Rasmus Ischebeck’s solutions in Mathematica (by John Jowett).

Setup

Copy of Accelerator`ConstantsUnits package

To make this notebook self contained (and not rely on the package being installed for the user), this will be evaluated on initialisation.  No need to look at it.

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Notation`AutoLoadNotationPalette=False
BeginPackage["Accelerator`ConstantsUnits`",{"Notation`","Units`","PhysicalConstants`"}]
If[$VersionNumber>8,Print["This package should still work in Mathematica Version 9.\nIt implements a system for managing physical constants and units that is different and independent of the new scheme implemented in Version 9.\nIt may be adapted for compatibility at a later date."]]
ConstantsUnits::usage ="ConstantsUnits.m is a package that provides notations and values for constants occurring in accelerator physics.";
PeV::usage=TeV::usage=GeV::usage=MeV::usage= keV::usage="keV, MeV, GeV, TeV, PeV are units of energy and evaluate to (Kilo ElectronVolt), ... etc.";
lumiUnit::usage="lumiUnit as the conventional unit of colliding beam luminosity, \!\(\*SuperscriptBox[\(cm\), \(-2\)]\)\!\(\*SuperscriptBox[\(s\), \(-1\)]\).";
c::usage="c is a symbol for the speed of light in vacuum; it evaluates numerically with N[ ].";
ℏ::usage="ℏ is a symbol for the reduced Planck constant; it evaluates numerically with N[ ].";
e::usage="e is a symbol for the electron charge; it evaluates numerically with N[ ].";
Notation[(Subscript[r, e]) ⟺ rElectron];
rElectron::usage="\!\(\*SubscriptBox[\(r\), \(e\)]\) or rElectron is a symbol for the classical radius of the electron; it evaluates numerically with N[ ].";
Notation[ParsedBoxWrapper[SubscriptBox["α", "s"]]⟺ParsedBoxWrapper["alphas"]]
alphas::usage="\!\(\*SubscriptBox[\(α\), \(s\)]\) or alphas is a symbol for the fine structure constant; it evaluates numerically with N[ ].";
Notation[(Subscript[λ, e]) ⟺ lambdaElectron];
lambdaElectron::usage="\!\(\*SubscriptBox[\(λ\), \(e\)]\) or lambdaElectron is a symbol for the Compton wavelength of the electron; it evaluates numerically with N[ ].";
Notation[(Subscript[G, e]) ⟺ GElectron];
GElectron::usage="\!\(\*SubscriptBox[\(G\), \(e\)]\) or GElectron is a symbol for the anomalous magnetic moment of the electron; it evaluates numerically with N[ ].";
Notation[(Subscript[μ, 0]) ⟺ mu0];
mu0::usage="\!\(\*SubscriptBox[\(μ\), \(0\)]\) or mu0 is a symbol for the permeability of the vacuum; it evaluates numerically with N[ ].";
Notation[(Subscript[ϵ, 0]) ⟺ epsilon0];
epsilon0::usage="epsilon0 is a symbol for the permittivity of the vacuum; it evaluates numerically with N[ ].";
Notation[(Subscript[m, e]) ⟺ mElectron];
mElectron::usage="\!\(\*SubscriptBox[\(m\), \(e\)]\) or mElectron is a symbol for the mass of the electron; it evaluates numerically with N[ ].";
Notation[(Subscript[m, p]) ⟺ mProton];
mProton::usage="\!\(\*SubscriptBox[\(m\), \(p\)]\) or mProton is a symbol for the mass of the proton; it evaluates numerically with N[ ].";
Notation[(Subscript[m, n]) ⟺ mNeutron];
mNeutron::usage="\!\(\*SubscriptBox[\(m\), \(n\)]\) or mNeutron is a symbol for the mass of the neutron; it evaluates numerically with N[ ].";
Notation[(Subscript[m, d]) ⟺ mDeuteron];
mDeuteron::usage="\!\(\*SubscriptBox[\(m\), \(d\)]\) or mDeuteron is a symbol for the mass of the deuteron; it evaluates numerically with N[ ].";
Notation[(Subscript[m, υ]) ⟺ mMuon];
