Simulations of capillary discharges

1. Simulations of capillary discharges Prague, November, 2012

2. INTRODUCTION • Owing to an extreme simple construction of the capillary discharges and to a broad range of their applications the capillaries have attracted a great deal of attention. • The main constitutive element of the capillary is a channel made in an insulator material and ended in the longitudinal direction by the two electrodes. An external low-inductance electric circuit produces the voltage between the electrodes and generates the electric current pulse in the channel. • The current heats the wall material and generates the magnetic field. As a result the prefilled capillary plasma and the plasma ablated from the wall is pinched causing the shock wave to propagate towards the axis and to form there the dense, high temperature short living filament. • The plasma filament parameters can correspond to the conditions for the lasing in the XUV and X-ray region. This is connected with one of the main applications of the capillary discharges.

3. Another important application of the capillary discharges is related to their property to form on the long time scale evolution inside the capillary the plasma density profile with a density minimum at the axis which provides the ultra-short laser pulse guiding (Ehrlich, et al., 1996). In this case the capillary plays a role an optical waveguide for a laser pulse (Tajima, 1985) providing its propagation over a distance greater than the defocusing length. • The interaction of intense laser radiation with plasmas is important • in a wide variety of applications such as the generation of coherent • short-wavelength radiation through high-harmonic generation , • X-ray lasers, gamma ray generation, and the development of novel • plasma-based accelerators. • For such applications the laser-plasma interaction length is • limited fundamentally by diffraction to lengths of the order of the Rayleigh range • where is the laser waist size. • In order to increase the distance over which the intensity of the laser pulse is maintained at a value close to that at its focus, it is necessary to channel the laser pulse in some manner.

4. Table-top soft x-ray lasers Electric current pulse Schematic view of setup Simulation Pin-hole images of the soft-x-ray emitting region On axis spectra of Cd capillary discharge plasma emission from 13.2 nm line of Ni-like Cd J. J. Rocca, Rev. Sci. Instr. 70, 3799 (1999)

5. A number of important applications, such as short-wavelength lasers and novel schemes for particle acceleration, involve the interaction with plasmas of ultra-short laser pulses with peak intensities in the range • The laser-plasma interaction length is limited by diffraction to distances of order the Rayleigh range. • The propagation of an intense laser pulse through a partially ionized plasma can also be limited by refrection. • In order to increase the distance over which the intensity of the laser pulse is maintained at a value close to that at its focus, it is necessary to channel the laser pulse in some manner. For example, it can be guided inside the initially performed hollow or prefilled with plasma narrow channel.

6. 1.0 GeV Beam Generation in Laser Interaction with Capillary Plasma Laser: 1.5 J/pulse Density: 4x1018 cm-3 Capillary: 312 mm diameter and 33 mm length 1 GeV beam: a0 ~ 1.46 (40 TW, 37 fs) CAPILLARY Peak energy: 1000 MeV Divergence(rms): 2.0 mrad Energy spread (rms): 2.5% Charge: > 30.0 pC W.P.Leemans et al, Nature Physics, 418 (2006)

7. Capillary discharges are an attractive method for forming a plasma waveguide. • In fast z-pinch discharges the very fast rising current causes the plasma to be pinched, driving a strong shock wave towards the axis. The collapsing annulus of highly ionized plasma can form a transient plasma channel. The channel may be formed in either initially-evacuated capillaries or gas filled capillaries. Hosokai et al. Opt.Lett., 2000, Fauser and Langhoff, 2000. • The major disadvantage of this approach is that the plasma channel exists for only a few nanoseconds which places severe restrictions on the allowable timing jitter between the start of the capillary discharge and the arrival of the laser pulse.

8. Physical model We use the approximation of two-temperature, one-fluid MHD. Owing to the large length-to-radius ratio of the capillary a one-dimensional approximation is considered . The following phenomena are essential for capillary discharge in evacuated channel: the interaction of the plasma with a wall, and the ablation of the wall material accompanied by the ionization of the produced gas the interaction of the magnetic field with the plasma (pinch effect). Important dissipative processes: electron and ion thermal conductivities, Joule heating, Nernst and Ettinghausen effects, radiation losses, viscosity. It is also necessary to incorporate the degree of ionization both into the equation of state and into the dissipative coefficients.

