Dose-voltage dependence of coaxial Bremsstrahlung diodes

Nuclear Instruments and Methods in Physics Research B34 (1988) 347-356 North-Holland, Amsterdam

DOSE-VOLTAGE T.W.L.

SANFORD,

DEPENDENCE J.A. HALBLEIB,

Sandia National Laboratories,

OF COAXIAL

BREMSSTRAHLUNG

J.W. POUKEY,

P. 0. Box 5800, Albuquerque,

347

C.E. HEATH

DIODES

and R. MOCK

NM 87185, USA

Received 27 January 1988 and in revised form 24 May 1988

The relation i) a IV2.65 is widely used to estimate the on-axis radiation-dose rate for flash X-ray sources, b,, as a function of diode current, I, and voltage, V, 1 m downstream of an optimized bremsstrahhmg target. This relation is valid only for pencil beams. In this paper, we show that for diodes having beams with finite spatial and angular extent, this relation can still be used if the power 2.65 is modified. Using particle-in-cell and radiation-transport codes, this modification is evaluated for a diode proposed for the 20-MeV HERMES III accelerator that is currently under construction. Predictions of the calculational model are compared with measurements obtained from experiments on the existing 3-MeV HELIA accelerator and are found to be in good agreement. These results are characteristic of finite-area coaxial diodes in general and show the trend of the deviation from 2.65 for such sources.

1. Introduction

High-power, pulsed, electron-beam diodes are used to produce intense bursts of X-rays via bremsstrahlung conversion in the anode. By fitting various experimental data, Martin [l] and Forster et al. [2] have shown that for small-beam radii the on-axis radiation dose rate, b,, measured 1 m downstream of the anode can be expressed by a relation of the form: b azv~,

20 Me.V ........ ,sMe”

(1)

where V is the voltage applied across the anode-cathode gap, Z is the diode current, and /3 equals 2.8 over a wide range of anode-converter materials. Martin [3] has reevaluated this relation and finds that p equals 2.65 for bremsstrahlung converters optimized to maximize the on-axis energy deposition. These authors point out that not only is this relation useful for predicting the forward radiation output but that the relation can be inverted to extract the diode voltage from a measure of both Z and b. Eq. (1) applies only to on-axis radiation. By fitting experimental data, Martin showed that the off-axis radiation could also be parameterized simply by using eq. (1) together with an exponential term to estimate the reduction in dose rate with angle [4]: t, a zvs,-

eq. (2), using p = 2.65, provides an excellent description of our calculational results for angles less than about 10” [6]. For larger angles, however, the angular distribution is sensitive to the exact converter composition, and we find that eq. (2) significantly underestimates the

ve/z.t

(2)

Here, V is measured in M.V and 0 is the angle of the radiation relative to the initial direction measured in radians. Using the electron-photon Monte Carlo transport code CYLTRAN [5], we have calculated the expected dose from pencil beams as a function of angle (fig. 1). For voltages between 8 and 15 MV, we find that 0168-583X/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Fig. 1. Comparison of calculated radial and angular dose profiles in a plane 25 cm downstream of a pencil beam incident on the 20-MeV optimized converter for incident electron energies of 5, 10, 15, and 20 MeV. All curves have been normalized to the peak value of the 20-MeV histogram.

T. W.L. Sanford et al. / Dose-voltage dependence of coaxial bremsstrahlung diodes

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calculated dose. In this region, eq. 2 is not expected to give a precise description of the radiation. By allowing /I to vary with angle, however, we find that eq. (1) can again be used to parameterize the calculations (fig. 2). Within typically better than 10% accuracy, it provides a compact description of the calculated radiation field with voltage. Importantly, for diodes generating beams with finite spatial and angular extent, we still find that eq. (1) can be used to provide a parameterization of the radiation field with voltage. For these cases, p becomes a function of the spatial location. In this paper, we calculate /I for a proposed coaxial diode for the 20-MV, 800~kA HERMES III accelerator [7,8] assuming a fixed electron distribution and fiied converter at the anode independent of voltage. These calculations illustrate the magnitude of the variation in p that can be expected from finite-size sources having space and angle distributions that change little with voltage. In reality, however, the electron flow in the diode is dependent on the voltage. To illustrate the magnitude such a dependence can have on p, we have also evaluated ~3 for this diode using the flow expected from a realistic voltage waveform [9]. The HERMES III accelerator is being constructed to provide an intense source of pulsed bremsstrahlung radiation over a 500 cm2 area with good uniformity for the simulation of the effects of gamma radiation. The diode being constructed for HERMES III [lo] and used

