29. Proton Emission Unit

Figure 29.1: The Proton Emission unit directory tree.

When activated, the Proton Emission unit generates protons within the domain from nuclear fusion reactions, follows the generated protons through the domain and detects them on specific detector screens. Protons are deflected by Lorentz forces inside the domain due to presence of electric and magnetic fields. The function ProtonEmission controls the generation and translation of the protons inside the domain as well as their recording on the detector screens. For each cell in the domain the average electric and magnetic fields are used and the electric and magnetic components do not change within each cell. Each cell emits a certain number of protons (depending on temperature and fusion reaction used) statistically inside a spherical cone characterized by an opening half-angle and a direction. The half-angle of the emission cone can range between 0 and 180 degrees, the latter implying emission of the protons into the complete surrounding sphere. For each detector an optional pinhole can be placed between the domain and the detector screen. Currently the proton emission code assumes emission of protons without altering the temperature and the chemical composition of the cell. This is an approximation valid only for low reaction rates.

29.0.1 Proton Generation

Protons are generated by fusion reactions in domain regions of high temperature. Depending on the type of nuclei present, several fusion reactions can take place in a cell. Currently there are two proton generating fusion reactions implemented in FLASH:

$$D + D \longrightarrow T (1.01 MeV) + p (3.03 MeV),$$ (29.1)

$$D + ^3He \longrightarrow ^4He (3.6 MeV) + p (14.7 MeV).$$ (29.2)

Nuclear reaction cross sections $\sigma$ are direct measures of probablity of nuclear reactions. Let $\Phi_{P0}$ be the initial flux of projectile nuclei P onto a homogeneous thin sheet of nuclei S and $\Phi_P$ the outcoming flux of projectile nuclei from the sheet. The flux of P is the number of P's per unit area per unit time. The difference in flux $\Phi_{P0}-\Phi_P$ due to nuclear reactions between P and S (there are other phenomena that lead to a reduction in flux, for example scattering) is the number of reactions that happened in the sheet per unit area per unit time and is directly proportional to the thickness $x$ of the sheet, the initial flux of P's and the nuclei number density $n_S$ of S's in the sheet:

$$\Phi_{P0}-\Phi_P = \sigma_S\;n_Sx\Phi_{P0}.$$ (29.3)

The proportionality constant $\sigma_S$ has dimensions of area and is defined as the nuclear reaction cross section for S. Since always $0\leq\Phi_P\leq\Phi_{P0}$, the dimensionless quantity $(\Phi_{P0}-\Phi_P)/\Phi_{P0}$ denotes the probability of the nuclear reaction to happen in the sheet. The quantity $n_Sx$ can be viewed as the number of S's per unit area of the sheet. Hence $1/n_Sx$ is the area in the sheet covered by each S. Rewriting the above equation we have:

$$\sigma_S = \left(\frac{1}{n_Sx}\right)\left(\frac{\Phi_{P0}-\Phi_P}{\Phi_{P0}}\right).$$ (29.4)

The cross section of S is thus equal to the area of each S in the sheet times the probability of nuclear reaction occuring in the sheet. It is thus the effective area of each S in the sheet that leads to a nuclear reaction. $\sigma_S$ is dependent on the relative velocity of each P towards the S's in the sheet. Let us denote this velocity by $v$ and let us further consider only sheet sections in which all incident P's have the same $v$. The sheet thickness will be traversed by each P in time $t$ and we have $vt=x$. Substituting this into the above expression for $\sigma_S$ and rearranging gives:

$$\sigma_S(v)v = \left(\frac{1}{n_St}\right)\left(\frac{\Phi_{P0}-\Phi_P}{\Phi_{P0}}\right).$$ (29.5)

The quantity $\sigma_S(v)v$ is called the reactivity of S and represents the probability of reaction per unit density of S per unit time in the sheet. Let us multiply the expression on the r.h.s. by $n_P/n_P$, where $n_P$ is the number density of incident P's onto the sheet. We have, after grouping together:

$$\sigma_S(v)v = \left(\frac{1}{n_Sn_P}\right)\left(\frac{\Phi_{P0}-\Phi_P}{\Phi_{P0}}\right) \left(\frac{n_P}{t}\right).$$ (29.6)

