CALCULATION OF NONLINEAR TUNE SHIFT USING BEAM POSITION MEASUREMENT RESULTS

International Journal of Modern Physis A Vol. 24, No. 5 (2009) 974 986 World Sientifi Publishing Company CALCULATION OF NONLINEAR TUNE SHIFT USING BEAM POSITION MEASUREMENT RESULTS PAVEL SNOPOK, MARTIN BERZ and CAROL JOHNSTONE University of California, Riverside, CA, USA Mihigan State University, East Lansing, MI, USA Fermi National Aelerator Laboratory, Batavia, IL, USA snopok@gmail.om berz@msu.edu jj@fnal.gov The alulation of the nonlinear tune shift with amplitude based on the results of measurements and the linear lattie information is disussed. The tune shift is alulated based on a set of speifi measurements and some extra information whih is usually available, namely that about the size and partile distribution in the beam and the linear optis effet on the partiles. The method to solve this problem uses the tehnique of normal form transformation. The proposed model for the nonlinear tune shift alulation is ompared to both the numerial results for the nonlinear model of the Tevatron aelerator and the independent approximate formula for the tune shift by Meller et al. The proposed model shows a disrepany of about 2%. 1. Introdution Finding the nonlinear tune shift depending on the position of the partile in the beam might be an elaborate task, beause the nonlinear omponent of the dynamis is not known to the desired preision or beause there are reasons to doubt the orrespondene between the model and the mahine optis. At the same time, there is still a way to find the tune shift, if there is a set of speifi measurements and some extra information whih is usually available, namely that about the geometry of the beam (its size and partile distribution) and the linear effet of the optis on the partiles (in the form of a one-turn linear transfer matrix). The tune of a system is one of the most important harateristis of the dynamis of partiles. For linear systems, the tune stays onstant, while in the nonlinear ase it might hange, mainly depending on the position of the partile in the beam (the so-alled tune shift with amplitude), but also depending on other parameters of the system. Consider the problem of evaluation of the tune shift with amplitude in the nonlinear ase using some extra information obtained by the speifi kind of mea- 974

Calulation of Nonlinear Tune Shift using Beam Position Measurement Results 975 Amplitude of the entral partile 3 x 10 3 2 1 0 1 2 3 0 2000 4000 Number of turns 6000 8000 Fig. 1. Measurement results: horizontal position of the enter of mass over a number of turns and its envelope. Fig. 2. Behavior of partiles in normal form oordinates. surements. All the proposed methods have been tested on the Tevatron aelerator model 1 and measurements, 2 but the algorithm for finding the tune shift with amplitude is appliable to other mahines. In fat, the algorithm should stay valid for any other synhrotron, as long as one an proeed with a linear normal form transformation. The normal form transformation is the ore of the method. 3 Throughout the artile the new set of oordinates, after the normal form transformation is applied to the pair of transversal phase spae oordinates (x, p x /p 0 )or(y, p y /p 0 ), is denoted (t +,t ). Here x and y are horizontal and vertial positions, respetively, of the partile under onsideration, p x and p y are the horizontal and vertial omponents of the momentum, respetively, p 0 is the referene momentum. The horizontal and vertial planes are assumed to be unoupled. 2. Calulation Results versus Measurement Results Suppose that one only has the information on the linear omponent of the dynamis of the partiles in the aelerator. Assume that there is some extra information available: the size of the beam, the partile distribution type and also the results of the speial type of measurements of the beam position. A orretor is introdued into the aelerator optis to kik the beam in the horizontal or vertial diretion. One the strength of the orretor is known, the displaement of the enter of the beam an be found. After the orretor is turned on and off instantaneously, the amplitude of the beam enter of mass dereases due to the filamentation of the beam, not the damping, as the motion is sympleti. The position of the enter of mass of the beam is then registered after eah turn of the partiles. One sample of the measurement data for the horizontal position is shown in Fig. 1. In the normal form oordinates the initially displaed beam behaves in a very similar fashion, whih allows to restore the information about the nonlinear tune. The normal form transformation is a nonlinear hange of oordinates, suh that

