Numerical Modeling of Materials Processing Applications of a Pulsed Cold Cathode Electron Gun

J.I. Etcheverry

Depto. de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina

O.E. Martínez

Depto. de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina

N. Mingolo

Depto. de Física, Facultad de Ingeniería, Universidad de Buenos Aires, Paseo Colón 850, 1063 Buenos Aires, Argentina

Abstract

A numerical investigation of the application of a pulsed cold cathode electron gun to materials processing is performed. A simple semi-empirical model of the discharge is used, along with backscattering and energy deposition profiles obtained by a Monte Carlo technique, to evaluate the energy source term inside the material. Numerical solution of the heat equation with the calculated source term gives useful information on melting and vaporization thresholds, melted radius and depth, and on the dependence of these variables with processing parameters such as operating pressure, initial voltage of the discharge and cathode – sample distance. Numerical results for stainless steel are presented, showing the need of several modifications on the experimental design in order to achieve a better efficiency.

I. INTRODUCTION

Electron beams are an essential tool in many applications of great practical importance, such as surface treatments, electron lithography, etc. These applications usually rely on the deposition of the electron beam energy in a thin layer adjacent to the surface of the treated sample.

Electron beams are usually obtained by means of hot cathode electron guns. However, several types of cold cathode glow discharges have been used to produce continuous wave [1,2] as well as pulsed electron beams [3–6]. Among the investigated applications of the electron beams produced by this kind of glow discharges are annealing [7–9], thin film deposition [10–12], and the etching of diamond films [13]. One advantage of cold cathode glow discharges is its less stringent vacuum requirements, that allows to obtain high energy processing with low temperature chemical kinetics.

In particular, Ranea Sandoval et al. [5] reported a cold cathode electron gun that is operated in a pulsed regime, at accelerating potentials up to 60 kV, and yielding current densities of about 20 A/cm^{2}. Mingolo et al. [14] applied this cold cathode electron gun to materials processing, showing amorphization in samples of Mg_{70}Zn_{30} after irradiation with pulses of an energy greater than an specific threshold.

In a previous work [15] we presented a simple semi–empirical model of the discharge of this kind of electron gun. The model showed good agreement with experimental results, and predicted interesting phenomena related to the self focusing of the electron beam. This self focusing allows to vary the dose of energy given to the material by adjusting the cathode– sample distance. The model also predicts that the dynamic change of the focusing distance during the electron beam pulse allows to obtain energy pulses much shorter than both the current and voltage pulses. This effect is produced, for some processing conditions, by the rapid increase of the irradiated area with time.

In that work we established also some analytical estimates of energy and initial voltage thresholds for melting of the irradiated sample. The validity range of these thresholds estimates was limited to small electron energies (at least for light materials), and irradiated areas much smaller than the cathode area. Also, the results were valid only for the central region of the irradiated spot. In this work, we will relieve these limitations by a comprehensive numerical study of the heat equation in the material and of the way the electron beam energy is deposited in the sample.

As the stopping power for the electrons and the scattering do not depend significantly on the state of the irradiated material (liquid or solid) [16], the study of the energy deposition process does not require the simultaneous analysis of the evolution of the material (temperature, phase changes, etc.). Hence, the analysis of the interaction between the electron beam and the material will be carried out in two consecutive steps: the deposition of the beam energy inside the material and afterwards the heating of the material produced by the incoming energy.

The reflectivity and the energy dissipation curves for each material have been obtained by means of a Monte Carlo technique [17]. These techniques have been shown to be a valuable and accurate tool for electron beam energy deposition calculations [18–21], and have been chosen here as a method without adjustable parameters that can give reliable results even for complex materials.

The thermal problem consists in the solution of the heat equation inside the material, with the previously calculated incoming energy flux. It has been solved numerically to obtain typical temperature profiles, maximum temperatures and melted pool radius and depths for different processing parameters (cathode – sample distance, peak voltage of the discharge, gas pressure, etc.).

For clarity, we devote separate sections to summarize the discharge model, the employed Monte Carlo method, the identification of the thermal problem, the numerical solution technique, and finally to discuss the numerical results.

II. MODEL OF THE COLD CATHODE ELECTRON GUN DISCHARGE

The cold cathode electron gun we are considering is described in detail by Ranea Sandoval et al. [5]. It can produce pulses in the range 0.2 μs – 30 μs, with high currents up to about 900 A at accelerating voltages of about 65 kV. It consists of a 7.5 cm diameter aluminum cathode enclosed in a dielectric shield. This cathode is introduced in a low pressure environment (usually an inert gas) and is connected to a high negative voltage source. The electron gun and its mounting circuit are depicted in Figure 1. With the spark-gap open, the high voltage source V_{0} charges the capacitor C. When the spark-gap (T) is triggered, the capacitor discharge through the electron gun produces a high voltage pulsed glow discharge.

