intensity; during the current decay the temperature decreases giving the minimum of the intensity; and the further temperature decay after the finish of the current gives rise to the strong emission again. The reader will understand that this figure is quite universal, i.e. the condition of n(l) = nz= 1 m~ 3 is not a strong constraint in understanding real situations.

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187

FIG 5.20 Calculated distribution of the total population, eq. (4.20), for «3D levels together with the ionizing plasma component and the recombining plasma component resolved, (a) For peak A; (b) for peak B. (Calculation by M. Goto.)

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Appendix 5A. Establishment of the coUisional-radiative rate coefficients In Appendix 4B we examined the transient characteristics of the excited-level populations approaching the final stationary state. We worked out the conditions under which the QSS approximation is valid, or we can use the population coefficients, r0(p) and r^p). We likewise have to justify eqs. (5.1)-(5.5) for the description of ionization and recombination of our plasma in the second sense. We take the example of the ionizing plasma treated in Appendix 4B. The temporal development of excited-level populations has been shown in Fig. 4B.1. As has been noted, at early times the level populations are simply accumulating. This means that the ground state has a depopulating flux, while it has no returning populating flux. Figure 5A. 1 shows the temporal development of the net depletion rate coefficient from the ground state. It starts with the sum of the excitation rate coefficients to all the levels and the ionization rate coefficient. This corresponds to the first two terms of the r.h.s. of eq. (5.3). With the course of time the excited level populations develop, and they finally reach the stationary state. The returning flux to the ground state gradually increases, so that the net depletion flux decreases.

FIG 5A. 1 Temporal development of the effective rate coefficients for depletion of the ground-state population and that for production of ions, corresponding to Fig. 4B.1. Both the rate coefficients tend to SCR after all the excited levels come to QSS. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.)

ESTABLISHMENT OF CR RATE COEFFICIENTS

189

This process corresponds to the development of the third negative term in the r.h.s. of eq. (5.3). At about t= l(T7s, or at tres, this transient state is over, and the net depletion rate coefficient tends to the collisional-radiative (CR) ionization rate coefficient, SCR. This figure also includes the net production rate coefficient of ions. It starts with the direct ionization rate coefficient. With the accumulation of the excitedlevel populations it also tends to the CR ionization rate coefficient. We remember that the time constant of depletion of the ground-state atoms is 1CT4 s in this example. So, the CR ionization rate coefficient is established much faster. We may draw two conclusions: 1. The CR ionization rate coefficient lies somewhere between the ionization rate coefficient and the sum of the rate coefficients for excitation and ionization, all from the ground state. The actual value of the rate coefficient depends on ne and Te of the plasma. 2. As in the case of the excited-level populations the response time ties gives the time for the validity of using SCR to describe the effective ionization rate of the plasma.

FIG 5A.2 Temporal development of the effective rate coefficients for depletion of ions and for production of the ground-state population, corresponding to Fig. 4B.4. Both the rate coefficients tend to aCR after all the excited levels come to QSS. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.)

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Figure 5A.2 shows a similar plot for the recombining plasma treated in Appendix 4B. Corresponding population developments have been given in Fig. 4B.4. In the present figure, the effective depletion rate coefficient of the ions and the effective production rate coefficient of the ground-state atoms are shown. If we include recombination into very high-lying levels, at the start, the former quantity diverges, because, as is seen in Fig. 4B.7, the three-body recombination rate coefficient becomes very large for large p. The production rate coefficient of the ground-state atoms is the direct radiative recombination rate coefficient (see Fig. 4B.7). This corresponds to the first two terms in eq. (5.4). Note that, for the ground state, under our present condition the tree-body recombination rate is much smaller than the radiative recombination rate. With the course of time both the net rate coefficients tend to the collisional-radiative (CR) recombination rate coefficient. At about t = tres the CR recombination rate coefficient is established. Thus, in this case again, the response time for excited-level populations gives the validity of using aCR to express the effective recombination rate of the plasma. We again remember that the depletion time constant of the protons is 10~2 s in this example, much longer than the time for establishing the CR recombination rate coefficient. Appendix SB. Scaling law In Appendix 3A we have seen the scaling properties of atomic parameters of hydrogen-like ions against the nuclear charge z. The speed and energy of plasma electrons also scaled by the scaling of the atomic parameters. Therefore, Te scaled according to z2: Te/z2 is the reduced temperature. In Chapter 4, we introduced the rate equation and the collisional-radiative model, and investigated the populations of excited levels. In the present chapter we examined the processes of ionizationrecombination and the ionization balance of a plasma. In this appendix, we investigate the scaling law which enables us to scale various quantities which appear in the CR model for hydrogen-like ions with respect to those for neutral hydrogen. We also point out the limitation of the scaling laws. We want the rate equation, eq. (4.2), to become independent of z. We adopt the scaling laws of the rate coefficients as introduced in Appendix 3A into this equation. Then it is obvious that the electron density should scale as

Then, the time scales as

Unfortunately, this scaling is inconsistent with the scaling for the bound electron, eq. (1.4), i.e. z~2 scaling. It is readily seen that, if we adopt the above scaling laws, eq. (4.2) becomes independent of z except for the last line representing

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191

recombination. If we further require that the recombination follow the scaling, the ion density should scale according to

If we adopt this scaling the conservation of particles is violated in the recombination processes. The scalings of the collisional-radiative coefficients are obvious:

We consider the ionization balance, eq. (5.2):

or eq. (5.28). This leads to the same scaling as eq. (5B.3). This is the reason why we used the quantity [z4nz/n(l)] in the discussion of ionization balance. See eq. (5.30) and Fig. 5.2, for example. We now turn to recombination of the plasma, eq. (5.2) with eq. (5.5):

In this case the conservation of particles, nz, should be valid. Then, the time for recombination scales according to

being inconsistent with eq. (5B.2). These inconsistencies concerning recombination lead to an interesting exception of the scaling law for Te as discussed in Section 5.4 and shown in Fig. 5.9. We again note another limitation of our scaling law; as Fig. 3.11 shows, the cross-section values for excitation and ionization near the threshold do not follow a simple scaling law, so that our results, eqs. (5B.4), (5B.4a), (5B.5), and (5B.5a), become less valid for low temperatures. *Appendix 5C. Conditions for establishing local therniodynaniic equilibrium In Chapter 2, we introduced local therniodynaniic equilibrium, which is abbreviated to LTE. As we have seen in Chapter 4 the populations in high-lying

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Rydberg levels are strongly coupled with the continuum state electrons, and they tend to be in thermodynamic equilibrium with the continuum electrons more easily. Partial LTE is the state that these levels are actually in thermodynamic equilibrium, or that these levels have Saha-Boltzmann populations. Low-lying levels, however, may depart from this equilibrium relation. There is thus the lower-bound level above which the higher-lying levels are in LTE. In the case that this lower-bound level comes down to the ground state, this situation is called complete LTE. We have considered this problem already in the text. In this appendix, we treat this problem further, with the aim of providing numerical expressions for conditions for establishing partial and complete LTEs. Partial LTE We start with the expression for the excited-level populations, eq. (5.32a),

where b(p) is the population normalized by its Saha-Boltzmann value. See eq. (5.32). We now define LTE for level p to be

Our problem is to find the lower boundary level which satisfies this definition in a particular plasma in the first sense, and to a certain degree, in the second sense. We have calculated rQ(p) and r\(p) for neutral hydrogen in Chapter 4, as given in Table 4.1, and examined the properties of the excited-level populations for a wide range of «e and Te. Similar calculations have been done for hydrogen-like ions, z = 2 or ionized helium, and z = 26 or 25 times ionized iron. We use the reduced electron temperature and the reduced electron density

respectively, in this appendix. We first consider recombining plasma. The definition of a recombining plasma in Section 5.4 is expressed as 6(l)<6i B (l) in eq. (5C.1). Here, IB stands for ionization balance. We look at Figs. 5.10-5.12: these figures show examples of the excited-level populations as given by r0(p) and by eq. (5C. 1) with b(l) = 6iB(l). We called the former situation a purely recombining plasma, and the latter was an ionization balance plasma. The populations of our recombining plasma lie somewhere between these two limiting distributions, depending on the instantaneous value of 6(1). The former plasma is nothing but the recombining plasma component which we examined in detail in Section 4.3 and Section 4.4. The latter has been treated in Section 5.4, and will further be considered later on. Figure 1.10(b) for a purely recombining plasma suggests that the region of TQ(P) ~ 1 is bound by Griem's boundary and Byron's boundary: the former is concerned mainly with ne

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and the latter with Te. Figures 5.11 and 5.12 also give the idea of how r0(p) tends asymptotically to 1. Figure 5C. 1 shows the lower boundary level />R with TJ as a parameter as determined from the numerical calculations ofr0(p) for z =1,2, and 26. The "dip" near O«2-3 x 105 K is a rather exceptional situation for high temperature; r0(p) — 1 holdsevenfor very low«eas discussed in Section 4.3. Since this situation violates the spirit of LTE, we ignore this dip. In fact, for very low density, even our assumption of the statistical distribution among the different / levels is no longer valid (see Fig. 4A.2), so that this special situation should be taken with some care. The region in the left-bottom corner, i.e. />R cannot become small for low temperature, comes from Byron's boundary. In this figure are plotted the numerically fitted expressions for the conditions of LTE; the thin solid curve is

and the thin dotted line is

FIG 5C.1 The principal quantum number of the lower boundary level />R for establishment of partial LTE in recombining plasma. The thick curves are the result of numerical calculation. z=l; z = 2; z = 26. The eq. (5C.3); eq. (5C.4). (Quoted thin lines are numerical formulas. from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.)

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FIG 5C.2 The principal quantum number of the boundary level />R in the 77 — O plane for recombining plasma. The region to the right of a curve is the region of density and temperature in which the higher-lying levels than the boundary level are in LTE. This figure is constructed from the three cases (z=l, 2, and 26) in Fig. 5C.1, and is therefore approximate. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.) Both conditions should be met simultaneously. It is noted that eq. (5C.3) is essentially the same as eq. (4.29) or even eq. (4.29b), and that eq. (5C.4) is the same as eq. (4.56). The small differences, e.g. instead of the 3We in the denominator of eq. (4.56), eq. (5C.4) adopts 2kT&, are partly due to the difference in the definitions of LTE; it is more stringent in this appendix, eq. (5C.2), than eq. (4.55) which gives Byron's boundary level. Figure 5C.2 is another plot of Fig. 5C.1: the region of temperature and density is shown in which levels p >/>R are in LTE with/> R as a parameter. In order for a level to be in LTE, the radiative transition processes concerning this level should be predominated over by the competing collisional transitions and can be neglected. Equation (5C.3), or eq. (4.25) with the equality sign replaced by an inequality sign, expresses this condition. If this condition is met the problem becomes the relationships among the collisional transition processes. We have seen that among the collisional transitions from a level, the dominant ones are excitation or deexcitation to the adjacent level, eq. (4.6). Figure 5C.3 is a schematic diagram

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FIG 5C.3 Schematic diagram of the four possible cases of the dominant populating and depopulating processes of level p in which we are interested, under high-density conditions where the radiative transitions are neglected. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.) for the dominant collisional populating and depopulating processes concerning level p which is under consideration. This figure depicts the four possible schemes which can be realized in dense plasmas. For the recombining plasma, in which the population flux originates from the continuum state electrons, scheme (b) or (d) is possible. In order for the level p to be in LTE, scheme (d), eq. (4.36), should be excluded. Equation (5C.4) is the expression for this condition. We now come to the ionization balance plasma in which b(l) in eq. (5C.1) is given values &IB(!)- We investigated this problem in Section 5.4; for high temperature, starting from low density with b(p) ~ <9(10), with an increase in «e it tends to 1. An example is seen in Fig. 5.10. Figure 5C.4 shows a similar plot to Fig. 5C.1 for the ionization balance plasma. The sharp lowering of pi# with the decrease in Te for low densities is the result of the near LTE populations for low-density and lowtemperature ionization balance plasma as discussed in Section 5.4. Neutral hydrogen (z = 1) and hydrogen-like helium (z = 2) show this lowering, but hydrogen-like iron (z = 26) does not. No reason can be found that the arguments concerning the near LTE populations in Section 5.4 do not apply for such high-z ions. The computer code may not yet be refined enough in the present calculation. This lowering, however, violates the spirit of LTE as the case of the dip in the recombining plasma. We ignore these special cases again. We obtain from numerical fitting

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IONIZATION AND RECOMBINATION OF PLASMA

FIG 5C.4 The boundary level for partial LTE in an ionization balance plasma, corresponding to Fig. 5C.1. The explanation is the same as for Fig. 5C.1, except that the thin solid curve is for eq. (5C.5). (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.) This lower boundary is shown with the thin solid line in Fig. 5C.4. It is interesting to note that this expression is quite close to eq. (4.29); this equation gives />G = 345eao67/?7CU18.* Again the minor difference in the two expressions is due partly to the difference in the definitions, LTE on one hand and Griem's boundary level on the other. Figure 5C.5 is another plot of Fig. 5C.4. In this case, the lowering of/>rB for low G has been ignored, and approximate boundaries are determined from z = l , 2, and 26. If we compare Fig. 5C.4 (or Fig. 5C.5) with Fig. 5C.1 (or Fig. 5C.2) for a recombining plasma, we recognize a substantial difference: at high temperature the addition of the ionizing plasma component makes it difficult for levels to enter into LTE, while for low temperature the addition makes it easy for them. The first point is easily understood from Fig. 5.10. The recombining plasma component Professor Griem gives in his new book, which was mentioned in the references of Chapter 1, a similar criterion, />« 8436°' 059 /tf' ns . The numerical factor was about half in his book of 1964.

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FIG 5C.5 The same as Fig. 5C.2, but for an ionization balance plasma. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.) alone is quite close to LTE, so that the additional population hinders the total population to be in LTE. In order for the population to return to the LTE values the ionizing plasma component should become sufficiently small. The lowtemperature case is not straightforward. We start with

This equation is rewritten as

In the high-density limit we have the factor [acR/ne]/[Z(l)Sci(]= 1. As Fig. 5.4 shows, with a decrease in «e from K^°+, [acR/«e] keeps its high-density limit value until «e reaches eq. (5.25). We now examine SCR and r\(p) for low-lying levels, especially p = 2. Figure 4.11 shows that, in the high-density limit, the levels p
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IONIZATION AND RECOMBINATION OF PLASMA

deviates from it for ne < nf+, levels 3

—!). By omitting the common factors we have, from eq. (3.29),

and

In the case that level (p— 1) is in Boltzmann equilibrium with level q, we have

It is obvious from the statistical weights, the energy differences, and the oscillator strengths shown in Fig. 3.4, that we have

Therefore, the dominant populating mechanism of level p is stepwise excitation (p— 1) —> p so long as level (p—1) is in Boltzmann equilibrium with lower-lying levels. See Fig. 5C.3(c). At high density, starting from the high-density limit all the populations of excited levels are controlled by the population of the lower boundary level in Boltzmann equilibrium. This level is level 2 down to ne at which level 3 deviates from Boltzmann equilibrium with level 2. In this low-temperature case, in the high-density region the dominant contribution to ionization is the ladder-like excitation-ionization as suggested by the p~6 distributions in Fig. 4.11. These populations, and thus the ionization flux, are controlled by the population of the low-lying level p = 2 or 3. Thus, it is natural that SCR behaves much the same as the population r^p), especially ^(2). We expect that

for p = 2, 3 , . . . , pB, Numerical calculations actually confirm that the relationship (5C.10) is quite accurate. As seen in Fig. 4.3, for this low temperature, the LTE populations in the highdensity limit [n(p)/Z(p)nzne] = r0(p) + r\(p) = 1 consists mainly of the second term, the ionizing plasma component, for low-lying levels. With a decrease in ne, the decrease in ri(p) (Table 4.1(a)) is compensated in eq. (5C.6) almost exactly by the decrease in SCR according to eq. (5C.10), keeping the same value of the second term. The decrease in TQ(P) as seen in Fig. 4.19 does not affect much the population, because the first term is quite small for these low-lying levels. See Fig. 4.3.

