New designs and a new analysis technique have been developed for an existing compact charged-particle spectrometer on the NIF and OMEGA. The new analysis technique extends the capabilities of this diagnostic to measure arbitrarily shaped ion spectra down to 1 MeV with yields as low as 106. Three different designs are provided optimized for the measurement of DD protons, T3He deuterons, and 3He3He protons. The designs are highly customizable, and a generalized framework is provided for optimizing the design for alternative applications. Additionally, the understanding of the detector’s response and uncertainties is greatly expanded upon. A new calibration procedure is also developed to increase the precision of the measurements.

Measuring charged-particle spectra is of great importance to both inertial confinement fusion (ICF) and high-energy-density-physics (HEDP) experiments on the NIF1 and OMEGA2 laser facilities. It is also crucial for neutron-recoil spectrometers because they infer neutron spectra directly from the measurement of charged-particle spectra.3–5 This expands the relevance to the Z facility6 where neutron-recoil spectrometers are projected to be implemented.5 From both neutron and charged-particle spectra, one can infer various plasma properties, such as areal density (ρR), ion temperatures (Tion), or asymmetries.7–12 Over decades, several CR-39 based spectrometers have been developed and fielded on all three facilities for this reason.3,9–13 Among these are the step range filters (SRFs), originally designed specifically for measuring fusion protons from DD reactions,12 

D+DT(1.01MeV)+p(3.02MeV).
(1)

The SRFs have an important role at the NIF and OMEGA due to their unique coverage range of low yields and low energies. Figure 1 shows the entire coverage of the SRFs and other charged-particle diagnostics for protons on both facilities. In addition to this, the SRFs are a non-fixed diagnostic with a highly simplistic design. This makes it possible to field roughly 20 per experiment on both NIF and OMEGA, each in different locations around the experiment.

The SRFs will also play an important role on the Z facility through a new non-magnetic neutron-recoil spectrometer which has recently been developed.5 This spectrometer infers a neutron spectrum from the measurement of low-energy (<4 MeV) elastically scattered protons. Because the system is non-magnetic, this measurement can only be accomplished using an SRF again due to the low particle energies.

While important, SRFs have had a handful of limitations in their early iterations. Early designs had limited energy coverage (1–3 MeV) and the analysis used on them only worked for Gaussian shaped DD-proton spectra.12 In this work, a new design geometry has been developed with multiple advantages over previous designs. This new design extends the energy coverage of a given SRF while also minimizing calibration complexity. Additionally, a new analysis technique has been developed for measuring arbitrarily shaped charged-particle spectra with energies above roughly 1 MeV. When using this technique, SRFs are uniquely useful for measuring spectra that are continuous down to 0, such as the protons from 3He3He reactions,

He3+He3α+p + p,
(2)

or the protons from T3He reactions,

T +3Heα+n + p,
(3)

whose lower energy behavior cannot be measured by other diagnostics. In the case of neutron-recoil spectrometers on the Z, this also allows for the measurement of down-scattered neutrons. Additionally, this technique can be applied to other charged-particles, such as deuterons from the T3He reaction

T +3Heα(4.8 MeV)+D (9.5 MeV).
(4)

This paper is structured as follows: Section II discusses the new SRF designs and their capabilities as well as prescribes a basic procedure for designing SRFs for alternative applications. Section III discusses the new analysis method and highlights improvements over previous work. Section IV gives a detailed description of the SRF instrument response, and Sec. V describes a new more accurate calibration procedure. Finally, Sec. VI highlights examples of data obtained with the new SRF diagnostic.

Broadly speaking, an SRF is simply a CR-39 detector with some number of filters (>1) in front of it. As such, the exact design of any single SRF can be highly customized with ease depending on the intended application. Additionally, an SRF can, in general, be applied to any CR-39 detector geometry, making it highly versatile. In this section, the basic procedure for designing an SRF will be discussed and a few designs will be presented toward the end. The remainder of this paper will largely focus on these designs, but it is important to note that all of the theory and capabilities discussed within can be generalized to alternative designs if needed.

