] T. The Mulassis Normalization factor. Omni-directional flux to Current [1]: Flux in Energy range of the simulation:

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1 he Mulassis Normalization factor he Mulassis normalization factor, which is set using the /analysis/normalise macro, is the scale factor that converts the simulation fluence to the real world fluence such that the analysis outputs can be provided in terms of real world values. he normalization factor to be used in Mulassis is composed of two components:. a geometric factor, converting incident flux to a current through a surface and 2. the number of particles in the energy range under consideration. he normalization factor is the product of the two values. Omni-directional flux to Current []: Consider a spatially uniform fluence φ(,θ,ω) of radiation in space, differential in particle kinetic energy, obliquity θ and azimuth ω (e.g. φ in units of MeV - cm -2 sr - ). Assume this fluence to be isotropic, i.e. independent of θ and ω, so that φ(,θ,ω) = φ(). he total π-fluence is simply 2π dω φ π ( ) = φ( ) d(cosθ ) = πφ ( ) () 0 he number of particle crossing a surface boundary per unit area per unit solid angle, at an angle θ with respect to the normal to the boundary is φ() cos θ, hence the use of the term cosine-law current in describing an isotropic fluence incident on a plane area. he total current crossing in through the boundary is then (with u = cos θ) 2π dω φ( ) udu = j ( ) = πφ( ) (2) 0 0 he total π-fluence and the total current have the same units (e.g. MeV - cm -2 ) but from Eqs () and (2) are related according to φ ( ) = j( ) (3) π In general, the outputs of Monte-Carlo calculations are inherently normalized to unit incident current, i.e. j(), as in the case of Mulassis where the incident particles are simulated from θ=0 to π/2 [3]. Flux in Energy range of the simulation: he flux of particles in the energy range under consideration in the simulation is given by integrating the differential particle spectrum over the energy limits of the simulation, or Φ min ( ) ] = φ d () Where Φ is the number of particles in the range to. In the case where an integral spectrum is provided, then it is a simple matter of subtracting the integral flux at the maximum energy in the simulation from the flux at the minimum energy in the simulation.

2 Normalization Factor: he normalization factor, or current in the energy range, for a differential flux/fluence spectrum is then a combination of Equations (3) and (), or: ] ] max = j( ) d = d = Φ J φ( ) Application to various spectral Forms: Linear in Energy: Differential energy spectrum is defined as: dφ = A + B d Normalization factor is: n (5) (6) A 2 = + B (7) 2 Power Law: Differential in energy spectrum is defined as: d φ = A d Normalization factor is: A B + B B+ (8) (9) Exponential: Differential in energy spectrum is defined as: Normalization factor is: d φ = d SPENVIS Environment Spectra B Ae (0) B ABe () he integral particle spectra, i.e. the flux from, are provided, so it is a simple matter of subtracting the flux at the maximum energy in the spectrum from the flux of the minimum energy in the spectrum, and dividing by, e.g. ( Φ] Φ] ) (2) NOE: while the particle spectrum can be included in the histogram as absolute or physical values, e.g. in terms of #/cm²/s/mev, the GPS normalizes the histogram to provide a probability function. hus the relative values are used within the GPS and not the absolute/physical values.

3 Slab Geometry Considerations: he outputs, e.g. dose or fluence, provided by a slab geometry simulation is an area output, i.e. it will always have the units cm -2, e.g. the dose would be in Rads cm -2. Spherical Shell Considerations: While the geometric considerations of a spherical shell normalization factor are already accounted for in the Mulassis code and the above relationships are directly applicable with no modification, in general the slab shielding normalization factor should be further modified by integration over the surface of the source sphere. For isotropic sources this factor becomes: n 2 sphere = nslab πr (3) Where R is the radius from which the source particles are generated. Source particle generation from θ min θ max he above relationships hold when the source particles are generated in a zenith range from 0 to π/2. When the solid angle of the source particles is limited to a range θ min to θ max, then the integration over the angles gives an additional factor of N angles : φ max max N angles = dφ dθ cosθ sinθ (a) φ θ θ min min 2 2 ( φ φ )( sin θ sin θ ) N = (b) angles max min Examples: In the following examples, a slab shielding geometry has been used with mm of Aluminium shielding 00 µm of Silicon. he incident particles are assumed to be isotropic, i.e. a cosine law angular distribution has been used and incident protons (primaries) generated in the simulation. Power Law Example From Equation 8, the definition of the differential power law spectrum, where A=.09E09 p + /cm 2 /MeV, B=-3.0 and we are interested in particles in the range from 0 MeV to 250 MeV, it follows from Equation 9 that: SHIELDOSE: 0.2 Rads Mulassis: 0.6 (±0.002) Rads A B + B+.09E E E06cm max MeV 0MeV 2 \2 [ ] min (5)

