FERMILB-TM- Compensation of Dogleg Effect in Fermilab Booster Xiaobiao Huang Λ and S. Ohnuma y July, Introduction The edge focusing of dogleg magnets in Fermilab Booster has been causing severe distortion to the horizontal linear optics. The doglegs are vertical rectangular bends, therefore the vertical edge focusing is canceled by body focusing and the overall effect is focusing in the horizontal plane. The maximum horizontal beta function is changed from.7m to 6.9m and maximum dispersion from.9m to 6.m. Beam size increases accordingly. This is believed to be one of the major reasons of beam loss. In this technote we demonstrate that this effect can be effectively corrected with Booster's quadrupole correctors in short straight sections (QS). There are QS correctors which can alter horizontal linear optics with negligible perturbation to the vertical plane. The currents of correctors are determined by harmonic compensation, i.e., cancellation of dogleg's harmonics that are responsible for the distortion with that of QS correctors. By considering a few leading harmonics, the ideal lattice can be partly restored. For the current dogleg layout, maximum fi x is reduced to.6m and maximum D x is reduced to.9m. This scheme can be useful after the dogleg in section # is repositioned. In this case it can bring fi x from.9m down to 7.7m, D x from.7m to.m. Harmonic Compensation The edge focusing effect is a quadrupole effect. The integrated quadrupole strength is [] tan ffi ffi tan ffi k l = = (.) ρ L where ffi is the entrance or exit angle with respect to the normal direction of the edge, L is the length of the magnet and ρ = L=ffi the bending radius. Λ Dept. of Physics, Indiana University, Bloomington, IN, 7 y Dept. of Physics, University ofhawaii at Manoa, Honolulu, HI 968
The p'th harmonic half-integer stopband integral is defined as [] J p = ß I fik(s)e jpffi ds = ß X i fi i [k l] i e jpffi i (.) where [k l] i is the integrated quadrupole strength at location i. The perturbation to beta function is [] fi(s) fi(s) = ν X p= J p e jpffi ν (p=) (.) For off-momentum particles, there is also a dipole effect because of dispersion. The equivalent dipole strength is B Bρ = [k l]d p p (.) where p p is momentum deviation. The minus sign indicates a weaker bending strength for positive momentum deviation. For a unit value of p p, the integer stopband integral is [] I pfi B f n = ßν Bρ e jnffi ds = Xp fii [k l] i D i e jnffi i (.) ßν The change to dispersion function is then [] D = x p co = fi(s) p=p i X n= ν f n ν n ejnffi (.6) There are magnets for each dogleg but each magnet has focusing effect at only one edge. Table () lists related parameters of doglegs. dogleg # ffi L(m) k l (m ) old.6.7.86 new..7..9876.7.9 Table : dogleg parameters. ' old' and '' are current layout. ' new' is dogleg after repositioning. ccording to equations (.) and (.6), the contribution of a harmonic to the changes of beta function or dispersion is weighted by a factor. Figure () shows magnitude of the weighted harmonics J p =(ν p =) and f n =(ν n )
. J n, n=:.... 7 x f n, n=: 6 Figure : The weighted harmonics for current dogleg layout. Top, jj pj=(ν p =). Bottom jf nj=(ν n ). β x : subtract = b, harmonics = r D: subtract = b, harmonics = r Figure : Changes to beta amplitude fi x and dispersion D x due to current dogleg layout. Blue curve is by subtracting MD output with and without doglegs. Red one is obtained using equations (.) and (.6). Top, fi x. Bottom, D x.
.8 J n, n=:.6....8.6.. x f n, n=: Figure : The weighted harmonics for repositioned dogleg layout. Top, jj pj=(ν p =). Bottom jf nj=(ν n ). 8 β x : subtract = b, harmonics = r 6 6. D: subtract = b, harmonics = r... Figure : Changes to beta amplitude fi x and dispersion D x due to repositioned dogleg layout. Blue curve is by subtracting MD output with and without doglegs. Red one is obtained using equations (.) and (.6).
