On the distribution function of electron spectra from hot laser plasmas

The emission of electrons from hot plasmas generated in the interaction of ultra-short (and ultra-high intensity) laser pulses with matter is often characterized by the so-called ‘hot electron temperature’. In this article it is shown that this number is not unambiguous. The reason is the following: to assign a temperature to an electron spectrum, it is necessary to describe the spectrum with a distribution function. However, different types of distribution functions are in use, e.g. the Boltzmann or Maxwell distribution, leading to different electron temperatures in spite of providing nearly the same form of the electron spectrum. For this reason, the main characteristics of all these distribution functions are presented in this article and compared. Depending on the distribution function used, the value of the hot electron temperature varies by up to 30% and in extreme cases by more than a factor of four. This fact should always be kept in mind when comparing values of hot electron temperatures. In addition, the reasons for using equilibrium distributions to describe the characteristics of laser-produced electrons—although probably no thermodynamic equilibrium is prevailing—are discussed.


Introduction
In the past decades the intensity of short-pulsed laser beams has continually increased.Since long it is possible to produce light intensities up to 10 21 W cm −2 and more using shortpulsed laser systems (100 fs = 10 −13 s and shorter) [1][2][3][4][5][6].Due to the interaction of such laser pulses with matter, plasmas are produced with electron temperatures of up to several MeV [5,6].On the other hand, more and more short-pulsed lasers with rather low or medium intensities in the order of 10 11 W cm −2 up to 10 14 W cm −2 are used for materials processing [7][8][9][10][11][12][13][14][15][16][17].In all cases, the so-called hot electron temperature is used to characterize the electron emissions.For the determination of this Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.electron temperature, it is necessary to assume a certain distribution function of the emitted electron spectrum in order to fit this function to the measured electron spectrum.Throughout the literature, several different distribution functions are used.These are usually equilibrium functions, although no good reasons for assuming a thermodynamic equilibrium are given.This contradiction is discussed in detail in the beginning of this article.Afterwards, the main characteristics of the different distribution functions are summarized while in the last part it is shown that most of them can be fitted to each other by modifying their temperature and absolute amplitude.

Electrons and ions in thermal equilibrium or not
It is well known that the production of energetic particles in laser-produced plasmas is most efficient when a laser pulse preceding the main pulse is present-especially in the highintensity regime [18][19][20].This pre-pulse is about 10 −4 smaller in intensity than the main pulse, namely 10 15 W cm −2 up to 10 16 W cm −2 .Nevertheless, this intensity is sufficiently large to produce a pre-plasma with an electron temperature in the sub-keV range [21] resulting in an electron density n el of up to 10 21 cm −3 [22].This pre-plasma is usually generated from about a picosecond (10 −12 s) [4] up to a nanosecond (10 −9 s) [23] before the main laser pulse arrives.During this time, collisions between the electrons and ions take place and thus a thermal equilibrium begins to form.The time τ el that is necessary to establish complete thermal equilibrium is given for the electrons by [24] τ el = 330 fs • ( T el 0.1 keV with T el the temperature of the electrons, ln Λ the coulomb logarithm that takes values between 5 and 15 [24] (here estimated as 10), and n el the electron density of the plasma.Using the values given above yields τ el = 33 fs.After this small amount of time, the electron component of the pre-plasma represents an ideal gas in thermal equilibrium.According to Kittel [25], the energy distribution of such particles is given by a classical Maxwell distribution for particles with three degrees of freedom: with f (E)•dE the number of particles with a particle energy between E and E + dE, k Boltzmann's constant, and T the thermodynamic temperature of the ideal gas.This assumption of a Maxwell distribution for the electron component in a preplasma is often made for calculations performed to simulate the interaction of the main laser pulse with the pre-plasma [26,27].As can be seen by the results of equation ( 1), this assumption is well justified in case the pre-plasma has a sufficiently high electron density.In case of an underdense plasma (n el ≪ 10 21 cm −3 ) this is not true any longer, as reported by Ditmire [28]: E.g. n el = 10 16 cm −3 yields τ el = 3.3 ns.The physical reason is that the collision frequency decreases with decreasing particle density.
For the ion component of the pre-plasma the following applies: Due to the small mass of the electrons compared to the mass of the ions, at first the electrons are accelerated by the electro-magnetic field of the laser light.Therefore, the heating of the ions takes place via collisions with the electrons.The time that is necessary to establish thermal equilibrium of the ions with the electrons is about 1000 times longer [24] than the time given in equation (1).Therefore, it depends on the time duration between the pre-pulse and the main pulse (see above), whether the ion component of the pre-plasma is or is not in thermal equilibrium with the electrons at the time of the arrival of the main laser pulse.
The duration of the interaction of the main laser pulse with the plasma is quite short compared to the thermalization time of electrons at temperatures in the order of 1 MeV instead of about 0.1 keV due to the pre-pulse (see above).For n el = 10 21 cm −3 , equation (1) yields τ el = 33 µs (3.3•10 −5 s).This time is quasi endless compared to the duration of the main laser pulse: about 100 fs (10 −13 s).Therefore, during the presence of the main laser pulse no interaction takes place between the particles of the plasma.They are only coupled via collective interaction mechanisms heating the electrons: resonant absorption of the laser light in the plasma.
In summary, there are generally no good reasons to assume that high-intensity laser-heated plasmas are in a state of thermal equilibrium, neither the whole plasma (ions and electrons) nor the electron component alone; if at all the electron component of the pre-plasma is in thermal equilibrium at the time of the main laser pulse.Nevertheless, the experimental data of measured electron and ion spectra almost always show characteristics of a Boltzmann-like distribution function: An exponential decrease in the electron number with increasing electron energy.For that reason, throughout the literature, distribution functions with that characteristic, e.g.equation (2), are used to fit the measured data.These fits deliver the parameter temperature that is used to characterize the particle emission of the laser produced plasma and thus the plasma itself.Although physically, there is no justification to speak of a thermodynamic temperature (as was discussed above), this method has become very useful and is widely accepted.In the next section, an overview of the different distribution functions being used is given.