4 able. Mulassis macro file required to run the Power law simulation. /geometry/layer/delete 0 /geometry/material/add SiliconCarbide Si-C 3.200E+00 /geometry/layer/shape slab /geometry/layer/add 0 Aluminium 7.000E+00 mm /geometry/layer/add Silicon.000E+02 mum /phys/scenario em /analysis/normalise.360e+06 cm2 /analysis/fluence/particle/add proton /analysis/file PowerLawExample /analysis/fluence/unit cm2 /analysis/fluence/type OMNI /analysis/fluence/add 2 /analysis/fluence/energy/default /analysis/fluence/angle/default /analysis/dose/add 2 /analysis/dose/unit rad /geometry/update /gps/particle proton /gps/energytype Pow /gps/alpha E+00 /gps/emin.000e+0 MeV /gps/emax 2.500E+02 MeV /gps/angtype cos /gps/mintheta 0.000E+00 deg /gps/maxtheta 9.000E+0 deg /control/execute display5.mac /event/printmodulo 0000 /run/beamon Exponential in Energy Spectrum Example From Equation 0, the definition of the differential exponential in energy spectrum, where A=.99E09 p + /cm 2 /MeV, B=-7. MeV and we are interested in particles in the range from 0 MeV to 250 MeV, it follows from Equation that: SHIELDOSE: 22 Rads Mulassis: (±2.6) Rads ABe B.99 E09 ( 7.) e ( 3.6E0) e.87e09 cm e 250MeV 0MeV 0 7. (6)

5 able 2. Mulassis macro file required to run the exponential in energy simulation # some comments /geometry/layer/delete 0 /geometry/layer/shape slab /geometry/layer/add 0 Aluminium 7.000E+00 mm /geometry/layer/add Silicon.000E+02 mum /phys/scenario em /analysis/file ExponentialInEnergy /analysis/dose/add 2 /analysis/dose/unit rad /geometry/update /gps/particle proton # Power law in Energy for Integral flux spectrum # from 2003/0/26 event # F = A*exp(E/B) # A = 3.6E0 # B = -7. # df/de = (A/B)*exp(E/B) # AB =.99E09 /gps/energytype Exp /analysis/normalise.87e09 cm2 /gps/ezero -7. /gps/emin 0 MeV /gps/emax 250 MeV /gps/angtype cos /gps/mintheta 0.000E+00 deg /gps/maxtheta 9.000E+0 deg /event/printmodulo 0000 /run/beamon SPENVIS Environment Spectrum Example Having run the SPENVIS models to produce an Integral trapped proton spectrum, as seen in able 3 below, the normalization factor is calculated from equation 2 as: ( Φ] Φ] ) ( Φ] Φ] ) 0.MeV.75E + cm 00MeV (.8990E E + 08) 2 (7) he source particle type and spectrum are then defined in SPENVIS/Mulassis as:

6 Figure. Inputs to the SPENVIS/Mulassis source particle generation. And the final dose in a slab of silicon 00 microns thick behind a mm thick slab of aluminium is 2.8 krads (±2.2 krads), while SHIELDOSE calculates krads.