LOCAL THERMODYNAMIC EQUILIBRIUM

199

For temperatures around the optimum temperature, the situation is different as we have noted concerning Fig. 5.12: for this low density, the populations are close to LTE, while, the relative contributions from the recombining plasma component and the ionizing plasma component is quite different from that in the high-density limit. We take level p = 2 for an example. In this temperature range, eq. (5C.10) is still valid in the density region close to nf. This is because ionization is controlled by the ladder-like excitation-ionization starting from/? = 2 at around this density. With a decrease in ne from the high-density limit, r0(2), the first term of eq. (5C.6), decreases substantially. At the same time r:(2) decreases, but this decrease is compensated by a decrease in SCR, as in the case of low temperature. In the present temperatures, however, (aCR/»e) increases (see Fig. 5.4), leading to an increase in the second term. This substantial increase compensates, or even overcompensates, the decrease in the first term. The resulting population tends to be higher than the LTE population for lower densities as has already been noted in the text. We now turn to the ionizing plasma, i.e. 6(1) > &IB(!)- It is clear that, if our plasma is purely ionizing, i.e. nz = 0, and lacks the first term of eq. (5C. 1), no LTE populations can exist. In many practical situations, we encounter plasmas which are ionizing, yet the first term is substantial to a certain degree. In these cases, in Figs. 5.10 and 5.11, for example, the populations would lie somewhere above the thick curves, or the second term tends to be larger than "it should be". Thus, the second term tends to make the level populations larger than the LTE values, preventing the levels from entering into LTE. This extreme situation is schematically illustrated by Fig. 5C.3(a): the population flux coming from the lower-lying level is too large, resulting in the ladder-like excitation-ionization. Under certain conditions, e.g. 6(1) is larger than £>IB(!) only by a small amount, some levels may be in scheme (b) and they can be in LTE. An example has been shown in Fig. 5.13: this plasma is slightly ionizing, i.e. 6(1) ~ lOO^iB(l), and levels p > 10 are in LTE. Thus, the condition of LTE includes, besides ne and Te, some constraint about the overall balance of ionization of the atoms or ions under consideration. The conditions for LTE in the present ionizing plasma reduce to r0(p) ~ 1 and

The first condition has been taken care of already, eqs. (5C.3) and (5C.4). In practical situations where LTE becomes an issue in this class of plasma, i.e. high density and temperature, these conditions are almost always met. Thus, the most crucial condition is eq. (5C.11). We now remember that, for high density and temperature, r\(p) is well approximated by eq. (4.35) or ri(p)=p~6. See Fig. 4.10. Within this approximation, eq. (5C.11) is rewritten as

This set of conditions for establishment of LTE involves several parameters. It would be illustrative to show the condition for certain particular cases. Table 5C. 1

200

IONIZATION AND RECOMBINATION OF PLASMA

TABLE 5C. 1 The critical level p\ for establishment of partial LTE in an ionizing plasma. Numbers in brackets denote powers of 10. 4 21 3 (a) 6 = 3.2x 10 K, 77 =10 mz=l z=2 z = 26 6IB(1) = 8.53[1] 6m(l) = 3.78[l] 6m(l) = 2.58[l]

eq. (5C.12)

10 104 106

2.6 6.4 13.2

3.2 6.8 14.8

b(l)

4 23 3 (b) e = 3.2x!0 K, 77 = 10 mz=l z =2 z = 26 6 m (l)=1.33[0] 6 m (l) = l-22[0] Ml) = l-71[0]

eq. (5C.12)

10 103 10s

2.2 4.9 10.1

2.2 4.7 10.0

b(l)

21 3 s (c) 6 = 5.12x 10 K, 77 =:10 mz=l z =2 z = 26 Ml) = l-49[2] 6 m (l)=1.35[2] Ml) = l-31[2]

eq. (5C.12)

10 10s 107

4.2 9.4 21.5

4.7 10.0 21.7

b(l)

5 23 3 (d) e = 5.12x!0 K, 77 =: 10 m" z=l z =2 z = 26 Ml) = 2.09[0] Ml) = 2.12[0] Ml) = 2.24[0]

b(l) 2

3

10 103 10s

2.0 4.4 9.8

2.9 7.2 15.1

2.2 5.3 11.1

4.3 9.7 22.5

2.0 4.5 10.1

3.2 7.3 16.2

2.3 5.3 11.6

4.3 9.8 22.6

2.0 4.5 10.1

eq. (5C.12) 22 4.7 10.0

5 19 3 (e) e = 2.56x!0 K, 77 =10 mz=l z =2 z = 26 Ml)=1.88[4] Ml) = 2.12[4] Ml) = l-82[4]

eq. (5C.12)

10 107 109

7.0 15.6 36a

10.0 21.7 46.8

b(l)

4 23 3 (0 O = 1.6x 10 K, 77 =10 mz=l z=2 z = 26 Ml) = l-60[0] 6 m (l)=l-25[0] 6 m (l) = l-15[0]

eq. (5C.12)

10 103 10s

2.7 6.0 12.2

2.2 4.7 10.0

b(l) s

1

Extrapolated.

7.2 16.5 39.0

2.6 6.5 13.7

7.3 17.0 41.0

2.7 6.5 14.5

LOCAL THERMODYNAMIC EQUILIBRIUM

201

gives several examples of the lower boundary level p\. The boundary determined from the numerical calculations for z= 1, 2, and 26 are compared with eq. (5C.12) Complete LTE In the text we examined the dependences of SCR and «CR on «e, and concluded that nf+ gives the region in which these rate coefficients take their high-densitylimit values. Thus, nf+ gives the lower boundary of complete LTE, perhaps except for a small numerical factor as discussed below. This boundary was given from eqs. (5.19) and (5.20), the comparison between the radiative and collisional transitions terminating on the ground state. This boundary density is larger than Griem's boundary for/> 0 = 2. In fact, in the discussions after eq. (5.21a), we have seen that ri^+ almost coincides numerically with Griem's boundary, eq. (4.29a) extended to/>Q= 1- Figure 5C.6 shows the lower boundary ?ys for complete LTE as determined from numerical CR model calculations for z= 1, 2, and 26. A substantial difference for different z is seen. Our numerical expression includes z

FIG 5C.6 Boundary electron density for establishment of complete LTE for an ionization balance plasma. The thick curves show the results of numerical calculation. : z = 1; : z = 2; : z = 26. The thin curves are : eq. (5C.13) with eq. (5C.14); — .. — . . — : 10,4(2, I)/ ^(2.1); — . — .—: 10/3(l)/a(l). (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.)

202

IONIZATION AND RECOMBINATION OF PLASMA

with

Figure 5C.6 compares the above formula with the result of the numerical calculations. In this figure the boundaries determined by eq. (5.19) and eq. (5.20) with p = 2 are given, where a factor 10 has been multiplied to be consistent with the present definition of LTE, eq. (5C.2). The plasma density of Fig. 1.7 is barely outside of the above criterion for complete LTE. In this case, however, the plasma is optically thick toward Lyman lines which terminate on the ground state (see Fig. 1.6), and the effective A coefficients are substantially reduced in eq. (5.20). Therefore, the above criterion should be relaxed substantially. Boltzmann equilibrium A short remark is given here on the condition for the establishment of the Boltzmann distribution, eq. (2.3), of an excited-level population with respect to the ground-state atom density. The case of an ionization balance plasma has been examined already. We now consider an ionizing plasma. The population kinetic scheme should be case (c) in Fig. 5C.3, with the chain of this relationship continuing down to the ground state. We have seen an example in the close of Section 4.2, Fig. 4.11. In this example, in the limit of high density the Boltzmann distribution with the ground state is established up to p = 5. We remember that this upper boundary for the levels comes from the difference between the scheme (c) and scheme (a) in Fig. 5C.3. Thus, for this distribution to be realized, the density should be high and the temperature should be low, so that r^p)^ 1. When we remember that the condition for complete LTE comes from eq. (5.20), the same condition for density, eq. (5C.13), applies to the present problem. For temperature, the left-bottom corner of Fig. 5C.1 gives an approximate region for this distribution. More quantitatively, the upper bound given by the thin dotted line is reduced by a factor of 2, i.e. p < 5 for O = 103 K and p = 2 for O = 104 K. This factor of 2 again partly comes from the present definition of LTE. If we examine these parameters it is concluded that an ionizing plasma having a high density, eq. (5C. 13) and low temperature as given above is too extreme to be practically possible. Appendix 5D. Optimum temperature, emission maximum, and flux maximum For an ionization balance plasma we introduced the optimum temperature, i.e. the temperature at which the ionization ratio [nz/n(l)] is unity. As Fig. 5.16 shows the emission line intensity is strong at around this temperature (1.35eV). But if we look at this figure more closely, we realize that the temperature at which the intensity takes a maximum is slightly higher in this example of neutral hydrogen. Furthermore, the ionization-recombination flux takes a maximum at still higher

OPTIMUM TEMPERATURE

203

temperature (1.7 eV). At this latter temperature [nz/n(lj\ is 11.5, much larger than 1. We consider this problem now. As in Fig. 5.16, we pose a constraint,

In the ionization balance plasma the ionization ratio is given by eq. (5.28):

First, we consider the ionization-recombination flux, ScRn(l)ne = acRnzne, and look for a temperature at which this flux takes a maximum:

It is straightforward to rewrite eq. (5D.3), under the constraint of eqs. (5D.1) and (5D.2), as

Figure 5.8 shows SCR and «CR as functions of Te. For low density, SCR is well approximated by S(l), eq. (5.6), which may be given by eq. (3.35a). It is straightforward to show that the slope of S(l) against Te is given by R/kTe, with the neglect of the small Te dependence of G. Figure 5.16 shows that the flux maximum takes place at Te= 1.7eV. The slope at this temperature should be 13.6/1.7 = 8. However, the approximation eq. (3.35a) is too crude for the present quantitative discussion. See Fig. 3.11(c): eq. (3.35a) corresponds to the dashed straight line. Actually, the slope of SCR is about 10-12 as seen in Fig. 5.8. The slope of «CR is about —1. See also eq. (5.15) and Fig. 5.3. Thus eq. (5D.4) gives the ionization ratio [nz/n(l)] of 10-12. This is consistent with the value 11.5 as we have seen in Fig. 5.16. As Fig. 5.15 suggests the emission efficiency of ionization, or the number of photons emitted per ionization event, has a substantial temperature dependence at low temperature: the efficiency is higher at lower temperature. The emission efficiency of recombination has a small but opposite temperature dependence. These facts make the peak of the emission line intensity shift slightly to a lower temperature than the peak of the ionization-recombination fluxes. We have now to consider the problem of whether the emission maximum can be lower than the optimum temperature or not. For this purpose, we consider, at the optimum temperature Teo at which nz = «(1), whether the emission line intensity has a positive slope or a negative slope. Under the same constraint, eq. (5D.1), we consider the temperature derivative of [C(l, 3)n(l)]. We assume that, in the vicinity of Teo ~ 1.35 eV, the excitation rate coefficient is proportional to (Te/Teo)^ and the

204

IONIZATION AND RECOMBINATION OF PLASMA

ionization rate coefficient is proportional to (Te/Teo)a, Then, it is rather straightforward to show that if 2/3/ (a + 1) is larger than 1, the slope of the line intensity is positive. From the above argument and from Fig. 3.1 l(c), it is obvious that both the slopes, a and j3, of the rate coefficients have similar magnitudes of around 15. Thus, the line intensity has a positive slope at Teo so that it should take a maximum at a temperature higher than the optimum temperature. Thus, we are able to understand the slight differences among the optimum temperature, the emission maximum, and the flux maximum. We have examined above the example of neutral hydrogen. As we noted in Section 5.4 (Fig. 5.9), the optimum temperature, Teo/z2, becomes high for high-z ions. For z = 26, hydrogen-like iron for example, Fig. 5.9 gives Teo/z2 ~ 2 x 105 K. In Fig. 5.8, at this reduced temperature the slopes of SCR and «CR are approximately 1 and —1, respectively. Equation (5D.4) suggests that, in this case, the optimum temperature and the maximum flux temperature coincide with each other. We further note that j3 for the excitation rate coefficient is around 1 and a is already noted above. Thus, the slope of the emission intensity is about null. Then, the emission maximum also coincides with the optimum temperature. References

The discussions of this chapter is based on: Fujimoto, T. 1979a /. Phy. Soc. Japan 47, 265. Fujimoto, T. 1979b /. Phy. Soc. Japan 47, 273. Fujimoto, T. 1980a /. Phy. Soc. Japan 49, 1561. Fujimoto, T. 1980b /. Phy. Soc. Japan 49, 1569. Fujimoto, T. 1985 /. Phy. Soc. Japan 54, 2905. Fujimoto, T. and McWhirter R.W.P. 1990 Phys. Rev. A 42, 6588. Hirabayashi, A., Nambu, Y., Hasuo, M., and Fujimoto, T. 1988 Phys. Rev. A 37,77. Appendix 5A is based on: Sawada, K. and Fujimoto, T. 1994 Phys. Rev. E 49, 5565. Appendix 5C is based on: Fujimoto, T. and McWhirter R.W.P. 1990 Phys. Rev. A 42, 6588. Appendix 5D is based on: Goto, M., Sawada, K., and Fujimoto, T. 2002 Phys. Plasmas 9, 4316.

6

CONTINUUM RADIATION Figure 1.3, which was a spectrum of neutral helium in a recombining plasma, shows continuum radiation, underlying the series lines, or in the shorterwavelength region, extending from the series lines. Figure 1.7 also shows a prominent continuum. In this chapter we investigate the characteristics of the continuum radiation. 6.1 Recombination continuum We consider the radiative recombination process as schematically depicted in Fig. 6.1, which is essentially the same as Fig. 3.7; an electron having energy e is captured by an ion in its ground state "1" in ionization stage z, and an ion (atom) in levelp is formed in ionization stage (z—1). A photon is emitted which carries the released energy away. Since, in a plasma, the energy of the electrons are distributed over the continuum from zero to high energy, the energies or the frequencies of the emitted photons are distributed from the threshold to high energy (frequency). Thus, this spectrum is continuous, extending from the threshold to higher frequencies. We call this continuum radiation the recombination continuum, In Section 3.2, we introduced the radiative recombination cross-section, crej,(e), and its explicit expression was given by eq. (3.18) for the case in which a fully stripped ion captures an electron to produce a hydrogen-like ion after recombination. The radiated power of the recombination continuum in

FIG 6.1 Radiative recombination of an electron having energy e with the groundstate ion z to form an "atom" in level p. A spontaneous transition q^>p is also shown. Level q is allocated the energy width /zAz/.

206

CONTINUUM RADIATION

FIG 6.2 (a) Schematic illustration of the recombination continuum and the accompanying line emissions. The intensity of the line q—>p, fP^(i/)di/, is replaced by P^(v)/^v. (b) Boltzmann plot of the discrete-level populations and its extension to the continuum electrons. The three points in negative energy correspond to the three emission lines in (a) and the point in positive energy corresponds to the recombination continuum in (a). frequency width dz/ from a plasma having ion density nz(l) and electron density «e is given by

where/(e)de is the energy distribution function of the continuum electrons and the energy width de is equal to hdv. See Figs. 6.1 and 6.2(a). In the literature, instead of the radiative recombination cross-section, the photoionization cross-section is given as standard data. By using Milne's formula, eq. (3.17), we rewrite eq. (6.1) in terms of the photoionization cross-section, where the Maxwell distribution,

CONTINUATION TO SERIES LINES

207

eq. (2.3a), is assumed for/(e)de:

The recombination continuum radiation for recombination of a fully stripped ion z to form a hydrogen-like ion (z— 1) in level p is explicitly given from eq. (6.2) and the photoionization cross-section, eq. (3.13):

with

In deriving the above equation (6.3) we have used the relation hv = e + z2R/p2, See Figs. 6.1 and 6.2(a). Thus, on the assumption that the Gaunt factor is unity, the slope of the recombination continuum spectrum gives Te. 6.2 Continuation to series lines The radiated power of a transition line /> is given as (see Figs. 6.1 and 6.2(a))

We consider the radiated power averaged over the frequency width Az/ that corresponds to the energy "territory" allocated to level q. See Figs. 6.1 and 6.2(a). Then, the middle expression of the above equation may be written as /^(z/jAz/, where P^(y) is the line intensity averaged over Az/. We now remember that the transition probability is expressed in terms of the absorption oscillator strength, eq. (3.1), and that the latter is related to the photoabsorption cross-section through eq. (3.9):

In the same spirit as above, we replace the l.h.s. of this equation by the averaged cross-section (crpiq(y)} times Az/. Similar quantities have been considered already in deriving eq. (3.9b) for the case of hydrogen-like ions. The transition probability is then given as

208

CONTINUUM RADIATION

If the upper level q is in LTE, its population is given by eq. (2.7) or eq. (2.1 a):

By substituting eqs. (6.6) and (6.7) into eq. (6.5) we readily have

This quantity is expressed by the square in Fig. 6.2(a). This square has the same area as the hatched area of the emission line. Equation (6.8) is identical to eq. (6.2) except for the exponential factors and the cross-sections. The difference between dv and Az/ is trivial. As suggested from the similarities between eqs. (3.9) and (3.10a), with an increase in energy of the upper level, from the discrete levels into the continuum, the photoabsorption cross-section, (crpiq(v)}, would change smoothly across the ionization limit to the photoionization cross-section apf(v). We saw an example of this smooth transition in Fig. 3.6. See also eqs. (3.9b) and (3.13), and Fig. 3.3 for the Gaunt factors for hydrogen-like ions. In an emission spectrum like Fig. 1.3, we find isolated lines, eq. (6.5) or eq. (6.8), at low energies, the recombination continuum, eq. (6.3) or eq. (6.2), at high energies and, between them, a transition from the line to continuum, or even the quasi-continuum, near the series limit. Thus, we may conclude that, if all the above assumptions are justified, the emission intensity of series lines, averaged over the frequency width, changes continuously along the series lines toward the quasi-continuum, and across the series limit to the recombination continuum. Since the electrons of the upper levels considered above are loosely bound (E< 0) or have a slightly positive energy (E> 0) they may be affected easily by the plasma environment. Then the atomic properties established for an isolated atom may be modified in the plasma. Thus, some of the above assumptions, e.g. eqs. (3.9) and (6.6) valid for an isolated atom, may be modified in a dense plasma. These points will be addressed in Chapter 9. Extension of the Boltzmann plot For the purpose of determining Te as well as nz(T)ne of the plasma, level populations per unit statistical weight, nz_i(q)/gz_i(q), are plotted against the energy of the level E(q) or x(<J)- We call this figure the Boltzmann plot. Examples have appeared in Figs. 4.20(b), 4.25, 5.12, 5.13, and 5.19(b). If level q is in LTE, its population is given by eq. (6.7); the slope of the fitted line gives Te and the intercept at x(q) = 0, the ionization limit, gives nz(l)ne. As we have seen above the intensities of emission lines continue smoothly across the ionization limit to the recombination continuum. We should thus be able to extend the Boltzmann plot into the transition region including the

CONTINUATION TO SERIES LINES

209

quasi-continuum, and further to the recombination continuum. The quantity given in the Boltzmann plot is nz--i(q)/gz-i(q), which is determined from eq. (6.5). In Fig. 6.2(b) three points are plotted in the schematic Boltzmann plot; these points correspond to the three emission lines depicted in (a). If level q is in LTE, eq. (6.5) reduces to eq. (6.8). Thus, nz_\(tf)/gz_\(tf) is expressed as

Likewise, a quantity which corresponds to the above quantity may be defined from eq. (6.2) for the recombination continuum:

In Fig. 6.2(b) a point is indicated which is calculated from the value of Pf(v) in (a) through eq. (6.10). The above procedure is followed in analyzing Fig. 5.18(b) and the "populations" of the continuum states are plotted on Fig. 5.19(b). They actually lie on the straight line, and the determination of Te and nz(l)ne, with the recombination continuum being taken into account, is more reliable than a plot only of the populations of the discrete levels. We consider recombination continua terminating on the Rydberg levels. For the purpose of illustration we consider a hydrogen-like ion. We ignore for a while the accompanying series lines. As Fig. 6.3 shows, for a particular frequency v, we have several lower levels possible for radiative recombination. The lowest level

FIG 6.3 Radiative recombination processes to several levels. pmin is the lowest level relevant to the photon energy hv. The final "level" may be in the continuum state.