10. System of equations

11. Dissipative coefficients

12. areobtained in the so called two-polynominal approximation Expressions for are determined with the accuracy of a few percent, which is quite sufficient because the accuracy of the Landau approximation for the collision integral is of the order of The values of are calculated by Braginskii for for certain multiples of which is not very convenient when w runs through a continuous set of values. For example, values of and differ by more than a factor of 2. We take into account the possible considerable difference between and otherwise the accuracy of the resulting system of equations is incorrectly reduced.

13. Neutral particles

14. Equation of state and degree of ionization For the equation of state and the ionization degree, the approximation of local thermodynamic equilibrium is used separately for the electron and ion components. - is the chemical potential of an ideal free-electron gas -the ionization potential of a mean ion. 1<z<Z/2 -- Sommerfeld's formula in the Thomas-Fermi model for the ion shell Z/2<z<Z -- the formula for the hydrogen-like ionization potential is chosen, taking into account thescreening of the ion electric field by $(Z-z-1)$ electrons the simplified Saha formula is used, taking only neutral and once ionized atoms .

15. Thermodynamic properties V – удельный объем одного иона

16. Boundary conditions Free boundary

17. Различные типы динамики капиллярного разряда (физические процессы, которые определяют тип динамики плазмы) Динамика плазмы в капиллярных разрядах зависит от нескольких параметров: • Радиуса капилляра • Параметров внешней электрической цепи • Материала стенок • В случае заполненного капилляра – начального давления заполняющего капилляр газа, его атомного номера и веса • 1. Роль магнитного поля: • а) основную роль играет пинч-эффект • б)влиянием магнитного поля можно пренебречь • 2. Испарение стенок капилляра

18. Пинчевой капиллярный разряд Радиус Начальная плотность Ar Капилляр из полиацетата Электрический ток Столкновительное возбуждение неоноподобных ионов аргона Рокка и др. 1994

19. Моделирование пинчевого капиллярного разряда t> 20нс 30-40% I около стенки <50% I по аргоновой плазме 5% I по керну

20. Comparison of the simulated trajectories of plasma elements with the experimentally observed radius of radiative plasma

21. Radial distribution of the plasma parametersatt=40 ns 1.The increase of on the axis. 2. I is separated in two spatially distinct components: near the axis, on the perepheiry of the discharge. 3.The plasma density drop corresponds to the boundary between Ar and ablated plasma

22. Radial distribution of plasma parameters in the kernel

23. Trajectories of elements of argon plasma and of ionized ablated material for different

24. Conclusion • Kernel, close to the axis is likely to be the place, where the amplification can take place. It is uniformly filled with hot dense plasma. • It is not pinch effect due to the lack of electric current in the center of the channel. • There is no MHD instabilities typical to Z-pinches So the plasma behavior in the kernel can be described in the frame of hydrodynamics. • Electric current distribution is nonuniform. A fraction, flowing near the wall causes heating and ablation of the wall material. Another fraction, localized near the axis, causes plasma acceleration and compression at the initial stage of discharge.

25. New type of the capillary discharge waveguide in the gas-filled channel was investigated bySpence and Hooker, Phys.Rev. E, 2000. The current pulsehad a peak of 250 A and a duration of order 200 ns. A hydrogen-filled capillary waveguide was used to guide laser pulses with peak intensities of greater than . through 20- and 40-mm long capillaries with pulse energy transmitions of 92% and 82%, respectively. • It is important to understand the mechanism by which the guiding electron density profile is formed. We performed MHD simulations of the plasma dynamics of a hydrogen-filled capillary discharge. The mechanism of formation of the guiding electron density profile is found to be very different from that of Z-pinch capillary discharge.