An SRF can be loosely defined by its detector geometry, number of physical filters (Nf), and number of unique filter windows (Nw). A filter window is simply a spatial region on the CR-39 detector covered by one or more filters. Often, SRF filters are overlapped for fabrication purposes, meaning an individual window can have anywhere between 1 and all of the filters in front of it. An example of a SRF with Nf = 7 and Nw = 7 (a 7F7W SRF) can be seen in Figs. 2(a)2(d) and a 5F9W SRF can be seen in Figs. 2(e)2(h). Both of these designs use 5 cm diameter CR-39 detectors to be compatible with the geometry and mounting hardware of the existing Wedge Range Filter (WRF) spectrometers.9,11 Ideally, an SRF will also have a background filter specifically designed to range out any potential signal. In the designs depicted in Fig. 2, this is accomplished by a 3 mm thick aluminum frame in the front of the stack of filters. This filter is important for background subtraction and does not count toward the >1 filter requirement of an SRF.

Designing an SRF requires choosing appropriate window thicknesses for a given measurement. In general, the goal of a particular measurement can be defined by a particle type and detectable energy range E[EminSRF,EmaxSRF]. CR-39 detectors have 100% detection efficiency for some E[EminCR39,EmaxCR39]; the exact values vary14 but are unimportant for the following discussion. Every filter arrangement in each window will downshift particles with some energy range Ein[Emini,Emaxi] such that energies of the particles when incident on the CR-39 are Eout[EminCR39,EmaxCR39]. For example, if one were to consider protons and assume EminCR39=1 MeV and EmaxCR39=3 MeV, then 10 μm Ta would cause proton energies Ein ∈ [1.9, 3.6] MeV to be detectable on the CR-39. In practice, the exact values of EminCR39 and EmaxCR39 are dependent on processing details, analysis decisions, and even the intrinsic properties of an individual piece of CR-39. The values chosen here are reasonable estimations.

Choosing the thinnest and thickest filter arrangement for a window in an SRF is simply an exercise of setting Eminthin=EminSRF and Emaxthick=EmaxSRF. For example, if the goal was to cover all proton energies between 2 and 12 MeV, the thinnest and thickest window regions might be 10 and 240 μm of Ta, respectively. This is because for 10 μm Ta, detectable proton energies are Ein ∈ [1.9, 3.6], while for 240 μm Ta, they are Ein ∈ [11.3, 12.0]. These numbers again assume EminCR39=1 MeV and EmaxCR39=3 MeV.

The only requirement for the remaining windows is that there exists some overlap in the energy coverage between subsequent windows. This requirement ensures that the analysis can search for self-consistency between window regions. To meet this requirement, we must choose some thickness ti+1 such that

Emini+1=Emini+fEmaxiEmini,
(5)

where ti+1 is the thickness of the next thickest filter after filter i and f is some fraction between 0 and 1.

Low values of f ensure more overlap, while high values reduce the number of required windows. Additional windows add robustness and flexibility to the fitting process. However, more windows will decrease the area and thus counts seen by any individual window. A general rule of thumb is to ensure that the number of counts in all windows exceeds 103, but this is rarely the limiting factor in practice. More practically, analysis, manufacturing, and calibration all get more complicated as Nf is increased, which often drives the desire to simplify.

Returning to the example of designing an SRF that can detect protons between 2 and 12 MeV, if we again assume EminCR39=1 MeV and EmaxCR39=3 MeV and take f to be 0.67, then Eq. (5) calculates that we should use a total of 15 windows with roughly 16.5 μm of Ta between each step. Again though, one may choose to use more or less windows depending on what is practical so long as some energy overlap exists between the windows.

The 5F9W SRF shown in Figs. 2(e)2(h) innovates on the SRF formula by achieving nine unique filter regions with only five physical filters. This greatly simplifies both the manufacturing and calibration processes without sacrificing on the number of windows. Additionally, all of the designs fielded to date only require three unique filter thicknesses, which simplifies calibration even further since multiple filters can be taken from the same stock (thus having roughly the same thickness). Three different designs using the 5F9W geometry are summarized in Table I.