210

CONTINUUM RADIATION

pmin is given from

Figure 6.4 shows schematically the recombination continua spectrum. The total radiated power of the recombination continua is given from eq. (6.3):

For the short-wavelength, or high-frequency, region of v > z2R/h, Fig. 6.4 suggests that, in the summation in eq. (6.11), only the first term of /> m i n =l is predominant and other terms may be neglected. For the long-wavelength region of v
FIG 6.4 Radiative recombination continua for several lower levels as given by eqs. (6.3) or (6.11).

FREE-FREE CONTINUUM - BREMSSTRAHLUNG

211

where gbf has been assumed to be unity. It should be noted that this simple eq. (6.12a) is based on the neglect of the series lines, and is a rather poor approximation to the actual spectrum. The same criticism is true for Fig. 6.4: in this calculation only the recombination continuum, eqs. (6.2), (6.3), and (6.11), is included and the accompanying quasi-continuum and the series lines are neglected. However, in real spectra, as seen in Figs. 1.3 and 5.18(b), pure recombination continua as shown in Fig. 6.4 constitute only a part of the spectrum. The sharp threshold is absent. 6.3 Free-free continuum - Bremsstrahlung In Fig. 6.3, if we move the final level p up and go across the ionization limit, we have final states in the continuum. Yet use of eq. (6.11) should be justified if we replace the Gaunt factor gbf by gff, the free-free Gaunt factor. Then, the range of integration of eq. (6.12) is from —oo to 0. By assuming g{{= 1, we have

By replacing the summation on the r.h.s. of eq. (6.11) with eq. (6.13) we obtain the spectral intensity distribution of the Bremsstrahlung continuum PB(i/)di/. A salient feature of the Bremsstrahlung is that the wavelength, or frequency, dependence is, except for the difference in the Gaunt factors, exactly the same as that of the recombination continuum. Now we combine the recombination continuum and the Bremsstrahlung. For the short-wavelength region of v>z2R/h, the continuum intensity is given approximately by

Equation (6.14) indicates that the nature of the continuum depends on the temperature: at very high temperatures of kTe ;$> 3z2R ~ z2 x 40 eV, the second term in the square brackets is larger than the first term. Thus, the Bremsstrahlung dominates over the recombination continuum, and vice versa at low temperatures. For the long-wavelength region of v z2R, the Bremsstrahlung gives the dominant contribution. The reader can gain some ideas in Fig. 1.2 about the temperature range in which the Bremsstrahlung is stronger and the range in which the recombination continuum is stronger. Actually, for magnetically confined plasmas, e.g. "tokamak, helical", the Bremsstrahlung radiation is used for the purpose of determining the density of impurity ions

212

CONTINUUM RADIATION

present in the hydrogen (deuterium) plasma. In the latter range the plasmas that actually emit strong recombination continua are limited in a rather low temperature range. The reason is considered below. From eq. (6.3) the recombination continuum intensity at the threshold is approximately given as

where we have collected all the z-dependences here, including those in /. For a recombination continuum to be strong, the factors on the r.h.s. of eq. (6.15) should be large: i.e. high nz and ne, and low Te. See Fig. 5.18(a) and (b). In laboratory spectroscopy, there are two classes of plasmas which emit strong recombination continua. The first is the arc discharge plasmas which have rather high nz and ne with temperatures about 1 eV or a bit higher. The plasma of Fig. 1.7 happens to be very similar to this class. Another class is the afterglow plasmas. In this case the temperature is rather low, less than 1 eV, sometimes less than 0.1 eV. In such a case, even if nz and ne are rather low we can observe a recombination continuum. Figure 1.3 is an example. In this case nz and «e are about 1020m~3 and lower than those in Fig. 5.18(b) by an order, but Te is about 0.15 eV and lower by a factor of 3, and the recombination continua are strong enough to be observed. It may be interesting to note that the r.h.s. of eq. (6.15) contains the factor z8. See also eq. (5B.8). This suggests that, for the same reduced temperature and density, a recombination continuum from high-z ions tends to be strong in comparison with that from singly ionized ions like the present examples. This may be the reason why the recombination continuum of ions is observed from the recombining laser-produced plasma which is quite small in volume. Reference

A part of the discussions in this chapter is based on: Cooper, J. 1966 Rep. Prog. Phys. 29, 35.

*7

BROADENING OF SPECTRAL LINES Spectral lines emitted by ions (atoms) in a plasma are not strictly monochromatic. Rather, they are broadened and have finite widths. They may even be shifted from the original positions, too. A remarkable example is shown in Fig. 1.7. Other examples are seen in Fig. 1.3: in the long-wavelength region, the 23P-«3D lines are substantially broadened as contrasted to the adjacent accompanying 2 P n S lines. This feature is the origin of the nomenclature "the diffuse series" for the former lines and "the sharp series" for the latter. In this chapter, we examine these phenomena. Since this subject requires theoretical treatment that is quite involved and complicated, and which are outside the scope of this book, we restrict ourselves to a rather elementary level of discussion. In the formulation below, we use angular frequency LO instead of frequency v to express the profile of the spectral line. This choice is more natural, as will be understood below. The symbol n is used to designate a level or the principal quantum number of the level, while /> or q are used to designate a state within a level. Doppler broadening When an emitter of radiation of angular frequency LOO is moving with velocity component v toward the observer, the radiation is detected with a frequency LO, where

When the emitting ions have a Maxwell velocity distribution, eq. (2.2) with m and Te replaced, respectively, by the ion mass Mi and the ion temperature T-v the observed profile of the line has the Gaussian shape

where AwD is called the Doppler (half) width and is given by

which corresponds to the most probable speed vp. See Section 2.1. The full width at half maximum (FWHM) is given as

214

BROADENING OF SPECTRAL LINES

In terms of the wavelength, the FWHM is expressed as

where A0 is the central wavelength. Other kinds of broadening stem from changes, quasi-static or temporal, in atomic states of the emitter ions. Photons that constitute a spectral line are emitted when the ions make a transition from an upper level to a lower level. The energy of a photon is the difference in energies of these levels, to each of which we assign a definite energy, for a while. Let the angular frequency of this photon be UJQ, In a plasma, ions and electrons constituting the plasma produce an electric field, called the plasma microfield, at the location of the emitter ion. The atomic states of the emitter ion are perturbed by this microfield, and so are the energies. In the case of the linear Stark effect, for example, the energies shift in proportion to the electric field strength. Accordingly the photon energy, or frequency, is changed. So, our task would be to understand the characteristics of this microfield induced in collisions; the microfield may be quasi-static or temporally changing, or even a short pulse. Then, the frequency shift and thus the overall spectral line broadening would result as the collection of responses of the ions to these perturbations. 7.1 Quasi-static perturbation Holtsmark field Suppose the temporal changes of the plasma microfield are so slow that we may regard this field as quasi-static. We begin with the simplest model, the binary approximation, of this field. The assumptions are: 1. perturbers are uncorrelated among them and with the emitter, so that they are distributed randomly in space; and 2. only the perturber located nearest to the emitter exerts the field on the emitter and no other perturbers affect it. Let the perturber density be N, The probability that a perturber is found in a small volume element dv is given as

The probability that no perturbers are found within the same volume is The corresponding probability for a finite volume v is

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215

The probability that no perturber is found in a sphere of radius r centered at the emitter and that one perturber, the nearest one, is found in the spherical shell with inner and outer radii (r, r + dr) is

The average distance between the adjacent perturbers pm is given by

The probability dP(r) of finding the nearest perturber at r is expressed in terms of An,

The strength of the field at the emitter exerted by a perturber at a distance r having a charge Ze is

The normal field strength is defined as

The probability that the emitter is subjected to field F is

or

with (3 = F/Fo, and J0°° WE((3)d(3 = 1. Strictly speaking, dP(F) in eq. (7.9) is not the same as dP(r) in eq. (7.7) so that a different notation should be used. However, for the sake of avoiding unnecessary complications, we use the same notation here. In actual situations, in which the binary approximation above is not valid, the field at the emitter is the vector sum of the fields produced by all the perturbers. If we take this fact into account, or if we remove assumption 2 above, the field

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distribution function W^((3) is modified, especially for small (3 where many distant perturbers contribute to the field at the emitter. The Holtsmark distribution applies to this situation:

In this case, a slightly different (0.2%) definition of the normal field strength from eq. (7.8) is adopted:

Figure 7.1 shows this distribution function as the curve with a = 0. The quasi-static distribution applies mainly to the fields produced by ion perturbers. The ions repel each other by the Coulomb repulsion, and assumption 1. above will be violated, too. The Holtsmark distribution should thus be modified. Taking the ion correlation into account automatically includes the shielding of the perturber charges by the plasma particles, the Debye shielding. Calculations of the distribution functions including these effects are reported in literatures.

FIG 7.1 The Holtsmark field distribution (for a = 0). /3 = F/F0, where F0 is the normal field strength, eq. (7.8). Distribution functions with the ion correlation effects being taken into account are shown for a > 0, where a = pm/Ri). Here pm is defined by eq. (7.6), and RD by eq. (7.11). (Quoted from Hooper Jr. 1968; copyright 1968, with permission from The American Physical Society.)

FIG 7.2 Linear Stark shifts of hydrogen levels of n = 2 and 3, and the Balmer a transition lines connecting these levels. The solid and dotted lines connecting the Stark split levels indicate, respectively, the TT and cr components, where, when observed from the direction perpendicular to the applied electric field, the former components are polarized in the direction of the electric field and the latter perpendicular to it. The relative intensities and positions of these lines are given, and they are compared with the experimentally observed Stark spectrum. In this calculation, the fine structure splitting is included for the « = 2 levels. (Calculated by M. Goto, and the experiment quoted from Mark and Wierl, 1929.)

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An example is shown in Fig. 7.1, where the parameter a = pm/Ro represents the strength of the shielding. Here, the Debye radius is given as

It is noted that Z= 1 is assumed in this calculation. Another effect appears when the emitter is an ion, instead of a neutral atom as is implicitly assumed so far. In this case, the emitter repels perturber ions, again violating assumption 1. The distribution functions for z = l are found slightly shifted (by about /3^0.1) to the lower field strengths than those in Fig. 7.1. As has been noted in the last paragraph before Section 7.1 the energies of the upper and lower states of an emitter are shifted by the Stark effect. In the case of the linear Stark effect, the energy shift of the states is proportional to the field strength, and thus each of the Stark components of the spectral line shifts accordingly. An example of the linear Stark splitting is shown in Fig. 7.2 for the case of the hydrogen (2-3) line, i.e. the Balmer a line, in a homogeneous electric field. Since, in a plasma, the field strengths are distributed, the spectral line is broadened. In the case of the linear Stark effect the line profile directly reflects the field distribution function, e.g. Fig. 7.1. We will examine this problem further in detail in Section 7.4. 7.2 Natural broadening We start with a classical electric dipole oscillator having resonance frequency UJQ as the model of the emitter ion. The radiation emitted by this oscillator is given by exp(iw0?). If the oscillation is stationary, it is obvious that this spectral line is monochromatic. In reality, however, the atomic system interacts with the radiation field or the vacuum, and excited atomic states are no longer strictly stationary. Rather, they make spontaneous transitions, and the oscillator and thus the radiation decay with time. We express this damped oscillator by

where 7 represents the decay rate of the energy of the oscillator. The spectral line profile is given from the Fourier transform,

We readily obtain a profile having a Lorentzian distribution

where the profile has been normalized to 1.

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219

In the last paragraph before Section 7.1, we started with the assumption that definite energies are assigned to the upper and lower levels of the transition. At this point, we remove this assumption: at least the upper level has an energy width corresponding to eq. (7.14). 7.3 Temporal perturbation - impact broadening Autocorrelation Junction We assume a stationary atomic oscillator. This oscillator may be perturbed by collisions with plasma particles. Owing to the Stark effect, the electric field induced at the emitter during a collision gives rise to a frequency shift Aw, the magnitude of which depends on the field strength and the response of the oscillator to the electric field. In the course of the collision event, the electric field increases from zero, takes a maximum and decreases. The accumulation of the frequency shifts over the duration of the collision results in a shift of the phase from the original phase in the absence of the collision. Let rj(t) be the total phase shift accumulated from time 0 to t. Then the state of the oscillation at time t is expressed by

The line profile is given by the Fourier transform similar to eq. (7.13),

Since the collisions are random (stochastic) and stationary, the Fourier transform can be replaced by another Fourier transform:

where we have defined the autocorrelation function

It may be interesting to see that, if we apply the above procedure to the case of natural broadening, i.e. eq. (7.12), we obtain the same profile as eq. (7.14). In the autocorrelation function, we eliminate the e1"0' factor, to have (j>(s):

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The time average, eq. (7.19a), can be replaced by an ensemble average over statistically identical atomic oscillators, which is expressed as ( • • • ) :

A: Adiabatic collisions -perturbation to the phase We assume the situation that the duration of a collision is short compared with the time interval between successive collisions, or the mean free time. Then, the details of the temporal change of the frequency shift Aw during the collision are insubstantial. Rather, the dominant effect of the collision would come from the resultant phase shift brought about by the collision. The emitter experiencing successive collisions is subjected to the phase shifts over these collisions. We now calculate the autocorrelation function <j)(s). For an increase As in the time interval s we have a change in <j)(s)

where A?y is the additional phase shift induced by collisions during the time interval As. Since collisions are assumed statistically independent, A?y is independent of TJ. Then, we have

Let P(p, v)dpdv be the probability of collisions in unit time which induce the phase shift 77, where p and v are, respectively, the impact parameter and the relative speed of the collision. Here the impact parameter is defined such that the target ion is assumed stationary, and the perturber particle is incident along a straight line. The distance of this line to the target is called the impact parameter. Then, we have

where f(v) is the velocity distribution function like eq. (2.2). We now rewrite eq. (7.23) by introducing cross-sections,

By changing A to d in eqs. (7.22) and (7.23a), we integrate eq. (7.22) to obtain

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221

By substituting eq. (7.24) into eq. (7.17) with eq. (7.19), we readily obtain

with and

We have again arrived at the Lorentzian profile with the FWHM 7. In the present case, however, the central frequency is shifted by A. We now examine the physical meaning of eq. (7.26). The cross-sections are written down from eq. (7.23); namely, for the real part

where, for the purpose of simplicity, we have omitted the velocity dependence and P(p, v). For the purpose of illustration we assume here that the phase shifts r/ are very large and random. Then the cross-section becomes ar = 2irfp dp. The width is proportional to N, or inversely proportional to the mean free time. These facts suggest that the broadening is caused by the disruption of the phase of the atomic oscillation, or of the wave train of radiation, by collisions. We may call this the collisional relaxation of coherence. Equation (7.27) correctly accounts for the coherence relaxation by collisions including those with small r/. The shift is given by the imaginary part

We now consider the case of small r/. The sign of 0-j and thus that of A are the same as that of the averaged r/. Their magnitudes are also proportional to each other. If we remember that a phase shift is the accumulation of the frequency shifts over the duration of the collision, a positive phase shift after the collision is induced by positive frequency shifts during the collision, and vise versa. Thus, we may understand that the shift of the line corresponds to frequency shifts averaged over collisions. We may expect that, starting from very distant collisions, or from a very large impact parameter, with a decrease in the impact parameter the phase shift rj induced by a collision would increase. The impact parameter at which the phase shift rj | is one radian is called the Weisskopf radius and denoted by p0. Collisions with impact parameters smaller than the Weisskopf radius may be called strong collisions. In the above illustration of the broadening and the shift, we considered strong collisions and weak collisions, respectively. In this section, we have assumed an atomic oscillator. In reality, the emitted radiation is due to a transition of the ion (atom) from the upper state p to the

222

BROADENING OF SPECTRAL LINES

lower state q. In quantum mechanics, an energy eigenstate p evolves with time according to exp[ix(p)t/K\, where x(p) is the ionization potential of this state. (Remember our convention that x(P) is taken positive.) The phase of the oscillation of the classical atomic oscillator eluj°' corresponds to the difference between the phases of the upper state, \(p)tjh, and the lower state, \(cf)tjh. The optical coherence is defined as the phase correlation between the upper and lower states. The phase change induced to the classical atomic oscillator by a collision, which we considered above, corresponds to a phase perturbation in the upper state or in the lower state or in both of them. In terms of quantum mechanics, the impact broadening is interpreted as the relaxation of optical coherence between the upper and lower states. In the above, we have considered adiabatic collisions; that is, the atomic state of the emitter ion is perturbed during a collision and the ion returns to its original state after the collision. If some other kinds of collisions give rise to relaxation of optical coherence then these collisions also contribute to broadening. We mention them briefly below. B: Collisional transition within the same level - disruption of the phase An atomic level having an angular momentum / is degenerate, or consists of several states having the same energy, with (2J+ l)-fold degeneracy. These degenerate states are called magnetic substates, each specified by a magnetic quantum number mj. In a collision by a perturber particle, a transition may take place between two magnetic substates in the same level. The wave train of the emitted radiation may continue, but the information of the phase may be lost in the transition. This kind of collisions also contributes to the relaxation of optical coherence and thus to broadening. The phase correlation among the magnetic substates is called the Zeeman coherence. Relaxation of Zeeman coherence due to transitions among the magnetic substates constitutes an interesting area of research, but this is outside the scope of this book. C: Inelastic transitions - termination of the wave train In some cases, a collision may result in a change of the atomic level, excitation or deexcitation, from the original upper level or the lower level or even both. In this case, the wave train terminates and the optical coherence is completely destroyed. This kind of collisions leads to broadening of the spectral line connecting the original upper and lower levels. In Section 7.2 and also in Section 7.3, we assumed eq. (7.12) to express the phenomenon of radiative decay of an excited ion. However, the spontaneous transition is also a stochastic process like collisions. Strictly speaking, we have to treat the radiative decay in a similar way to that for collisions. If we proceed in this

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223

way, we will find that the result is the same as the foregoing one, eq. (7.14). It may be noted that the natural broadening is also regarded as due to the relaxation of optical coherence. In this case, the relaxation is equivalent to the decay: the coherence decays along with the upper-level population. Criterion between quasi-static broadening and impact broadening In Section 7.1 we introduced the quasi-static field which would result in line broadening, and we have been concerned with the impact broadening in this section. We now consider which aspect, quasi-static or impact, is more important in particular situations. As we have seen in eq. (7.26) or eq. (7.27), roughly speaking, the impact broadening width 7 is given by the mean free time r between successive collisions,

For the purpose of comparison of the relative importance of the two broadening mechanisms, it would be natural to compare 7 with the quasi-static broadening width, say Aw. We now introduce the parameter h