A very thin high electric space charge region (less than 1 cm), known as the cathode sheath, develops adjacent to the cathode. Practically all the discharge voltage drops in this region, producing electric fields of several tens of kV/cm that accelerates ions towards the cathode, and electrons in the opposite direction to form an electron beam [22]. The electrons are emitted by the cathode when it is bombarded by the accelerated ions and fast neutral atoms. The electrical field free region appearing farther from the cathode (negative glow) occupies almost all the cathode – anode region (tens of centimeters) [5]. In this region the self induced magnetic field strongly focuses the electron beam, with a focusing distance typically around 20 cm to 60 cm depending on the discharge parameters (current, voltage, etc.). The sample can be located within this region, without significantly disturbing the discharge. By choosing the precise location of the sample, the diameter of the treated area can be adjusted, typically between 1 mm and 1 cm.

In work [15] we have shown, based on evidence from refs. [5,14], that current i(t) and voltage drop over the gun v(t) can be adequately calculated by assuming the relationship:

i(t) = av(t)^{k}p^{m} k = (3 ± 0.15), m = (2.2 ± 0.2)  (1)

where p is the gas pressure, and a is a constant that depends on the oxidation of the cathode, on the repetition rate of the discharge, on the O_{2} partial pressure, etc., and that it is determined experimentally. For the electron gun circuit shown in Figure 1, the following expressions are obtained:

v(t) = v_{0} (1 + \frac{ap^{m}(k − 1)v_{0}^{k−1}t}{C})^{− \frac{1}{k−1}}  (2)
i(t) = ap^{m}v_{0}^{k} (1 + \frac{ap^{m}(k − 1)v_{0}^{k−1}t}{C})^{− \frac{k}{k−1}}  (3)

In reference [15] we have shown that the approximate shape of the electron beam can be deduced from the assumptions that the cathode sheath is very thin compared to the focusing distance of the electron beam, that the electric field is negligible in the negative glow region, and that electrons acquire an energy approximately equal to ev(t) in the cathode fall. With these assumptions, the radial shape of the electron beam can be estimated from the trajectory of an electron initially at the edge of the beam, with initial velocity given by the non relativistic expression u_{0} = \sqrt{2ev/m}, and accelerating radially inwards due to the magnetic field generated by the electron beam. Using cylindrical coordinates, with z = 0 at the cathode surface, the electron velocity is represented by u = (\frac{dr}{dt}, 0, \frac{dz}{dt}), and the non relativistic Newton equations for the electron give (m and e are the electron mass and charge, respectively, and μ_{0} is the vacuum magnetic permeability):

m\frac{d^{2}r}{dt^{2}} = −\frac{μ_{0}ie}{2πmr} \frac{dz}{dt}  (4)
m\frac{d^{2}z}{dt^{2}} = \frac{μ_{0}ie}{2πmr} \frac{dr}{dt}  (5)

where the magnetic field B = (μ_{0}i)/(2πr) was calculated using Ampère theorem. The initial conditions for an electron at the edge of the beam are:

r(0) = r_{0}  ṙ(0) = 0 z(0) = 0 ż(0) = u_{0}  (6)

where r_{0} is the cathode radius, and u_{0} is the initial velocity (assumed to be normal to the cathode surface).

In ref. [15] we chose to approximate the solution of these equations, in order to obtain explicit expressions for the magnitudes of interest. Here we will arrive to an exact implicit relation yielding z(r). To compute the irradiated spot radius, we will have to numerically invert this relation.

From equations (4), (5) we trivially deduce the conservation of the electron energy, ṙ^{2} + ż^{2} = u_{0}^{2}. Using this fact, and changing the independent variable from t to r, we have:

ṙ(r) = \sqrt{u_{0}^{2} − (u_{0} − A log \frac{r_{0}}{r} )^{2}}  (7)
ż(r) = u_{0} − A log \frac{r_{0}}{r}  (8)

where A = \frac{μ_{0}ie}{2πm}. The equation for the trajectory is immediately obtained:

\frac{dz}{dr} = \frac{u_{0} − A log(r_{0}/r)}{\sqrt{u_{0}^{2} − (u_{0} − A log(r_{0}/r))^{2}}}  (9)

