| Parameter Description | Argument Order | Argument Name | Units | Default Value |
|---|---|---|---|---|
| Length | 1 | L | cm | |
| Current Density | 2 | j | kA/cm**2 |
The letters "F" and "f" denote the lithium lens. The first parameter is the length of the lens in cm. The second parameter is the electric current density in kA/cm/cm. It can be also used for an estimate of the lattice function change due to linear contribution of the beam-beam effects or the electron beam space charge in the electron cooler. For these cases, one needs to make a correction for the electric field effect.
Examples: F1 L[cm]=10 j[kA/cm**2]=88.72665The example below represents a description of the Fermilab lithium lens. This description includes all important physical phenomena in the lens: (1) non-linearity of lithium lens focusing due to the skin-effect and (2) particle scattering in the berylium window and in the lithium (see also M- axial symmetric multipole and T-transverse random particle scattering).
#-------------------------------------------------- # Lens description in the math header $LensCurrent= 517.719904; => 517.719904 $PulseLength=360e-6; => 0.00036 # usec, duration of half sinusooidal pulse $LensDelay=45; => 45 # deg, delay relative to the wave crest $LensL=15; => 15 # cm, effective length of the lens $LensR=1; => 1 # cm, radius of the lens $DTi=0.1; => 0.1 # cm, thickness of titanium contaner $Nlp=9; => 9 # number of pieces to split lens for non-linear tracking $RoLi=11.45e-6; => 1.145e-05 # Ohm cm, resistivity of lithium $RoTi=42e-6; => 4.2e-05 # Ohm cm, resistivity of titanium $LensLp=$LensL/$Nlp/2; => 0.833333333 $LensI0=$LensCurrent*$RoTi/($RoTi+2*$RoLi*$DTi/$LensR); => 490.951365 $LensJ0=$LensI0/($PI*$LensR*$LensR); => 156.274673 $Feff=0.5/$PulseLength; => 1388.88889 $SigmaR=$c*$c * 1e-9 / $RoLi; =>7.84939021e+16 $DeltaSkin=$c/(2*$PI*sqrt($SigmaR*$Feff)); => 0.456971508 $KsiMax=sqrt(2)*$LensR/$DeltaSkin; => 3.09475216 $x=$KsiMax*$KsiMax; => 9.57749093 $FiKsiMax= $KsiMax / (2*sqrt(2)) * (1 + $x*(-1/8+$x*(-1/192+$x*(1/9216+$x/737280)))); => -0.6217 $FrKsiMax= $KsiMax / (2*sqrt(2)) * (-1 + $x*(-1/8+$x*(1/192+$x*(1/9216-$x/737280)))); => -1.7895 $Fr2=sqrt($FiKsiMax*$FiKsiMax + $FrKsiMax*$FrKsiMax); => 1.8944391 $Psi=atan($FiKsiMax/$FrKsiMax) + $PI/4; => 1.11976553 $si=sin($PI*$LensDelay/180-$Psi); =>-0.328171637 $co=cos($PI*$LensDelay/180-$Psi); => 0.944618111 $KsiNorm=sqrt(2)/$DeltaSkin; => 3.09475216 $x=$KsiNorm*$KsiNorm; => 9.57749093 $M1= $KsiNorm*$co/2/$Fr2; => 0.771563189 $M3=-$KsiNorm*$si*$x*3/8/$Fr2; => 1.92543689 $M5=-$KsiNorm*$co*$x*$x*5/16/$Fr2; => -44.233878 $M7= $KsiNorm*$si*$x*$x*$x*35/128/$Fr2; => -128.783313 $M9= $KsiNorm*$co*$x*$x*$x*$x*63/256/$Fr2; => 3195.28114 $LensJ=$LensJ0*$M1; => 120.575785 $GL1=.200*$LensI0/($LensR*$LensR)*$M1*$LensL/$Nlp; => 126.266667 $GL3=.200*$LensI0/($LensR*$LensR)*$M3*$LensL/$Nlp; => 315.098623 $GL5=.200*$LensI0/($LensR*$LensR)*$M5*$LensL/$Nlp; => -7238.89426 $GL7=.200*$LensI0/($LensR*$LensR)*$M7*$LensL/$Nlp; => -21075.4478 $GL9=.200*$LensI0/($LensR*$LensR)*$M9*$LensL/$Nlp; => 522909.213 $ScatLi =1000*sqrt(($LensL+0.5*$LensR)/$Nlp/155.)*13.6/$E0; => 0.179195734 $ScatBe=1000*sqrt(0.7/35.5)*13.6/$E0; => 0.238717229 $ScatTot=1000*sqrt(($LensL+0.5*$LensR)/155.+1.4/35.5)*13.6/$E0; => 0.634800623 $LiLens="fLens tLi mLens3 mLens5 mLens7 mLens9 fLens"; begin lattice. Number of periods=1 hLens tBe $LiLens $LiLens $LiLens $LiLens $LiLens $LiLens $LiLens $LiLens $LiLens tBe hLens IAP2START oDR700A end lattice # begin list mLens3 Order:m=-3 Bm*L[kG/cm**(m-1)]=$GL3 mLens5 Order:m=-5 Bm*L[kG/cm**(m-1)]=$GL5 mLens7 Order:m=-7 Bm*L[kG/cm**(m-1)]=$GL7 mLens9 Order:m=-9 Bm*L[kG/cm**(m-1)]=$GL9 tLi Rms angle[mrad]=$ScatLi 1/L*dL/dx[1/cm]=0 Tilt[deg]=0 tBe Rms angle[mrad]=$ScatBe 1/L*dL/dx[1/cm]=0 Tilt[deg]=0 fLens L[cm]=$LensLp j[kA/cm**2]=$LensJ # end list