Previous work12 established a forward-fit procedure to the track densities measured behind each window nmeasi. In the previous analysis, one would assume that some spectrum Sin defined by some set of parameters p was incident on the SRF. This spectrum is then down-ranged using Monte Carlo models, such as SRIM15 or MCNP,16 through each window to create a set of CR-39 incident spectra {Sout}. The expected track density in a given window is then given by

nmodeli=14π20dEoutSouti(Eout,p)ηCR39(Eout),
(6)

where is the CR-39’s distance from the particle-source and ηCR-39(Eout) is the CR-39 detection efficiency to particles with energy Eout. The reduced chi-squared for the choice Sin is then

χred2=1NwNpi=1Nwnmodelinmeasi2σni2,
(7)

where Nw is the number of windows, Np is the length of p, and σni is the statistical uncertainty on the measurement nmeasi.

One limitation of this method is its dependence on ηCR-39(Eout), which is not well understood and has been shown to vary between individual pieces of CR-39.14 In practice, it is often approximated by the boxcar function

ηCR39(Eout)H(EoutEmin)H(EoutEmax),
(8)

where H is the Heaviside step function and Emin and Emax are the minimum and maximum detection energies.

In reality, both Emin and Emax depend on how the CR-39 is processed and analyzed and on the intrinsic properties of the individual piece. This uncertainty was not previously considered in Ref. 12.

A new analysis technique inspired by the analysis of WRF data9,11 has been developed to circumvent this problem. This technique relies on the energy–diameter relationship of the tracks in the CR-39. After taking data, every window region will have a large set of diameter tracks {D}i. We circumvent the efficiency uncertainty by only considering diameters that correspond to particle energies guaranteed to satisfy E ∈ [Emin, Emax]. In practice, this means sub-selecting tracks with diameters that are sufficiently large. A general rule of thumb is to consider all tracks with diameters D ∈ [0.5Dmax, 0.9Dmax], where Dmax is the largest significant diameter.14 

The analysis starts by assuming a particular relationship fDvE(E) that converts particle energies to track diameters uniquely. Suitable forms for this relationship are discussed in Ref. 14. Each window i then has its diameter sets converted into CR-39 incident energy sets {Eout}i using this relationship. These energies are then reverse-ranged using stopping-power tables to produce the set of SRF-incident energies {Ein}i. When binned, these sets represent the inferred spectra that were incident on each individual window. Because, in practice, each window will see the same spectrum, all inferred window spectra should be consistent with one another in the spectral regions where they overlap. This criterion is ultimately what is used to test the validity of the initial choice of fDvE(E). A basic flow chart for this algorithm is shown in Fig. 3. Once a good choice for fDvE(E) is found, the final inferred spectrum is just the average of all of the window-inferred spectra. Visual demonstrations of this method are shown in Figs. 4 and 5.

Care needs to be taken with how self-consistency is determined between the window-inferred spectra. Having different filters means that each window sees different portions of the original spectrum. This means each window-inferred spectrum will, in general, have different moments that cannot be directly compared. In general, sub-regions of spectral overlap need to be identified and compared. How this is done is somewhat arbitrary, but the following goodness parameter has been found to be effective,

χ2=1NEjNw(j)inij2inij2inij,
(9)

where NE is the number of energy bins with Nw(j) > 1, Nw(j) is the number of filters that measure data between energy nodes Ej and Ej+1, and nij is the density of counts between energy nodes Ej and Ej+1 for window i. This χ2 is a mean-normalized variance of count densities between each window averaged over every energy bin. It effectively compares the absolute magnitude of each energy node between all windows that happen to measure data there.

The detector response of any spectrometer is a crucial component of its performance. Detailed information about the shape of a spectrum can only be resolved if the response is well understood. For the SRFs, spectral broadening occurs via three main mechanisms: diameter blurring during the scanning process, artificial broadening in the analysis from poor filter-thickness calibration, and broadening due to energy straggling in the filters.

When the CR-39 is scanned by an optical microscope, variations in the diameters are observed. This diameter-blurring mechanism causes spectra inferred from diameter distributions to be wider than the original source. The exact cause of this diameter blurring is not well understood but is thought to be partially related to varying focusing during the scan. The magnitude of this effect was determined from an experiment described in the  Appendix and is summarized in Fig. 6 for various etch times and mean diameters.