Defining r^{∗} = r_{0} exp(−u_{0}/A), and integrating between r and r_{0} we finally get:

z(r) = ∫_{r}^{r_{0}} \frac{dr log(r/r^{∗})}{\sqrt{u_{0}^{2}/A^{2} − log^{2}(r/r^{∗})}} = r^{∗} ∫_{0}^{√\overline{u_{0}^{2}/A^{2}−log^{2}(r/r^{∗})}} exp(\sqrt{\frac{u_{0}^{2}}{A^{2}} − s^{2}})ds  (10)

Notice that z(r) depends on t through u_{0}(v) and A(i) and the time dependence of the voltage and the current. Hence the electron beam shape depends on t, through the dependence on t of the voltage and the current. Figure 2 shows a typical spatial shape for the electron beam, calculated from (10), altogether with the approximate expression of reference [15].

III. INTERACTIONS ELECTRON BEAM – MATERIAL

Once the electron energy and the electron beam current and shape are known, the energy flux impinging against the material surface can be calculated. In order to describe the energy source inside the material, it is necesary to compute the reflectivity (energy carried away from the material by backscattered electrons) and the energy deposition profile. Backscattered energy and depth dose functions can be obtained for selected materials from extensively tabulated data, or from available empirical or semi-empirical formulae (see [23,24] and references therein).

However we chose to compute both the reflectivity and the depth dose curve, for each material and electron energy, by means of a well stablished Monte Carlo procedure. The main advantage of this approach is the flexibility to model more complex targets (alloys, layered materials, etc.). The Monte Carlo techniques have already proven their ability to produce accurate results for many of the phenomena involved in the irradiation of materials with electron beams, particularly in the energy region of interest here (tens of keV) [18–20,25–31].

A more detailed account of the Monte Carlo technique is given in [17], we will summarize here the most relevant features for completeness. The technique is similar to the one used by Murata et al. [18], and is based on assuming that the trajectory of an electron of energy in the keV range can be approximated by successive straight paths followed by elastic collisions against nuclei. The length of the straight segments is given an exponential probability distribution, with mean value the mean free path at the given electron energy. The electron looses energy in each segment, and the energy loss is calculated by a corrected Bethe formula for the stopping power [32]. The elastic collisions are modelled by means of a screened Rutherford cross section [18,25]

Each trajectory was followed until either the electron leaved the material (backscattered electron), or its energy fell under some threshold (500 eV in the simulations here presented). Performing statistics over 20000 trajectories, we obtain the percentage of backscattered energy (reflectivity), and the energy deposition profile, for the given material and incoming electron energy.

This approach is of very simple implementation, and gives reliable results for a wide range of materials and electron energies [18,19,24,26,27]. The principal limitations have been already investigated. For instance, Kotera et al. [19] show that for heavy elements the main source of error is the screened Rutherford cross section, particularly at small electron energies (1-10 keV). Murata et al. [27] analyze the importance of the production of secondary electrons in the energy dissipation curve; for small energies (several keV) and light elements Valkealahti et al. [26] show that the screened Rutherford cross section underestimates the elastic cross section, and produces slighthly wider energy dissipation curves. Also, Bethe stopping power formula is not valid for electron energies in the keV range and has to be replaced by a more accurate expression [19,33].

Figure 3 shows the obtained reflectivity R. Notice that it depends almost linearly on the energy of the incoming electron, and much strongly on Z (see Figure 4). In order to obtain a practical formula for the reflectivity, we fitted a quadratic expression to the data obtained for Al, Fe, Cu, Ag, Au and various energies. The resulting expression is:

R = −3.259 + 0.8744Z + 0.02633E
−0.004509Z^{2} + 0.0001885ZE + 0.00009694E^{2}  (11)

where E is the energy of the incoming electrons (in keV) and Z is the atomic number of the material.

The penetration depth (range) of the electrons in the material will be useful for length rescaling. The actual value for a given material and initial energy of the incident electrons could be obtained from the Monte Carlo calculations. However, we chose to use the Tabata, Ito, Okabe formula [32], valid in the range 0.3 keV – 30 MeV, and tested for several materials (R is given in kg/m^{2}):

R = a_{1} [\frac{1}{a_{2}} ln [1 + a_{2}(γ − 1)] − \frac{a_{3}(γ − 1)}{1 + a_{4}(γ − 1)^{a_{5}}} ]  (12)
a_{1} = 2.335A/Z^{1.209}  a_{4} = 1.468 − 1.180 10^{−2}Z
a_{2} = 1.78 10^{−4}Z  a_{5} = 1.232/Z^{0.109}
a_{3} = 0.9891 − 3.01 10^{−4}Z