26. Hydrogen capillary (nonpinching discharge) Alumina capillary filled with hydrogen Three stages of the plasma evolution 1. Magnetic field penetrates the plasma on a time scale of 1ns. Pinching of the plasma is negligibly weak. The plasma is heated and ionized locally. Radial distributions of plasma parameters are homogeneous. 50ns 2. A redistribution of the plasma temperature and density occurs during t=50-80ns. Thermal conduction becomes significant. 3. Quasi-state equilibrium at a given electric current The plasma temperature has its maximum on the axis, because the Ohmic heating is balanced mainly by thermal conduction to cold wall. This results in an axial minimum in the electron density profile.

27. Temporal evolution of the axial electron temperature and density After t=80ns the axial electron density is constant. The axial electron temperature slowly decreases with time. At t=60ns Experimental value

28. The measured and calculated electron density profiles at t=60ns t=55 ns t=60 ns t=65 ns Measured profile corresponds to an average electron density profile over several ns. Allowing for errors in measurement of the time t of approximately ns, and averaging over the duration of the probe pulse used in the measurements, the simulations are in good agreement with the measurements.

29. For practical applications of H-filled capillary discharge waveguides, it is important that the capillary has a long lifetime. The thermal flux due to electron thermal conduction Partial ionization of the wall and heating of the free electrons The lattice heating by energy exchange with free electrons Direct heating of the lattice due to ion-lattice thermal conduction plays only a secondary role. For t=100-150ns, the degree of ionization of atoms in the wall material in a layer depth. no melting or disruption Lattice temperature = In experiment after discharge pulses the increase in capillary diameter was

30. Simple model of plasma equilibrium For t>80 ns the following conditions are fulfilled: • There is no screening of the axial electric field, and consequently it is uniform across the capillary • The magnetic field pressure is much less than the plasma pressure, and hence the plasma pressure can be considered to be constant across the capillary • The electrons are unmagnetized • There is no difference between the electron and ion temperatures The equation for heat flow boundary conditions

31. Assuming Coulomb logarithms to be constant, electron thermal conductivity electric conductivity we consider Assuming that Introducing a new variable and new function determined by we obtain The unique nontrivial solution of this equation is found numerically.

32. Assuming Coulomb logarithms to be constant, electron thermal conductivity electric conductivity we consider Assuming that Introducing a new variable and new function determined by we obtain The unique nontrivial solution of this equation is found numerically.

33. Assuming Coulomb logarithms to be constant, electron thermal conductivity electric conductivity we consider Assuming that Introducing a new variable and new function determined by we obtain The unique nontrivial solution of this equation is found numerically.

34. The pressure is constant across the capillary The electron density profile The axial electron density

35. t=100ns simulation simple model The equilibrium state of the capillary discharge depends only on the electric current, the capillary radius, the total mass of hydrogen per unit length of the capillary. for 3-5mm long capillaries The time scale of outflow

36. Laser pulse guiding inside the capillary A Gaussian beam will propagate through a plasma channel with a parabolic density profile of the form with a constant spot size - the classical electron radius From the equilibrium model is determined from a parabolic fit to the measured electron density profile

37. The plasma pressure is always much greater than the magnetic pressure, and as such the pinch effect can be neglected. • Three stages of the plasma evolution have been identified. • There is a good agreement between the simulated density profile and measured in experiment. • The results of MHD simulations allowed us to formulate simple model. • The matched spot size depends only on the capillary radius, the initial ion density and the mean ion charge • The evolution of a hydrogen filled capillary discharge waveguide is different compared with previously discussed.

38. Capillary discharge in evacuated channel • Plasma inside the channel is created from the ablation and ionization of the wall material. • The initial conditions: after the threshold voltage is reached, a strongly nonuniform surface breakdown happens at the wall. • During this stage 3D-effects and effects of plasma quasineutrality disturbance play role. • This stage is short in time and does not affect plasma parameters at a later time. • For a correct description of the discharge, it is actually sufficient to allow only for the fact that the discharge begins at the wall surface. • It happens when the formula for a plasma electric conductivity is used both for high and low temperatures.

39. Evacuated polyacetal capillary Capillary radius The capillary is of polyacetal The capillary plasma is perturbed by a current pulse Shin et al, Phys.Rev.E, 50, 1376, 1994

41. The electric current flows in the vicinity of the channel axis during the first stage of the discharge, then the radial distribution of the electric current density becomes smooth.