We can conclude that, in the case of h 1 quasi-static broadening is dominant. For impact broadening we may assume that r ~ Pm/v, where v is the average relative speed of the perturber. We have noted in deriving this relationship the fact that the perturbers are charged particles, and the effects are long-range Coulomb fields. For the quasi-static perturbation, the electric field strength at the emitter is proportional to r~ , where r is the distance to the nearest perturber. A typical value of r would be pm. The frequency shift in the linear Stark effect is proportional to the electric field strength and in the quadratic Stark effect it is proportional to the square of the field strength. Thus, we may write

The statement that a collision with impact parameter of the Weisskopf radius p0 induces a phase shift of 1 radian may be expressed as

where we introduce the typical value of the frequency shift Aw\y for this Weisskopfradius collision. We may further assume that

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Combining the above approximate relationships, we rewrite eq. (7.29) as

or by using eq. (7.6)

In the above, small numerical factors have been dropped. The quantity p^N is a measure of the number of perturber particles inside the sphere of Weisskopf radius. Thus, the criterion between the dominance of impact broadening, K 3 1, has another interpretation. 7.4 Examples Hydrogen-like ions - linear Stark effect Hydrogen-like ions, with which we have been mainly concerned in earlier chapters, have energy-level structures in which different / states having the same principal quantum number have almost the same energy. See Fig. l.ll(b). We will ignore here small energy separations from the fine structure splitting which is explicitly shown for n = 2 states in Fig. 7.2. The one-electron system with nuclear charge z in an electric field is analytically solved in parabolic coordinates. The quantum numbers specifying the states are: n, the principal quantum number; KI and HI, the parabolic quantum numbers; and m, the magnetic quantum number. An example of the energy splitting of levels and the Stark splitting of the spectral line is shown in Fig. 7.2 for neutral hydrogen n = 2 and 3 levels and the Balmer a line connecting them. Suppose we apply an electric field F in the z-direction. The energy shift of a Stark state p is given by the first-order perturbation from the interaction Hamiltonian, H' = er • F, and the unperturbed wavefunction f/>°,

where (p \ z \p) is the matrix element, or the expectation value, of the z-component of the electron position vector r in the direction of the electric field. See eq. (3.3) and Table 3.1(a). Since KI + n2 < n — 1 (n l5 «2 > 0), the maximum shift is given by eq. (7.30) with (n\ — «2) replaced by (n — 1). For the purpose of a rough estimate, we assume large n and adopt for a typical frequency shift

EXAMPLES

225

It may be interesting to remember that (n2a0/z) is the radius of the classical orbit, eq. (1.2). We now consider eq. (7.29). Suppose F is given by F0 of eq. (7.8) and T ~ Pm/v. With the neglect of a small factor we have

It is readily seen that, in many cases of practical interest, we have for ion perturbers hi ;$> 1. Thus, the quasi-static picture is relevant for ion broadening. On the other hand, for electron perturbers, we have he
with the parameter

where n and n' are the principal quantum numbers of the upper and lower levels. The profile of TH(X), which will be called the Holtsmark profile, is shown in Fig. 7.3 as y = Q; this function is normalized to 1. The far wing is proportional to (v — uj0)~3- This is the direct consequence of the Holtsmark distribution, Fig. 7.1 (a = 0). It is seen that the FWHM is AwH = &ABF0, or

It is noted that SFQ is essentially the same as Aw of eq. (7.31). This line profile is on the assumption that the Stark shifts are distributed rather uniformly over the

226

BROADENING OF SPECTRAL LINES

FIG 7.3 The Holtsmark profile function for ion broadening TK(x), eq. (7.33), and the Stark profile function T(x,y), eq. (7.38), of hydrogen-like ions, where TK(x) is identical with T(x, 0). (Based on Sobelman et al., 1981.) frequency. The Balmer a line, as shown in Fig. 1.7, approximately meets this requirement (see Fig. 7.2). However, the Balmer (3 line as well as the Lyman (3 line, for example, lacks the unshifted component in the Stark splitting. Thus, they show a central dip in the profile, which is the signature of ion broadening. This feature is barely seen in Fig. 1.7. Electron broadening As we have seen, impact broadening is relevant to electron collisions. In this case, however, the dominant broadening mechanism is non-adiabatic, i.e. transitions among the Stark states in a level, similar to class B in Section 7.3. Since the perturbation is of rather long range, collisions with large impact parameters give the main contribution, where the upper bound of the range of perturbation is limited by the Debye shielding. The first-order perturbation is sufficient, and the magnitude of the perturbation is closely related with the spatial extent of the wavefunctions as was the case in the ion broadening above. The resulting line profile turns out to be of Lorentzian shape, except for the line center where the positions of the Stark split components of the line are located at different frequencies. The broadening parameter 7e corresponding to the FWHM of the Lorentzian profile is given by

EXAMPLES

227

where the Weisskopf radius is approximately given by

and RD is the Debye radius given by

Here /(«,«') is a coefficient representing the strength of the interaction, or a parameter related with the spatial extent of the wavefunctions. The definition of the Debye radius here is slightly different from eq. (7.11) besides the allowance for multiply charged perturbers. The term 0.215 accounts for the contribution from strong collisions having impact parameter smaller than p0. For transitions with large values of n and a small value of n', I(n, n') is approximately given by n4/z2, The actual broadening is a combination of ion broadening and electron broadening. The resulting profile may be represented by modifications of the Holtsmark profile, eq. (7.33), with the parameter ^/e/SFQ,

A few examples of this profile are shown in Fig. 7.3, where the case of ^/e/SFQ = 0 corresponds to the Holtsmark profile. For ^/e/SFQ ;$> 0, electron broadening becomes dominant, and the profile tends to Lorentzian. For moderate-density plasmas, it turns out that the electron broadening is rather unimportant, i.e. 7e/#F0< 1- This conclusion may look inconsistent with our assumption of /2e<§; 1 below eq. (7.32). This is because of the gross underestimate of 7 ~ v/pm as an approximation of 7e as given by eq. (7.36) in these situations. Figure 1.7 shows an example of Stark-broadened hydrogen lines in a very dense plasma. In the case of the Balmer j3 line, for example, ^/e/SFQ is about 3, and the electron broadening is dominant. Still the central dip remains, which is the signature of the ion broadening as noted before. For lines of principal importance, e.g. lower members of the Lyman and Balmer series lines, detailed calculations and comparison with experiments have been reported. The lines in Fig. 1.7 are fitted by these detailed data. Readers are referred to the relevant literature. Non-hydrogen-like ions - quadratic Stark effect For non-hydrogen-like ions in which different / levels have different energies, the effect of an electric field is different from that for degenerate hydrogenlike ions, i.e. the quadratic Stark effect. Figure 1.4 shows an example of the

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FIG 7.4 The Stark effect for neutral helium lines of (a) the 21P-41S, P, D, F transitions and (b) the 23P^3S, P, D, F transitions. The transition strength is expressed as the brightness of the traces. Since the lower levels are almost unaffected, the shifts and the intensity changes are due entirely to the perturbation to the upper levels. (Calculated by M. Goto.) non-hydrogen-like energy-level structure, i.e. neutral helium. Figure 7.4 shows an example of the Stark shift patterns of lines; i.e. for neutral helium (a) the 21P-41S, P, D, F lines and (b) the 23P-43S, P, D, F lines. The first-order perturbation gives rise to the wavefunction mixing

where p and q denote the states, and the superscript 0 means the unperturbed state and energy. The interaction Hamiltonian H' = er • F is the same as in eq. (7.30). Thus, the perturbation is strong between levels /<->(/± 1) having a large electric dipole matrix element ( q \ z \ p ) and having a small energy separation. See Table 3.1 (a) for the former (although this is for hydrogen, the radial wavefunctions

EXAMPLES

229

and the matrix elements are much the same for helium) and Fig. 1.4 for the latter. In the present example, for allowed transitions to 2P, the wavefunctions of the upper states, 4S or 4D, could be mixed into the 4P or 4F wavefunctions. Other n levels have smaller "overlap" of the wavefunctions and are energetically too far to affect substantially the n = 4 levels. As a result, the "forbidden transitions", 2P-4P and 2P-4F, become partially allowed. In this figure, the transition strength is expressed as the brightness of the trace. It is actually seen that these transitions become substantially strong in finite field strengths. The intensity is determined by the degree of mixing, eq. (7.39). The 4S wavefunction does not mix substantially with the larger L states because of its large energy distance from these levels. See Fig. 1.4 and eq. (7.39). It is noted that the lower level, 2:P or 23P, is only slightly perturbed, so that the Stark effect in this figure is virtually the perturbations incurred to the upper levels. The energy shift results from the second-order perturbation: the quadratic Stark effect. The energy perturbation to state p is given by

As eq. (7.39a) suggests, the energy of state p is "repelled" by other states that are connected by the dipole matrix element, in the opposite direction. See eq. (3.3). The strength of this repulsion is proportional to the square of the matrix element and inversely proportional to the energy separation between these states. In the example of Fig. 7.4, this feature is seen for the S states, and, to a certain extent, the P states. Note the opposite locations of the 4P state with respect to the 4D and 4F states for the singlet and triplet systems, as seen in Fig. 1.4. When the Stark shift becomes comparable to the energy separation between the states interacting with each other, eq. (7.39a) is no longer valid, and the energy shift deviates from quadratic. Rather the Stark effect tends to the linear Stark effect, as in the case of hydrogen. This transition is seen with the 4:P states. The D and F states have very small energy separations so that the transition takes place at very small field strengths, and the Stark effect virtually starts as the linear Stark effect. We now confine ourselves to the strictly quadratic Stark effect. Since we assume that the electric field is in the z-direction, the matrix element is Fe(q\z\p). The energy shift, or the frequency shift, is given by

By adopting the normal field strength, eq. (7.8), for the typical field strength of F we may rewrite this equation as

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where the energy separation in the denominator has been normalized by the Rydberg constant multiplied by z2. Here, it is remembered that z is the effective core charge felt by the optical electron and that Z is the charge of the perturber. We now calculate the criterion parameter h, eq. (7.29). We assume that the dominant perturbation comes from one level, and that \(q \ z \p)\2 is of the order of (lQl-lQ3)(a0/z)2 as suggested from the integrals of the position vector as shown in Table 3.1(a). In many cases, E(p,q)/z2R is of the order of lO^-lCP2. Thus, we have

For ion perturbers, we have hi < 1 in many cases. For high density and low temperature, hi could be larger than 1. For electrons he is much smaller than one. Impact broadening is important. In the following, for both kinds of perturbers we assume impact broadening. As is easily seen, e.g. in eq. (7.26), impact broadening is approximately proportional to the average speed, so that electrons are the dominant perturbers. In the following we consider only electron broadening. By following a quantum calculation similar to the procedure that led to eq. (7.25), we arrive at the Lorentzian distribution which is exactly the same as eq. (7.25). On the assumption that only one perturbing state q is responsible for the broadening and shift, the width and shift are expressed by

where the parameter j3 is defined by

and the cross-sections are given by

It is readily seen from eq. (3.2) that the first factor of the r.h.s. of eq. (7.43) is \/g(q)/3 \(p\r q) \ in atomic units, which is of the order of lO1^2. See Table. 3. l(a). The multiplication factors /'(/?) and /"(/?) are shown in Fig. 7.5. It is obvious from eqs. (7.42) and (7.44) that the effects of ion collisions are approximately Z4//3(m/A/)1//6 times the effects of electron collisions. Thus, ion

EXAMPLES

231

FIG 7.5 The parameters for the broadening and shift in quadratic Stark broadening. See eqs. (7.42a) and (7.42b). (Based on Sobelman et al, 1981.) contributions are of the order of 10% of the electron contributions. In the case that /?;$> 1, the collision mechanism is adiabatic, class A in Section 7.3, and for /?<§; 1, it is inelastic, class C. In Fig. 7.4 we have seen that, for the S state, the quadratic Stark effect is approximately valid for the whole range of the electric field in this figure. On the other hand, for the P, D, and F states the splitting feature changes with increase in the field strength. Thus, the nature of line broadening is expected to be quadratic or linear, depending on the typical field strength and the line. Since levels with n > 5 have higher / levels, i.e. G, H , . . . , levels, all of which have very small quantum defects, broadening tends to be linear. Figure 7.6 shows an example of experimental observations of Stark broadening. The spectrum is of neutral helium in a plasma produced by shock wave heating. The Stark broadening profiles corresponding to Fig. 7.4(a) and (b) are seen. The normal field strength is of the order of 107 V/m under this experimental condition. The singlet lines are not obvious. The triplet lines show the typical features of the Stark effects: 1. We may regard the 22P—43D, F line to be rather close to the linear Stark broadening as noted above. 2. On the wing of this broadened intense line, the "forbidden" line 23P 43P appears. See Fig. 7.4(b). 3. Another line 23P—43S is only slightly broadened. It is suggested for this line that /2 e l; the ion broadening is quasi-static. The asymmetric profile reflects the frequency shift due to the quadratic Stark effect; see

FIG 7.6 An example of observed Stark broadening. This is for neutral helium lines. The triplet lines show typical features of Stark broadening. This figure is reproduced from photographic film, so that the ordinate is approximately the logarithm of the actual intensities. (QuotedfromOkasakae^a/., 1977, with permission from The Physical Society of Japan.)

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233

Fig. 7.4(b). Unfortunately, however, this spectrum is recorded on photographic film, so that the ordinate is nonlinear, and is time integrated over the plasma formation and decay. Thus, we should restrict ourselves to qualitative discussions. Inglis-Teller limit For hydrogen-like ions, the FWHM is approximately given by eq. (7.35). We consider the situation in which we observe series lines with a small n' with varying n, e.g. the Lyman lines or the Balmer lines. With an increase in n the width increases according to eq. (7.35), while the separation between the adjacent lines decreases according to eq. (1.5). When the line width becomes equal to the line separation, we expect that the adjacent lines cannot be resolved. Rather, these overlapping lines form a quasi-continuum, which was mentioned in the previous chapter. Examples are seen in Figs. 1.7, 1.3, and 5.18(b). Remember here that, although the latter examples are for neutral helium, the upper states are «3D, which are rather close to the hydrogen-like states as noted above. We thus expect to have a certain relationship between the plasma density (ion density) and the principal number n of the last line that is discernible as a line. We call this limit the Inglis-Teller limit. The relationship is given as

where TV is in [m ]. We apply this method first to Fig. 1.7. We assign n = 6. Then we have NK 3 x 1023 m~3. This should be equal to ne, which is quite close to the value derived from the overall fitting of the spectrum. Figures 5.18(b) and 1.3 give the last discernible line with n= 11 and n=\l, respectively. These numbers give values « e «3 x 1021 m~ 3 and 1 x 1020 m~3, respectively. 7.5 Voigt profile As we have seen above, natural broadening and impact broadening, or relaxation of optical coherence, led to a line profile with a Lorentzian shape; see eqs. (7.14) and (7.25). Even the Holtsmark profile is rather close to a Lorentzian profile, as seen in Fig. 7.3. We sometimes encounter situations in which the ions emitting the line radiation with the Lorentzian profile are in thermal motion, having a Maxwell velocity distribution and producing Doppler broadening, eq. (7.2), provided the line were monochromatic. The line profile we observe in this case is thus the result of convolution of the Lorentzian profile with the Gaussian profile. For the purpose of simplicity we rewrite (LJ — LJQ) in eq. (7.14) or (LJ — UQ — A) in eq. (7.25) as uj. The Lorentzian profile is

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The Gaussian profile is

Convolution of eq. (7.46) over eq. (7.47) yields

with the parameter

The profile of eq. (7.48) is called the Voigt profile and a above is the Voigt parameter. Figure 7.7 shows examples of the Voigt profiles along with the Gaussian profile (a = 0) and the Lorentzian profile (a = oo), all having the same HWHM. It is seen that even for small a values the line wing is heavily affected by the Lorentzian profile. This feature has significant consequences in radiation transport of the line as discussed in the next chapter.

FIG 7.7 Examples of the Voigt profiles along with the Gaussian profile (a = 0) and the Lorentzian profile (a = oo), where all the profiles have FWHM = 2.

REFERENCES

235

Note the following: superposition of the Gaussian profiles with the Doppler widths Aw D i, Aw D 2 ,... results in a Gaussian profile with Doppler width AwD, where

Superposition of Lorentzian profiles with FWHM 71; 72,... results in a Lorentzian profile with FWHM

References

The standard textbook of line broadening is: Griem, H.R. 1974 Spectral Line Broadening by Plasmas (Academic Press, New York). The discussions in Section 7.4 are largely based on: Sobelman, 1.1., Vainshtein, L.A., and Yukov E.A. 1981 Excitation of Atoms and Broadening of Spectral Lines (Springer, Berlin). The discussion in Section 7.3 about the relationship between the collision broadening and the coherence relaxation is experimentally verified in: Hirabayashi, A., Nambu, Y., Hasuo, M., and Fujimoto, T. 1988 Phys. Rev. A 37,83. The figures are taken from: Mark, H. and Wierl, R. 1929 Z. Phys. 55, 156. Hooper, C.F. Jr. 1968 Phys. Rev. 165, 215. Okasaka, R., Shimizu, M., and Fukuda, K. 1977 /. Phy. Soc. Japan 43, 1708.

*8

RADIATION TRANSPORT Until now, we have assumed that photons emitted in a plasma leave it without being absorbed. In real situations, photons emitted by atoms (ions) in transition q top (Fig. 3.1) could be absorbed before leaving the plasma by atoms in levelp which are present in some other locations in the plasma. In the case that the population of atoms p is substantial, the effect of absorption cannot be neglected, or it even controls temporal changes and spatial distributions of n(q) in the plasma. In the case that level p is the ground state of atoms or ions, this is especially the case. In this chapter we deal with the phenomena related to absorption of photons. Here the term atom is used instead of ions. This is because, in many cases in which radiation transport is important, atoms play the major role rather than ions. 8.1 Total absorption

Before treating the collective process of emission and absorption of photons in a plasma, we consider a simple situation: a medium containing atoms in level p absorbs white background light by making transition p —> q. A good example of this situation is the Fraunhofer absorption lines in the solar spectrum. For the purpose of simplicity we assume the situation of one dimension as depicted in Fig. 8.1. In the region x > 0 there is a uniform medium. From the —^-direction, radiation is incident on the medium at x = 0 and continues to travel in it. Let the intensity of the incident radiation at frequency v be 7° per unit frequency interval (dj/= Us"1]) per steradian. This quantity is the intensity, or the spectral radiance, introduced at the beginning of Section 2.2. Let the intensity at x be Iv(x). The absorption property of the medium is specified by the absorption coefficient /cjm"1] at frequency v in the absorption line profile of transition p^q. An example of nv is schematically shown in Fig. 8.2(a). The change in the intensity of the radiation in the medium is given by

FIG 8.1 Geometry of one-dimensional radiation absorption. In the region x > 0 we have a uniform absorbing medium.