Here γ = 1/\sqrt{1 − (v/c)^{2}}, where v, c are the electron and the light velocities respectively. For a compound, the atomic number Z and the atomic weight A in expression (12) must be replaced by their effective values:

Z_{ef} = ∑_{i} w_{i}Z_{i} A_{ef} = Z_{ef} (∑_{i} w_{i} \frac{Z_{i}}{A_{i}} )^{−1}  (13)

where A_{i}, Z_{i}, w_{i} are the atomic weight, atomic number, and weight fraction of the i th. component of the material. For instance, for AISI 304 stainless steel (Z=26, A=55.84, density 8.02 g/cm^{3}) at electron energies 20 keV, 30 keV, 40 keV, and 50 keV, the ranges are 1., 2., 3.2, and 4.7 μm, respectively. For a lighter element, Al (Z=13, A=27, density 2.42 g/cm^{3}), the corresponding ranges are 3.4, 6.8, 11.2, and 16.5 μm.

Figure 5 shows the dependence of the deposited energy density vs. depth curves, as the material is changed. The 50 keV electron beam impinges normally on the material. Figure 6 shows the dependence of the deposited energy vs. depth curves on the electron energy (for a given material). In both Figures depths are normalized to the range of the incident electrons in the given material as obtained from equation (12), and densities are normalized to unit area.

The depth dose curves are usually represented by means of shifted gaussian curves [16,26], in spite of the fact that they are clearly not symmetrical around the maximum. This gaussian fit is preferred in many cases because it has a simple interpretation in terms of direct and diffusive contributions [24]. As the main concern here is the accurate approximation of the Monte Carlo results, we chose to fit the deposited energy density vs. depth profiles with the expression:

f(z/R) = (a\sqrt{z/R} + b) exp (−c(z/R)^{2})  (14)

giving much better agreement than the usual gaussian or cubic fits.

The coefficients corresponding to electrons of energies in the range 10–100 keV impinging on Fe are shown in Table I. Notice the similarity of the coefficients for energies greater than about 30 keV.

IV. THERMAL PROBLEM

The thermal problem under consideration essentially consists of the solution of the heat equation inside the material, with the appropriate energy source term. A subtlety is that when the electron beam is initially focused on the sample, the processing time is very short, of the order of a nanosecond. Accordingly, the thermal diffusion length is very short and could be somewhat smaller than the penetration depth of the electron beam. On the other side, if the electron beam is not initially focused, the processing times are of the order of the characteristic discharge times of the capacitor (several tens of microseconds). Under these conditions, the thermal diffusion length in metals is much longer than the penetration depth of electrons (for electrons in the 20–30 keV range). In order to use realistic meshes to solve the heat equation, it is then necessary to alternate between two different numerical strategies: in the first case we include the incoming energy as a source term inside the material, and in the second case we include it as a boundary flux. This will produce small differences in the computed magnitudes at the transition from one strategy to the other, but will allow great savings in computational work.

The material is assumed to fill a t = 0 the region 0 ≤ z ≤ z_{1}, and the electron beam impinges on the surface z = 0. Let T (r, z, t) stand for the temperature of the material, κ(T) denote the temperature dependent thermal conductivity, and ρ, c_{p} be the density and specific heat respectively.

As explained before, if the range of the electrons is of the same order than a typical diffusion length, this flux is included as a source term in the heat equation written in cylindrical coordinates with boundary condition of zero thermal flux in the surface z = 0:

ρc_{p} \frac{∂T}{∂t} = \frac{1}{r} \frac{∂}{∂r} (rκ(T)\frac{∂T}{∂r} ) + \frac{∂}{∂z} (κ(T)\frac{∂T}{∂z} ) + Q(r, z, t)  (15)
\frac{∂T}{∂z} (r, 0, t) = 0  (16)

If the electron range is much smaller than the typical diffusion length, the energy flux is included as a boundary condition, to avoid having to use a mesh fine enough to resolve the source term:

ρc_{p} \frac{∂T}{∂t} = \frac{1}{r} \frac{∂}{∂r} (rκ(T)\frac{∂T}{∂r} ) + \frac{∂}{∂z} (κ(T)\frac{∂T}{∂z} )  (17)
−κ\frac{∂T}{∂z} (r, 0, t) = Φ(r, t)  (18)

Both the initial temperature and the temperature of the lateral and rear surfaces are supposed to be equal to the room temperature, T_{a} = 300K.

The material is allowed to melt and to vaporize, with latent heats of melting and vaporization λ_{f} and λ_{v}. We denote with T_{f}, T_{v} the melting and vaporization temperatures. However, we do not include an analysis of the gaseous phase, and will therefore restrict ourselves to the cases where no etching takes place.