976 P. Snopok, M. Berz & C. Johnstone after the transformation the dynamis of the partiles is represented in a very systemati way. The details of the transformation algorithm an be found in [3]. The most important part for this study is that after the normal form transformation all the partiles follow onentri irles with angular veloity depending on the amplitude. This is the key fat allowing to establish a onnetion between the nonlinear tune shift with amplitude and the behavior of the beam. The funtion onneting the initial and final oordinates of the partiles after one full revolution (alled the transfer map) has the form: ( ) os 2πµ(r) sin 2πµ(r) M =, (1) sin 2πµ(r) os2πµ(r) where the tune µ(r) an be represented in the following form: µ = µ 0 + 1 r 2 + 2 r 4 +... (2) Here µ 0 is a onstant linear tune, 1, 2 are the oeffiients of the higher order terms in the expansion of the dependene of the tune µ on the partile s amplitude in the normal form oordinates, where the amplitude is defined to be r = (t + ) 2 +(t ) 2 for the partile with normal form oordinates (t +,t ). Figure 2 shematially shows the positions of four partiles with initial positions hosen along some fixed polar angle, after several turns. Partiles annot leave their orresponding irles, but the rotation frequenies are different for different radii. Assume that the outer partiles move faster than the inner partiles. In this partiular ase the outer partile will leave the inner partile behind in the phase. As a result of suh a redistribution of partiles, the enter of mass of the beam initially displaed from the origin shifts toward the origin of the oordinate system and then osillates around it. In other words, the amplitude of the enter of mass of the beam in the normal form oordinates dereases until it reahes a stable value. As it is assumed that an aurate linear lattie desription is available, one may use the linear normal form transformation, for whih the information on the linear dynamis is suffiient, to obtain the information on the distribution of the beam in the linear normal form oordinates after the kik. The linear normal form transformation is disussed in great detail in [3]. In new oordinates all the partiles follow irles with the same angular frequeny µ 0. Hene, the linear transformation does not provide any information on the oeffiients in the expansion (2) desribing the nonlinear tune shift. At the same time the linear transformation is suffiient to obtain an approximate initial distribution of the beam in the normal form oordinates. Sine the tune of the partile an also be viewed as the limit of the total phase advane divided by the number of turns when the number of turns goes to infinity, the average tune for a large number of turns is the same in both sets of oordinates, as the ontribution of the normal form transformation and the inverse normal form transformation beomes negligible. Hene, if the nonlinear tune of the partile in

Calulation of Nonlinear Tune Shift using Beam Position Measurement Results 977 the normal form oordinates is found, then the nonlinear tune of that partile in the original oordinates is the same. As a rule, 1 r 2 is the dominating term in the expansion (2). Aelerator designers try to avoid high order nonlinearities, unless there is a speifi need of them. Hene, finding the oeffiient 1 is the most important part. Later, if there are multiple measurements available, the oeffiient 2 ould be attempted to be found as well. If the transfer map M from Eq. (1) is known, one an trak the behavior of partiles for arbitrary many turns. That, in turn, allows us to find the number of turns orresponding to the moment when the amplitude of the enter of mass is at the half of its value after the kik, N 1/2. This establishes the onnetion between 1 and N 1/2.ThenumberN 1/2 an be found from the measurements (Fig. 1). The general sheme for establishing the onnetion between 1 and N 1/2 is as follows: (1) The outer partile of the beam having the amplitude R rotates with a frequeny µ(r), and hene, after 1/ µ(r) µ 0 turns this partile phase advane is 2π bigger (or smaller, depending on the sign of 1 ) than that of the partile lose to the origin; assume that the kik is weak enough and the beam is not displaed too far from the origin. In fat, in most ases the kik is suh that the origin is still inside the part of the phase spae overed by the beam. (2) R depends on the strength of the kik and the initial partile distribution, the value of R an be found as the maximum of the deviations of partile positions from the origin after the linear normal form transformation, that is, all the omponents to find R are known. (3) The value of 1 is not known, but one an always fix a ertain 1 and using the form of the transfer map (1) obtain the value of N 1/2 as a funtion of 1 and R. (4) One the algorithm for finding N 1/2 ( 1 ) is established, it an be used multiple times to obtain the orret value of the oeffiient 1 for a known value of N 1/2 inferred from the measurements as disussed above; it is a typial one-parameter optimization problem. Hene, the problem under onsideration has been redued to establishing a dependene of N 1/2 on various values of 1 and R. Depending on the initial distribution of partiles this an be a ompliated task, whih is not possible or not feasible to solve analytially to obtain an expliit expression for 1 = 1 (N 1/2 ). This problem is addressed below and solved numerially for various distributions. 3. Setor Approximation, Uniform Partile Distribution Let our initial distribution be uniform in the setor bounded by the two radii,r 2 and two angles ϕ 1,ϕ 2 (Fig. 3). This is a very simple and unrealisti ase, but it is instrutive to onsider it first in order to obtain basi formulas. After N turns eah partile of this distribution will have the phase advane of θ N (r) =2πNµ(r) =2πN(µ 0 + 1 r 2 + 2 r 4 )

