F-Lithium lens

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]=88.72665
The 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