Assuming that the current density is homogeneous inside the electron beam, the energy flux incoming to the material by unit surface can be calculated as

Φ(r, t) = (1 − R)i(t)v(t)\frac{1}{πr^{2}(t)}Θ(r(t) − r)  (19)

where r(t) is the irradiated spot radius at time t, calculated by inverting relation (10) for the given i(t), v(t) values, and where Θ(·) is the Heaviside function. The reflectivity R is calculated according to expression 11.

As the electron beam lateral diffusion inside the material is negligible compared to typical irradiated spot sizes, the energy density deposited into the material in the unit time can be calculated as:

Q(r, z, t) = (1 − R)i(t)v(t)\frac{1}{πr^{2}(t)} \frac{1}{R}f(z/R)Θ(r(t) − r)  (20)

where R is the electron range in the material, and f is the normalized energy dissipation curve calculated by the Monte Carlo technique (14).

V. NUMERICAL TECHNIQUE

The first step of the numerical solution of the model is the Monte Carlo calculation of the reflectivity and energy deposition profiles for the given material and various electron energies. The reflectivity and the energy deposition profiles for each electron energy are then calculated from expressions (11) and (14).

The next step consists on the calculation of the voltage and the current at time t (given by expressions (2) and (3)), and of the irradiated spot radius at that time. This is accomplished by numerically inverting the defining relation (10):

d = z(r(t)) = r^{∗} ∫_{0}^{√\overline{u_{0}^{2}/A^{2}−log^{2}(r(t)/r^{∗})}} exp(\sqrt{\frac{u_{0}^{2}}{A^{2}} − s^{2}})ds  (21)

where d is the cathode–sample distance, and A and u_{0} depend on t through the dependence of v(t) and i(t) on t. The inversion is done using a Brent type routine [34]. According to whether or not the typical electron penetration depth is negligible compared to a typical thermal diffusion length, the boundary energy flux (19) or the heat source inside the material (20) are then computed.

At last, the heat equation (15) or (17) is solved, by means of the lines method consisting in the discretization to second order of the spatial terms, and the integration in time of the resulting system of ordinary differential equations ( [35]). Time integration was achieved by means of a multistage Runge-Kutta method of second order, specifically designed to handle the mildly stiff problems that usually appear in the semi-discretization of parabolic equations.

Latent heats of melting and vaporization (λ_{f} and λ_{v} respectively) were taken into account by using a standard regularized expression for the enthalpy. The great advantage of this method is that it does not require to follow explicitly the free boundaries (liquid-solid or liquid-vapor interfaces). The left side of the heat equation is replaced by the expression:

(ρc_{p} + λ_{f}δ_{ν}(T − T_{f}) + λ_{v}δ_{ν}(T − T_{v})) \frac{∂T}{∂t}  (22)

where δ_{ν} is an approximation to the Dirac δ function:

δ_{ν}(T) = \frac{3}{4} (1 − (\frac{T}{ν} )^{2})  (23)

In the calculations below ν = 30K.

VI. NUMERICAL RESULTS

In what follows, the parameter a used to describe the voltage and current evolution during an electron beam pulse is given the value a = 0.063 A/kV^{3}/torr^{2.2}. This value reflects typical data from Mingolo et al. [14] data. Only results concerning stainless steel samples will be shown here (composition 70 % Fe, 18-20 %Cr, 8-10 %Ni, 0.08 %C), a more complete treatment of some other materials will appear elsewhere.

Figure 7 shows typical temperature profiles obtained for an initial voltage v = 24.8kV , and a cathode sample distance d = 30 cm. Subplots a) to e) correspond to time t = 2, 6, 10, 14, and 18 μs respectively. Notice that the temperature profiles are almost uniform in the central region of the irradiated spot. The effect of the defocusing of the electron beam is clearly seen, because the radial spreading from the central uniform region is much greater than the diffusion in z, and is almost entirely due to the increase of the irradiated area produced as the current and voltage decrease in time.

Figure 8 shows the corresponding evolution of the voltage drop (8- a), current (8- b), irradiated spot radius (8- c), energy density flux (8- d), melted radius (8- e), maximum achieved temperature (8- f), and melted depth (8-g).