mMuon::usage="\!\(\*SubscriptBox[\(m\), \(υ\)]\) or mMuon is a symbol for the mass of the muon; it evaluates numerically with N[ ].";
Notation[(Subscript[m, Z]) ⟺ mZ];
mZ::usage="\!\(\*SubscriptBox[\(m\), \(Z\)]\) or mZ is a symbol for the mass of the Z-particle; it evaluates numerically with N[ ].";
Notation[(Subscript[m, W]) ⟺ mW];
mW::usage="\!\(\*SubscriptBox[\(m\), \(\(W\)\(\\\ \)\)]\)or mW is a symbol for the mass of the W-particle; it evaluates numerically with N[ ].";
Notation[(Subscript[a, ∞]) ⟺ aBohr];
aBohr::usage="\!\(\*SubscriptBox[\(a\), \(∞\)]\) or aBohr is a symbol for the Bohr radius; it evaluates numerically with N[ ].";
Notation[(Subscript[r, p]) ⟺ rProton];
rProton::usage="\!\(\*SubscriptBox[\(r\), \(p\)]\) or rProton is a symbol for the classical radius of the proton; it evaluates numerically with N[ ].";
Notation[(Subscript[Z, 0]) ⟺ Z0];
Z0::usage="\!\(\*SubscriptBox[\(Z\), \(0\)]\) or Z0 is a symbol for the impedance of free space; it evaluates numerically with N[ ].";
Notation[(Subscript[σ, T]) ⟺ sigmaThomson];
sigmaThomson::usage="\!\(\*SubscriptBox[\(σ\), \(T\)]\) or sigmaThomson is a symbol for the Thomson cross section; it evaluates
numerically with N[ ].";
Notation[(Subscript[σ, SB]) ⟺ sigmaStefanBoltzmann];
sigmaStefanBoltzmann::usage="\!\(\*SubscriptBox[\(σ\), \(SB\)]\) or sigmaStefanBoltzmann is a symbol for the Stefan-Boltzmann constant; it evaluates numerically with N[ ].";
Notation[(Subscript[μ, B]) ⟺ muBohr];
muBohr::usage="\!\(\*SubscriptBox[\(μ\), \(B\)]\) or muBohr is the Bohr magneton; it evaluates numerically with N[ ].";
Notation[(Subscript[μ, N]) ⟺ muNuclear]
muNuclear::usage="\!\(\*SubscriptBox[\(μ\), \(N\)]\) or muNuclear is the nuclear magneton; it evaluates numerically with N[ ].";
ToFundamentalSI::usage="toFundamentalSI[f] converts the quantity f to fundamental SI units.  Generally this helps to simplify the expressions of the units.";
ClassicalProtonRadius::usage="ClassicalProtonRadius is the value of the classical radius of the proton.";
DimensionCheck::usage="DimensionCheck[f] returns an expression showing the dimensional consistency of f, based on the internal database of symbols and units maintained by the ConstantsUnits package.\nThe alternative form DimensionCheck[f,symbolsUnits] uses a list of {symbol,unit} pairs given as second argument instead of the internal database.";
NumericalValueUnits::usage="NumericalValueUnits[f] interprets f as the product of a numerical value fn and symbolic units fu and returns {fn,fu}.";
PhysicalUnits::usage="PhysicalUnits[f] interprets f as the product of a numerical value fn and symbolic units fu and returns fu.";
NumericalValue::usage="NumericalValue[f] interprets f as the product of a numerical value fn and symbolic units fu and returns fn.";
IntroduceSymbol::usage="IntroduceSymbol[symbol,usagetext,units] introduces a symbol, registers its units for purposes of dimensional analysis and defines a usage message for it.";
SymbolsUnits::usage="SymbolsUnits[] returns a list of all symbols and units registered in the internal database.";
PhysicalUnitsPlot::usage="PhysicalUnitsPlot[f,{x,xmin,xmax}] works like Plot but removes the physical units of f and x before plotting.  Further, it creates labels for the axes from the symbolic forms of f and x and indicates their units.  It has all options of Plot except AxesLabel.";
nuclearMassGeV::usage="nuclearMassGeV[{Z,A}] returns the mass of a nucleus in GeV."