in the present evaluation employs an indentation in the anode (fig. 3). The primary purpose of the indentation is to control the electrons such that they impact the anode converter at near normal angles, producing a uniform radiation pattern downstream [ll]. A byproduct of our calculations shows that the indentation can also significantly sharpen the risetime of the radiation pulse and reduce its width. This sharpening is caused by the loss of the low-energy electrons associated with the rising and falling portion of the voltage pulse to the side anode-wall adjacent to the converter. For the simulation of nuclear radiation effects such sharpening is often desirable [12]. Knowledge of p for the near-field exposures, together with eq. (1) and the V and I pulse shapes, permits estimation of the temporal behavior of the near-field radiation pulse. In this paper, we demonstrate this calculation for the 3.2-MV, 150&A HELIA accelerator [13]. This accelerator has formed the pulsed-power test bed for the HERMES III accelerator. In the paper, the HELIA calculations are compared with PIN-detector measurements of the radiation and are shown to be in good agreement. This agreement, together with other comparisons of the calculated spatial radiation pattern [14] and diode impedance [15] with experiment gives us confidence in our model and calculational technique. As an example, we also evaluate the temporal radiation pulse expected from HERMES III. For the finite-area sources of the HERMES III and HELIA accelerators, the spatial and angular distributions of electrons incident on the anode converter were

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T. W.L.. Sanford et al. / Dose-uoltage dependence of coaxial bremsstrahlung diodes

calculated at discrete voltages using the electromagnetic particle-in-cell code, MAGIC [16], to model diode performance as a function of time. These coupled spatial-angular distributions were used as input to the CYLTRAN transport code for prediction of the subsequent radiation dose profiles in CaF, thermoluminescent dosimeters (TLDs). For most of these calculations, the efficient next-event-estimator approximation [17,18] was used to estimate the doses. The resulting doses were then fit to eq. (1) in order to determine p. Before presenting results for the HERMES III and HELIA accelerators, we discuss the reduction in /3 that is expected for finite-area sources relative to pencil beams and show that our calculations reproduce the empirical relation of Martin [3]. This discussion permits comparison of the /I obtained from the finite-area sources with those obtained from pencil beams and provides insight into the variation in /? calculated for the former. This paper expands on the work reported earlier at the 1987 Particle Accelerator Conference [19].

2. /3 for a pencil beam source The efficiency for producing bremsstrahlung radiation in the forward direction increases superlinearly with the kinetic energy of the electrons incident on the converter [17] because the radiation becomes more and more concentrated in the forward direction and because of the rapid rise in bremsstrahlung cross sections with energy. Fig. 1 shows predictions of the CYLTRAN code using the next-event-estimator method to illustrate the relative reduction in the angular width of the resulting radiation pattern for normally incident electrons over the range 5-20 MeV from a converter optimized [17] at 20 MeV. This converter is composed of 4.81 g/cm2 of Ta, followed by 5.24 g/cm2 of graphite, and 0.832 g/cm2 of Kevlar [20]. Experimentally, the Ta is used to efficiently convert the electron energy to radiation energy, the graphite is used to absorb the residual electrons without significantly attenuating the radiation produced in the Ta, and the Kevlar is used to prevent debris generated in the target from going forward. Because the width of the radiation narrows with increasing energy [eq. (2)], the non-forward radiation fluence does not increase as rapidly with energy or, equivalently, with diode voltage as does the forward radiation fluence. Accordingly, the power dependence, 8, of eq. (1) decreases with. increasing angle, 8, of the radiation relative to the incident electron direction. For the optimized 20-MeV converter, the decrease that we calculate (fig. 2) is plotted as the upper curve in fig. 4 for pencil beams. In the forward direction, /3 equals 2.89; it is reduced to 1.88 a 43”.