The quantity $(\Phi_{P0}-\Phi_P)n_P/\Phi_{P0}t$ represents the number of P's in the sheet that reacted per volume unit per time unit. It is known as the volumetric reaction rate between P and S (number of reactions per volume per time) in the sheet. If we swap the particles with the target nuclei in the sheet, i.e. if we consider the P's stationary in the sheet and the S's moving in the opposite direction, then $v$ is the same and we have:

$$\sigma_P(v)v = \left(\frac{1}{n_Sn_P}\right)\left(\frac{\Phi_{S0}-\Phi_S}{\Phi_{S0}}\right) \left(\frac{n_S}{t}\right),$$ (29.7)

where now $\Phi_S$ refers to the flux of S's in the opposite direction. The volumetric reaction rate $R_{PS}$ between P and S must be identical in both pictures, hence:

$$R_{PS} = \left(\frac{\Phi_{P0}-\Phi_P}{\Phi_{P0}}\right) \left(\frac{n_P}{... ...t) = \left(\frac{\Phi_{S0}-\Phi_S}{\Phi_{S0}}\right) \left(\frac{n_S}{t}\right)$$ (29.8)

and therefore the nuclear cross sections $\sigma_S(v)$ and $\sigma_P(v)$ are equal and will be referred to as the nuclear reaction cross section $\sigma(v)$. The volumetric reaction rate is

$$R_{PS} = n_Pn_S\;\sigma(v)v,$$ (29.9)

where $v$ is the relative velocity between P's and S's. This equation allows for a comparison to reaction rates of 2nd order chemical reactions. The number densities $n_P$ and $n_S$ of the nuclei play the role of concentrations of reactants. The quantity $\sigma(v)v$ is analogous to the chemical reaction constant, which in case of nuclear reactions depends on the approach velocity of the nuclei and their effective interaction area. If both nuclei P and S are the same, then using both $n_P$ and $n_S$ separately in the rate equation would overcount the number of reactions by 2. If there are $n$ identical nuclei in a certain volume, one can only form $n(n-1)/2$ distinct pairs, which for large $n$ becomes $\approx n^2/2$. The volumetric reaction rate including the possibility for identical nuclei reads

$$R_{PS} = \frac{n_Pn_S}{1+\delta_{PS}}\;\sigma(v)v,$$ (29.10)

where $\delta_{PS}$ is 1, if P and S are identical and 0 elsewhere.

In a cell with a definite temperature $T$, the relative velocities between the interacting nuclei vary and one must use a velocity distribution function $f(v)$ (most common used is Maxwellian). The volumetric nuclear reactivity $\sigma(v)v$ must therefore be replaced by an averaged reactivity

$$\langle \sigma v\rangle = \int_0^{\infty}\sigma(v)f(v)v\;dv$$ (29.11)

and the volumetric reaction rate becomes

$$R_{PS} = \frac{n_Pn_S}{1+\delta_{PS}}\;\langle \sigma v\rangle.$$ (29.12)

The dimensions of $\langle \sigma v\rangle$ are usually given in cm$^3$s$^{-1}$. It can be calculated from experimentally determined cross sections assuming a Maxwellian velocity distribution of the nuclei at a certain temperature and integrating over the entire velocity range. Since these integrations are time consuming, the values of $\langle \sigma v\rangle$ as a function of temperature are conveniently fitted using simple functional forms. For both above reactions, FLASH uses the functional fit as provided by Bosch and Hale (1992):

$$\langle \sigma v\rangle = C_1{\eta}^{-5/6}\xi^2\exp(-3\eta^{1/3}\xi),$$ (29.13)

where

$$\xi = C_0/T^{1/3}$$ (29.14)

$$\eta = 1 - \frac{C_2T+C_4T^2+C_6T^3}{1+C_3T+C_5T^2+C_7T^3}.$$ (29.15)

The values of the coefficients from $C_0$ through $C_7$ and the valid temperature range of the fit can be found in Atzeni and Meyer-Ter-Vehn (2004).