978 P. Snopok, M. Berz & C. Johnstone Fig. 3. Uniform partile distribution in the setor. (orders up to 4 are taken into aount). Hene, the partile with radius <r<r 2 loated on the front (bak) line of the distribution will have a phase differene of θ N (r) =2πN(µ(r) µ( )) with respet to the inner partile. To find the entroid of the resulting planar figure, bounded by two radii,r 2 and two urves given by φ 1 +θ N (r), φ 2 +θ N (r), <r<r 2 three integral formulas are used: S = rdrdθ; x = 1 r 2 os θdrdθ; y = 1 r 2 sin θdrdθ. S S For the ase under onsideration: x (N) y (N) ϕ2+θ N (r) ϕ 1+θ N (r) ϕ2+θ N (r) ϕ 1+θ N (r) r 2 os θdθdr; r 2 sin θdθdr. (3) Hereafter, x (N) and y (N) are the oordinates of the beam enter of mass in the normal form oordinate system (t +,t ). 3 Let us simplify the form of the last two integrals. Without loss of generality one an assume ϕ 1 = ϕ 2 = ϕ (the angle an be hanged as only the radius is the quantity of interest). In addition to that the oordinate θ is hanged to ψ + θ N (r). Then one has dθ = dψ, and the integrals transform to: x (N) y (N) ϕ+θn (r) ϕ+θ N (r) r 2 os θdθdr ϕ+θn (r) ϕ+θ N (r) r 2 sin θdθdr ϕ ϕ ϕ ϕ r 2 os(ψ + θ N (r))dψdr; (4) r 2 sin(ψ + θ N (r))dψdr. (5)

Calulation of Nonlinear Tune Shift using Beam Position Measurement Results 979 After integrating with respet to ψ, under the remaining integral one an use the some trigonometri identities ultimately obtaining x (N) y (N) r 2 {sin(ϕ + θ N (r)) sin( ϕ + θ N (r))} dr = 2 S sin ϕ r 2 os θ N (r)dr, r 2 { os(ϕ + θ N (r)) + os( ϕ + θ N (r))} dr = 2 S sin ϕ r 2 sin θ N (r)dr. These integrals annot be found analytially due to the polynomial nature of the argument θ N. Even if one assumes θ N r 2, the result is a ompliated expression given in terms of Fresnel funtions r 2 1 ( os θ N (r)dr = R2 sin(2πn 1 R2 2 4πN ) sin(2πn 1 R1 2 )) 1 1 ( ) S(2(N 8π(N 1 ) 3/2 1 ) 1/2 R 2 ) S(2(N 1 ) 1/2 ), r 2 sin θ N (r)dr = 1 ( R2 os(2πn 1 R 4πN 2) 2 os(2πn 1 R1) 2 ) 1 1 ( ) C(2(N 8π(N 1 ) 3/2 1 ) 1/2 R 2 ) C(2(N 1 ) 1/2 ), (6) (7) where S(x) = x 0 sin( π 2 t2 )dt, andc(x) = x 0 os( π 2 t2 )dt. 4 The shape of the graphs of Fresnel funtions explains the behavior of the beam enter of mass shown in Fig. 1: both C(x) and S(x) osillate around 1 2 as x with slowly dereasing amplitude. Hene, the differene of the two C or S funtions osillates around zero, provided the arguments are proportional, whih is the ase in Eqs. (7). To illustrate this the graph of the funtion S(1.5x) S(x) is shown in Fig. 4. To alulate the integrals (4) (5) in the general ase, numerial integration methods should be employed. For this study the adaptive Simpson quadrature method 5 is used. The main issue with the setor approximation is that it is not sharp enough, and only works for beams whih are displaed by the transversal kik in suh a way that the whole beam is away from the origin. To handle the situation with a beam that rosses the origin, and to be more preise with the onlusions about the entroid, attention should be paid to the exat shape and position of the beam after the kik. 4. Elliptial Beam, Uniform Distribution Let us assume that the partiles in the beam are distributed uniformly (we will onsider the general ase later), and the beam has an elliptial shape. Then after