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Fig. 4. Calculated values of p as a function of 8, the angle of the radiation with respect to the direction of the incident electrons, for a pencil beam. The open-circle data correspond to /I calculated using the converter optimized for 20-MeV electrons (fixed) (fig. 2). The solid-square data correspond to p calculated using converters optimized at each incident electron energy (variable).

In the upper curve of fig. 4, at 19= 0, p exceeds the value obtained by Martin because the converter thickness was not optimized at each incident electron energy. The lower curve in fig. 4 is p, calculated when the converter is optimized at each incident electron energy. In this case, /I is reduced from that obtained using the fixed 20-MeV-optimized converter. This reduction is easy to understand. As the electron energy is decreased below 20 MeV, selfabsorption of radiation decreases for converters that are optimized for the specific source energy relative to the selfabsorption for electrons of that same energy in converters optimized for 20-MeV electrons. Thus, more radiation is extracted. In the forward direction, our calculated power of 2.62 for this variable converter is in excellent agreement with the 2.65 empirically determined by Martin.

3. fl for the HERMES III source For finite-size electron-beam sources, the radiation on-axis in the near field is no longer due to just radiation generated at zero angle. Instead, the radiation is a composite of that produced at many angles relative to the incident electron. Based on the above discussion, which shows that /I decreases with increasing angle, we therefore expect /3 to be reduced for finite-size sources.

T. W.L. Sanford et al. / Dose-voltage dependence of coaxial bremsstrahlung diodes

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VblV) Fig. 6. Calculated dose versus voltage for the on-axis (1.5 < p < 3.0 cm) X-ray dose 10 and 45 cm downstream of the converter for the finite-area HERMES III source shown in fig. 5 and a Ta/C converter optimized for 20-MeV electrons.

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As an example of the magnitude of the reduction in p expected from a finite-size source, we have calculated fl expected from a typical source proposed for the HERMES III accelerator using the electron flow pattern expected at peak voltage. The MAGIC code was used to simulate the steady-state flow in an indented-anode diode [ll] chosen to produce a beam at the converter that provides a relatively uniform radiation pattern at peak voltage. Fig. 3 shows the essential features of that diode, along with a graphical representation of that flow. Assuming a fixed A-K voltage of 20 MV, the MAGIC calculation yielded the coupled spatial-angular distribution of the electrons at the converter. The distribution is specified in terms of a radially dependent (a) current density, (b) mean angle of incidence, and (c) RMS variation of the angular distribution. The first two of these are shown in fig. 5; the RMS variation of the angular distribution at a given radius is typically +25% of the magnitude of the mean angle at that radius. This information defined the electron source used in CYLTRAN calculations at 5,10,15, and 20 MeV. Radiation due to the small electron losses to the side anode wall that is adjacent to the converter was not included. After sampling a source position from (a), the source direction was sampled from a Gaussian defined by (b) and (c). The two-dimensional X-ray energy deposition profiles predicted by CYLTRAN for a Ta/C converter optimized for 20-MeV incident electrons were used to deduce /3. As can be seen from fig. 6, the results of our calculations for the finite-size HERMES III beam show that,

