Capillary Discharge Parameter Assessment for X-ray Laser Pumping

This paper assigns optimum capillary discharge characteristics with respect to reaching the maximum emission gain on wavelength l = 18.2 nm and corresponding to Balmer α transition H-like carbon. The computer modelling of the capillary discharge evolution is carried out using the NPINCH programme, using a one-dimensional physical model based on MHD equations. The information about the capillary discharge evolution is processed in FLY, FLYPAPER, FLYSPEC programmes, enabling the population to be modelled on specific levels during capillary discharge.


Introduction
Non-stationary plasma of a fast capillary electrical discharge was studied as a potential active medium for a soft X-ray laser.The aim was to achieve an optimum set of characteristics for lasing at 18.2 nm wavelength H-like carbon C 5+ in an alumina capillary.Optimum in the sense of reaching a maximum gain value for each set of parameters.
Capillary pinch dynamics is determined by many selected parameters: capillary geometry (radius and capillary length), the substance of the capillary, the initial filling pressure, and the electric current time dependance.This dependance, in particular, is given by an electric circuit which is joined to the capillary.
A capillary discharge Z-pinch acting as a medium for a soft X-ray laser uses ASE, the "Amplified Spontaneous Emission" effect, and electron-collisional recombination pumping.The main variable for ASE is the gain, and the most important goal of this paper is to find specific parameters of a capillary to obtain the maximum gain.
Electron-collisional recombination, sometimes referred to as "three body" recombination, is the inverse process to electron-collisional ionization from excited levels.The combined recombination and radiative downward cascading process is illustrated by the equation It is assumed that the ions exist plentifully in the initial state, which is one stage of ionization higher than that of the lasing ions.Note that this is not a self-contained population-replenishment scheme.Hence, replenishment depends upon further reionization, which is provided in our case by the release of the next current pulse.
It was assumed that ablation from inner walls has only an imperceptible effect, and so it was not in-cluded in the calculations.The capillary discharge plasma quantities were calculated by means of the NPINCH [2] code under the one-dimensional, twotemperature, one-fluid MHD approximation.The output data was then processed by means of the FLY code, which enables the creation of a history file of the population on the individual levels along the capillary axis.On the basis of the populations history, the gain and the gain history can be calculated.

Optimization
Optimization itself involves searching the maximum in the four-dimensional space of the parameters.Two parameters relate to the current-impulse shape, the third parameter is initial density (or pressure), and the fourth is the radius of the capillary.
The right choice in the range of parameters is of fundamental importance.
The results of our study should be instrumental for experimenters in the CAPEX-U [4] project.The range of parameters therefore has to be related to their experimental equipment, with the view to verifying the results.Consequently, the choice of the first parameter was obvious because an alumina capillary r = 1.6 mm in radius is used in CAPEX-U.The choice of the current pulse had to coincide with the source in CAPEX-U.The electric current in a capillary was fitted from measuring by function.
The chosen parameters are therefore associated with equation 2: current peak value I max and dI/dt |t=0 .Two cases were calculated I max ≈ 30 kA and I max ≈ 60 kA, on behalf of dI/dt |t=0 the area within the range of value 0.5 • 10 12 -2.5 • 10 12 A/s was chosen.The last parameter is initial density N, according to the theoretical analysis and in order to achieve the gain maximum, the maximum electron density for carbon has to be N e ≈ 3 • 10 20 .

Gain evaluation
The selected parameters for each case served as input data for NPINCH.A very useful output of the calculation was the dens.datfile, which contained a time history of electron and ion density, and electron temperature.This file futher served as input for the FLY and the FLYPAPER code, and by means of these we obtained the time histories of populations on set levels N 3 for level n = 3 and N 2 for level n = 2 H-like carbon ions.Inversion function F [3] is represented by Equation 3, where the statistical weights g u /g l are for our transition 9/4, and N l , N u presents the lower and upper laser state population densities.
The cross-section with corresponding coefficients for our transition λ ul = 13.381 8 • 10 −7 cm, A ul = 5.72 • 10 10 s −1 can be illustrated as And finally, G after cross-section substitution can be illustrated as For each calculation, we listed the parameter data, information about the maximum value and the time of reaching gain, the pinch time (the time when the maximum of electron density N e is reached).

I max = 30 kA
The results for I max = 30 kA, are in Fig. 1.The maximum gain value was achieved at dI/dt |t=0 = 1 • 10 12 A/s, initial carbon density 1.4 • 10 17 cm −3 and G max ≈ 0.05 cm −1 .The calculated data resembled the Gauss function, so they were fitted by this function, and the deviation between the calculated data and the fitted function was minimal.The most information is shown in Fig. 2. The best gain was achieved when the time of reaching current maximum was equal to the pinch time.Now we can take a closer look at a particular case to show the history of certain variables.The parameters of a capillary discharge N 0 = 4.0 • 10 17 cm −3 , dI/dt |t=0 = 2 • 10 12 A/s, r 0 = 1.6 mm and I max = 60 kA have been chosen.We will plot the curve for an individual quantity, such as electron density, electron temperature, inversion of population, and gain.After that, a rapid drop in these quantities follows due to adiabatic expansion.The electron density drops by nearly two orders in 5 ns and the electron temperature drops by nearly 90 eV.These are ideal conditions for creating population inversion at the desired levels.
Figure 8 shows that at time 46.5 ns, 2.5 ns after pinch time, both the inversion function and gain had an above zero value.At time 47.2 ns, the gain maximum G max = 3.2 cm −1 is reached, and just about one ns later the inversion function also reaches its maximum and starts to fall steadily.However, the gain begins to drop very rapidly, due to rapid density decay of the whole mass (both electrons and ions, which are lasing) along to axix.It follows from the gain behaviour that the hypothetical laser pulse could be 2 ns in length.
How does the initial density influence the pinch behaviour?With increasing initial density the maximum density also increases.At the same time, the pich time is delayed.The effect is totally contrary to the electron temperature.As the initial density increases the maximum electron temperature value decreases.

Conclusion
The optimal conditions for achieving maximum gain for initial density and electric current pulse shape have been obtained for two different current maxima, I max = 30 kA, I max = 60 kA and capillary radius r = 1.6 mm.Figures 2 and 4 show clearly that the time of achieving the current maximum and the pinch time have to be synchronized for optimal current differentiation at zero time, so that both times will be almost identical.

Fig. 1 :Fig. 2 :
Fig. 1: Maximum Gmax against initial density for various current derivations at Imax = 30 kA.The data is fitted by the Gauss function

Fig. 3 :Fig. 4 :
Fig. 3: Maximum Gmax against initial density for various current derivations at Imax = 60 kA.The data is fitted by the Gauss function

Table 1 :
Parameters of pulse shape for Imax = 30 kA