TOTAL ABSORPTION

237

FIG 8.2 Formation of absorption and emission lines, (a) Profile of the absorption coefficient, (b) Formation of an absorption line. The optical thickness near the line center increases to reach almost complete absorption, (c) A similar plot for an emission line. The black-body radiation with the excitation temperature limits the maximum of the emission intensities near the line center. or

Since we assume a uniform medium eq. (8.la) is readily solved:

Figure 8.2(b) shows examples of the absorption line profiles. The quantity KVX is called the optical thickness or the optical depth and sometimes denoted as rv,

In the case of small optical thickness rv<^_\, which we call the optically thin case, we retain the first two terms of the expansion of the exponential function in eq. (8.2):

This situation is shown in Fig. 8.2(b) for the weak absorption cases. In the opposite case of rv ;$> 1, the optically thick case, the incident background radiation is almost completely absorbed

This situation is seen in Fig. 8.2(b) near the line center for strong absorption cases. The absorption coefficient is given from the absorption cross-section, introduced by eq. (3.9) or (3.9a):

238

RADIATION TRANSPORT

This quantity may be expressed in terms of Einstein's B coefficient and the absorption line profile P'(v), which is normalized to 1 over the line. The line shape of Fig. 8.2(a) is nothing but this profile.

or, corresponding to eq. (3.9),

Thus, the absolute magnitude of j'rvdv is proportional to n(p)fx, Figure 8.2 shows schematically (a) the absorption coefficient with the Voigt profile and (b) the development of the absorption line profile, eq. (8.2), for increasing x, normalized by the background, Iv(x)/I%. For small absorption or small x, the absorption profile is exactly the same as that of nv, eq. (8.3). With the increase in x the medium becomes optically thick near the line center, and the absorption profile becomes broader. In the case of sufficiently large x, eq. (8.5) becomes valid near the line center. Since the Voigt profile has a broad wing coming from the Lorentzian profile as shown in Fig. 7.7, when the medium is optically very thick near the line center, the line develops broad wings where the medium changes from optically thick to thin. The strong absorption lines in Fig. 8.2(b) show this feature. As a measure of the amount, or the strength, of absorption of the line by the medium, the total absorption or the equivalent width is introduced:

which is nothing but the area of the absorption in Fig. 8.2(b). This quantity ma; be expressed in terms of wavelength in units of [nm].

where KA is the absorption coefficient corresponding to nv. The merit of this quantity is that, in practical observations, this is independent of the instrumental function, or resolution, of the spectrometer. Figure 8.3 shows an example of the total absorption against the n(p)fx value, or the thickness of the medium. These curves are called the curve of growth. In the case that the medium is optically thin over the line it is obvious from eqs. (8.4) and (8.3) that the total absorption is proportional to eq. (8.7a) multiplied by the dimension of the medium x,

TOTAL ABSORPTION

239

FIG 8.3 (a) The curve of growth of the equivalent width against the increase in the number of absorbing atoms. 2Aw1/2 is the FWHM of the Doppler broadening, eq. (7.3a), and the parameter a is the Voigt parameter as defined by eq. (7.49). (b) An example of the curve of growth constructed from the observed equivalent widths of the solar Ti and Fe lines. (Quoted from Wright, 1948.) This linear dependence is seen in Fig. 8.3 in the left-bottom corner. With an increase in the medium thickness the total absorption increases. When the medium becomes optically thick at the line center eq. (8.5) begins to apply there, and the total absorption grows more slowly than in the optically thin case (see Fig. 8.2(b)). In such a case, roughly speaking, the total absorption is determined by the separation between the line wings at which KVX is about 1. See Fig. 8.2(b). If the absorption profile is Gaussian, as Fig. 7.7 suggests, the total absorption increases slowly. This corresponds to the case of a = 0 in Fig. 8.3(a), where a is the Voigt parameter, eq. (7.49). The slope is approximately 0.1 against the medium thickness, or the n(p)fx value. As in the case of Fig. 8.2(b), we sometime encounter situations in which the absorption profile has a Voigt profile, eq. (7.48); in this

240

RADIATION TRANSPORT

case the Lorentzian wing becomes important at higher thicknesses of the medium; see Figs. 7.7 and 8.2(b). Then, the slope becomes 0.5, as shown in Fig. 8.3 for several a values. This is the case when the Lorentzian broadening is due to natural broadening, which is unlikely in laboratory plasmas, or Stark broadening with a constant FWHM. In some other cases,/and x are kept constant, and n(p) varies, and further, the broadening of the line is proportional to n(p), i.e. the resonance broadening for the case that level p is the ground state; in these cases the slope is 1 at high densities. This case is not shown, though. The reason why this is so is that, with an increase in n(p), the FWHM grows in proportion to n(p), so that the optical thickness at the line center is independent ofn(p). See eq. (8.7a). Figure 8.3(b) shows an example of observed equivalent widths: the absorption lines are neutral metal atom lines in the solar atmosphere. 8.2 Collision-dominated plasma We now replace the above absorbing medium in Fig. 8.1 with a plasma which is again assumed to be uniform. The ions (atoms) in it may emit line radiation corresponding to the transition p <— q as well. The emission property is expressed by the emission coefficient i]v\Wm^3 sr^1 s]. Equation (8.1) is modified to include the emission process,

OR

Since we assume a uniform plasma eq. (8.10a) is readily solved:

The emission coefficient is expressed as

where P(v) is the emission line profile normalized to 1. In the following we assume P(v) = P'(y)', this is equivalent to assuming complete redistribution of frequency. This assumption is that the atom, when it emits a new photon, has lost its memory of the frequency of the photon it absorbed. We also add to eq. (8.7) the induced emission process to yield

We define the source function

COLLISION-DOMINATED PLASMA

241

which is rewritten from eqs. (8.12) and (8.13) as

In deriving the last line we used the relationships (2.12) between Einstein's A and B coefficients. We express the population ratio [n(p)/g(p)]/[n(q)/g(q)] q in terms of the temperature; see eq. (2.3). We define the excitation temperature Tex for the level p and q populations; see eq. (2.13). Then we have

where we have used the definition of the black-body radiation, eq. (2.14). Equation (8.16) is called Kirchhoff's law. Thus, equation (8.11) is written as

We may replace KVX by TV. In many cases of practical interest, the incident radiation is virtually absent. Then eq. (8.17) reduces to

Figure 8.2(c) schematically shows the development of an emission line with increasing value n(p)fx. This figure corresponds to Fig. 8.2(b). It is readily recognized that eq. (8.18) is complementary to eq. (8.2). In the optically thin case, or rv
If our plasma is optically thin over the line, the profile of the emission line is the same as that of nv which is shown in Fig. 8.2(a). This is also true for the cases of small absorption profile in Fig. 8.2(b). In this case, the total line intensity which is the intensity, eq. (8.19), integrated over the line profile

is proportional to the upper-level population and the dimension of the plasma. This is the situation which we assumed up to the preceding chapters. It should be

242

RADIATION TRANSPORT

noted that eq. (8.19a) is essentially the same as eq. (4.1). Note that we assume onedimensional geometry here. In the optically thick case, or T^> 1, we have

Figure 8.2(c) shows schematically the development of the emission line profile with increasing n(p)fx value. It is noted that, with the increase in the thickness, the peak intensity increases and tends to that of the black-body radiation, eq. (8.20), at the excitation temperature, but it never exceeds that. In actual situations, we sometimes encounter an inhomogeneous plasma - the central part of the plasma may have higher excitation temperature than the surrounding plasma. This is the case for a positive column of discharge plasmas. In such a case we have to integrate eq. (8.10a). For the purpose of understanding what happens in such a case, we consider a simple example: a plasma having two layers as depicted in Fig. 8.4; the layer 1 plasma has thickness x\ and excitation temperature Tl5 which is higher than T2, the temperature of the layer 2 plasma having thickness x2. We observe the radiation emitted by this composite plasma from the layer 2 side. It is easily understood that, even if another low-temperature plasma layer on the left, or the opposite side to the observation direction, were present it would contribute little to the observed radiation, so that the actual plasmas can be well modeled by this two-layer model. We further assume that the absorption coefficients are the same for both layers. This situation is realized with many plasmas when level p is the ground-state atom. An example is a discharge plasma including mercury in the fluorescent lamp. It is straightforward from eqs. (8.18) and (8.17) to obtain Iv = ^(TiXl-expC- KVXI}} exp(-K^2) + BV(T2)[\ - exp(-K^2)].

(8.21)

Figure 8.5 shows schematically the profile of the emission line. The first term is shown with the dashed curve, and the second term with the lower dash-dotted curve. The resulting profile is shown with the solid curve. A central dip develops; this phenomenon is called self-reversal.

FIG 8.4 A model of the geometries of an inhomogeneous plasma.

COLLISION-DOMINATED PLASMA

243

FIG 8.5 The emission line profile from an inhomogeneous plasma. Self-reversal develops. We now consider a more general case where the absorption coefficient and the emission coefficient may change along the line of sight; we integrate eq. (8.10a) to obtain

with

Self-reabsorption Suppose we observe a plasma with an optical system, i.e. a lens and a spectrometer (Figure 8.6). Let the intensity observed at a certain frequency be I\. (If we use a lens and a spectrometer, the quantity we actually measure corresponds to eq. (4.1) with units of [W]. However, we simplify the situation here and deal with the spectral radiance having units of [W m~ 2 sr^1 s].) We place a concave mirror on the opposite side of the plasma and focus the image of the plasma on the plasma itself. The observed intensity with this mirror installed is 72. We assume the plasma to be uniform. Then, this situation is a special case of the situation of Fig. 8.4 and eq. (8.21), where x2 = xi, and T2= T\. In the case that the plasma is optically thin the intensity 72 is, if we neglect the finite reflection efficiency of the mirror and the absorption losses by the plasma container, twice I\. If the plasma is sufficiently optically thick, 72 is equal to I\. See eq. (8.20). In other words, we can determine

244

RADIATION TRANSPORT

FIG 8.6 Geometry for the self-reabsorption experiment.

FIG 8.7 An example of line-absorption A^ against the increase in the medium thickness or the number of absorbing atoms. The abscissa is the optical thickness at the line center for Doppler broadening, KOX = e2n(p)fp!qx/4mcs0AujY)\/fK', see eq. (8.35) later. The parameter a is the Voigt parameter. the optical thickness of the plasma at this frequency from

In many cases, we cannot resolve the profile of an emission line by using, say, a spectrometer. Rather we observe the total line intensity integrated over the profile

RADIATION TRAPPING

245

We define the line absorption in a similar way to eq. (8.24):

An example of line absorption is shown in Fig. 8.7. This is for the Voigt profile; a is the Voigt parameter, eq. (7.49). It is to be noted that, with an increase in the plasma thickness, the line absorption tends to 1 for the pure Gaussian profile (a = 0). For the Voigt profile it never tend to 1. The reason is understood from the explanation already given in Section 8.1. It tends to a constant value, 0.586. In more general cases when the excitation temperature and the absorption coefficient vary over the plasma we can readily generalize eq. (8.25). 8.3 Radiation trapping We consider a medium, not necessarily a plasma, which consists of uniformly distributed atoms having two levels p and q; level p may be the ground state. The geometrical shape of the medium may be an infinite slab, an infinite cylinder or of more complicated shapes. As the initial condition, excited atoms in level q are prepared with a certain spatial distribution. For example, in the case of a cylindrical medium the initial excitation may be limited in the thin axial region: a pencil-shaped excitation. This situation is actually realized when we excite the atoms with a beam of pulsed laser light or with an electron beam. At time zero these excited atoms begin to make transitions p <— q, emitting photons of this transition line. A photon may travel over a certain distance until it is absorbed by an atom in level p at some other location in the plasma. The excitation of the first atom is thus transferred by the photon to another location. This emission-absorption process may be repeated several times. As a result of this chain of processes, the spatial distribution of the excited atoms q is redistributed in the medium. At the same time, photons that do not suffer absorption escape the plasma with a certain probability, and the excitation is lost with a certain rate from the medium. The spatial distribution of the upper level atoms and the loss of the excitation are controlled by these repeated emissionabsorption processes. This phenomenon is called radiation trapping. In the early twentieth century the phenomenon of radiation trapping, or of the migration of excitation, was considered to be analogous to diffusion. The term diffusion is sometimes still used even now. In both cases, excitation in the present context is transferred over space in steps. In each step in the latter (i.e. diffusion) it travels by a distance of the order of the mean free path. By repeating many steps it spreads in space and finally it may escape from the medium. The diffusion coefficient is a parameter to quantify how fast the diffusion takes place. Later it was realized that radiation trapping is very different; we assume a medium having an absorption coefficient of the line like in Fig. 8.2(a) and its optical thickness at the line center over the medium is substantially larger than unity, i.e. an optically thick medium. We assume this situation throughout in the following discussions. Since

246

RADIATION TRANSPORT

the emission probability has the same profile as that of the absorption coefficient, an emitted photon almost always has a frequency in the region of the line core, where the medium is optically thick. This photon is readily absorbed by an atom located close to the original atom. Since the optical thickness is large, even after many repetitions of this process, excitation can spread over only a short distance. So far, the phenomenon is similar to diffusion. In this optically thick medium, however, the far wings of the line are optically thin. In one stage of this emissionabsorption chain, a photon may be emitted in the far wing, even with a small probability. This photon has a small chance of being absorbed in its path in the medium and escapes with a high probability. This escape is almost independent of the location of the photon emission. This is the reason why photons escape the optically thick medium. It is readily understood that the phenomenon of radiation trapping is substantially different from that of diffusion. For radiation trapping we cannot define a quantity like the diffusion coefficient. If we calculate the mean free path of photons it diverges. Radiation trapping is thus essentially a non-local phenomenon and needs a different treatment. From the above argument, we can infer the emission line profile observed from an optically thick medium. As will be seen in the following subsection, the upperlevel population, or the excitation temperature, is high in the central region, where it is difficult for the emitted photons to escape, and low in the periphery region, where the photons easily escape. The profile of the emission line is formed by accumulation of photons that succeeded in escaping the medium. Photons in the optically thin far wing come from all the regions of the medium, so that it reflects the properties averaged over the medium. In the near wing where the optical thickness is KVR^ 1, where R is the typical dimension of the medium, e.g. the radius in the case of a cylinder, the photons carry information collected over the medium. The intensity is high because it reflects the central high temperature. Near the line center, photons emitted in the central region are immediately absorbed. We can observe photons coming only from the thin (thickness d) periphery region of the medium where the optical thickness measured from the boundary is nvd< 1. In this region the excitation temperature is low so that the black-body radiation intensity, eq. (8.20) with eq. (8.16), is low. In other words, we "see" the region of the medium within K^^ 1 from the boundary. As a result we have a line which develops the self-reversal profile, which is similar to Fig. 8.5; Note that the intensity of the strong peaks is closely related with the black-body radiation with the higher excitation temperature. Eigenmode analysis For simplicity we denote the population n(q) as n, n(p) as N, and A(q,p) as A. We assume n<^N, and TV is constant over the volume. The time development of the population n at location r is given by

RADIATION TRAPPING

247

where G(r',r)dr' is the probability that a photon emitted at r' is absorbed at r. Thus the second term represents the production rate of the population n at r from the population at r' through a single emission-absorption process. This function is given by

with p = | r' — r , where T(p) is the transmission probability. This is the probability that a photon traverses a distance p without being absorbed, and is given by

where the emission profile P(v) is defined by eq. (8.12) and KV is given by eq. (8.13). Here again we assume the complete redistribution of frequency, i.e. P(v) = P'(v). The integrand of eq. (8.28) is schematically shown in Fig. 8.8(a) for several cases of p; the area under the curve gives the transmission probability, which is schematically shown in Fig. 8.8(b). The induced emission, the second term of eq. (8.13), has been neglected because we assume low excitation, n(q)
FIG 8.8 (a) The integrand of eq. (8.28) for several cases of the traversing distance p; (b) transmission probability.