T. W.L., Sanford et al. / Dose-voltage dependence of coaxial bremsstrahhmg 30.

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on the axis and near the source, B is reduced from 2.89 for a pencil beam (fig. 4; upper curve) to about 2.2. Data points in fig. 6 are for 1.5 < p < 3.0 cm. Fig. 7 shows the variation in /3 as the position along the axis is increased from the near field to the far field. The variation in fi as a function of radial distance p from the beam axis at z = 10, 25, and 40 cm is shown in fig. 8. The increase in fi with increasing z shown in fig. 7 is expected, because at larger distances from the converter, the fluence is more forward directed. The above calculations do not take into account the change in electron flow in the diode as a function of voltage. They only show the variation in dose rate due to changes in the bremsstrahlung and related cross sections with electron energy for a fixed electron flow pattern. Curve A of fig. 9 shows the voltage pulse that is expected at the HERMES III diode calculated using a circuit-analysis code [9]. Using this pulse, MAGIC was configured to generate the time-dependent change in electron flow with voltage. The flows at four voltages are shown in fig. 10. The geometry of the calculations correspond to that being constructed for HERMES III [lo]. In a manner similar to the earlier calculation at 20 MV, each of the flow patterns generated by MAGIC at 3, 7, 11, 14.5, 17.4, 19, 21.5, and 23 MV during the rising portion of the voltage pulse and at 23, 20.5, 18, 15.5, 12.5, and 9 MV during the falling portion of the pulse were used as input to CYLTRAN. All flow incident on the anode downstream of z = 60 cm was used in the simulation. Differences between the rise and fall

Fig. 9. (a) Voltage pulse expected at the Hermes III diode, (b) expected radiation pulse averaged over 572 cm* at z = 10 cm when changes in electron flow and losses to side anode-wall with time are included in calculation, (c) expected radiation pulse averaged over 572 cm* at z =lO cm when using the time-independent source of figs. 3 and 5, where no account of changes in electron flow and losses to side anode-wall with time are included in the calculation.

are expected because the Z? and b terms in Maxwell’s equations are not completely negligible. The inductive voltage has an opposite sign during the rise and fall of the pulse. Additionally, the system in the simulation remembers its earlier state to some degree. In fig. 11, the resulting output doses for each of the patterns were fit to eq. (1) keeping Z constant in order to obtain /? separately for the rising and falling portions of the voltage pulse. The doses correspond to averages over 572 cm* (p I 13.5 cm) just downstream of the converter (z = 10 cm). In the calculations, we took account of the transport of electrons hitting the side anode-walls. At lower voltages, the electron losses to the side-walls are greater because of changes in the radial forces acting on the beam [14]. In the calculations, as in the diode being fabricated, the side-walls were made of range thick graphite (13 g/cm*). Graphite was chosen to both minimize the damage from the incident electrons and to minimize the subsequent radiation. The converter was similar to the optimized Ta/C converter at 20 MeV described earlier. Surrounding both the graphite side-walls and converter was a 1 cm thick aluminum vacuum chamber. In fig. (ll), note the significant increase in /I over that calculated without including the change in flow and resulting losses to the side anode-walls as shown in fig. (6). During the rising portion of the voltage pulse the dose rate scales as ZV4.09 and during the falling portion and ZV3.14. As with the previous, fixed source

352

T. W.L., Sanford et al. / Dose-voltage dependence ofcoaxial bremsstrahlung

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z (cm) Fig. 10. Calculated electron flow patterns in the indented-anode diode being constructed for HERMES III [lo] at four voltages V=7 MV, corresponding to the rising portion of the voltage pulse shown as curve A in fig. 9 at: (A) V -0, I-O, Z-co,(B) I= 193 kA, Z = 36.2 &?,(C) V = 14.5 MV, Z = 402 kA, Z = 36.1, and (D) V = 21.5 MV, Z = 729 k.4, Z = 29.5 D. Here, V refers to the voltage applied between the cathode tip and converter, and Z refers to the current incident on the anode for I greater than -60 cm.

case, J3 varies only slightly across the z = 10 cm face changing from 3.77 at 1 5 p I 3 cm to 4.23 at 12 I p s 13.5 cm during the rise. The difference in voltage dependence between the rising and falling portion of the pulse reflects the dependence of the flow pattern on time and shows that such changes may need to be considered if accurate estimates of dose rates versus voltage are to be obtained for a given diode.