Notice that the irradiated area increases a factor 50 (due to the factor 7 increase in the irradiated spot radius), while the current and voltage suffer only a slight decrease. This makes clear, as shown in [15], that in these conditions the electron beam defocusing is the major responsible for the decrease in the incoming energy flux. Also it shows the stated decrease in the effective irradiation time: the intensity decays several hundreds of nanoseconds, while the characteristic decay times for voltage and current are of the order of several tens of microseconds. As a consequence, the efficiency of the melting process is very small, because the major part of the energy is spent heating an area much wider than the area where melting is achieved. The actual value of the efficiency depends on the processing parameters, but is always small when strong defocusing of the electron beam occurs. For instance, for the given example only less than 0.55 J of the 28 J of the beam energy are actually used to melt the material.

Figure 8-e also shows an interesting fact: the melted zone spreads for a while after the almost complete decay of the incoming energy flux, due to the heat transfer from the very hot central region. It is also noticeable that in depth spreading of the melted region continues up to much longer times (Figure 8 - g). The practically constant temperature intervals correspond to melting and vaporisation of the sample.

Figure 9 shows maximum achieved temperatures, as a function of initial discharge voltage, for several cathode–sample distances. Notice the previously mentioned small jump at the transition from the boundary flux to the source term numerical strategies.

A fact evident from Figure 9 is that maximum temperature curves are steeper at greater voltages over the gun (at the same sample – cathode distance). Also, it is very interesting to note that the slope of the maximum temperature curves at a given temperature (melting point, for instance) is smaller at smaller cathode – sample distances and higher initial voltages. This implies that sensitivity against slight variations of current is minimized using the highest available voltage at the smallest distance that still allows reaching the desired temperature, and can be explained by the fact that at higher voltages the irradiated area is larger, and the electron beam radial profile (see Figure 2) is less steep as the focus moves away.

The values of the physical constants ρ, c_{p}, κ, etc., used in the numerical simulations are summarized in Table II.

VII. CONCLUSIONS

We have developed a detailed numerical model of the application of a cold cathode electron gun to materials processing. The exposed results prove the usefulness of the mixed deterministic and stochastic methods for the analysis of the complex phenomena involved in this application, and constitute a valuable tool both for guiding experimental research and to obtain insight on the influence of parameters not easily accessible to direct experimental study.

The presented model yields information on maximum achieved temperatures, melting and vaporization thresholds, melted pool radius and depths, etc., for a wide range of processing conditions. The obtained results extend those obtained with the previous analytical model [15], including fully three dimensional analysis, accurate treatment of the source term inside the material, and a more accurate study of the spatial shape of the electron beam. Results for stainless steel were presented here as an example, a more detailed analysis of several materials will appear elsewhere.

As an important fact, it is worth to notice that for typical processing parameters producing melting of the sample (see for instance Figures 7, 8) the fraction of the used energy actually employed in the melting area is very small, yielding efficiencies of a few percent. This is clearly seen in Figure 8, where the energy flux diminishes much faster than the discharge current and voltage drop. This observation suggests that the electron gun design, or the discharge circuit should be modified in order to obtain a better efficiency.

Another important conclusion is that sensitivity against current and voltage fluctuations is reduced at higher voltages and smaller cathode– sample distances.

TABLES

TABLE I. Best fit coefficients corresponding to expression 14 of deposited energy density normalized to unit area vs. depth normalized to the range. Material: iron.

Energy (keV)  c  a  b
10  6.214  5.145  0.5879
30  4.566  3.936  0.5771
50  4.204  3.609  0.5962
70  4.028  3.415  0.6219
90  3.953  3.281  0.6559

TABLE II. Physical constants for AISI 304 stainless steel ( [36]).

Material  ρ(g/cm^{3}) c_{p}(J/gK) κ(J/msK) T_{f}(K) T_{v}(K) λ_{f}(J/cm^{3}) λ_{v}(J/cm^{3})
Stainless Steel  8.02  0.449  30  1800  2750  1940  30000

REFERENCES

[1] R.A. Dugdale. J. Mater. Sci. 10, 896, (1975).

[2] J.J. Rocca, J.D. Meyer, M.R. Farrell, G.J. Collins. J. Appl. Phys. 56, 790 (1984).

[3] G. W. McClure. Phys. Rev. 124, 969 (1961).

[4] B.B. O’Brien Jr. Appl. Phys. Lett. 22, 503 (1973).

[5] H.F. Ranea–Sandoval, N. Reesor, B.T. Szapiro, C. Murray, J.J. Rocca. IEEE Trans. on Plasma Sci. Vol. PS-15, No. 4, 361 (1987).

[6] J.Christiansen, C.Schultheiss. Z. Phys. vol A290, 35 (1979); W.Benker, J. Christiansen, K.Frank, H. Gundel, W. Hartmann et al. IEEE Trans. Plasma Sci, vol 17, 754 (1989).