7 able 3. SPENVIS rapped Proton Integral spectrum from a 2 year GO orbit ( 300 x km, 0º inclination, 0º RAAN, 0º Arg. Perigee, orbit from // :00:00U) using AP8- MAX. Energy (MeV) otal mission average flux (/cm2/s) Integral proton spectra otal mission fluence (/cm2) Mission segment Average flux (/cm2/s) Segment fluence (/cm2) E E E E E E+5.777E E E E E E E E E E E E+.9327E E E E+.88E E E E+.037E E E E E E E E+.3656E E E E E E E E E E E E E E E E+2.552E E E E E E E+0.959E E+0.959E E E+2.335E E E+0 5.3E+.769E+0 5.3E E E E E E E E E E E+0.270E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+08

8 able. SPENVIS rapped Proton Differential spectrum from a 2 year GO orbit ( 300 x km, 0º inclination, 0º RAAN, 0º Arg. Perigee, orbit from // :00:00U) using AP8- MAX. Energy (MeV) otal mission average flux (/cm2/mev/s) Differential proton spectra otal mission fluence (/cm2/mev) Mission segment Average flux (/cm2/mev/s) Segment fluence (/cm2/mev) E E E E E E E E E E+5.657E E E E E E E E E E E E+5.8E E E E E E E E E E E E+.082E E E+06.30E E+06.30E E E+3.387E E E E E E E E E E E+0.78E E+0.78E E E E E E E+.6233E E E E E E E E+0.55E E E E+0.2E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-0.96E+07.75E-0.96E E E E E+05

9 able 5. he corresponding Mulassis Macro file to run the example simulation, derived from ables 3 and. /geometry/layer/delete 0 /geometry/layer/shape slab /geometry/layer/add 0 Aluminium 7.000E+00 mm /geometry/layer/add Silicon.000E+02 mum /phys/scenario em /analysis/normalise.750e+ cm2 /analysis/fluence/particle/add proton /analysis/file Spenvis_GO_RP /analysis/fluence/unit cm2 /analysis/fluence/type OMNI /analysis/dose/add 2 /analysis/dose/unit rad /geometry/update /gps/particle proton /gps/energytype Arb /gps/histname arb /gps/emin.000e-0 MeV /gps/emax.000e+02 MeV # # # he following is from the SPENVIS Differential flux # spectrum. /gps/histpoint.000e E+08 /gps/histpoint.500e E+08 /gps/histpoint 2.000E-0.66E+08 /gps/histpoint 3.000E E+07 /gps/histpoint.000e E+07 /gps/histpoint 5.000E-0.5E+07 /gps/histpoint 6.000E E+07 /gps/histpoint 7.000E E+07 /gps/histpoint.000e e+07 /gps/histpoint.500e E+06 /gps/histpoint 2.000E E+06 /gps/histpoint 3.000E E+05 /gps/histpoint.000e e+05 /gps/histpoint 5.000E E+0 /gps/histpoint 6.000E E+0 /gps/histpoint 7.000E E+0 /gps/histpoint.000e E+03 /gps/histpoint.500e+0.55e+03 /gps/histpoint 2.000E+0.2E+02 /gps/histpoint 3.000E E+0 /gps/histpoint.000e+0 2.7E+0 /gps/histpoint 5.000E+0.677E+0 /gps/histpoint 6.000E E+00 /gps/histpoint 7.000E E+00 /gps/histpoint.000e e+00 /gps/histpoint.500e E+00 /gps/histpoint 2.000E E+00 /gps/histpoint 3.000E+02.75E-0 /gps/histpoint.000e E-02 /gps/arbint Lin /gps/angtype cos /gps/mintheta 0.000E+00 deg /gps/maxtheta 9.000E+0 deg /event/printmodulo /run/cputime 3600 /run/beamon SPENVIS rapped environment spectrum with θ max =0 Using the same particle spectrum from the previous example, we want to calculate the dose on a 00 µm silicon sphere inside of an aluminium spherical shell mm thick. Because the target area is greatly reduced, most simulation particles will miss the target completely. By limiting the direction in which the particles are fired to, e.g. θ=0 to 0, instead of the default range of 0 to 90, the efficiency of the simulation can be greatly improved.

10 he additional components to the normalization factor to be included are:. the angular difference, a factor of sin 2 θ max, and 2. the additional π of particle generation space. his yields a normalization factor of: N=π sin 2 (0 ) n spenvis N=π E+ cm -2 N=.80E+ cm -2 References:. Seltzer, S.M, Conversion of Depth-Dose Distributions from Slab to Spherical Geometries for Space Shielding Applications, IEEE rans. Nucl Sci, Vol NS- 33, No 6, Dec 986,