980 P. Snopok, M. Berz & C. Johnstone Properties of Fresnel Funtions S(x),S(1.5x),S(1.5x) S(x) 0.6 0.4 0.2 0 0.2 0.4 0.6 S(x) S(1.5x) S(1.5x) S(x) 0.8 10 5 0 5 10 x Fig. 4. S(1.5x) S(x) funtion graph. Fig. 5. = d ρ>0. Fig. 6. = d ρ<0. the transformation to the normal form oordinates, the beam has an elliptial shape again, and the axes of the transversal setion of the beam are equal. Then the boundary urve for the beam in the normal form oordinate pair is a irle, and the parametri representation for it an be found in the form of the equations for two half-irles: (r, ϕ 1 (r)), (r, ϕ 2 (r)). Without loss of generality it an be assumed that the resulting irle has its enter on the horizontal axis, with the oordinates (d, 0), where d>0 is known (similar to the previous setion: the angular position of the distribution of partiles does not matter, sine we are ultimately interested in the distane to the origin from the enter of the distribution whih is invariant from the angle). Let ρ be the radius of the beam, then the beam lies between = d ρ and R 2 = d+ρ, where it is often the ase that the radius is less than zero, whih means that the origin (0,0) is inside the beam (Fig. 5 6). Both d and ρ parameters an be found by applying the linear normal form transformation to the displaed beam boundaries. Below it will be shown that the two ases > 0and < 0 an be treated in a uniform way. For the moment let us assume that > 0.

Calulation of Nonlinear Tune Shift using Beam Position Measurement Results 981 Fig. 7. The intersetion of two irles. Similar to the previous setion, the entroid of the beam has the oordinates x (N) S = ϕ(r)+θ N (r) ϕ(r)+θ N (r) rdθdr; r 2 os θdθdr; y (N) = 1 r 2 sin θdθdr; S ϕ(r)+θ N (r) the only differene being that ϕ is now a funtion of r. To simplify the integral expression, some additional information is needed on the intersetion of two irles, as we are integrating along the ars of a irle and the boundary urve is also a irle. Let us onsider two irles: the first one entered at the origin (0, 0) and having a radius r, and the seond one entered at (d, 0) and having a radius ρ (Fig. 7). This setup gives { x 2 + y 2 = r (x d) 2 + y 2 or x = r2 ρ 2 + d 2, (9) = ρ 2d whih yields os ϕ(r) = x r = r2 ρ 2 + d 2 ( r 2 ρ 2 + d 2 ), ϕ(r) = aros. Using the resulting expression for ϕ and trigonometri identities, similar to what is done in the previous setion, one obtains =2 ϕ(r)+θ N (r) =2 r 2 r 2 os θdθdr =2 r 2 sin ϕ(r)osθ N (r)dr ( r r 2 2 ρ 2 + d 2 ) sin aros os θ N (r)dr ( ) r2 ρ 1 2 + d 2 2 os θ N (r)dr, (8) (10)

