Help & Methodology Reference
The Space RHA Proton Dose & Flux Calculator is a browser-based tool for proton irradiation test planning. It converts between beam current and particle flux, computes the ionizing dose rate in silicon using energy-dependent stopping power, and derives irradiation times needed to reach a target total ionizing dose (TID).
The tool is designed for radiation hardness assurance (RHA) engineers planning proton beam exposures at accelerator facilities. Given a proton energy and either a beam current or a desired flux, it calculates the corresponding dose rate in rad(Si)/s, krad(Si)/min, and Gy(Si)/s. Additional panels compute the time required to accumulate a specified total dose and the total dose delivered by a given proton fluence.
The default stopping power source is the Bethe–Bloch formula with Sternheimer density correction for silicon, which is accurate above approximately 20 MeV. For lower energies, the tool accepts user-supplied LET tables (e.g., from the NIST PSTAR database) with log-log interpolation.
The tool computes the mass electronic stopping power (LET) of protons in silicon using the PDG (Particle Data Group) form of the Bethe–Bloch equation with Sternheimer density-effect correction [1]. The total mass stopping power in MeV·cm2/g is:
The maximum kinetic energy transferable to a free electron in a single collision is:
At high energies, the electric field of the incident proton polarizes the medium, reducing the effective energy loss below the uncorrected Bethe–Bloch prediction. The density correction δ(βγ) accounts for this effect and is computed using the parameterization from Sternheimer, Berger, and Seltzer [3] with the following silicon-specific constants:
| Parameter | Symbol | Value (Si) |
|---|---|---|
| Mean excitation energy | I | 173 eV |
| First break point | x0 | 0.2014 |
| Second break point | x1 | 2.8715 |
| Fit coefficient | a | 0.1492 |
| Fit exponent | m | 3.2546 |
| Offset constant | C̄ | 4.4355 |
| Low-energy correction | δ0 | 0.14 |
The density correction is evaluated in three regimes based on x = log10(βγ):
The calculator supports two primary input modes, selectable via radio buttons at the top of the tool:
Enter a beam current (in A, mA, μA, nA, or pA) and beam area. The tool computes the particle flux as:
Enter a desired particle flux (in p/cm2/s) and beam area. The tool computes the beam current required to produce that flux:
The beam cross-sectional area can be specified in three ways:
| Shape | Input | Area Calculation |
|---|---|---|
| Circular | Diameter (cm or inches) | A = π(d/2)2 |
| Rectangular | Width × Height (cm or inches) | A = w × h |
| Direct Area | Area value (cm2 or in2) | As entered |
The tool includes presets for common proton irradiation facilities that automatically configure the beam geometry. For example, the UC Davis Crocker Nuclear Laboratory (CNL) preset sets a circular beam with an 8 cm diameter (area ≅ 50.27 cm2). Selecting a preset overrides the manual beam shape inputs; selecting "Custom" restores manual entry.
Once the flux and stopping power are known, the ionizing dose rate in silicon is computed from the fundamental relationship between particle flux, mass stopping power, and absorbed dose rate:
The conversion factor arises from the definition of the rad (1 rad = 100 erg/g = 6.242 × 107 MeV/g) and relates the product of flux and mass stopping power to absorbed dose rate:
The tool also displays dose rate in convenient derived units:
| Unit | Conversion from rad/s |
|---|---|
| krad/min | Ḋ × 60 / 1000 |
| Gy/s | Ḋ / 100 |
The Bethe–Bloch formula produces the mass stopping power in units of MeV·cm2/g, which is independent of the material density. For some applications it is useful to know the linear stopping power — the energy deposited per unit path length — which the tool displays in keV/μm.
The conversion from mass stopping power to linear stopping power is straightforward:
The linear stopping power is particularly useful for estimating how much energy a proton deposits within a thin sensitive volume of known thickness, and for comparing with tabulated data that may be expressed in keV/μm rather than MeV·cm2/g.
The Bethe–Bloch formula becomes increasingly inaccurate below approximately 20 MeV, where it underestimates the stopping power because shell corrections and charge-state effects (not included in the standard PDG form) become significant. For proton energies below ~20 MeV, the tool provides a custom LET table input that accepts externally sourced stopping power data.
Custom LET tables should be used whenever the proton energy of interest is below approximately 20 MeV. The tool displays a warning indicator when Bethe–Bloch is selected at low energies. Common scenarios include irradiations at low-energy proton facilities (e.g., 2–15 MeV medical cyclotrons) and dose calculations for degraded-energy beams.
