Implemented Collisions

Binary Collisions

These binary interactions have currently been implemented:

• Elastic collision of hard spheres SphSphSphSph

  • functions: dsigmadt_SphSphSphSph sigma_SphSphSphSph

• Photon pair production from electron positron annihilation ElePosPhoPho

  • functions: dsigmadt_ElePosPhoPho sigma_ElePosPhoPho

• Electron-positron pair production from photon pair annihilation PhoPhoElePos

  • functions: dsigmadt_PhoPhoElePos sigma_PhoPhoElePos

• Electron(or Positron)-Photon scattering (Compton Scattering) ElePhoElePho

  • functions: dsigmadt_ElePhoElePho sigma_ElePhoElePho

Emission Interactions

These emissive interactions have currently been implemented:

• Synchrotron (cyclotron) emission
  • functions: SyncKernel

Adding New Binary Collisions

Users may add their own binary interaction cross sections to the /src/commom/DifferentialCrossSectionFunctions.jl file of DiplodocusCollisions.jl. Functions should be named in the following format: "sigma_name1name2name3name4" and "dsigmadt_name1name2name3name4" where the names are three letter abbreviations of the particles involved (see Particles, Grids and Units). The named pairs name1name2 and name3name4 should be in alphabetical order. Both the total cross section and differential cross sections must be provided for a single interaction.

All cross sections are to be defined in terms of the Mandelstram variables $s=(p_1^{\mu }+p_2^{\mu }{)}^2$, $t=(p_1^{\mu }-p_3^{\mu }{)}^2$ and $u=(p_2^{\mu }-p_3^{\mu }{)}^2$. To maintain accuracy of cross sections and avoid DivZero issues when momenta is small compared to the mass of the particles (at Float64 precision), each Mandelstram variable in the cross sections should be split into two components:

• s=sSmol+sBig where $sBig=(m_1+m_2{)}^2$

• t=tSmol+tBig where $tBig=(m_1-m_3{)}^2$

• u=uSmol+uBig where $uBig=(m_2-m_3{)}^2$

The "Big" part typically cancels with terms in the cross sections, leading to better accuracy. Therefore the function should defined as follows:

function sigma_name1name2name3name4(sSmol::Float64,sBig::Float64,tSmol::Float64,tBig::Float64,uSmol::Float64,uBig::Float64)
    ... 
end

function dsigmadt_name1name2name3name4(sSmol::Float64,sBig::Float64,tSmol::Float64,tBig::Float64,uSmol::Float64,uBig::Float64)
    ... 
end

where all of sSmol, sBig, tSmol, tBig, uSmol, and uBig must be included in the function definition irrespective of if they actually appear in the cross section. It is also important to define a normalisation for the cross sections to avoid emission and absorption terms becoming small compared to the Float64 minimum.

WARNING

Total cross sections may typically be defined/derived in textbooks and other sources to include division by $1/2$ if output states are identical. This factor should NOT be included here as this factor is included separably in the code.

Internal Collision Functions

DiplodocusCollisions.sigma_SphSphSphSph Function

sigma_SphSphSphSph(sSmol,sBig)

returns the total cross section for the binary interaction of hard spheres with normalised masses (wrt electron mass) $m_1,m_2,m_3,m_4=m_{\text{Sph}}$. Normalised by $\pi R_{Sph}^2={\sigma }_T$.

$$\sigma =\frac{1}{2}$$

Arguments

• sSmol::Float64 : s - sBig

• sBig::Float64 : (m1+m2)^2

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DiplodocusCollisions.dsigmadt_SphSphSphSph Function

dsigmadt_SphSphSphSph(sSmol,sBig,tSmol,tBig,uSmol,uBig)

returns the differential cross section for the binary interaction of hard spheres with normalised masses $m_1,m_2,m_3,m_4=m_{\text{Sph}}$. Normalised by $\pi R_{Sph}^2={\sigma }_T$.

$$\frac{d\sigma }{dt}=\frac{1}{s-4m_{\text{Sph}}^2}$$

Arguments

• sSmol::Float64 : $s-sBig$

• sBig::Float64 : $(m_1+m_2{)}^2=4m_{\text{Sph}}^2$

• tSmol::Float64 : $t-tBig$

• tBig::Float64 : $(m_3-m_1{)}^2=0$

• uSmol::Float64 : $u-uBig$

• uBig::Float64 : $(m_2-m_3{)}^2=0$

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DiplodocusCollisions.sigma_ElePosPhoPho Function

sigma_ElePosPhoPho(sSmol,sBig)

returns the total cross section for electron positron annihilation to two photons. Berestetskii 1982 (88.6). Masses and momenta are normalised by the rest mass of the electron $m_{\text{Ele}}$ and the cross section is normalised by ${\sigma }_T$.

$${\sigma }_{e^{+}{e}^{-}\to \gamma \gamma }=\frac{3}{4s^2(s-4)}((s^2+4s-8)\log \left(\frac{\sqrt{s}+\sqrt{s-4}}{\sqrt{s}-\sqrt{s-4}}\right)-(s+4)\sqrt{s(s-4)})$$

Arguments

• sSmol::Float64 : $s-sBig$

• sBig::Float64 : $(m_1+m_2{)}^2=4\therefore s=sSmol+4$

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DiplodocusCollisions.dsigmadt_ElePosPhoPho Function

dsigmadt_ElePosPhoPho(sSmol,sBig,tSmol,tBig,uSmol,uBig)

returns the differential cross section for electron positron annihilation to two photons. Berestetskii 1982 (88.4). Masses and momenta are normalised by the rest mass of the electron $m_{\text{Ele}}$ and the cross section is normalised by ${\sigma }_T$.