248

RADIATION TRANSPORT

FIG 8.9 Examples of the temporal development of the relative population distributions of excited atoms in radiation trapping under optically thick conditions. An infinite cylinder and Lorentzian profile are assumed. The parameter is the time given in units of the time constant of the fundamental decay mode, T0 = (g0A)~l. (Quoted from Golubovskii and Lyagushchenko, 1975.) The solution of the integro-differential equation (8.26) may be expressed as

Equation (8.26) reduces to an eigenvalue problem

This has been solved for several cases. An example is given in Fig. 8.10; the radial eigenfunction
The eigenvalues are arranged in the order g0 < gi < g2 < • • • . In the present example, the eigenvalue for the lowest mode is approximately given by

where KP is the absorption coefficient at the line center and R is the radius of the cylinder. From eqs. (7.46) and (8.7a) we have

RADIATION TRAPPING

249

FIG 8.10 The spatial profiles of the eigenfunctions of eq. (8.29) in several lowest modes. An infinite cylinder with radius r = 1 and the Lorentzian profile. (Constructed from the table of Golubovskii and Lyagushchenko, 1975.) The quantity go is called the escape factor for the fundamental decay mode. Figure 8.11 shows the relative magnitudes of the eigenvalues for higher modes. The above examples of the numerical calculation shown in Fig. 8.9 are interpreted as follows. The spatial profile of the initial excitation is expressed as a superposition of the eigenfunctions. Each eigenmode population decays with its characteristic decay rate. Higher modes have shorter time constants. After a sufficiently long time all of the higher-mode populations have died out, and only the fundamental mode survives. Then, the shape of the population distribution no longer changes, and the population and the emitted radiation intensity decay with the common rate g$A. For the geometry of an infinite slab, the expression of go is slightly different from eq. (8.32), but the relative magnitudes of the eigenvalues for higher modes are almost the same as the case of the cylinder as seen in Fig. 8.11. In the case of a Gaussian profile, eq. (7.47), for an infinite cylinder with radius R, an approximate expression for g0 is given as

where KO is the absorption coefficient at the line center. From eqs. (7.47) and (8.7a) we have

250

RADIATION TRANSPORT

FIG 8.11 Escape factor of the higher modes normalized by that of the lowest decay mode. For diffusion the first higher mode has a rate nine times that of the fundamental mode. Figure 8.11 shows the relative magnitudes of the eigenvalues for the Gaussian profile. Another formula fits more accurately the numerical calculation over a wider range of optical thickness:

Figure 8.12 shows the comparison of eq. (8.34a) with a numerical calculation and also eq. (8.34). We have noted the essential difference of the radiation trapping phenomenon from the diffusion. An important consequence lies in Fig. 8.11. For diffusion, the effective decay rate of the next higher mode is nine times that of the fundamental decay mode, for an infinite slab, for example. Thus, the higher modes would die out rather rapidly. On the other hand, in the case of radiation trapping the decay time constants of higher modes, especially in the case of the Lorentzian profile, are not much larger than that of the fundamental decay mode: gi is only 1.6 times larger than g0. This means that, starting from an arbitrary initial population distribution it takes a long time to reach the final distribution. In Fig. 8.9, it is seen that time t = 4/(g0A) is not long enough to reach the final profile. Escaping photon analysis We treat the situation as assumed in eq. (8.26). Suppose an excited atom is produced at time 0. The probability of that atom to survive at time t and make a

RADIATION TRAPPING

251

FIG 8.12 The escape factor, eq. (8.34a), compared with eq. (8.34) (dashed line) and the result of numerical calculation (solid line, Phelps, A.V., 1958 Phys. Rev. 110, 1362). transition during the time interval dr is given by

where A stands for, as before, the transition probability [s l]. Suppose there are n° excited atoms at t = 0. Among them ao«° atoms are assigned to emit photons that escape the medium without being absorbed. The number of these photons corresponding to eq. (8.36) is

Another fraction a^n0 of atoms emit photons that are absorbed once before escaping the medium. The number of these escaping photons from the medium at t is given by the convolution of a^A exp(—At) with p(t),

This quantity has a maximum att=l/A. Similarly, a2n° atoms emit photons that are absorbed exactly twice before escaping. The corresponding function S2(f) is the result of convolution of a function similar to eq. (8.38) with p(f).

252

RADIATION TRANSPORT

FIG 8.13 Examples of the coefficients an for an infinite cylinder and Gaussian profile. The parameter A0 is (l/k0R) with radius R. (Quoted from Wiorkowski and Hartmann, 1985; copyright 1985, with permission from The Optical Society of America.) Likewise, the number of photons that escape from the medium after being absorbed n times is given as This function has a maximum at t = n/A. The overall photon number function is the sum of the Sn(f)'s:

The set of values of an depends on various parameters, e.g. the geometry and the optical thickness of the medium, the absorption line profile, the initial spatial excitation profile. Figure 8.13 shows an example of the expansion coefficients. Equation (8.41) corresponds to the population function, eq. (8.29), in the eigenmode analysis. After a sufficiently long time, the latter tends to the fundamental decay mode with the effective decay probability g$A. Equation (8.41) should decay with the same effective rate. Thus, for sufficiently large n the coefficients should follow

Appendix 8A. Interpretation of Figure 1.5 We have now reached the point where we can interpret on a sound footing the spectra in Fig. 1.5, which were for ionized helium.

INTERPRETATION OF FIGURE 1.5

253

It is known that the helium plasma of Fig. 1.5 has electron density about 1020 m~3. See the close of Section 7.4. This density is virtually equal to the groundstate ion density, «(1). We assume «(1) = 1020 m~3, the ion temperature 7~i = 104 K, and the radius of the plasma R = Q.Q5 m. Table 8A.I gives, for each transition, the wavelength, the FWHM of the Doppler broadening as given by eqs. (7.3) and (7.3a), where the angular frequency has been converted to frequency, and the FWHM of the Stark broadening as given by eq. (7.35), where z = 2 and Z = l . It is seen that the former broadening is predominant, especially for the lower members of the lines for which the effect of radiation trapping is strong. We neglect the latter broadening in the following discussions. This table contains the absorption oscillator strength, the absorption coefficient at the line center, and the corresponding optical thickness over the plasma radius. We now assume that the effect of radiation trapping is approximately described by the effective reduction of the transition probability by the amount of the escape factor of the fundamental decay mode, go. We use Fig. 8.12 or eq. (8.34a) to obtain the escape factor, which is given in the last column of this table. Table 8A.2 gives for Fig. 1.5(a) and (b) the apparent recorded intensity in relative units, where the obvious saturation effect of the detector for the 1-2 transition in (a) has been corrected for: by assuming the same observed profile in (a) and (b), the peak of the 1-2 line in (a) was enhanced by a factor of 2.1. We correct for the reduction of the observed line intensity due to radiation trapping by dividing the intensity by the escape factor. The result is given in the next column. The transition probability is given by standard tables or partly in Fig. 3.5(a). Finally the upper-level population per unit statistical weight is given. These population distributions for (a) and (b) are plotted in Figs. 8 A. l(a) and (b), respectively. Since the measurement is relative, the populations are given on a relative scale. In these figures, the calculated population distributions for the ionizing plasma component, Fig. 4.5, and those for the recombining plasma component for low temperature, Fig. 4.20(a), are given with the curves. Since the TABLE 8A. 1 For each transition this table lists the wavelength, FWHM of the Doppler broadening, FWHM of the Stark broadening, the oscillator strength, the absorption coefficient and the optical thickness at the line center, and the escape factor for the fundamental decay mode. Transition

A (nm)

2A^1/2 (s ')

A^H (s ')

1-2 1-3 1-4 1-5 1-6 1-7 1-8

30.38 25.63 24.30 23.73 23.43 23.26 23.15

7.04(11) 8.35(11) 8.81(11) 9.02(11) 9.13(11) 9.20(11) 9.24(11)

6.8(9) 1.8(10) 3.4(10) 5.4(10) 7.9(10) l-l(H) 1-4(11)

fi,n 4.16(-1) 7.91(-2) 2.9(-2) 1.39(-2) 7.8(-3) 4.8(-3) 3.2(-3)

k0 (m ')

k0R

1.47(2) 2.3(1) 8.2(0) 3.83 2.12 1.3 0.86

7.35 9.4(-2) 1.15 4.3(-l) 0.41 0.7 0.19 0.8 0.11 0.9 0.065 0.93 0.04 0.97

go

254

RADIATION TRANSPORT

TABLE 8A.2 For each transition this table lists the signal peak (corrected for saturation), the value corrected for the radiation trapping effect, and the upperlevel populations per unit statistical weight. Transition

1-2 1-3 1-4 1-5 1-6 1-7 1-8

(a)

$(«,!)

$/go

15 3.2 1.0 0.3 -

159 7.4 1.4 0.38

(b)

n(n)/g(n)

4.25 0.74 0.34 0.18

$(«,!)

$/go

n(n)/g(n)

4.5 4.6 5.3 6.5 3.4 2.0 0.9

47.9 10.7 7.6 8.1 3.8 2.1 0.9

1.28 1.07 1.87 3.95 3.22 2.84 1.83

experiment is for hydrogen-like helium, the nuclear charge is z = 2. We thus have to compare the experiment with the calculation for reduced electron density «e/27 ~ 1018 m~3. See the scaling law, eq. (5B.1). It is seen that, for both cases, the experimental population distribution fits well into the calculated distributions, being consistent with our speculation made in Chapter 5; Fig. 1.5(a) is the spectrum of an ionizing plasma and Fig. 1.5(b) is for a recombining plasma. In the above collisional-radiative model calculation the effect of radiation trapping is not taken into account. For the transition 1-2, Table 8A.I suggests that the effective transition probability should be reduced by a factor of 10, so that the actual population would be larger than the present calculation by a certain amount. In view of this inconsistency, the agreement of the experimental populations with the calculation is surprisingly good. It is noted that we did not adjust the electron temperature in the calculation. It is Te = 22 x 1.28 x 105 K for Fig. 8 A.I (a) and 22 x 103 K for Fig. 8A.l(b). See eq. (3A.9). In the former case of the ionizing plasma, it is known that the overall population distribution is rather insensitive to electron temperature so long as the temperature is high. So, we cannot say anything definite except that this plasma is a high-temperature ionizing plasma with density at around ne K 1020 m~3. In the latter case of the recombining plasma, roughly speaking, the density is determined at the position of the peak population, p = 5-6, and the temperature is reflected in the population distribution of the higher levels. It is suggested that this latter plasma happens to have Te K 4 x 103 K and ne K 1020 m~ 3 or a bit higher. The Balmer lines in Fig. 1.7 are broadened predominantly by Stark broadening, in contrast to the above. These lines are fitted with the profiles from accurate calculation for the Stark broadening for each line. It is found that the Balmer a line has substantial optical thickness at the line center, and the peak intensity is reduced by about a factor of three under this condition. This reduction of intensity is incorporated in Fig. 1.7.

REFERENCES

255

FIG 8A.1 Population distributions derived from Fig. 1.5. (a) For Fig. 1.5(a). The line intensities, after various corrections, result in the relative population distribution plotted with the closed circles. The curves are the calculated population distributions reproduced from Fig. 4.5 for an ionizing plasma, (b) For Fig. 1.5(b). The curves are reproduced from Fig. 4.20(a) for a lowtemperature recombining plasma.

References

Parts of the discussions in this chapter are based on: Holstein, T. 1947 Phys. Rev. 72, 1212. Holstein, T. 1951 Phys. Rev. 83, 1159.

256

RADIATION TRANSPORT

Mitchell, A.C.G. and Zemansky, M.S. 1934 Resonance Radiation and Excited Atoms (Cambridge University Press, London). Wiorkowski, P. and Hartmann, W 1985 Opt. Comm. 53, 217. The figures are taken from: Fujimoto, T. 1979 /. Quant. Spectrosc. Radial. Transfer 21, 439. Fujimoto, T. and Nishimura, Y. 1985/. Quant. Spectrosc. Radial. Transfer 34,217. Golubovskii, Yu.B. and Lyagushchenko, R.I. 1975 Opt. Spectr. 38, 628. Wright, K.O. 1948 Pub. Dom. Astrophys. Obs. 8, 1.

*9

DENSE PLASMA When ions (atoms) are immersed in a dense plasma, they experience an environment which is substantially different from those encountered in ordinary situations. Since the density is high some of the plasma particles may come so close to the ion core that they substantially perturb the original potential exerted by the ion core to the bound electrons, resulting in modifications to the motions of these electrons. Thus the properties of the electronic state of the ions may be modified. The Stark broadening of the transition line treated in Chapter 7 falls into this category. Electron collisions are so frequent that qualitatively new features may appear in the population kinetics of the ion. Various features which result from the collection of collisional and radiative processes, as treated in Chapters 4 and 5, may be regarded as belonging to this category. In this chapter we review some other salient features of ions, or those manifested by them, immersed in dense plasmas. As a measure of the criterion for a "dense plasma" we can consider several densities in comparison with the properties of ions: geometrical and dynamical. One geometrical quantity of the plasma would be the mean distance between the ions, pm, as defined by eq. (7.6), and another would be the typical distance over which the Coulomb force is effective, RD, as defined by eq. (7.11) or eq. (1.31 a). If the dimension of the atomic wavefunction, as given by eq. (1.2), of the ion under consideration becomes non-negligible compared with either of the above distances, the plasma may be regarded substantially dense for this ion. The critical density would be of the order of 1028z2 (m~3) for the ground state. When excited ions are concerned, because of the larger electron orbit radius, the critical density would be substantially lower. We will see this to be actually the case later on. The dynamical quantity would be a typical excitation rate by electron impact for this ion in this plasma, in comparison with the inverse residential time of the ion, or the transition probability. A criterion of the reduced density of ne/z7 ~ io 17 ~ 20 m~ 3 has been encountered already in Chapters 4 and 5 for singly excited ions. See Fig. 1.10, for example. Another criterion, which becomes important in this chapter, concerns doubly excited ions: the autoionization probability is of the order of 1013 s^1, which is almost independent of the z of the ion. Thus, the criterion in this case would be ne/z3 ^ io 23 ~ 26 ni~3. The reader may have some ideas about the «e regions in Fig. 1.2 where the high-density effects play a significant role. 9.1 Modifications of atomic potential and level energy In this section, unless otherwise stated, we will consider a plasma made of protons and electrons. Thus, ions with which we will be concerned are protons. In this plasma some of the protons may have a bound electron, i.e. they form hydrogen

258

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atoms in discrete states. It will be shown later that the distinction between the bound discrete states and the continuum states is rather ambiguous in a dense plasma. We first focus our attention on one of the ions having a bound (optical) electron. Since the plasma is dense and the plasma particles are in thermal motion, the chances are high that a plasma particle comes close to the ion core, even inside the bound electron orbit, and exerts an electric field on the electron. Thus the bound electron experiences various perturbations. If we average these perturbations over time, the net effect would be regarded as screening or shielding of the ion core charge by the plasma particles. This is nothing but the Debye shielding introduced by eq. (7.11) or eq. (1.31 a). The effective potential felt by the optical electron is thus approximated by the Debye-Hiickel potential,*

where the Debye radius or the Debye length, eq. (7.11), is

where «i is the ion density. Owing to the modification of the Coulomb potential the wavefunctions, as given in Fig. 3.2, and thus the energy eigenvalues, as given in Fig. l.ll(b), are modified. Intuitively speaking, as a result of the "narrowing" of the Coulomb potential, eq. (9.1), the discrete states of the electron are "squeezed". The energy is "pushed up", and high-lying bound levels are lost into the "continuum" states. Thus, the number of discrete states becomes finite. If the temperature of the plasma is low or the density is high, the picture of "shielding" becomes inadequate. Rather, the ion sphere model is more appropriate. It is customary to express the mean distance between the ions as RO, in place of pm as defined by eq. (7.6). We follow this convention here. Several versions of the model potential have been proposed. One of them is

For small r, this potential behaves similarly to eq. (9.1), and at r = RQ the potential tends to zero. This picture may be interpreted as: each proton has its own territory with radius R0, and an electron having a positive energy migrates to the neighboring territories of other ions. In this model, again, the discrete state energies are pushed up. * Since the ion core is charged positive, the electric potential is positive. However, we will be concerned with the motion of an electron, which is attracted by the ion core. In this regard, it is more convenient to express the potential as negative. We adopt this convention in this chapter.

ATOMIC POTENTIAL AND LEVEL ENERGY

259

FIG 9.1 Shifts of level energies in a dense plasma, (a) Positions of several levels of hydrogen-like ions normalized by the original ionization potential of each level. The abscissa is the Debye length (times z) in atomic units. Debye-Hiickel model. (Quoted from Rogers et al, 1970; copyright 1970, with permission from The American Physical Society.) (b) Level positions of hydrogen-like Ne . Ion sphere model. (Quoted from Yamamoto and Narumi, 1983, with permission from The Physical Society of Japan.)

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An example of the calculations of the level energies is shown in Fig. 9.1; (a) is for a hydrogen-like ion calculated according to the Debye-Hiickel potential, where the energy position is normalized to its original ionization potential. It is seen that, with an increase in density, or a decrease in the Debye length, the levels are pushed up and, further, engulfed into the continuum. Figure 9.1(b) is the result of calculation for hydrogen-like neon (z= 10) based on the ion sphere model. Two features are noted in this figure: 1. The shifts of level energies are approximately parallel, so that, if we look at this figure in such a way that the energy of the ground Is state is fixed, then the energy level structure of the low-lying levels stays almost the same and, with an increase in density, the ionization limit comes down. This phenomenon is called the pressure ionization or the lowering of the ionization potential. The magnitude of the lowering for the Debye-Hiickel potential is given by

FIG 9.2 (Continued)

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261

FIG 9.2 (a) Spectra from an aluminum plasma produced by laser irradiation. Plasma polarization shifts of hydrogen-like A111+ Lyman S and Lyman e lines are seen, (b) Comparison of the observed shifts of the Lyman 6 line with the results of several model calculations. (Quoted from Renner et al., 1998; with permission from IOP Publishing.)

