The occurrence probability of events due to large energy depositions in a device - like those commonly referred to as single event effects (SEE) – is usually expressed by means of the cross section as a function of LET – or mass electronic stopping power or mass restricted energy loss - of the device material (e.g., see Sects. 11.4-11.4.9 of [Leroy and Rancoita (2016)] and references therein). The functional form for expressing the SEE cross section for a device is commonly provided by the Weibull function.

Furthermore, as discussed in this , one can re-express that device cross section as a function of the energy of incoming particles. In this way, the overall events occurrence can be estimated using the entire range of particle spectral fluence. In fact, nowadays the spectral fluences of low- and high-Z cosmic rays were observed up to very high energies (e.g., see ).

The following link allows one to access to the determination of Weibull function parameters for device cross section due to a large energy deposition (e.g., SEE cross section):

- Weibull function parameters for device cross section due to a large energy deposition

### How to use this calculators for determinatio of Weibull function parameters for device cross section

This tool allows one to determine the Weibull function parameters for device cross section due to a large energy deposition by means of a fit of the data provided by the user.

The input parameters and options for the tool are described below. When the input form has been completed, pressing the "CALCULATE" button will start the calculation and open the "Results" page (allow for pop-up in your browser settings). The result page will be also linked at the bottom of the calculator page.

### Input Parameters:

- Minimization method

- SEE device cross section data as a function of LET or mass electronic stopping power or mass restricted energy loss

- fixed parameters for Weibull function fit (optional)

- Plotted cross section options

In case the user chose to plot the cross section as a function of energy further input parameters have to be provided:

- Energy limits

- Incident particle

- Device material

- Energy Loss Type

### Minimization method

The fit with the Weibull function is performed by means of the TMinuit class of ROOT data analysis framework. In the calculator panel, using the pull down menu, the user can select the minimization method that will be used to fit the provided data:

- MIGRAD+HESSE for Cross section data with experimental errors. In case no errors are provided, fixed percentual errors are used to optimize the resulting fit.

- MIGRAD for Cross section data without experimental errors

### SEE cross section data

This section defines the points of the SEE Cross section in cm^{2}/device as a function of the stopping power (or LET) in MeV cm^{2} mg^{-1}. In case MIGRAD+HESSE minimization is select, the errors on SEE Cross section is also needed. If this is not provided a fixed 0.1% error is applied to all data points.

The input format is one point per line (Stopping power - Cross section - Cross section error , separated by a space or tab); it is also possible to copy and paste values.

Input data with a null value for the cross section are not used in the fitting procedure, but to deterrmine the minimum value of the "Stopping power (or LET) Threshold - *x _{0}*"

### Fixed parameters (optional)

Data provided by the user is fitted with the Weibull function:

where *x* is the stopping power (or the restricted energy loss) in MeV cm^{2} mg^{-1}, *A* is the saturation cross section, *x _{0}* is the stopping power (or LET) threshold parameter,

*P*is the width parameter and

_{W}*s*is a dimensionless exponent.

User can fix the saturation cross section

*A*and the stopping power threshold

*x*. If the input fields are empty no parameter is fixed.

_{0}

### Plot cross section options

The cross section can be plotted as a function of stopping power (or LET) as provided by the users or can be plotted as a function of the energy of incoming particles. In this case data related to the stopping power is needed to convert the stopping power to energy.

### Energy limits

This section define the energy limits of the conversion from stopping power to energy. Minimum energy must greater the 1 eV. In case of SRIM electronic stopping power, the maximum energy has to be lower than 5 GeV/amu.

### Incident Particle

In the calculator panel, using the pull down menu, the user can select the species of the incident particle, either a proton or one of the elemental ions.

The user can also modify the mass (in amu) of the incident particle (e.g., for all isotopes one can refer to this page): the default mass is that of the most abundant isotope (MAI), except for proton and alpha particle masses. Further information are available at the following webpage.

### Device Material

In the section "Device Material Selection" it is possible to specify an **User Defined** device material or a predefined **Compound**. User can also select the device as a **gas**, this is allowed only for single element and natural gas target (H, He, N, O, F, Ne, Cl, Ar, Kr, Xe, Rn).

The stopping power in target gases is usually higher than that in an equivalent solid target. The Gas/ Solid correction disappears for higher velocity ions with energies above 2 MeV/amu. But at lower velocities the effect can be quite large - almost a 2 times change in stopping bacause of the Phase effect near the Bohr velocity, 25 keV/amu.

in the User Defined section individual elements can be selected as well as the composition of the device material choosing the number of elements in the compound. The required parameters for each element are:

- Atomic number (Z)/Chemical symbol

- Stoichiometric index or element fraction

*Electronic Stopping Power for User Defined Compounds*

Electronic Stopping Power for User Defined Compounds can be determined by means of Bragg's additivity rule, i.e., the overall Electronic Stopping Power in units of MeV cm^{2}/g (i.e., the mass electronic stopping power) is obtained as a weighted sum in which each material contributes proportionally to the fraction of its atomic weight. For instance, in case of a GaAs medium ones obtains (e.g., Eq. (2.20) at page 15 in [ICRUM (1993)]):

where and *A*_{Ga} [*A*_{As}] are the Electronic Stopping Power (in units of MeV cm^{2}/g) and the atomic weight of Gallium [Arsenic], respectively.

For few compounds belonging to the ICRU list the parameters employed for the energy loss formula (including those for the densiity effect) are reported in Table II of Sternheimer et al. (1984).

Further discussion can be found in the help of the SR-framewrok electronic stopping (see also results page).

### Energy Loss type

When “Cross section as a function of” is selected as “Energy [MeV]”; The user can select 3 different energy loss types:

- SRIM Electronic Stopping Power

- SR-Framework Electronic Stopping

- SR-Framework Restricted Energy Loss

When restricted energy loss is selected, the Effective detectable energy W_{0} can be modified: the current default value is 100 keV.

For the conversion of the cross section as a function of the incoming particle energy, the input cross section is assumed to be provided as a function of the corresponding type of the selected energy loss (i.e., SRIM Electronic stopping power, SR-Framework Electronic stopping, SR-Framework Restricted energy loss). One has to remark that SRIM Electronic stopping power and SR-Framework Stopping power agree within 5% (or better) up to the maximum energy obtained from SRIM Module.exe (i.e., 5 GeV/amu), see discussion here.

As discussed at page 70 of [Leroy and Rancoita (2016)] for all incoming particle energy for which the maximum transferred energy W (see Eq. (1.29) of [Leroy and Rancoita (2016)]) is lower than or equal to W_{0} there is no difference in between the SR-Framework electronic stopping power and the restricted energy loss (see Figure 1).

Figure 1. Energy limit (in MeV/amu) below which there is no difference in between the SR-Framework electronic stopping power and the restricted energy loss as a function of W_{0}.

### Result

The result page contains the input cross section data as function of the stopping power (or LET) plotted along with the Weibull fit function. Parameters of the Weibull function are provided. The table reports the input data with the fitted function values.

When the plot as a function of energy is selected further plot are shown:

- Stopping power as a function of energy.

- Fitted cross section as a function of energy.

The table reports the cross section as a function of the energy.