$$\frac{d{\sigma }_{e^{+}{e}^{-}\to \gamma \gamma }}{dt}=-\frac{3}{s(s-4)}({(\frac{1}{t-1}+\frac{1}{u-1})}^2+(\frac{1}{t-1}+\frac{1}{u-1})-\frac{1}{4}(\frac{t-1}{u-1}+\frac{u-1}{t-1}))$$

Arguments

• sSmol::Float64 : $s-sBig$

• sBig::Float64 : $(m_1+m_2{)}^2=4\therefore s=sSmol+4$

• tSmol::Float64 : $t-tBig$

• tBig::Float64 : $(m_3-m_1{)}^2=1\therefore t=tSmol+1$

• uSmol::Float64 : $u-uBig$

• uBig::Float64 : $(m2-m3{)}^2=1\therefore u=uSmol+1$

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DiplodocusCollisions.sigma_PhoPhoElePos Function

sigma_PhoPhoElePos(sSmol,sBig)

returns the total cross section for photon-photon annihilation to electron-positron pair. Masses and momenta are normalised by the rest mass of the electron $m_{\text{Ele}}$ and the cross section is normalised by ${\sigma }_T$.

$${\sigma }_{\gamma \gamma \to e^{+}{e}^{-}}=\frac{3}{2s^3}((s^2+4s-8)\log \left(\frac{\sqrt{(}s)+\sqrt{(}s-4)}{\sqrt{(}s)-\sqrt{(}s-4)}\right)-(s+4)\sqrt{s(s-4)})$$

Arguments

• sSmol::Float64 : $s-sBig$

• sBig::Float64 : $(m_1+m_2{)}^2=0\therefore s=sSmol$

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DiplodocusCollisions.dsigmadt_PhoPhoElePos Function

dsigmadt_PhoPhoElePos(sSmol,sBig,tSmol,tBig,uSmol,uBig)

returns the differential cross section for photon-photon annihilation to electron-positron pair. (Inverse proceess of electron positron annihilation to two photons). Masses and momenta are normalised by the rest mass of the electron $m_{\text{Ele}}$ and the cross section is normalised by ${\sigma }_T$.

$$\frac{d{\sigma }_{\gamma \gamma \to e^{+}{e}^{-}}}{dt}=-\frac{3}{s^2}({(\frac{1}{t-1}+\frac{1}{u-1})}^2+(\frac{1}{t-1}+\frac{1}{u-1})-\frac{1}{4}(\frac{t-1}{u-1}+\frac{u-1}{t-1}))$$

Arguments

• sSmol::Float64 : $s-sBig$

• sBig::Float64 : $(m_1+m_2{)}^2=0\therefore s=sSmol$

• tSmol::Float64 : $t-tBig$

• tBig::Float64 : $(m_3-m_1{)}^2=1\therefore t=tSmol+1$

• uSmol::Float64 : $u-uBig$

• uBig::Float64 : $(m_2-m_3{)}^2=1\therefore u=uSmol+1$

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DiplodocusCollisions.sigma_ElePhoElePho Function

sigma_ElePhoElePho(sSmol,sBig)

returns the total cross section for electron-photon (Compton) scattering. Berestetskii 1982 (86.16). Masses and momenta are normalised by the rest mass of the electron $m_{\text{Ele}}$ and the cross section is normalised by ${\sigma }_T$.

$${\sigma }_{e\gamma \to e\gamma }(s)=\frac{3}{4(s-1)}[(1-\frac{4}{(s-1)}-\frac{8m_e^4}{{(s-1)}^2})\log \left(s\right)+\frac{1}{2}+\frac{8}{s-1}-\frac{1}{2s^2}]$$

Arguments

• sSmol::Float64 : $s-sBig$

• sBig::Float64 : $(m_1+m_2{)}^2=1\therefore s=sSmol+1$

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DiplodocusCollisions.dsigmadt_ElePhoElePho Function

dsigmadt_ElePhoElePho(sSmol,sBig,tSmol,tBig,uSmol,uBig)

returns the differential cross section for electron-photon scattering (Compton) scattering. Berestetskii 1982 (86.6). Masses and momenta are normalised by the rest mass of the electron $m_{\text{Ele}}$ and the cross section is normalised by ${\sigma }_T$.

$$\frac{d{\sigma }_{e\gamma \to e\gamma }}{dt}(s,t)=\frac{3}{(s-1{)}^2}[{(\frac{1}{s-1}+\frac{1}{u-1})}^2+(\frac{1}{s-1}+\frac{1}{u-1})-\frac{1}{4}(\frac{s-1}{u-1}+\frac{u-1}{s-1})]$$

Arguments

• sSmol::Float64 : $s-sBig$

• sBig::Float64 : $(m_1+m_2{)}^2=1\therefore s=sSmol+1$

• tSmol::Float64 : $t-tBig$

• tBig::Float64 : $(m_3-m_1{)}^2=0\therefore t=tSmol$

• uSmol::Float64 : $u-uBig$

• uBig::Float64 : $(m_2-m_3{)}^2=1\therefore u=uSmol+1$

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DiplodocusCollisions.SyncKernel Function

SyncKernel(p3v,p1v,m1,z1,B)

Returns the emission rate for a single photon $p3v$ state emitted by a charged particle in state $p1v$ with charge $z1$ relative to the fundamental charge and mass $m1$ relative to the mass of the electron, in a uniform magnetic field $B$.

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