The recommended source for custom stopping power data is the NIST PSTAR database [2], which provides total electronic stopping powers for protons in a wide range of materials over energies from 1 keV to 10 GeV. PSTAR values incorporate shell corrections, the Barkas effect, and the Bloch correction, making them significantly more accurate than bare Bethe–Bloch at low energies.
To use PSTAR data: visit the NIST PSTAR web interface, select silicon as the target, generate a table of total stopping power (MeV·cm2/g) vs. proton kinetic energy (MeV), and paste the two-column data into the custom table text area. Lines beginning with # are treated as comments. The tool accepts tab-separated, comma-separated, semicolon-separated, or space-separated values.
When custom table data is active, the tool uses log-log (power-law) interpolation between data points, which is the standard interpolation method for stopping power data. Given two adjacent tabulated points (E1, LET1) and (E2, LET2), the interpolated LET at energy E is:
Outside the range of the custom table, the tool performs log-log extrapolation using the slope defined by the two nearest data points. A minimum of two data points is required for interpolation to be active.
Given a target total dose Dtarget and the computed dose rate Ḋ, the irradiation time is simply:
The tool accepts the target dose in rad, krad, Mrad, or Gy, and displays the result in the most convenient time unit (microseconds through days) depending on the magnitude.
Given a total proton fluence Φ (in p/cm2), the total absorbed dose is:
This is the same conversion factor used for the dose rate, applied to fluence rather than flux. The tool displays the result in rad, krad, and Gy, along with the time required to deliver that fluence at the currently configured flux.
The Quick Reference Table at the bottom of the tool displays, for a set of standard proton energies (1, 2, 5, 10, 20, 50, 100, 200, 500, and 1000 MeV), the following quantities:
| Column | Description |
|---|---|
| Energy (MeV) | Proton kinetic energy |
| LET (MeV·cm2/g) | Mass stopping power from the active LET source |
| LET (keV/μm) | Linear stopping power in silicon |
| Flux for 1 krad/min | Particle flux (p/cm2/s) needed to deliver 1 krad(Si)/min |
| Current for 1 krad/min | Beam current needed at the active beam area to deliver 1 krad(Si)/min |
The target dose rate of 1 krad/min (= 16.667 rad/s) is used as a convenient reference because it is a common dose rate for proton TID testing. The required flux is computed by inverting the dose rate formula:
The reference table updates dynamically when the beam area or LET source changes. The current column scales linearly with beam area, so users can quickly see how beam size affects the required current at each energy.
Above approximately 20 MeV, the PDG Bethe–Bloch formula with Sternheimer density correction reproduces the silicon stopping power to within 1–2% compared to evaluated databases such as NIST PSTAR and ICRU Report 49 [4]. In this regime the formula is reliable for dose rate estimation without additional corrections.
Below approximately 20 MeV, the standard Bethe–Bloch formula progressively underestimates the stopping power. Several effects not included in the basic formula become significant:
Shell corrections: At lower velocities the assumption that all target electrons participate equally in energy loss breaks down. Inner-shell electrons contribute less when the proton velocity approaches the orbital velocity of K-shell electrons.
Barkas correction: A Z13 correction term accounting for the difference in stopping power between particles and antiparticles (charge-sign dependence).
Bloch correction: Higher-order terms beyond the first Born approximation that become important when the proton velocity is comparable to the orbital velocities of target electrons.
These effects are incorporated in the NIST PSTAR database but not in the tool's built-in Bethe–Bloch implementation. Users working at low energies should use the custom LET table with PSTAR data.
The stopping power used in this tool is the electronic stopping power only. Nuclear (elastic and inelastic) energy loss is not included. For protons above a few hundred keV, the nuclear stopping power is negligible compared to electronic stopping (<0.1%), so this omission does not affect dose accuracy. However, nuclear reactions can produce secondary particles (neutrons, gammas, recoil ions) that deposit additional dose locally. This secondary dose contribution is not captured by the simple flux × LET model and is generally small for thin targets.
The tool computes the mean energy loss per unit path length. In practice, the actual energy loss fluctuates from proton to proton (energy straggling). For thin absorbers where the total energy loss is a small fraction of the proton energy, the dose distribution follows a Landau or Vavilov distribution rather than a simple delta function at the mean. The tool does not model straggling and reports dose based on mean stopping power only.
The flux calculation assumes that the beam current is uniformly distributed over the specified beam area. Real proton beams are typically Gaussian or have other non-uniform profiles. The flux reported by the tool represents the area-averaged flux. Local dose rates at the beam center will be higher than the reported average, and dose rates at the beam periphery will be lower. Facilities typically specify a uniformity requirement (e.g., ±10% over the irradiation area), and the tool's results should be interpreted within that context.