for the current dogleg layout. Figure () shows changes to fi x and dispersion D x according to equations (.), (.), (.) and (.6) with comparison to the same quantities obtained from subtraction of MD output. We can see the 7th of f n and th of J p dominate the two spectra, respectively. Figure () and () are for the new dogleg layout. For the i'th QS quadrupole, given a current of +, we can also get its stopband harmonics spectrum Jp i and fn. i To cancel the p'th half-integer harmonic J p of doglegs, we want to pick up those quadrupoles whose J p term is large and points along the direction of J p of doglegs in the complex plane. The same is true for f n. Define efficiency i p = jj p i j J i p jj p j cos( ffii a)=re (.7) J p B i n = jf i n j jf n j cos( ffii b)=re f i n f n (.8) where ffi i a ang ffi i b are angles between J i p and J p, f i n and f n respectively. Now our goal is to minimize the total harmonics of doglegs and quadrupoles for both J p and f n. In choosing a set of quadrupole current toachieve this, we study the efficiency table to determine which quadrupoles are to be used and calculate the needed current values. Then we update the total weighted harmonics spectrum and repeat. Quadrupoles having large efficiency for the leading harmonic of one spectrum (J p or f n ) and efficiency of the same sign for the leading harmonic of the other spectrum are of our interest. This process involves a lot of "eyeball inspection" and thus is not very efficient. To automate the process, we found that a bit-by-bit approach is needed. The procedure is Step sort the magnitude of the weighted overall harmonics of J p and f n to descending order and locate the most important harmonics, the 7th of f n and the th of J p, for example. Step sort the i to descending order Step for each of the first 6 sorted i, if it has the same sign as Bi 7,change the current ofthei'th QS by adding./ i. (i.e. cancel % of J with this quadrupole) Step sort the B7 i to descending order Step for each of the first 6 sorted B7 i, if it has the same sign as i,change the current of the i'th QS by adding./b7 i. (i.e. cancel % of f 7 with this quadrupole) Step 6 go to step It is important for us to make only a small change to the leading harmonic and recalculate the total harmonics in each iteration. Otherwise the solution
would be incorrect because of the interference among quadrupoles. When the leading harmonics get closer to the next biggest one, we may consider more than 6 QS's and require the efficiencies have the same sign with that of the next biggest one, too. The process stops when no quadrupoles can be found to reduce the leading harmonic without increase another hamonic to same level. It can quickly bring the weighted harmonics down to a lower level. lso maximum beta function and dispersion become smaller. Figure () and (6) shows the total weighted harmonics after compesation for the current layout and the new layout..8 J n, n=:.7.6..... 7 x f n, n=: 6 Figure : The weighted harmonics after compsensation, jj pj=(ν p =) and jf nj=(ν n ) for current layout The solution obtained by harmonic compensation might notbethebest one possible. local maximum search in the -dimension space is then necessary. The search starts from the solution obtained by harmonic compensation, then change each quadrupole until it won't bring the objective function down. The obejective is the maximum horizontal radius of the beam defined as r fix ffl x R x = ß + D p x (.9) p where ffl x is the 9% horizontal emittance. Typical values ffl x =ßm mrad and p p = ±:% are used. This can often improve the solution slightly. Result For both the current lattice and the one with respositioned dogleg, we can improve the linear optics properties with the scheme described in the 6
last section. We show the results by making comparsions between cases. case case case case Ideal lattice, no doglegs Current lattice, with two doglegs, no compensation New lattice, dogleg repositioned, no compensation Current lattice, with two doglegs, with compensation case New lattice, dogleg repositioned, with compensation For each case, we compare the maxima of fi x, fi y, D x and R x as defined in equation (.9). The quatities are listed in Table (). case max(fi x) max(fi y) max(d x) R x (mm).68.6.9. 6.9.6 6..86.88..8 8..6.6.9. 7.66.6.. Table : compare maxima of fi x, fi y, D x (in meters) and R x. The original and compensated horizontal beta function and dispersion are shown in Figure (7) and (8) for the two lattice layouts. For both cases, we can see that the regularity of the ideal lattice function is partly restored by compensation.. J n, n=:....... x f n, n=: Figure 6: The weighted harmonics after compsensation, jj pj=(ν p =) and jf nj=(ν n ) for new layout 7
The quadrupole current values for case andcase are well below the limit. The tune shifts are small. Q y = 6:78 and Q x = 6:778 for case. Q y = 6:779 and Q x = 6:7996 for case. Tunes can be nearly independently adjusted by modifying quadrupole corrector C current itunel and itunes. compare β x 7 compare D x 6 Figure 7: Comparison of horizontal beta function and dispersion for the current dogleg layout. Red curve is original and black one is compensated. Top, fi x. Bottom, D x. compare β x compare D x.... Figure 8: Comparison of horizontal beta function and dispersion for the new dogleg layout with repositioned dogleg. Red curve is original and black one is compensated. Top, fi x. Bottom, D x. 8
