Implemented Collisions

Binary Collisions

These binary interactions have currently been implemented:

Binary Interaction	Abbreviation	Internal Functions (cross sections)
Elastic collision of spheres	SphSphSphSph	dsigmadt_SphSphSphSph sigma_SphSphSphSph
Photon pair production from electron-positron annihilation	ElePosPhoPho	dsigmadt_ElePosPhoPho sigma_ElePosPhoPho
Electron-Positron pair production from photon pair annihilation	PhoPhoElePos	dsigmadt_PhoPhoElePos sigma_PhoPhoElePos
Electron(or Positron)-Photon scattering (Compton scattering)	ElePhoElePho	dsigmadt_ElePhoElePho sigma_ElePhoElePho

Emission Interactions

These emissive interactions have currently been implemented:

Emissive Interaction	Abbreviation	Internal Functions (emissivity kernel)
Synchrotron(cyclostron) emission of photons by a charged particle (Name1)	SyncName1Name1Pho	SyncKernel

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 : $(m1+m2{)}^2=4\therefore s=sSmol+4$

• tSmol::Float64 : $t-tBig$

• tBig::Float64 : $(m3-m1{)}^2=1\therefore t=tSmol+1$

• 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 : $max((m_1+m_2{)}^2,(m_3+m_4{)}^2)=4\therefore s=sSmol+4$

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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 process 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 : $max((m_1+m_2{)}^2,(m_3+m_4{)}^2)=4\therefore s=sSmol+4$

• tSmol::Float64 : $t-tBig$

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

• uSmol::Float64 : $u-uBig$

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

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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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