In this equation we have assumed the charge of our ion to be ze. The ion sphere model gives a different value for the lowering depending on the details of the model. 2. Even so, there remains a small difference in the shifts of the levels. This difference results in a small shift of the spectral line connecting a pair of the levels. This phenomenon is called the plasma polarization shift. An example of the experimental observations of this shift is shown in Fig. 9.2 for a laser-produced aluminum plasma. Figure 9.2(a) shows emission spectra of hydrogen-like aluminum Lyman 6 (1-5) and Lyman e (1-6) lines. With a decrease in the distance from the target surface, or an increase in the plasma density, the peak of both of these lines shifts toward the longer-wavelength direction. Figure 9.2(b) compares the observed shifts with the results by several calculations. 9.2 Transition probability and collision cross-section The shift of level energies was a result of the modification to the "shape" of the wavefunctions (Fig. 3.2) caused by a change in the potential. Another consequence of this modification is a change in the oscillator strength (see eqs. (3.2) and (3.3)) and thus the radiative transition probability, eq. (3.1). Figure 9.3(a) shows for neutral hydrogen an example of the changes in the transition probabilities against

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a change in the Debye length. All of the probabilities decrease with the increase in the plasma density. It should be remembered here, however, that the level energies were also modified as we have seen above. Figure 9.3(b) shows another example of calculations by the ion sphere model; for hydrogen the energy positions of excited levels with respect to the ground state are given by the horizontal positions of symbols (+) and the lowering of the ionization potential is given with the vertical lines. Note that these results are consistent with Fig. 9.1 (a). It is seen that, with the decrease in the Debye length, the shifts of the level energies result in a decrease in the energy separation between the levels. We now remember eq. (3.9b), the photoabsorption cross-section averaged over the energy width that is allocated to one upper level. In calculating the quantity corresponding to eq. (3.9b) for a dense plasma we may expect that the decrease in the transition probabilities, Fig. 9.3(a), or the corresponding decrease in the absorption oscillator strength,

FIG 9.3

(Continued)

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263

FIG 9.3 (a) Changes in the radiative transition probabilities of neutral hydrogen in a dense plasma. (Quoted from Roussel and O'Connell, 1974; copyright 1974, with permission from The American Physical Society.) (b) Changes in the energy positions of excited levels (with respect to the ground state) and the lowering of the ionization potential of hydrogen in a dense plasma. The quantity corresponding to eq. (3.9b) is given by the vertical positions of symbols (+). The overall absorption spectrum depends little on the plasma density. Ion sphere model. (Quoted from Hohne and Zimmermann, 1982; with permission from IOP Publishing.) might be compensated by the decrease in the energy width. Figure 9.3(b) shows the photoabsorption cross-section, or the absorption oscillator strength, averaged over the energy width allocated to each upper level by the vertical positions of the symbols (+). This figure also shows eq. (3.9b), the corresponding quantity for

264

DENSE PLASMA

the low-density limit, with the smooth curves, and as its extension, eq. (3.13), the photoionization cross-section, for energies above the "ionization limit". It is remarkable that the overall features of the absorption cross-sections for photoexcitation and photoionization depend little on the plasma density, especially for plasmas with a Debye length larger than 2QaQ. This is in sharp contrast to the strong dependence of the boundary energy between photoexcitation and photoionization, or the lowering of the ionization potential, on the plasma density. We saw that the Debye shielding modifies the atomic wavefunctions. The same shielding would affect collision processes, i.e. in a collision process the charge of the incident particle, an electron or an ion, is shielded by the nearby plasma particles, reducing the effect of this Coulomb field on the ion which is acted upon by this incident particle. Thus, the excitation cross-section, for example, may be reduced in a dense plasma. Figure 9.4 shows an example of calculations for excitation of the ground-state hydrogen-like ion: (a) is for excitation of ls^2s, and (b) is for ls^2p. Both the collision strengths, eq. (3.25), and thus the cross-sections, decrease with a decrease in the Debye length. However, it is noticed that the plasma effect is more salient for the excitation of 2p than for 2s. This is interpreted as: in excitation of an optically allowed transition, collisions with large impact parameters are relatively important, while for excitation of an optically forbidden transition collisions with small impact parameters are dominant. Even an exchange of the incident electron and the target electron could take place when the two electrons come close. Thus, the Debye shielding affects the former collisions more strongly than the latter. So far, we have relied on the picture of "screening" or "shielding" to express the effect of a dense plasma environment. However, the real situation for an ion in a dense plasma may be substantially different. The perturbation by plasma particles is strongly time dependent, and sometimes even violent, for instance. It is thus questionable whether the above picture describes properly the real situations. Unfortunately, a realistic treatment of a dense plasma environment involves very complicated procedures, e.g. correlation and coherence between the collisions cannot be neglected and collisions involving more than two particles become important. Only a limited number of attempts have been made in this direction until now. An alternative approach is to treat a transition of ions as induced by a fluctuating electric field originating from the thermal motion of plasma particles. An example of such calculations is shown in Fig. 9.5. This is for a proton-electron plasma with Te = 340 eV and ne = 1029 m~3. The transition of a hydrogen-like ion 2Si/2^2P 3 / 2 is calculated for Ne9+ or Ar17+ ions immersed in this plasma. Figure 9.5(a) shows the frequency spectrum of plasma density fluctuations weighted by the generalized oscillator strength of the transition. The abscissa is the angular frequency normalized by the electron plasma frequency. The frequency corresponding to the energy difference between the lower and upper levels is shown

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265

FIG 9.4 Dependence of the excitation cross-section (collision strength) on the plasma density. Hydrogen-like ions. The abscissa is the collision energy normalized by 2z2R. (a) Is —> 2s transition; (b) Is —> 2p transition. (Quoted from Hatton et al., 1981; with permission from IOP Publishing.) with the arrows. Figure 9.5(b) shows the temporal development of the upper-level population «(2P3/2) for the initial condition of «(2Si/ 2 )=l and K(2P3/2) = 0 at t = 0. No transitions other than 2Si/2^2P 3 / 2 are considered in this calculation. The effect of coherent excitation is manifested in the case of Ne9+. Also shown are

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FIG 9.5 Excitation of 2S i /2 —> 2P3/2 of hydrogen-like ions in a dense plasma with re = 340eV and « e =10 29 m~3. (a) Frequency spectrum of plasma density fluctuations weighted by the generalized oscillator strength of the transitions, (b) Temporal development of the upper-level populations (solid lines). Dashed lines show the solution of the rate equation. (Quoted from Kitamura et al., 2000; copyright 2000, with permission from The American Physical Society.) the solution of the rate equation with the dashed lines; this is equivalent to assuming the Born approximation for the excitation cross-section. 9.3 Multistep processes involving doubly excited states

In Section 3.5 we noted that any of the levels of an ion (not an atom in this case), irrespective of whether they are the ground state or an excited state, are

MULTISTEP PROCESSES WITH DOUBLY EXCITED STATES

267

accompanied by a Rydberg series of excited levels of an ion in the adjacent lowerionization stage. See Figs. 3.17(a) and 9.6. Suppose that the singly excited ion is a hydrogen-like excited level, say, the 2p level. This "normal" level is the ionization limit of the doubly excited levels of, in this case, helium-like (2p,«/') with the spectator electron «/'. As we have seen in Section 3.5 these doubly excited levels play an important role in the recombination process of the ground-state ion, Is in this case, through dielectronic recombination. In a dense plasma other features emerge. DL excitation and deexcitation If an ion in a doubly excited level, say helium-like (2p,«/'), is immersed in a dense plasma, a collisional transition may take place before this ion spontaneously decays through autoionization, (2p, «/')—> ls + e(£7), eq. (3.45), or a stabilizing radiative transition, (2p,n/')->(ls,n/ / ) + /"', eq. (3.48). See Fig. 3.17(a). If n is larger than 2 the core electron, 2p in this example, is more tightly bound than the spectator electron «/', so that electron collisions would induce a transition more likely in the latter electron. As we assumed in our discussion of singly excited ions in a plasma, we assume here again that the doubly excited ions with different /' are populated according to their statistical weights, so that n is enough to specify the level. We thus denote a doubly excited ion with a core electron p and a spectator electron q as (p, q). Here p or q means the principal quantum number. In the following we simply write pq instead of (p, q). See Fig. 9.6. If the temperature is not very low, the most likely collision process is excitation pq + e -^p(q + 1) + e, as in the case of a singly excited level, eqs. (4.9) and (4.30). In Chapter 4 we divided the excited level population into two components: the ionizing plasma component and the recombining plasma component. We proceed here in a similar spirit; in the ionizing plasma in Fig. 9.6 the ground-state ion has a population but the singly excited ion p has no population, and in the recombining plasma, the population of the ground state ion 1 is absent. We thus divide the population of the doubly excited ions into the corresponding two components. What we are interested in here is the "ionization" process and the "recombination" process through these doubly excited levels rather than the population of these levels itself. We may thus adopt a rather crude approximation. The former process consists of the series of processes

See Figs. 4.8, 4.9, and 1.10(a). In eq. (9.5b) we omit "e" which induces the ladderlike transitions. This series of processes, eqs. (9.5a) and (9.5b), is a collective process which reduces to net excitation, 1 + e —>/> + e. Thus, in a dense plasma, the

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FIG 9.6 Schematic energy-level diagram of the ground state 1 and an excited state p of hydrogen-like ions and helium-like Rydberg states Iq converging to the ground state 1 andpq converging top. Levelpq is populated by dielectronic capture and depopulated by autoionization, radiative transition, including the stabilizing transition and collisional excitation.

direct excitation is supplemented by this indirect excitation through the doubly excited levels. Figure 9.6 schematically illustrates this process. We call this new excitation process dielectronic capture ladder-like (DL) excitation. For recombining plasma, very high-lying doubly excited levels should be in LTE with respect to the "ion" density n(p) and electron density ne, or we should have n(pq) = Zp(pq)n(p)ne, where Zp(pq) is given by eq. (2.7) with respect to level p, i.e. the statistical weight g(p) for the ion and the ionization potential measured from the position of level p. See Figs. 4.17-4.25 and 1.10(b). This population balance may be expressed as

where we have omitted "e" again. This ion may autoionize:

MULTISTEP PROCESSES WITH DOUBLY EXCITED STATES

269

This series of processes is nothing but net deexcitation p + e —> 1 + e. Thus again, the direct deexcitation process is supplemented by this indirect multistep process, which is called DL deexcitation. We now estimate the magnitudes of the rates for these processes. We take an example of hydrogen-like neon (z= 10) and consider excitation and deexcitation Is—2s and Is—2p. We confine ourselves to high temperature. We define (extended) Griem's boundary level pqo for the doubly excited levels. This boundary is given by an equation similar to eq. (4.25) with a modification that, on the l.h.s., the autoionization probability, Aa(pq), and the probability for the stabilizing transition, AT(pq—> Iq), are added. See Fig. 9.6. We thus determine extended Griem's boundary which behaves in a similar manner to that in Fig. 1.10(a). The "ionization potential", or the principal quantum number, of extended Griem's boundary level is given in Fig. 9.7(a). We first consider the DL excitation. Suppose an electron is dielectronically captured by an ion in level 1 to form one of the levels pq, eq. (9.5a), for which q > qo- See Fig. 9.6. According to the discussion developed in Chapter 4 concerning the ladder-like excitation-ionization, we may expect that the spectator electron q may be further excited in the ladder-like excitation chain and finally "ionized" to leave the core ion in level p. We remember that, for singly excited ions, the CR ionization rate coefficient is approximated by eq. (5.8). In the same spirit, the effective rate coefficient for the DL excitation CDL(1, p), may be expressed by

where r&(pq) is the dielectronic capture rate coefficient as defined by eq. (3.43) and given by eqs. (3.46) and (3.47). Thus, we have

This may be approximated to

It is noted that the Saha-Boltzmann coefficient in eq. (3.46), written as Z^pq) here, refers to the ground state 1, and is related to Zp(pq) by

FIG 9.7 (a) Extended Griem's boundary level in the doubly excited levels 2sq and 2pq of Ne9+ against ne. (b) Horizontal lines: the direct excitation rate coefficient (referred to the l.h.s. ordinate) and the deexcitation rate coefficient (r.h.s. ordinate) of Ne9+ for the ls-2s transition and that for the ls-2p transition for Te = 106 K. : approximate DL excitation or deexcitation rate coefficient given by eq. (9.10) or eq. (9.12); and — • — • —: results of numerical CR model calculations of the DL excitation and deexcitation rate coefficients, respectively. (Quoted from Fujimoto and Kato, 1985; copyright 1985, with permission from The American Physical Society.)

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271

By using eqs. (3.42)-(3.44a) we may rewrite eq. (9.8) as

where o-ex(lz^>pz) is the excitation cross-section extrapolated below the excitation threshold. See Fig. 3.11. This equation is just the extrapolation of the excitation rate coefficient from the threshold down to the extended Griem's boundary energy. See eq. (3.28). This energy is given in Fig. 9.7(a), and from eq. (9.10) we obtain the DL excitation rate coefficient. The result is shown with the solid line in Fig. 9.7 (b). This figure shows the excitation (and deexcitation) rate coefficient for the direct process with the horizontal lines for a particular temperature of this figure. Under the condition of this figure, the DL excitation rate coefficient both for Is-2s and Is — 2p becomes larger than that for the direct excitation in higherdensity regions. Figure 9.7(a) shows that with an increase in electron density the boundary level 2s(/G comes down faster than 2pqG. This is because the former levels lack the stabilizing transition. The autoionization probability is of similar magnitude for both the 2sq and 2pq levels. This latter statement is understood as follows; the autoionization probability is approximately given by the threshold value of the direct excitation cross-section (Fig. 3.11: on the ordinate scale they are 0.45 and 1.9, respectively, for 2s and 2p) divided by the statistical weight of the doubly excited level, eq. (3.47) (1:3 for 2sq, and 2pq), Thus the resulting autoionization rates are only slightly smaller for 2sq^ Is than those for 2pq^ Is. We now turn to the dielectronic capture ladder-like deexcitation. We saw in Section 4.3 that the levels lying higher than Griem's boundary are in LTE with respect to the density of the next ionization stage ion and electron density, eq. (4.47). At high temperature, Byron's boundary plays no role. See Fig. 1.10(b) and eq. (4.56). We assume a similar situation for the doubly excited levels: levels pq with q>qo have LTE populations with respect to n(p) and ne, and those with q
By using eq. (3.47) with eq. (9.9) we approximate eq. (9.11) as

Thus, the rate coefficient for DL excitation, eq. (9.10), and that DL deexcitation obey the principle of detailed balance, eq. (3.31) or (3.31a). The solid lines in

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Fig. 9.7(b) also show the DL deexcitation rate coefficients, which are referred to the right-hand side ordinate. Figure 9.7(b) includes the results of detailed numerical calculations similar to those for the rate coefficients of collisional-radiative (CR) ionization and recombination, as done in Chapter 5. They agree reasonably well with the above approximations.

CR recombination of excited ions In the above subsection, we assumed high temperature. We now consider the situation of low-temperature recombining plasma, i.e. we start with level p ions in a low-temperature plasma. The electron density may be high. In Chapters 4 and 5, we reviewed the features of the low-temperature recombining plasma by taking the example of hydrogen. As Fig. 1.10(b) schematically shows Byron's boundary, eq. (4.55) or eq. (4.56), plays an important role; when the electron density is high so that Griem's boundary lies below Byron's boundary, excited levels are divided by this latter boundary. Levels lying higher than this boundary are strongly coupled with each other and thus to the continuum state electrons, so that their populations are in LTE with respect to the ion density and electron density. Lower-lying levels than Byron's boundary are in the flow of ladder-like deexcitation. See Fig. 4.24. Another important feature was that for higher electron density for which pG B holds, the flux of the CR recombination is controlled by the collisional deexcitation flux through Byron's boundary level, eq. (5.26). See Fig. 5.6. The situation should be similar for doubly excited levels. For simplicity we consider the high-density limit first. We may conclude as follows. We consider populations in the levels pq with q > gB. These populations may be lost by autoionization, and this loss is replenished by recombination of the ions p, resulting in DL deexcitation, as we have seen above. In the present case, the lower end of the integration of eq. (9.12) should be replaced by E^, the energy of Byron's boundary pq#. Another flux of electrons flows downward through Byron's boundary pq#. This flux of electrons out of the group of levels pq with q > q# is compensated by the influx of electrons from the ion p, again. In other words, both of these fluxes contribute to depopulation of the ion p. It is noted that the magnitude of the second out-flux may be approximated by the CR recombination rate coefficient for hydrogen-like ions, which should be rather close to that for neutral hydrogen as shown in Fig. 5.4. We here assumed low temperature so that pB is large. An approximate calculation of these processes are made for lithium-like aluminum ions (the ground-state configuration is Is22s) concerning research with an x-ray laser. An example of the results is given in Fig. 9.8. This figure is similar to Fig. 9.7; the direct deexcitation rate coefficient is shown for 3d —> 2s and 3p —> 2s by the horizontal lines and extended Griem's boundary and Byron's

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273

FIG 9.8 Deexcitation rate coefficient of lithium-like aluminum ions in the recombining phase in a dense plasma. Horizontal lines: the direct deexcitation rate coefficient for transitions between the singly excited lithium-like aluminum levels. Thin lines: extended Griem's boundary and Byron's boundary for 3pq and 3dq doubly excited levels. the DL deexcitation rate coefficient for 3d —> 2s; for 3p —> 2s; — • — • — the CR recombination rate coefficient which is common to the low-lying levels. (Quoted from Kawachi and Fujimoto, 1990; copyright 1990, with permission from The American Physical Society.)

boundary for the doubly excited (3dn/') and (3p«/') levels are shown with the thin lines. The DL deexcitation rate coefficient is given with the thick solid and dashed lines. The CR recombination rate coefficient is given with the dash-dotted line. In this low electron temperature (remember that the reduced

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temperature in the present example is r e /z 2 ~450K, where z is 10), the last loss mechanism is predominant at higher densities. Figure 9.9 compares the calculated population distribution over the singly excited levels in a Boltzmann plot by the conventional CR model and that with these additional recombination and all the deexcitation processes included. The difference is substantial, especially for low-lying levels. This figure also includes the result of experimentally determined populations, indicating good agreement with the calculation which includes all the above processes. Thus, this figure demonstrates the validity of the above theory.

FIG 9.9 The populations of a recombining lithium-like aluminum plasma in the Boltzmann plot. A: result of calculation by the conventional CR model. O: result of the CR model calculation with the DL deexcitation and CR recombination processes included. 0A: experiment. (Quoted from Kawachi et al., 1999; with permission from TOP Publishing.)

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Decrease and disappearance of resonance contributions to excitation cross-section In Section 3.5 we introduced the resonance contributions to the excitation crosssection by taking the ls^2s transition in hydrogen-like ions as an example. See eq. (3.51) and Fig. 3.21. See also Fig. 9.10. Figure 9.11 is another plot of Fig. 3.21; the resonance contributions, which appear as sharp peaks in the latter figure, have been averaged over energy, and in the present figure they appear as smooth crosssections added on the top of the direct excitation cross-section. The energy range of Fig. 3.21 is from 1.02 to 1.21 keV. Contributions from the (3s, ri), (3p,«), and (3d, K) levels are shown separately. It would be natural to assume that, in a dense plasma, the intermediate state, (3p,«) for example, may suffer electron collisions before it autoionizes. Compare eq. (3.51) with eq. (9.5): the first lines are common and, in a dense plasma, instead

FIG 9.10 Schematic energy-level diagram of hydrogen-like Is, 2s, and 3p levels with the accompanying doubly excited helium-like levels. The doubly excited levels 3p« are populated by the dielectronic capture from the ground state. It is depopulated by autoionization to the ground state and to 2s, the latter of which results in the resonance contribution to the excitation cross-section for Is —> 2s, stabilizing radiative transition and collisional excitation which results in the DL excitation of Is—>3p.