ConstantsUnits::badarg = "You called `1` with argument `2`!"
Begin["`Private`"]
minGodelPrimeIndex = 10;
ClassicalProtonRadius=Convert[ElectronCharge^2/((4 π VacuumPermittivity) ProtonMass SpeedOfLight^2),Meter]
ToFundamentalSI[f_]:=PowerExpand[SI[f]//.Units`Private`$ToFundamental,{Kilogram,Meter,Second,Candela}]
DimensionCheck[f_,symbolsWithUnits_List]:=Expand[f//.MapIndexed[(#1[[1]]->Prime[minGodelPrimeIndex+First[#2]] #1[[2]]&),Union[symbolsWithUnits]]];
DimensionCheck[f_]:=DimensionCheck[f,Accelerator`ConstantsUnits`Private`$SymbolsUnits]
PeV=Peta ElectronVolt;TeV= Tera ElectronVolt;GeV=Giga ElectronVolt;
MeV=Mega ElectronVolt;keV=Kilo ElectronVolt;eV=ElectronVolt;
lumiUnit = 1/((Centi*Meter)^2*Second);
N[c]=SpeedOfLight;
N[e]=ElectronCharge;
N[ℏ]=PlanckConstantReduced;
N[mElectron]=ElectronMass;
N[mProton]=ProtonMass;N[mNeutron]=NeutronMass;N[mDeuteron]=DeuteronMass;
N[mMuon]=MuonMass;
N[alphas]=FineStructureConstant;
N[rElectron]=ClassicalElectronRadius;
N[rProton]=ClassicalProtonRadius;
N[lambdaElectron]=ElectronComptonWavelength;
N[lambdabarElectron]=N[ElectronComptonWavelength/(2 π)];N[aBohr]=BohrRadius;N[sigmaThomson]=ThomsonCrossSection;N[sigmaStefanBoltzmann]=StefanConstant;
N[GElectron]=If[$VersionNumber>=4.1,(-ElectronGFactor-2)/2,ElectronGFactor-1]
N[mu0]=Convert[VacuumPermeability,Henry/Meter]
N[epsilon0]=Convert[VacuumPermittivity,Farad/Meter]
N[mZ]=Convert[N[91187.6`5.999999999999999  (GeV/1000)/(SpeedOfLight^2)],Kilogram]
N[mW]=Convert[N[80403.`5.000000000000002  (GeV/1000)/(SpeedOfLight^2)],Kilogram]
N[muBohr]=Convert[N[(ElectronCharge PlanckConstantReduced)/(2 ElectronMass)],Joule/Tesla]
N[muNuclear]=Convert[N[(ElectronCharge PlanckConstantReduced)/(2 ProtonMass)],Joule/Tesla]
N[Z0]=Convert[N[VacuumPermeability SpeedOfLight],Ohm]
nuclearMassGeV[{z_,a_}]:=Convert[(IsotopeData[{z,a},"AtomicMass"]/Quantity[1,"AtomicMassUnit"] AMU-z ElectronMass )SpeedOfLight^2//N,GeV]
NumericalValueUnits[Times[a_/;NumericQ[a],b__]]:={a,Times@b};
NumericalValueUnits[a_/;NumericQ[a]]:={a,1};
NumericalValueUnits[a_]:={1,a}
NumericalValue[f_]:=First[NumericalValueUnits[f]]
PhysicalUnits[f_]:=Last[NumericalValueUnits[f]]
SetAttributes[PhysicalUnitsPlot,HoldAll];
Options[PhysicalUnitsPlot]=Options[Plot];
PhysicalUnitsPlot[f_,{var_,varmin_,varmax_},opts___?OptionQ]/;PhysicalUnits[ToFundamentalSI[varmin]]==PhysicalUnits[ToFundamentalSI[varmax]]:=
	Module[{vunits,funits,nvar},
		vunits=PhysicalUnits[N[varmin]];funits=PhysicalUnits[N[f/.var->varmin]];
		fnew=f/.var->nvar vunits;
		If[FreeQ[funits,Plus],Plot[Evaluate[fnew/funits],{nvar,ToFundamentalSI[varmin/vunits],ToFundamentalSI[varmax/vunits]},AxesLabel->{StandardForm[HoldForm[var] [vunits]],StandardForm[HoldForm[f][ funits]]},opts],Print["Units of ordinate appear to be inconsistent."]]]
$SymbolsUnits=({#1,PhysicalUnits[N[#1]]}&)/@ToExpression[Names["Accelerator`ConstantsUnits`*"]]
$SymbolsUnits=Select[$SymbolsUnits,First[#1]=!=Last[#1]&& !(NumericQ[First[#1]/Last[#1]])&]
IntroduceSymbol[sym_Symbol,usagetext_String,unit_]:=($SymbolsUnits=Append[Select[$SymbolsUnits,(First[#1]=!=sym)&],
{sym,unit}];
sym::"usage"=ToString[sym]<>" "<>usagetext<>" (units: "<>ToString[unit]<>")")
SymbolsUnits[]:=$SymbolsUnits
(* Sin/: Sin[x_]^2 := 1 - Cos[x]^2 *)
(* Protect[ Evaluate[protected] ] *)
End[ ]
Protect[Evaluate[$Context <> "*"]]
EndPackage[ ]
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Needs["Accelerator`ElectronRings`"]