982 P. Snopok, M. Berz & C. Johnstone and hene x (N) y (N) S = ( ) r2 ρ r 1 2 + d 2 2 dr; ( ) r2 ρ 1 2 + d 2 2 os θ N (r)dr; ( ) r2 ρ 1 2 + d 2 2 sin θ N (r)dr. r 2 r 2 The integrand is not simplified further here, as the non-uniform density beam ase will be onsidered in the next setion, whih only makes the integrand more ompliated, thus not allowing any simplifiation of the general form. Let us onsider a speial ase of = d ρ<0 with the layout orresponding to Fig. 6. For 0 <r<, the intersetion of the two irles in (9) is purely imaginary, and hene the whole ontour (r, ϕ [ π, π)) belongs to the beam, and one an assume for suh r that ϕ goes from π to π. This is the approah used later for the non-uniform distribution. (11) 5. Elliptial Beam, Normal or Arbitrary Distribution Assume that the beam distribution is normal in both diretions in every pair of oordinates, and eah two diretions are independent. As the beam is round in the normal form oordinates, the varianes σ x and sigma y in both eigen-diretions are the same, i.e. σ = σ x = σ y, and hene the resulting density of the bivariate distribution is defined by the formula f(x, y) = 1 ( ((x d) 2 2πσ 2 exp + y 2 ) ) 2σ 2, as the mean values for the distribution are d and 0. Note that this formula is only valid for the initial distribution, when θ N =0,andafterN turns θ N should be subtrated from the value of the angle. A typial partile distribution after various number of turns is shown in Fig. 8. In the ase of the non-uniform distribution the expressions for S, x,andy are essentially the same as in Eqs. (11), exept that now the integrand is ompliated by the additional fator of f(r os(θ θ N ),rsin(θ θ N )). The term θ N is introdued to always take the density of the initial normal distribution: x (N) y (N) S = ϕ(r)+θ N (r) ϕ(r)+θ N (r) ϕ(r)+θ N (r) rf(r os(θ θ N ),rsin(θ θ N ))dθdr; r 2 os θf(r os(θ θ N ),rsin(θ θ N ))dθdr; r 2 sin θf(r os(θ θ N ),rsin(θ θ N ))dθdr. (12)

Calulation of Nonlinear Tune Shift using Beam Position Measurement Results 983 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 5 0 5 (a) initial distribution (b) after 1000 turns () after 3000 turns Fig. 8. Partile distribution in the beam, the ross indiates the enter of mass of the beam. 6. Numerial Experiment Results All the numerial results of this setion are based on the Tevatron model available on the Fermi National Aelerator Laboratory website. 1 All the alulations are performed using the arbitrary order ode COSY INFINITY 6 written by Martin Berz, Kyoko Makino, et al. at Mihigan State University. The soure ode for the lattie is in the format of the MAD programming environment, 7 for whih a onverter 8 to COSY INFINITY is readily available. The alulation method desribed above allows one to find the dependene r = r(n, 1, 2 ) for elliptial beams with an arbitrary partile distribution, the only requirement being that the initial distribution density funtion is known. Having this data available and employing various optimization methods, one an find the orret values of 1 based on one partiular measurement or both oeffiients 1 and 2, provided that measurements for different kik strengths are available. Let us ompare the results of the proposed algorithm to the values obtained by traking the nonlinear model of the Tevatron aelerator. We use the beam position monitor (BPM) measurement results 2 similar to those shown in Fig. 1. The number of turns after whih the amplitude of the enter of mass falls down to half of its value varies depending on the BPM. Taking the average over the total of 115 reliable BPMs, one obtains that N 1/2 = 1021. A parameter fitting proedure results in the expeted value of 1 = 2511 for the initial beam amplitude after the kik of r =0.24 10 3. Taking into aount that µ 0 =0.585, one obtains µ µ 0 + 1 r 2 =0.585 1.4463 10 4. (13) To oneive how lose the obtained value of 1 is to the realisti value of the tune shift with amplitude, a omparison was performed in COSY using the nonlin-