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FIG 9.11 Another plot of Fig. 3.21; the excitation cross-section for Is —> 2s, plus the resonance contributions through the doubly excited levels 3s«, 3p«, 3d«. These contributions are lost owing to the development of the DL excitation process in these doubly excited levels. The corresponding increase in the direct excitation of, say ls^3p, is expressed as extrapolation of the excitation crosssection below the excitation threshold down to extended Griem's boundary level. Similar extrapolation is given to the direct ls^2s excitation crosssection. (Quoted from Fujimoto and Kato, 1987; copyright 1987, with permission from The American Physical Society.) of the second line of eq. (3.51), eq. (9.5b) becomes dominant. We define extended Griem's boundary again for the doubly excited levels, say (3p,ri). Then the flux of dielectronic capture into the levels lying higher than this boundary will enter into the ladder-like excitation chain to result in the DL excitation of 3p. Thus, the corresponding part of the resonance contribution to the excitation cross-section Is —> 2s is lost. Figure 9.11 illustrates this situation. For a certain electron density the part of the resonance cross-section with energies higher than the energy of extended Griem's boundary is lost. For each of the contributions, (3s, ri), (3p,ri), and (3d,ri),the energy positions of extended Griem's boundary levels are given, and the part of the resonance contributions at higher energies are lost. This part instead contributes to the DL excitation of the core singly excited level. In this figure, the excitation cross-section Is —> 3p is extrapolated to this energy, eq. (9.10), and the excitation cross-section Is —> 2s is also extrapolated below the threshold to

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277

account for the DL excitation introduced in the preceding subsection. It is noted that, in the electron densities considered in this figure, ne < 1028 m~3, the energylevel structure and the excitation cross-section are almost unaffected. See Fig. 9.1 (b) for the former and Fig. 9.4 for the latter: RQ > 100a0 for the present example. 9.4 Density of states and Saha equilibrium Density of states In Chapter 2, in considering thermodynamic equilibrium, we derived the SahaBoltzmann equilibrium relationship. In doing so, we implicitly assumed an isolated atom. In the case of a hydrogen atom, the statistical weight of a level with principal quantum number p is 2p2. Thus, the number of states of a level over one principal quantum number (remember Fig. 1.11 (a)) is

The energy width corresponding to one principal quantum number is

See eq. (1.5). This quantity is understood as the energy width allocated to this level p, which is 2p2-fold degenerate. Thus, we may define the number of states in a unit energy interval, or the density of states, g(p)dp/dE, or

where we use the rydberg units of energy. In Fig. 9.12 the smooth curve in the negative energy region and the dotted curve connecting with it, represents eq. (9.13). It is obvious that eq. (9.13) diverges toward the ionization limit, or zero energy. This is a natural consequence of our assumption of isolated atoms which has an infinite number of Rydberg levels. In the preceding chapters the continuum states of electrons were approximated as free states. An example is the Maxwell distribution of electron energies, eqs. (2.2) and (2.2a). In this case, the density of states is given from eq. (2.5a) with ge = 2 and g(l) = 1 as

This density of states is inversely proportional to electron density, and is shown in Fig. 9.12 in the positive energy region with the dash-dotted curves for several values of ne. A strong discontinuity is seen at the zero energy. This is the result of

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FIG 9.12 Density of states of neutral hydrogen in the energy region close to the ionization limit. connecting to in negative energy: for an isolated hydrogen atom, eq. (9.13); — • — • — for free electrons, eq. (9.14); in positive energy, extending across the ionization limit to negative energies: the density of states on the basis of the ion sphere type model, eqs. (9.22)-(9.26b). (Quoted from Shimamura and Fujimoto, 1990; copyright 1990, with permission from The American Physical Society.) our approximations which are inconsistent each other: i.e. an isolated atom for negative energies and free electrons for positive energies. We now remember that our atoms and ions are in a plasma; we assume a plasma made of protons and electrons. In reality, the electron states with negative energies are affected by plasma particles as we have seen in Section 9.1, and the electrons with positive energies move in the Coulomb potentials of protons under the influence of other plasma particles. If we intend to resolve the difficulty above,

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279

these effects should be properly taken into account. As we have noted already in Section 9.2, this poses enormous difficulties. Instead, we adopt here a rather crude model. We start with a model potential of the ion sphere type for an electron:

and treat the electron motion classically. To be consistent with eq. (9.1), we are using here the convention that an electron is regarded to have a positive charge e. We first consider an electron having a high energy E so that its trajectory is virtually a straight path inside the sphere. Let p0 be the momentum and r0 be the distance of the closest approach to the proton. We may define the angular momentum quantum number semiclassically

Then we may assign to this angular momentum the number of states 2(27 + 1), i.e. the number of the directions of the angular momentum (space quantization) times that of the electron spin. We have the transit time

We assign the energy width to this electron state

To help justify this reasoning, see the discussions around eq. (1.6). From the above we may define the density of states for this angular momentum. The density of states for the present E is given by the summation of that over /:

where lc is the cut-off angular momentum which is defined by eq. (9.16) with r0 replaced by R0, For sufficiently high energy, eq. (9.19) may be approximated to

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DENSE PLASMA

Here we have used eq. (7.6) for R0 = pm and E =pQ/2m2, We have arrived at an expression that is exactly the same as eq. (9.14). Thus, by following the above procedure we are able to obtain the "correct" density of states of free electrons. For lower energies, we follow similar procedures; instead of the straight path we adopt hyperbolic trajectories for positive energies and elliptic trajectories for negative energies. We calculate the density of states from the transit times of the electron over the sphere. We define the units of energy

and the dimensionless energy

We rewrite eqs. (9.13) and (9.14) in the form

with

Then \X\ 5//2G(o), which may be called the reduced density of states, is independent of «e. Figure 9.13 shows the reduced density of states of eqs. (9.13) and (9.14) with the thin dashed line for negative energy and the dash-dotted line for positive energy, respectively. For positive energy, X> 0, the electron trajectory is hyperbolic as shown in Fig. 9.14(a). From arguments similar to those leading to eq. (9.19a) we obtain an analytical expression

The result is plotted on Fig. 9.13 with the solid line for X > 0. It is noted that this tends to a finite value at the null energy. For slightly negative energies, — 1 < X< 0, the major axis of the elliptic electron orbit is so large that the orbit extends outside the ion sphere. The circular orbit is absent because its radius is larger than the ion sphere radius, R0, See Fig. 9.14(b). By following a similar procedure we obtain the expression

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281

FIG 9.13 Reduced density of states. See text for details. (Quoted from Shimamura and Fujimoto, 1990; copyright 1990, with permission from The American Physical Society.) which is plotted in Fig. 9.13 with the solid line for —1 < X < Q . This curve continues smoothly from the curve, eq. (9.24), at the null energy. At X=—\ the radius of the circular orbit is equal to the ion sphere radius. This is seen from eq. (1.2), i.e. n = ^/Ro/a0, and the energy is given from eq. (1.1), i.e. E(ri) = —Ra0/R0, For still lower energies, —2<X<—1, elliptic orbits with high ellipticity extend out of the sphere. At the same time, elliptic orbits with small ellipticity and circular orbit stay within the sphere. See Fig. 9.14(c). Thus, the density of states in this energy region consists of two parts, with the former contribution

which is plotted in Fig. 9.13 with the dotted line. The latter contribution is given as

The sum of these contributions, i.e. eq. (9.26), is plotted with the solid line. This continues smoothly from eq. (9.25) at X=—\. It should be noted that the former orbits extending out of the sphere may be regarded as continuum states. This is because in actual situations no solid wall is present at r = RQ. Rather, the electron arriving at RQ crosses the boundary to enter into the territory of the adjacent ion,

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FIG 9.14 Trajectories of electrons with respect to the ion sphere for various energy regions. and it repeats this migration. For the lowest energy region of X < —2, all the orbits stay within the sphere (Fig. 9.14(d)) and we have G(a) = I , or the density of states is nothing but eq. (9.13), as shown with the solid line in Fig. 9.13. Again, the functions are continuous at X= —2. The same plot as the above is also given in Fig. 9.12 with the solid lines. In this figure the energy positions of several discrete levels are given with the arrows. For « e =10 25 m~3, for example, X=—\ corresponds to n ~ 7(~2v/5). For —2<X<—\, both the discrete states with periodic orbit motions and the continuum states whose orbit extends out of the sphere exist for 5 < n < 7(~v / 2 • 5). See Fig. 9.14(c). The energy X= —2 may be regarded as continuum lowering or the lowering of the ionization potential, which has been discussed at the beginning of this chapter. In our model this is given as

Compare this result with eq. (9.4).

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283

In the above, we followed the semiclassical procedure, i.e. classical electron orbits, quantization of angular momenta and space quantization, and the energy width given by the transit time. By adopting an alternative procedure, i.e. the Bohr-Sommerfeld quantization to the electron orbits, we can obtain a result that is exactly the same as the above. For details, the reader is referred to the paper in the references at the end of this chapter. Correction to the Saha-Boltzmann and Saha relationships In Section 2.1 we derived the thermodynamic equilibrium relationship between the "atom" density and the "ion" density, eqs. (2.7) and (2.9). In the process of derivation we assumed the Maxwell distribution of electrons. As we have seen above, this "free electron states" model is increasingly worse for plasmas with higher densities. The ionization potential also suffers lowering in these plasmas. We thus have to modify eq. (2.7). As Fig. 9.12 indicates the density of states of the "ion" should be increased in positive energies, and the continuum states with negative energies should be included. For the "atom" only the discrete states should be counted. In other words, the correct procedure to calculate the "atom" and "ion" densities is: in Fig. 9.12, for a particular electron density, we take a function of density of states and "fill" this function with electrons according to the Boltzmann distribution. We then enumerate the total number of electrons distributed over the discrete states and the number of electrons in the continuum states. Here we assume our plasma to be electrically neutral. We thus obtain the density ratio of "atoms" and "ions". This ratio is expressed as a correction to eq. (2.9). This correction is expressed as a correction factor O(re, «e) to eq. (2.7):

where «(1) is the atom density in the ground state, «i is the "ion" density and Z(l) is the original Saha-Boltzmann coefficient without the lowering of the ionization potential included. Figure 9.15 shows the correction factor for several temperatures. For low temperatures, the correction is large as expected.* We now define the coupling parameter1" F, which is the ratio of the average energy of the Coulomb potential to the average thermal energy of ions,

* We give here the correction factor to eq. (2.7), not to eq. (2.9). If we want to enumerate the total number of "atoms" including those in excited levels, it is straightforward to do that. However, for low temperature, say Te = 10 K, excited-level populations are almost negligible. For high temperature, say Te = 6 x 104 K, the ionization ratio [«z/«(l)] is about 104. See Figs. 5.2 and 5.8. In this latter case, the number of neutral atoms is extremely low irrespective of whether we include excited levels or not. t This parameter is sometimes called the plasma parameter. We do not adopt this nomenclature, because we call ne and Te the plasma parameters.

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FIG 9.15 The correction factor to the Saha-Boltzmann relationship, eq. (2.7). The corresponding values given by Ebeling et al. (1976) Theory of Bound States and lonization Equilibrium in Plasmas and Solids (Akademie-Verlag, Berlin) are also given. (Quoted from Shimamura and Fujimoto, 1990; copyright 1990, with permission from The American Physical Society.)

It is noted that this parameter is nothing but AX/Win our model. See eq. (9.27). It may be shown that our correction factors in Fig. 9.15 are plotted with a single curve as a function of F as shown in Fig. 9.16. In this figure the function exp(—F) is also plotted; this function is based on the simple assumption that the original state density for free electrons is shifted by AX, eq. (9.27). See Fig. 9.12. It is seen that our correction factor is approximated very well by this simple expression. In Chapter 2 we mentioned the Saha equilibrium. We noted the difficulty of divergence of the partition function. This was due to the summation of the statistical weights over the excited levels to infinite principal quantum number. As we have seen above, the difficulty of divergence is resolved by the present model of the density of states.

REFERENCES

285

FIG 9.16 The correction factor as a function of the coupling parameter. It compares well with the simple expression for the lowering of the ionization potential. (Quoted from Shimamura and Fujimoto, 1990; copyright 1990, with permission from The American Physical Society.) References

The discussions in Section 9.3 are based on: Fujimoto, T. and Kato, T. 1985 Phys. Rev. A 32, 1663. Fujimoto, T. and Kato, T. 1987 Phys. Rev. A 35, 3024. Kawachi, T. and Fujimoto, T. 1997 Phys. Rev. E 55, 1836. The discussions in Section 9.4 are based on: Shimamura, I. and Fujimoto, T. 1990 Phys. Rev. A 42, 2346. The figures are taken from: Hatton, G.J., Lane, N.F., and Weisheit, J.C. 1981 /. Phys. B 14, 4879. Hohne, F.E. and Zimmermann, R. 1982 /. Phys. B 15, 2551. Kawachi, T., Ando, K., Fujikawa, C., Oyama, H., Yamaguchi, N., Hara, T., and Aoyagi, Y. 1999 /. Phys. B 32, 553. Kitamura, H., Murillo, M.S., and Weisheit, J.C. 2000 Phys. Plasmas 7, 3441. Renner, O., Salzmann, D., SondhauB, P., Djaoui, A., Krousky, E., and Forster, E. 1998 /. Phys. B 31, 1379. Rogers, F.J., Graboske Jr. H.C., and Harwood, D.J. 1970 Phys. Rev. A 1, 1577. Roussel, K.M. and O'Connell, R.F. 1974 Phys. Rev. A 9, 52. Yamamoto, K. and Narumi, H. 1983 /. Phy. Soc. Japan 52, 520.

INDEX

absorption coefficient 236 absorption cross-section 39, 237 adiabatic collisions 220 autocorrelation function 219 autoionization 64 Bethe limit 54 black-body radiation 25, 241 Boltzmann distribution 22, 26, 92, 109, 202 Boltzmann plot 121, 132, 208 Bremsstrahlung 211 BYRON 10, 105, 109, 119, 129 Byron's boundary 129, 133, 144, 164, 165, 192, 194, 272 cascade 99, 101 charge exchange 74 collision strength 56 collisional-radiative ionization rate coefficient 150, 189 collisional-radiative model 95 collisional-radiative recombination rate coefficient 150, 190 complete LIE 24, 192, 201 continuum radiation 205 corona equilibrium 100, 135, 167 corona phase 99, 100 CRC phase 114, 123, 126 cross-section 41, 48, 61, 72 curve of growth 238 Debye radius 218, 227 Debye-Hiickel potential 258 deexcitation cross-section 56 dense plasma 257 density of states 24, 277 dielectronic capture 64, 267 dielectronic capture ladder-like deexcitation 271 dielectronic capture ladder-like excitation 268 dielectronic recombination 64, 67 diffusion 245 dipole matrix element 31 Doppler broadening 213 effective collision strength 59 effective principal quantum number 20, 111 Einstein's A and B coefficients 26

electron broadening 226 emission coefficient 240 equivalent width 238 escape factor 249, 253 excitation cross-section 48, 66, 71, 73, 78, 264, 275 excitation temperature 241 exponential integral 47, 115, 123, 161 first Bohr radius 14 forbidden transition 31, 54, 229, 264 free-free continuum 211 Gaunt factor 37, 43, 45, 211 Gaussian profile 213, 233 GRIEM 10, 103, 105, 118, 119 Griem's boundary 103, 111, 120, 128, 130, 131, 139, 145, 153, 159, 161, 164, 192, 196, 269, 273, 276 Grotorian diagram 3, 19 Holtsmark field 214 Holtsmark profile 225, 233 impact broadening 219, 223, 230 impact parameter 220, 264 Inglis-Teller limit 233 ion broadening 225, 231 ion sphere model 258 ionization balance 167, 176, 180, 186, 195, 202 ionization cross-section 59, 61, 62 ionization flux 156, 179, 182 ionization ratio 154, 167, 174 ionizing plasma 137, 177, 182, 184, 188, 199, 202, 254, 267 ionizing plasma component 9, 10, 95, 96, 171, 173, 175, 177, 181, 186, 253 Kirchhoff s law 241 Klein-Rosseland formula 57, 79 Kramer's formula 37 ladder-like deexcitation 127 ladder-like excitation-ionization 104, 109 line absorption 245 linear Stark effect 218, 224, 231 Lorentzian profile 218, 221, 233 lowering of ionization potential 260, 282

INDEX lowering of ionization limit 154, 159, 164 LIE 24,92, 118, 143, 191 Maxwell distribution 22, 48, 57, 77, 283 Milne's formula 46 minus sixth power distribution 105, 129, 183 natural broadening 218, 240 normal field strength 215, 231 optical coherence 222 optical depth 237 optical electron 14 optical thickness 237, 245, 252 optically allowed transition 31, 53, 264 optically forbidden transition 31, 54, 229, 264 optimum temperature 169, 181, 202 oscillator strength 31, 34, 37, 45, 53, 77, 86, 101, 161, 264 overbarrier model 76

287

recombining plasma 142, 177, 182, 184, 190, 192, 267, 272 recombining plasma component 9, 10, 95, 111, 120, 171, 174, 175, 177, 181, 186, 253 reduced electron density 15, 192, 254 reduced electron temperature 15, 77, 192 relaxation of coherence 222 relaxation time 85, 89, 140, 142, 144, 146 resonance contribution 71, 275 resonance line 20, 48 response time 138, 146 Rydberg constant 14 Rydberg level 20

partial LIE 24, 192, 200 photoionization 42, 45 Planck's distribution 27 plasma 13 plasma microfield 214 plasma polarization shift 261 population coefficient 92 population inversion 121, 126 pressure ionization 260

Saha equilibrium 25, 284 Saha-Boltzmann distribution 24, 92 satellite line 67 saturation phase 98, 102, 104, 109, 116, 118, 121, 125, 126 scaling law 2, 15, 76, 190 self-reversal 242, 246 spectator electron 64, 267 spectrum 3 spherical harmonic 32 stabilizing transition 67 Stark effect 218, 228 statistical populations 18, 83, 134 statistical weight 19, 97, 111, 120, 277 Stefan-Boltzmann law 27 sum rule 34

quadratic Stark effect 223, 227, 229, 231 quantum cell 17, 23 quantum defect 20, 231 quasi-static broadening 214, 223 quasi-steady state (QSS) 91, 136, 145

three-body recombination 63, 79, 118, 142, 145, 160, 190 total absorption 238 transient time 138 transit time 279 transition probability 31, 40, 77, 262

radiation trapping 245 radiative decay 88, 100, 114, 126, 136, 140, 144 radiative recombination 42, 45, 68, 78, 114, 123, 125, 145, 164 Rayleigh-Jeans law 27 recombination continuum 205, 211 recombination flux 157, 179, 181

Voigt parameter 234, 245 Voigt profile 233, 245 Weisskopf radius 221 Wien's displacement law 29 Zeeman coherence 222