Overview of electron distribution functions
For the description of measured electron spectra from laserproduced plasmas, usually the superposition of two parts with different temperatures is used [26,29].The two parts are assigned to different acceleration mechanisms and the higher temperature is called the 'hot electron temperature'.As this 'hot electron temperature' is almost independent of the lower temperature part of the spectrum, for the following comparison of distribution functions it is sufficient to use only one part.
All functions under consideration are normalized to an area of unity: -The classical Maxwell distribution f Max,clas , see equation ( 2), is supposed to be the correct one in case of a thermal equilibrium prevailing, see above.-The relativistic border case of the Maxwell distribution f Max,bord is sometimes used when quite high electron temperatures are observed [30]: -The relativistically correct Maxwell distribution [31] f Max,relat , that covers the two border cases given in equations ( 2) and ( 3), is quite seldomly used [32].The reason may be that it is mathematically not simple.It reads with m el the electron rest mass, c the speed of light in vacuum, and K 2 (x) the modified Bessel function of the second order.-The Boltzmann distribution f Boltz is most often used because it is the simplest one: -The modified Maxwell distribution f Max,mod , also sometimes used in the literature, is given by [33][34][35][36][37].
The normalization constant C is determined using integral No. 3.478 from Gradstein and Ryski [38] and leads to with Γ the gamma function and µ ∈ {1 … 2.5}: µ = 1 is the border case of the classical Maxwell distribution and µ = 2.5 represents the so-called super-Gaussian distribution.This distribution results from calculations: The more dominant the absorption of the laser light in the plasma due to collisions, the higher the value of µ [34].Although the collisional absorption only dominates at laser intensities up of 10 15 W cm −2 , this distribution function is considered here.
In figure 1, the comparison of the considered distribution functions is shown.For their characterization, the quotient of the mean electron energy Ē and the temperature in terms of energy k•T are given in the legend.This quotient is independent of the temperature itself, except for the relativistic correct distribution.This one changes its shape with the electron temperature: At the temperature k•T = 100 keV, which is much smaller than the energy equivalent to the electron rest mass, m el c 2 = 511 keV, it is quite similar to the classical Maxwell distribution and at the temperature k•T = 1000 keV, which is larger than m el c 2 , it is quite similar to the relativistic border case of the Maxwell distribution, as expected.
As discussed above, the different distribution functions are assigned to different heating mechanisms.In addition, the resulting electron temperature is used to characterize the emission from laser-produced plasmas.Therefore, it is desirable to determine which distribution function was present when an electron spectrum was measured.When attempting this, the problem described in the next section arises.
Although not included in this study, it shall be noted that several more distribution functions are discussed-with different dependencies of the effective electron temperature.Some of these dependencies are: • the degree of plasma inhomogeneity [36,39] • the influence of the laser radiation itself [40,41] • as well as electromagnetic instabilities [42].This short is, however, is very likely not complete.

Mutual representation of different distribution functions
The measurement of an electron spectrum only provides the shape and absolute height of the distribution function.From this shape, the type of distribution function has to be determined, resulting in a temperature.However, a fixed shape of a distribution function (e.g. the classical Maxwell distribution) can be represented by almost all the other distribution functions by changing their temperature and amplitude, as shown in figure 2. The differences of the presented distribution functions are largest at very low electron energies.Therefore, the measurement of the electron radiation, especially at very low electron energies (below 200 keV), has to be very precise.However, this is exactly what cannot be achieved by most of the electron spectrometers that are able to measure the very short-pulsed radiation emitted from laser-produced plasmas.In conclusion, it cannot be decided whether, e.g. a Maxwell distribution with a temperature of 100 keV or a modified Maxwell distribution (µ = 1.5) with a temperature of 270 keV is present.

Conclusions
For the reasons given above, it is always necessary to state which type of distribution function was used to assign an electron temperature.Doing so, it is always possible to relate the temperature of a measured electron spectrum to a temperature that would have been determined using another distribution function.

Figure 1 .
Figure 1.Comparison of distribution functions according to equations (2)-(6) for an electron temperature of 100 keV (a) and 1000 keV (b).For the modified Maxwell distribution, two cases for µ were chosen: µ = 1.5 and 2.5.All functions are normalized to unity area.

Figure 2 .
Figure 2. The classical Maxwell distribution can be well represented by most of the other distribution functions by changing their (see legend) and absolute height.