42. The radial distributions of the plasma parameters at t = 200 ns Typical plasma parameters distribution in the quasi-equilibrium state, when there is both mechanical equilibrium and thermal quasi-equilibrium (Joule heating is balanced by the heat outflow due to thermal conductivity and radiation losses). Consequently, after the transition stage, the Capillary plasma is in equilibrium.

43. MHD instabilities Our model gives somewhat overestimated plasma temperature and underestimated density of the discharge plasma In these conditions different ideal and dissipative MHD instabilities can be expected to occur inside the channel. These instabilities can lead to the excitation of MHD turbulence, and a change in the transport process. We investigated the possible influence of an anomalous transport on the parameters of the considered capillary discharge. By incorporating the additional turbulent transport as well as the correspondent Joule heating we decrease the plasma temperature and increase the plasma density.

44. Conclusion • The capillary discharge fills the channel with plasma of uniform density and slightly nonuniform temperature. • Plasma is in quasi-equilibrium. • The situation is unstable from the point of view of typical MHD instabilities of Z-pinches.

45. THE RESULTS OF MHD SIMULATIONS Capillary diameter The capillary plasma is perturbed by a current pulse • t=5-10ns - the fast plasma compression • the ablation of the wall is due to the heat • flux from the central region • 3. the quasi-equilibrium stage: mechanical equilibrium (the Ampere force is balanced by gradient of the plasma pressure) thermal quasi-equilibrium (Joule heating is balanced by the heat outflow due to thermal conductivity and radiative energy losses) Janulevich et al., J. Opt. Soc. Am. B-Opt. Phys. 20, 215, 2003.

46. The radial distributions of electron density and temperature for different moments of time • The density profile is concave with the minimum on the axis, which insures the • pump laser guiding. • 2. The density on the axis increases with decrease of the capillary diameter. • For 1mm capillary • For 0.5 mm capillary

47. Another important application of the capillary discharges is related to their property to form on the long time scale evolution inside the capillary the plasma density profile with a density minimum at the axis which provides the ultra-short laser pulse guiding (Ehrlich, et al., 1996). In this case the capillary plays a role an optical waveguide for a laser pulse (Tajima, 1985) providing its propagation over a distance greater than the defocusing length. • The interaction of intense laser radiation with plasmas is important • in a wide variety of applications such as the generation of coherent • short-wavelength radiation through high-harmonic generation , • X-ray lasers, gamma ray generation, and the development of novel • plasma-based accelerators. • For such applications the laser-plasma interaction length is • limited fundamentally by diffraction to lengths of the order of the Rayleigh range • where is the laser waist size. • In order to increase the distance over which the intensity of the laser pulse is maintained at a value close to that at its focus, it is necessary to channel the laser pulse in some manner.

48. In the plasma waveguides a plasma is formed with a transverse electron density profile with a minimum on the axis of propagation, corresponding to a transverse refractive index profile that decreases with radius providing a focusing effect. A lowest order Gaussian beam will propagate through a plasma channel with a constant spot size is the classical electron radius The laser matched waist size Pondermotive and relativistic effects are neglected. The plasma channel is not further ionized by the propagating pulse.

49. The fast-Z-pinch (Hosokai et al.,2000), gas-filled (Spence et al., 2001), and ablation (Ehrlich et al., 1996, Levin et al., 2005) capillaries have been used for the laser guiding and electron acceleration. • The high quality GeV range energy electron beams have been obtained in the high intense laser interaction with plasmas generated inside the capillary filled with hydrogen-gas (Leemans et al., 2007, Karsch et al., 2007, Rowlandset al., 2008) and inside the ablative capillary (Kameshima et al., 2008).

50. Our aim is to study the long time operation of the capillary in the experiments on the laser electron acceleration in the plasma formed by a discharge inside the ablative capillary. • The experiment set up and the results on the capillary operation. • 2. The results of magneto hydrodynamics (MHD) simulations of • the capillary discharge laser pulse guiding • 3. The relativistic electron acceleration • 4. Discharge effect on the capillary wall erosion. .