The tracing of the protons through the domain is analogous to the one used for proton imaging. Currently only $(x,y)$ pairs are recorded for each proton on each detector screen. Time resolved proton emission is not yet implemented, i.e. the protons exit the domain in one time step once they are created.

29.0.2 Proton Detector Screens

The setup of the proton detector screens and the proton recording technique is the same as for the proton imaging unit (section 28.0.4). Option for additional pinholes is provided as well. As there are no beams in this unit, the protons are recorded on the nearest detector screen in 3D space. If protons do not hit any detector screen, they will be lost. As the protons are not associated with a particular beam and detector, there is no option for recording offscreen protons.

29.0.3 Proton Emission Boxes

In order to allow for more simulation flexibility, the proton emission code is equipped with the possibility of selecting active proton emission regions (boxes) inside the domain. If no such boxes are specified, the entire domain is active, i.e. protons will be generated from the entire domain. If one or more emission boxes are given, protons will only be generated from inside the box. Proton emission boxes are allowed to overlap in space, but are not allowed to be completely outside the domain boundaries. Each emission box is characterized by its rectangular bounding box coordinates (lower left and upper right corners).

29.0.4 Usage

To include the use of the Proton Emission unit, the following should be included into the setup line command:  

+protonEmission [pem_maxDetectors=<number> pem_maxEmissionBoxes=<number> threadProtonTrace=True]

The +protonEmission shortcut handles all the logistics for properly including the ProtonEmission unit. The default settings are: maximum number of detectors = 1, maximum number of emission boxes = 1 and no use of threading during tracing of protons through the domain, which can be changed by the following three setup variables:

• pem_maxDetectors: The maximum number of proton emission detectors for the simulation.
• pem_maxEmissionBoxes: The maximum number of proton emission boxes for the simulation.
• threadProtonTrace: Enables threading during proton domain tracing.

The runtime parameters for the proton emission unit are very similar to those for the proton imaging. Setup and placement of detector parameters are identical to the proton imaging detectors.

29.0.4.1 Proton Emission Detectors Runtime Parameters

The following are the runtime parameters for the proton emission detectors. The _n at the end of each runtime parameter characterizes the detector number.

• pem_numberOfDetectors: The number of proton emission detectors that are going to be used.
• pem_detectorCenter[X,Y,Z]_n: The global 3D components of the detector center position vector ${\bf C}$.
• pem_detectorNormal[X,Y,Z]_n: The local 3D components of the detector normal vector ${\bf n}$.
• pem_detectorSideLength_n: The side length (cm) of the detector square screen.
• pem_detectorSideTiltingAngle_n: The tilting angle (degrees) of two sides parallel to the detector screen unit axis ${\bf u}_y]$ with respect to the tilting axis.
• pem_detectorSideTiltingAxis_n: The global tilting axis ('x','y' or 'z') for the detector screen.
• pem_detectorPinholeRadius_n: The radius of the pinhole. If $\leq 0$ no pinhole is used.
• pem_detectorPinholeDist2Det_n: The pinhole center distance from the detector center, useful only if a pinhole radius $>0$ was specified.

29.0.4.2 Proton Emission Boxes Runtime Parameters

The following are the runtime parameters for the proton emission boxes. The _n at the end of each runtime parameter characterizes the emission box number.

• pem_numberOfEmissionBoxes: The number of proton emission boxes that are going to be used.
• pem_emissionBoxCornerL[X,Y,Z]_n: The global 3D components of the emission box lower (left) corner.
• pem_emissionBoxCornerU[X,Y,Z]_n: The global 3D components of the emission box upper (right) corner.