1

2

sunRadiation={Prad->3.828 (10^(26 ))Watt,sunSurfaceArea->6.09 (10^(12 ))(((Kilo Meter))^2),sunEarthDistance->1.496 Kilo Meter,sunAngularSize -> 32. (Degree/60),sunAge->4.6 (10^(9 ))Year,sunPhotonEnergy->2 ElectronVolt,sunVisibleBandwidth->0.5 (1000 pertenthpercent) };

Convert[sunPhotonEnergy/.sunRadiation,Joule]

Photon flux from sun, estimate

sunPhotonFlux=ToFundamentalSI[(Prad/sunPhotonEnergy)/.sunRadiation]

sunBrilliance=(sunPhotonFlux/(sunSurfaceArea sunVisibleBandwidth 4 π (Radian^2) ))/.sunRadiation

Convert[sunBrilliance, ((Hertz )/((((Milli Meter))^2)(((Milli Radian))^2)))]

3

globatron={Eb->5. PeV,ρ->8000. Kilo Meter}

Bgtron=Convert[N[(Eb/(c (e ρ)))/.globatron],Tesla]

γgtron=ToFundamentalSI[N[((5 PeV)/((Subscript[m, p]) (c^2)))]]

Synchrotron radiation loss per turn of a ring, constant bending radius ρ

U0=((4 (Eb^4) π (Subscript[r, e]))/(3 (c^6) (Subscript[m, e]^3) ρ))

Synchrotron radiation energy loss for protons

U0/.{(Subscript[m, e])->(Subscript[m, p]),(Subscript[r, e])->(Subscript[r, p])}

Convert[U0/.{(Subscript[m, e])->(Subscript[m, p]),(Subscript[r, e])->(Subscript[r, p])}/.globatron//N,ElectronVolt]

As a fraction of beam energy

Convert[(U0/Eb)/.{(Subscript[m, e])->(Subscript[m, p]),(Subscript[r, e])->(Subscript[r, p])}/.globatron//N,1]

Convert[N[U0/. globatron],ElectronVolt]

Convert[(U0/Eb) /.globatron//N,1]

This machine would not work.

4

pc=Convert[N[(Sqrt[(Eb^2)-(Subscript[m, e]^2) (c^4)])/. Eb->1 MeV+(Subscript[m, e]) (c^2)],MeV]

ToFundamentalSI[N[   1 MeV+(Subscript[m, e]) (c^2)]]

5

Solution a)-c)

undulator={undulatorL->5.1 Meter,λu->15 Milli Meter,Bt->1.2 Tesla,g->10. Milli Meter,ku->2π/λu}

ring={Eb->3 GeV,Ib->500 Milli Ampere,ϵx->1. Nano Meter,ϵy->1. Pico Meter,βx->3.5 Meter,βy->2. Meter,αx->0.,αy->0.,γ->(Eb/((Subscript[m, e]) (c^2)))}

B0=(Bt/(Cosh[π g/λu]))/.undulator

Ku=ToFundamentalSI[N[((e B0)/((Subscript[m, e]) c ku))//. undulator]]

λ=ToFundamentalSI[N[((λu (1+((Ku^2)/2)))/(2(γ^2)))//. ring//. undulator]]

Convert[λ,Angstrom]

Solution d)-e)

undulatorN=(undulatorL/(λu))/.undulator//ToFundamentalSI

bandwidth=(1/(n undulatorN))/.n->1

Q[n_,K_]:=((1+(K^2)/2)/n)F[n,K];
F[n_,K_]=(((n^(2 ))(K^2))/(((1+(K^2)/2))^2))(((BesselJ[((n+1)/2),Y]-BesselJ[((n-1)/2),Y]))^2)/.Y->((n (K^2))/(4(1+(K^2)/2)))

flux=ToFundamentalSI[(((1.43 (10^14) Hertz) undulatorN Ib Q[1,Ku])/Ampere)/. ring]

F[1,Ku]

Ku

Plot[Table[Q[n,x],{n,1,5}],{x,0,5}]

Solution f)-

σx=Convert[(Sqrt[βx ϵx])/.ring,Micro Meter]

γx=((1+((αx)^2))/(βx))/.ring

σxp=Convert[(Sqrt[γx ϵx])/.ring,Micro Radian]

σy=Convert[(Sqrt[βy ϵy])/.ring,Micro Meter]

γy=((1+((αy)^2))/(βy))/.ring

σyp=Convert[(Sqrt[γy ϵy])/.ring,Micro Radian]

Solution g)-

σr=Convert[(1/(4 π))(Sqrt[λ undulatorL])/.undulator,Micro Meter]

σrp=Convert[(Sqrt[(λ/undulatorL)])/. undulator,Micro Radian]

σxeff=Convert[(Sqrt[((σx)^2)+((σr)^2)]),Micro Meter]

σxpeff=Convert[(Sqrt[((σxp)^2)+((σrp)^2)]),Micro  Radian]

σyeff=Convert[(Sqrt[((σy)^2)+((σr)^2)]),Micro Meter]

σypeff=Convert[(Sqrt[((σyp)^2)+((σrp)^2)]),Micro  Radian]

Brilliance=Convert[(flux/(4 (π^2) σx σxp σy σyp  ))//N,(Second^(-1))(((Meter Radian))^(-2))]

Convert[Brilliance,(Second^(-1))(((Milli Meter Milli Radian))^(-2))]