984 P. Snopok, M. Berz & C. Johnstone Normal form amplitude 2 1 Calulated amplitude Modeled amplitude 0 0 1000 2000 3000 4000 5000 Number of turns Fig. 9. Calulation results and the omparison with the nonlinear model. ear model of the Tevatron available at the offiial lattie page at Fermilab. 1 The model reflets hanges made to the aelerator as of November 2005, while the measurement results are dated January 2006. Hene, there are strong reasons to assume that the model distribution traking should yield results omparable to the alulated value of the tune shift, and therefore, the measurements. The COSY alulation shows that the expeted value of 1 for the nonlinear model should be 2541, whih means the alulated value found by applying the algorithm differs from the model value by 1.2%. At the same time, only the information about the distribution of the partiles in the beam, the size of the beam, and the linear dynamis was used to find the nonlinear tune shift. Neessary additional information was extrated from the measurements. Figure 9 shows the graphs of the alulated amplitude with 1 = 2511 and the model amplitude with 1 = 2541. The slight differene between the graphs an be explained not only by using different 1 s, but also by the fat that the fourth order term 2 r 4 in the expansion of µ has not been taken into aount. Also the nonlinear model represents an approximation to the real mahine s optis. At the same time, the similarity of the graphs allows to onlude that the model represents the real mahine quite aurately, at least for the low order nonlinearities affeting the tune shift (mainly the sextupole ontent of the ring). Also, the validity of the approah studied is perfetly supported by the independent alulations done earlier. There is an estimate of the nonlinear tune shift 9 based on the approah by R.Meller et al., 10 given by the following formula: µ µ 0 κa 2 1, κ, (14) 4πN 1/2 where A is the amplitude of the enter of mass of the beam, measured in σ units of the beam under onsideration. This formula is derived for the beams with a normal distribution of partiles, and it represents a good approximation when the transversal kik is relatively weak (A is not too muh greater then 1), and the

Calulation of Nonlinear Tune Shift using Beam Position Measurement Results 985 underlying dynamial system is weakly nonlinear with a quadrati term being the main ontribution to the tune shift. It is noteworthy that Eq. (14) is onneted to the initial distribution, the strength of the transversal kik and the transfer map of the system under onsideration via A and N 1/2. The omparison of the value κa 2 from Eq. (14) to the value of 1 r 2, obtained by the alulation using the algorithm desribed above, leads to the following result: κ =7.96 10 5, A =1.36, µ µ 0 + κa 2 =0.585 1.4723 10 4, (15) that is, the differene between the values obtained using different approximations in Eqs. (13) and (15) is 1.75%. 7. Summary The orrespondene is found between the first and the most important term in the expansion of the nonlinear tune shift with amplitude and the BPM measurement results after the beam is kiked transversely. This orrespondene an be found by fitting the parameter 1. To be able to find the orret value of the parameter it is neessary to have the information on the behavior of the amplitude r of the enter of mass. This amplitude an be found in the most general ase using Eq. (12). A method for the alulation of the nonlinear tune shift with amplitude was tested on the Tevatron BPM measurement results and ompared to the nonlinear model alulations as well as the independent approximation formulas. In both ases the disrepany was within 2%, whih an be onsidered a very good result onsidering that only the information on the one-turn linear transfer map and the geometry of the beam has been used, while the lak of information on the nonlinear behavior was ompensated by a single BPM measurement with one partiular perturbation (kik) strength. After the oeffiient 1 has been found, one might try finding the oeffiient 2 if multiple measurement results are available. On the other hand, in the ase of the Tevatron, 2 r 4 is 2 orders smaller than 1 r 2, so in this partiular study there was no attempt made to find 2. Referenes 1. M. A. Martens. Tevatron lattie page. http://www-ap.fnal.gov/~martens/tev_ lattie/tev_lattie.html. 2. Y. Alexahin. Private ommuniation. 3. M. Berz. Modern Map Methods in Partile Beam Physis. Aademi Press, San Diego, 1999. Also available at http://bt.pa.msu.edu/pub. 4. M. Abramowitz and I. A. Stegun (Eds.). Handbook of Mathematial Funtions with Formulas, Graphs, and Mathematial Tables, pages 300 302. New York: Dover, 1972. 5. W. M. MKeeman. Algorithm 145: Adaptive numerial integration by Simpson s rule. Communiations of the ACM, 5(12):604, 1962. 6. K. Makino and M. Berz. COSY INFINITY version 9. Nulear Instruments and Methods, A558:346 350, 2005.

986 P. Snopok, M. Berz & C. Johnstone 7. Methodial aelerator design. http://mad.home.ern.h/mad/. 8. The MAD to COSY Converter. http://osy.pa.msu.edu/onverters/mad2osy. 9. M. G. Minty and F. Zimmermann. Measurement and Control of Charged Partile Beams, pages 52 53. Partile aeleration and detetion. Springer, 2003. ISBN 3-540- 44187-5. 10. R. E. Meller, A. W. Chao, J. M. Peterson, S. G. Peggs, and M. Furman. Deoherene of kiked beams. Tehnial Report SSC-N-360, SSCL, 1987.