4. #8 for the HELIA source The theoretical model was tested by evaluating j3 for a planar-anode diode on the HELIA accelerator (141. Specifically, we used the calculated /3 together with eq. (1) and the measured voltage and current waveforms for a given shot on HELIA to predict the relative b wave-

form as measured in a Si-PIN diode located 2.5 m downstream of a planar-anode converter made of 8.75 g/cm’ thick graphite. The PIN diode was estimated to have a temporal resolution of less than 3 ns [21]. For this calculation we used the measured voltage pulse shown in fig. 12 as input to MAGIC in a manner similar to that described for the HERMES III time-dependent simulation. In this case, however, we used the MAGIC distributions evaluated at 0.39,0.62, 1.08, 1.33, 1.87, 2.19, 2.85 and 3.5 MV for only the rising portion of the voltage pulse, which is adequate in the present case. Electron flow patterns at four voltages are shown in fig. 13. In fig. 14, the simulated angle of incidence at the converter, averaged over the entire converter, is shown. The angle varied from about X0 at 0.5 MV to about 30 ’ at 3 MV. As before, we used the MAGIC distributions for each of the given voltages as input to

T. W.L. Sanford et al. / Dose-voltage dependence of coaxial bremsstrahlung

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353 of

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of the Si-PIN detector. The combined geometry of the X-ray source region, including X-ray production in the side wall of the anode from z = -6 cm forward, together with the detector was used in the transport calculations, as opposed to the next-event-estimator approximation used earlier. The results are plotted in fig. 15. They show that the increased pinch angle of the beam at the planar-anode converter with increased voltage reduces the effective /3 from the 2.8 expected for normal incidence (fig. 4, upper curve) at the position of the detector. This reduction occurs because for those X-rays emitted toward the detector, the mean emission angle, as measured relative to the directions of the radiating electrons, increases with voltage. As the emission angle increases the bremsstrahlung intensity drops [fig. 1 and eq. (2)]. Using 1.8 for p, we expect that the relative variation in the radiation pulse, b,, should follow IV’.*. Fig. 16 is a comparison of the detector signal with the predicted dose-rate time history based on this power law and the measured Z-V waveforms (fig. 12). Agreement is excellent. However, because of the delay between the turn-on of the current with respect to the voltage, the comparison is not sensitive to the exact power of V. This lack of sensitivity follows from the observation that the bulk of the variation in the leading edge of the voltage waveform occurs when little current is flowing. Furthermore, the trailing edge of the voltage pulse fell precipitously due to self-breaking switches in the pulse-forming lines [13]. Consequently, the time over which the voltage was changing was small, and the dependence on p of the predicted radiation output was masked by experimental resolution. We find, in fact, that IV - the power pulse gives an equally consistent description of the relative shape of the radiation pulse. The insensitivity of the comparison in fig. 15 to p for our experimental conditions helps to explain why other experiments with similar conditions have often found an unexpectedly good correlation between the shapes of the power pulse and the radiation pulse. energy

Fig. 11. Calculated dose averaged over 572 cm2 (0 I p I 13.5 cm) at z = 10 cm versus voltage for the source shown in fig. 10: (A) during the rising portion of the voltage pulse shown in curve A of fig. 9, (B) during the falling portion of the voltage pulse shown in curve A of fig. 9.

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Fig. 12. Measured voltage (V) and current (I) wave forms for a typical HELIA shot. The voltage at the diode has been corrected for the inductance between the measuring locations and the diode [8,14].

The discussion of the last two sections is now applied to estimating the expected radiation pulse averages over the 572 cm* area (p < 13.5 cm) just downstream of the HERMES III converter. If we assume that the impedance of the diode is constant, independent of voltage (which is true for our steady-state MAGIC simulations [ll], but not quite accurate for the above time-dependent simulation (fig. lo), we can easily estimate the shape of the radiation pulse using the expected voltage pulse (curve A of fig. 9) together with i>(r) a Z(OVt)a a v(r) Bt’. , here /3 equals 4.09 during the

354

T. W.L. Sanford et al. / Dose-voltage dependence of coaxial bremsstrahlung diodes

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Fig. 15. Calculated dose at the location of the Si-PIN detector on-axis and 2.5 m downstream of the graphite Converter versus voltage for the finite-area HELIA source shown in fig. 13.