PPENDIX Local Beta Bump and Its pplication in the Booster Under some circumstances, we maywant to suppress (or increase) the beta function at one location keeping it intact elsewhere. Here we showhow this can be done with three quadrupoles. One is placed at the location and the other two beside it, one upstream and one downstream, respectively. For the normalized betatron phase-space coordinates Y, P, which are related to the usual coordinates y, y by the transfer matrix is Y P M(s js )= p = fi = ff= p fi p fi y y cos ψ sin ψ sin ψ cos ψ (-) where ψ is the phase advance from s to s. It is easy to see that the transfer matrix for a thin quadrupole is where k = ± B Bρ M = fi[k l] (-) with ` ' for horizontal plane and `+' for vertical plane. Recalling that the linear optics functions fi, ff and fl transfer according to [] @ fi ff fl with M = @ we get @ fin ff N fl N = M @ fi ff fl M M M M M M M M + M M M M M M M M = @ c cs s cs c s cs s cs c for a section with c =cosψ and s = sin ψ, and @ fin+ ff N+ fl N+ = @ fi[k l] (fi[k l]) fi[k l] @ fin ff N fl N @ fin ff N fl N (-) (-) (-) (-6) for a thin quadrupole. Superscript N indicates properties in the normalized coordinates (normalized to the unperturbed lattice), + indicates the downstream side and indicates the upstream side. 9
In normalized coordinates, the unperturbed lattice has (fi N ;ff N ;fl N )= (; ; ) everywhere. Let's label parameters at the three quadrupoles with subscript,, with for the one at upstream, for the one at downstream and in between. By applying the quadrupole correctors, we want to meet three conditions fi N = r fi N = ff N+ = In condition, r is a pre-set value to specify how much change we want to make atlocation(e.g..9, to suppress fi to 9% of its original value) and the other two conditions are required so that the perturbation is confined in the range between quadrupole and. ccording to equation (-6), a thin quadrupole will not change fi N at its location. Butitwillchange ff N to ff N fi[k l]fi N and fl N will be changed according to +ff = fifl. Thus at the downstream side of quadrupole, we have @ fin ff N+ fl N+ = @ x +x (-7) with x = fi [k l]. Combining with equation (-), the condition fi N = r becomes s x +c s x +( r) = (-8) in which c =cosψ,s =sinψ. lso we letc =cosψ and s = sin ψ. Equation (-8) has roots of x in which wepickuptheone with smaller absolute value. Quadrupole turns (fi N ;ffn ;fl N ) to (fi N;X ; +X ) with X = ff N+ = ff N x fi N. The latter is then transfered to quadrupole to meet condition fi N =, which yields s r X c s X +( s r + c r ) = gain we choose the solution with smaller absolute value, x = ffn X fi N (-9) (-) Now we can calculate ff N with equation (-) Condition fi N ff N = c s (r X )+(c s )X (-) = and condition ffn+ = ff N x fi N = together lead to x = ff N (-)
The integrated strengths of quadrupole,, can thus be determined with x,x and x known. s an example, we apply this local beta bump scheme to Fermilab Booster. To change linear optics of the horizontal plane effectively, we use QS quadrupoles. Since the solution to equation (-8) is of the form x = q ( c ± r s s ) (-) and to equation(-9) X = r s (c ± r q r s ) (-) r>s ; must be ensured for a meaningful solution to exist. For the Booster, the phase advance over one section is about degrees. So if we want to suppress fi x at section n, we have to use QS at section n-, n and n+. For the current dogleg layout, with all quadrupole correctors turned off, the maximum of fi x is 6.9m at S. To suppress % of it, the procedure described above requires ffl iqs9=.69 ffl iqs=-.7 ffl iqs=.67 fi x at S becomes.7m, which is 9.% of the unperturbed value. Figure (9) shows the change of fi x and dispersion throughout the ring. There is little change to fi x out of the range from QS9 to QS. However, inside it, at QS and QS, fi x is increased. nd dispersion is perturbed globally. We can apply this method together with harmonic compensation to constrain dispersion. The method can still be useful despite the two side effects. Since our knowledge about the unperturbed lattice is not perfect, we are interested in the sensitivity of the locality to errors in beta functions and phase advances. Monte Carlo simulation is carried out for it. For the above example the result is shown in Figure () and Figure (). When the error of phase advance is in the level of degrees, the global perturbation is roughly % of the desired local bump. References [] S. Y. Lee: cclerator Physics, World Scientific, Chapter
β x at QSs. change to dispersion... Figure 9: Changes to fi x bump. Top, fi x. Bottom, D x. and dispersion due to QS9, QS, QS set up for local beta.6 index =, ratio =.9.....6.8.. β /β as error of β.8 magnitude of β /β globally.7.6..... 6 error of ψ in degree; k r b=.,.,. ( β/β) Figure : Magnitude of global fi x fi indicating the locality error. To suppress % of fi at S of the current layout. Top, consider fi error only. Bottom, both phase and fi considered, with fi fi =:; :; : for black, red and blue curves, respectively
. index= (s), local beta bump with error.... beta error., phase error.*pi Figure : Changes to fi with (blue) or without (red) error in the case of suppressing % of fi at S of the current layout. Phase error is degrees and fi fi =: