3.1.

The Saha-Boltzmann model

The Saha-Boltzmann model⁹ is the standard model used to calculate ionization balance within a plasma. The principal expression for this model gives the ratio between a charge state Z_i with density n_i and the level above (Z_i + 1) with density n_i+i as

where Λ is the thermal de Broglie wavelength, given by

The Saha-Boltzmann ratio is a tried and tested piece of physics, its only drawback is the premise on which it is based; that the plasma be in equilibrium. In the case of a laser-produced plasma this is an approximation.

3.2.

Accounting for perturbation by photo-ionization

The Saha-Boltzmann ratio considers the two levels it relates to be in equilibrium with each other. In the case when the lower level is effected by photo-ionization, the balance of processes that achieves this equilibrium must be adjusted. Assuming the pulse time of the laser is on order of nanoseconds, we can conclude equilibrium is reached and add photo-ionization to the list of processes we consider.

In an unperturbed plasma, the collisional processes that pertain to changes in ionization are the collisional ionization rate and the three-body recombination rate. The collisional ionization rate is the rate at which electron-ion collisions result in ionization of the ion. The rate of ionization K_ci of a given level i in units of cm³s^–1 is given by the Lotz approximation¹⁰

where k_BT is in electron volts, I_e(y) is the exponential integral

and y is the reduced ionization energy, given by y = E_ion/k_BT.

The rate coefficient K_3br for the reverse process, three-body recombination, can be found by use of detailed balance and the Saha-Boltzmann ratio⁷. In equilibrium K_ci and K_3br will be balanced like so;

After some rearrangement, we can replace the ratio ni/n_(i+1) with equation 5 to find

Figure 1a shows a three level atomic system where these processes are in balance.

Figure 1:

Example 3-level systems showing the populating and de-populating processes for an ionization level i both (a) without photo-ionization and (b) with photo-ionization.

In a plasma driven by a laser with photon energy above the photo-ionization threshold, we can add photoionization into equation 9 to give a new level dependent balance between the populating and de-populating processes – see figure 1b. For example, in the case where photo-ionization is possible only for the neutral and the 1+ ion we have:

where Φ is the incident radiation flux.

3.3.

Continuum lowering

In the high density limit, the Saha-Boltzmann model loses accuracy due to the introduction of perturbative effects. One such effect is continuum lowering, where the ionization potential of the ions is reduced by the close proximity of other ion potentials. This effect has been quantified by Stewart and Pyatt¹¹, who give a correction factor to the ionization such that

The distance a_c is given by either the Debye length λ_D, or the ion sphere radius r₀ given by

The choice that gives the smallest correction should be chosen as the model reconciles two previous suggested models; the ion-sphere model and the Debye-Hückel model¹² (the former being applicable in the low temperature, high density limit and the latter applicable in the high temperature, low density limit). There are other models that predict the effects of continuum lowering, the most notable being that of Ecker and Kröll¹³, however the Stewart-Pyatt model was selected as it is widely utilized and is applicable in our high density, low temperature limit. This continuum lowering correction can also be applied to the rate-equation model. We need not modify the model any further, but should include any transitions that have become possible with the reduction in ionization energy.

4.1.

Reference values

The first step towards producing time-resolved measurements was producing a set of reference curves to relate temperatures to energy content values. These energy content values were calculated by simply adding the energy in the free electrons and ions together:

This dependence on the ion populations n_i, which are calculated by an iterative process, makes calculating U_i from T_e relatively simple, but this is not the case for the converse. This is why reference curves have been implemented, to allow simple transitions between the temperature and energy content.

4.1.1

Saha-Boltzmann vs. rate equation

Initially, reference curves for the Saha-Boltzmann and rate equation models were produced for comparison. As the population ratios are dependent on both the plasma temperature and electron density, an iterative algorithm must be implemented. The population distribution for each temperature was calculated first using a trial electron density. The projected ion densities were normalized such that the total species density was equal to the solid density and a projected electron density was extracted from these values. The difference between the projected electron density and the initial density was then halved and added to the lower of the two, and the process was repeated until the correction required was below a chosen tolerance (the bisection method). In the case of the rate equation model, this is complicated slightly by the contribution of the n_(i+2) level to the calculation of the _n(i+1) level. To rectify this, a second iteration is included. For each trial electron density, the population distribution is calculated using trial populations for n_(i+2) first, and then by the populations just predicted.

This is repeated until the change in the populations is below a given tolerance, and the outer iteration over the electron density can progress.

Initial focus was on irradiances in the region of those seen experimentally^3;4;5 and the difference between the two models will increase with irradiance, so a comparison at the peak irradiance of ≈1 × 10¹⁰ W cm^–2 was conducted first, shown in figure 2. At these low irradiances, the two models overlap perfectly, with negligible contribution from photo-ionization. This implies that for the irradiances seen in experiment the rate equation model is applicable but offers no improvement over the Saha-Boltzmann model.

Figure 2:

Average ionization and energy density of aluminium as a function of temperature for the Saha-Boltzmann and rate equation models with an irradiance of 1 × 10¹⁰ Wcm^–2.

At higher irradiances, we see the differences in the models increase – see figure 3 – however significant changes are not seen below 1 × 10¹⁵ Wcm^–2. Significant deviation from the Saha-Boltzmann model at these temperatures indicates the plasma is no longer in equilibrium. Irregularities in the continuity of the higher irradiance curves are likely caused by instability in the iteration method.

Figure 3:

Average ionization and energy density of aluminium as a function of plasma temperature for the rate equation model at irradiances from 1 × 10¹³ Wcm^–2 to 1 × 10¹⁶ Wcm^–2.

4.1.2

The effects of continuum lowering

As with the rate equation model, reference curves for the Saha-Boltzmann model and a corrected form using the Stewart-Pyatt model were produced. These are shown in figure 4. The difference in the two models is relatively small, peaking at around 0.2 for Z_i, but the impact this will have on time-resolved measurements is unclear. Figure 5 shows the effect on the absorption coefficients. This is more significant as K_ib is proportional to and K_pi will now have contributions from higher ionization levels. This will result in more rapid heating of the target. The point at which photo-ionization drops below 10% of the total absorption shifts significantly from 18.5 eV to 26.7eV but K_ib dominates from 10 eV regardless.

Figure 4:

Average ionization and energy density of aluminium as a function of plasma temperature for the Saha-Boltzmann model with and without the Stewart-Pyatt ionization energy correction.

Figure 5:

Absorption coefficients for inverse bremsstrahlung K_ib, photo-ionization K_pi, and the combination of the two K_tot against plasma temperature for both the corrected (w/CL) and uncorrected (SB) Saha-Boltzmann model in aluminium.

4.1.3

Continuum lowering in the rate equation model

The effects of continuum lowering on the rate equation model have also been investigated. The same algorithm was implemented, with the ionization energies corrected using the Stewart-Pyatt model. The results for 1 × 10¹⁰ W cm^–2 are shown in figure 6. The correction has affected the Saha-Boltzmann and rate equation models equally, as would be expected, generating the same consistency between the models as before the correction.

Figure 6:

Average ionization and energy density of aluminium as a function of plasma temperature for the Stewart-Pyatt corrected Saha-Boltzmann and rate equation models with an irradiance of 1 × 10¹⁰ Wcm^–2.

Results for multiple higher irradiances are shown in figure 7. The effect of continuum lowering is present for all irradiances, with reduced impact at higher irradiances.

Figure 7:

Average ionization and energy density of aluminium as a function of plasma temperature for the Stewart-Pyatt corrected rate equation model at irradiances from 1 × 10¹³ Wcm^–2 to 1 × 10¹⁶ Wcm^–2.

4.2.

Single cell variant

Using the models detailed in section 3, a simple time-resolved scenario was constructed. A 10 nm slice of material, of unit area, was irradiated with time-dependent pulses of 46.9 nm radiation. The cell was assigned a temperature and energy content, and absorption coefficients were calculated accordingly. These were then used to calculate the reduction in intensity across the cell, and therefore the deposited energy. Finally, the energy content was updated and compared to the reference values to find the new cell temperature. This was repeated for each time step with the next irradiance value in the time-dependent pulse. As there were no calculations performed in each time step, it was trivial to run both the Saha-Boltzmann model and the Stewart-Pyatt correction simultaneously.

A sine-squared pulse of 1.2ns duration with a peak irradiance of 3 × 10¹⁰ Wcm^–2 was modelled initially, again to best match experimental parameters. The predicted temperatures and ionizations for both models are shown in figure 8. The small changes introduced by continuum lowering in temperature and ionization, married with the larger change in the absorption coefficient, have led to a significant increase in the final temperature, from 37.7 eV to 54.0 eV, and ionization, from 3.36 to 4.48, of the test cell. This is likely to have strong implications in the spatially-resolved simulations.

Figure 8:

Temporal evolution of the plasma temperature and average ionization of a 10nm aluminium cell for the uncorrected and Stewart-Pyatt corrected Saha-Boltzmann model.

4.3.

Density variations

Thus far the effects of temperature on the various models have been well considered, but the density has remained constant at n_solid. The spatially-resolved simulations in section 4.4 are hydrostatic and do not consider plasma expansion, so the effects of density are key to determining the validity of these results. As a simple test, a cell at 10% of n_solid was examined under the same conditions as before, the results of which are shown in figure 9.

Figure 9:

Temporal evolution of the plasma temperature and average ionization of a 10 nm aluminium cell at solid density and at 10% solid density for the uncorrected and Stewart-Pyatt corrected Saha-Boltzmann model.

The decrease in density has reduced both the final temperature and the final ionization for both models as expected. The most significant change however, is in the absorption coefficient – shown in figure 10. The total absorption coefficient for the plasma has reduced by an order of magnitude at early times and by nearly two at late times. This indicates that in the case of an expanding plasma, the lower density plume will absorb little of the incoming radiation and energy deposition will be dominated by the material at close to solid density. From this, we can conclude that the hydrostatic model has a good chance of being valid, with the plume contributing little to the absorption. These low density results are also of interest in the ionization of gas jet targets by extreme ultraviolet laser pulses.

Figure 10:

Temporal evolution of the total absorption coefficient of a 10 nm aluminium cell at solid density and at 10% solid density for the uncorrected and Stewart-Pyatt corrected Saha-Boltzmann model.

4.4.

Multi cell variant

Expanding the simulations to include spatial resolution was simple. The algorithm for each cell remained largely unchanged, but an additional calculation of the irradiance reaching each cell was added between them. This was calculated from the energy deposited in the previous cell. An example temperature distribution is shown in figure 11a. By varying the peak irradiance and integrating over the pulse we can produce a graph of fluence plotted against ablated depth – see figure 11b. Determining how to measure the ablation depth requires some thought. The simplest choice is a minimum plasma temperature or minimum ionization; melting temperature, boiling temperature, and 10% ionization are sensible choices. A minimum ionization was selected as the most likely condition for a complete solid-plasma transition.

Figure 11:

(a) Spatial distribution of plasma temperature for a peak irradiance of 3 × 10¹⁰ W cm^–2 and (b) predicted ablated depths as a function of incident fluence for an aluminium target.

Two main conclusions present themselves from figure 11b; there appears to be a maximum achievable ablation depth, and the ablated depth approaches this value rapidly (below 10 J cm^–2).