T. W. L. Sanford et al. / Dose-voltage dependence of coaxial bremstrahlung

diodes

355

the indented pulse relative to the planar pulse is due to the radiation generated in the side graphite wall in the indentation leaking through the front graphite converter, and adding to the radiation generated from the converter.

6. Summary

-11 10

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TIME (nsl Fig. 16. Comparison of radiation pulse measured by the Si-PIN detector on the HELIA experiment for the shot shown in fig. 12 with the shape of that predicted using the power law, 1, CCIV’.*, where the Z and V wave forms were those measured and where the exponent was obtained from the MAGIC/CYLTRAN simulations (fig. 15).

rise and 3.14 during the pulse fall. The calculated radiation pulse is shown as curve B of fig. 9, where it can be compared with the voltage pulse. As expected, the 21 ns FWHM of the radiation pulse is considerably reduced from the 33 ns FWHM of the primary voltage pulse. If the variation in the flow and the losses to the side anode-wall are not taken into account, p would equal about 2.2. The resulting radiation pulse would have a FWHM of 23 ns with the shape shown as curve C of fig. 9. The impact on the rise time and fall time of the radiation pulse is considerable when the variation in flow is included, but, because of the fast rise time and fall time of the voltage pulse relative to its width, the impact on the radiation is only to reduce the FWHM of the radiation by 12%. For slower rise times, however, the impact could be significant. Comparison of radiation measurements taken on HELIA with both planar and indented anodes [14] confirm the reduction in pulse width when indenting the anode. Fig. 17 shows a comparison between the radiation measured in the on-axis %-PIN detector for the planar-anode diode just discussed (figs. 13 and 16) with that measured when the indentation in the diode is optimized to simultaneously transmit the beam at peak energy and produce a uniform radiation pattern [14]. Note the 34% reduction of the FWHM in going from the 26.8 ns for the planar anode to the 17.3 ns for the indented one. We believe that the reduced rise time for pulse

p in the relation b a ZYp, which is used to evaluate converters, dose rates downstream of bremsstrahlung has been calculated for both pencil beams and finite-size sources. The expression has been shown to provide a compact description of the radiation as a function of diode voltage over the range 5-20 MV. For pencil beams, our calculations agree with the empirical relation of Martin, giving /I = 2.62 for on-axis radiation from the variable converter that is optimized at each energy. This agreement gives us confidence in the CYLTRAN calculations. For finite-size sources, p is reduced from that expected from a pencil beam. The fixed-source example of HERMES III shows that fi values ranging from 2.2 near the converter to 2.8 in the far field are possible. The HERMES III calculations, which took into account the explicit voltage dependence of the flow, illustrate the importance of including such effects if an accurate dose-rate/voltage correlation is desired for a given source. Lastly, a byproduct of these calculations shows the potential of using an indentation in the anode to shar-

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Fig. 17. Comparison of measured radiation pulse in the on-axis Si-PIN detector on HELIA for (A) planar-anode diode (fig. 16) and (B) indented-anode diode. The converter in both diodes was 8.75 g/cm’ of graphite and the side anode wall for the indented-anode diode was graphite (141.

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T. W.L. Sanford et al. / Dose-voltage dependence of coaxial bremsstrahlung diodes

pen the rise time of the radiation quently reduce its width.

pulse and to subse-

Acknowledgements We would like to thank W. Beezhold, J.R. Lee, K.R. Prestwich, J.J. Ramirez, and D.E. Hasti for useful disand M.G. Mazarakis for reviewing the cussions, manuscript. Lastly, we would like to thank M. Song and G. Albright for preparing the paper. This work was supported by the US Department of Energy under contract DE-AC04-76DPOO789.

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