[7] C.A. Moore, J.J. Rocca, G.J. Collins, P.E. Russell, J.D. Geller. Appl. Phys. Lett. 45, 169 (1984).

[8] N.J. Ianno, J.T. Vedeyen, S.S. Chan, B.G. Streetman. Appl. Phys. Lett. 39, 622 (1981).

[9] C.A. Moore, J.J. Rocca, T. Johnson, G.J. Collins, P.E. Russell. Appl. Phys. Lett. 43, 290 (1983).

[10] D.C. Bishop, K.A. Emery, J.J. Rocca, L.R. Thompson, H. Zarnani, G.J. Collins. Appl. Phys. Lett, 44, 598 (1984).

[11] L.R.Thompson, J.J. Rocca, P.K. Boyer, K. Emery, G.J. Collins. Appl. Phys. Lett. 43, 777 (1983).

[12] M. Hobel, J. Geek, G. Linker, C. Schultheiss. Appl. Phys. Lett. 56, 973 (1990).

[13] Koji Kobashi, Shigeaki Miyauchi, Koichi Miyata, Kozo Nishimura, J.J. Rocca. J. Mater. Res. 11, 2744 (1996).

[14] N.Mingolo, J.J.Rocca. J. Mater. Res. Vol. 7 (5) 1096 (1992).

[15] J.I.Etcheverry, N.Mingolo, J.J. Rocca, O.E.Martínez. IEEE Trans. Plasma Sci. 25, 3, 427 (1997).

[16] M. von Allmen. Coupling of beam energy to solids. Laser and electron beam processing of Materials, Academic Press, New York, 1980.

[17] J.I.Etcheverry, O.E.Martínez. Proceedings of the First Argentina-USA Bilateral Symposium on Materials Science and Engineering, Anales de la Asociación Química Argentina, 84 (3), 221, (1996).

[18] K.Murata, T.Matsukawa, R.Shimizu. Jap. J. Appl. Phys. 10, 6, 678 (1971).

[19] M.Kotera, K. Murata, K. Nagami. J. Appl. Phys. 52 (2), 997 (1981).

[20] T. Matsukawa, R. Shimizu, K.Harada, T. Kato. J. Appl. Phys. 45 (2), 733 (1974).

[21] M. Yasuda, H.Kawata, K.Murata, K.Hashimoto, Y.Hirai, N.Nomura. J. Vac. Sci. Technol. B 12 (3), 1362 (1994).

[22] S.A. Lee, U.A. Andersen, J.J. Rocca, M. Marconi, N.D. Reesor. Appl. Phys. Lett. 51 (6), 409 (1987).

[23] T.E.Everhart, P.H.Hoff. J. Appl.Phys.42 (13), 5837 (1971).

[24] K.A. Valiev. The physics of submicron lithography. Plennum Press, 1992.

[25] B. Grosswendt, E. Waibel. Nucl. Inst. and Meth. 155, 145 (1978).

[26] S. Valkealahti, J. Schou, R. M. Nieminen. J. Appl. Phys. 65 (6), 2258 (1989).

[27] K. Murata, D.F. Kyser, Ch. H. Ting. J. Appl. Phys. 52 (7), 4396 (1981).

[28] D. Liljequist. J. Appl. Phys. 62 (2), 333 (1987).

[29] D. Liljequist, M. Ismali. J. Appl. Phys. 62 (2), 342 (1987).

[30] D. Liljequist, M. Ismali. J. Appl. Phys. 68 (7), 3061 (1990).

[31] J. Baró, J. Sempau, J.M. Fernández Varea, F. Salvat. Nucl. Inst. Meth. in Phys. Res. B 84, 465 (1994).

[32] Tsoulfanidis. Measurement and Detection of Radiation, Mc GrawHill, NY, 1983.

[33] C.J. Tung, J.C. Ashley, R.H. Ritchie. Surf. Sci. 81, 427 (1979).

[34] Brent, R. P. Algorithms for minimization without derivatives. Englewood Cliffs, N.J. (1973), Prentice-Hall.

[35] C.A.J.Fletcher. Computational Techniques for Fluid Dynamics. Springer Series in Computational Physics, Springer Verlag 1988.

[36] CRC Handbook of Chemistry and Physics, 73 Ed., 1992-1993, CRC Press, Boca Raton, Ann Arbor, London, Tokyo.

Figure captions

Figure 1: Schematic diagram of the electron gun ( [14]) and the discharge circuit. The 90 nF capacitor, initially charged at V_{0}, is discharged across the electron gun after triggering of the spark-gap (T). R1=20 MΩ, and R2=25 Ω.