As seen in Fig. 6, relative diameter broadening is invariant to etch time and mean diameter, staying constant at σblur/D = 8.1 ± 0.5%. How this translates to spectral broadening is somewhat complicated due to the fact that many diameters are used in the analysis and by the fact that the relation between track-diameters and particle energy varies from CR-39 to CR-39.14 A simulated example of the fully integrated effect is shown in Fig. 7. To demonstrate how this effect varies with fDvE, a full mapping of the broadening with different diameter–energy relations is shown in Fig. 8.

If the SRF filter thicknesses are not properly calibrated, additional artificial broadening can be induced in the analysis. This occurs when the individual window spectrum cannot be made to agree due to the discrepancy between the real thicknesses and the assumed thicknesses. An example of this on simulated data is shown in Fig. 9.

An additional side-effect of poor filter thickness calibration is how it interacts with the analysis procedure discussed in Sec. III. The analysis attempts to compensate for poor filter thickness values by finding an fDvE(E) that can best match window spectra to one another. This choice of fDvE(E) will, in general, not match the true best function derived from using the correct filter thickness values. This itself also impacts the width of the inferred spectra by changing the slope of the diameter–energy relationship characterized by the c-parameter discussed in Ref. 14. In some cases, this can actually narrow the inferred spectra. The exact effect is sensitive to the differential between the assumed thicknesses and the true thicknesses and thus is difficult to capture. To test this, several simulated DD-p 5F9W SRFs were exposed to a normally distributed proton source with μ = 3.0 MeV and σ = 100 keV. The filter thicknesses of each SRF were randomly perturbed about their true values with a relative σ of 15% and then analyzed using the nominal values. The inferred mean energies and spectral widths were then tallied, the results of which are summarized in Fig. 10.

As can be seen in Fig. 10, random perturbations on the filter thicknesses shift the inferred mean energy symmetrically about the true value. The inferred widths, however, are generally wider than the true value. This shift in the distribution is the broadening due to poor filter-thickness calibration. We note again that a non-negligible portion of the distribution actually has narrower inferred widths. The exact values for the distributions shown in Fig. 10 are also sensitive to the original source spectrum and SRF design. As a result, we recommend that this exercise be repeated per application/SRF design to determine how accurately the filter thicknesses need to be known.

Finally, we note that the analyzer has full access to the spectra inferred for each individual window, making it possible to determine when this broadening mechanism is occurring. This, perhaps, opens the door to in situ filter calibrations whenever it becomes clear that the filters are not properly calibrated. In general, this cannot be a sufficient replacement to true calibrations. This is well illustrated by the example given in Fig. 9. Here, all four of the window spectra fall short of the true mean value of 3.0 MeV due to all of the assumed filter thicknesses being too thin. While it is clear that the filters are poorly calibrated relative to each other, there is no signature in the data to illuminate where the true mean value should lie. Still, altering the filter thicknesses to get better agreement would give a better estimate of the spectral width. That said, we note again that changes in fDvE(E) also alter the widths of the window spectra. For these reasons, we simply recognize that in situ calibrations have a promising potential but are subject to details that go beyond the scope of this paper.

Additional broadening can occur simply due to energy straggling in the SRF filters. Energy straggling is the broadening in particle energies due to statistical variations in small-angle collisions throughout the filter medium. As a result, mono-energetic particles incident onto an SRF will have non-mono-energetic distributions once they are incident on the CR-39 detector.

To investigate this broadening, an SRF was modeled with MCNP6.16 The SRF was exposed to several mono-energetic proton and deuteron lines of varying energies and the resultant CR-39 incident spectra behind each window were recorded. These spectra were converted to diameter distributions using the c-Parameter model discussed in Ref. 14. These diameter distributions were analyzed using the technique established in Sec. III. The widths of the resultant spectra reflect the broadening caused by straggling. The results of this exercise are shown in Fig. 11.

As can be seen in Fig. 11, the broadening due to energy straggling is generally low when compared to the diameter blurring mechanism shown in Fig. 8(a). It is, however, more significant than the filter uncertainty mechanism shown in Fig. 10(b). In general, diameter blurring is the dominant broadening mechanism of the three.

The spectral accuracy of the SRF is partially dictated by the accuracy to which the areal density (ρL) of each individual filter is known. Generally, filter thicknesses of order 10 μm are used, which can be difficult to accurately characterize. Often, manufacturers provide thicknesses within 10%–15% tolerances, which can result in large energy uncertainties, as shown in Fig. 10.

Micrometer measurements are often used to reduce this uncertainty, but even this method is limited. Micrometers provide no information about the density (ρ) of the material in question, so one often has to assume it. Additionally, micrometers can often give inconsistent results when working with thinner materials. Regardless, this method is still valuable when finer characterizing techniques are unavailable.

One technique for calibrating the SRFs is through fully integrated spectral measurements. The basic methodology for this technique is to measure well known spectra with a fully assembled SRF. If a single SRF with Nf filters measures Nf unique spectra, then the filter areal densities should, in principle, be uniquely quantified. It should be specifically noted that Nf, not Nw, is what dictates the required number of measurements. This fact motivates designing SRFs with filter overlaying to maximize Nw for the measurement accuracy while minimizing Nf for calibration simplicity.

Recently, this technique was applied to the DD-p 5F9W SRF design described in Table I using the MIT-HEDP Accelerator Facility.17 Four unique DD-proton spectra were generated by using varying laboratory angles and external filters. While the design has five total filters, only four of them measured these spectra and thus were calibrated. A cartoon of the exact calibration setup is shown in Fig. 12, and the resultant spectra are shown in Fig. 13. Additionally, sample calibration results are shown in Fig. 14.

After the data are obtained, the calibration is simply a process of finding the set of areal densities that minimize the function

ϵ=ϵE2+ϵσ2,
(10)

where ϵE is the error in the mean energy given by

ϵE2=iESRFiEi+δEi(fDvE)Ei+δEi(fDvE)2,
(11)

where ESRFi is the mean energy of the ith source as determined by the SRF, Ei is the true source mean energy, and δEi is the energy shift caused by diameter broadening. ϵσ is the error in the spectral width and is similarly given by

ϵσ2=iσSRFiσi2+δσi(fDvE)2σi2+δσi(fDvE)22,
(12)

where σSRFi is the standard deviation of the ith source as determined by the SRF, σi is the true source standard deviation, and δσi is the combined broadening caused by the diameter broadening and energy straggling. Note that in our case, δσi>σi and Eq. (12) approximately reduces to a comparison of the expected broadening and not of the actual source width. Despite that, Eq. (12) is still a valuable constraint due to the filter uncertainty broadening mechanism discussed in Sec. IV. It should be noted that the weighting of Eq. (10) has no theoretical basis and is just an ad hoc function that was found to give good calibration results shown in Fig. 15.

This is essentially an optimization problem with Nf free parameters. The dimensionality of this problem can become problematic as Nf gets too large. In such cases, Monte Carlo approaches are recommended to decrease computational requirements.

It is important to note that the minimization of Eq. (11) or Eq. (12) is not sufficient for calibration. There are degenerates in the areal density parameter space that allow matching the mean energies or the spectral widths with areal densities far away from the true values. This is well demonstrated in Fig. 15. Here, simulated data were generated using known filter areal densities and then several random filter configurations were used to analyze the data in an attempt to re-derive the correct configuration. As can be seen, minimization of ϵE and ϵσ alone was not sufficient to derive the correct configuration. The combined error (ϵ) was necessary.

In this section, we present and discuss various experimental examples of SRF data that has been analyzed with the technique discussed in Sec. III.

First, we start with an example of data taken from the MIT-HEDP Accelerator Facility.17 A 5F9W SRF measured DD protons ranged through 15 μm of aluminum. This spectrum was also measured using an AMETEK SBD for comparison. The results of this experiment are shown in Fig. 16.

As can be seen in Fig. 16, the SRF does an excellent job of matching the mean energy inferred by the SBD. On the other hand, the SRF spectrum is wider than the SBD spectrum due to the various broadening mechanisms discussed in Sec. IV. Here, the total broadening is roughly 230 keV, which is of the order of (albeit noticeably lower than) the results plotted in Fig. 8.

Next, we show an example of data taken from the OMEGA laser facility (Fig. 17). This figure shows a DD-proton spectrum emitted from an implosion measured by both a DD-p 5F9W SRF and CPS2. SRF and CPS2 are in good agreement on the absolute yield and the mean energy. Again, we see that the SRF spectrum is broader than the CPS spectrum corresponding to a minimum instrumental SRF broadening of 357 keV. Again, this is of the order expected from Fig. 8, but in this case, it is a bit larger.