29.0.4.3 Proton Emission General Runtime Parameters

• pem_appendOldDetectorFiles: If set to true, emission protons will be added to old existing detector files, provided old and new detector file names match.
• pem_cellStepTolerance: This factor times the smallest dimension of each cell is taken as the positional error tolerance for the protons during their path (parabolic or Runge-Kutta) tracing through each cell.
• pem_cellWallThicknessFactor: Controls the (imaginary) thickness of the cell walls to ensure computational stability of the proton emission code. The cell thickness is defined as this factor times the smallest cell dimension along all geometrical axes. The factor is currently set to $10^{-6}$ and should only very rarely be changed.
• pem_detailedTiming: If set to true, detailed timing info for several stages during the proton emission will be provided in the application logfile.
• pem_detectorFileNameTimeStamp: If set to true, a time stamp will be added to each detector file name. This allows for time splitting of detectors.
• pem_detectorXYwriteFormat: Controls formatted ascii output of the proton $(x,y)$ pairs on the detector screens (default 'es20.10').
• pem_emissionAmplificationFactor: Amplifies the proton emission count of each cell by this factor.
• pem_emissionConeHalfApexAngle: The half apex angle (degrees) of each cell emission cone.
• pem_emissionConeCenter[X,Y,Z]: The global 3D components of the emission cone center. This is for directional purposes only. Each cell sends an emission cone with the specified half appex angle to this center location.
• pem_ignoreElectricalField: If true, the electrical field in each cell is ignored.
• pem_ignoreMagneticField: If true, the magnetic field in each cell is ignored.
• pem_maxProtonCount: The maximum number of emitted protons that can be created on one processor per batch in the domain.
• pem_opaqueBoundaries: If true, the protons do not go through cells marked as opaque boundaries.
• pem_printDetectors: If true, it prints detailed information about the proton detectors to a file with name $<$basename$>$ProtonEmissionDetectors.txt, where $<$basename$>$ is the base name of the simulation.
• pem_printMain: If true, it prints general information regarding the proton emission setup to a file with name $<$basename$>$ProtonEmissionMain.txt, where $<$basename$>$ is the base name of the simulation.
• pem_printProtons: If true, it prints detailed information about the all protons generated during the simulation. Protons are 'generated' on the domain surface and the info is written to several files labeled by batch number BID, processor rank number PID and a time stamp. Each processor writes its own file(s) with name(s): $<$basename$>$printProtonsBatch$<$BID$>$Proc$<$PID$>$.txt, where $<$basename$>$ is the base name of the simulation. Use of this feature is reserved ONLY for debugging purposes and is currently limited to 10 batches and 100 processors per time stamp. Usage of a larger number of batches/processors during a simulation does not abort the run, but protons on batches with BID $> 10$ and processors with PID $> 99$ are simply ignored and not printed. Users other than code developers should not activate this feature.
• pem_printProtonSources: If true, it prints general information regarding the proton emission setup to a file with name $<$basename$>$ProtonEmissionMain.txt, where $<$basename$>$ is the base name of the simulation.
• pem_protonDeterminism: If true, the Grid Unit will be forced to use the sieve algorithm to move the proton particle data. Forcing this algorithm will result in a slower movement of data, but will fix the order the processors pass data and eliminate round off differences in consecutive runs.
• pem_RungeKuttaMethod: Specifies which Runge Kutta method to use for proton tracing. Current options are: 'CashKarp45' (order 4, default), 'EulerHeu12' (order 1), 'BogShamp23' (order 2), 'Fehlberg34' (order 3) and 'Fehlberg45' (order 4).
• pem_screenProtonBucketSize: Sets the bucket size for flushing screen protons out to disk.
• pem_useParabolicApproximation: If true, the code traces protons parabolically through cells for low $B$ / high $v$ combinations (section 28.0.1.2).
• useProtonEmission: If false, no proton emission will be performed, even if the code was compiled to do so. Bypasses the need to rebuild the code.
• threadProtonTrace: If true, proton tracing through a block is threaded. This runtime parameter can only be set during setup of the code.