Figure 2: Calculated beam shape for a typical discharge. The numerical solution and the results of the Taylor approximation are shown. Voltage drop: 22.5 kV, current: 50 A, a=0.063, r_{0} = 3.75 cm.

Figure 3: Percentage of backscattered energy, calculated by the Monte Carlo technique, vs. incident electron energy, for various materials. Solid line corresponds to the best fit using expression 11. Circles: Au, squares: Ag, plus: Fe, diamonds: Al.

Figure 4: Reflectivity and backscattering coefficients, as functions of Z, for 30 keV electrons. Plus: backscattering coefficient, diamond: reflectivity. Solid lines correspond to best fits using expression a_{0} + a_{1}Z + a_{2}E + a_{20}Z^{2} + a_{11}EZ + a_{02}E^{2}

Figure 5: Deposited energy density normalized to unit area vs. depth normalized to the electron range. Beam energy: 50 keV. Number of electrons: 20000.

Figure 6: Deposited energy density normalized to unit area, vs. depth normalized to the electron range. Material: iron. Beam energies: 10 keV, 50 keV, 100 keV. Number of electrons: 20000.

Figure 7: Typical temperature profiles (K) calculated for an AISI 304 stainless steel sample. Initial voltage drop v = 24.8kV. Cathode sample distance d = 30 cm. Pressure: 290 mtorr.

Subplots a) to e) correspond to time t =2, 6, 10, 14 and 18 μs respectively.

Figure 8: Calculated (a) voltage drop, (b) current, (c) irradiated spot radius, (d) energy flux, (e) melted radius, (f) maximum temperature, and (g) melted depth. Initial voltage drop of 24.8kV. Pressure: 290 mtorr. Cathode–sample distance: 30 cm. Material: stainless steel.

Figure 9: Maximum calculated temperature values for an AISI 304 stainless steel sample, as a function of initial voltage of the discharge, and for several cathode sample distances. Pressure: 290 mtorr. Cathode sample distances: continuous line: 20 cm, plus: 25 cm, stars: 30 cm, crosses: 35 cm, circles: 40 cm.

FIGURES

FIG. 1. Schematic diagram of the electron gun ( [14]) and the discharge circuit. The 90 nF capacitor, initially charged at V_{0}, is discharged across the electron gun after triggering of the spark-gap (T). R1=20 MΩ, and R2=25 Ω.

FIG. 2. Calculated beam shape for a typical discharge. The numerical solution and the results of the Taylor approximation are shown. Voltage drop: 22.5 kV, current: 50 A, a=0.063, r_{0} = 3.75 cm.

FIG. 3. Percentage of backscattered energy, calculated by the Monte Carlo technique, vs. incident electron energy, for various materials. Solid line corresponds to the best fit using expression 11. Circles: Au, squares: Ag, plus: Fe, diamonds: Al.

FIG. 4. Reflectivity and backscattering coefficients, as functions of Z, for 30 keV electrons. Plus: backscattering coefficient, diamond: reflectivity. Solid lines correspond to best fits using expression a_{0} + a_{1}Z + a_{2}E + a_{20}Z^{2} + a_{11}EZ + a_{02}E^{2}

FIG. 5. Deposited energy density normalized to unit area vs. depth normalized to the electron range. Beam energy: 50 keV. Number of electrons: 20000.

FIG. 6. Deposited energy density normalized to unit area, vs. depth normalized to the electron range. Material: iron. Beam energies: 10 keV, 50 keV, 100 keV. Number of electrons: 20000.

FIG. 7. Typical temperature profiles (K) calculated for an AISI 304 stainless steel sample. Initial voltage drop v = 24.8kV . Cathode sample distance d = 30 cm. Pressure: 290 mtorr. Subplots a) to e) correspond to time t =2, 6, 10, 14 and 18 μs respectively.

FIG. 8. Calculated (a) voltage drop, (b) current, (c) irradiated spot radius, (d) energy flux, (e) melted radius, (f) maximum temperature, and (g) melted depth. Initial voltage drop of 24.8kV. Pressure: 290 mtorr. Cathode–sample distance: 30 cm. Material: stainless steel.

FIG. 9. Maximum calculated temperature values for an AISI 304 stainless steel sample, as a function of initial voltage of the discharge, and for several cathode sample distances. Pressure: 290 mtorr. Cathode sample distances: continuous line: 20 cm, plus: 25 cm, stars: 30 cm, crosses: 35 cm, circles: 40 cm.