Next is a demonstration of a 7F7W SRF being used to measure T3He deuterons generated from an implosion experiment performed on the NIF. In this example, the deuterons were measured by several SRFs and MRS. Figure 18 shows these data as inferred by the two diagnostics.

In Fig. 18, we see that both diagnostics capture the general shape of the spectrum, a Gaussian with a low energy tail due to the areal density evolution of the implosion. The SRF-measured spectrum has an artificial high energy tail because it also measured a T3He proton continuum that sits beneath the deuteron signal. The MRS is not subject to this issue due to its magnetic system separating the two signals. The mean energies are in good agreement, with differences thought to be due to asymmetries along the spectrometers’ lines of sight. Once again, the SRF signal is broadened relative to the MRS signal. Here, the broadening must be at least 839 keV. This large broadening is expected for deuterons due to the low c-Parameter required to describe the diameter–energy relationship. Additional broadening is thought to come from the T3He proton background.

Finally, Fig. 19 shows an example of 3He3He protons from an implosion experiment on the NIF being measured by a WRF and a 5F9W SRF. This example well demonstrates the newly developed analysis’s ability to recover arbitrary spectral shapes. As can be seen in the figure, the SRF and WRF are in good agreement.

In this paper, we have greatly expanded the capabilities of the SRF spectrometer. A new 5F9W SRF design was developed with three configurations for measuring common charged-particle spectra at the NIF and OMEGA. Additionally, a procedure for designing custom SRFs was established.

A new analysis technique for measuring arbitrary spectra was developed. This technique is extremely versatile and allows for measurement of higher order features not previously possible with the SRF spectrometers. This includes continuous non-peaked spectra (such as 3He3He-p spectra) as well as spectra with skews and long down-scatter features. This capability is not only advantageous for charged-particle measurements but is also crucial for the measurement of down-scattered neutrons in a proposed Z neutron-recoil spectrometer.5 

This work has extensively explored the response of the SRF spectrometers. Three different broadening mechanisms were identified: diameter blurring, filter thickness uncertainties, and filter energy straggling with diameter blurring generally being the dominant mechanism.

A new calibration technique for the SRFs has been developed and demonstrated in this work. This calibration procedure allows for more precise mean energy measurements than previously possible.

Finally, these improvements and new capabilities were all demonstrated on real experimental data from the NIF and OMEGA. This provides additional confidence in all of these developments.

The authors sincerely thank the OMEGA and NIF operations staff who supported this work and Bob Frankel, Ernie Doeg, and Michelle Valadez for processing the CR-39. This material is based upon work supported by the Department of Energy, National Nuclear Security Administration under Center of Excellence Award No. DE-NA0003868, the National Laser Users’ Facility under Award No. DE-NA0003539, and LLE under Award No. 417532-G. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

To quantify the effect of diameter blurring, a single SRF was exposed to 3 MeV DD protons at the MIT-HEDP Accelerator Facility. This SRF had Nw = 4 windows with aluminum filters of areal densities of 56.7, 110.7, 151.2, and 178.2 mg/cm2. Prior to the experiment, each filter was calibrated individually using a SBD for the measurement of the energy downshifts. A cartoon of the experimental setup is shown in Fig. 20. The resulting SBD spectra and SRF diameter distributions for each of the filters are shown in Fig. 21.

With the data shown in Fig. 21, it is possible to come up with relationships between the track diameters and particle energies using models described in Ref. 14. With these fits established, one can calculate how much broadening to expect in the diameter distributions from sources other than diameter blurring. The sources include the finite width of the fusion source [full-width-half-maximum (FWHM) of 130 keV] and energy straggling in the filters. The effects of straggling were determined using TRIM15 and are shown in Table II. The contribution of diameter blurring is then

σblurD=1Dσmeas2σother2,
(A1)

where σmeas is the standard deviation of the diameter distribution, σother is the calculated standard deviation from the finite fusion source width and energy straggling, and D is the mean diameter of the distribution. The results of this calculation for all filters and various etch times are shown in Fig. 6.

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